Factorizing Wormholes in a Partially Disorder-Averaged SYK Model

In this paper, we introduce a"partially disorder-averaged"SYK model. This model has a real parameter that smoothly interpolates between the ordinary totally disorder-averaged SYK model and the totally fixed-coupling model. For the large $N$ effective description, in addition to the usual bi-local collective fields, we also introduce a new additional set of local collective fields. These local fields can be understood as"half"of the bi-local collective fields, and in the totally fixed-coupling limit, they represent the"half-wormholes"which were found in recent studies. We find that the large $N$ saddles of these local fields vanish in the total-disorder-averaged limit, while they develop nontrivial profiles as we gradually fix the coupling constants. We argue that the bulk picture of these local collective fields represents a correlation between a spacetime brane and the asymptotic AdS boundary. This illuminates how the half-wormhole saddles emerge in the SYK model with fixed couplings.

dynamics of the model in the low-temperature limit. The dual of the Schwarzian mode in the gravity theory is well understood as the boundary graviton of the Jackiw-Teitelboim (JT) gravity [12,13] in the context of nearly AdS 2 /CFT 1 correspondence [14,15,16,17].
The relationship between a boundary ensemble average and a bulk gravity theory becomes more evident in the recent calculation of the JT gravity partition function at a non-perturbative level, defined by summing over all higher genus topologies. For instance, it was shown that the non-perturbative JT gravity partition coincides with a random Hermitian matrix integral [18]. In this correspondence, the central role is played by Euclidean wormholes in the gravitational path integrals. For instance, in the presence of multiple conformal boundaries, the natural rule to define semi-classical gravitational path integral is to include all Euclidean manifolds consistent with the boundary condition. This includes Euclidean wormhole geometries where several conformal boundaries get connected, resulting in the random matrix integral, as mentioned above. It was also shown that even in a very simple topological model of gravity, once we include all possible topologically distinct manifolds respecting the boundary condition, its boundary naturally realizes an ensemble average [19]. Its generalization to supersymmetric cases can be found in [20].
One manifestation of the emergence of ensemble averages in the presence of wormholes is the non-factorization of partition functions. Let us consider the simplest example, where we have two disjoint conformal boundaries. Then the gravitational path integral receives contributions from wormholes connecting the two boundaries. This is puzzling from the boundary field theory point of view, because on these two disjoint boundaries, its partition function gets factorized [21]. The resolution is to regard its dual as an ensemble of theories living on the boundaries. The relation between ensemble averages and the physics of wormholes was extensively discussed a few decades ago [22,23], and recent discussions mainly focus on its implications for holographic duality [24,25,26,27,28,29,30,31,32,33,34,35] and the black hole information paradox [36,37,38,39,40,41,42].
Given these developments, a natural question arises: How do we get a bulk gravity description of a fixed boundary field theory without averaging ? We are particularly interested in the fate of Euclidean wormholes in such a setup. There have been several attempts toward this goal. For example, it was shown in [43] that by introducing "eigenbranes" in the bulk of JT gravity, one can fix several eigenvalues of the dual random matrix. This question was studied in [33] for the SYK model with one time point (i.e., zero-dimensional SYK model) with a fixed coupling constant. There, in the calculation of a product of partition functions Z L Z R without averaging, two types of large N saddles are found. One is the usual wormhole saddles which connect two boundaries L and R, and give the averaged two-point function Z L Z R over the random coupling. In addition to these, there is another type of saddles called "half-wormholes". From a bulk gravity point of view, these saddles can be interpreted as geometries starting from a conformal boundary and ending somewhere in the bulk. Quite remarkably, the contributions of these half-wormholes are essential for Z L Z R to get factorized. This idea is further developed in [44,45,46,47,48,49,50,51] for more details.
To further investigate these ideas, wormholes vs. half-wormholes and ensemble averaging vs. fixed-coupling (or theory), in this paper, we introduce a partially disorder-averaged SYK model. This model has a real parameter σ that smoothly interpolates between the ordinary totally disorder-averaged SYK model (σ = ∞) and the totally fixed-coupling model (σ = 0). This is archived by modifying the probability distribution of the coupling constant. In the σ = ∞ limit, the probability distribution is reduced to the ordinary Gaussian form. In contrast, in the σ = 0 limit, the probability distribution becomes a product of delta functions that enforces each component of the coupling constant to a fixed external value. This type of interpolation between the totally disorder-averaged and fixed-coupling cases is previously studied in the context of matrix integrals in [45].
One convenient way to study the conventional SYK model in the large N limit is introducing bi-local collective fields {G(τ 1 , τ 2 ), Σ(τ 1 , τ 2 )} depending on two boundary times. This formalism has a further advantage for studies of two-point (or higher-point) functions of the partition functions. When each time corresponds to each boundary theory, the bi-local fields have a natural bulk interpretation for the corresponding gravitational configurations, including Euclidean wormholes in a suitable setup. Below we will argue that in the large N analysis of our deformed SYK model, we need to introduce additional local collective fields {G σ (τ ), Σ σ (τ )}. These local fields are analogs of "half-wormholes" because they depend only on one boundary time. We will study in detail the behavior of these local fields {G σ (τ ), Σ σ (τ )} as we change the deformation parameter σ, from σ = ∞ whose low energy part has a complete gravitational description in terms of JT gravity, to the totally fixed coupling model σ = 0 whose gravitational description we would like to know better.
The remainder of the paper is organized as follows. In section 2, we define a partially disorder-averaged SYK model by modifying the probability distribution of the coupling constants. We show that this correctly interpolates between the totally disorder-averaged and totally fixed coupling case.
In section 3, we study one-point function of the partition function after this partial disorder averaging. For the large N effective description, in addition to the usual bilocal collective fields, we also introduce a new additional set of local collective fields. We explain that these local fields can be understood as the "half" of the bi-local collective fields. Before studying the one-dimensional model, in section 3.1, we study the zerodimensional model (i.e., SYK model with one time point). We show that the partially disorder-averaged partition function is proportional to the "hyperpfaffian" of the external coupling together with a coefficient that vanishes in the total disorder-averaged limit while giving a finite value in the total fixed coupling limit. Then, we go back to the one-dimensional model. After reviewing the totally disorder-averaged case (which is wellstudied in the literature) in section 3.2, we study the slightly fixed coupling case in section 3.3. We show that for the large N collective field description, the effect of slightly fixed coupling is given by the same form as the chemical potential contribution in the complex SYK model. In section 3.4, we explain a possible bulk picture of the new local collective fields and in section 3.5 we comments on the general σ (i.e. general partial disorder-averaged) case.
In section 4, we study two-point function of the partition functions after the partial disorder averaging. In section 4.1, focusing on the zero-dimensional model, we study the wormhole and half-wormhole saddles in detail. We also comment on the case when we add explicit interactions between the left and right SYK's in section 4.2. Our discussion on the Brownian SYK model is given in section 4.3.
We study the transition of the spectral density as well as the spectral form factor from the totally fixed coupling limit to the totally disorder-averaged limit in section 5. Our conclusions and further discussions are presented in section 6. Supplemental computations and discussion are gathered in appendices.

Partial Disorder Average
In this section, we define our partially disorder-averaged SYK model, which smoothly interpolates between the ordinary totally disorder-averaged SYK model and the totally fixed coupling model by introducing a real parameter σ. The original SYK model [1,2] is defined by the Hamiltonian where the one-dimensional Majorana fermions satisfy The coupling J i 1 ···iq is random, and drawn from Gaussian ensemble with the vanishing mean value J i 1 ···iq = 0. Its Lagrangian is given by Now we introduce a partial disorder averaging by deforming the probability distribution of the random coupling as Here J (0) i 1 ···iq is the external fixed coupling, and the parameter σ controls how much we take disorder averaging. This type of partial disorder averaging was studied in [45] for a matrix model. As in [45], σ = ∞ corresponds to the total disorder-averaging coincides with the ordinary SYK model, while σ → 0 corresponds to totally fixing the coupling constant J i 1 ···iq = J (0) i 1 ···iq . In the following, it is convenient to introduce a new set of effective couplings by The normalization factor N σ is defined by 9) and this gives us (2.10) For the usual total-disorder-averaged SYK model, this factor does not play any crucial role, but for our partially disorder-averaged SYK model, we have to keep at least nontrivial σ-dependence coming from this normalization factor. For example, in the totallyfixed-coupling limit (σ → 0), combined with this normalization factor, the probability distribution gives delta functions:

Partially Disorder-Averaged Partition Function
Having specified the model of our interest, in this section we study the partially disorderaveraged partition function of this model in the large N limit, by introducing collective fields. We see that to study the partially disordered model with finite σ, in addition to ordinary bi-local fields {G(τ 1 , τ 2 ), Σ(τ 1 , τ 2 )} in the totally averaged theory at σ = ∞, it is convenient to introduce an additional set of collective fields denoted by {G σ (τ ), Σ σ (τ )}. We will explain a possible bulk picture of these local collective fields in section 3.4 and for the two-point function we study in section 4, they eventually correspond to the degrees of freedom of "half-wormhole". The original partition function with fixed coupling J i 1 ···iq is given by Taking the partial disorder averaging of this partition function with respect to the probability distribution (2.4), we obtain The normalization factor N σ is canceled out by the Gaussian integral of the coupling. We emphasize that even though we integrated J i 1 ···iq to obtain Z J , this still can be the totally fixed-coupling partition function in the σ → 0 limit. Indeed, we can see that in the totally fixed-coupling limit (σ → 0), the second term vanishes, while the last term survives because J 2 /σ 2 → 1. The result is nothing but the totally fixed-coupling action (2.3). On the other hand, in the totally disorder-averaged limit (σ → ∞) the last term vanishes and the remaining first two terms are the usual disorder-averaged SYK effective action.
We also note that besides the last term, the original SYK effective action has the global O(N ) gauge symmetry: χ i → j O ij χ j . However, this global O(N ) symmetry is now broken by the last term by fixing the coupling constant J (0) i 1 ···iq . In addition, this term contains a factor i q/2 , so in order to keep the action real, it looks like we need a complex J (0) i 1 ···iq , such that the entire last term is still real. This type of complex coupling is similar to the one discussed in [31,46], but the precise relation is unclear.

Fixed-couplings as an external source
Here we will explain how to introduce collective fields in our partially disorder-averaged SYK model. For the first line in (3.3), we introduce a set of bi-local fields (G LR , Σ LR ) as usual. To introduce a new set of collective fields for the second line in (3.3), we will give the following interpretation of the external fixed couplings.
Let us now assume a specific form of the external coupling J (0) i 1 ···iq . We choose it to be proportional to the totally anti-symmetric tensor ε i 1 ,···iq for i k = 1, · · · , N , with ε 1,2,··· ,q = 1, ε 2,3,··· ,q+1 = 1, e.t.c. From now on, we interpret the second term in (3.3) as the couplings between the fermions χ i in the SYK model and the external fermionic sources θ i by decomposing the anti-symmetric tensor ε i 1 ,···iq into the products of θ i , where the coefficient was chosen for later convenience. 1 This variable θ i can be interpreted as non-dynamical fermions living in another universe. We discuss this interpretation in Appendix A. After using this decomposition, we have the effective action with J σ defined in (2.6). Now the term depending on the fixed couplings is rewritten as a local term similar to the usual bi-local term in the first line, which enables us to introduce the collective fields to this term. Now we are considering the SYK model coupled to the external fields θ i . We will analyze this model instead of the original partially disorderaveraged SYK model. After all computations, we replace all the dependence of θ i by the original fixed-couplings J (0) i 1 ···iq , then we expect to recover the result in the original partially disorder-averaged SYK model. Now it is useful to employ the following Hubbard-Stratonovich trick: We insert the above identity to the partially disorder-averaged partition function (3.2) with S eff [χ] given by (3.5). Rewriting all terms in S eff [χ] (3.5) by the bi-local G and the local G σ fields, we can perform the fermion integrals (see Appendix B for detail) and obtain Before studying this effective action in detail, let us first consider the meaning of the local field G σ we introduced here in addition to the usual SYK bi-local fields. If we consider the combination of G q σ (τ 1 )G q σ (τ 2 ), this is written as where we used the decomposition of J This relation implies that we can think of the local field G σ (τ ) as the "half" of the bi-local field G(τ 1 , τ 2 ). Furthermore, one can regard the anti-symmetric variable θ i as fermionic degrees of freedom living in the surface at which the bulk spacetime ends. Since the new variable G σ (τ ) is the propagator between the original SYK fermions χ i and the new ones θ i , this also suggests that G σ (τ ) corresponds to the correlation between the boundary and the interior of the bulk. These interpretations will be useful in the following discussion. Since all terms in the effective action (3.8) are proportional to N , we can use large N saddle-point evaluation. Variation of each field leads to the following saddle-point equations δG : where the inverse is defined by From (3.11), in order to keep the time translation symmetry, the only possible solution for the local field Σ σ is a time-independent constant solution. Such a constant term in (3.11) has the same structure as the chemical potential contribution in the complex SYK model as we will discuss more in detail in section 3.3.

Zero-dimensional SYK
Before studying this model in detail, let us first consider the zero-dimensional case discussed in [33,44]. This model is not dynamical, but it might capture some important topological contributions. The treatment of this case is slightly different from the onedimensional case we discussed above. This is because in the effective action (3.3), the second term identically vanishes in this case, and of course, the kinetic term also does not exist: (3.16) Therefore, in this case, we cannot introduce the bi-local fields, which are identically zero, so at most, we can introduce the local fields only: Hence, in this case, after integrating out the fermions (see Appendix B for detail) we have where the log(−Σ σ ) term comes from expanding the linear in the fermion term in the Hubbard-Stratonovich trick (3.17). Now we can simply integrate out both fields as where for the last line, we assume N is a multiple of q. For the total-disorder-averaging limit (σ → ∞), we have J σ → 0 (2.8). Therefore, in this limit we have Z J = 0. On the other hand, for finite σ > 0, we have a nontrivial partition function, which is proportional to the "hyperpfaffian" of the external coupling [44]: where p = N/q and A's denote ordered q-subsets A = i 1 < · · · < i q . For the third line, we used (3.4) and assumed N ∈ 8Z to eliminate the factor coming from the i q/2 . Therefore, we find Going back to the expression of (3.20), for the numerical coefficient, besides the J σ dependence, we find in the large N approximation that We can also evaluate this partition function in the large N saddle-point approximation. From the effective action, we have the large N saddle-point equations and there are q solutions: where m = 1, 2, · · · , q. Evaluating the on-shell action for these large N saddle-point solutions, we find We obtained the same J σ -and N -dependence as the above exact computation. The remaining sum can also be evaluated as where we put p = N/q and for the first equality we assumed p = Z, but for the second equality we took p → Z and evaluated by using l'Hôpital's rule. The 1/ √ q difference from the exact result (3.23) is accounted by the one-loop contribution [33]. To evaluate this contribution, we expand the fields around the saddle-point solutions: Then, the quadratic action is written as where in the second line we change the variables byG σ = iJ q δΣ σ to make the Gaussian integral convergent. Therefore, the one-loop contribution is given by Z This precisely agrees with (3.23).

Total disorder-averaging
Now let us come back to the one-dimensional model (3.8). In the totally disorder-averaging limit (σ → ∞), we have J σ → 0. This means that the variation of G σ (3.13) sets Σ σ = 0.
Therefore, the remaining dynamics is described by the bi-local fields {G, Σ}, which is nothing but the effective system of the ordinary SYK model. Combining (3.11) and (3.12), the Schwinger-Dyson equation for G(τ 1 , τ 2 ) is now obtained as The free solution, which is obtained by setting J = 0, is found as On the other hand, the IR solution, which is obtained by eliminating the kinetic term, is given by for zero temperature case where ∆ = 1/q and This solution can be determined by using the scaling ansatz and the Fourier transform .

(3.36)
Now we realize that the IR effective action is invariant under the conformal transformation τ → f (τ ) together with the bi-local field transformation For the IR solution (3.34), a general transformation gives (3.39) In particular, the finite temperature solution can be obtained by this transformation with f (τ ) = (π/β) tan(πτ /β) as (3.40) Therefore, we can only keep J σ corrections and ignore the terms proportional to J 2 . In this limit, we start from the zero-th order (in J σ ), where the free bi-local sector is given by Σ(τ 1 , τ 2 ) = ∂ τ 1 δ(τ 12 ) and Σ σ = 0. To get the first order, inverting the equation (3.14), we find Therefore, equating with (3.13), the differential equation for G σ (τ ) is found as 45) and the solution of this equation is where c is an integration constant. From (3.44), we also find Focusing on the leading contribution (in J σ ) for this Σ σ , the last term in (3.11) gives a constant contribution for Σ(τ 1 , τ 2 ). Fourier transforming the bi-local time to frequency space, such a constant contribution gives a contribution Σ(ω) = µδ(ω) + · · · , where the ellipsis denotes contributions from other terms. Such a contribution localized at ω = 0 is the same structure as a chemical potential of the complex SYK model 2 [52,53,54]. Hence, the saddle-point equation for G(τ 1 , τ 2 ) becomes This is nothing but Green's equation for a free fermion with a chemical potential µ, and the solution is given by where β is the inverse temperature. We suppose the above chemical potential contribution also influences on the IR region as in the complex SYK model. For the IR region, we stay at the strong coupling limit J T . On top of that, we would like to keep the leading term of J σ as a small parameter and neglect the sub-leading contribution of J. All this means that in this subsection we study the regime of Including the potential contribution, the saddle-point equation for G(τ 1 , τ 2 ) can be written as This equation is the saddle-point equation of the complex SYK model with a chemical potential µ. We will show that the same structure also appears in another formalism of the slightly fixed coupling SYK model in Appendix C. It is very interesting that even though we started with Majorana fermions, after slightly fixing the random couplings, we obtain the induced chemical potential. This might be related to the branes discussed in [18,43,48] in order to fix the boundary theory.
In the IR limit, the kinetic term and the chemical potential do not contribute (since they are delta functional local in Σ(τ 1 , τ 2 ) [52,54]) and the saddle-point solution is given by at zero temperature. In this case, the coefficient is defined by , (3.53) and the "spectral asymmetry" E is related to the chemical potential and it cannot be determined in the IR limit, but it requires interpolation to the UV solution [52,53,54].

Bulk interpretation of G σ and Σ σ
In this subsection, we consider a bulk interpretation of the local fields {G σ , Σ σ }. Before considering these fields, let us first remind ourselves about the scalar field coupled to JT gravity in the (near) AdS 2 [14,15,16,17]. The bulk scalar field is described by the action on the Poincare coordinates The boundary action is given by The bulk field is now expressed in terms of the bulk-to-boundary propagator (3.57) where the boundary source is defined by Using this expression and evaluating the boundary action (3.56), one finds If we consider the conformal transformation of the boundary τ → f (τ ), we also have [14,15,16,17] (3.61) Now we will see that the same effective action is obtained for the fluctuation of the Σ σ field in our partially disorder-averaged SYK model. 4 To see this, we consider 1/N , fluctuations of the fields around the totally disorder-averaged IR saddle-point solutions: where G IR σ = Σ IR σ = 0. Substituting these expansions into the action (3.8), this generates several interactions among the fluctuations and the background. The term we are interested in right now is the interaction between the background of the bi-local fields and the fluctuations of the local fields. This term is given by Using the explicit form of the background solution (3.34), we find The leading interaction term between the Schwarzian mode and the local field δΣ σ is given by simply transforming G IR → G IR f (3.39) as This form of the boundary action precisely agrees with (3.61). This implies that the bulk interpretation of the local field δΣ σ is a boundary source of extra bulk fields. Then, it is natural to interpret the dual field δG σ as the expectation value of the corresponding boundary operator.

Comments on general σ case
In this subsection, we make some comments on the general σ case. As we mentioned below (3.15), the time translation symmetry of Σ(τ 1 , τ 2 ) requires from (3.11) that the only possible solution for the local field Σ σ is a time-independent constant solution. Then, as we explained in section 3.3, such a constant term in (3.11) has the same structure as the chemical potential contribution in the complex SYK model, and the solution of G(τ 1 , τ 2 ) is given by (3.49) for free case or by (3.52) for the IR case. Given these facts, we can evaluate the integral in (3.14) for example for G free as . (3.66) This seems to indicate that we can still have a nontrivial time-dependent solution for G σ (τ ) even though we should have a time-independent constant solution for Σ σ . At the same time, we also have to make the solutions consistent with the other equation (3.13). This seems quite nontrivial for the general σ case. It would be very interesting to study this general σ case more in detail, but we leave this to future work.
4 Two-point Function Z L Z R J In this section, we study the partially disorder-averaged two-point partition functions in the large N limit. Before the partial disorder averaging, we have the two-point function of the partition functions Using the duplicated Hubbard-Stratonovich transformation we now find with Similar to (3.11) -(3.14), the large N saddle-point equations are The large N saddle-point equations are a little complicated to solve, so let us first consider the zero-dimensional case again. In this case, we only have {G LR , Σ LR } for the bi-local fields. For the following discussion, it is more convenient to combine the two couplings into one effective coupling that appears in the interaction term between the local and bi-local sectors. This can be implemented by rescaling the Hubbard-Stratonovich fields as After all of these rescalings, the effective action of this case reads where we introduced the coupling between the local and bi-local sectors by This coupling behaves λ σ ∼ 1 2 (J 0 /σ) 2/q as σ → 0 i.e., in the totally fixed-coupling limit and λ σ ∼ 1 2 (JJ 0 /σ 2 ) 2/q as σ → ∞ i.e., in the total disorder-averaged limit. Here from the action, we excluded the constant contribution − 2N q log J coming from the rescaling of Σ LR , so that the two-point function of the partition functions is now written as Now the saddle-point equations are written as Let us first solve for {G σ , Σ σ } in terms of G LR . From (4.10), we can find one trivial solution and q(q − 2) nontrivial solutions 5 with n = 1, 2, · · · , q(q − 2). The corresponding solutions of Σ a σ are given by the first equation of (4.10). The nontrivial solutions are valid for q > 2. In the rest of this subsection, we focus on q > 2 case, and we will present a discussion for q = 2 in Appendix D. We will call the first trivial solution {G L σ , G R σ } = {0, 0} "wormhole" saddle-point solution, while the other q(q − 2) nontrivial solutions are analogous to the "half-wormhole" saddle-point solutions. 6 The reasoning for these names will be explained below.
For the half-wormhole saddles, we look for solutions that behave differently from the above wormhole solutions in the totally fixed-coupling limit (σ → 0). Hence, in the σ → 0 limit for the first equation, we look for solutions satisfy Σ LR ≈ λ σ Σ L σ Σ R σ . This is solved as , (4.13) Figure 2: The half-wormhole saddles in the totally fixed-coupling limit for q = 12. and (4.14) Notice that we have 2q independent solutions similarly to the wormhole solutions. Since in the totally fixed-coupling limit (σ → 0), we have λ −1 σ ∝ σ −2/q → 0, we have G LR → 0 in this limit (σ → 0). In contrast to the wormhole solutions, this means that we have no correlation between the two SYK systems (L and R) and they are analogous to the half-wormhole saddle-point solutions found in [33] and discussed further in [44]. These half-wormhole solutions are depicted in Figure 2.
On the other hand, in the disorder-averaged limit (σ → ∞), the coupling λ σ becomes small, and G LR becomes large. Therefore, the approximation Σ LR ≈ λ σ Σ L σ Σ R σ we used is no longer valid in this limit, and we should treat as the extra term λ σ Σ L σ Σ R σ as a perturbation around the wormhole solutions.

On-shell action
In this subsection, we study the σ-dependence of the on-shell action of the saddle-point solutions found above.
Let us first consider the on-shell action by taking a derivative with respect to explicit λ σ dependence as in [4] For the wormhole solutions, it is obvious that Let us also study the value of the on-shell action directly (without taking derivative with respect to λ σ ). For the wormhole solutions, this is given by For the half-wormhole solutions at small σ, we have G LR ≈ 0. Therefore, it is given by We can see that the λ σ dependence agrees with that obtained by integrating (4.17).
For σ → ∞, we have only wormhole contributions. Therefore, combining with the prefactor in (4.8), the two-point function is given by where we used J → J in the σ → ∞ limit, and the summation (3.28). On the other hand, for σ → 0, we have J → 0. Therefore, the wormhole contributions, which do not have σ dependence, do not contribute. For the half-wormhole contributions, the σ dependence coming from (4.19) precisely cancels with the σ dependence coming from the prefactor in (4.8) and we find We can see that the J and J 0 dependence of the two-point function precisely agree with the results of [44] in both limits. Also the above σ → 0 result precisely agrees with the square of (3.23) with θ 2 i = 1/2 in the σ → 0 limit. Furthermore, at σ = 0, we can compute the two-point function directly (which is presented in Appendix E) and the result agrees with the above computation. The large N saddles contributing to this totally fixed-coupling two-point function look different from the conclusion of [33,44], but this is simply due to the difference of the "linked" half-wormhole and the "unlinked" half-wormhole, as we will explain further in section 6.

Adding a coupling
Let us make some comments what happens if we introduce a coupling between L and R systems in zero-dimensional SYK model (4.22) according to [26,33]. The saddle-point equations become If we consider the strong coupling limit near the totally fixed coupling region such that 1 λ σ µ, the interaction term can be neglected. In this case, the half-wormhole saddle and the wormhole saddle degenerates, and we have a solution with G L,R σ ≈ 0, Σ L,R σ ≈ 0. Therefore, even near the fixed coupling region, the wormhole contribution becomes dominant. This is consistent with the conclusions [33,44].

Brownian SYK
Next, we study the Brownian SYK case discussed in [24]. From the effective action (4.4), the Brownian SYK corresponds to reducing the bi-local fields to local fields. Furthermore, in this reduction, the diagonal fields in the a index reduce to constants due to the anticommutation relation of the fermion (2.2), so we only need to keep {G LR , Σ LR }. Therefore, the effective action is now given by In general, it is still hard to search for time-dependent saddle-point solutions. Hence, let us now consider static (time-independent) saddle-point solutions. As explained in [24], in this case, we have where for the rightmost equation, we take the low-temperature limit (i.e., large β) and the plus/minus sign depending on the real part of Σ LR . Then, the large N saddle-point equations are We can easily see that the equations for {G a σ , Σ a σ } (4.29) are structurally identical to the 0d SYK case (4.10). Therefore, the wormhole and half-wormhole solutions are given by where we also used G LR = ±1/2. For the wormhole solution {G L σ , G R σ } = {0, 0}, the remaining equations (4.28) are solved as These wormhole solutions agree with the ones found in [24]. 7 We emphasize that the wormhole solutions are obtainable without continuing to Lorentzian time, but simply taking the low-temperature limit in Euclidean time. For the half-wormhole solutions, the Σ LR solution is obtained as The on-shell actions for these solutions are computed as for the wormhole solutions and (4.34) for the half-wormhole solutions. For the total disorder-average limit (σ → ∞), in order to keep the large β approximation (4.27) valid, we need βJ 2 large. Therefore,both the wormhole and the half-wormhole solutions contribute equally to the disorder-averaged two-point function in this limit. This behavior is different from the one found in the zero-dimensional case in section 4.1. Again, we expect that this difference is associated with the difference of the "linked" and "unlinked" half-wormhole, as we will explain further in section 6.

Spectrum
In this section, we study the spectrum of the partially disorder-averaged SYK model. To see the spectrum, it is more convenient to incorporate the σ-dependence into the Hamiltonian itself, such that the probability distribution of the newly defined random coupling has the original distribution form. The σ-dependence in the probability distribution can be eliminated as by introducing a new coupling constant The Hamiltonian of the SYK model can be written in terms ofĴ i 1 ···iq as 8 In the σ → 0 limit, the Hamiltonian becomes independent ofĴ i 1 ···iq , thus the average over the random coupling is performed trivially as The partition function Z(it) J of the partially disorder-averaged SYK model in the σ → 0 limit simply becomes that in the SYK model with a fixed coupling J (0) where E J (0) are the eigenvalues of the Hamiltonian H J (0) . Then the density of states, Figure 4: A log-log plot of the spectral form factor g(t) with inverse temperature β = 0.0001, and σ = 0.03(blue), 0.3(orange), 3(green) in the partially disorder-averaged SYK model. We take N = 8, q = 4 and J = 1, and take 50000 samples for averaging. We have chosen the same value of the external coupling as Figure 3.
which is given by the Fourier transform of Z(it) J , becomes a sum of the delta functions In Figure 3, we numerically computed the density of states for various σ. In the σ → 0 limit, the spectrum has sharp L = 2 N/2 delta-function-like peaks as expected.
We also computed the σ-dependence of the spectrum form factor which probes the discreteness of the spectrum as depicted in Figure 4. We can observe erratic oscillations at late times for small values of σ, which is characteristic of the SYK model with a fixed coupling [5].

Conclusions and Discussions
In this paper, we studied a partially disorder-averaged SYK model. We introduced a partially disorder-averaged SYK model by modifying the probability distribution of the coupling constants with a real parameter σ. The probability distribution becomes the usual Gaussian form in the σ → ∞ limit while in the σ → 0 limit, it becomes a product of delta functions which enforces each component of the coupling constant to a fixed external value. Given this partial disorder averaging, we studied the one-and two-point functions of the partition functions as well as spectral density and the spectral form factor. For the large N effective description, in addition to the usual bi-local collective fields, we introduced a new additional set of local collective fields. We explained that these local fields can be understood as the "half" of the bi-local collective fields.
Bulk dual to the half-wormhole saddles For the study of the two-point function of the zero-dimensional model partition function, we found that it contains the wormhole and half-wormholes as large N saddles, and their configurations change as we gradually vary σ. A natural question is what is the gravity dual to the half-wormhole for each value of σ. As we saw in section 4.1, in the totally disorder-averaged limit σ = ∞, the half-wormhole and the wormhole saddles degenerate, so in this sense, the dual geometry is just a smooth wormhole geometry drawn as the left picture in Figure 5. Such geometry can be found in the one-dimensional model [24] as well as in the dual JT gravity set-up for example in [18].
Let us consider the half-wormhole saddles in the slightly fixed coupling region. As we explained section 4.1, in this region, the half-wormhole saddle can be obtained by considering a perturbation around the wormhole saddles by adding the small effect of the half-bi-local fields. The interpretation of this perturbation in the gravity side is considered in section 3.4. We found that adding the small effect of the half-bi-local fields G σ corresponds to having a nontrivial classical configuration of the bulk field dual to G σ on the original wormhole background. As we also pointed out in Appendix A, the G σ can be interpreted as the correlation between the fermions χ i on the boundary and other fermioninc degrees of freedom corresponding to θ i , which represents the fixed couplings, living on "another universe". We interpret the gravity dual of "another universe" as a bulk brane, whose structure strongly depends on the microscopic details of the bulk gravity theory. Such a "UV-sensitive brane" might appear in the low-energy effective gravity from the condensation of some stringy modes in string theory as pointed out in [51]. We leave it as an interesting research direction for the future.
Similar proposals are also mentioned in [45,46]. The intuitive picture can be drawn as the middle picture of Figure 5. In this picture, the spacetime ends at the bulk brane placed in the middle of the background wormhole geometry. G σ represents the correlation between the spacetime boundary and this brane. Since we have a nontrivial profile for the bi-local field G LR in this parameter region, the left and right boundaries are still connected by the wormhole. We imagine that as we gradually lower the value of σ, the "gash" made by the brane is getting bigger, and in the totally fixed-coupling limit σ = 0, the wormhole will be eventually torn into two pieces (right picture in Figure 5). This is consistent with the fact that the half-wormhole saddle in this limit represents no correlation between the left and right boundaries, i.e., G LR = 0 while there is a nontrivial profile for the half-bilocal fields G σ and Σ σ . This is also consistent with the bulk picture of the half-wormhole in [33]. The middle corresponds to the slightly fixed-coupling parameter region, where the half-wormhole saddles can be considered as a small perturbation of the wormhole saddle. The right picture represents the half-wormhole in the totally fixed-coupling limit, where the correlation between the left and the right boundaries totally vanishes.

Linked and unlinked half-wormholes
We analyzed the saddle point solutions for Z L Z R J for large and small values of σ. We found that there exist both wormhole and half-wormhole saddles except for the totally disorder-averaged limit σ = 0, where these two types of the saddles degenerate. In the totally disorder-averaged limit, only the half-wormhole saddles contribute to Z L Z R i.e., the wormhole saddles and the half-wormhole saddles contribute in the same order to Z L Z R . Here "linked half-wormhole" means the half-wormhole saddle in the collective field description where both wormhole and half-wormhole saddle exist. On the other hand, in [33] they introduced another collective field description without wormholes. In this description, the half-wormholes are called "(unlinked) half-wormholes" to distinguish them from the linked half-wormholes. The half-wormhole saddles expressed as the nontrivial profile of the half-bi-local fields, i.e., G σ , Σ σ = 0 discussed in this paper, would correspond to their unlinked half-wormhole saddles. In this paper, when writing the effective action for the collective fields (4.6) in 0d SYK, we rescaled them to have nontrivial profiles of the wormhole solutions even in the strictly totally fixed coupling limit. It fits more into the situation of [33] given the collective fields without rescaling (4.4). As you can see, since J σ → J 0 andJ → 0 in the totally fixed-coupling limit, we can simply integrate over the bi-local field G LR , forcing us to Σ LR (τ 1 , τ 2 ) = 1 2 Σ L σ (τ 1 )Σ R σ (τ 2 ). As a result, the effective action can be written solely with half-bi-local fields (as well as bi-local fields of the type G LL , G RR , Σ LL , Σ RR ). In this collective field description, it is obvious that there are only half-wormholes described by the half-bi-local fields and no wormholes represented by the solutions G LR = 0, Σ LR = 0 with G L,R σ = Σ L,R σ = 0. This situation corresponds to the collective field description with no wormhole saddles discussed in [33].

Future directions
In this paper, we studied a partially disorder-averaged Majorana SYK model. Many generalizations of the SYK model were studied by various authors. Some examples include the complex SYK model [52,53,54], supersymmetric SYK model [56,57,58], SYK model with global symmetries [59,60], and higher dimensional generalization of the SYK model [61,62,63,64]. It would be interesting to study a partially disorder-averaged version of these generalizations of the SYK model.
The initial motivation to introduce disorder in the original Sachdev-Ye model [1,65,66] was to simulate zero-temperature quantum phase transition between the quantum disordered spin-liquid phase and the magnetically ordered spin-glass phase. It would be interesting to study whether our partially disorder-averaged model brings any further insights into this aspect.
The original motivation to consider the half-wormholes was to resolve the tension between the factorization of the partition functions and the existence of the wormholes in the bulk in the AdS/CFT context. We found that the wormhole solutions persist even in the fixed coupling limit, but they have zero measure in the path integral. Instead, the half-wormhole solutions emerge as the "dissolved"-wormholes, and they dominate the partition function. It would be interesting to study whether a similar thing could happen in the gravitational path integral. function, we find Now we freeze the dynamics of χ R i and denote the non-dynamical fermions as θ i ∝ χ R i . Therefore, now we can interpret the above two-point function as a one-point function of Z L with external fermionic sources θ i : where we denoted χ i ≡ χ L i and also adjusted the proportionality constant of θ i ∝ χ R i to give the precise second term. This effective action precisely agrees with (3.5), and in this sense, we can interpret the external variable θ i as non-dynamical fermions living in another universe.

B Integration over the fermions
In this appendix, we present some details for the integration over fermions. This is different from the usual Gaussian integrals of Grassmann variables due to the external variables θ i (3.4) and its anti-commutation relation To implement this anti-commutation relation in the path-integral, we can regard this θ i as N -dimensional Dirac matrices constructed from the Pauli matrix (for example see [67]). With this representation of θ i , we first consider the fermion integration in the one-point function (of the one-dimensional model) studied in section 3: For the kinetic term, using the delta function δ(G − N −1 i χ i χ i ), we rewrite it in terms of the bi-local field G(τ 1 , τ 2 ), so that it does not appear in the fermion integral. As usual Majorana fermion integrals, we introduce complex N/2 fermions c i and also complex N/2 external variables ϕ i by Then, the above integral is given by Exponentiating this into the effective action and sifting Σ( For the zero-dimensional case in section 3.1, we do not have the bi-local fields. Therefore, the fermion integral is simply Finally, let us consider the fermion integration for the two-point function in the zerodimensional SYK model studied in section 4.1. Since we have where we used θ 2 = 1/2, the integrating out To be precise, we should regard G a σ as a matrix as However, as long as we keep the large N limit, we can treat G a σ as a usual c-number.

C Another formalism for slightly fixed-coupling
In this appendix, we will show that the same structure we saw in section 3.3 and 3.4 also appears in another formulation of the slightly fixed-coupling SYK model. For simplicity, in this appendix, we focus on the q = 4 case. For this purpose, instead of (3.7), we introduce another Hubbard-Stratonovich trick Using this trick for the partially disorder-averaged partition function and performing the Gaussian integral for χ i , we obtain where the trace is now taken both for the indices i, j and the bi-local time (τ 1 , τ 2 ) with If this ∆ ij is completely diagonal for the indices i, j, we recover a factor of N in front of the trace term, but for a general ∆ ij , we miss this factor of N . The extra local fields {G σ , Σ σ } represents a non-singlet sector of the global O(N ) symmetry. In general, it is hard to understand the non-singlet sector of a large N theory [68,69]. In the context of AdS 2 /CFT 1 , these fields are probably related to the open string degrees of freedom attached to the boundary (see section 6.2 of [4]). Even though we miss the factor of N in front of the trace term and the second line of the above action, let us here pretend that we can use the large N saddle-point evaluation. Variation of each field leads to saddle-point equations δG : Σ(τ 1 , τ 2 ) = J 2 G(τ 1 , τ 2 ) 3 , (C.5) where we defined two kinds of inverse. The first one is inverse for the bi-local time: ∆ ik (τ 1 , τ 2 ) ∆ −1 kj (τ 1 , τ 2 ) = δ ij , (for fixed τ 1 , τ 2 ) . (C.10) We first note that the saddle-point solution of the diagonal part of G σ must be given by where for the second equality, we used the equal-time anti-commutation relation of the fermion (2.2). Let us now consider the following external coupling J ijkl is the totally anti-symmetric tensor without any diagonal piece. Namely, if any of the indices coincide, A (0) ijkl = 0. Then, from (C.7) the index structure of Σ σ ij is decomposed into symmetric and antisymmetric parts as Σ σ ij (τ ) = δ ij Σ σ dig (τ ) + Σ ant ij (τ ) , (C. 13) and the equation (C.7) is reduce to Using this expression of Σ σ ij into (C.4), we can see that the diagonal components of ∆ ij are independent of the index i Therefore, the summation in (C.6) is trivially taken and gives a factor of N , which cancels with the 1/N coefficient. Hence, we can simply invert ∆ −1 (τ 1 , τ 2 ) and this equation is now written as δ(τ 1 − τ 2 ) = dτ 3 ∆ Dig (τ 1 , τ 3 )G(τ 3 , τ 2 ) . (C.17) The modification from the ordinary SYK is the J σ term in (C.16), which acts as a mass or chemical potential term.
D Two-point function of q = 2, 0d SYK In this appendix, we study the q = 2 case of the 0d SYK model discussed in section 4.1.
For q = 2, we have the saddle-point equations From (D.2), we find equations for G L σ as Therefore, the trivial solution is {G L σ , G R σ } = {0, 0}, which can be understood as the "wormhole" saddle-point solution again. The nontrivial solution can be found only if In this case, {G L σ , G R σ } are not yet fixed by the equations (D.2). Since in the totally fixed-coupling limit (σ → 0), we have λ −1 σ ∝ σ 2/q , this solution behaves as G LR → 0 in this limit. Therefore, this nontrivial solution is understood as the "half-wormhole" saddle-point solution.
For the wormhole case, the solution of the bi-local sector is again given by (4.12) as {G LR , Σ LR } = {±1, ±1}. For the half-wormhole case, solving (D.1), we find The on-shell actions are obtained for the half-wormhole solutions with n = 1, 2.
E Direct computation of Z L Z R J at σ = 0 In this appendix, we directly solve the two-point function (4.8) at σ = 0. Because the prefactor J 2N/q → σ 2N/q in the σ → 0 limit, while the coupling in the effective action (4.6) is λ σ → σ −2/q in this limit, so that the action gives a divergence. To get a finite non-zero result, we have to expand the interaction term of (4.6) perturbatively and consider the N -th order term (which gives λ N σ ). Therefore, the two-point function at σ = 0 is given by Now we can see that each set of {G, Σ} decoupled from each other. Therefore, we can perform the integrals separately. For the bi-local sector, we have while for the local sector, we have Therefore, combining these integral results, we find that the two-point function at σ = 0 is given by where for the last line, we used the large N approximation besides the factor 2 −N J 2N/q 0 . This result agrees with the saddle-point evaluation (4.21).