The local SYK model and its triple-scaling limit

The SYK model Sachdev and Ye (1993); Kitaev (2015), a quantum mechanical model of fermions with random couplings, has garnered a lot of attention recently. The emergence of a conformal symmetry in the low energy theory and the pattern of the symmetry breaking gives rise to a (pseudo) Goldstone mode with a Schwarzian action. A similar pattern of symmetry breaking and the associated Goldstone mode with a Schwarzian action arises in the two dimensional JT gravity, and this leads to the correspondence between the SYK model and JT gravity in the low energy limitMaldacena and Stanford (2016); Polchinski and Rosenhaus (2016). Another non-trivial matching between the SYK model and the bulk JT gravity is the growth of the out-of-time-ordered correlators (OTOC) for matter fields which saturate the maximal Lyapunov bound indicating that the SYK model is a chaotic model Maldacena et al. (2016).

A natural question that arises is about the generality of the low-energy sector of the SYK model. To this end, different variants of the model have been studied, such as the supersymmetric version Fu et al. (2017), models with reduced randomness Xu et al. (2020). For SYK models, typically the low energy analysis is done by a saddle point approximation in the large $N$ limit, where $N$ is the number of flavours of fermions. However, the saddle point expansion breaks down for large $q$ ($q>\sqrt{N}$), where $q$ is the size of the interactions in the Hamiltonian (see appendix A of Anegawa et al. (2023)); $H\sim\psi^{q}$. The limit of $q\sim\sqrt{N}$ is the so-called double-scaling limit. In this limit, chord diagram technique Erdős and Schröder (2014); Berkooz et al. (2017) is suited for the analysis. It has also been shown that the double-scaled SYK model in the low-energy limit, also leads to a Schwarzian theoryLin (2022) which in turn means that the Lyapunov exponent is again maximal.

Another interesting variation of these models is regarding the randomness. The original SYK model had ${}_{N}C_{q}$ number of random couplings. It is important to understand if the interesting properties of the low energy spectrum of SYK model survive as we vary the number of random couplings. A version of this question has been studied in Xu et al. (2020) as sparse SYK models. In this work, the random couplings were made probabilistic, i.e., turned on or off with some probability, and this way, the effective number of couplings were made to vary. If $p$ is the probability for any of the couplings to be turned on, (which we take to be the same for all the couplings), then the effective number of couplings is $N_{\text{eff}}=p\times{}_{N}C_{q}$. It was shown that as long as the effective number of couplings $N_{\text{eff}}$ is sufficiently large, more precisely $N_{\text{eff}}=N^{\alpha}$ with $\alpha>1$, the model has the same saddle point in the low energy limit as the full SYK model. For $\alpha\leq 1$, the saddle point equations change drastically and become more complicated to solve. This means that if the number of random couplings in the SYK model is reduced to $\mathcal{O}{(N)}$, the saddle point is expected to change Xu et al. (2020); Anegawa et al. (2023).

An analysis of a model with $\mathcal{O}(N)$ couplings is not explored analytically as far as we are aware. Our plan in this short paper is to consider one such model of SYK with just $N$ random couplings and analyze it. We consider the following model;

where $J_{i,i+1,\dots,{i+q-1}}$ are the random couplings. Note that the indices in $J_{i,i+1,\dots,{i+q-1}}$ are continuous from $i$ to $i+q-1$. Imagine that we arrange the $N$ fermions on a circle. Then the interaction term in eq. (1) can be thought of as the coupling of fermions with only nearest-neighbor interactions ranging from $\psi_{i}$ to $\psi_{i+q-1}$. As can be seen from the above Hamiltonian there are only $N$ number of random couplings. We call this model as local SYK model to contrast with the conventional SYK model.

The conventional SYK has a good large $N$ limit in which the melon diagrams dominate. However, there is no good large $N$ saddle point for the local SYK model for any fixed value of $q$, as can be easily seen from the Feynman diagrammatics. Hence, one approach is to rely on numerical analysis Anegawa et al. (2023). Another possibility which we explore in this paper is to consider a large $q$ limit, along with large $N$. In fact, we find that the model is amenable to an analytical analysis in this limit. In particular, we explore a triple-scaling limit of this model and we find that an appropriate such limit is given by

The main result of our work is to show that there exists a good triple scaling limit mentioned above and the chord-diagram technique used for the fully random SYK model carries over to our model in a straighforward fashion. Furthermore, restricting to low energies, we find a Schwarzian-like behaviour. In particular, we show, by considering matter operators which schematically are of the form $M\sim\psi^{q}$ with different set of random couplings, that the OTOC of such operators has a Lyapunov exponents which saturates the chaos bound Maldacena et al. (2016). The reason for our emphasis on the triple-scaling limit will be clear when we discuss the chord diagram technique in the next section.

To begin with, let us review the method of chord-diagrams to compute the partition function in conventional SYK Berkooz et al. (2017). This involves computing the moments of the Hamiltonian by expanding the partition function as

where

and $\langle\dots\rangle_{J}$ indicates that the quantities are disorder-averaged over the random couplings $J$. For the evaluation of this term, we think of this term as $k$ copies of Hamiltonian on a circle, corresponding to trace. Each Hamiltonian term in eq. (4) has ${}_{N}C_{q}$ random couplings. Consider one particular choice of $k$ random couplings, one from each Hamiltonian. Since the distribution for the random coupling $J$’s are Gaussian, we perform Wick contractions among the random coupling $J$’s for the particular choice of couplings under consideration. This contraction among the $k$ couplings can be represented in the form of a diagram with chords, where a chord between two points represents the contraction between the associated random couplings. Thus, the evaluation of the term in eq. (4) entails the evaluation of various such chord diagrams corresponding to different choice of $k$ random couplings and different ways of contractions among them.

For the Wick contraction, $k$ must be positive and even integer and the Wick contracted indices of $J$ must be the same. This means that the fermion terms associated with this pair of random couplings must be the same. The important point is that this diagrammatic method of calculation for $m_{k}$ works not only for the conventional SYK model but also for the local SYK model defined in (1) as well. This is because the random variables $J$ in local SYK are also drawn from a Gaussian distribution. However as can be seen from the diagram, chords in general intersect. These intersections can be understood as follows. For contraction between the fermion terms, they must be bought next to each other. In this process, they have to cross other fermion terms and pick up additional factors of $(-1)$ due to Grassmann nature of the fermions $\psi_{i}$. This crossing of fermions is captured by the intersection of chords. Let us label a set of fermions as

If we anti-commute two sets of fermions $\Psi^{q}_{I_{1}}$ and $\Psi^{q}_{I_{2}}$ past each other, we will get a factor of $(-1)^{f}$ where $f$ is the number of common indices between $I_{1}$ and $I_{2}$.

By taking into account these $(-1)^{f}$ factors, we can evaluate $m_{k}$ explicitly following Berkooz et al. (2017) using the chord-diagram technique. For that purpose, we need to evaluate the averaged value of this $(-1)^{f}$ factor, which turns out to be different between conventional SYK model and local SYK model.

Conventional SYK case

For conventional SYK case, the probability of obtaining $f$ fermions in common in the two sets is given by

Let us explain the above result. First, pick $q$ fermion labels from the $N$ available fermions for the $\Psi_{I_{1}}$. The number of ways of doing this is ${}_{N}C_{q}$. Then we pick the fermions in the second set for the $\Psi_{I_{2}}$ where $f$ of them are common. For that, one has to pick $f$ fermions from $q$ fermions in $\Psi_{I_{1}}$ and then pick the rest of the $q-f$ fermions for the second set from the $N-q$ fermions, i.e., excluding the one that are in the first set $\Psi_{I_{1}}$. This yields ${}_{q}C_{f}\times{}_{N-q}C_{q-f}$ ways. Thus, up to an overall normalization constant denoted as $A$, $p(f)$ is given by

The constant $A$ can be obtained by imposing the condition that probabilities sum up to unity.

So, using eq. (8) in eq(7), we get (6).

Let us now compute the appropriate scaling of $q$ with $N$ in the large $N$ limit. In the limit limit $N\gg q\gg f$, by using the Stirling’s formula

we obtain

Thus, we see that the $p(f)$ becomes

Thus, we see from the above that the appropriate combination that appears is $q^{2}\over N$ and hence we need to keep this quantity fixed, which in the literature is called the double-scaling limit

Writing the answer in eq. (11) in terms of $\lambda$ defined in this way, we get

This is Poisson distribution.

Now, the average value of the phase factor obtained by commuting two sets of fermions is given by

Local SYK case

For local SYK model, the probability of obtaining $f$ fermions in common in the two sets is given by

We have implicitly assumed that $q<\frac{N}{2}$ in writing the above result. The reason for this is that we will be ultimately interested in the limit where $\frac{q}{N}\rightarrow 0$. The above result eq. (15) can be understood as follows. First, pick a set $q$ fermion labels (should be continuous) from the $N$ available fermions for the $\Psi_{I_{1}}$. The number of ways of doing this is $N$, which is the number of choices for the first fermion. Then we pick the fermions for the $\Psi_{I_{2}}$ where $f$ of them are common. For $f=0$, we have $(N-2q+1)$ choices, and for $f=1$ till $f=q-1$, we have only two choices. Finally for $f=q$ we have unique choice. Thus, taking into account the overall normalization constant, we obtain (15).

Similarly we can now compute $\langle(-1)^{f}\rangle$. We find

Therefore we take the following double-scaling limit in the local SYK model

With this, ${\bf q}$ becomes

Let us compare the $p(f)$ and ${\bf q}$ between the case of conventional SYK case (13) (14) and local SYK case (15) and (18). They do not match. However if we take furthermore triple-scaling limit

q matches in the leading order in $\lambda$. In fact in this paper we focus on this triple-scaling limit since it is only in this limit that we can obtain Schwarzian action and the OTOC exhibit a Lyapunov exponent saturating the chaos bound.

Let us note one crucial difference between the double-scaling limit in conventional SYK and the local SYK. In conventional SYK, where $\lambda=\frac{q^{2}}{N}$ is fixed to any finite value with $N\rightarrow\infty$, the probability of triple intersections vanish and so we can treat any pair of fermions independently of other fermion sets in the evaluation of a trace. However, this independence of intersections of fermion pairs is true in local SYK only in the triple-scaling limit where $\lambda=\frac{2q}{N}\ll 1$. Let us elaborate this with a simple example. Consider the trace

where $I_{1},I_{2},I_{3}$ are three arbitrary sets of fermions of the type mentioned in eq. (1). Naively, we would assign a factor of $\bf{q}^{3}$ for this term for the three exchanges of fermion sets. An explicit evaluation of this can be done by computing

where $p(m,n,r)$ is the probability of configuration of three sets of indices $I_{1},I_{2},I_{3}$ with the number of common indices between them as in the second line of eq. (21) above. We find, in the limit of eq. (17),

which is not exactly the same as ${\bf{q}}^{3}$ in the double-scaling limit. But in the triple-scaling limit in which we take $\lambda\rightarrow 0$ the above result can be approximated as $1-3\lambda\simeq{\bf q}^{3}$ upto corrections of $\mathcal{O}({\lambda^{2}})$ which are small and hence can be ignored. In the appendix AA, we explicitly do the calculation of a general trace of fermion sets. We confirm that, to leading order in $\lambda$ in the triple-scaling limit, it is consistent to use ${\bf q}$, eq. (18), for exchange of any pair of fermions sets as if they independent. Hence the technique of chord diagrams only works in the triple-scaling limit for the local SYK model.

Having obtained the factor q for the intersection of two chords in the chord diagram, let us now consider the evaluation of the term eq. (4). To obtain a value for this term we need to determine the value of the two point function of the random variable $J$. Let us parametrize this two-point function in terms of $\sigma$ as

The $N$-scaling of $\sigma$ in the two point function for the random variable $J$ above, can be fixed by the requirement that the leading term in $m_{k}$ is independent of $N$ as we show below. Let us elaborate on this in both conventional SYK model and local SYK model.

Conventional SYK case

A general term contributing to $m_{k}$ in eq. (4) is of the form

where $k$ is even and we use the notation

Using the two point function to compute eq. (24), we find

The factor $(J^{2}\sigma)^{\frac{k}{2}}$ is simply due to the contractions of $J$’s in eq. (24) in pairs using eq. (23). The factor of $\left({}_{N}C_{q}\right)^{\frac{k}{2}}$ is because there are ${}_{N}C_{q}$ choices for indices $I$ in each $J_{I}$ and the exponent $\frac{k}{2}$ is appropriate to the case when all the $\frac{k}{2}$ pairs are distinct pairs. The function $F(k,{\bf q})$ is due to the combinatorics of chord diagrams and the intersection of chords and this does not scale with $N$ in the double-scaling limit. The dots denote the subleading terms which are suppressed by factors of ${}_{N}C_{q}$ compared to the leading term, which happens when not all of the $\frac{k}{2}$ pairs of $J$’s are distinct. Thus, if we choose $\sigma$ to be such that

then in the large $N$ limit, we can obtain a finite answer, with only the leading term in eq. (26) giving a non-vanishing contribution.

Local SYK case

In local SYK, the $N$-scaling of the random variable $J$ is different from the conventional SYK. To understand this, we can now run the same argument as eq. (27). In the local SYK case, the analog of eq. (26), is given by

since there are $N$ choices for $I$ in each $J_{I}$ to contract in local SYK model. Thus, the appropriate scaling of the $\sigma$ in this case is given by

With all the pieces in place, let us now discuss the computation of the moments eq. (4). From these results, we can further take a limit, called the triple-scaled limit, where we obtain the Schwarzian action. As mentioned earlier, possible contractions in eq. (4) can be represented as possible ways of drawing chords with $k$ insertions on the circle, with one such configuration shown in FIG. 1. We cut the circle at any point (red dot in the circle in FIG. 1, the choice of where to cut the circle will be irrelevant) turning the circle into a straight line, see FIG. 2 below.

All the contractions will now be enclosed in the region between the two cuts. So, there are no chords before the first node (the node immediately after the cut = the left red circle) or after the last node (the node just before the cut = the right red circle). We now evaluate the quantity eq. (4) in the auxiliary Hilbert space called the chord Hilbert space, which is just the Hilbert space of state labelled by a non-negative integer, labelling the number of open chords. So, the quantity in eq. (4) is an transition element from $|0\rangle$, the zero chord state to $\langle 0|$ to the final zero chord state. Let $v^{i}$ be the number of chords of open chords after $i$ nodes. As shown in Berkooz et al. (2017), the quantity $v^{i}$ follows a recursion relation which can be written as

where $T$, the transfer matrix, is given in the chord basis by

In obtaining the above matrix for $T$, the inner product for the chord number states is taken to be

Thus the answer for $m_{k}$ can immediately be written as

It then follows that

The operator $T$ can be written in terms of chord creation and annihilation operators $\alpha,\alpha^{\dagger}$ defined by

and thus $T$ becomes

where $W$ is an operator acting on chord number states as

It is easy to check that

However, since $T$ is not real symmetric, it won’t be a Hermitian matrix. So, to remedy this and make $T$ symmetric, we renormalize the states as follows. Let $\tilde{n}$ be the renormalized chord number state defined as

where

The annihilation and creation operators are appropriately redefined as

in terms of which

with the renormalized $T$ given by

Written in terms of the original operators $\alpha,\alpha^{\dagger}$, it reads

Again, it is easy to check that

and thus the matrix of $\tilde{T}$ is symmetric. Rewriting in terms of the variable

we get

where $k$ is the conjugate momentum to $l$. Now considering the limit

we find

Let us remind the reader that from eq. (30), the operator $T$ (or $\tilde{T}$) is an evolution operator along the boundary i.e., along the nodes after cutting the circle. Taking large $l$ in the triple-scaling limit eq. (48) makes the boundary time almost continuous. Thus, it can be associated with a conjugate Hamiltonian by

where $H_{\text{bdy}}$ is conjugate to boundary time. Therefore we can read off $H_{\text{bdy}}$ as

which is the action for the Schwarzian theory written in Lin (2022).

Let us make a few more comments before we turn to OTOCs. Even though the chord intersection factor $\bf{q}$ is not exactly the same for the local and non-local SYK and differ at $\mathcal{O}{(\lambda^{2})}$ in the limit of small $\lambda$, the steps leading to eq. (49) are still valid in the local SYK case as well. Heuristically, the limit of $l$ satisfying eq. (48) can be thought of as restricting the range of energies in the double scaled SYK to a window analogous to the case of fixed $q$ SYK where the Schwarzian theory emerges, $\frac{1}{N}\ll\frac{E}{J}\ll 1$.

We are interested in computing a quantity of the form

where the operators $M_{1},M_{2}$ are matter operators whose details will be discussed shortly. The operator $M_{i}(t)$ is obtained from $M_{i}(0)$ by doing a time evolution with the Hamitonian $H$. The matter operators, considered in conventional SYK, are also made of $\psi^{q}$ terms with random couplings

where $\tilde{J}$ are also random variables drawn from a Gaussian distribution. The OTOC calculation in non-local SYK is more amenable for matter operators of the form $\psi^{q}$ where $q$ is also taken to scale as $\sqrt{N}$. Similarly, in the local SYK model under consideration, as we will see, it will be easy to analyze operators of the form

As before, we restrict ourselves to operators of sizes $q_{i}$ such that

The evaluation of the OTOC, eq. (52) requires the evaluation of terms of the form

To compute the OTOC, and in particular to evaluate terms as above eq. (56), we need to evaluate the appropriate factors of $\langle(-1)^{f}\rangle$ when two fermion sets of unequal lengths cross each other. Consider two fermion sets of length $q_{1},q_{2}$. For the conventional SYK this factor is similar to eq. (14) given by

The computation of the analogous factor for local SYK requires a more careful analysis which we shall do now. Without loss of generality we take the case $q_{1}\leq q_{2}$. First, we pick the $q_{1}$ fermions out of $N$ fermions. The number of ways this can be done is $N$. Then we pick the second set of $q_{2}$ fermions from the set of $N$ flavours. For this set to have $f=0$ fermion in common, the number of ways of picking the second set is $(N-q_{1}-q_{2}+1)$. For $f=1$ fermion in common, the number of ways is only two, either the first or the last fermion of the first set can be in common. So is the case till $f=q_{1}-1$. For the case of $f=q_{1}$, since $q_{2}>q_{1}$, the number of ways of picking the second set is $q_{2}-q_{1}+1$. We can now translate these into probabilities by including a proportionality factor which can be fixed by requiring probabilities add up to unity. Thus, we have

Requiring that probabilities add up to unity, we get

We can now compute $\langle(-1)^{f}\rangle$. We find

From the above we see the factor upon exchanges of a set of $q_{1}$ fermions with another set of $q_{2}$ fermions gives a value that depends on whether $q_{1}$ is even or odd. For the chord diagram technique to work, we need to restrict to the matter operators such that

We thus see that the double scaling analysis for the local SYK model imposes a significant constraint on the size of the operators. But note that this is not too constraining in the sense that the size of the operators $q_{1}$ and $q_{2}$ need not be exactly same but only that ${\absolutevalue{q_{2}-q_{1}}\over N}\rightarrow 0$ in the limit of $N\rightarrow\infty$. Thus, we have only a universal factor given by

where ${\absolutevalue{q-q_{1}}\over N}\rightarrow 0,{\absolutevalue{q-q_{2}}\over N}\rightarrow 0$. Thus, again we see that the double-scaling limit for the matter operators in the case of local SYK corresponds to

Since, the OTOC calculation involves the evaluation of terms as in eq. (56), which involves the intersection of matter operators and also the Hamiltonian chords, we require

where $q_{H},q_{1},q_{2}$ are the sizes of the Hamiltonian, matter operators $M_{1},M_{2}$ respectively. With this understanding the OTOC can be evaluated following Berkooz et al. (2019). The only important difference is that the factor of $(-1)^{f}$ is the same for the intersection of any type of chord unlike the conventional SYK. The triple-scaling limit can then be taken by taking the parameter $\lambda=\frac{2q}{N}$ to be small. We indeed get the same Lyapunov exponent as in the case of the double-scaling limit of the conventional SYK model given by

The above result is valid in the range of temperatures

The constraint on the temperature in terms of $\lambda$ is the statement that we restrict to low energies (upper inequality) and the consistency a of saddle point analysis in deriving the result eq. (65), see Berkooz et al. (2019).

Before we end this section, let us make a few comments. Although, we focused in this section on the OTOCs, one can analyze the two-point function itself which is technically simpler compared to OTOC. The analysis, in the triple-scaling limit, again parallels the case of conventional SYK Berkooz et al. (2017) with just the parameter ${\bf q}$ of double-scaling limit, eq. (14) of conventional SYK replaced by the parameter of local SYK, eq. (18).

In this paper, we considered a variant of the SYK model with reduced number of random couplings. In particular, we considered a model with nearest neighbour fermion couplings, with the number of random couplings being $N$. We outlined how a double-scaling limit of such a model can be obtained by taking $q,N\rightarrow\infty$ with the ratio $\frac{q}{N}$ held fixed. The triple-scaling limit of this model corresponds to further taking $\frac{q}{N}\rightarrow 0$. We argued that the low energy limit of this triple scaled local SYK model also leads to a Schwarzian theory. This is surprising at first since the model is very sparse and would not be expected to give rise to a Schwarzian theory since it would then mean that the OTOCs would have a maximal Lyapunov exponent. A naive expectation would be that since the model is a highly sparsed version of the full SYK model, one may possibly obtain a Lyapunov growth for OTOCs which would be much smaller than the maximum value of $2\pi T$. However, in the double scaled limit, the size of the fermion terms $q$ is also taken to grow proportional to $N$ thus making the model ‘highly’ connected in some sense, which lead to the maximal Lyapunov exponent. We have also argued that, by considering appropriate matter operators, that the OTOCs indeed have the maximal Lyapunov exponent.

For the local model under consideration, the existence of a maximal Lyapunov exponent suggests that the low energy limit involves “time-reparametrization” mode from an appropriate emergence and breaking of the conformal symmetry in the low energy limit as happens in the conventional SYK model. It would be interesting to understand the properties of this conformal fixed point, in particular, the scaling dimension of fermion operator at this fixed point. The sparse version of the SYK model have been studied in the literature Xu et al. (2020). In Xu et al. (2020), the authors show that for fixed $q$, in the large $N$ limit, the low energy fixed point for sparse model of SYK is the same as the fully random SYK for that value of $q$. This results holds true so long as the model is sufficiently random that the effective number of random couplings is $N^{\alpha}$ with $\alpha>1$. This analysis can be extended to a system with $\mathcal{O}(N)$ effective number of random couplings but by considering the large $q$ limit Anegawa et al. (2023). In particular, this is argued to be true even for $q\sim N^{\rho}$, where $0\leq\rho<\frac{1}{2}$. At $\rho=\frac{1}{2}$, the saddle point analysis in terms of $G,\Sigma$ can also be done with $\lambda=\frac{q^{2}}{N}\rightarrow 0$. To compare with the current investigation, the model we study has $\mathcal{O}{(N)}$ number of couplings with $q\sim\mathcal{O}{(N)}$. It would be interesting to understand the relation between the $G,\Sigma$ method for $q\sim{N}$ and contrast it the chord-diagram analysis presented above. Recall that, in the conventional SYK, the $G,\Sigma$ analysis can be done for both finite $q$ and large $q$ (even in the double-scaled regime). But for the Local SYK model, we do not expect a $G,\Sigma$ analysis to lead to a Schwarzian theory for finite $q$. However, since we find an analytically solvable limit of double scaling regime for local SYK, we can expect that a $G,\Sigma$ field analysis can also be done in this regime, where $q$ scales as $N$.

Finally, let us make one more comment. The analysis of the local SYK model considered in this paper in the double, triple-scaling limit is specific to this model. For example, if we further truncate the couplings of local SYK to half by setting $J_{i,i+1\dots}$ to vanish, say when $i\mod 2=0$, the fermion couplings that are retained commute with each other. So, even in the double-scaling limit ${\bf q}=1$. Thus, the analysis leading to the Schwarzian and the derivation of the Lyapunov exponent, in the triple-scaling limit, is no longer valid. So, even though, the number of random couplings is $\mathcal{O}{(N)}$, more precisely in this case $\frac{N}{2}$, the conclusions are significantly different. This shows that the details depend on the specific random couplings and not just on the number of them.

Acknowledgments