Half-wormholes and ensemble averages

Abstract

Recently, the concept of half-wormholes is introduced to give a resolution to the factorization puzzle in holography and help understand better the relation between ensemble average theories and gravity in the bulk. Half-wormholes are proposed to be the contributions to the gravitational path integral that correspond to fluctuations of each individual theory around the average of the whole ensemble of theories. In this paper, we further explore the extent to which the half-wormhole interpretation is applicable. In particular, to further demonstrate that the half-wormhole interpretation is not merely a feature of a specific theory but is a general feature of ensemble average theories, we examine various models, including different enriched 0-dimensional SYK-like models, the 1-dimensional Brownian SYK model and its generalization. To further demonstrate that the half-wormhole interpretation applies to more general probability distributions apart from the zero-mean Gaussian distribution, we consider random couplings with other non-trivial moments. Specifically, introducing a non-trivial mean value to the random coupling renders the spectral correlators to exhibit both disconnected saddles and connected saddles. The inclusion of higher-order moments leads to new “multi-linked half-wormhole” saddles. We also clarify the distinctions between the unlinked half-wormhole and the linked half-wormhole in our modified Brownian SYK model.

1 Introduction

The AdS/CFT correspondence [1,2,3] provides a non-perturb-ative definition of quantum gravity. An important lesson from the recently progress in understanding the black hole information paradox is that a summation of different configurations in the semi-classical gravitational path integral is crucial to probe some quantum mechanical properties of the system, such as the Page curve [4,5,6,7], the late-time behavior of the spectral form factor [8, 9], and correlation functions [10, 11], see also a recent review [12]. However, the inclusion of spacetime wormholes leads to an apparent factorization puzzle [13]; a holographic computation of the correlation functions of field theory partition functions living on different boundaries gives non-factorized results, i.e. $\langle Z_{L}Z_{R}\rangle \ne \langle Z_L\rangle \times \langle Z_R\rangle$, which is in tension with the general expectation on the field theory side. This revitalizes the hypothetical connection between wormholes and ensemble averages [14,15,16,17], and motivates an appealing conjectural duality between a bulk gravitational theory and (the average of) an ensemble of theories on the boundary [8, 18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64], whose prototype is the by-now well known duality between the two-dimensional Jackiw-Teitelboim (JT) gravity [65, 66] and the Schwarzian sector of the Sachdev-Ye-Kitaev (SYK) model [67,68,69], or more directly the random matrix theories [8, 18]. These results suggest that solving the factorization problem could shed light on the microscopic structure of quantum gravity that are not universal and hence cannot be captured by the ensemble-averaged quantities [70, 71]. In [72], the factorization problem is carefully studied in a toy model introduced in [52], where it is shown that the (approximate) factorization can be restored if other half-wormhole contributions are included. In the dual field theory analysis, these half-wormhole contributions are identified with non-self-averaging saddle points in each individual theory of the ensemble. This idea is explicitly realized in a 0-dimensional “one-time” SYK model in [73], followed by further analyses [74,75,76,77,78,79,80,81]. An explicit connection between the gravity computation in [72] and the field theory computation in [73] is proposed in [59].

In this paper, we explore the extent to which the half-wormhole interpretation is applicable.

In Sect. 2, we first review the computation in [73] and provided another way to derive the same result. Our discussion is based on a detailed Lefschetz thimble analysis, which is independent of the argument given in [73]. The Lefschetz thimble analysis gives a systematic way to justify how to identify the correct set of saddle points to be included in the path integral. This serves as a very non-trivial cross-check of the result in [73] and a preparation for the later parts of this paper.

In Sect. 3, we consider a 0d SYK model whose random couplings are drawn from a probability distribution with a non-zero mean. The motivation to study this model is the following. The 0d SYK model studied in [73] is simple enough to explicitly demonstrate the contribution from half-wormholes, but some crucial properties of the more familiar 1d SYK model are missing in this 0d toy model. The most notable consequence of this is that the averaged partition function $\langle z\rangle$ of the 0d toy model in [73] is zero, so there is no disconnected contribution to any spectral correlation functions at all. This is quite different from the original 1d SYK model and hence raises a question if the conclusion obtained in the 0d model in [73] applies to the more familiar 1d SYK and other similar models. Our analysis in this section is a first step towards a thorough study of this question and a resolution of the factorization in that case. Our analysis shows that in the presence of a non-zero mean value of the random couplings, the structure of half-wormhole-like contributions is much richer, and in particular new types of non-self-averaging contributions become important and should be considered in the saddle point analysis of the spectral correlation functions.

In Sect. 4, we consider other generalized 0d SYK models whose random couplings are drawn from more general probability distributions other than the zero-mean Gaussian distribution. The motivation for this study is the following. First of all, a significant feature in the analysis of [73, 77] is the Gaussian property of the randomness in the ensemble average model. On the other hand, the gravitational analysis [72] of a topological model [52] reveals that other connected half-wormhole configurations, which contain more than two half-wormholes linking together (see discussion around Figure 18 and in section 6.1 of [72]), play a crucial role in understanding the factorization problem in the gravity theory. This raises the question of whether there are similar multi-boundary linked-half-wormhole contributions in the boundary ensemble average theories. This motivates us to consider random couplings drawn from continuous distributions with non-trivial higher moments, which is a natural origin of non-trivial interconnections between the different factors of the spectral correlation functions. Furthermore, the results of the gravitational analysis in the toy model [52] suggest that a good boundary dual description could involve an ensemble average of different theories with Poisson distributions. A related analysis in [82] shows that random variables drawn from Poisson distributions have a natural connection to gravitational systems. Ensemble-averaged theories involving an average over uniform distributions on the moduli space [36,37,38,39,40, 46, 62, 63] are shown to have clear connection to gravitational system. These motivate us to consider 0d SYK models with random couplings drawn from discrete distributions. Our computation shows that in theories with non-trivial higher moments, there are very rich structures of the non-self-averaging contributions to the spectral correlation functions, and the inclusion of higher moments yields new “multi-linked half-wormhole” saddles in addition to the original two-linked half-wormhole saddle. All these new saddle points should be taken into account in order to solve the factorization puzzle in these models. In addition, we find that when the random couplings are Poisson distributed, the multi-linked half-wormhole contributions are all suppressed in the large-N limit; contributions with disk and cylinder topology are good enough to solve the factorization puzzle.

In Sect. 5, we study the 1d Brownian SYK model and its generalizations. Since the SYK model is originally defined in 1d, the computation in this section clearly helps further explore the important question of whether the half-wormhole interpretation of the non-self-averaging contributions to various spectral correlators applies to models in 1-dimensional spacetime. By an explicit computation, we confirm that there is indeed a similar decomposition of the partition function into the averaged contribution and a punctured-disk-like non-self-averaging contribution. This agrees with our expectation obtained from the computation in the 0d model in Sect. 3. The results in this section also provide direct evidence of the wide applicability of the half-wormhole type interpretation of the non-self-average contributions in general ensemble average theories.

2 SYK at one time point: the cylinder model

In this section, we study the half-wormhole contributions in the toy 0d SYK model that can be considered as the usual 0+1d SYK model at a single instant of time. We first briefly review the previous results in [73] and also in [74, 77]; in Sect. 2.3.2 we provide a detailed study of the various saddle points via a Lefschetz-thimble analysis, which is also a useful preparation for the analysis of the other models in this paper.

2.1 SYK model with one time point

As in [73], we are interested in the following Grassmann integral¹

$$\begin{aligned} z=\int \text {d}^N \psi \exp (\text {i}^{q/2} J_{i_1\dots i_q}\psi _{i_1\dots i_q}), \end{aligned}$$

(1)

where $\psi _{i_1\dots i_q}=\psi _{i_1}\psi _{i_2}\dots \psi _{i_q}$ and $\psi _i$ are Grassmann variables. The number z can be understood as the partition function of $0+0$ dimensional analogue of SYK model. The random couplings $J_{i_1\dots i_q}$ is drawn from a Gaussian distribution

$$\begin{aligned}{} & {} \langle J_{i_1\dots i_q}\rangle =0, \end{aligned}$$

(2)

$$\begin{aligned}{} & {} \langle J_{i_1\dots i_q}J_{j_1\dots j_q}\rangle =t^2\delta _{i_1j_1}\dots \delta _{i_qj_q},\quad t^2=\frac{(q-1)!}{N^{q-1}}. \end{aligned}$$

(3)

We sometimes use the collective indies A, B to represent a series of q indices to simplify our notation

$$\begin{aligned} A=\{i_1<\dots < i_q\},\qquad J_A\psi _A\equiv J_{i_1\dots i_q}\psi _{i_1\dots i_q}. \end{aligned}$$

(4)

Integrating out the Grassmann variables directly gives²:

$$\begin{aligned} z{} & {} =\int \text {d}^N \psi \exp (\text {i}^{q/2}J_A \psi _A) \end{aligned}$$

(5)

$$\begin{aligned}{} & {} =\sum '_{A_1<\dots <A_p} \text {sgn}(A)J_{A_1}\dots J_{A_p}\,, \end{aligned}$$

(6)

where $p=N/q$ and the expression (6) is the hyperPfaffian $\text {Pf}(J)$ of the $J_{i_1\ldots i_q}$ hypermatrix.

Before diving into the technical details, let us first outline the computation we will perform in this section. The ensemble theory (1) could be regarded as an effective description of a dual gravitational system. However, explicit computation uncovers a “factorization” problem of the spectral correlators (correlation functions of the partition functions), namely

$$\begin{aligned} \langle z^2\rangle =\langle z_L z_R\rangle \ne f_L( z_L)f_R(z_R). \end{aligned}$$

(7)

where we have trivially rewritten $z^2$ to $z_L z_R$ to emphasis that the two copies of z are independent to each other, and $f_L$, $f_R$ are some functions. This leads to a puzzle: the correlation functions of the partition functions of two different theories are expected to factorize into two factors each only depends on one of the z’s, but this is in contradiction with the above equation. In this section, we review a proposal [73] to resolve the puzzle and provide an independent computation by the Lefschetz-thimble method to support the results there. The main conclusion is that in the path integral of the spectral correlators, apart from the wormhole saddle that gives $\langle z_L z_R\rangle$, one should also include the contribution from another saddle point, which is referred to as the half-wormhole saddle, into the path integral. Then we have approximately

$$\begin{aligned} z^2\approx \langle z_l z_R\rangle +\text {half-wormhole saddle}, \end{aligned}$$

(8)

where the “half-wormhole saddle” is denoted by $\Phi (0)$ in the rest of this section.

2.2 The ensemble averaged quantities

We first consider the ensemble averaged quantities $\langle z^2\rangle$ which is defined as

$$\begin{aligned} z^2{} & {} =z_L z_R \end{aligned}$$

(9)

$$\begin{aligned}{} & {} =\int \text {d}^N \psi ^L \text {d}^N \psi ^R \exp \left\{ \text {i}^{q/2} J_A\left( \psi _A^L+\psi _A^R\right) \right\} , \end{aligned}$$

(10)

$$\begin{aligned} \langle z^2\rangle{} & {} =\int \text {d}^{2N} \psi \exp \left\{ \frac{N }{q}\left( \frac{1}{N} \psi _i^L \psi _i^R\right) ^q\right\} , \end{aligned}$$

(11)

where we have assumed that q and N are even, and L, R labels the two copies of z on the left-hand-side. In the following, we would like to compute this quantity by saddle-point analysis. In the 0+1d SYK model, the analog of this quantity is the “spectral form factor” (SFF) $\langle Z(\beta +iT)Z(\beta -iT)\rangle$. It is known that both a “disk” saddle point and a “wormhole saddle” contribute to the SFF; the disk saddle is responsible for the decay of the SFF at the early time and the wormhole saddle is responsible for the linear increasing period called the “ramp” of the SFF in a relatively later time regime.

However, in the 0d SYK model there is no time. Moreover, since $\langle z\rangle =0$, only a wormhole saddle is possible to exist. In the following, we will confirm the existence of the wormhole saddle by comparing the exact evaluation of $\langle z^2\rangle$ and its saddle point approximation. The exact values of (11) can be computed by introducing a G variable

$$\begin{aligned} \langle z^2\rangle= & {} \int \text {d}^{2N}\psi \int _{{\mathbb {R}}}\text {d}G \nonumber \\{} & {} \times \delta \left( G-\frac{1}{N}\sum _{i=1}^N \psi _i^L \psi _i^R\right) \exp \left( \frac{N}{q}G^q\right) \end{aligned}$$

(12)

$$\begin{aligned}= & {} N^{-N}\int _{{\mathbb {R}}}\text {d}G\exp \left( \frac{N}{q}G^q\right) (-\partial _G)^N \delta (G) \end{aligned}$$

(13)

$$\begin{aligned}= & {} \frac{N!(N/q)^{N/q}}{N^N (N/q)!} \end{aligned}$$

(14)

$$\begin{aligned}= & {} e^{-(1-\frac{1}{q})N}\sqrt{q}\left( 1+\frac{1-q}{12N}+\mathcal {O}\left( \frac{1}{N^2}\right) \right) , \end{aligned}$$

(15)

where in the last step we expand to the next-to-leading order of 1/N.

Next we derive the same result (15) from a saddle point approximation. We start by rewriting the $\delta$ function in (121314)

$$\begin{aligned} \delta \left( G-\frac{1}{N}\sum _i \psi _i^L \psi _i^R\right) =\int \text {d}\Sigma \,e^{\text {i}\Sigma \left( G-\frac{1}{N} \psi _i^L \psi _i^R\right) }, \end{aligned}$$

(16)

and then deform the contour of the integration along which the $g,\sigma$ variables, defined by

$$\begin{aligned} \Sigma =\text {i}e^{-\text {i}\frac{\pi }{q}}\sigma ,\quad G=e^{\text {i}\frac{\pi }{q}} g, \end{aligned}$$

(17)

are real. This modification ensures the convergence of the integral. The resulting effective action is

$$\begin{aligned} \langle z^2\rangle&=\int _R \text {d}g \int _R \frac{\text {d}\sigma }{2\pi /N}\nonumber \\&\quad \times \exp \left\{ N\left( \log \left( \text {i}e^{-\frac{\text {i}\pi }{q}}\sigma \right) -\text {i}\sigma g-\frac{1}{q}g^q\right) \right\} , \end{aligned}$$

(18)

$$\begin{aligned}&\equiv \int _R \text {d}g \int _R \frac{\text {d}\sigma }{2\pi /N}e^{NS}\,. \end{aligned}$$

(19)

The saddle point equations of this path integral are

$$\begin{aligned}{} & {} -\text {i}\sigma -g^{q-1}=0,\quad g^q=-1, \end{aligned}$$

(20)

$$\begin{aligned}{} & {} \rightarrow \quad g=e^{\frac{(2m+1)\text {i}\pi }{q}},\quad m=0,\dots ,q-1. \end{aligned}$$

(21)

All of them give the same on-shell action

$$\begin{aligned} \langle z^2\rangle _s=\frac{N}{2\pi } e^{-\left( 1-\frac{1}{q}\right) N}. \end{aligned}$$

(22)

To match with the exact result (15) we need to add in contributions from fluctuations around each of these saddle points. For simplicity let us take $q=4$ and focus on one of the saddle points

$$\begin{aligned} \sigma _s=g_s=-(-1)^{\frac{3}{4}},\quad \langle z^2\rangle _{s}=\frac{N}{2\pi }e^{-\frac{3}{4}N}. \end{aligned}$$

(23)

Expanding the exponent around this saddle

$$\begin{aligned} \sigma =\sigma _s+x,\quad g=g_s+y, \end{aligned}$$

(24)

to the 4th order

$$\begin{aligned} S_2{} & {} \sim -\frac{3}{4}+\frac{3\text {i}x^2}{2}-\text {i}xy-\frac{\text {i}y^2}{2}\nonumber \\{} & {} \quad +\left[ (-1)^{3/4}x^3+\frac{(-1)^{3/4}}{3}y^3\right] \epsilon +\frac{y^4-x^4}{4}\epsilon ^2, \end{aligned}$$

(25)

where we have added $\epsilon \equiv 1$ to keep track of the expansion, then expanding $\exp (S_2)$ to the second order of $\epsilon$ and finally evaluating the integral (18) to this order directly gives the contribution from this saddle up to 2-loop as

$$\begin{aligned} \langle z^2\rangle _{\text {saddle}+\text {loop}}= e^{-\frac{3}{4}N}\frac{1}{2}\left( 1-\frac{1}{4N}\right) . \end{aligned}$$

(26)

Adding contributions from all the 4 saddles we arrive at

$$\begin{aligned} \langle z^2\rangle _{\text {saddle}+\text {loop}}=2e^{-\frac{3}{4}N}\left( 1-\frac{1}{4N}\right) , \end{aligned}$$

(27)

that agrees with (15) at 2-loop order. These saddles are named as the wormhole saddles because in the 0+1d SYK model, they have a gravity dual which can be viewed as a wormhole.³

At first glance, it may be surprising that we need to add all complex saddle points (which are not along the integral contour along the real axis) to obtain the correct result. However this can be explained and justified with the method of Lefschetz thimbles which we discuss in Sect. 2.3.2 with some technical details reviewed in  Appendix A. The method of Lefschetz thimbles is a way to determine which saddle points should be considered when there are multiple saddle points in the integral domain. In short, for each saddle point we can associate a steepest descent path which is called the Lefschetz thimbles and if the thimble intersects with the chosen integral contour then the corresponding saddle point should be included.

2.3 The non-averaged quantities

Now we try to compute the non-averaged quantity (10) in the saddle point approximation. Following [73], we rewrite $z^2$ as an integral

$$\begin{aligned}{} & {} z^2=\int _R \text {d}\sigma \, \Psi (\sigma ) \Phi (\sigma ), \end{aligned}$$

(28)

$$\begin{aligned}{} & {} \Psi (\sigma )=\int \frac{\text {d}g}{2\pi /N}\exp [N(-\text {i}\sigma g-1/q g^q)], \end{aligned}$$

(29)

where the coupling dependent piece $\Phi$ is

$$\begin{aligned} \Phi (\sigma ){} & {} =\int \text {d}^{2N}\psi \exp \left\{ \text {i}e^{-\frac{\text {i}\pi }{q}} \sigma \psi _i^L\psi _i^R+\text {i}^{q/2}J_A\left( \psi _A^L+\psi _A^R\right) \right. \nonumber \\{} & {} \quad \left. -\frac{N}{q}\left( \frac{1}{N}\psi _i^L\psi _i^R\right) ^q \right\} \,. \end{aligned}$$

(30)

This expression (28) is derived by inserting the trivial identity $1=\int \text {d}G \delta (G-1/N \sum _i \psi _i^L\psi _i^R)$ and rotating the contour. In this form, the ensemble average of $\langle z^2\rangle$ is entirely attributed to the ensemble average of $\Phi (\sigma )$

$$\begin{aligned} \langle z^2\rangle =\int _R \text {d}\sigma \, \Psi (\sigma ) \langle \Phi (\sigma )\rangle \,, \end{aligned}$$

(31)

since the $\Psi (\sigma )$ does not depend on the random couplings.

The integral region of (28) can be divided into two subregions depending on whether $\Phi (\sigma )$ is self-averaging or not. By self-averaging we mean the fluctuations around the average value is small in the large N limit

$$\begin{aligned}&\Phi (\sigma )- \langle \Phi (\sigma )\rangle \approx 0\,, ~\Leftrightarrow ~~ \langle \left( \Phi (\sigma )- \langle \Phi (\sigma )\rangle \right) ^2\rangle \approx 0\,,\nonumber \\&~\Leftrightarrow ~~ \langle \Phi (\sigma )^2\rangle \approx \langle \Phi (\sigma )\rangle ^2\ . \end{aligned}$$

(32)

If the wormhole saddle points (2021) are in this subregion, we then know that the result of the integral (28) is self-averaging, namely it can be approximated as $\langle z^2\rangle \approx \langle z^2\rangle _{\text {wormhole-saddle}}$. In this simple model, both $\langle \Phi (\sigma )\rangle$ and $\langle \Phi (\sigma )^2\rangle$ can be computed exactly. The $\langle \Phi (\sigma )\rangle$ can be directly obtained as

$$\begin{aligned} \langle \Phi (\sigma )\rangle =(\text {i}e^{-\frac{\text {i}\pi }{q}}\sigma )^N. \end{aligned}$$

(33)

To compute $\langle \Phi (\sigma )^2\rangle$, we introduce $\sigma _{AB}$ and $g_{AB}$ analogous to (17)

$$\begin{aligned} g_{AB}=e^{-\text {i}\frac{\pi }{q}}G_{AB}\,,\quad G_{AB}=\frac{1}{N} \sum _{i}\psi _i^A\psi _i^B\,, \end{aligned}$$

(34)

where we label the L, R in one of the $\Phi (\sigma )$ (30) by $L=1,R=2$, and $L',R'$ in the other of the $\Phi (\sigma )$ by $L'=3,R'=4$. Then the combination (AB) is one of $\{(13),(14),(23),(24)\}$. Each $g_{AB}$ is paired with a $\sigma _{AB}$ whose subscript has the same meaning as in $g_{AB}$. Then similar to the computation we used to get (15), $\langle \Phi (\sigma )^2\rangle$ can be done exactly

$$\begin{aligned}{} & {} \langle \Phi (\sigma )^2\rangle \nonumber \\{} & {} =\int _R\frac{\text {d}^4 \sigma _{AB} \text {d}^4 g_{AB}}{(2\pi /N)^4} \exp \left\{ N\left[ \log (-e^{-\frac{2\text {i}\pi }{q}}(\sigma ^2\right. \right. \nonumber \\{} & {} \left. \left. \quad +\sigma _{14}\sigma _{23}-\sigma _{13}\sigma _{24})) -\text {i}\sigma _{AB}g_{AB}-\frac{1}{q}g_{AB}^q\right] \right\} , \end{aligned}$$

(35)

$$\begin{aligned}{} & {} =(-e^{-\frac{2\text {i}\pi }{q}})^N\hspace{-4mm}\sum _{n_1+n_2+n_3=\frac{N}{q},n_i\ge 0}\hspace{-2mm}\frac{N!}{N^{2q(n_2+n_3)}}\nonumber \\{} & {} \quad \times \left( \frac{N}{q}\right) ^{2(n_2+n_3)}\frac{\sigma ^{2qn_1}(qn_2)!(qn_3)!}{(qn_1)!(n_2!)^2(n_3!)^2}, \end{aligned}$$

(36)

It can be organized into a polynomial in $\sigma$

$$\begin{aligned} \langle \Phi (\sigma )^2\rangle= & {} (-e^{-\frac{2\text {i}\pi }{q}})^N\left( \sigma ^{2N}+\frac{2N!q!}{(N-q)! q^2 N^{2q-2}}\sigma ^{2N-2q} \right. \nonumber \\{} & {} \left. +\dots +e^{2N\frac{1-q}{q}}2q\right) \end{aligned}$$

(37)

$$\begin{aligned}\sim & {} (-e^{-\frac{2\text {i}\pi }{q}})^N\left( \sigma ^{2N}+\frac{2(q-1)!}{q N^{q-2}}\sigma ^{2N-2q}\right. \nonumber \\{} & {} \left. +\dots +e^{2N\frac{1-q}{q}}2q\right) , \end{aligned}$$

(38)

where the phase factor is trivial whenever q divides N. Apparently when $q>2$ and $\sigma ^{2q}N^{q-2}\gg 1$ we have

$$\begin{aligned} \langle \Phi (\sigma )^2\rangle \approx \langle \Phi (\sigma )\rangle ^2, \end{aligned}$$

(39)

and the result is self-averaging.

In this simple example, we can perform an exact computation to get the results. However, in other models there is not a similar exact computation so it is important to understand how to reach the same conclusion by saddle approximation.

2.3.1 The saddle points analysis: $\sigma \ne 0$, the trivial saddle

The saddle point equations of the integral (3536) are

$$\begin{aligned}{} & {} g_{AB}^{q-1}=-\text {i}\sigma _{AB},\quad -\text {i}g_{13}=\frac{\sigma _{24}}{f},\quad \text {i}g_{14}=\frac{\sigma _{23}}{f},\nonumber \\{} & {} \text {i}g_{23}=\frac{\sigma _{14}}{f},\quad -\text {i}g_{24}=\frac{\sigma _{13}}{f}, \end{aligned}$$

(40)

where $f\equiv \sigma _{14}\sigma _{23}-\sigma _{13}\sigma _{24}+\sigma ^2$. The above set of equations has a trivial solution $\sigma _{AB}=g_{AB}=0$, which we call the “trivial saddle”. The trivial saddle point value of $\langle \Phi (\sigma )^2\rangle$ is

$$\begin{aligned} \langle \Phi (\sigma )^2\rangle _{\text {trivial}}=\left( \frac{N}{2\pi }\right) ^4 \left( -e^{-\frac{2\text {i}\pi }{q}} \sigma ^2\right) ^N=N^4\langle \Phi (\sigma )\rangle ^2\ , \end{aligned}$$

(41)

and the 1-loop fluctuations around the trivial saddle points is $1/N^4$ suppressed. Therefore the contribution up to 1-loop level is

$$\begin{aligned} \langle \Phi (\sigma )^2\rangle _{\text {trivial}+1\text {loop}}=(-e^{-\frac{2\text {i}\pi }{q}} \sigma ^2)^N=\langle \Phi (\sigma )\rangle ^2, \end{aligned}$$

(42)

which says the trivial saddle always agrees with the first term in (3738). If this saddle dominates the integral the quantity $z^2$ is self-averaging.

2.3.2 The saddle points analysis: $\sigma \ne 0$, the wormhole saddle

However, there could be other non-trivial solutions to the saddle point equation (40) with $\sigma _{AB}\ne 0$. From the equations of motion we obtain

$$\begin{aligned}{} & {} x^{q-2}=y^{q-2}, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} (x^{q-1}-y^{q-1}+\sigma ^2)^{2}=x^{q-2}=y^{q-2}, \end{aligned}$$

(44)

$$\begin{aligned}{} & {} g_{13}^q=g_{24}^q,\quad g_{23}^q=g_{14}^q \end{aligned}$$

(45)

where

$$\begin{aligned} x=g_{13}g_{24},\quad y=g_{14}g_{23}. \end{aligned}$$

(46)

It is easy to check that solutions of the above equation satisfy $x=ye^{\frac{2m\pi \text {i}}{q-2}}$, and for each choice of m there are $2q^2$ solutions of $g_{ab}$.

For simplicity, we again focus on the $q=4$ case, where there are only two classes of solutions $x=\pm y$.

$\bullet$ When $x=y$ we find 32 non-trivial saddles. The on-shell action of them are all the same

$$\begin{aligned} \langle \Phi (\sigma )^2\rangle _{\text {non-trivial}}^+=N^4\langle \Phi (\sigma )\rangle ^2=\langle \Phi (\sigma )^2\rangle _{\text {trivial}}, \end{aligned}$$

(47)

where the factor $N^4$ comes from the measure of (3536). However, the one-loop fluctuations around these non-trivial saddle points amount to one-eighth of the fluctuations from the trivial saddle, with a value of $1/(8N^4)$.Although when the 1-loop effects are taken into account, we observe that the trivial saddles give larger contributions. But the contributions from the non-trivial saddles are also significant. Therefore naively we should add the contributions from all these saddle point, however, if we were adding the contributions from both the trivial and non-trivial saddles the answer would exceed the exact value (3738). This suggests that only a subset of these saddle point contributions should be included in the integral expression of $\langle \Phi (\sigma )^2\rangle$. Indeed, through a straightforward Lefschetz-thimble analysis, as reviewed in  Appendix A (see also [83]), we conclude that only trivial saddles are needed. The detailed reasoning is as follows.

Fig. 1

Anti-thimble on the $\sigma _{13}$ plane (left) and the $\sigma _{24}$ plane (right)

To start with, we choose the real part of the action (3536) as a Morse function

$$\begin{aligned} h&\equiv \Re (S)\nonumber \\&=\sum _{abj} \left( -\frac{g_{abj}^{4}}{4}+\frac{3 g_{ab1}^{2}g_{ab2}^{2}}{2}+g_{ab1}\sigma _{ab2}+g_{ab2}\sigma _{ab1}\right) \nonumber \\&\quad +\frac{1}{2}\log \left( (\sigma _{142}\sigma _{231}+\sigma _{141}\sigma _{232}-\sigma _{132}\sigma _{241}-\sigma _{131}\sigma _{242})^{2}\right. \nonumber \\&\quad \left. +(1+\sigma _{141}\sigma _{231}-\sigma _{142}\sigma _{232}-\sigma _{131}\sigma _{241}+\sigma _{132}\sigma _{242})^{2}\right) \,, \end{aligned}$$

(48)

where we have set $q=4$ for simplicity and rescaled $\sigma$ to 1 since we are interested in the case $\sigma \ne 0$. The $g_{abi}$ and $\sigma _{abj}$ are the real and imaginary parts of the field $g_{ab}$ and $\sigma _{ab}$

$$\begin{aligned} g_{ab}=g_{ab1}+\text {i}g_{ab2},\quad \sigma _{ab}=\sigma _{ab1}+\text {i}\sigma _{ab2}\ . \end{aligned}$$

(49)

The downward flow equations of the Morse function are

$$\begin{aligned} \frac{\text {d}g_{abj}}{\text {d}t}=-\frac{\partial h}{\partial g_{abj}},\quad \frac{\text {d}\sigma _{abj}}{\text {d}t}=-\frac{\partial h}{\partial \sigma _{abj}}\ . \end{aligned}$$

(50)

The end point of each anti-thimble is one of the saddles, labeled by c, at $g_{abj}^{c}$ and $\sigma _{abj}^{c}$, which leads to the following boundary conditions of the flow equation

$$\begin{aligned} \lim _{t\rightarrow +\infty }g_{abj}=g_{abj}^{c},\quad \lim _{t\rightarrow +\infty }\sigma _{abj}=\sigma _{abj}^{c}\ . \end{aligned}$$

(51)

We can then solve the flow equation and obtain the Lefschetz anti-thimbles going through each saddle point. If they intersect with the original integration contour the saddle point contributes to the integral.

For example in Fig. 1 we illustrate examples of the anti-thimbles of the saddle point

$$\begin{aligned}{} & {} g_{13}=1,\quad g_{24}=-1,\quad g_{14}=(-1)^{3/4},\quad g_{23}=(-1)^{1/4}, \end{aligned}$$

(52)

$$\begin{aligned}{} & {} \sigma _{13}=\text {i},\quad \sigma _{24}=-\text {i},\quad \sigma _{14}=(-1)^{3/4},\quad \sigma _{23}=-(-1)^{1/4}\,, \end{aligned}$$

(53)

which do not intersect with the original integration contour, namely the real axis. This means the contribution of this saddle should not be included in the integral.

Examples of anti-thimbles of another saddle point

$$\begin{aligned}{} & {} g_{13}=-(-1)^{1/4},\quad g_{24}=(-1)^{3/4},\quad g_{14}=-1,\quad g_{23}=-1, \end{aligned}$$

(54)

$$\begin{aligned}{} & {} \sigma _{13}=(-1)^{1/4},\quad \sigma _{23}=(-1)^{3/4},\quad \sigma _{14}=-\text {i},\quad \sigma _{23}=-\text {i}\,, \end{aligned}$$

(55)

is shown in Fig. 2. Again they do not intersect with the real axis so the contribution from this saddle should not be included either.

Fig. 2

Anti-thimble on the $g_{13}$ plane (left) and the $g_{24}$ plane (right)

We can run this analysis over all the nontrivial saddles, and we find none of them contribute to the integral.

Fig. 3

The shaded region is where a non-trivial saddle in (56) dominates over the trivial saddle. The plot for the other two non-trivial saddles can be obtained from this plot by simple rotations

$\bullet$ When $x=-y$, there are also nontrivial saddle points, and a similar analysis of Lefschetz-thimbles demonstrates that they do not contribute to the integral either.

Actually, there is a quicker way to arrive at the same conclusion. We find that the on-shell actions corresponding to these saddle points are

$$\begin{aligned} \left( \frac{\sigma ^2}{2}\right) ^{\frac{N}{3}}e^{-N\pm \frac{3}{2}2^{\frac{1}{3}}N e^{\frac{2\text {i}m \pi }{3}}\sigma ^{\frac{4}{3}}},\quad m=0,\pm 1. \end{aligned}$$

(56)

However, these saddle points should be saddle points of the entire multi-dimensional integral including the integral over $\sigma$. As a result, this saddle should also satisfy the fall-off condition of the $\sigma$ integral, otherwise, they will not contribute to the $\sigma$ integral. Therefore we should only consider the decaying saddle points namely

$$\begin{aligned} \left( \frac{\sigma ^2}{2}\right) ^{\frac{N}{3}}e^{-N +\frac{3}{2}2^{\frac{1}{3}}N e^{\pm \frac{2\text {i}\pi }{3}}\sigma ^{\frac{4}{3}}},\quad \left( \frac{\sigma ^2}{2}\right) ^{\frac{N}{3}}e^{-N -\frac{3}{2}2^{\frac{1}{3}}N \sigma ^{\frac{4}{3}}}.\nonumber \\ \end{aligned}$$

(57)

We plot the region where these non-trivial saddle dominates over the trivial saddle in Fig. 3, and it is easy to observe from the figure that the wormhole saddle (2021) of $\langle z^2\rangle$, located at $|\sigma |=1$, is in the region where the trivial saddle dominates.

Another family of solutions to the equation of motion (40) has $x=0$ or $y=0$. On shell actions on these saddles behave as

$$\begin{aligned} \sigma ^{\frac{2N}{3}}e^{-N +\frac{3}{2}N e^{\pm \frac{2\text {i}\pi }{3}}\sigma ^{\frac{4}{3}}},\quad \sigma ^{\frac{2N}{3}}e^{-N -\frac{3}{2}N \sigma ^{\frac{4}{3}}}, \end{aligned}$$

(58)

whose dominant regions are similar to Fig. 3 and they are sub-leading compared with the trivial saddle.

Since the trivial saddle is on the original integration contour, putting all the results together we confirm that the path integral over $g_{ab}$ and $\sigma _{ab}$ can be approximated entirely by the trivial saddle point. Due to (42), we conclude that the wormhole saddle (2021) is in the self-averaging region.

2.3.3 The saddle points analysis: $\sigma = 0$, the half-wormhole saddles

The analysis in the above sub-sections concludes that the leading saddle point contributions to the $\Phi (\sigma )$ function are all proportional to positive powers of $\sigma$. However, this raises a puzzle: all these results vanish at $\sigma =0$, but from the exact result in (3738) we know

$$\begin{aligned} \langle \Phi (0)^2\rangle _{\text {exact}} \sim 2q e^{-\frac{3}{2}N}\ne 0\ . \end{aligned}$$

(59)

at $\sigma =0$. This indicates that there must be other saddle points, which are missed in the previous analysis due to being subleading at generic $\sigma \ne 0$, becomes important at $\sigma =0$. This is possible because in the large-N limit the $\Psi (\sigma )$ function is peaked at the origin, so other saddle points could give a large contribution near the origin. In this section, we thus focus on the $\sigma \sim 0$ region of the integration and look for new saddle point contributions.

In practice, we can apply the same analysis as in the previous section, except that now we evaluate at $\sigma \sim 0$. As expected, the trivial saddle gives

$$\begin{aligned} e^{N\log (\sigma )} \sim 0\ . \end{aligned}$$

(60)

At $\sigma =0$, the subleading non-trivial saddles (57) and (58) discussed in the previous section now have different on-shell values

$$\begin{aligned} \frac{e^{-\frac{3}{2}N}}{2^{N/2}},\quad e^{-\frac{3}{2}N}\ , \end{aligned}$$

(61)

respectively. So (58) dominates. Adding them up and including the 1-loop correction, the result agrees precisely with the exact solution (59)

$$\begin{aligned} \langle \Phi (0)^2\rangle =2q e^{-\frac{3}{2}N}\,. \end{aligned}$$

(62)

We can continue to carry out the sigma integral to get the contribution from this saddle to the $z^2$, since the saddle is supported at $\sigma =0$, this is easily done the result is simply $\Phi (0)$.

A general lesson we can learn from this computation is that the half-wormhole saddle points always exist. But most of the time they are hidden behind the leading saddles. Nevertheless, they become important whenever the leading saddle decreases faster, e.g. the $\sigma \sim 0$ region in this case.

With both the wormhole and the half-wormhole saddle contributions, we can now approximate

$$\begin{aligned} z^2\approx \langle z^2\rangle +\Phi (0)\ . \end{aligned}$$

(63)

The wormhole saddle is holographically dual to bulk worm-hole-like geometry, and the half-wormhole saddles are conjectured to be dual to half-wormhole-like configurations that are typically sub-dominant. This result indicates one way to resolve the factorization problem; when we consider bulk gravitational path integral, the factorization problem is caused by only considering the wormhole-like connected geometry, if other sub-dominant contributions, such as the half-wormhole geometries are also taken into account, the factorization property will be restored (so that the result is $z^2\equiv z_L z_R$ that factorizes).

3 Sourced one-time SYK: a disk-and-cylinder model

An important difference between the 0d-SYK model and the 1d-SYK model is that the averaged partition function $\langle z\rangle$ vanishes in the 0d model. From the bulk gravity point of view, this corresponds to the exclusion of the gravity configuration where a surface with the disk topology fills a single boundary in the bulk. as shown in [72]. JT gravity admits a limit where the bulk geometry can always be approximated by disks and cylinders. Therefore to understand if the discussion in the previous model is also applicable when disk topology is also allowed in the bulk, we consider a sourced 0d-SYK model where the random coupling is drawn from a Gaussian distribution ${{\mathcal {N}}}(u,t^2)$ with non-zero mean⁴

$$\begin{aligned} \langle J_A\rangle =J_A^0=u,\quad \langle J_A^2\rangle -\langle J_A\rangle ^2=\tau ^2 \frac{(q-1)!}{N^{q-1}}\equiv t^2. \end{aligned}$$

(64)

Since a non-zero expectation value of the couplings is equivalent to turning on a source term of the random couplings, we call this model “sourced” 0d-SYK model.

The ensemble averaged quantities can be computed directly by averaging over the couplings and integrating out the fermions

$$\begin{aligned} \langle z\rangle= & {} \text {PF}(J^0), \end{aligned}$$

(65)

$$\begin{aligned} \langle z^2\rangle= & {} \int \text {d}^{2N}\psi \nonumber \\{} & {} \times \exp \left( \text {i}^q t^2 \sum _A \psi _A^L \psi _A^R +i^{q/2}J_A^0\left( \psi _A^L+\psi _A^R\right) \right) \nonumber \\ \end{aligned}$$

(66)

$$\begin{aligned}= & {} \sum '_{A,B}\text {sgn}(A)\text {sgn}(B)\left( J_{A_1}^0J_{B_1}^0+\delta _{A_1 B_1}t^2\right) \dots \nonumber \\{} & {} \dots \left( J_{A_p}^0J_{B_p}^0+\delta _{A_p B_p}t^2\right) . \end{aligned}$$

(67)

Our main results about this model are

• 1. The self-averaging part of z does not persist; they are subdominant compared with the non-self-averaging contribution in the large N limit.

• 2. The half-wormhole contribution $\Phi$ can be improved so that the approximation $z^2\approx \langle z^2\rangle +\Phi$ is still good in this model.

3.1 The averaged quantities

Let us first compute the averaged quantities. We again proceed by looking for proper collective variables and establish a saddle point analysis that’s similar to the model discussed in the previous section.

The ensemble average of z is simply the ensemble average of the hyperPfaffian (6)

$$\begin{aligned} \langle z\rangle =\sum '_{A_1<\dots <A_p} \text {sgn}(A)J^0_{A_1}\dots J^0_{A_p}=m[p] u^p, \end{aligned}$$

(68)

where p is not summed over, $pq=N$ and the factor m[p] is defined as

$$\begin{aligned} m[p]=\frac{(pq/2)!}{p!((q/2)!)^p}. \end{aligned}$$

(69)

This result can alternatively be derived by introducing a collective variable

$$\begin{aligned} G=\sum _{i<j}\psi _i \psi _j, \end{aligned}$$

(70)

followed by a similar computation as we show in (15). Introducing the following collective variables

$$\begin{aligned}{} & {} G_{LR}=\frac{1}{N}\sum _i \psi _i^L \psi _i^R, \end{aligned}$$

(71)

$$\begin{aligned}{} & {} G_L=\frac{1}{N}\sum _{i<j}\psi _i^L \psi _j^L,\quad G_R=\frac{1}{N}\sum _{i<j}\psi _i^R \psi _j^R, \end{aligned}$$

(72)

we can compute the averaged quantity $\langle z^2\rangle$

$$\begin{aligned} \langle z^2\rangle =\sum _{k=0}^p c[k]\,t^{2k} \left( m[{p-k}]\,u^{p-k}\right) ^2, \end{aligned}$$

(73)

where m[p] is defined in (68) and the coefficient c[k] is

$$\begin{aligned} c[k]{} & {} =\frac{1}{k!}{ N\atopwithdelims ()q }{ N-q\atopwithdelims ()q }\dots {N-(k-1)q\atopwithdelims ()q} \end{aligned}$$

(74)

$$\begin{aligned}{} & {} =\frac{N!}{k! (q!)^k (N-kq)!}. \end{aligned}$$

(75)

The details of the derivations of $\langle z\rangle$ and $\langle z^2\rangle$ are presented in Appendix B. The averaged partition function (68) is proportional to $u^p$ because in each term of the hyperPfaffian there are no repeated $J_{A_i}$ so the result does not depend on t; rather, each random coupling has to “contract” with itself thus producing p copies of the factor of u. The polynomial expression of the averaged squared partition function (73) can be also understood from summing over the Feynman diagrams as shown in Fig. 5. It turns out that each diagram in Fig. 5 correpond to a term $z_2^{(k)}$ in (73), i.e.

$$\begin{aligned} \langle z^2\rangle =\sum _{k=0}^p z_2^{(k)}, \quad z_2^{(k)}=c[k]\,t^{2k} \left( m[{p-k}]\,u^{p-k}\right) ^2. \end{aligned}$$

(76)

Diagramatically, the $z_2^{(k)}$ come as follows. We first contract k pairs of $J_{A_i}$ (in the diagram the contraction is denoted by a blue line connecting $z_L$ and $z_R$) which gives the factor $t^{2k}$ and c[k] is the total number of different contractions. Each of the rest $J_A$ becomes $\mu$ in the average and they contribute a factor $\left( m[{p-k}]\,u^{p-k}\right) ^2$ (in the diagram the contraction is denoted by a red line connecting $z_L$ or $z_R$ with a red dot.).

In the large-N limit, we can find the dominant terms by computing the ratio⁵

$$\begin{aligned} r_k{} & {} =\frac{z_2^{(k)}}{z_2^{(k-1)}}\nonumber \\{} & {} =\frac{t^2 (-k+p+1) (-4 k+4 p+1) (-4 k+4 p+3)}{3 u ^2 (2 k (p-k)+k)} \nonumber \\ \end{aligned}$$

(77)

$$\begin{aligned} r_p{} & {} =\frac{t^2}{p u^2},\quad r_{1}\sim \frac{p^2 t^2}{ u^2}, \end{aligned}$$

(78)

here for simplicity we have chosen $q=4$. First we notice that $r_k$ decreases with respect to k, namely

$$\begin{aligned} r_{k}>r_{k+1} \end{aligned}$$

(79)

Therefore if $r_1 \ll 1$ i.e.

$$\begin{aligned} \frac{u}{t }\gg {p}, \end{aligned}$$

(80)

then the dominant term will be

$$\begin{aligned} \langle z^2\rangle \approx z_2^{(0)}=\left( m[{p}]u^{p}\right) ^2\ . \end{aligned}$$

(81)

In this case, we clearly have

$$\begin{aligned} \langle z^2\rangle \approx \langle z\rangle ^2\ . \end{aligned}$$

(82)

Since the geometric meaning of $z^{(0)}_2$ is two disconnected disks, the above result means in this regime, this “two-disk” saddle is dominant, which results in a self-averaging z due to (82). This behavior resembles the early-time characteristics of the SFF of the 0 + 1 SYK model.

On the other hand, if $r_p\gg 1$ i.e.

$$\begin{aligned} \frac{u}{t }\ll \frac{1}{\sqrt{p}} \end{aligned}$$

(83)

the dominant term will be $z_2^{(p)}$. Geometrically, this contribution corresponds to connected wormhole configurations. Therefore in this regime the wormhole saddle dominates, and z is non-self-averaging.

In the rest regime of the parameter $\frac{u}{t }$

$$\begin{aligned} \frac{1}{\sqrt{p}}\lesssim \frac{u}{t }\lesssim p, \end{aligned}$$

(84)

neither the disk nor the wormhole saddle point dominates. It suggests that there might be a new saddle point that contributes the most. It turns out when our toy model has a well-defined large N limit, the parameters u and t lie in this regime.

We now examine this result more carefully by a saddle point analysis. As we show in the Appendix B, by introducing the $G,\Sigma$ variable we can rewrite $\langle z\rangle$ as

$$\begin{aligned} \langle z\rangle _{}=\int _{{\mathbb {R}}}\text {d}G \int _{\text {i}{\mathbb {R}}}\frac{\text {d}\Sigma }{2\pi \text {i}/N}\Sigma ^{N/2} e^{u \text {i}^{q/2}\frac{N^{q/2}}{(q/2)! }G^{q/2}} e^{- N\Sigma G}\,. \end{aligned}$$

(85)

We again rotate the integral contour as in the model with zero mean

$$\begin{aligned} \Sigma \rightarrow \text {i}e^{-\text {i}\frac{2\pi }{q}} \sigma ,\quad G\rightarrow e^{\text {i}\frac{2\pi }{q}} g, \end{aligned}$$

(86)

which leads to the action:

$$\begin{aligned} \langle z\rangle _{}= & {} \int _{{\mathbb {R}}}\frac{\text {d}g \text {d}\sigma }{2\pi /N }\exp \left\{ \frac{N}{2}\left( \log (\text {i}e^{-\frac{2\pi \text {i}}{q}}\sigma )\right. \right. \nonumber \\{} & {} \left. \left. -2\text {i}\sigma g-\frac{2\mu }{q}g^{q/2} \right) \right\} , \end{aligned}$$

(87)

with

$$\begin{aligned}{} & {} \mu \equiv \text {i}^{q/2}u \frac{2{N}^{q/2-1}}{(q/2-1)!},\nonumber \\{} & {} \leftrightarrow \quad u=(-\text {i})^{q/2}\mu \frac{(q/2-1)!}{2 N^{q/2-1}}. \end{aligned}$$

(88)

The saddle point equations are

$$\begin{aligned}{} & {} \frac{1}{\sigma }-2\text {i}g=0,\quad -2\text {i}\sigma -\mu g^{q/2-1}=0,\nonumber \\{} & {} \rightarrow \quad \mu g^{q/2}=-1. \end{aligned}$$

(89)

Comparing (87) with (18) it is easy to find that to reproduce the exact result (76) we have to add the contributions from all the q/2 saddles. For the choices (64) and (88) we have

$$\begin{aligned} \frac{u}{t}\sim \frac{\mu }{\tau } \frac{(q/2-1)!}{\sqrt{(q-1)!}} N^{\frac{1}{2}}\sim \sqrt{p}, \end{aligned}$$

(90)

which exactly lies in the regime (84). To find the new saddle explicitly, we start from a path integral expression of $\langle z^2\rangle$

$$\begin{aligned} \langle z^2\rangle= & {} \int _R \text {d}^3 G_i \int _{\text {i}{\mathbb {R}}}\text {d}^3 \Sigma _i\, e^{\frac{N}{q}(\tau ^2 G_{LR}^q+\mu G_L^{q/2}+\mu G_{R}^{q/2})- N(\Sigma _i G_i)} \nonumber \\{} & {} \times \frac{1}{2}\left( (\Sigma _{LR}+\text {i}\sqrt{\Sigma _L\Sigma _R})^N+(\Sigma _{LR}-\text {i}\sqrt{\Sigma _L\Sigma _R})^N\right) \, \nonumber \\ \end{aligned}$$

(91)

whose detailed derivation is in  Appendix B. The saddle point equations are

$$\begin{aligned}{} & {} G_{L(R)}^{-1+\frac{q}{2}}=\frac{2}{\mu }\Sigma _{L(R)},\quad G_{LR}^{-1+q}=\frac{1}{\tau ^2}\Sigma _{LR}, \end{aligned}$$

(92)

$$\begin{aligned}{} & {} G_{L(R)}=\frac{\text {i}\Sigma _{R(L)}}{2\sqrt{\Sigma _L \Sigma _R}}\frac{f_+^{n-1}-f_-^{n-1}}{f_+^n+f_-^n}, \end{aligned}$$

(93)

$$\begin{aligned}{} & {} G_{LR}=\frac{f_+^{n-1}+f_-^{n-1}}{f_+^n+f_-^n}, \end{aligned}$$

(94)

where $f_\pm =\Sigma _{LR}\pm \text {i}\sqrt{\Sigma _L\Sigma _R}$. For simplicity, we choose $\tau ^2=\mu =1$. There are always two types of trivial solutions

$$\begin{aligned}&\text {wormhole solution}:&G_L=G_R=0,\nonumber \\{} & {} G_{LR}=e^{\frac{2\text {i}m\pi }{q}}, \end{aligned}$$

(95)

$$\begin{aligned}&\text {disconnect solution}:&G_{LR}=0,\quad G_{L}=e^{\frac{4\text {i}m_L \pi }{q}},\nonumber \\{} & {} G_{R}=e^{\frac{4\text {i}m_R \pi }{q}} \end{aligned}$$

(96)

with on-shell action respectively

$$\begin{aligned}{} & {} \text {wormhole solution}: \langle z^2\rangle _{\text {wh}}=e^{-N(1-\frac{1}{q})}e^{\frac{2\text {i}m \pi N}{q}} \end{aligned}$$

(97)

$$\begin{aligned}{} & {} \text {disconnect solution}:\nonumber \\{} & {} \qquad \qquad \langle z^2\rangle _{\text {dis}}={2^{-N}}e^{-N(1-\frac{2}{q})}{e^{\frac{4\text {i}m \pi N}{q}}}. \end{aligned}$$

(98)

Note that the ratio of these two contributions is

$$\begin{aligned} \frac{\langle z^2\rangle _{\text {wh}}}{\langle z^2\rangle _{\text {dis}}}=\left( 2e^{-1/q}\right) ^N, \end{aligned}$$

(99)

so when $q\ge 2$ the wormhole contributes more. We find another solution where only one of $(f_+)^N$ and $(f_-)^N$ survives in the large N limit. Assuming $f^N_-\rightarrow 0,N\rightarrow \infty$, (94) simplifies to

$$\begin{aligned}{} & {} G_{L(R)}=\frac{\Sigma _{R(L)}}{-2\text {i}\sqrt{\Sigma _R\Sigma _L}}\frac{1}{\Sigma _{LR}+\text {i}\sqrt{\Sigma _L \Sigma _R}}, \end{aligned}$$

(100)

$$\begin{aligned}{} & {} G_{LR}=\frac{1}{\Sigma _{LR}+\text {i}\sqrt{\Sigma _L \Sigma _R}}, \end{aligned}$$

(101)

which leads to

$$\begin{aligned} G_{LR}^q+G_{R}^{q/2}+G_{L}^{q/2}=1,\quad G_{R}^{q/2}=G_{L}^{q/2}. \end{aligned}$$

(102)

For the case of $q=4$, (9293) and (102) can be solved explicitly and gives the following contribution to the integral (91)

$$\begin{aligned}{} & {} \langle z^2\rangle _{\text {non-trivial}+}\approx e^{-0.63 N}e^{\frac{2m\text {i}\pi N}{4}}\nonumber \\{} & {} >\langle z^2\rangle _{\text {wh}}= e^{-0.75 N}e^{\frac{2m\text {i}\pi N}{4}}. \end{aligned}$$

(103)

We also checked that these solutions indeed satisfies $\lim _{N\rightarrow \infty } f_-^N = 0$. There are similar saddles that satisfy $f_+^N=0$. Therefore we conclude that in the large N limit the dominate saddles are the non-trivial ones.

3.2 The non-self-averaged contributions to z

Contrary to the model with zero means, we expect a non-vanishing “disk” saddle point in this $u\ne 0$ model, which gives $\langle z\rangle \ne 0$, in the path integral representation of z. Moreover, we will show that there are new saddle point contributions to z as shown in Fig. 4, analogous to the half-wormhole contribution to $z^2$ in the previous model with zero means, which we call the “punctured disk” (or “single half-wormhole”) saddles.

Fig. 4

A pictorial illustration of the “disk” and the “Punctured disk” saddle of z respectively

With the help of the collective variables (72), we insert the identity

$$\begin{aligned} 1= & {} \int _{-\infty }^\infty \text {d}G_h\int _{-\text {i}\infty }^{\text {i}\infty } \frac{N\text {d}\Sigma _h}{2\pi \text {i}}\nonumber \\{} & {} \times e^{-\Sigma _h(NG_h-\sum _{i<j}\psi _i \psi _j)+\frac{N\mu }{q}\left( G_h^{q/2}-\left( \frac{1}{N}\sum _{i<j}\psi _i \psi _j\right) ^{q/2}\right) }, \nonumber \\ \end{aligned}$$

(104)

into the non-averaged partition function z

$$\begin{aligned} z=\int \text {d}^N \psi \exp \left( \text {i}^{q/2} J_{i_1\dots i_q}\psi _{i_1\dots i_q}\right) . \end{aligned}$$

(105)

To make the integral well defined, we again rotate the contour by $\Sigma _h=\text {i}e^{-2\text {i}\pi /q}\sigma _h,G_h=e^{2\text {i}\pi /q}g_h$, then z can be cast into the form

$$\begin{aligned} z=\int _{-\infty }^\infty \frac{N\text {d}\sigma _h}{2\pi } \Psi (\sigma _h)\hat{\Theta }(\sigma _h), \end{aligned}$$

(106)

where the first factor is similar to (29)

$$\begin{aligned} \Psi (\sigma _h)=\int _{{\mathbb {R}}}\frac{\text {d}g_h}{2\pi /N} \exp \left[ N\left( -\text {i}\sigma _h g_h-\frac{\mu }{q} g_h^{q/2}\right) \right] , \end{aligned}$$

(107)

and the second factor is

$$\begin{aligned} \hat{\Theta }(\sigma _h)= & {} \int \text {d}^N \psi \exp \left[ \text {i}e^{-\frac{2\text {i}\pi }{q}}\sigma _h \sum _{i<j}\psi _i \psi _j \right. \nonumber \\{} & {} \left. +\text {i}^{q/2} J_A\psi _A-\text {i}^{q/2}u \sum _A\psi _A \right] . \end{aligned}$$

(108)

The function $\Psi (\sigma _h)$ is again peaked at $\sigma _h=0$, so a naive generalization of the proposal of the existence of the half-wormhole saddle suggests the approximation

$$\begin{aligned} z\approx \langle z\rangle +\Theta _1, \end{aligned}$$

(109)

where

$$\begin{aligned} \Theta _1= & {} \hat{\Theta }(0)=\text {Pf}(J-J^0)\nonumber \\= & {} \sum '_{A}\text {sgn}(A)(J_{A_1}-J_{A_1}^{0})\dots (J_{A_p}-J_{A_p}^{0}). \end{aligned}$$

(110)

To examine this approximation, we define the error function:

$$\begin{aligned} \text {Error}=z-\langle z\rangle -\Theta _1\, \end{aligned}$$

(111)

and compute variance of the error

$$\begin{aligned} \langle \text {Error}^2\rangle =\langle z^2\rangle -\langle z\rangle ^2+\langle \Theta _1^2\rangle -2\langle z\Theta _1\rangle . \end{aligned}$$

(112)

The quantities $\langle z^2\rangle , \langle \Theta ^2_1\rangle ,\langle z\Theta _1\rangle$ can be computed by summing over the Feynman diagrams in Fig. 5.

Fig. 5

Feynman diagrams for $\langle z^2\rangle , \langle \Theta _1^2\rangle ,\langle z\Theta _1\rangle$. Each black dot represents a factor of z, each red dot and the attached line represents a contraction with the $J_A^0$ source, and each blue line is a contraction of a pair of $J_A$. So the diagram containing k blue lines corresponds to $z_2^{(k)}$ in (76)

Recalling that $\langle z^2\rangle =\sum _{k=0}^{p} z_2^{(k)}$ which is given by summing over all the diagrams, $z_2^{(0)}=\langle z \rangle ^2$ which is given by the last diagram in Fig. 5, $z_2^{(p)}=\langle z^2\rangle _{\mu =0}$ which is given by the first diagram in Fig. 5 and $\langle \Theta _1^2\rangle =\langle \Theta _1 z\rangle =z_2^{(p)}$, we find

$$\begin{aligned}{} & {} \langle \text {Error}^2\rangle = \sum _{k=1}^{p-1} z_2^{(k)}. \end{aligned}$$

(113)

Based on our analysis in the previous section, this error is negligible only when the ratio of u to t is very small and z exhibits self-averaging, or when the ratio is very large and z is mostly non-self-averaging. However, within the regime defined by inequality (84), the error becomes non-negligible, rendering the naive half-wormhole proposal (109) invalid. The underlying reason for this failure is the emergence of a new saddle point when we tune the parameter u.

One possibility of what is happening in this parameter regime (84) is that a specific Feynman diagram in Fig. 5, denoted as $z_2^{(k)}$, will dominate the summation (76) in the large N limit. If this is the case, it is possible to find a dominant non-self-averaging contribution $\Theta ^{(k)}$, which we call the “punctured disk” to z such that

$$\begin{aligned} z\approx \langle z\rangle + \Theta ^{(k)}\,, \end{aligned}$$

(114)

where the value of k is determined by the value of the parameter u/t.

The non-trivial point is that if this is true, then the approximation of $z^2$ is in the precise form of the aforementioned proposal

$$\begin{aligned} \langle z^2\rangle \approx z_2^{(k)}+\langle z\rangle ^2\ . \end{aligned}$$

(115)

One proposal for the $\Theta ^{(k)}$ is

$$\begin{aligned} \Theta ^{(k)}=\int \text {d}^N\psi \frac{(\text {i}^{q/2}J_A^{(0)}\psi _A)^{p-k}}{(p-k)!}e^{\text {i}^{q/2}(J_B-J_B^{(0)}) \psi _B}. \end{aligned}$$

(116)

For examples

$$\begin{aligned} \Theta ^{(p-1)}{} & {} = \sum _A'\text {sgn}(A)(J_{A_1}-J_{A_1}^0)(J_{A_2}-J_{A_2}^0)\dots \nonumber \\{} & {} \quad \dots J_{A_i}^0\dots (J_{A_p}-J_{A_p}^0)\,, \end{aligned}$$

(117)

$$\begin{aligned} \Theta ^{(p-2)}{} & {} = \sum _A'\text {sgn}(A)(J_{A_1}-J_{A_1}^0)\dots J_{A_i}^0\dots \nonumber \\{} & {} \quad \dots J_{A_j}^0\dots (J_{A_p}-J_{A_p}^0)\,, \end{aligned}$$

(118)

$$\begin{aligned} \Theta ^{(0)}{} & {} =\langle z\rangle \,,\quad \Theta ^{(p)}=\Theta _1. \end{aligned}$$

(119)

Then from the Feynman diagrams in Fig. 5 it is not hard to find that

$$\begin{aligned} \langle {\Theta ^{(k)}}{\Theta ^{(k)}}\rangle =\langle \Theta ^{(k)} z\rangle = z_2^{(k)}. \end{aligned}$$

(120)

This result ensures that (115) is true.

We will present a further analysis from another approach to this model somewhere else.

3.3 The non-self-averaged contributions to $z^2$

In the previous section, we considered the non-self-averaged contribution $\Theta ^{(k)}$ to the partition function z. In this section, we study the non-self-averaged contribution to $z^2$ and try to understand its relationship with the $z_2^{(k)}$.

The result (114) immediately gives

$$\begin{aligned} z^2&= \left( \langle z\rangle + \Theta ^{(k)}\right) ^2=\langle z\rangle ^2+2 \langle z\rangle \Theta ^{(k)}+\left( \Theta ^{(k)}\right) ^2\ . \end{aligned}$$

(121)

In the previous section, we have shown that this relation leads to (115). Using (115), we can further rewrite this relation into

$$\begin{aligned} z^2&= \langle z^2\rangle -z_2^{(k)}+2 \langle z\rangle \Theta ^{(k)}+\left( \Theta ^{(k)}\right) ^2\ . \end{aligned}$$

(122)

This provides an approximation to $z^2$

$$\begin{aligned} z^2\approx \langle z^2\rangle +\Lambda ^{(k)}\,, \end{aligned}$$

(123)

where

$$\begin{aligned} \Lambda ^{(k)}=-z_2^{(k)}+2 \langle z\rangle \Theta ^{(k)}+\left( \Theta ^{(k)}\right) ^2 \end{aligned}$$

(124)

We thus observe that once the punctured disk contribution $\Theta ^{(k)}$ is known, all the higher boundary non-self-averaging contributions can be recursively determined. Additionally, a geometric interpretation of (124) is that the sum of connected two-boundary contributions, including the wormhole contribution $z_2^{(k)}$ and the linked half-wormhole $\Lambda ^{(k)}$ is the same as the sum of all non-self-averaging disconnected contributions, either one disk plus one punctured disk or two punctured disks. This can be shown in Fig. 6.

One might wonder whether $\Lambda ^{(k)}$ has a similar expression as the half-wormhole contribution in the model with zero mean that was introduced in [73] and recast in (63). In Appendix C, we demonstrate that this is not the case.

Fig. 6

A pictorial illustration of equation (124)

4 Modified SYK at one time point: beyond Gaussian approximation

The models considered in the literature so far only involve random couplings drawn from Gaussian distributions. On the other hand, SYK-like field theories with other kinds of random couplings are expected to have similar chaotic behaviors as the Gaussian SYK model does. So it is possible that in the low energy limit they also admit effective gravitational descriptions. In particular, explicit examples of field theories with random variables subjecting to other distributions includes the Poisson random variable appearing in the theory of [52, 73, 82]. Additionally, it is conjectured [22] that any 2-dimensional dilaton gravity theory possesses a dual random matrix description that is generally non-Gaussian. It is therefore interesting to consider random couplings beyond Gaussian distributions and check if there are other non-self-averaging contributions to these models. Separating the physical observables into the self-averaging and non-self-averaging parts is generically applicable in ensemble average theories, so we expect that there always exist half-wormhole-like non-self-averaging saddles for various different observables. We will demonstrate how it works in this section and further understand the relation between the different non-self-averaging quantities.

4.1 SYK at one time point: $\langle J_a\rangle =0,\quad \langle J_a^4\rangle _c\ne 0$

In this section, we consider theories whose random couplings have vanishing mean values and non-trivial quadrupole moments, namely

$$\begin{aligned} \langle J_A\rangle =0\,,\qquad \langle J_A^2\rangle = t^2\,, \qquad \langle J_A^4\rangle =v^4+3 \langle J_A^2\rangle ^2\ . \end{aligned}$$

(125)

Note that the introduction of non-vanishing $\langle J_A^4\rangle -3 \langle J_A^2\rangle ^2$ could potentially alter the outcome of $\langle z^4\rangle$, but not $\langle z^2\rangle$. As a result, we expect that the original half-wormhole proposal should be modified. The additional contribution to $\langle z^4\rangle$ can be attributed to a new wormhole saddle that has four boundaries. In addition, it is reasonable to believe that when $v<t$, this new wormhole saddle is negligible and the original half-wormhole saddle remains valid. We will confirm this through a direct calculation in the following section.

It is easy to compute the correlation functions of the partition function of this model

$$\begin{aligned} \langle z\rangle =0,\quad \langle z^2\rangle =\frac{N!}{p!(q!)^p}t^2. \end{aligned}$$

(126)

The quadupole moments of $J_A$ in (125) contributes nontrivially to $\langle z^4 \rangle$

$$\begin{aligned} \langle z^4\rangle= & {} \sum _{A,B,C,D}'\text {sgn}(A)\text {sgn}(B)\text {sgn}(C)\text {sgn}(D)\nonumber \\{} & {} \times \left\langle J_{A_1}J_{B_1}J_{C_1}J_{D_1}\cdots J_{A_p}J_{B_p}J_{C_p}J_{D_p} \right\rangle , \end{aligned}$$

(127)

which can be expanded

$$\begin{aligned}{} & {} \langle z^4\rangle =\sum _{k=0}^{p} c[k] n[N-qk] v^{4k} t^{4(p-k)}\equiv \sum _k z_4^{(k)},\nonumber \\{} & {} n[{N}]=\frac{N!}{(q!)^{2N/q}}\sum _{\begin{array}{c} n_1+n_2+n_3=N/q\\ n_i\ge 0 \end{array}}\frac{(qn_1)!(qn_2)!(qn_3)!}{(n_1!n_2!n_3!)^2},\nonumber \\ \end{aligned}$$

(128)

where c[k] defined in (7475) is the number of ways to choose k q-subsets out of N and n[N] is the multiplicities coming from the different Wick contractions. In particular, we get

$$\begin{aligned} \langle z^4\rangle _{v=0}=n[N] t^{4p}, \end{aligned}$$

(129)

which reduces to the result in [73]. To find the dominant term in the large N limit let us define the ratio

$$\begin{aligned}{} & {} \tilde{r}_k=\frac{z_4^{(k)}}{z_4^{(k-1)}}\sim \frac{v^4}{t^4}\frac{1-k+p}{k}\frac{4! (4p-kp)!}{(4p-4k+4)!}, \end{aligned}$$

(130)

$$\begin{aligned}{} & {} {\tilde{r}}_1\sim \frac{v^4}{t^4}\frac{1}{p^{2}},\quad {\tilde{r}}_p\sim \frac{\upsilon ^4}{t^4}\frac{1}{p} \,, \end{aligned}$$

(131)

where we have again taken $q=4$ for simplicity. We find that ${\tilde{r}}_k$ initially decreases and then increases as k increases. For $p>1$ we have ${\tilde{r}}_p>{\tilde{r}}_1$, so ${\tilde{r}}_p$ is the maximal value. Therefore If ${\tilde{r}}_p\ll 1$ i.e.

$$\begin{aligned} \frac{v^4}{t^4}\ll p, \end{aligned}$$

(132)

then the dominant term will be $z_4^{(0)}$ and the contributions from non-trivial higher moments, e.g. the nontrivial quadrupole moments proportional to v, can be ignored. Then the situation will be similar to the previous models with $v=0$. Namely, the half-wormhole saddle of $z^2$, when $\langle J_A\rangle =0$, can be written as

$$\begin{aligned} \Phi&=\sum '_{A,B}\text {sgn}(A)\text {sgn}(B)\left( J_{A_1}J_{B_1}-\delta _{A_1 B_1}t^2\right) \dots \nonumber \\&\quad \dots \left( J_{A_p}J_{B_p}-\delta _{A_p B_p}t^2\right) \,, \end{aligned}$$

(133)

such that

$$\begin{aligned} \langle \Phi ^2\rangle \approx \langle \Phi z^2\rangle \approx 2\langle z^2\rangle ^2, \end{aligned}$$

(134)

and

$$\begin{aligned} \langle \text {Error}^2\rangle= & {} \langle z^4\rangle -\langle z^2\rangle ^2+\langle \Phi ^2\rangle -2\langle z^2\Phi ^2\rangle \nonumber \\\approx & {} 3\langle z^2\rangle ^2-\langle z^2\rangle ^2+2\langle z^2\rangle ^2-4\langle z^2\rangle ^2=0, \end{aligned}$$

(135)

in the leading order of N as before.

Contrarily, if ${\tilde{r}}_p\gg 1$, $z_4^{(p)}$ can be the leading contribution, whose corresponding Feynman diagram is shown in Fig. 7.

Fig. 7

A schematic picture of $z_4^{(p)}$ with $p=4$, for generic p there are p purple dots in the middle. Each purple line represents a pair of identical $J_A$’s, and the purple dots are vertex coming from the non-trivial quadrupole moment proportional to v

Therefore, there will be no half-wormhole saddle anymore since the (two-mouth) wormhole saddles are not dominant.

One can consider more general distribution with all the cumulants to be non-vanishing. The analysis and the results will be similar. If v is very large then it is the four-way wormhole saddle that dominate. It is therefore possible to introduce a new ”four-linked-half-wormhole” saddle as we show in next section. However, if v is relatively small it is still the two-sided wormhole (with some legs as shown in Fig. 5) that dominates.

4.2 SYK at one time point: $\langle J_a\rangle =\langle J_a^2\rangle =\langle J_a^3\rangle =0$

In this section, we consider a special model where we could focus on the multi-linked half-wormhole saddles. We will show that the multi-linked half-wormhole saddles are not simply related to the two-linked-half-wormhole saddle. In this model the random coupling only have non-vanishing quadrupole moment

$$\begin{aligned} \langle J_a\rangle =\langle J_a^2\rangle =\langle J_a^3\rangle =0,\quad \langle J_a^4\rangle =v^4. \end{aligned}$$

(136)

Such a distribution could also be considered as an extremal limit of other distributions.

4.2.1 Averaged quantities: $\langle z^4\rangle$ and $\langle z^8\rangle$

Due to our special choice (136) the first non-vanishing averaged quantity is

$$\begin{aligned} \langle z^4\rangle= & {} \int \text {d}^{4N}\psi \nonumber \\{} & {} \times \exp \left( v^4\sum _{A_1<\dots < A_q}\psi _{A_1}^1\psi _{A_1}^2\psi _{A_1}^3\psi _{A_1}^4\dots \psi _{A_q}^1\psi _{A_q}^2\psi _{A_q}^3\psi _{A_q}^4 \right) \nonumber \\= & {} \int \text {d}^{4N}\psi \exp \left( \frac{v^4}{q!} \left( \sum _{i}^N\psi _{i}^1\psi _{i}^2\psi _{i}^3\psi _{i}^4\right) ^q\right) . \end{aligned}$$

(137)

Then we can introduce the $G,\Sigma$ trick

$$\begin{aligned} \langle z^4\rangle= & {} \int \text {d}^{4N}\psi \int \text {d}G\, \nonumber \\{} & {} \times \delta \left( G_4-\sum _{i}^N\psi _{i}^1\psi _{i}^2\psi _{i}^3\psi _{i}^4\right) \exp \left( \frac{v^4}{q!}G_4^q\right) \nonumber \\= & {} \int \text {d}^{4N}\psi \int \text {d}G\frac{\text {d}\Sigma }{2\pi \text {i}}\exp \left( -\Sigma \left( G_4-\sum _{i}^N\psi _{i}^1\psi _{i}^2\psi _{i}^3\psi _{i}^4\right) \right) \nonumber \\{} & {} \times \exp \left( \frac{v^4}{q!}G_4^q\right) =(\partial _{G_4})^N \exp \left( \frac{\upsilon ^4}{q!}G_4^q\right) \, |_{G_4=0}\nonumber \\= & {} \left( \frac{v^4}{q!}\right) ^{N/q}\frac{N!}{(N/q)!}=v^{4p}\frac{N!}{p!(q!)^p}. \end{aligned}$$

(138)

The computation of $\langle z^8\rangle$ is more involved

$$\begin{aligned} \langle z^8\rangle =\int \text {d}^{8N}\psi \exp \left( \frac{v^4}{q!} \left( \sum _{i}^N\psi _{i}^a\psi _{i}^b\psi _{i}^c\psi _{i}^d\right) ^q\right) , \end{aligned}$$

(139)

where

$$\begin{aligned} 1\le a<b<c<d\le 8. \end{aligned}$$

(140)

In the following we use the collective index $A'$ to label the 4-element string. Introducing antisymmetric tensors $G_{abcd}=G_{A'}$ and $\Sigma _{abcd}=\Sigma _{A'}$ as the collective field variables such that (139) can be expressed as

$$\begin{aligned} \langle z^8\rangle= & {} \int \frac{\text {d}G_{A'}\text {d}\Sigma _{A'}}{(2\pi \text {i})^{70}}(\text {PF}(\Sigma _{A'}))^N\nonumber \\{} & {} \times \exp \left( -\sum _{A'}\left( \Sigma _{A'}G_{A'}+\frac{v^4}{q!}G_{A'}^q\right) \right) \end{aligned}$$

(141)

$$\begin{aligned}= & {} \left( \sum '_{A'_1<A'_2}\text {sgn}(A')\partial _{G_{A'_1}}\partial _{G_{A'_2}}\right) ^N\exp \left( \frac{v^4}{q!}G_{A'}^q\right) |_{G_{A'}=0} \nonumber \\ \end{aligned}$$

(142)

$$\begin{aligned}\approx & {} \left( \frac{v^4}{q!}\right) ^{\frac{2N}{q}} \frac{N!^2}{p!^2}\frac{1}{2}{8\atopwithdelims ()4}=35\left( \frac{v^4}{q!}\right) ^{\frac{2N}{q}} \frac{N!^2}{p!^2}\,, \end{aligned}$$

(143)

where in the last line we have taken the large N limit. In this limit we have

$$\begin{aligned} \langle z^8\rangle \approx 35 \langle z^4\rangle ^2 \ . \end{aligned}$$

(144)

4.2.2 The un-averaged $z^4$

Following similar ideas as in the previous sections, we insert a suitable identity to the expression of $z^4$

$$\begin{aligned} z^4= & {} \int \text {d}^{4N}\psi \exp \left( \text {i}^{q/2}\sum _{A,i}J_A\psi _A^i\right) \int \text {d}G_4 \nonumber \\{} & {} \times \delta \left( G_4-\sum _{i}^N\prod _{a=1}^4\psi _{i}^a\right) \nonumber \\{} & {} \times \exp \left( \frac{v^4}{q!}\left[ G_4^q-\left( \sum _{i}^N\prod _{a=1}^4\psi _{i}^a\right) ^q\right] \right) \,. \end{aligned}$$

(145)

Rotating the contour as before we can rewrite $z^4$ as

$$\begin{aligned} z^4=\int \text {d}\sigma \Psi (\sigma )\hat{\Gamma }(\sigma )\,, \end{aligned}$$

(146)

where $\Psi (\sigma )$ is same as (29) and the second factor is

$$\begin{aligned} \hat{\Gamma }(\sigma )= & {} \int \text {d}^{4N}\psi \exp \left( \text {i}e^{-\frac{\text {i}\pi }{q}}\sigma \prod _a \psi ^a_i \right. \nonumber \\{} & {} \left. +\text {i}^{q/2}\sum _{A,a}J_A\psi ^a_A-v^4\sum _A \prod _a\psi _A^a\right) . \end{aligned}$$

(147)

Therefore we expect the half-wormhole saddle is given by

$$\begin{aligned} \Gamma= & {} \hat{\Gamma }(0)\nonumber \\= & {} \sum _{ABCD}\text {sgn}(A,B,C,D)\nonumber \\{} & {} \times \prod _{k=1}^p \left( J_{A_k}J_{B_k}J_{C_k}J_{D_k}-\delta _{A_k}^{B_k}\delta _{C_k}^{B_k}\delta _{C_k}^{D_k}v^4\right) \,, \end{aligned}$$

(148)

which satisfies

$$\begin{aligned} \langle \Gamma \rangle =0, \qquad \langle \Gamma ^2\rangle= & {} \langle \Gamma z^4\rangle \approx 34 \langle z^4\rangle ^2 \,, \end{aligned}$$

(149)

$$\begin{aligned} \langle ( z^4-\langle z^4\rangle -\Gamma )^2\rangle= & {} \langle z^8\rangle -\langle z^4\rangle ^2+\langle \Gamma ^2\rangle -2\langle \Gamma z^4\rangle \nonumber \\\approx & {} 0\,. \end{aligned}$$

(150)

We find clearly that the contribution from this four-linked-wormhole saddle is not equal to the square of (two-linked) half-wormhole saddle. Even though we derive it in the 0d-SYK toy model, it should exist in other SYK-like theory as long as the $G,\Sigma$ trick can be applied.

4.3 SYK at one time point: Poisson distribution

Up to now we have only considered random couplings with continuous probability distributions. It is also interesting to consider random couplings that take discrete values such as the Poisson distribution. Ensemble theory or theories with random coupling with Poisson distribution have been studied in [52, 59, 82]. A property of Poisson distribution is that all the cumulants are equal; for a model with a 2q-point interaction among 2N fermions, we define

$$\begin{aligned} \langle J^n\rangle _{c}=N\lambda \,, \quad \forall n\,, \end{aligned}$$

(151)

so that a large-N limit is well defined. Starting with action (1) we can compute the first few moments

$$\begin{aligned} \langle z\rangle&= \int \text {d}^{2N} \psi \, e^{N\text {i}^{q} \lambda \sum _A \psi ^1_A}, \end{aligned}$$

(152)

$$\begin{aligned} \langle z^2\rangle&=\int \text {d}^{4N} \psi \, e^{N\text {i}^{q}\lambda \sum _A(\psi _A^1+\psi _A^2)}e^{N\text {i}^{2q} \lambda \sum _{A}\psi _A^1\psi _A^2}, \end{aligned}$$

(153)

$$\begin{aligned} \langle z^3\rangle&=\int \text {d}^{6N} \psi \, e^{N\text {i}^{q}\lambda \sum _A(\psi _A^1+\psi _A^2+\psi _A^3)}\nonumber \\&\quad \times e^{N\text {i}^{2q} \lambda \sum _{A}(\psi _A^1\psi _A^2+ \psi _A^1\psi _A^3+\psi _A^2\psi _A^3)}e^{N\text {i}^{3q}\lambda \sum _A \psi _A^1\psi _A^2\psi _A^3}\ . \end{aligned}$$

(154)

For a generic k, we find

$$\begin{aligned} \langle z^k\rangle =\int \text {d}^{2kN}\psi e^{N\lambda \sum _A \sum _{n=1}^k \frac{1}{n!}(\text {i}^{q}\sum ^k_{i=1} (\psi _A^i))^n}. \end{aligned}$$

(155)

Formally we can define

$$\begin{aligned} {{\mathcal {Z}}}(\lambda ){} & {} \equiv \langle z^\infty \rangle \end{aligned}$$

(156)

$$\begin{aligned}{} & {} =\int \text {d}\psi \exp \left\{ N\lambda \sum _A \left( e^{\text {i}^{q}\sum _{i=1} \psi _A^i}-1\right) \right\} . \end{aligned}$$

(157)

We can compute these moments by integrating out the fermions directly

$$\begin{aligned} \langle z^n\rangle =\left\langle \text {Pf}(J_A)^n\right\rangle . \end{aligned}$$

(158)

However, the ensemble average of $\text {PF}(J_A)^n$ in the last expression is very complicated. We therefore look at the large N behavior of $\langle z^n\rangle$, which can be done by introducing G

$$\begin{aligned} G=\sum _{i<j}\text {i}\psi _i\psi _j\,, \end{aligned}$$

(159)

and $\Sigma$ as before and do a saddle point approximation. The $G,\Sigma$ expression of $\langle z\rangle$ is similar to (85B.54)

$$\begin{aligned} \langle z\rangle =\int \text {d}\Sigma \text {d}G (-\text {i})^{N}\Sigma ^{N} e^{N \text {i}^{{q}}\lambda \frac{ G^{{q}}}{{q}!}}e^{\text {i}N \Sigma G}. \end{aligned}$$

(160)

The saddle point equations are

$$\begin{aligned}{} & {} \Sigma G=\text {i},\quad \frac{\lambda }{(q-1)!}(\text {i}G)^q =1, \end{aligned}$$

(161)

whose solutions are

$$\begin{aligned}{} & {} \text {i}G=\left( \frac{ (q-1)!}{\lambda }\right) ^{1/q}e^{\frac{2m\pi \text {i}}{q}},\quad m=1,\dots ,q. \end{aligned}$$

(162)

The structure of these solutions is identical to that in [73] and the models discussed previously in this paper, so we should add up all these q saddle points contributions⁶

$$\begin{aligned} \langle z\rangle _{\text {Disk}}= & {} e^{-N(1-\frac{1}{q})}\left( \frac{N^{q}\lambda }{(q-1)!}\right) ^p\sum _m e^{\frac{2m\pi \text {i}}{q}} \end{aligned}$$

(163)

$$\begin{aligned}= & {} q e^{-N(1-\frac{1}{q})}\left( \frac{N^{q}\lambda }{(q-1)!}\right) ^p, \end{aligned}$$

(164)

where $p=N/q$ as before. Adding the 1-loop factor $1/\sqrt{q}$ we end up with the large-N behavior

$$\begin{aligned} \langle z\rangle _{\text {Disk}+1\text { loop}}=\frac{1}{\sqrt{q}} e^{-N(1-\frac{1}{q})}\left( \frac{N^{q}\lambda }{(q-1)!}\right) ^p. \end{aligned}$$

(165)

Other moments can be computed similarly. For example, to compute $\langle z^2\rangle$, we introduce three collective variables

$$\begin{aligned}{} & {} G_1=\sum _{i<j}\text {i}\psi _i^1\psi _j^1,\quad G_2=\sum _{i<j}\text {i}\psi _i^2\psi _j^2, \end{aligned}$$

(166)

$$\begin{aligned}{} & {} G_{12}=\sum _{i}\psi _i^1\psi _i^2 \end{aligned}$$

(167)

such that

$$\begin{aligned}{} & {} \text {i}^q\sum _A\psi _A^1=\frac{G_1^q}{q!},\quad \text {i}^q\sum _A\psi _A^2=\frac{G_2^q}{q!}, \end{aligned}$$

(168)

$$\begin{aligned}{} & {} \text {i}^{2q}\sum _A \psi _A^1\psi _A^2=\frac{G_{12}^{2q}}{(2q)!}. \end{aligned}$$

(169)

Imposing these relations with the help of a set of Lagrangian multiplier fields $\Sigma _1$, $\Sigma _2$ and $\Sigma _{12}$, the $\langle z^2\rangle$ can be expressed as

$$\begin{aligned} \langle z^2\rangle= & {} \int [\text {d}^3 G_i d^3 \Sigma _i] e^{N\frac{\lambda }{q!}(G_1^q+G_2^q+\frac{q!}{(2q)!}G_{12}^{2q})}e^{i N\sum _i(\Sigma _i G_i)}\nonumber \\{} & {} \times \int \text {d}^{2N}\psi e^{\frac{1}{2}{\Psi }M{\Psi }}, \end{aligned}$$

(170)

$$\begin{aligned}= & {} \int [\text {d}^3 G_i d^3 \Sigma _i] \sqrt{\text {det}[\Sigma _1\Sigma _2 A^2-\Sigma _{12}^2 I_{2N}]}\nonumber \\{} & {} \times e^{\frac{N\lambda }{q!}(G_1^q+G_2^q+\frac{q!}{(2q)!}G_{12}^{2q})}e^{i N \sum _i(\Sigma _i G_i)} \end{aligned}$$

(171)

$$\begin{aligned}= & {} \int [\text {d}^3 G_i d^3 \Sigma _i]i^{2N} \sum _{k=1}^{N} {2N \atopwithdelims ()2k}\Sigma _{12}^{2N-2k}(\Sigma _1\Sigma _2)^k\nonumber \\{} & {} \times e^{N\frac{\lambda }{q!}(G_1^q+G_2^q+\frac{q!}{(2q)!}G_{12}^{2q})}e^{N i \sum _i(\Sigma _i G_i)} \end{aligned}$$

(172)

where we have defined

$$\begin{aligned}{} & {} \Psi =\left( \psi _1^1,\dots ,\psi _{2N}^1,\psi _1^2,\dots ,\psi _{2N}^2\right) , \end{aligned}$$

(173)

$$\begin{aligned}{} & {} M=\begin{pmatrix} \Sigma _1 A&{}-\text {i}\Sigma _{12} I_{2N}\\ \text {i}\Sigma _{12} I_{2N}&{} \Sigma _2 A\\ \end{pmatrix}, \end{aligned}$$

(174)

$$\begin{aligned}{} & {} A=-A^T,\quad A_{ij}=1,\quad \forall i<j. \end{aligned}$$

(175)

The saddle point equations are

$$\begin{aligned}{} & {} \text {i}\Sigma _i+\frac{\lambda }{(q-1)!}G_i^{q-1}=0,\quad i=1,2, \end{aligned}$$

(176)

$$\begin{aligned}{} & {} \text {i}\Sigma _{12}+\frac{\lambda }{(2q-1)!}G_{12}^{2q-1}=0, \quad \sum _i \Sigma _i G_i=2\text {i}. \end{aligned}$$

(177)

This set of equations has multiple solutions. For example, the wormhole saddle is

$$\begin{aligned}{} & {} G_1=G_2=\Sigma _1=\Sigma _2=0, \end{aligned}$$

(178)

$$\begin{aligned}{} & {} G_{12}=\left( \frac{2 (2q-1)!}{\lambda }\right) ^{1/2q}e^{\frac{2m\pi \text {i}}{2q}}, \end{aligned}$$

(179)

$$\begin{aligned}{} & {} \langle z^2\rangle _{\text {WH}+1\text {loop}}=\frac{1}{\sqrt{2q}} e^{-2N(1-\frac{1}{2q})}\left( \frac{(2N)^{2q}\lambda }{2(2q-1)!}\right) ^p \end{aligned}$$

(180)

and the disconnected disk saddle is

$$\begin{aligned} G_{12}=\Sigma _{12}=0,{} & {} G_1=G_2=\left( \frac{ (q-1)!}{\lambda }\right) ^{1/q}, \end{aligned}$$

(181)

$$\begin{aligned} \langle z^2\rangle _{disc+1\text {loop}}= & {} \frac{1}{q}e^{-2N(1-\frac{1}{q})}\left( \frac{N^{q}\lambda }{(q-1)!}\right) ^{2p} \end{aligned}$$

(182)

$$\begin{aligned}= & {} \langle z\rangle _{\text {Disk}+1\text {loop}}^2. \end{aligned}$$

(183)

The ratio of these two saddles is

$$\begin{aligned} \frac{\langle z^2\rangle _{\text {WH}+1\text {loop}}}{\langle z^2\rangle _{\text {Disk}+1\text {loop}}}=\sqrt{\frac{q}{2}} \left( \frac{q!^2 2^{2q} }{e\lambda q (2q)!}\right) ^p\,. \end{aligned}$$

(184)

In the large N or $p=N/q$ limit, the wormhole saddle can dominate only when $\lambda < \frac{q!^2 2^{2q} }{e q (2q)!}\left( \frac{q}{2}\right) ^{\frac{1}{2p}}$. For $q=4$ and $N\rightarrow \infty$, this requires $\lambda < 0.336$, which is consistent with what we learned from our previous results. Indeed, for the wormhole or other connected saddle points to be dominant we would like to have non-negligible higher moments of the random coupling. For the Poisson distribution this means, apart from the factors of N to be compensated by powers of fermion contractions, $\lambda$ should be much larger than $\lambda ^2$.

Then a natural question is that in this limit how about other n-boundary wormhole saddles? In the following let us focus on a particular n-linked-wormhole saddles. When $n=2k$ is even, the situation is similar to the one in Sect. 125:

$$\begin{aligned}{} & {} \langle z^{2k}\rangle _{\text {connected}}\nonumber \\{} & {} \quad =\int \text {d}^{4kN} \psi \text {d}G\frac{\text {d}\Sigma }{2\pi }\exp \left( \text {i}N\Sigma \left( G-\sum _{i}^{2N} \prod _{a=1}^{2k}\psi _{i}^a\right) \right) \nonumber \\{} & {} \qquad \times \exp \left( N\frac{\lambda }{(2q)!}G^{2q}\right) \end{aligned}$$

(185)

$$\begin{aligned}{} & {} \quad =\int \text {d}G\frac{\text {d}\Sigma }{2\pi }(\text {i}\Sigma )^{2N} \exp \left( \frac{N\lambda }{{(2q)}!}G^{2q}+\text {i}N \Sigma G\right) \end{aligned}$$

(186)

where the collective variable G is

$$\begin{aligned} G=\sum _{i}^{2N} \prod _{a=1}^{2k}\psi _{i}^a. \end{aligned}$$

(187)

The expression (186) is of the same form as (160) so the saddle point approximation is

$$\begin{aligned}{} & {} \langle z^{2k}\rangle _{2k-\text {WH}+1\text {loop}}=\langle z^2\rangle _{2-\text {WH}+1\text {loop}} \end{aligned}$$

(188)

$$\begin{aligned}{} & {} \quad =\frac{1}{\sqrt{2q}} e^{-2N(1-\frac{1}{2q})}\left( \frac{(2N)^{2q}\lambda }{2(2q-1)!}\right) ^p. \end{aligned}$$

(189)

When $n=2k+1$ is odd, the situation is similar to the one of $n=1$:

$$\begin{aligned}{} & {} \langle z^{2k+1}\rangle _{\text {connected}}\nonumber \\{} & {} \quad =\int \text {d}^{(4k+2)N} \psi \text {d}G\frac{\text {d}\Sigma }{2\pi }\nonumber \\{} & {} \qquad \times \exp \left( \text {i}N \Sigma \left( G-\sum _{i<j}^{2N} \prod _{a=1}^{2k+1}\psi _{i}^a\prod _{a=1}^{2k+1}\psi _{j}^a\right) \right) \nonumber \\{} & {} \qquad \times \exp \left( \frac{N\lambda }{q!}G^q\right) \end{aligned}$$

(190)

$$\begin{aligned}{} & {} \quad =\int \text {d}G\frac{\text {d}\Sigma }{2\pi }(\text {i}\Sigma )^{2N} \exp \left( \frac{N\lambda }{{q}!}G^{q}+\text {i}N\Sigma G\right) , \end{aligned}$$

(191)

where the collective variable G is obviously defined as

$$\begin{aligned} G=\sum _{i<j}^{2N} \prod _{a=1}^{2k+1}\psi _{i}^a\prod _{a=1}^{2k+1}\psi _{j}^a, \end{aligned}$$

(192)

therefore the saddle point approximation is

$$\begin{aligned}{} & {} \langle z^{2k+1}\rangle _{(2k+1)-\text {HW}+1\text {loop}}=\langle z\rangle _{\text {Disk}+1\text {loop}}\nonumber \\{} & {} \quad =\frac{1}{\sqrt{q}} e^{-N(1-\frac{1}{q})}\left( \frac{N^{q}\lambda }{(q-1)!}\right) ^p. \end{aligned}$$

(193)

These higher n-linked-wormholes should be compared with the corresponding powers of the disk solution, and furthermore since $\langle z^2\rangle _{2-\text {WH}+1\text {loop}}\gg 1$ and $\langle z^2\rangle _{\text {Disk}+1\text {loop}}\gg 1$ due to large-N so

$$\begin{aligned} \langle z^{2k}\rangle _{2k-\text {WH}+1\text {loop}} \ll \left( \langle z^2\rangle _{2-\text {WH}+1\text {loop}}\right) ^{k}\,, \end{aligned}$$

(194)

and

$$\begin{aligned} \langle z^{2k+1}\rangle _{(2k+1)-\text {WH}+1\text {loop}} \ll \left( \langle z\rangle _{\text {Disk}+1\text {loop}}\right) ^{2k+1}\,, \end{aligned}$$

(195)

we conclude that all these multiple-linked wormholes with $k>0$ are suppressed. In other words, the ensemble of z can be approximated by a Gaussian when the ratio (184) is of order 1.

5 The modified Brownian SYK model

Given the above results in 0 dimension, it is interesting to check if similar stories hold in high-dimensional models. Therefore in this section, we look for the wormhole and half-wormholes saddles in 1d Brownian SYK models [9].

We first briefly review the Brownian SYK model. The Brownian SYK model is characterized by couplings that are only correlated at the same instant of time. Therefore after integrating over the coupling we end up with a local effective action. The quantity that is analogous to the partition function but with some information of real time evolution is

$$\begin{aligned} z&\equiv \textrm{Tr}\, U(T)\nonumber \\&=\int \mathcal {D} \psi _{i} \exp \left\{ -\text {i}\int _{0}^{T} d t\left[ -\frac{\text {i}}{2} \psi _{i} \partial _{t} \psi _{i}\right. \right. \nonumber \\&\left. \left. \quad +J_{i_{1} \ldots i_{q}}(t)\text {i}^{\frac{q}{2}} \psi _{i_{1} \ldots i_{q}}\right] \right\} \ . \end{aligned}$$

(196)

The random couplings of this model satisfy

$$\begin{aligned}{} & {} \langle J_A\rangle =0,\quad \langle J_A(t)J_B(t')\rangle =\delta (t-t')\delta _{AB}{{\mathcal {J}}}^2, \end{aligned}$$

(197)

$$\begin{aligned}{} & {} {{\mathcal {J}}}^2=2J\frac{(q-1)!}{N^{q-1}}, \end{aligned}$$

(198)

where the one-dimensional Majorana fermions is normalized by

$$\begin{aligned} \{\psi _i,\psi _j\}=\delta _{ij}. \end{aligned}$$

(199)

In the rest of this section we look for the linked half-wormhole contributions in a generalization of this model.⁷ In particular, we consider a generalized Brownian SYK model with non-vanishing mean value of the random couplings:

$$\begin{aligned}{} & {} \langle J_A\rangle =J_A^{(0)}=\mu , \end{aligned}$$

(200)

$$\begin{aligned}{} & {} \langle J_A(t)J_B(t')\rangle =\delta (t-t')(\delta _{AB}\tau ^2+\mu ^2), \end{aligned}$$

(201)

and in this section we use the convention $\{\psi _i,\psi _j\}=2\,h \delta _{i,j}$. A comparable model has been examined in [79] with a focus on the half-wormhole saddle. However, the key difference lies in the fact that in [79], the random coupling is expressed as a Grassmann number. In Appendix D, we also revisit this modified the (Brownian) SYK model for comparison purposes with our models.

Taking the disorder averaging of the coupling we obtain the averaged theory

$$\begin{aligned} \langle z(T)\rangle _J=&\int \mathcal {D}\psi \,e^{-S_{a}}\,, \end{aligned}$$

(202)

$$\begin{aligned} S_a=&\frac{1}{2}\int _0^T \text {d}t \sum _i^N \psi _i \partial _t \psi _i-\text {i}^{q/2} \int \text {d}t \sum _{A}J_A^{(0)}\psi _A\nonumber \\&-\frac{\tau ^2}{2}\int \text {d}t \left( \sum _{A}\psi _A^2\right) \end{aligned}$$

(203)

We can convert the effective Hamiltonian of the averaged theory as a spin system

$$\begin{aligned} \langle z\rangle _J&=\text {Tr}\left( e^{-T {{\mathcal {H}}}}\right) \,, \end{aligned}$$

(204)

$$\begin{aligned} {{\mathcal {H}}}&=-\text {i}^{q/2} \sum _A J_A^{(0)}\psi _A-\frac{\tau ^2}{2} \sum _A h^q \end{aligned}$$

(205)

$$\begin{aligned}&=-\text {i}^{q/2} \sum _A J_A^{(0)}\psi _A-\frac{\tau ^2}{2}{N \atopwithdelims ()q }h^q \ . \end{aligned}$$

(206)

When $\mu =0$, the averaged partition function is given by

$$\begin{aligned}{} & {} \langle z\rangle _J=e^{T\frac{\tau ^2}{2}{N\atopwithdelims ()q}h^q}\equiv 2^N e^{T E_0}, \end{aligned}$$

(207)

$$\begin{aligned}{} & {} E_0=\frac{\tau ^2}{2}{N\atopwithdelims ()q}h^q\sim \frac{\tau ^2}{2}N^q h^q. \end{aligned}$$

(208)

When $\mu \ne 0$, we have to evaluate the trace

$$\begin{aligned} \langle z \rangle _J{} & {} =e^{TE_0}\text {Tr}(e^{T \text {i}^{q/2}\mu \sum _A \psi _A}) \end{aligned}$$

(209)

$$\begin{aligned}{} & {} =e^{TE_0}\int \mathcal {D}^N \psi _i \exp (T \text {i}^{q/2} \sum _i \psi _i). \end{aligned}$$

(210)

However there is no simple expression for $\langle z\rangle$. We first consider the simplest case with $q=1$

$$\begin{aligned} I_f=\int \mathcal {D}^N \psi _i \exp (a \sum _i \psi _i). \end{aligned}$$

(211)

The idea is to transfer the Majorana fermions to Dirac fermions which have a well-defined rules of integrals. Assuming the total number of fermions is even $N=2K$ then we introduce K Dirac fermions as

$$\begin{aligned}{} & {} c_i=\frac{1}{2\sqrt{h}}(\psi _{2i-1}-\text {i}\psi _{2i}),\quad c_i^\dagger =\frac{1}{2\sqrt{h}}(\psi _{2i-1}+\text {i}\psi _{2i}),\nonumber \\{} & {} i=1,\dots , K\,, \end{aligned}$$

(212)

$$\begin{aligned}{} & {} \psi _{2i-1}=\sqrt{h}(c_i+c_i^\dagger ),\quad \psi _{2i}=\text {i}\sqrt{h}(c_i-c_i^\dagger )\,, \end{aligned}$$

(213)

which obey

$$\begin{aligned} \{c_i,c_j\}= & {} \{c_i^\dagger ,c_j^\dagger \}=0,\quad \{c_i,c_j^\dagger \}=\delta _{ij}\, \end{aligned}$$

(214)

The integration measure changes as

$$\begin{aligned} {{\mathcal {D}}}\psi _{2i}{{\mathcal {D}}}\psi _{2i-1}=2h{{\mathcal {D}}}c_i {{\mathcal {D}}}c_i^\dagger \,. \end{aligned}$$

(215)

Thus the integral can be evaluated as

$$\begin{aligned} I_1= & {} (2h)^K \int \prod _i{{\mathcal {D}}}c_i {{\mathcal {D}}}c_i^\dagger \nonumber \\{} & {} \times \exp \left( a \sum _i^K \sqrt{h}\left[ (1+\text {i})c_i+(1-\text {i})c_i^\dagger \right] \right) \end{aligned}$$

(216)

$$\begin{aligned}= & {} (2h)^K\left( 2 \cosh (\sqrt{2}ah)\right) ^K. \end{aligned}$$

(217)

Now we let us consider the case of $q=2$

$$\begin{aligned}{} & {} I_2(a)=\int \text {d}^N\psi _i \exp \left( \frac{a}{2}\sum _{i\ne j}\psi _i A_{ij}\psi _j\right) ,\nonumber \\{} & {} \text {with }\quad \left( A_{ij}=-A_{ij}=a,\quad i<j\right) , \end{aligned}$$

(218)

which looks like a Gaussian but we need to replace $\psi _i$ with $c_i$:

$$\begin{aligned} I_2= & {} (\sqrt{2h})^N \int \prod _i{{\mathcal {D}}}c_i {{\mathcal {D}}}c_i^\dagger \, e^{{{\mathcal {H}}}} \end{aligned}$$

(219)

$$\begin{aligned} {{\mathcal {H}}}= & {} \left( \text {i}a h\left( \sum _i^{K}[c_i^\dagger c_i-c_ic_i^\dagger ] +2 \sum _{i<j}[c_ic_j-c_i^\dagger c_j^\dagger ]\right) \right. \nonumber \\{} & {} \left. +2ah\sum _{i<j}[c_i^\dagger c_j+c_i c_j^\dagger ]\right) \end{aligned}$$

(220)

To get an idea how to compute this integral let us consider a simple case of $N=4$:

$$\begin{aligned}{} & {} \psi _1= \sqrt{h}(c_1+c_1^\dagger ),\psi _2=\text {i}\sqrt{h}(c_1-c_1^\dagger ),\nonumber \\{} & {} \psi _3=\sqrt{h}(c_2+c_2^\dagger ),\psi _4=\text {i}\sqrt{h}(c_2-c_2^\dagger ), \end{aligned}$$

(221)

$$\begin{aligned}{} & {} \sum _{i<j}\psi _i\psi _j= \text {i}h(c_1^\dagger c_1-c_1 c_1^\dagger +c_2^\dagger c_2-c_2 c_2^\dagger +2c_1c_2-2c_1^\dagger c_2^\dagger )\nonumber \\{} & {} \qquad \qquad \quad +2h(c_1^\dagger c_2+c_1c_2^\dagger ). \end{aligned}$$

(222)

We have four different states $|\Psi _i\rangle$:

$$\begin{aligned} |00\rangle ,\quad c_{1}^{\dagger }|00\rangle ,\quad c_{2}^{\dagger }|00\rangle ,\quad c_{1}^{\dagger }c_{2}^{\dagger }|00\rangle . \end{aligned}$$

(223)

So the operator $\sum _{i<j}\psi _i\psi _j$ can be written as a $4\times 4$ matrix:

$$\begin{aligned} \sum _{i<j}\psi _i\psi _j=\left( \begin{array}{cccc} -2 \text {i}h &{} 0 &{} 0 &{} -2 \text {i}h \\ 0 &{} 0 &{} -2 h &{} 0 \\ 0 &{} 2 h &{} 0 &{} 0 \\ -2 \text {i}h &{} 0 &{} 0 &{} 2 \text {i}h \\ \end{array} \right) \end{aligned}$$

(224)

with 4 eigenvalues $\{\pm \text {i}2 h, \pm \text {i}2\sqrt{2}h\}$ so path integral over $c_i$ and $c_i^\dagger$ can be computed as

$$\begin{aligned}{} & {} \sum _i \langle \Psi _i|e^{a \sum _i\psi _i\psi _j}|\Psi _i\rangle =2\left( \cos (2ah)+\cos (2\sqrt{2}a h)\right) . \nonumber \\ \end{aligned}$$

(225)

For example of $N=6$, the corresponding matrix is

$$\begin{aligned}{} & {} \sum _{i<j}\psi _i\psi _j\nonumber \\{} & {} =\left( \begin{array}{cccccccc} -3 \text {i}h &{} 0 &{} 0 &{} 0 &{} -2 \text {i}h &{} -2 \text {i}h &{} -2 \text {i}h &{} 0 \\ 0 &{} -\text {i}h &{} 2 h &{} 2 h &{} 0 &{} 0 &{} 0 &{} -2 \text {i}h \\ 0 &{} -2 h &{} -\text {i}h &{} 2 h &{} 0 &{} 0 &{} 0 &{} 2 \text {i}h \\ 0 &{} -2 h &{} -2 h &{} -\text {i}h &{} 0 &{} 0 &{} 0 &{} -2 \text {i}h \\ -2 \text {i}h &{} 0 &{} 0 &{} 0 &{} \text {i}h &{} 2 h &{} -2 h &{} 0 \\ -2 \text {i}h &{} 0 &{} 0 &{} 0 &{} -2 h &{} \text {i}h &{} 2 h &{} 0 \\ -2 \text {i}h &{} 0 &{} 0 &{} 0 &{} 2 h &{} -2 h &{} \text {i}h &{} 0 \\ 0 &{} -2 \text {i}h &{} 2 \text {i}h &{} -2 \text {i}h &{} 0 &{} 0 &{} 0 &{} 3 \text {i}h \\ \end{array} \right) \nonumber \\ \end{aligned}$$

(226)

which can be divided into two blocks. We get the eigenvalues by directly diagonalizing the matrix:

$$\begin{aligned} \pm 5\text {i}h,\quad \pm (2\sqrt{3}+1)\text {i}h,\quad \pm 3\text {i}h,\quad \pm (2\sqrt{3}-1)\text {i}h. \end{aligned}$$

(227)

Similarly for general N, we can write effective Hamiltonian defined in (219220) as

$$\begin{aligned} {{\mathcal {H}}}=\sum _{i\le j=1}^k\left( \alpha _{ij}c_i^\dagger c_j+\beta _{ij}c_i c_j^\dagger +\gamma _{ij}c_i^\dagger c_j^\dagger +\theta _{ij} c_i c_j\right) , \end{aligned}$$

(228)

with

$$\begin{aligned}{} & {} \alpha _{ii}=\text {i}h,\quad \beta _{ii}=-\text {i}h,\quad \alpha _{ij}=2h,\quad \beta _{ij}=2h, \end{aligned}$$

(229)

$$\begin{aligned}{} & {} \gamma _{ij}=-2\text {i}h,\quad \theta _{ij}=2\text {i}h,\quad \gamma _{ii}=0,\quad \theta _{ii}=0. \end{aligned}$$

(230)

This Hamiltonian is quadratic and famously can be diagonalized by the Bogoliubov and Valatin’s method [85, 86]. Explicitly we can do the transformation by taking an operator basis for the Hamiltonian

$$\begin{aligned} H=c^{\dagger }Mc \end{aligned}$$

(231)

where we have

$$\begin{aligned} c^{\dagger }=\left( c_{1}^{\dagger },c_{2}^{\dagger },\dots ,c_{1},c_{2},\dots \right) . \end{aligned}$$

(232)

In the simple case with $N=4$ the matrix can be expressed as

$$\begin{aligned} M=\left( \begin{array}{cccc} \text {i}h &{} h &{} 0 &{} -\text {i}h \\ -h &{} \text {i}h &{} \text {i}h &{} 0 \\ 0 &{} \text {i}h &{} -\text {i}h &{} h \\ -\text {i}h &{} 0 &{} -h &{} -\text {i}h \\ \end{array} \right) , \end{aligned}$$

(233)

we can directly take the diagonalization and get the eigenvalues

$$\begin{aligned}&\text {i}(1+\sqrt{2})h,\quad -\text {i}(1+\sqrt{2})h, \end{aligned}$$

(234)

$$\begin{aligned}&-\text {i}(1-\sqrt{2})h,\quad -\text {i}(-1+\sqrt{2})h. \end{aligned}$$

(235)

For simplicity we take the notation as

$$\begin{aligned} \lambda _{1}=\text {i}(\sqrt{2}+1)h,\quad \lambda _{2}=\text {i}(\sqrt{2}-1)h, \end{aligned}$$

(236)

then the resulting effective Hamiltonian becomes

$$\begin{aligned} H=\lambda _{1}\left( d_{1}^{\dagger }d_{1}-d_{1}d_{1}^{\dagger } \right) +\lambda _{2}\left( d_{2}^{\dagger }d_{2}-d_{2}d_{2}^{\dagger } \right) . \end{aligned}$$

(237)

To evaluate the trace we still take the states as (223) therefore we have

$$\begin{aligned} \text {Tr}(e^{H})=e^{-a(\lambda _{1}+\lambda _{2})}+e^{a(\lambda _{1}-\lambda _{2})}+e^{a(-\lambda _{1}+\lambda _{2})}+e^{a(\lambda _{1}+\lambda _{2})}, \end{aligned}$$

(238)

so we can recover the result (225). For general N the operator (228) can be expressed as a block matrix

$$\begin{aligned} M=\left( \begin{array}{cc} A+\text {i}hI_N&{}-\text {i}A \\ \text {i}A&{}A-\text {i}h I_N \end{array} \right) , \end{aligned}$$

(239)

with

$$\begin{aligned} A=\left( \begin{array}{cccc} 0&{}h&{}h &{}\cdots \\ -h&{}0&{}h &{}\cdots \\ -h&{}-h&{}0&{}\cdots \\ \vdots &{}\vdots &{}\vdots &{}\ddots \end{array} \right) \end{aligned}$$

(240)

The characteristic equation is

$$\begin{aligned}{} & {} \det \left( A+(\text {i}h-\lambda ))(A-(\text {i}h+\lambda )-H^2\right) \nonumber \\{} & {} =\det \left( (h^2+\lambda ^2)I_N-2\lambda A \right) \end{aligned}$$

(241)

$$\begin{aligned}{} & {} = (\lambda +h)^N+(\lambda -h)^N=0\,. \end{aligned}$$

(242)

So the eigenvalues are

$$\begin{aligned} \lambda _m=\text {i}h\tan \left( \frac{m \pi }{2 N}\right) ,\quad m=1,3,\dots ,N-1\,. \end{aligned}$$

(243)

then the Hamiltonian becomes

$$\begin{aligned} H=\sum _{i=1}^{N}\lambda _{i}\left( d_{i}^{\dagger }d_{i}-d_{i}d_{i}^{\dagger }\right) . \end{aligned}$$

(244)

and the trace will have the form

$$\begin{aligned} \textrm{Tr}(e^{H})&=\sum _{\sigma =\pm 1} e^{a\sum _{i=1}^{k}\sigma _{i}\lambda _{i}}=\sum _{\sigma }\prod _{i=1}^k e^{a\sigma _{i}\lambda _{i}} \end{aligned}$$

(245)

$$\begin{aligned}&=\prod _{i=1}^k \sum _{\sigma }e^{a\sigma _{i}\lambda _{i}}=2^k\prod _{i=1}^k\cosh \left( a\lambda _i\right) \,, \end{aligned}$$

(246)

Now let us consider the function

$$\begin{aligned} X_n=\sum _{1\le i_1<\dots i_n\le N}\psi _{i_1}\dots \psi _{i_N}. \end{aligned}$$

(247)

We would like to argue that in the large N limit, we have the approximation

$$\begin{aligned} n! X_{2n}\approx (X_2)^n, \end{aligned}$$

(248)

as we find for the 0-dimensional theory. Note that unlike the situation of the 0-dimensional theory, $\{X_n\}$ do not form a basis for $X_2^n$. For example, let us take $N=6$, there is indeed the identity

$$\begin{aligned} X_2^2=-15+2! X_4 \end{aligned}$$

(249)

but we find that

$$\begin{aligned} X_2^3= & {} 3!X_6+15 X_2+12(\psi _1\psi _2+\psi _1\psi _6+\psi _3\psi _4+\psi _4\psi _5 \nonumber \\{} & {} \quad +\psi _5\psi _6)-4(\psi _1\psi _4+\psi _2\psi _4+\psi _3\psi _6). \end{aligned}$$

(250)

Let us focus on the second last term in $X_2^n$

$$\begin{aligned}{} & {} X_2^n \approx \dots c_1 X_{2n-4}+n! X_{2n}, \end{aligned}$$

(251)

$$\begin{aligned}{} & {} c_1= (n-2)!{n \atopwithdelims ()2} {N \atopwithdelims ()2}, \end{aligned}$$

(252)

where $c_1$ is computed as follows. We need to pick 2 $X_2$ out of n and contract them, and the $(n-2)$ $X_2$’s remain not contracted and gives $(n-1)!$ $X_{2n-4}$. Notice that the subleading term is $X_{2n-4}$ instead of $X_{2n-2}$, since if we contract one fermion in $X_2$ to get

$$\begin{aligned} \psi _1\psi _2 \psi _1\psi _3 \mapsto \psi _3\psi _2, \end{aligned}$$

(253)

there is going to be another contraction that gives

$$\begin{aligned} \psi _1\psi _3\psi _1\psi _2 \mapsto \psi _2\psi _3. \end{aligned}$$

(254)

The two outcomes simply cancel with each other. The main conclusion of this computation is, given that $X_{2n} \sim N^{2n}$, the subleading terms can be safely neglected and approximate $X_{2n}$ by $X_2^n$. So in the large N limit, we can use the $G,\Sigma$ trick to compute the fermionic integral

$$\begin{aligned} I_q(a)= & {} \int \text {d}^N \psi _i \exp \left( a\sum _A \psi _A\right) \end{aligned}$$

(255)

$$\begin{aligned}\approx & {} \int \text {d}^N \psi _i \, e^{a\frac{G^{\frac{q}{2}}}{\frac{q}{2}!}} e^{\text {i}\sigma (G-\sum _{i<j}\psi _i\psi _j)}\text {d}G\text {d}\sigma \end{aligned}$$

(256)

$$\begin{aligned}= & {} \int \text {d}G \text {d}\sigma I_2(-\text {i}\sigma )e^{a\frac{G^{\frac{q}{2}}}{\frac{q}{2}!}}e^{\text {i}\sigma G} \end{aligned}$$

(257)

$$\begin{aligned}= & {} I_2(\text {i}\partial _{G})e^{a\frac{G^{\frac{q}{2}}}{\frac{q}{2}!}} |_{G=0}\,,\nonumber \\ A= & {} \{1\le a_1<\dots <a_q\le N\}. \end{aligned}$$

(258)

where the function $I_2$ is defined in (218). We can evaluate this expression and we expect the half-wormhole contributions to be similar as the 0-SYK model

$$\begin{aligned} z&\approx \langle z\rangle +\Theta \,, \end{aligned}$$

(259)

$$\begin{aligned} \Theta&=\int \text {d}^N \psi \, e^{ -\int _0^T \text {d}t\, \frac{1}{2}\sum _i^N \psi _i \partial _t \psi _i+\text {i}^{q/2} \int _0^T \text {d}t \sum _A (J_A-\mu ) \psi _A }\ . \end{aligned}$$

(260)

Indeed we find that in the late time this is a good approximation. The detailed analysis is similar to the Brownian SYK model as we have shown in [84], but it is not particularly illuminating, so we omit them here.

6 Discussion

In this paper we examine the half-wormhole proposal in various simple SYK-like models. We show that the structure of half-wormhole-like non-self-averaging contributions in the 0-dimensional SYK type models depends on the distribution of the couplings. When the distribution of the random couplings admits a non-vanishing mean value, there is a new saddle point, which we call the “punctured disk”, to the un-averaged partition function z. When the mean value of the coupling is very large then only the disconnected saddles dominate therefore the correlation functions automatically factorize. On the contrary, when the mean value is not very large compared with the other moments, the wormhole saddles contribute significantly to the path integral. In this case the factorization of spectral correlators can be restored by adding various half-wormhole-like non-self-averaging saddles. Moreover, when the random couplings satisfy a general distribution with non-trivial higher moments, new half-wormhole saddles exist and should be included in the path integral. In models where the random couplings are drawn from discrete distributions, such as the Poisson distribution, we greatly modified the conventional approach of introducing collective variables and provide explicit proposals for the expression of half-wormhole-like contributions.⁸ Additionally, we generalize the construction of half-wormhole saddles to the Brownian SYK model and confirm that non-self-averaging saddles also exist and help restore the factorization of the spectral correlators.

There are proposals in [72, 74, 77, 81] of the half-wormhole contributions to $z^2$ in the original 1d SYK model. It would be interesting to generalize our punctured disk saddle to the SYK model.

Appendices

Appendix A: Lefschetz thimbles

In this appendix, we review the method of Lefschetz thimble [83]. Suppose we would like to evaluate the integral

$$\begin{aligned} Z=\int _{\mathcal {M}_{{\mathbb {R}}}} \text {d}x^i e^S\,, \end{aligned}$$

(A.1)

where the integration contour is $\mathcal {M}_{{\mathbb {R}}}$. Then we complexify the manifold on which the integration is done to $\mathcal {M}_{\mathbb {C}}$. We choose $\Re (S)$ to be the Morse function, as we want to find the contours where S has a constant imaginary part. The saddle points of the integral are the critical points of the Morse function because of the Cauchy-Riemann equations. Around each critical point on $\mathcal {M}_{{\mathbb {C}}}$ we introduce a set of local coordinates $\{z_i\}$. The Morse flow is determined by the flow equations

$$\begin{aligned} \frac{\text {d}z^i}{\text {d}t}=-g^{i{\bar{j}}}\frac{\partial {\bar{S}}}{\partial {\bar{z}}^j},\quad \frac{\text {d}{\bar{z}}^{j}}{\text {d}t}=-g^{i{\bar{j}}}\frac{\partial S}{\partial z^{i}}\, \end{aligned}$$

(A.2)

where $g^{i{\bar{j}}}$ is Kähler metric and the bar means complex conjugate. We find

$$\begin{aligned} \frac{\text {d}(S-{\bar{S}})}{\text {d}t }=\frac{\partial S}{\partial z^i}\frac{\text {d}z^i}{\text {d}t}-\frac{\partial {\bar{S}}}{\partial {\bar{z}}^i}\frac{\text {d}{\bar{z}}^i}{\text {d}t}=0, \end{aligned}$$

(A.3)

which implies that the imaginary part of S is a constant along the flow. Each of the critical points is associated with a pair of flows, the thimble and the anti-thimble. The thimble is the “stable” direction such that the Morse function $\Re (S)$ decays along the thimble and the integral of $\exp (S)$ along the thimble converges. On the contrary, the anti-thimble is the “unstable” direction. Explicitly the boundary conditions for a particular critical point $p_{\sigma }$ are

$$\begin{aligned}&\lim _{t\rightarrow -\infty }z(t)=p_{\sigma },\quad \text {for thimbles}, \end{aligned}$$

(A.4)

$$\begin{aligned}&\lim _{t\rightarrow +\infty }z(t)=p_{\sigma },\quad \text {for anti-thimbles}. \end{aligned}$$

(A.5)

The main statement in [83] that we will use repeatedly is that the integral can be approximated by a weighted sum over integrals along the thimbles of each critical point

$$\begin{aligned} Z=\sum _i n_i \int _{\mathcal {J}_i} \text {d}t\, e^{S[t]}, \end{aligned}$$

(A.6)

where i runs over all the critical points, $\mathcal {J}_i$ is the Lefschetz thimble attaching to the $i^{\text {th}}$ critical point, and the weight $n_i$ is given by the intersection number between the anti-thimble and the original integration contour $\mathcal {M}_{{{\mathbb {R}}}}$.

1.1 Appendix A.1: some examples

To illustrate how this works, we first go through some simple examples.

1.1.1 Appendix A.1.1: The Gaussian function

Let us consider a simple example with

$$\begin{aligned} S=-x^2/2+\sigma x. \end{aligned}$$

(A.7)

The integral can be regarded as a zero-dimension theory with quadratic interaction and a complex source $\sigma$. The only critical point is at $x=\sigma =a+\text {i}b$. The flow equation is

$$\begin{aligned} \frac{\text {d}x}{\text {d}t}={\bar{x}}-(a-\text {i}b). \end{aligned}$$

(A.8)

Expressing $x=x_1+\text {i}x_2$, we get the following equations

$$\begin{aligned} \frac{\text {d}x_1}{\text {d}t}=x_1-a,\quad \frac{\text {d}x_2}{\text {d}t}=b-x_2. \end{aligned}$$

(A.9)

The general solution can be easily solved

$$\begin{aligned} x_{1}=a+c_{1}e^{t},\quad x_{2}=b+c_{2}e^{-t}, \end{aligned}$$

(A.10)

where $c_{1}$,$c_{2}$ are two undetermined constants. The boundary conditions for the thimble is

$$\begin{aligned} (x_{1},x_{2})\rightarrow (a,b),\quad t\rightarrow -\infty , \end{aligned}$$

(A.11)

while for the anti-thimble we have

$$\begin{aligned} (x_{1},x_{2})\rightarrow (a,b),\quad t\rightarrow +\infty , \end{aligned}$$

(A.12)

where (a, b) is the critical point. Then with these boundary conditions we can get the equations for the thimble and the anti-thimble respectively

$$\begin{aligned} x_{2}=b, \end{aligned}$$

(A.13)

$$\begin{aligned} x_{1}=a. \end{aligned}$$

(A.14)

We plot the thimble and the anti-thimble in this case in Fig. 8, where for simplicity we let $\sigma =1+\text {i}$.

We can also compare the saddle point solution with the exact result. The integral can evaluated as

$$\begin{aligned} \int \text {d}x e^{-x^{2}/2+\sigma x}=\sqrt{2\pi }e^{\sigma ^{2}/2}. \end{aligned}$$

(A.15)

While the saddle point solution gives

$$\begin{aligned} e^{\sigma ^{2}/2}, \end{aligned}$$

(A.16)

with the one-loop correction $\sqrt{2\pi }$ the saddle point analysis recovers the exact result.

Fig. 8

The red line denotes the thimble and the blue line denotes the anti-thimble. The anti-thimble intersects with the real line, so this saddle point contributes

1.1.2 A.1.2: The Airy function

A slightly less trivial example is the Airy action

$$\begin{aligned} Z_\lambda =\int _{-\infty }^\infty \text {d}x e^S,\quad S= \text {i}\lambda \left( \frac{x^3}{3}-x\right) . \end{aligned}$$

(A.17)

It is not hard to find that for real $\lambda$ there are three “convergent” regions, namely $\Re (S)<\infty$, on the complex x-plane:

$$\begin{aligned}{} & {} x=r e^{\text {i}\theta },\quad \frac{2n \pi }{3}\le \theta \le \frac{\pi }{3}+\frac{2n\pi }{3},\quad n=0,1,2. \end{aligned}$$

(A.18)

In each convergent region, the Airy integrand is exponentially small. As we vary $\lambda$ to complex values, we should deform the integration contour of x accordingly so that the integral remains converge. This gives an analytic continuation of $Z_\lambda$. The two critical points are located at $x=\pm 1$. The values of saddle points are

$$\begin{aligned} S_\pm =\mp \frac{2\text {i}\lambda }{3}. \end{aligned}$$

(A.19)

Since along the (anti-)thimbles, the imaginary part of S is a constant and

$$\begin{aligned} \Im (S_\pm )=\mp \frac{2 \Re (\lambda )}{3}. \end{aligned}$$

(A.20)

Therefore the two (anti-)thimbles associated with the two critical points will not intersect except for the case of $\Re (\lambda )=0$. The thimble which connects critical points is called the Stoke ray. Using the Lefschetz thimbles $\mathcal {J}_\pm$, we can rewrite the integral as

$$\begin{aligned} Z_\lambda =n_+ \int _{\mathcal {J}_+} \exp S+n_- \int _{\mathcal {J}_-} \exp S. \end{aligned}$$

(A.21)

To solve the thimbles, let us take $\lambda =1$, then the flow equations are

$$\begin{aligned} \frac{\text {d}x}{\text {d}t}=\text {i}({\bar{x}}^2-1). \end{aligned}$$

(A.22)

Expressing $x=x_1+\text {i}x_2$, we obtain

$$\begin{aligned} \frac{\text {d}x_1}{\text {d}t}=2 x_1 x_2,\quad \frac{\text {d}x_2}{\text {d}t}=x_1^2-x_2^2-1. \end{aligned}$$

(A.23)

We plot the anti-thimbles in Fig. 9

Fig. 9

The red line denotes the anti-thimble of $x=-1$ and the Green line denotes the anti-thimble of $x=1$.The grey regions are the convergent regions

Therefore for $\lambda =1$ both of the saddle points contribute. This result is expected since that the two critical points are already located on the real line.

The problem we met in the main text is better illustrated by the following toy model

$$\begin{aligned} {\tilde{Z}}=\int _{-\infty }^\infty \text {d}x \exp S, \quad S=\text {i}\left( \frac{x^3}{3}+x\right) . \end{aligned}$$

(A.24)

The integral is convergent and can be expressed by the Airy function

$$\begin{aligned} {\tilde{Z}}=\frac{2 \pi \text {Ai}\left( \frac{1}{\root 3 \of {3}}\right) }{\root 3 \of {3}}=0.83. \end{aligned}$$

(A.25)

We now try to compute the integral with saddle point approximation, where the saddle points are located at $x=\pm \text {i}$. The saddle point value, plus the 1-loop correction, of the integral at these two saddle points, ${\tilde{Z}}_\pm$ are the same, and the sum of them is larger than the exact evaluation of the integral

$$\begin{aligned} {\tilde{Z}}_+={\tilde{Z}}_-=0.733,\quad {\tilde{Z}}_++{\tilde{Z}}_->\tilde{Z}. \end{aligned}$$

(A.26)

This is exactly the situation we are encountering. In this toy model, it is easy to show that the anti-thimble associated with the saddle point $x=-\text {i}$ does not intersect with the real axis, Fig. 10, so the saddle point $x=-\text {i}$ does not contribute to the integral.

Fig. 10

The red line is the anti-thimble of $x=\text {i}$ which intersects with the real line while the Green line is the anti-thimble of $x=-\text {i}$ which does not intersect with the real line. The grey regions are the convergent regions

1.2 Appendix A.2: Multi-variable cases

Let us consider another example with two variables

$$\begin{aligned} Z=\int \text {d}\sigma \frac{\text {d}g}{2\pi } e^{S},\quad S=\log \sigma -\text {i}\sigma g-\frac{1}{2}g^2. \end{aligned}$$

(A.27)

The integral can be done directly to get

$$\begin{aligned} Z=0\ . \end{aligned}$$

(A.28)

There are two saddle points

$$\begin{aligned} g_\pm =\pm \text {i},\quad \sigma _\pm =\mp 1. \end{aligned}$$

(A.29)

with saddle point contributions to the integral (on-shell action)

$$\begin{aligned} Z_\pm =\mp \frac{1}{\sqrt{e}},\quad Z_-+Z_+=0\, \end{aligned}$$

(A.30)

Matching this with the exact solution suggests that $n_\pm =1$. Note that $\sigma _\pm =\mp 1$ are already on the real line so corresponding anti-thimbles always intersect with the original contour. The flow equations are

$$\begin{aligned} \frac{d\sigma }{dt}=-\frac{1}{{\bar{\sigma }}}-\text {i}{\bar{g}},\quad \frac{\text {d}g}{dt}=-\text {i}{\bar{\sigma }}+{\bar{g}}. \end{aligned}$$

(A.31)

Expressing $\sigma =\sigma _1+\text {i}\sigma _2$ and $g=g_1+\text {i}g_2$ we obtain the following differential equations

$$\begin{aligned}{} & {} \frac{\text {d}\sigma _1}{\text {d}t}+g_2+\frac{\sigma _1}{\sigma _1^2+\sigma _2^2}=0, \end{aligned}$$

(A.32)

$$\begin{aligned}{} & {} \frac{\text {d}\sigma _2}{\text {d}t}+g_1+\frac{\sigma _2}{\sigma _1^2+\sigma _2^2}=0, \end{aligned}$$

(A.33)

$$\begin{aligned}{} & {} \frac{\text {d}g_1}{\text {d}t}+\sigma _2-g_1=0,\quad \frac{\text {d}g_2}{\text {d}t}+\sigma _1+g_2=0. \end{aligned}$$

(A.34)

We find that indeed these two saddles should both be included. We plot the g-plane of the anti-thimbles in Fig. 11.

Fig. 11

Since $\sigma _\pm$ are already on the real line here we only plot the g-plane of the anti-thimbles. Clearly both of these two anti-thimbles intersect with the real axis so these two saddle points both contribute to the integral

Note that this example is special case of (18) with $q=2$.

1.3 $\bullet$ Flow equations in real coordinates

Sometimes it more convenient to use the real form of the flow equations (A.2). We start with the relations

$$\begin{aligned} \frac{\partial S}{\partial z}=\frac{1}{2} \frac{\partial S}{\partial x}+\frac{1}{2\text {i}} \frac{\partial S}{\partial y}, \end{aligned}$$

(A.35)

$$\begin{aligned} \frac{\partial {\bar{S}}}{\partial {\bar{z}}}=\frac{1}{2} \frac{\partial {\bar{S}}}{\partial x}-\frac{1}{2\text {i}} \frac{\partial {\bar{S}}}{\partial y}, \end{aligned}$$

(A.36)

where

$$\begin{aligned} z=x+\text {i}y. \end{aligned}$$

(A.37)

Then we evaluate the equation as

$$\begin{aligned} \frac{\text {d}z}{\text {d}t}+\frac{\text {d}{\bar{z}}}{\text {d}t}&=- \frac{\partial S}{\partial z}-\frac{\partial {\bar{S}}}{\partial {\bar{z}}}=-\frac{\partial \Re (S)}{\partial x}-\frac{\partial \Im (S)}{\partial y}, \end{aligned}$$

(A.38)

$$\begin{aligned} \frac{\text {d}z}{\text {d}t}-\frac{\text {d}{\bar{z}}}{\text {d}t}&=\frac{\partial S}{\partial z}-\frac{\partial {\bar{S}}}{\partial {\bar{z}}}=-\text {i}\frac{\partial \Re (S)}{\partial y}+\text {i}\frac{\partial \Im (S)}{\partial x}, \end{aligned}$$

(A.39)

where we work in the flat space. Recall the Cauchy–Riemann equation we get

$$\begin{aligned} \frac{\text {d}x}{\text {d}t}=-\frac{\partial \Re (S)}{\partial x},\quad \frac{\text {d}y}{\text {d}t}=-\frac{\partial \Re (S)}{\partial y}. \end{aligned}$$

(A.40)

To illustrate it we consider a special case in the Airy function

$$\begin{aligned} S=\text {i}\left( \frac{x^{3}}{3} +x \right) . \end{aligned}$$

(A.41)

In the complex plane its conjugate is

$$\begin{aligned} {\bar{S}}=-\text {i}\left( \frac{{\bar{x}}^{3}}{3} +{\bar{x}} \right) , \end{aligned}$$

(A.42)

and we can define the components

$$\begin{aligned} x=x_1+\text {i}x_2,\quad {\bar{x}}=x_1-\text {i}x_2. \end{aligned}$$

(A.43)

The flow equation in complex coordinates becomes

$$\begin{aligned} \frac{\text {d}x}{\text {d}t}=-\frac{\partial {\bar{S}}}{\partial {\bar{x}}}=\text {i}({\bar{x}}^{2}+1), \end{aligned}$$

(A.44)

which leads to the equations in real coordinates

$$\begin{aligned} \frac{\text {d}x_1}{\text {d}t}=2x_1x_2,\quad \frac{\text {d}x_2}{\text {d}t}=x_1^{2}-x_2^{2}+1. \end{aligned}$$

(A.45)

On the other hand we can get the equations with the real part of S:

$$\begin{aligned} \Re (S)=-x_2-x_1^{2}x_2+\frac{x_2^{3}}{3}. \end{aligned}$$

(A.46)

From the Eqs. (A.40) we can recover the two flow Eq. (A.45).

Appendix B: Details of the derivation of $\langle z\rangle$ and $\langle z^2\rangle$

This result can be derived by a recursive method with respect to p as shown in [84]. We choose a set of collective variables

$$\begin{aligned} G=\frac{1}{N}\sum _{1\le i<j\le N}\psi _i\psi _j. \end{aligned}$$

(B.47)

For simplicity of demonstration let us first consider the $q=4$ case it is easy to see

$$\begin{aligned} G^2=\frac{2!}{N^2}\sum _A \psi _A, \end{aligned}$$

(B.48)

then $\langle z\rangle$ can be rewritten as

$$\begin{aligned} \langle z\rangle _{}=\int _{{\mathbb {R}}}\text {d}G \int _{\text {i}{\mathbb {R}}}\frac{\text {d}\Sigma }{2\pi \text {i}/N} \text {d}^N \psi \, e^{-\frac{u}{2 }N^2G^2} e^{- \Sigma (NG-\sum _{i<j}\psi _i\psi _j)}. \nonumber \\ \end{aligned}$$

(B.49)

Now we can integrate the out the fermions to get

$$\begin{aligned} \int \text {d}^N \psi \, e^{\Sigma \sum _{i<j}\psi _i \psi _j}=(\Sigma )^{N/2} m[p]\, |_{(q=2)}=\Sigma ^{N/2}.\nonumber \\ \end{aligned}$$

(B.50)

Then (B.49) becomes

$$\begin{aligned} \langle z\rangle _{q=4}= & {} \int _{{\mathbb {R}}}\text {d}G \int _{\text {i}{\mathbb {R}}}\frac{\text {d}\Sigma }{2\pi \text {i}/N}\Sigma ^{N/2} e^{- \frac{ u N^2G^2}{2}}e^{-N \Sigma G}\, \nonumber \\= & {} N^{-N/2}(\partial _G)^{N/2}e^{-\frac{ u N^2 G^2}{2}}\, |_{G=0} \, \end{aligned}$$

(B.51)

$$\begin{aligned}= & {} \left( \frac{u}{2}\right) ^{N/4} \frac{(N/2)!}{(N/4)!}=m[p] u^p|_{q=4}\,. \end{aligned}$$

(B.52)

For general q, the proof is similar with the modification

$$\begin{aligned} \sum _A\psi _A=\frac{N^{q/2}}{(q/2)!}G^{q/2}\,. \end{aligned}$$

(B.53)

In summary, we have generalized the $G,\Sigma$ trick and derived an effective action to compute $\langle z\rangle$:

$$\begin{aligned} \langle z\rangle _{}=\int _{{\mathbb {R}}}\text {d}G \int _{\text {i}{\mathbb {R}}}\frac{\text {d}\Sigma }{2\pi \text {i}/N}\Sigma ^{N/2} e^{u \text {i}^{q/2}\frac{N^{q/2}}{(q/2)! }G^{q/2}} e^{- N\Sigma G}\,. \end{aligned}$$

(B.54)

By introducing the following collective variables

$$\begin{aligned}{} & {} G_{LR}=\frac{1}{N}\sum _i \psi _i^L \psi _i^R, \end{aligned}$$

(B.55)

$$\begin{aligned}{} & {} G_L=\frac{1}{N}\sum _{i<j}\psi _i^L \psi _j^L,\quad G_R=\frac{1}{N}\sum _{i<j}\psi _i^R \psi _j^R, \end{aligned}$$

(B.56)

the averaged quantity $\langle z^2\rangle$ can be expressed as

$$\begin{aligned} \langle z^2\rangle= & {} \int _R \text {d}^3 G_i \int _{i {\mathbb {R}}}\text {d}^3 \Sigma _i\, e^{\frac{N}{q}(\tau ^2 G_{LR}^q+\mu G_L^{q/2}+\mu G_{R}^{q/2})- N(\Sigma _i G_i)}\nonumber \\{} & {} \times \int \text {d}^{2N}\psi e^{\frac{1}{2}{\Psi }M{\Psi }}, \end{aligned}$$

(B.57)

$$\begin{aligned}= & {} \int _R \text {d}^3 G_i \int _{i {\mathbb {R}}}\text {d}^3 \Sigma _i\,e^{\frac{N}{q}(\tau ^2 G_{LR}^q+\mu G_L^{q/2}+\mu G_{R}^{q/2})- N(\Sigma _i G_i)}\nonumber \\{} & {} \times \sqrt{\text {det}[\Sigma _L\Sigma _RA^2+\Sigma _{LR}^2]}, \end{aligned}$$

(B.58)

$$\begin{aligned}= & {} \int _R \text {d}^3 G_i \int _{i {\mathbb {R}}}\text {d}^3 \Sigma _i\, \sum _{m=0}^{N/2} {N \atopwithdelims ()2m}( \Sigma _{LR})^{2m}\nonumber \\{} & {} \times (\text {i}^2 \Sigma _L\Sigma _R)^{\frac{N}{2}-m}e^{\frac{N}{q}(\tau ^2 G_{LR}^q+\mu G_L^{q/2}+\mu G_{R}^{q/2})} e^{- N(\Sigma _i G_i)}, \nonumber \\ \end{aligned}$$

(B.59)

where we have defined

$$\begin{aligned} \Psi= & {} \left( \psi _1^L,\dots ,\psi _{N}^L,\psi _1^R,\dots ,\psi _{N}^R\right) , \end{aligned}$$

(B.60)

$$\begin{aligned} M= & {} \begin{pmatrix} \Sigma _L A&{} \Sigma _{LR} I_{N}\\ -\Sigma _{LR} I_{N}&{} \Sigma _R A\\ \end{pmatrix}, \end{aligned}$$

(B.61)

$$\begin{aligned} A= & {} -A^T,\quad A_{ij}=1,\quad \forall i<j. \end{aligned}$$

(B.62)

Using the same tricks of integration by part, it can be evaluated exactly as

$$\begin{aligned} \langle z^2\rangle= & {} N^{-N}\sum _{k=0}^p{N\atopwithdelims ()kq}(\partial _{G_{LR}})^{kq}(\text {i}^2\partial _{G_L}\partial _{G_R})^{\frac{N-kq}{2}}\nonumber \\{} & {} \times e^{\frac{N}{q}(\tau ^2 G_{LR}^q+\mu G_L^{q/2}+\mu G_{R}^{q/2})}|_{G_i=0} \end{aligned}$$

(B.63)

$$\begin{aligned}= & {} N^{-N}\sum _{k=0}^p \text {i}^{N-kq}{N\atopwithdelims ()kq}\frac{(kq)!}{k!}\left( \frac{N\tau ^2}{q}\right) ^k\left[ \frac{(\frac{q(p-k)}{2})!}{(p-k)!}\right] ^2\nonumber \\{} & {} \times \left( \frac{N \mu }{q}\right) ^{2p-2k} \end{aligned}$$

(B.64)

$$\begin{aligned}= & {} \sum _{k=0}^p c_k m_{p-k}^2t^{2k}u^{2p-2k}, \end{aligned}$$

(B.65)

where m[p] is defined in (68) and the coefficient c[k] is

$$\begin{aligned}{} & {} c[k]=\frac{N!}{k! (q!)^k (N-kq)!}. \end{aligned}$$

(B.66)

Appendix C: A naive expression of the half-wormhole contribution and its failure

Inspired by our analysis of the punctured disks for z, we can insert another two copies of identities (104) in $z^2$

$$\begin{aligned}{} & {} z^2=\int \text {d}\sigma _w \text {d}\sigma _{h_L}\text {d}\sigma _{h_R}\Psi (\sigma _w,\sigma _{h_L},\sigma _{h_L})\hat{\Lambda }(\sigma _w,\sigma _{h_L},\sigma _{h_L}), \nonumber \\ \end{aligned}$$

(C.67)

$$\begin{aligned}{} & {} \Psi (\sigma _w,\sigma _{h_L},\sigma _{h_L})=\Psi (\sigma _w)\Psi (\sigma _{h_L})\Psi (\sigma _{h_R})\,, \end{aligned}$$

(C.68)

$$\begin{aligned}{} & {} \hat{\Lambda }(\sigma _w,\sigma _{h_L},\sigma _{h_L})\nonumber \\{} & {} =\int \text {d}^{2N} \psi \exp \left[ \text {i}e^{- \frac{2\text {i}\pi }{q}}\sigma _{h_L} \sum _{i<j} \psi ^L_{ij} +\text {i}e^{-\frac{2\text {i}\pi }{q}}\sigma _{h_R} \sum _{i<j} \psi ^R_{ij}\right. \nonumber \\{} & {} \left. \quad +\text {i}e^{\frac{\text {i}\pi }{q}}\sigma _w \psi _i^L \psi _i^R+\text {i}^{q/2} J_A(\psi _A^L+\psi _A^R)\right. \nonumber \\{} & {} \left. \quad -\text {i}^{q/2}u \sum _A(\psi _A^L+\psi _A^R)-\text {i}^q t^2\psi _A^L\psi _A^R \right] , \end{aligned}$$

(C.69)

where we have introduced three pairs of $G,\Sigma$ variables

$$\begin{aligned}{} & {} G_w=\frac{1}{N}\psi _i^L\psi _i^R, \end{aligned}$$

(C.70)

$$\begin{aligned}{} & {} G_{h_L}=\frac{1}{N}\sum _{i<j} \psi ^L_{ij},\quad G_{h_R}=\frac{1}{N}\sum _{i<j} \psi ^R_{ij}, \end{aligned}$$

(C.71)

and rotated the contour as before. As before, the function $\Psi$ is highly peaked around $\Psi (0,0,0)$ so one may expect that there is a half-wormhole saddle point

$$\begin{aligned} \Lambda&=\hat{\Lambda }(0,0,0)\nonumber \\&=\sum '_{A,B}\text {sgn}(A)\text {sgn}(B)\nonumber \\&\quad \times \prod _{k=1}^p\left( (J_{A_k}-J_{A_k}^0)(J_{B_k}-J_{B_k}^0)-\delta _{A_k B_k}t^2\right) \,, \end{aligned}$$

(C.72)

whose average manifestly vanishes $\langle \Lambda \rangle =0$ and it further satisfies $\langle \Lambda ^2\rangle =\langle z^2\Lambda \rangle =2 {z_2^{(p)}}^2$. If we naively follow the discussion in [73] and propose the following approximation

$$\begin{aligned} z^2\approx \langle z^2\rangle +\Lambda \end{aligned}$$

(C.73)

then the error can be evaluated as

$$\begin{aligned} \bigg \langle \left( z^2-(\langle z^2\rangle +\Lambda )\right) ^2\bigg \rangle= & {} \langle z^4\rangle -\langle z^2\rangle ^2-2\langle z^2\Lambda \rangle +\langle \Lambda ^2\rangle \nonumber \\> & {} 2\langle z^2\rangle ^2-2{z_2^{(p)}}^2\approx 2(z_2^{(k)})^2, \nonumber \\ \end{aligned}$$

(C.74)

where we have used

$$\begin{aligned} \langle z^4\rangle > 3\langle z^2\rangle ^2,\quad \langle z^2\rangle \approx z_2^{(k)}. \end{aligned}$$

(C.75)

Therefore, the average of the error square and $\langle z^2\rangle ^2$ are in the same order in the large N limit, which meaning the approximation (C.73) is not accurate.

Appendix D: Random coupling from product of Grassmann variables $J_A^{(0)}=J\prod _i\theta _{A_i}$

A modified SYK-like model dubbed as partially disorder-averaged SYK model is proposed in [81]. In this model, the random coupling ${\widetilde{J}}_A$ consists of two pieces

$$\begin{aligned} {\widetilde{J}}_A=J_A+J_A^{(0)} \end{aligned}$$

(D.76)

where $J_A$ is the standard random coupling of the SYK model while $J_A^{(0)}$ is specially chosen as

$$\begin{aligned} J_{i_i\dots i_q}^{(0)}= \text {i}^{q/2} q!\mu \, \theta _{i_1}\dots \theta _{i_q},\quad \text {with}\quad \{\theta _i,\theta _j\}={2}\delta _{ij} \end{aligned}$$

(D.77)

so we can think of it as coupling the fermions $\psi _i$ in the original model with some background Majorana fermions $\theta _i$ (or non-dynamical fermions living in another universe [81]). Note that $J_A^{(0)}$ is not a c-number which is different from our models studied in the previous section.

1.1 D.0.1: 0d model

Let us first consider the 0-dimensional model to see the difference explicitly. In this case the integral (1) can be written as

$$\begin{aligned} z= & {} \int \text {d}^N \psi \exp \left( \text {i}^{q/2}\sum {\widetilde{J}}_{i_1\dots i_q}\psi _{i_1\dots i_q}\right) \, \nonumber \\= & {} \int \text {d}^N \psi \exp \left( \text {i}^{q/2}\sum _A J_A\psi _A+\mu \left( \sum _i \theta _i\psi _i\right) ^q \right) .\nonumber \\ \end{aligned}$$

(D.78)

The averaged quantity $\langle z\rangle$

$$\begin{aligned} \langle z\rangle =\int \text {d}^N \psi \exp \left( \mu \left( \sum _i \theta _i\psi _i\right) ^q\right) \,, \end{aligned}$$

(D.79)

can be computed in two ways. One can integrate out the fermions $\psi _i$ directly. The result is

$$\begin{aligned} \langle z\rangle{} & {} = \frac{\mu ^{N/q} N!}{(N/q)!} \int \text {d}^N\psi (\theta _1\psi _1)\dots (\theta _N \psi _N) \end{aligned}$$

(D.80)

$$\begin{aligned}{} & {} =\left[ \prod _i \theta _i\right] \frac{\mu ^{N/q} N!}{(N/q)!}\equiv \left[ \prod _i \theta _i\right] {\mathfrak {m}}_p\, . \end{aligned}$$

(D.81)

Note that z is not a c-number and depends on the background fermions living in other universe. Here we will not think of this as a problem but a feature since the model is not exactly the original SYK model. Alternatively we can compute this average quantity by the $G,\Sigma$ trick:

$$\begin{aligned}{} & {} G_\sigma =\sum _i \theta _i\psi _i\,,\nonumber \\{} & {} \langle z\rangle =\int \text {d}^N \psi \int \text {d}G_\sigma \frac{\text {d}\Sigma _\sigma }{2\pi } e^{\text {i}[\Sigma _\sigma ( G_\sigma -\sum _i \theta _i\psi _i)]}e^{ \mu G_\sigma ^q} \nonumber \\{} & {} \qquad = \left[ \prod _i \theta _i\right] \int \text {d}G_\sigma \frac{\text {d}\Sigma _\sigma }{2\pi }\Sigma _\sigma ^N e^{\text {i}\Sigma _{\sigma } G_{\sigma }+\mu G_\sigma ^q} \end{aligned}$$

(D.82)

$$\begin{aligned}{} & {} \qquad = \left[ \prod _i \theta _i\right] (\partial _{G_\sigma })^N e^{\mu G_\sigma ^q}|_{G_\sigma =0} \end{aligned}$$

(D.83)

$$\begin{aligned}{} & {} \qquad =\left[ \prod _i \theta _i\right] \frac{\mu ^{N/q}N!}{(N/q)!}. \end{aligned}$$

(D.84)

One can also use the effective action (D.82D.83D.84) to derive the large N result of (D.79) as shown in [81]. We will not repeat that analysis here. Instead, we would like to consider the half-wormhole saddle of z

$$\begin{aligned} z\approx \langle z\rangle +\Theta \end{aligned}$$

(D.85)

as we did in last section. The subtlety is that as we stressed z is not a c-number so the approximation (D.85) is in the sense

$$\begin{aligned} \langle [z-(\langle z\rangle +\Theta )]^2\rangle \approx 0, \end{aligned}$$

(D.86)

which is a c-number due to (D.77) is small. Let us proceed by computing the averaged quantity $\langle z^2\rangle$

$$\begin{aligned} \langle z^2\rangle= & {} \int \text {d}^{2N}\psi \exp \left( {\frac{\tau ^2}{q!}} \left( \sum _i \psi _i^L\psi _i^R\right) ^q \right. \nonumber \\{} & {} \left. +\mu \left( \sum _i\theta _i \psi _i^L\right) ^q+\mu \left( \sum _i\theta _i \psi _i^R\right) ^q\right) , \end{aligned}$$

(D.87)

$$\begin{aligned}= & {} \int \text {d}^{2N}\psi \sum _k \left( \frac{\tau ^2}{q!}\right) ^k \nonumber \\{} & {} \times \sum _{{i_1<\dots< i_{kq}}}(\psi ^{LR}_{i_1}\dots \psi ^{LR}_{i_{qk}})\frac{(qk)!}{k!} \left( \frac{(N-kq)!}{(N/q-k)!}\right) ^2\mu ^{2p-2k} \nonumber \\{} & {} \times \sum _{j_1<\dots < j_{N-qk}\ne \{i\}}(\theta _{j_1}\psi _{j_1}^L)\dots (\theta _{j_{N-qk}}\psi ^L_{j_{N-kq}}) (\theta _{j_1}\psi _{j_1}^R)\nonumber \\{} & {} \dots (\theta _{j_{N-k}}\psi ^R_{j_{N-kq}}), \end{aligned}$$

(D.88)

$$\begin{aligned}= & {} \int \text {d}^{2N}\psi \sum _k \left( \frac{\tau ^2}{q!}\right) ^k\mu ^{2p-2k}\frac{(qk)!}{k!} \left( \frac{(N-kq)!}{(N/q-k)!}\right) ^2\nonumber \\{} & {} \times {N\atopwithdelims ()kq}\psi ^{LR}_1\dots \psi _N^{LR}, \end{aligned}$$

(D.89)

$$\begin{aligned}= & {} \sum _k \left( \frac{\tau ^2}{q!}\right) ^k\mu ^{2p-2k}\frac{(qk)!}{k!} \left( \frac{(N-kq)!}{(N/q-k)!}\right) ^2{N\atopwithdelims ()kq}, \end{aligned}$$

(D.90)

$$\begin{aligned}= & {} \sum _k^p \tau ^{2k}\mu ^{2(p-k)}c_k{\mathfrak {m}}_{p-k}^2\equiv \sum _k {\mathfrak {z}}_2^{(k)}\,, \end{aligned}$$

(D.91)

where $c_k$ of defined in (7475) and ${\mathfrak {m}}_p$ is defined in (D.81). The result (D.87D.88D.89D.90) is in the same form of (B.65). So the analysis of the half-wormhole saddle will be similar; we insert the a suitable identity to (D.78)

$$\begin{aligned} z= & {} \int \text {d}^N \psi \exp \left( \text {i}^{q/2}\sum {\widetilde{J}}_{i_1\dots i_q}\psi _{i_1\dots i_q}\right) \int \text {d}G_{\sigma } \nonumber \\{} & {} \times \delta \left( G_\sigma - \sum _{i}\theta _i\psi _i\right) \exp \left( \frac{\mu }{q!}\left( G_\sigma ^q-\left( \sum _{i}\theta _i\psi _i\right) ^q\right) \right) , \nonumber \\ \end{aligned}$$

(D.92)

$$\begin{aligned}= & {} \int \text {d}^N \psi \frac{\text {d}\Sigma _\sigma \text {d}G_\sigma }{2\pi \text {i}}\exp \left( \text {i}^{q/2}\sum _A J_A\psi _A+\Sigma _\sigma \sum _i\theta _i\psi _i\right) \nonumber \\{} & {} \times \exp \left( -\Sigma _\sigma G_\sigma +\frac{\mu }{q!}G_\sigma ^q\right) \,. \end{aligned}$$

(D.93)

Following the arguments below (104) one can obtain the half-wormhole saddle⁹

$$\begin{aligned} \Theta =\left[ \prod _i \theta _i\right] \int \text {d}^N \psi \exp \left( \text {i}^{q/2}\sum _A J_A\psi _A\right) \,. \end{aligned}$$

(D.94)

Then it is easy to find that the half-wormhole saddle satisfies

$$\begin{aligned} \langle \Theta \rangle =0,\quad \langle \Theta ^2\rangle =\langle \Theta z\rangle ={\mathfrak {z}}_2^{(p)}\,, \end{aligned}$$

(D.95)

so the approximation (D.85) will be sufficient if ${\mathfrak {z}}_2^{(p)}$ is the dominant term in (D.91) as we have shown in last section. When ${\mathfrak {z}}_2^{(p)}$ is not the dominant term we have to consider the contribution of fluctuation of $\Sigma _\sigma$. To finish our analysis of the half-wormhole saddle for z, let us redo the computation of $\langle z^2\rangle$ with the $G,\Sigma$ trick. We need introduce three G variables

$$\begin{aligned}{} & {} G_{LR}=\frac{1}{N}\sum _i \psi _i^L \psi _i^R, \end{aligned}$$

(D.96)

$$\begin{aligned}{} & {} G_L=\frac{1}{{N}}\sum _i\theta _i\psi _i^L,\quad G_R=\frac{1}{{N}}\sum _i\theta _i \psi _i^R \end{aligned}$$

(D.97)

then $\langle z^2\rangle$ can be written as

$$\begin{aligned} \langle z^2\rangle= & {} \int \text {d}^{2N}\psi \prod _{a} \text {d}G_{a}\prod _a \frac{\text {d}\Sigma _a}{2\pi \text {i}/N } \nonumber \\{} & {} \times \exp \left( \frac{N}{q}\left( t G_{LR}^q+u G_L^q+u G_R^q\right) \right) \nonumber \\{} & {} \times \int \exp \left( -\Sigma _{LR}\left( NG_{LR}-\sum _i\psi _i^L \psi _i^R\right) \right) \nonumber \\{} & {} \times \exp \left( -\Sigma _{L}\left( NG_{L}-\sum _i\theta _i\psi _i^L \right) \right) \nonumber \\{} & {} \times \exp \left( -\Sigma _{R}\left( NG_{R}-\sum _i\theta _i\psi _i^R \right) \right) , \end{aligned}$$

(D.98)

$$\begin{aligned}= & {} \int \left[ \prod _{a} \text {d}G_{a} \frac{\text {d}\Sigma _a}{2\pi \text {i}/N}\right] \nonumber \\{} & {} \times \exp \left( N\left( \frac{t}{q} G_{LR}^q+\frac{u}{q} G_L^q+\frac{u}{q} G_R^q-\sum _a\Sigma _a G_a\right) \right) \nonumber \\{} & {} \times \left( \Sigma _{LR}+\Sigma _L\Sigma _R\right) ^N\,, \end{aligned}$$

(D.99)

where in order to have a well-defined large N scaling we have introduced

$$\begin{aligned} t=\frac{\tau ^2}{(q-1)!}N^{q-1},\quad u=q\mu N^{q-1}. \end{aligned}$$

(D.100)

The saddle point equations are

$$\begin{aligned}{} & {} tG_{LR}^{q-1}=\Sigma _{LR},\quad u G_L^{q-1}=\Sigma _L,\quad u G_R^{q-1}=\Sigma _R \,, \end{aligned}$$

(D.101)

$$\begin{aligned}{} & {} G_{LR}=\frac{1}{\Sigma _{LR}+\Sigma _L\Sigma _R},\quad G_L=-\frac{ \Sigma _R}{\Sigma _{LR}+\Sigma _L\Sigma _R}, \end{aligned}$$

(D.102)

$$\begin{aligned}{} & {} G_R=-\frac{ \Sigma _L}{\Sigma _{LR}+\Sigma _L\Sigma _R}\,. \end{aligned}$$

(D.103)

The obvious solutions are the “wormhole” saddles with

$$\begin{aligned} G_L=G_R=\Sigma _L=\Sigma _R=0, \end{aligned}$$

(D.104)

which corresponds to ${\mathfrak {z}}_2^p$. There are also other saddles corresponding to other ${\mathfrak {z}}_2^k$. For the simplest case $q=2$, these solutions can be written explicitly. The “wormhole” saddles are

$$\begin{aligned}{} & {} G_L=G_R=\Sigma _L=\Sigma _R=0, \end{aligned}$$

(D.105)

$$\begin{aligned}{} & {} G_{LR}=\pm \frac{1}{\sqrt{t}},\quad \Sigma _{LR}=\pm \sqrt{t}, \end{aligned}$$

(D.106)

$$\begin{aligned}{} & {} \langle z^2\rangle _{\text {WH}}= e^{-\frac{N}{2}}t^{N/2}, \end{aligned}$$

(D.107)

which do not depend on $\mu$ and the other four solutions are

$$\begin{aligned} \Sigma _L= & {} \Sigma _R=u G_L =u G_R =\pm \sqrt{\frac{u^2-t}{u}}, \end{aligned}$$

(D.108)

$$\begin{aligned} \Sigma _{LR}= & {} \frac{t}{u},\quad G_{LR}=\frac{1}{u} \end{aligned}$$

(D.109)

$$\begin{aligned} \Sigma _L= & {} u G_L=-\Sigma _R=-u G_R=\pm \sqrt{\frac{u^2-t}{u}}, \end{aligned}$$

(D.110)

$$\begin{aligned} \Sigma _{LR}= & {} -\frac{{t}}{u},\quad G_{LR}=-\frac{1}{u}\,, \end{aligned}$$

(D.111)

$$\begin{aligned} \langle z^2\rangle _{\text {new}}= & {} e^{-\frac{N}{2}(2-\frac{t}{u^2})}u^N \end{aligned}$$

(D.112)

Apparently when $u \rightarrow \infty$, $\Sigma _{LR},G_{LR}\rightarrow 0,$ then we expect that in this limit the dominant saddle will correspond to ${\mathfrak {z}}_2^0$ since in this limit saddle point value does not depend on t. Comparing these two saddle values we find

$$\begin{aligned}{} & {} \frac{\langle z^2\rangle _{\text {WH}}}{\langle z^2\rangle _{\text {new}}}=\exp \left( \frac{N}{2}\left( 1-x+\log x\right) \right) \le 1, \end{aligned}$$

(D.113)

$$\begin{aligned}{} & {} x=\frac{t}{u^2} \,. \end{aligned}$$

(D.114)

Note that when $x=1$ such that $\langle z^2\rangle _{\text {WH}}=\langle z^2\rangle _{\text {new}}$ the new saddle just reduces to the wormhole saddle. Therefore it implies that the new saddle always dominates.

This new saddle is named as “unlinked half-wormhole” in [81] to distinguish it from the half-wormhole saddle which was found in [73]. One interpretation of this new saddle is that it is the analogue of the disconnected saddle in this model; indeed, we do not find other disconnected saddle with $G_{LR}=0$, $\Sigma _{LR}=0$ and $G_{L/R}, \Sigma _{L/R}\ne 0$, in addition, this saddle is present only when $u\ne 0$, and this saddle is more and more important as u increases.

The analysis of the half-wormhole saddle for $z^2$ will be similar so we will not repeat here.

1.1.1 D.0.2: 1d model

Now we come back to the 0+1d model that is a variant of the Brownian SYK model. Let us begin by deriving the wormhole saddle of $\langle z^2 \rangle$¹⁰

$$\begin{aligned}{} & {} z_L z_R = \int \text {d}^{2N}\psi \exp \left\{ -\int _{0}^T \text {d}t\frac{1}{2}\sum _i\left( \psi _i^L\partial _t \psi _i^L+\psi _i^R\partial _t \psi _i^R\right) \right. \nonumber \\{} & {} \qquad \quad \left. +\text {i}^{q/2}\int _0^T\text {d}t\sum _A {\tilde{J}}_A \left( \psi _A^L+\psi _A^R\right) \right\} \end{aligned}$$

(D.115)

$$\begin{aligned}{} & {} \langle z_L z_R\rangle \nonumber \\{} & {} =\int \text {d}^{2N} \psi \exp \left\{ -\int _{0}^T \text {d}t\frac{1}{2}\sum _i(\psi _i^L\partial _t \psi _i^L+\psi _i^R\partial _t \psi _i^R) \right. \nonumber \\{} & {} \quad +\left. \int _{0}^T \text {d}t \left( {\frac{\tau ^2}{q!}} \left( \sum _i \psi _i^L\psi _i^R\right) ^q+\mu \left( \sum _i\theta _i \psi _i^L\right) ^q \right. \right. \nonumber \\{} & {} \left. \left. \quad +\mu \left( \sum _i\theta _i \psi _i^R\right) ^q\right) +\tau ^2E_0T\right\} \, . \end{aligned}$$

(D.116)

$$\begin{aligned}{} & {} =\int \text {d}^{2N} \psi \left[ \prod _{a} \text {d}G_{a} \frac{d \Sigma _a}{2\pi \text {i}}\right] \exp \left\{ -\int _{0}^T \text {d}t\frac{1}{2}\sum _i(\psi _i^L\partial _t \psi _i^L \right. \nonumber \\{} & {} \left. \quad +\psi _i^R\partial _t \psi _i^R)+\tau ^2TE_0 +\int _{0}^T \text {d}t \left( \frac{\tau ^2}{q!}G_{LR}^q+\mu G_L^q \right. \right. \nonumber \\{} & {} \left. \left. \quad +\mu G_R^q -\sum _a\Sigma _a G_a+\sum _i\bigg [\Sigma _{LR}\psi _i^L\psi _i^R+\Sigma _L \theta _i\psi _i^L\right. \right. \nonumber \\{} & {} \left. \left. \quad +\Sigma _R \theta _i\psi _i^R\bigg ]\right) \right\} , \end{aligned}$$

(D.117)

where $E_0={ N\atopwithdelims ()q }$ is the constant term coming from $\psi _A^{L(R)} \psi _A^{L(R)}=(-1)^{\frac{q}{2}}$. As explained in [73], we can focus on the time-independent saddles then the fermions can be simply integrated out. The result is¹¹

$$\begin{aligned}{} & {} \langle z_L z_R\rangle \nonumber \\{} & {} =e^{T \tau ^2 E_0}\int \left[ \prod _{a} \text {d}G_{a} \frac{\text {d}\Sigma _a}{2\pi \text {i}}\right] e^{TN(\frac{t}{q}G_{LR}^q+\frac{u}{q} G_L^q+\frac{u}{q} G_R^q -\sum _a\Sigma _a G_a)}\nonumber \\{} & {} \quad \times \left[ \cosh \left( T\sqrt{\Sigma _L^2+\Sigma _R^2-\Sigma _{LR}^2}\right) \right] ^N. \nonumber \\{} & {} =\int \left[ \prod _{a} \text {d}G_{a} \frac{\text {d}\Sigma _a}{2\pi \text {i}}\right] e^{\tau ^2 TE_0}e^{S_{eff}} \end{aligned}$$

(D.118)

For general T, the saddle equation is very hard to solve due to the complexity of $\cosh$ function. However the equations simplify in the large T limit because of the following approximations

$$\begin{aligned}{} & {} \log \bigg (\cosh \bigg (T\sqrt{\Sigma _L^2+\Sigma _R^2-\Sigma _{LR}^2}\bigg )\bigg ) \nonumber \\ {}{} & {} \approx \pm \text {i}T \sqrt{\Sigma _{LR}^2-\Sigma _{L}^2-\Sigma _{R}^2}\,. \end{aligned}$$

(D.119)

Then in this limit the effective action becomes

$$\begin{aligned} S_{eff}={} & {} TN\left( \frac{t}{q}G_{LR}^q+\frac{u}{q} G_L^q+\frac{u}{q} G_R^q -\sum _a\Sigma _a G_a\right) \nonumber \\{} & {} \pm \text {i}NT\sqrt{\Sigma _{LR}^2-\Sigma _{L}^2-\Sigma _{R}^2}\,, \end{aligned}$$

(D.120)

and corresponding saddle point equations are

$$\begin{aligned}{} & {} tG_{LR}^{q-1}=\Sigma _{LR}, \end{aligned}$$

(D.121)

$$\begin{aligned}{} & {} u G_L^{q-1}=\Sigma _L,\quad u G_R^{q-1}=\Sigma _R, \end{aligned}$$

(D.122)

$$\begin{aligned}{} & {} G_{LR}=\pm \frac{\text {i}\Sigma _{LR}}{\sqrt{\Sigma _{LR}^2-\Sigma _{L}^2-\Sigma _{R}^2}}\,, \end{aligned}$$

(D.123)

$$\begin{aligned}{} & {} G_L=\mp \frac{\text {i}\Sigma _{L}}{\sqrt{\Sigma _{LR}^2-\Sigma _{L}^2-\Sigma _{R}^2}}, \end{aligned}$$

(D.124)

$$\begin{aligned}{} & {} G_R=\mp \frac{\text {i}\Sigma _{R}}{\sqrt{\Sigma _{LR}^2-\Sigma _{L}^2-\Sigma _{R}^2}}\,. \end{aligned}$$

(D.125)

So the wormhole saddle still presents [9]

$$\begin{aligned}{} & {} G_L=G_R=\Sigma _L=\Sigma _R=0,\quad G_{LR}=\pm \text {i}, \end{aligned}$$

(D.126)

$$\begin{aligned}{} & {} e^{S_{eff}}\Big |_{\text {WH}}=e^{\text {i}^qTN \frac{t}{q}}. \end{aligned}$$

(D.127)

The unlinked half-wormhole saddle is:

$$\begin{aligned}{} & {} G_{LR}=\Sigma _{LR}=0,\quad G_L=\sin \alpha ,\quad G_R= \cos \alpha , \end{aligned}$$

(D.128)

$$\begin{aligned}{} & {} e^{S_{eff}}\Big |_{\text {unlink}}=e^{TN \frac{u}{q}(\cos ^q\alpha +\sin ^q\alpha )}\nonumber \\{} & {} \le e^{S_{eff}}\Big |_{\text {unlink},\alpha =0,\pi /2}=e^{TN \frac{u}{q}}\, \end{aligned}$$

(D.129)

where the relation

$$\begin{aligned} G_L^2+G_R^2-G_{LR}^2=1, \end{aligned}$$

(D.130)

is fulfilled and $\alpha$ satisfies

$$\begin{aligned} \cos \alpha =\pm \frac{\cos ^{q-1}\alpha }{\sqrt{\cos ^{2q-2}\alpha +\sin ^{2q-2}\alpha }}. \end{aligned}$$

(D.131)

In the late time $(T\rightarrow \infty )$, there is indeed a wormhole saddle so it possible to include a linked half-wormhole saddle for z. We also assume that the half-wormhole saddle is time independent since the wormhole saddle is time independent. Then the analysis is completely same as the one for the 0-dimensional model. So the half-wormhole saddle will be given by

$$\begin{aligned}{} & {} \Theta =\left[ \prod _i \theta _i\right] \int \text {d}^N \psi \exp \bigg (T\text {i}^{q/2}\sum _A J_A\psi _A\bigg )\,, \end{aligned}$$

(D.132)

$$\begin{aligned}{} & {} \langle \Theta ^2\rangle \approx \langle \Theta z\rangle \approx \langle z_L z_R\rangle |_{\text {Wormhole saddle}}\,. \end{aligned}$$

(D.133)