Complex Sachdev-Ye-Kitaev model in the double scaling limit

We solve for the exact energy spectrum, 2-point and 4-point functions of the complex SYK model, in the double scaling limit at all energy scales. This model has a $U(1)$ global symmetry. The analysis shows how to incorporate a chemical potential in the chord diagram picture, and we present results for the various observables also at a given fixed charge sector. In addition to matching to the spectral asymmetry, we consider an analogous asymmetry measure of the 2-point function obeying a non-trivial dependence on the operator's dimension. We also provide the chord diagram structure for an SYK-like model that has a $U(M)$ global symmetry at any disorder realization. We then show how to exactly compute the effect of inserting very heavy operators, with formally infinite conformal dimension. The latter separate the gravitational spacetime into several parts connected by an interface, whose properties are exactly computable at all scales. In particular, light enough states can still go between the spaces. This behavior has a simple description in the chord diagram picture.


Introduction
The Sachdev-Ye-Kitaev (SYK) model [1][2][3][4][5] is a quantum mechanical model (0 + 1 dimensions) of N fermions with random all-to-all interactions, which can nevertheless be studied analytically in the large N limit. The SYK model exhibits various interesting properties. In particular, it is a highly chaotic model, with a maximal quantum Lyapunov exponent at low temperatures [6]. Letting χ i be N Majorana fermions, i = 1, · · · , N , such that {χ i , χ j } = 2δ ij , the Hamiltonian of the SYK model is given by (1.1) p is a parameter in the model which sets the length of the all-to-all interactions in the Hamiltonian, and the J i 1 ···ip are the random couplings. The model can be analyzed using Feynman diagrams. At large N , the dominant diagrams are the so-called melonic diagrams. The sum of all those diagrams is described by a set of Schwinger-Dyson (SD) equations [1][2][3][4][5], and at low energies the model has a conformal regime in which the Schwinger-Dyson equations can be solved. At low energies, the equations are invariant under a symmetry of time reparametrizations, which is broken spontaneously (and also explicitly in the full theory). Beyond the strictly conformal regime, the modes which describe this reparametrization symmetry are lifted by the Schwarzian action (see also [7][8][9][10][11][12][13][14]). Various correlation functions in the Schwarzian theory have been found in [7][8][9]11].
From the holographic point of view, the Schwarzian action is equivalent to Jackiw-Teitelboim gravity in the bulk [11,12,15,16]. Recently, it was shown that the partition function of Jackiw-Teitelboim gravity (on surfaces of any genus) is described by a particular double-scaled random matrix model [17], and relations of random matrix theory and chaos to the SYK model were discussed in [18,19].
The techniques above are applicable when the value of p is taken to be a constant independent of N , such as p = 4, or in the large p limit (as long as N → ∞ first). However, an interesting scaling was recently studied, known as double-scaled SYK [18,[33][34][35], where p is taken to scale as √ N . In this limit the model can be solved exactly at all energy scales using combinatorial tools, and it is rigidly controlled by a quantum group symmetry which replaces the conformal symmetry at all energy scales.
In solving double-scaled SYK, the combinatorial description of the partition function, as well as of correlation functions, is given in terms of chord diagrams (which are reviewed in section 3). The summation of those chord diagrams can be performed analytically. In particular, the 4-point function, which encodes the quantum Lyapunov exponent, was found exactly [35]. At low energies and in the limit p 2 /N → 0, it indeed agrees with the result [9] found from the Schwarzian theory.
The double scaled limit is in a sense also much more universal as many different microscopic models reduce to similar chord diagrams prescription; for example, one can replace the Majorana fermions by Pauli matrices (which commute between different sites) and obtain the same set of chord diagrams, or consider the SUSY model [36] which results in modified rules, still within the chord diagrams framework. It also suppresses spin glass phases in the theory [37]. Moreover, at low energies and when p 2 /N → 0, the model is still described by the Schwarzian theory, which means that it is still equivalent to Jackiw-Teitelboim gravity living in two-dimensional AdS space.
In this paper, we study the version of the SYK model, where the fermions are complex rather than real such that there is a global symmetry, in the double-scaling limit. This is a variant of the original Sachdev-Ye model [1], which can be written in terms of complex fermions, with four-fermion interactions.

Outline and summary of new results
We start in section 2 where we review the definition of the complex SYK model and mention some known results about it. In particular, the model was studied in the literature in the large p limit (N taken to infinity first and then p is taken to be large), and we will make contact with these results. Indeed, this large p limit is obtained as a limit of double-scaled SYK as p 2 /N → 0. As the combinatoric approach to doublescaled SYK reduces the calculation of various observables to a description in terms of a summation over chord diagrams, we derive the value assigned to each chord diagram in generic double-scaled SYK theories with complex fermions in section 3. The result is simply given in terms of rules assigning to every chord and pairs of chords a particular value. Then we use these rules to evaluate observables in the theory at all energies. In section 4 we calculate the canonical and the grand canonical partition functions of the theory, where in the latter a chemical potential for the global U (1) symmetry is turned -2 -on. In addition, we find in section 5 the exact partition function in every sector of a particular fixed charge, providing a refined information related to the charge. We verify that in the limit p 2 /N → 0 these results reduce to the large p results from the literature. In section 6 we consider another SYK model with a global symmetry, namely a U (M ) symmetry, and use similar tools to get the partition function with chemical potential for the various Cartan generators.
We then go on to study correlation functions. In section 7 we find the full two-point function (in a fixed chemical potential and in a fixed charge sector). We go to the limit exhibiting a conformal behavior, as well as get small corrections to it. As a check, we match to the spectral asymmetry factor in the theory. We also consider an analogous measure of asymmetry defined by the two-point function with these corrections, exhibiting a slightly different behavior; in particular, it is not simply determined by the charge of the operator, but depends also on its dimension. In section 8 we calculate the four-point function. In particular, this gives the Lyapunov exponent in the limit of small p 2 /N and we match this with the literature. In section 9 we discuss the effect of very heavy operators. These are operators that in the low energy limit do not go to operators of a finite conformal dimension, but rather formally have an infinite dimension. They have a significant effect of separating spacetime into two spaces, as in [38], and the chord diagram picture provides a simple way to see this. We also discuss how very light states can still go between these separated spaces. We finish with several appendices containing further details to which we refer from the main text.
2 Review of the complex Sachdev-Ye- Kitaev model In this section we briefly review the complex SYK model. In subsection 2.1 we discuss the definition of the model and in subsection 2.2 we review some key results about it that we will make contact with, following [39].

Definition of the model
The complex Sachdev-Ye-Kitaev model [40] is a quantum mechanical model of N complex fermions ψ i andψ i (where i = 1, 2, ..., N ) with random all-to-all interactions. The fermions satisfy ψ i ,ψ j = 2δ i j , ψ i , ψ j = ψ i ,ψ j = 0, i, j = 1, · · · , N , (2.1) In the last term above, I and I denote an index set I = {i 1 , i 2 , · · · , i p } consisting of p distinct indices i 1 < i 2 < · · · < i p . Fermions with capital indices stand for the following product of the components Note that we have reversed the ordering inψ I . This notation will turn out to be convenient later. The couplings J are Gaussian complex random variables satisfying (J (no sum), (2.4) where J is a normalization constant for the disorder. 1 The model possesses a U (1) global symmetry that acts on the fermions as follows (2.5) The associated conserved charge is the fermion number defined as We will also use the specific charge defined by which takes values in the range −1/2 < Q < 1/2. This will be useful when comparing to existing results in the literature. This model admits a non-trivial double scaling limit in which In this paper we explore the complex SYK model in this limit. But first we present a short summary of known results in the usual, fixed p, large N complex SYK model. 1 Our notations are related to those of [39] as follows: p = q/2 , ψ i = √ 2f i , J i1···ip j1···jp here = 1 2 p J i1,i2,··· ,iq there , and for N p, J here = √ 2N 2 p J there . Note that by J 2 there we mean the one in Eq. (C14) of [39] (that is the one used in the quoted results) which appears to us to differ from the one in Eq. (1.2) there (the former being half the latter).

Thermodynamics
In the limit where N is taken to infinity first, at fixed p, followed by a zero temperature limit T → 0, the canonical free energy, F , 2 of this model has the following low temperature expansion [39] (2.9) In the above expression, E 0 (Q) is a non-universal ground state energy and S(Q) is the universal zero-temperature entropy (universal in the sense that it is independent of the 'UV' details of the theory; for example adding higher order fermion interaction terms to the Hamiltonian does not change the result). For generic p the analytic form of the ground state energy is not known. However, it can be computed analytically in a large p expansion and has been found to be The universal zero-temperature entropy S(Q) is a symmetric function of the U (1) charge Q and has been computed analytically for any p > 2. In a large p expansion, the expression for S(Q) takes the following form Because of the non-universality of E 0 (Q), the thermodynamic grand potential, Ω = −T log Z(µ, T ), has both universal and non-universal pieces. The universal part of Ω has been computed in [39] for generic p from the G, Σ action. In the large p limit, the analytic expression for Ω has been found to be 3 In the above expression v is the solution of the equation (2.13)

Two-point function
A quantity that plays an important role in the complex Sachdev-Ye-Kitaev model is the so called 'spectral asymmetry' factor whose thermodynamic definition is given by the charge derivative of the entropy 4 E = 1 2π (2.14) This factor reflects an asymmetry in the spectral function A(ω) which is defined as where G(ω) is the Green's function of a single fermion, which in terms of the Euclidean time τ is defined as follows (note there is no summation over i in the following equation) In the above expression T τ specifies the τ ordering, Z(µ, β) is the grand canonical partition function and K is the sum of the Hamiltonian and the fermion number operator As shown in [39], assuming conformal invariance in the IR, the Green's function G(ω) in the frequency domain (also in the presence of a chemical potential) at zero temperature takes a scaling form which can then be plugged in the Schwinger-Dyson equations, fixing the dimension ∆ = 1/(2p) and C. In the above, z is the complexified frequency. In the τ domain, one obtains the following ratio When the net-charge is zero (Q = 0), E vanishes, and otherwise E acquires a non-zero value which in the large p limit is given by (2.14).
In this paper, we calculate these various quantities in the double-scaled complex SYK model. We will find that at the leading order in large p, our results match with those in [39].
In the large-N double scaled SYK model, the natural operators to consider are those that are made out of a string of fundamental fermions whose length is of the order of √ N . For such operators that carry a U (1) charge σ, the ratio of the Green's functions (2.19) gets modified so that we have σ times the spectral asymmetry. In fact, in our analysis we will be able to calculate subleading in N corrections to the ratio of the coefficients (of the time dependence) in the 2-point functions, finding a different behavior where there is also a dependence on the dimension of the operator. However, it should be stressed that the relation to the spectral asymmetry only holds in the conformal regime, which is what we indeed get at leading order in N . The subleading corrections go beyond the scaling regime, and enter in a slightly different measure of asymmetry that we consider.

Computation using chord diagrams
Using chord diagrams, we now present a method for solving the complex Sachdev-Ye-Kitaev model in the double scaling limit The method boils down to the calculation of the traces of generic products of complex fermions, and then taking into account the Gaussian structure of the couplings. We begin in subsection 3.1 by briefly reviewing the chord diagram method for solving the SYK model with Majorana fermions. The generalization to the complex SYK will then be clearer. In subsection 3.2 we work out the formulas for traces of fermions relevant for the complex SYK, and in subsection 3.3 we work out their simplified large N limit. The main result of this section is given in (3.11) or (3.12). The large N rules are summarized in fig. 4. In the next sections we use these rules to compute the partition functions and correlation functions.

Chord diagrams and the real SYK model
The real SYK model is a quantum mechanical model of interacting Majorana fermions whose Hamiltonian is given by (1.1). If we write the Hamiltonian in a short-hand -7 -notation as then the moments of the Hamiltonian are given by The average over the Gaussian random coefficients J I j is given by Wick's theorem, instructing us to sum over all pairings of the k index sets I j . This is represented combinatorially by chord diagrams: we draw a circle, on which we mark k nodes, corresponding to the Hamiltonian insertions. The nodes are connected in pairs by chords, representing the Wick contractions. An example of a particular chord diagram is shown in fig. 1. Figure 1: An example of a typical chord diagram.
For every chord diagram, we are left with a trace over the fermions tr(χ I 1 χ I 2 · · · χ I 1 · · · ), where each index set I j appears twice. This trace is then evaluated simply by commuting the different χ's, so that eventually contracted χ's are next to each other, in which case we can use χ 2 i = 1 and the trace becomes trivial. In the chord diagrams, this amounts to disentangling the intersections, so that each intersection, as a consequence of the fermion anti-commutation relations, gives (−1) |I j ∩I j | where I j , I j are the index sets corresponding to the intersecting chords, and |I j ∩ I j | is the size of their (set) intersection. The size of the intersection |I j ∩ I j | is a random variable (since we sum over the I j 's), which in the large N double scaled limit (3.1) follows a Poisson distribution with mean p 2 /N [33]. Weighting by the Poisson probability distribution, one finds that each chord intersection contributes Triple intersections, i.e., configurations in which there are 3 chords with I 1 ∩ I 2 ∩ I 3 = ∅ are negligible in the large N limit. Therefore, the combinatorial problem that one gets is to sum over all chord diagrams, with a weight that depends on the number of pairwise crossings of chords, i.e., where π denotes a chord diagram, and cr(π) is the number of pairwise crossings in the diagram. These combinatorial sums can be evaluated using a transfer matrix technique, which captures all energy scales in the model, and gives rise to a q-deformation of the Schwarzian action [34,35]. In a similar manner, one can extend this construction to correlation functions.

Chord diagrams and the complex SYK model
The generalization to the charged SYK model is as follows. First of all, we note that after carrying out the average over the (Gaussian distributed) couplings in the random Hamiltonian (as will be done in section 4), we get a sum over traces of the form In this trace, the indices of a pair of consecutiveψψ are contracted with those of another ψψ pair (as in · · ·ψ I ψ I · · ·ψ I ψ I · · · ). However, in the remainder of this section we will allow for a more general arrangement as shown in (3.9) since it will be useful for calculating correlation functions of generic operators which we eventually do in section 7.
As a starting point we will represent the complex fermions using N -dimensional gamma matrices. We will use the following conventions for the Pauli matrices and write the complex fermions ψ i as the following tensor product -9 -Forψ i we simply replace σ + → σ − in the above formulas. Using this representation we can now work out the trace of a product of complex fermions. Fermions with capital indices, ψ I andψ J , stand for the appropriate product of components as indicated in (2.3). Let p j denote the length of the index set I j . Here we have taken the length to be generic, because this generalization is needed when we compute correlation functions. As mentioned, the quantity that we would like to evaluate in this section is a slight generalization of (3.6) where now at each point any ψ orψ is allowed (not necessarily alternating), with a total of 2k insertions, such that each of the k indices I j appears in one ψ and oneψ. The trace is defined to be normalized 5 as tr 1 = 1. This object is represented by a chord diagram: as reviewed above, this is a circle or a line (which is equivalent, by cutting open the circle at a point), on which 2k nodes are marked, such that pairs of nodes are connected by chords. Since we have here two kinds of insertions (ψ andψ), the chords are oriented, so that each chord has a direction; let us choose a convention where the arrow goes from a ψ insertion to aψ insertion (see fig. 2).

I1¯ I1 I2
· · · Ii · · · Figure 2: Chord diagram representation of the generic trace (3.9). The chords are oriented to go from a ψ insertion to aψ insertion having the same index set I.
The basic idea to evaluate (3.9) is the following. We will consider each component i of the tensor product structure in the representation (3.8). The contribution of ψ i to this component is simply σ + , each ψ j<i contributes σ 3 , while each ψ j>i gives no contribution; forψ it is the same, with σ + → σ − . 6 All the σ 3 commute among themselves and give only a sign, so the first step is to determine the sign and by this eliminate all the σ 3 's. Then the remaining strings of σ ± is straightforward to evaluate, giving either a vanishing result, or a power of two. 5 In the representation (3.8), the trace in each tensor product factor is normalized in this way. 6 Here the convention for the definition of ψ I andψ I in (2.3) is useful, since the evaluation becomes easier using the identities σ a 3 σ + = σ + and σ − σ a 3 = σ − (for any integer power a).
-10 -More explicitly, let us consider two indices 1 ≤ j < k ≤ N . For simplicity let us discuss first the case where each of them appears in a distinct single chord only -this is actually the case for most of the indices appearing in chords. The case where these two chords intersect is shown in figure 3a. Consider the k'th component of the tensor product representation. The chord containing the k'th index gives one σ + and one σ − , while the other chord containing the j'th index gives two σ 3 as shown in the figure (note that we can consider each i < k independently because they all give σ 3 factors that commute). That is, we have something of the form tr [· · · σ + · · · σ 3 · · · σ − · · · σ 3 · · · ]. Using the algebra of the σ matrices, we can get rid of the σ 3 insertions, resulting in a factor of (−1). If the chords were not intersecting, then using σ 2 3 = 1 we would get no sign.  fig. 3b we draw specific orientations for concreteness, but the argument in the text is independent of the orientations of the chords.
More generally, consider two indices j < k such that each of them can appear in any number of chords. As before, since the σ 3 's commute, we can consider each chord containing j in turn. As shown in figure 3b, when we consider the k'th tensor product factor, by inserting σ 2 3 = 1 in intermediate steps, and using the fact that σ 3 σ ± σ 3 = −σ ± , each intersection of a chord containing j with a chord containing k gives a (−1). Now we would like to combine all the signs from all the tensor product factors. Take a pair of chords J, K that intersect. For any j ∈ J and k ∈ K, if j < k we saw that when looking at the k'th tensor product factor, we get a (−1), and similarly for j > k we get a (−1) from considering the j'th tensor product factor; but if j = k there are no σ 3 's and no (−1). As a result, by eliminating all the σ 3 's, we find a sign which where p j = |I j | and p ij = |I i ∩ I j |. After this step, we are left only with factors of σ ± . If we have two consecutive σ + or σ − then this just gives zero since σ 2 ± = 0. This just corresponds to the fact that if we have two consecutive ψ's orψ s with common indices, we immediately get zero. Otherwise, consider again a particular index i with the corresponding tensor product factor. If it appears in no chord, then the trace in this tensor product factor is simply tr 1 2 = 1 (recall that the trace is normalized to one). If it appears in a single chord, then we have tr (σ + σ − ) = tr 2 0 0 0 = 1 (or tr(σ − σ + ) which is the same here and below). If the index i appears in m number of chords, then we get tr ((σ + σ − ) m ) = 2 m−1 .
As a result, a general oriented chord diagram with fixed indices I j , that is expression (3.9), equals unless when restricted to any particular i the ψ andψ do not appear in an alternating form, in which case the value of the trace is just 0. Let us also introduce the notation s ijk··· which denotes the number of sites that appear in I i , I j , I k and so on, but in no other set. Note the difference between the two symbols p ij and s ijk··· . They are related by p ij = s ij + k s ijk + · · · . In terms of s ijk··· we can write (3.11) as This result holds for any value of N and p i .

Large N simplification
A simplification occurs in the large N double scaling limit. Consider the same trace object as above, associated to a particular oriented chord diagram, but now summing over the indices k j=1 N p j −1 (3.13) The combinatorial prefactor turns counting of events in the sum into probabilities of those events. As was shown in [33], in the large N limit, the intersections are independently Poisson distributed p ij = |I i ∩ I j | ∼ Pois p i p j N , and there are no triple (or higher) intersections with probability that goes to 1 in the large N limit. Therefore, in (3.12), the sum over m is now restricted only to m = 2.
Thus, we should consider independently pairs of chords I i , I j . First, we need to make sure that we do not get a vanishing result. For an index i appearing only in a single chord, it appears in one ψ I and oneψ I and therefore it does not vanish. Any index i appearing in two chords (a higher number of chords can be neglected in the large N limit as was just mentioned) will necessarily give a vanishing result if the two chords are intersecting (just because one chord gives a σ + and a σ − , and there is one end of the other chord between these two, such that no matter whether it is a σ + or a σ − we would have σ 2 ± = 0). By this logic, a non-zero p ij is allowed only in the first four possibilities appearing in figure 4. For each such pair of chords (that are not intersecting and therefore get no (−1) factors) we need to sum over the number of elements p ij with the Poisson probability distribution; their contribution from (3.11) is For all the other combinations of pairs of chords, we must have p ij = 0 and therefore they contribute simply e −p i p j /N (3.15) from the Poisson distribution. In addition, if they intersect, they also give a factor of (−1) p i p j from (3.11) (for which as mentioned, p ij = 0 necessarily in the large N limit). These rules are summarized in fig. 4 for convenience. Using these rules we can now calculate physical observables in the double scaled complex SYK model. In the ensuing sections we calculate the partition function and correlation functions of generic operators.

Partition function
In this section we obtain the canonical and grand canonical partition functions.

Canonical partition function
For simplicity, we first apply the rules that we already found in order to calculate the ensemble averaged partition function (without a chemical potential) Chord configuration Value Value in the presence of a chemical potential µ The calculation proceeds by expanding out the above quantity in moments then evaluating each moment using the result from the previous section, and then resumming the moments. In the last equality, we carried out the ensemble average to obtain a sum over oriented chord diagrams (abbreviated as CD). This is the result for even k. For odd k the moments vanish due to the ensemble average. Note the simplification here in comparison to (3.13), in that theψψ are contracted in pairs. Pictorially, the corresponding oriented chord diagrams consist of adjacent pairs of chords of opposite orientation, so that each such pair can be replaced by a single unoriented chord; we will refer to such an unoriented chord as an H-chord (since each end of it corresponds to a single Hamiltonian insertion). This is demonstrated in figure 5.
where λ = p 2 /N is what we keep fixed as N → ∞. This is just as in the real SYK where for every intersection one gets a factor of e −2p 2 /N , whereas in the complex SYK model this factor gets modified to e −4p 2 /N . In the next subsection we give an analytic expression for the partition function (evaluating the sum over chord diagrams) in the more general case when having a chemical potential.

Grand canonical partition function
We now move on to calculate the ensemble averaged grand canonical partition function of the complex SYK model which is given by where Q is defined in (2.6). As before we will use the moment method and compute (at this stage, without any restriction on µ) note that we are not expanding the charge term. We will see, however, that resummation is difficult for finite µ = 0 that is independent of N , since the sum over k diverges. We note that the divergence originates from very large values of k, where the computation is not reliable. To remedy this we can either (i) scale µ appropriately with N , or (ii) go to the fixed charge basis. We will do each of those below. Computing the individual moments uses techniques similar to the ones used before. The moments now read (4.6) - 16 -In the representation (3.8), the chemical potential term is represented by (4.7) Similarly to section 3.3, let us provide the rules for computing oriented chord diagrams (that will also be valid beyond the partition function) in the large N limit, where now we allow the presence of a chemical potential.
Earlier, the first step was to apply the σ 3 's resulting in a sign. This clearly remains the same (we do not use the trace here, so this is unaffected by the presence of a chemical potential). Non-zero p ij of intersecting chords give vanishing contribution to the trace and therefore the only signs we get are (−1) p i p j for intersecting chords. In the partition function and the correlation functions that we consider here, all intersections are in pairs, so overall the sign is plus.
Then, for every site i, if it does not appear in any chord, we simply have tr e µ e −µ = cosh(µ). If it appears in at least one chord, then if the first appearance is in a ψ (that is, the first chord in which i appears goes to the right) we get tr 2 0 0 0 e µ e −µ = e µ times 2 to the power of the number of chords it appears in minus one; if the first appearance is in aψ (the first chord goes to the left), we would get instead of e µ , a factor of tr 0 0 0 2 e µ e −µ = e −µ . In the large N limit there are no triple intersections of the index sets I j , and therefore the only options we have are summarized by • Each index i appearing in no chord gives cosh(µ), • Each index i appearing in 1 chord going to the right gives e µ , • Each index i appearing in 1 chord going to the left gives e −µ , • Each index i appearing in 2 chords going to the right gives 2e µ , • Each index i appearing in 2 chords going to the left gives 2e −µ .
For simplicity, let us start first with the case that all the indices in the various I j are distinct. We can first assign a value of cosh(µ) N which would be the case if there were no chords. An index in a chord, if it goes to the right, should be assigned e µ -17 -instead of cosh(µ) so that a chord going to the right gets (e µ / cosh(µ)) p i (and µ → −µ if it goes to the left). For the partition function, there are the same number of chords of each orientation, the e ±µ factors cancel, and altogether we simply get an extra factor of (cosh(µ)) N −kp in m k (µ) relative to the case without the insertion of a chemical potential. Now let us see how this counting changes for the general case, where the I j are not necessarily distinct. Recall that in the large N limit, the different intersections are independent and they do not overlap, so we can consider each p ij separately. The allowed non-zero p ij were given in figures 4a, 4b, 4c and 4d. Let us start with figure 4a. In order to allow an intersection of the corresponding two index sets, we need to take two indices, one from each of those two chords, and turn them into one index which is common to the two chords, and one which does not appear in any chord. In the counting of the previous paragraph, the two indices were together assigned the value of one (since they are of opposite orientation). However, now they should be assigned a value of 2e −µ cosh(µ) (where the cosh(µ) corresponds to the additional index not appearing in any chord). Therefore, the additional rule with respect to the previous paragraph, is to assign each index of this type the value of 2e −µ cosh(µ). Similarly, each index in the second, third, and fourth pair of figure 4 should be assigned a value of 2e µ cosh(µ), 2e −µ cosh(µ), 2e µ cosh(µ) respectively.
All that is left is to weight those possibilities by the Poisson distribution. For example, for the case of fig. 4a we have and similarly for the rest. For the pairs of chords that must have p ij = 0 we simply get exp [−p i p j /N ]. These rules for evaluating a chord diagram in the presence of a chemical potential are summarized in fig. 4, and can be used in the calculation of various observables. 8 We shall now specialize to the partition function, and recast this into the language of (the unoriented) H-chords. We have already enumerated the possibilities for allowed p ij > 0, corresponding to figures 4a, 4b, 4c, and 4d, in terms of H-chords. First, for every one of the H-chords (and there are k/2 of those), we get the pair of figure 4a. 8 As mentioned, these rules are general (valid for any arrangement of complex fermions in the trace) and in particular hold for the N = 2 supersymmetric SYK model analyzed in [36]. The large N analysis there corresponds to a scaling µ ∼ 1/ √ N which is mentioned below in subsection 4.3.1. In this case, the chemical potential enters only in the values assigned to the entire diagram and single chords (rules in figs. 4g-4i), with no contribution of the chemical potential coming from pairs of chords.
Therefore for every such pair of chords we have (setting p i = p j = p in (4.8)) For intersecting H-chords there are no p ij > 0 allowed. There areκ H non-intersecting pairs, of the forms shown in figure 6. For each of those, there are two pairs corresponding to two of figures 4a-4d as mentioned before, one giving again exp . Putting everything together one obtains (where recall that κ H is the number of intersections of H-chords, andk H is the number of pairs of H-chords that do not intersect). The ensemble averaged grand canonical partition function is obtained formally by resumming m k (µ) There are two new features of this result when compared to the case µ = 0. The first is that for each intersection of the H chords we get the factor q(µ) ≡ e −4λ cosh 2 (µ) (4.12) which explicitly depends upon the chemical potential. A second feature is that for every pair of H chords that do not intersect we got a factor of e 4λ sinh 2 (µ) . Due to this factor a term like e λ sinh 2 (µ) 2 k 2 appears in the expression of the moments. This term changes the game in two ways. The first is that the presence of a term of the form c k 2 makes it difficult to rewrite m k (µ) as dEρ(E)E k . The second is that this term blows up as k → ∞, preventing us from resumming the series in a straightforward way. In the next section we consider fixed charge sectors where this problem does not occur, but for now we will proceed with a fixed chemical potential.
Otherwise, apart from these new ingredients, the sum over chord diagrams has the same form as in the double-scaled SYK model with Majorana fermions. Therefore, the analytic evaluation of this sum over the (unoriented) chord diagrams appearing in the moments (4.10) can be done exactly in the same manner (for a review see section 2.3 of [35]). This leads to the following result where the q-Pochhammer symbol is defined by and (a 1 , a 2 , · · · ; q) n stands for the product j (a j ; q) n (similarly the ± notation means the product of the corresponding terms with each sign). In the λ → 0 limit, we expect this result for the partition function to agree with that in [39]. This is indeed verified in appendix A.

Partition function at different scalings of the chemical potential
In general, one should specify how µ scales with N . We saw that for 0 = µ ∈ R independent of N , the sum over k in the partition function naively diverges. In this subsection we consider various scalings of µ of the form with α > 0 andμ fixed. In this case the effect of the k 2 term is subleading and can be ignored, and therefore there is no problem in evaluating the sum (up to subleading corrections in N ). In the following we consider several possibilities for α.

α = 1/2
For α = 1/2 we find that (in the expression below we have kept terms only up to order where q = e −4λ and Z(β, µ = 0) is the partition function of the complex SYK model with zero chemical potential.

α = 1/4
There is another interesting scaling for the chemical potential in which α = 1/4. In this case we find that the partition function takes the following form (again keeping terms only up to order O(N 0 )) The main change is that the range of energies has changed, as can be seen from the coefficient of β in the exponent (the extremal values of the energy are given at cos(θ) = ±1). Note that for α > 1/4 the change in the range of the energies is subleading in N . The non-trivialμ dependent finite (N independent) change in the energies happens only for α = 1/4. Indeed, for α < 1/4 the energies are suppressed exponentially ∼ exp −#N positive power . The overall factor in front counts the effective number of states and will also appear in correlation functions. It can therefore be normalized away.

Canonical partition function at fixed charges
In the previous section, we coupled the chemical potential to the U (1) charge operator (2.6). The state |0 , satisfying ψ i |0 = 0 (note that it is not the ground state) has U (1) charge −N/2. Then the states of charge n areψ i 1 · · ·ψ i n+N/2 |0 , and the full range of charges of states in the Hilbert space goes from −N/2 to N/2 with spacings of a unit charge. For simplicity we take N to be even. In this section we extract the canonical partition function in the fixed charge ensemble by defining z = e 2µ = e iχ (5.1) and projecting on the fixed charge sector where γ is a closed contour encircling the origin. To be precise, in the convention used here, the charge (corresponding to Q) is (−n). Note that for an imaginary chemical potential, the sum over k converges. This is because in the expression of the moments, the sign in front of the k 2 term in the exponent is now negative. With this issue solved, we can use the moment method to compute a finite fixed charge partition function.
In the ensuing subsection we estimate the χ-integral of the moments m k (χ) by a large N saddle point.

Large N evaluation
We will assume that the charges n scale as N α for some α > 0. Then, we can approximate the χ-integral of the moments m k (χ) using a saddle point where we have defined the following functions The saddle point, for large n and N , is encoded in the term We have kept the n dependence since we allow for n to increase with N as well (the most we can have is n ∝ N ). 9 The saddle point is then given by Plugging this back into (5.5), keeping k, λ fixed, and p = √ λN → ∞, we obtain In addition, in (5.3) we have an explicit factor of which was evaluated at the saddle point. We notice that the bothersome k 2 terms in the last two expressions cancel exactly. At the saddle point, the function f (n) and the parameter q(n), which are now functions of the charge, are Putting everything together, we have the following result for the integral in Since n is an integer going from −N/2 to N/2, we have that 4n 2 /N 2 ≤ 1 which implies that the log is well defined. More importantly, the expression for the integral is of the form m k = dθρ(θ; n)E(θ; n) k from which we can read the range of energies and the full density of states of the model, for each charge sector n separately. A few comments are in order 1. The main observation is that the model has the q-Gaussian density of states as in the Majorana SYK model, for each charge sector, but with a rescaled energy E, and -more interestingly -a renormalized intersection weight q(n) that depends on the charge density. We expect this to carry over to the computation of any correlation function, as we will verify in sections 7 and 8. In particular, when n/N → ±1/2 then q(n) → 0. This means that the model becomes more and more strongly coupled and, at least in terms of the macroscopic density of states and correlation functions, becomes more and more like the standard unitary or orthogonal RMT ensembles (i.e., anything that can be computed using a single trace in the Hilbert space at time scales that do not scale like N , or correspondingly, at the level of the analogue of free probability theory).
In the gravity dual this should probably be interpreted as stronger and stronger coupling in the bulk. This is to be expected since when the charge is large, the electric flux backreacts more strongly on spacetime. Here we also see that when chords intersect in the bulk the suppression factor (proportional to q(n) nominally) becomes larger, until in the large charge limit q(n) → 0 and particles are not allowed to cross.

2.
A second comment has to do with the range of energies in the charge n sector, which is The results here apply for any scaling of the charges n ∼ N α with 0 < α ≤ 1. The case of α = 3/4 is special because in this case the range of energies is finite. Defining a fixed r as the range of the energies in the charge n ∼ N 3/4 sector is 3. Finally, the prefactor in (5.10) is easy to understand. Using the fact that π 0 dθ 2π (q, e ±2iθ ; q) ∞ = 1, the total number of states in the charge n sector is (in the N → ∞ limit) given by 10 (5.14) We know that the exact number of states of charge n (that isψ i 1 · · ·ψ i n+N/2 |0 ) is given by N n + N/2 ; (5.15) using Stirling's formula for N ! and (N/2 ± n)! in the large N limit, this binomial coefficient becomes the same as (5.14).
From Eq. (5.10) (after inserting the factor of 2 N that comes from restoring the normalization of the trace) and Eq. (5.2), we find that the partition function in the canonical ensemble reads where (using the definition (2.7)) The range of charges −N/2 ≤ n ≤ N/2 translates to −1/2 ≤ Q ≤ 1/2, which is the convention of [39].

5.2
Fixed charge canonical ensemble in the limit λ → 0 Our interest in this section is the limit λ → 0 at a fixed Q of (5.16). This limit was computed in [34] (see Eq. (4.13) there) where we should substitute instead of λ there the valueλ = 4λ 1−4Q 2 . We find that the free energy F (T, Q) = −T log Z(T, Q) has the following low temperature expansion in the λ → 0 limit We can compare this result with the findings of [39]. The canonical free energy of the complex SYK model has a low temperature expansion (2.9). We find from (5.18) that These agree with (2.10) and (2.11). Thus, the large p limit of the model (appendix C of [39]) is indeed reproduced by λ → 0.
where an unconstrained summation over all upper and lower indices is to be understood. This model appears also in, e.g., [42,43]. For Hermiticity of the Hamiltonian we demand that (J The Hamiltonian is, therefore, just a product of SU (N ) currents with random coefficients.
Depending on symmetry properties of the tensor J, the model has many variants. A-priori, the only symmetry property of the tensor J i 1 ···ip j 1 ···jp is invariance under exchange of pairs (i m , j m ) ↔ (i n , j n ), so that exchanging separately say i m ↔ i n does not produce the same interaction term. However, for simplicity, we will consider a single coupling for every set of sites i 1 < · · · < i p and j 1 < · · · < j p . Other variants can be treated using a similar set of tools, and the results slightly differ. Concretely, the Hamiltonian that we consider is where an unconstrained summation over the upper and lower flavor index α is to be understood. Correspondingly, we assume that the variance of the random coupling J is: Our goal in this section is to calculate the ensemble averaged partition function in the limit of large N but finite M and finite λ = p 2 /N . In the above equation, Q α , defined as (tanh µ α · sechµ α ) 2 . (6.9) 3. For every pair of H-chords that intersect, there is a factor of (6.10) In the above formulas, A(µ) is the following function of µ α (6.11) With these rules, it is now straightforward to write down the generic moment which is found to be . (6.12) In the above expression, the sum in the second line is over all unoriented chord diagrams (as in figure 1) with k/2 H-chords. As a consistency check, setting M = 1 reproduces the result of the U (1) model (4.10).
-28 -For general M with no chemical potentials, (6.12) becomes This takes the same form as in Majorana SYK, where the effective q in this model is 14) The factor of M N in the denominator corresponds to the the total number of fermions in the system.

Two-point function
In this section we will compute two point functions in the complex SYK model, and the first issue is to discuss what are the natural operators in the theory. We will be interested in operators of the form The coefficients (J (M ) ) I I are again taken to be random Gaussian variables with variance (J (M ) ) These coefficients are uncorrelated with the random coefficients J I I in the Hamiltonian. Given an operator M , we have denoted by p M the number of ψ's in the operator, and byp M the number ofψ's. We will refer to the sum p M +p M as the "size" of the operator. The difference p M −p M determines the charge. In this section, we will take the size of the operator to be double scaled as well, or more precisely We will refer to such operators as double scaled random operators.
In section 8 we will compute 4-point functions of such operators and in section 9 we will discuss "longer" operators. Before computing the two point function we turn to explain why this is the right class of operators.

Why double scaled random operators?
The rationale for requiring random couplings was discussed in [34,35]. Let us briefly review the arguments there (and then somewhat rephrase them).
1. "Single trace" probes should be in the same statistical class as the energy momentum tensor.
Consider an AdS black hole, which we think about as some core set of degrees of freedom governed by a suitable random Hamiltonian. Note that an AdS 2 might appear as an IR of an altogether different UV theory, and the degrees of freedom with which we describe the black hole may or may not be simply related to the degrees of freedom of that UV theory. We then probe the black hole using the available bulk probes, such as single trace operators or their analogues. The Hamiltonian in quantum mechanics, or the local energy-momentum tensor in higher dimensions, is one such operator. Probing with the full Hamiltonian does not provide any more information beyond the partition function, but the local energy momentum operator does (if the theory is higher dimensional). Since the full Hamiltonian is a random operator when acting on a suitable set of degrees of freedom describing the BH, we can expect that the local energy momentum operator will also be effectively described by some random (local) operator acting on these d.o.fs.
But the local energy-momentum tensor is just one operator in a tower of single trace operators with which we can probe the system. In N = 4 SYM we can use its primary tr(X 2 ) to probe the black hole, or we can just as well use any other of the tr(X n ) operators. If the former is a random operator on the states of the black hole, we can expect that all single trace operators would be random operators. If we have some idea about the statistical ensemble of the Hamiltonian, we can expect that the ensemble for all the other single trace operators will be of a similar nature. This lands us on the proposal for the observables above, after we allow for a more general charge and mass.

Universality of the observables
It is important to note that there is actually a large number of models with different microscopic details, yet with a double scaling limit which reduces to the same set of chord partition functions (for example, the model originally discussed in [33]). This is consistent with expectations from gravity, where, if one is interested in the AdS 2 part of space-time, one glues it to an external region in order to break conformal invariance. There is a broad range of possibilities for such different UV spaces (one usually thinks about it in the other direction -AdS 2 appears as the IR near horizon limit of many different backgrounds). The probes which are legitimate in the entire theory are really defined on the boundary of spacetime, i.e., the boundary of the UV region, which is not strict AdS 2 physics. Different random microscopic models, or different embeddings of AdS 2 in bigger spaces, come with an altogether different set of particles and hence observables. So on top of the Hamiltonian we are actually instructed to be able to define a large set of observables (1) which are independent of the microscopics of the AdS 2 model, (2) for which we freely specify quantum numbers such as mass and spin, and (3) which obey factorization of correlation functions. Our class of random operators is precisely like that.

Double scaled random operators as consistent truncations
We can also phrase this choice of operators in the language of consistent truncations. By a consistent truncation one means truncating the set of fields of GR/String theory to a smaller set of fields, setting all the others to zero. We require that the fields that we kept close under the equations of motion. I.e., none of the fields that we have set to zero will have a tadpole for any configuration of the fields that we kept (at least for those that satisfy the equations of motion). Suppose we are given a random Hamiltonian and as many as we want (but finite) number of random operators of any length, all with random coefficients. We will refer to these operators as the basic set of operators. Suppose we consider any additional operator, with random coefficients (with zero mean) which are drawn independently of any of the coefficients in the basic set. It is obvious then that the 1-point function of the additional operator is zero, in any trace which contains as many insertions of the Hamiltonian and basic set operators as we want. So in any correlator of operators from the basic set we will not excite any quanta of any independent random operator outside the basic set, which is just the statement that independent random operators can be consistently truncated. Going back to the picture above that different realizations come with different particles (observables), we can just refer to them as different consistent truncations of a richer theory.

Two-point function in a fixed chemical potential
The two point function that we will be interested in is T τ specifies the τ ordering of the product and K, defined in (2.17), is the sum of the Hamiltonian and the fermion number operator. The notation · · · J M stands for the average over the random operators and · · · J,J M for the average over the ensemble of both the Hamiltonian and operator couplings. The calculation of the two-point function proceeds by calculating the moments as follows. Define the moments Then the 2-point function for τ > 0 is and for τ < 0 we just take the result for (7.6), plug τ → −τ , exchange p M ↔p M , and multiply by (−1) p M +p M . Performing the average over the disorder, the moments become (we denote k = This time, in addition to the H-chords that comprise a pair of oriented chords with opposite orientation, there are also two oriented chords representing the I index and J index contractions of the external operators; we distinguish them by drawing those as dashed chords (see figure 7 for an example). Note the two dashed chords correspond to index sets of different sizes in general. Nevertheless, we can just as well assign a (dashed) unoriented chord, to be referred to as an M -chord, corresponding to this pair of oriented operator chords. (7.8)

Two-point function in a fixed charge sector
In this subsection we compute the 2-point function in the model, where the external states in the trace have a given fixed charge (−n). In order to do that, we change the time evolution of the operators in the previous subsection from K to the usual Hamiltonian H, and keep K only in the overall exponential inside the trace. That is, we start with tr e −βK e τ H M e −τ HM and perform the same projection to a fixed charge as in the partition function. This results simply in the omission of the exp [2τ µ(p M − p M )/β] term in (7.5).
As done for the partition function, in order to perform fixed charge projection we -33 -first change variables 2µ = iχ, which results in 11 As before we calculate dχ e −inχ m k 1 ,k 2 (χ) (where n is an integer assuming N is even) via a saddle point approximation, along the lines of section 5.1. The terms in the exponent that scale with a positive power of N are This has a saddle point (at least for non-finite n) at where X is defined to be Plugging the saddle point expression (7.11) for χ back in (7.10) gives (here and below). There is also the term in m k 1 ,k 2 (χ) The 2-point function at a fixed charge sector must be finite when we sum over k (as the 2-point function itself is finite). Indeed, the k 2 terms in (7.13) and (7.15) cancel exactly.
Completing the saddle point calculation with the second derivative term, and performing some simplifications, we find the sum over chord diagrams is the same as in the 2-point function analysis in [34,35], giving for the moments in the fixed charge sector where the quantities S M , A, B 1 , B 2 are defined as follows From the expression for the moments (7.18) we obtain the result for the 2-point function of operators of size ∼ O(N 1/2 ) in a charge (−n) sector (keeping terms to the order in which we are working) J,J M (multiplied by (−1) for a fermionic operator M ) which is obtained from (7.23) by the replacement p M ↔p M and τ → −τ as mentioned above. In the following subsection, we evaluate the integrals in (7.23) in the conformal regime.

Conformal limit
The expression (7.23) is the two-point function of the operator M in a fixed charge sector for a generic value of p 2 /N . In this subsection we consider the p 2 /N → 0 limit corresponding to the usual SYK within the conformal regime (7.24) In the real SYK case, the coupling J M S used in [5] is related to the J used in [35] by J M S = √ λJ (as q → 1). Since here J is always accompanied by an exp [p 2 /(2N )] factor, we define J = 1 − q(n)Je p 2 /(2N ) . (7.25) This definition of J depends on the charge, and is simply a convenient definition; it is not meant to provide an alternative definition for the coupling in the complex SYK model. The p 2 /N → 0 limit is described conveniently using the variables y related to θ through θ i = π − λ(n)y i . Under the parametrizatioñ when λ(n) → 0 we have the following limits for the q-Pochhammer symbols (as in [35]) In this limit, the expression for the two-point function becomes · exp 2Jτ cos(λy 1 ) λ(n) e A+B 1 + 2J(β − τ ) cos(λy 2 ) λ(n) e A+B 2 .
(7.32) If 1 y 2 1/λ(n), then we can expand the cosine functions. There is a saddle point for the y 2 integral in this regime. The y 2 dependent piece of the exponent is The saddle point is at . (7.34) We look at the range 1 βJ 1/λ, τ J 1/λ (7.35) (corresponding to low temperatures but not comparable to N ) so that the conditions on y 2 above are indeed satisfied. Doing the saddle point estimation for the y 2 integral -38 -together with going to real time τ = it and changing the integration variable ω = ω β 2π , we get 12 The last integral is the one obtained in [10] where the time is renormalized. In the limit of low temperature we have t β, so the last line becomes (7.37) and in total we find This result exhibits a conformal behavior, plus small corrections to the scaling form, coming from the second line in (7.38). As explained in [39] and reviewed in (2.19), the spectral asymmetry factor can also be obtained through the Green's function of a single fundamental fermion in a conformal regime where t > 0, for M a single fermion. (7.39) As mentioned above, G(t < 0) is obtained by exchanging p M ↔p M and t → −t. Let us allow for the moment |n| to scale up to the maximal N/2, and remember that p M ∼ O(N 1/2 ).
Before concentrating on the conformal regime, let us consider a slightly different measure of asymmetry obtained from correlation functions. Specifically we take the same ratio of correlation functions as in (7.39), with generic operators, but consider the t → 0 limit using the IR correlation function (7.38). This gives (note that To evaluate this carefully, we note that (7.41) There are several regimes, depending on the charge n: 1. For large charges, by which we mean n ∼ O(N α ) with α > 3/4, we have that e A+B 2 → 0 strongly,; however, the e −(A+B 2 ) terms diverge strongly with N . 2. Otherwise, for moderate charges, by which we mean n ∼ O(N α ) with α ≤ 3/4, the term e A+B 2 is finite, but then e In order to understand this behavior we note the following. If we concentrate on the scaling n ∼ O(N 3/4 ), the successive k'th moments are reliable, since there are no k dependent terms that grow or decay with N . Moreover, there are pieces that go as N −1/4 which can be trusted since we only dropped terms that scale as N −1/2 (as p, p M , andp M are of this size) in our approximations in analyzing the large N double-scaled limit. Therefore, we now concentrate on moderate charges.
The argument of the exponent is proportional to the charge of the operator (p M − p M ). While in a conformal regime this is the only dependence on the operator [39,40], here we also get a dependence on the dimension (or size) of the operator (p M +p M ). Finally, we go back to the definition in (7.39). As mentioned, the second line in (7.38) has a time dependence that gives a correction to the scaling behavior. While we will see below that these time dependent terms should be present in the correlation function physically, for N → ∞ they go to zero (for any charge n) and we obtain a conformal behavior. Extrapolating to the case of a 2-point function of a single fermion p M = 1,p M = 0, we have (at the order that we can trust for moderate charges) 13 This is the leading contribution for large p of the result (2.14) that was found in [39].

Verifying the time dependent terms
One may be puzzled about the result ( n (while that of E n is (−n)). We found that the range of energies at a given charge is given by (5.11) (being most reliable for charges that scale as N 3/4 ; however we even allow here the general case n ∼ O(N ) so that it will imply the other scalings). Since we are at low temperature, the dominant contributions from states of energies E n and E m will come from the lowest energies at the given charges. Therefore the difference between the lowest energies from the range (5.11) at these charges is given by We should compare this energy difference to the coefficient of (it) in the exponent in the second line of (7.38) (being the dominant time behavior at low temperature) which is We see that it matches exactly with the energy differences. This verifies that this exponential dependence on t in the correlator (which goes to zero as N −1/4 for n ∼ O(N 3/4 )) must indeed be there.

.1 Four-point function of neutral operators
In section 7 we considered the random operator M , and we can just as easily allow several such operators (as in [35]). In doing this, we introduce a flavor index A and consider the operators In this section we calculate the 4-point function of neutral operators in the presence of a chemical potential (and then mention fixed charge sectors). Taking two flavors of operators M 1 , M 2 , there are two channels that we can consider, namely M 1 M 1 M 2 M 2 and M 1 M 2 M 1 M 2 (in this schematic notation we omit the time dependence, and by expectation value mean disorder averaged and thermal expectation value). The 4-point function of an operator from one flavor will decompose into these channels. The latter channel, referred to as the crossed one, is the one encoding the quantum Lyapunov exponent and we concentrate on it.
As we are restricting to neutral operators we have p M 1 =p M 1 and p M 2 =p M 2 . As before, the crossed 4-point function is determined by the moments, which are given by The moments reduce to a sum over chord diagrams (abbreviated CD in the equation).
We mark the pair of M 1 operators and connect them by a dashed chord (as before, it is a single chord in the unoriented case), and similarly for the pair M 2 . These two dashed chords cross in this channel. In the sum over chord diagrams we are instructed to sum over all possible configurations of solid chords. See fig. 8 for an example of a chord diagram. The evaluation of these moments again follows from the rules of fig. 4. In appendix C we count the appearances of every rule of fig. 4 according to the unoriented chord diagram, where similarly to before we denote by κ H the number of intersections of H-chords, κ HM 1 the number of intersections between H and M 1 chords, and similarly for M 2 .
-43 -Combining the ingredients in appendix C, we get In order to make sense of this (so that there is no problem from the k 2 term as discussed before), we can take for example µ ∼ O(N −1/4 ) and drop O(N −1/2 ) corrections, leaving us with For many purposes we can ignore constants common to all the moments, as for example they do not affect the Lyapunov exponent. This expression for the 4-point function is then the same as in real SYK [35] (see Eq. (2.11) there) with the replacements Similarly to subsection 7.3, starting from (8.4) we can go to a fixed charge sector. With the appropriate prefactor common to all the moments (that we do not quote), the 4-point function is the same as in the real double-scaled SYK (for the final expression -44 -for the 4-point function see [35]), with the replacements (recall Q = −n/N )

Lyapunov exponent
As before, we can make contact with the usual large N SYK (in which p is independent of N ) by taking the p 2 /N → 0 limit. In the double scaled SYK model with real fermions, it was found that the chaos exponent has the following dependence on q [34] Plugging the relations (8.6) (in the regime of small p 2 /N the exponent in the expression for J real SYK can be set to 1), we find the following Lyapunov exponent This agrees with the result in [43] (after translating eq. (4.9) there to the conventions here). Alternatively, from (8.7) and (8.8), one can read off the Lyapunov exponent in a fixed charge Q (8.10)

Heavy Operators
In this section we discuss how to use heavy operators to disconnect spacetimes, and how light enough particles can still go between them. More generally, given that we have good control over double-scaled SYK models at any energy scale, we can provide precise answers to questions that mix the UV and IR degrees of freedom in the theory. In such a setting one excites the theory using a high energy operator, and then examines the -45 -response of the low energy gravitational background. Generally, such operators might be singular objects in the language of the IR degrees of freedom. These include, for example, singularities, the 'end of the world brane' (as in [44]) and processes that glue universes such as in [38]. But with control over the full theory we can carry out "precision measurements" of such objects. In this section we will discuss some exact results for the simplest of such objects and study the behavior of massive operators, whose dimension (to the extent that it can even be defined in the language of the low energy theory) is parametrically large, or even infinite. Since for neutral operators, the canonical correlation functions in complex SYK take the form of those in Majorana SYK, we will discuss this in the ordinary Majorana SYK model [35], and relate it to [38].
The situation is therefore that we are interested in the Hamiltonian (1.1) consisting of p fermions, and an additional operator with p M fermions, which satisfy and we use the definitions Note that the notations in this section, such as p and λ, stand for quantities in the Majorana SYK model and should be distinguished from those in previous sections.
The similarity in the analysis shows how different physical systems are described by a similar chord diagram description. For a simple gravitational interpretation we then take λ, E → 0, 14 but we still have the freedom of what to do with λ M (or q M ). Generally, since the scaling dimension of the Hamiltonian is 1 in the IR, then the scaling dimension of an operator of length p M will be ∼ p M /p = λ M /λ. In the limit of λ → 0 and p M /p fixed, the operator becomes a conformal operator of dimension p M /p. Such an operator does not have a radical effect on spacetime -it bends the trajectory of the "boundary particle" but in a controlled and computable fashion. A more interesting set of questions happens when λ M is not taken to zero at the same time as λ, so q M does not go to one, but rather q M is taken to be fixed. In this case one ends up with an operator whose conformal dimension formally tends to infinity, or more precisely, one cannot assign to it a conformal dimension.
Nevertheless, exact computations in the presence of such objects are just as straightforward. We will see that their effect, as seen within the low energy limit, is to split the AdS 2 space into two spaces, touching at an interface, as argued in [38]. If λ M is taken to be even larger, this interface can shrink to a point. We will also see that, even though that point is singular (from the point of view of the low energy theory), one can compute how correlation functions "go through it". Hence, we can have a large amount of control over the transmission through this space-time singular locus. In this limit, q M simply controls how many quanta of correlations go through the singularity.
The limit that we will be interested in here is thus It is rather intuitive to understand what happens in this limit. Recall the basic structure of chord diagrams for the 2-point function, fig. 9. The intersections of chords of the Hamiltonian with the additional chord associated with the operator are assigned a weight q n M where n is the number of Hamiltonian chords crossing the operator chord. In the limit of q M → 0 the 2-point function splits into two independent regions consisting of only Hamiltonian chords, i.e., spacetime splits into two. Finite q M therefore controls, in a very simple way, how the two spaces are partially connected -in the limit q M → 0 they disconnect and in the limit q M → 1 they connect (with M becoming a weakly coupled particle on spacetime). In subsection 9.1 we make more precise the notion of these partially connected spaces, and in subsection 9.2 we turn on probe operators in those spaces by studying the 4-point function, which describes a particle going from one space to the other. -47 -

Disconnecting universes with massive operators
Consider first the basic formula of a 2-point function for Majorana SYK with coupling J defined by J 2 i 1 ···ip J = N p −1 J 2 (with J i 1 ···ip the coefficients in (1.1)) as in [34,35] 15 where the measure factor w(θ) is defined as usual as We would like to evaluate this expression in the limit where λ, E → 0 and q M fixed. We first go to convenient low energy parameters y i defined by to obtain (see Eq. (5.12) in [35] for the limit of the measure) where W (y 1 , y 2 ) is a function of y 1 , y 2 given by W = log(2 sinh(2πy 1 )) + log(2 sinh(2πy 2 )) + 2J log 1 + q 2 M q 2k − 2q M q k cos(λ(y 1 − y 2 )) . (9.9) In the limit λ → 0, we can further approximate the y 1 and y 2 integrals via a saddle point approximation. The saddle point equations are , (9.10) and a similar equation with (y 1 ↔ y 2 , β − τ → τ ). We approximate each of the sums using an integral, which is possible when λ → 0, giving ∞ k=0 2λq M q k sin (λ (y 1 ± y 2 )) 1 + q 2 M q 2k − 2q M q k cos (λ (y 1 ± y 2 )) ≈ 2 tan −1 q M sin (λ (y 1 ± y 2 )) 1 − q M cos (λ (y 1 ± y 2 )) . (9.11) In a similar manner we can approximate The saddle point equations become 14) (and we require that 1 J M S β, J M S τ 1/λ for y 1,2 to be in the right range). We should plug the above saddle point expressions into the y 1 and y 2 integrals. To do this we need to first expand cos(λy i ) inside the log in the expression for W (y 1 , y 2 ): 16 -49 -In the above we approximate the k sum with an integral and using (9.14) we get Putting it all together, the result of the saddle point estimate of the y i integrals is Similarly, the partition function is 17 where approximately Together we have the final result We get that the un-normalized 2-point functionG = Z(β)G is just (up to a β and τ independent constant) the product of thermal partition functions with inverse temperatures The effect of non-vanishing finite q M = 1 is to increase the values of β 1 and β 2 by the same amount ∆(q M ). This effect can also be seen in the standard SYK model by using the saddle point equations (2.9) and (2.10) of [38]. In that paper the k i are our y i and the shift in τ i is controlled by 1/ , with being the scaling dimension of the operator whose two-point function is under consideration. This is pictorially represented in figure 10. The formulas, however, do not agree exactly in all the regimes of /C in their language vs. q M in our language.
∆(q M ) Figure 10: Pictorial representation of the two-point function of a massive operator. The black curves represent the thermal circle deformed by the presence of this heavy operator (indicated by red dots). ∆(q M ) represents the geodesic distance between the two insertion points.
9.1.1 Comments on the q M → 0 limit, or touching space-times The limit q M → 0 corresponds to even heavier operators. In this limit, the distance ∆(q M ) → 0. This means that the spaces only touch at a point. Otherwise, time evolution in each space in general proceeds on its own. More precisely, this is true for any probe operator of fixed conformal dimension (but light enough particles can go between the two universes, i.e. particles close to the BF bound in the AdS language). For finite mass operators, in general, if we put several probe operators in the two spaces, we can consider two sets of chord diagrams -the "within universes" diagrams where operators (and Hamiltonians) are contracted within each part of the trace separately, and "between universes" diagrams where operators (and Hamiltonians) are also contracted between the two parts of the trace. In terms of figure 9 the first class of diagrams have chords that stay within region I or region II, and the second class have chords going from I to II. The latter are suppressed by at least q M relative to the first. So for q M → 0, the evolution of all operators happens within each spacetime.
This can also be understood in terms of a putative convergence of operators in our statistical class to the standard large N limit of β-ensemble random matrices. Recall that [33] if one computes the moments of double scaled operators, then they converge to those of the semi-circle when p/ √ N → ∞. The operator M (recall that we are now in the Majorana SYK model) is given by (9.1), and it has N p M independent degrees of freedom. For p M ∼ N/2, it has the same order of magnitude of parameters as a full random 2 N/2 × 2 N/2 Hermitian matrix, just written in a different basis of operators. In this case its correlators satisfy planarity constraints and cannot intersect any other contractions of matrices. In other words, long operators tend to act as β-ensembles random matrices, and the latter split spacetime into fragments. We see that (at the level of macroscopic observables computed in a single trace), the same is true even for much smaller operators p M ∼ A √ N when A is large.
In the next subsection we will make the computation above more precise by inserting an additional light operator on each side. We will see that we can compute the leading transmission through the zero size neck quite easily. In particular we will see which operators make it between the universes easily and which get stuck at the singularity.

Connecting universes
In this section we consider the 4-point function of two heavy operators M M and two light operators M L . From Eq. (4.11) of [35] we have the expression for the crossed -52 -four-point function as follows L ; q) n (q; q) n Q n (cos θ 1 |q L e ∓iθ 2 ; q)Q n (cos θ 4 |q L e ∓iθ 3 ; q) and Q n are the Al Salam-Chihara polynomials defined in terms of the basic hypergeometric series r φ s through Q n (cos(θ)|a, b; q) = (ab; q) n a n 3 φ 2 q −n , ae ±iθ ab, 0 ; q, q . (9.25) We now focus on the limit λ → 0 , λ M → ∞ , λλ M fixed large (9.26) such that we go to a gravity regime (the limit on λ) and we can arrange the connection between the spaces in an expansion in q M . We then also take the limit λ L → 0 , λ L λ M ≡ λ 2 LM fixed, not small , (9.27) such that the new light field can go between the spaces without suppression. When q M = 0 only the n = 0 term (which is one) survives in the last line of (9.23) and the 4-point function factorizes into a product of two 2-point functions (9.28) -53 -This result can be interpreted as follows. The presence of the massive operator M M for which λ M → ∞ such that q M → 0, creates a background where the thermal circle which was of length β gets deformed into two osculating circles of lengths see fig. 11. The light operator q L then probes this background and measures the boundary lengths β 1 and β 2 of these two spaces. This is seen in the expression for the crossed four-point function which factorizes into a product of two thermal two-point functions with inverse temperatures β 1 and β 2 . The main point, however, is that this correlator is finite even in the limit q M → 0. In this limit, no Hamiltonian chords go from one space to the other so the two of them gravitationally decouple. Nevertheless light states (corresponding to operators whose dimension is close to zero, which is allowed in quantum mechanics), can go between the spaces quite easily.
-54 -A Grand potential in the λ → 0 limit In this section we will calculate the grand potential Ω = −T log Z(T, µ) in the λ → 0 limit and compare it to the large p result in [39]. We will work in the scaling µ =μ N 1/4 , and keep only terms up to O(N 0 ). The partition function in this scaling was found in (4.17) and is (recalling λ = p 2 /N ) where q(µ) = exp(−4λ (cosh µ) 2 ) = exp −4λ + O(N −1/2 ) so q = e −4λ . Note that we reinstated a factor of 2 N to the partition function, which was implicit before because of the normalization tr 1 = 1 that was used. We want to evaluate the θ integral in the above expression for which we use the methods developed in [34,35]. The q → 1 limit of the measure factor can be simplified by change of variables θ = π − 4λy For notational convenience, let us define the quantity Then, the θ-integral in (A.1) becomes 1 we can expand the cosine in the above equation The y-integral can then be approximated by a saddle point. We find the saddle point is at y * = (π √ 1 − q)/(16βJδλ 2 ) and the saddle-point estimate of the y-integral is The above expression is the q → 1 limit of the expression in the second line of (A.1).
Combining with the terms from the first line and noting that (q; q) 3 ∞ ≈ exp (−π 2 /(8λ)), we obtain the following low temperature expansion of Ω(T,μ) We now compare this result to the large p result in [39], that was quoted in (2.12), where v there can be expanded andJ is defined in (2.13). With the scaling µ =μ N 1/4 , we find that taking the large N limit first and then doing the low temperature expansion gives We see that this indeed matches with (A.7) (up to smaller O(λ 0 , log(λ)) terms).

B Details of the U (M ) model
Let us introduce the following notation The next step is to correct this result for non-zero mutual intersection of the site indices {I i }. In the large-N limit the intersections among the index sets occurs independently in pairs with a Poisson distribution and contribution from triple and higher intersections are subleading which can therefore be ignored. So at the level of chord diagrams, at large-N the dominant contribution to a diagram comes from all possible pairs of oriented chords. These configurations are shown in figure 12. However, these configurations have a hanging flavor index at the nodes which are not contracted. In a chord diagram, every oriented chord in figures 12 is accompanied with a chord of opposite orientation and the flavor indices of these two chords are contracted separately at each node. Therefore we have to simultaneously consider the contribution to the trace from four oriented chords for every pair of unoriented chords and two oriented chords for every unoriented chord. The distinct pairs of unoriented chords and its corresponding oriented version is shown in the figure 13.
Once we know the contribution of each of the configurations above, we then need to know how many such configurations are there in a given chord diagram. In a chord β 1 ) for which the trace is non-zero. For each case, the trace is easy to calculate. The basic rules are: (1) for every pair ψ iaαa andψ iaαa (in this particular order) there is a contribution of e µα a / cosh µ αa (2) for every pairψ iaαa and ψ iaαa (in this particular order) there is a contribution of e −µα a / cosh µ αa (3) for every quadruple ψ iaαa ...ψ iaαa ...ψ iaαa ...ψ iaαa there is a contribution of 2e µα a / cosh µ αa and finally (4) for every quadrupleψ iaαa · · · ψ iaαa · · ·ψ iaαa · · · ψ iaαa there is a contribution of 2e −µα a / cosh µ αa . With these rules, the result of the trace is (α 1 =α 1 ) =(β 1 =β 1 ) 1 (cosh µ α 1 cosh µ β 1 ) 2 + (α 1 =β 1 ) =(β 1 =α 1 ) δ j 1 (cosh µ α 1 cosh µ β 1 ) 2 + α 2e −µα cosh µ α e 2µα (cosh µ α ) 2 + δ j 1 (· · · ) (B.15) where by δ j 1 we mean that j 1 necessarily should intersect at least one of the other sets (such as I 1 , I 3 , I 4 ). However, the probability of δ j 1 happening in fact goes to zero at large N ; there is a finite probability that a set of size p intersects another set of size p, but imposing that a single site intersects a set of size of the order of p is already suppressed (going as 1/ √ N ). Therefore we can ignore the δ j 1 terms in the above expression. Since the site index set I has distinct indices which are ordered, this procedure can be repeated for every i a and k b in I 1 ∩ I 3 and the net correction factor to (B.12) due to non-vanishing p 13 is C(µ) p 13 (all the corrections here and below are with respect to (B.12)) C(µ) ≡ 1 A(µ) 2   (α 1 =α 1 ) =(β 1 =β 1 ) 1 (cosh µ α 1 cosh µ β 1 ) 2 + α 2e µα cosh µ 3 α   . (B.16) Next, we look at indices in I 2 ∩ I 4 . The calculation is identical to the previous case and the net result is simply C(−µ) p 24 .
Finally, we look at the indices in I 1 ∩ I 4 and I 2 ∩ I 3 . Again the calculation is the same as in the previous cases except that there is vanishing contribution from the case α 1 = α 1 = β 1 = β 1 . So let us define Then, the correction factors that we get from non-vanishing p 23 and p 14 is D p 14 +p 23 . Doing the sum over the p ij (B.20)
(B.21) The calculation of this trace is identical to that in the previous case and gives the same result. Since the result from these two diagrams is the same we can conclude that for every pair of the unoriented H chords that do not intersect we have a factor of exp 4λ A(µ) 2 α (tanh µ α sechµ α ) 2 . (B.22)
(B.23) First of all we note that the net sign of this chord configuration is positive on account of the fact that there are four oriented chord intersections. Again, if there are no intersections (all p ij = 0), we get a factor of A 2p . For non-empty intersections we get A 2(p− p ij ) times corrections which is evaluated in the same manner as in the subsection B. The only difference now is that for every index in an intersection (say i a = k b for example when looking at indices in I 1 ∩ I 3 ), whenever all the associated flavor indices -63 -are the same (that is α a = α a = β b = β b ) we always get a zero no matter which pair of chord intersection we are looking at. However, when flavor indices are not all equal we get a factor of D p ij . Since there are four such p ij 's after summing over the p ij variables we get a contribution of exp(4λ(D − 1)) . (B.24) The factor of D − 1 can be simplified a bit For individual chords, we have the following trace structure: tr ψ I 1 A ψ I 2 A · · ·ψ I 2 A ψ I 1 A · · · e −µrQ r .

(B.27)
If there are no intersections I 1 ∩ I 2 = ∅ (that is p 12 = 0), we get a factor of A p . For I 1 ∩ I 2 = ∅, this factor receives corrections that we evaluate next. For simplicity, we first look at p 12 = 1 in which we have i a = j b for some i a ∈ I 1 and j b ∈ I 2 . Unlike the previous diagrams, there are two distinct cases a = b and a = b. For the diagrams we considered previously, the case a = b is the same as a = b because we could relabel the various summation indices freely for the reason that we were looking at those oriented chords intersections that do not end in the same Hamiltonian. When we consider intersections between two oriented chord intersections that end in the same Hamiltonian, we are constrained by the flavor summation indices and the case a = b distinct from the case a = b. Below, we look at these two cases in turn. First, consider the case a = b. Without loss of generality say a = b = 1. The fermions outside the intersection contribute A(µ) p−1 . For the remaining fermions we have α 1 β 1 tr(· · ·ψ i 1 α 1 ψ i 1 α 1 · · ·ψ i 1 β 1 ψ i 1 β 1 · · · e −µrQ r ) .
From the above expression, to get a non-zero trace we see that the contracted i 2 index restricts α 2 = α 2 and the contracted j 1 index restricts α 1 = α 1 . When α 1 = α 2 the contribution to the trace is (cosh µ α 1 ) −2 (cosh µ α 2 ) −2 and when α 1 = α 2 the contribution to the trace is 2e −µα 1 / (cosh µ α 1 ) 3 . Therefore for the case a = b we have the result As a check, we see that for M = 1 where there is no difference between the cases a = b and a = b and therefore the two results give the same answer. Next let us estimate how many cases are there like a = b and a = b. Since a and b take values 1, 2, · · · , p, there are a total of p 2 pairs of which p are of a = b type and p(p − 1) are of a = b type. This implies that the probability that a = b is 1/p and a = b is (p − 1)/p. Therefore, for large p, cases like a = b are much more likely to occur than cases like a = b.
For generic p 12 , since each site index sets have distinct indices, this procedure can be repeated independently for every pair ({i a , i b }, {j a , j b }) with i a and j b in I 1 ∩ I 2 and the net correction factor due to non-vanishing p 12 is where, we have used the definition in (B.16). Summing over p 12 , we get that for every H-chord, there is an associated factor of exp (λ(C(−µ) − 1)) .