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arXiv:1905.04203 · Theoretical and Mathematical Physics

Revealing Nonperturbative Effects in the SYK Model

I. Ya. Aref’eva, I. V. Volovich, M. A. Khramtsov

Published Nov 1, 2019• 6 citations
arXivHTMLPDF

Abstract

In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.

DOI
10.1134/s0040577919110059

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In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.

Työväentutkimus Vuosikirja•Jan 1, 2017•0 citations
In the large-N limit, we study saddle points of two SYK chains coupled by an interaction that is nonlocal in Euclidean time. We study the free model with the order of the fermionic interaction q = 2 analytically and also investigate the model with interaction in the case q = 4 numerically. We show that in both cases, there is a nontrivial phase structure with an infinite number of phases. Each phase corresponds to a saddle point in the noninteracting two-replica SYK. The nontrivial saddle points have a nonzero value of the replica-nondiagonal correlator in the sense of quasiaveraging if the coupling between replicas is turned off. The nonlocal interaction between replicas thus provides a protocol for turning the nonperturbatively subleading effects in SYK into nonequilibrium configurations that dominate at large N. For comparison, we also study two SYK chains with local interaction for q = 2 and q = 4 . We show that the q= 2 model has a similar phase structure, while the phase structure differs in the q = 4 model, dual to the traversable wormhole.
Abstract We study the SYK model in the large N limit beyond the replica-diagonal approximation. First we show that there are exact replica-nondiagonal solutions of the saddle point equations for q = 2 for any finite replica number M.In the interacting q = 4 case we are able to construct the numerical solutions, which are in one-to-one correspondence to the analytic solutions of the quadratic model. These solutions are singular in the M → 0 limit in both quadratic and quartic interaction cases. The calculations of the on-shell action at finite integer M show that the nondiagonal replica-symmetric saddles are subleading in both quadratic and quartic cases. We also study replica-nondiagonal solutions of the SYK in the strong coupling limit. For arbitrary q we show that besides the usual solutions of the replica-diagonal saddle point equations in the conformal limit, there are also replica-nondiagonal solutions for any value of M (including zero). The specific configurations that we study, have factorized time and replica dependencies. The corresponding saddle point equations are separable at strong coupling, and can be solved using the Parisi ansatz from spin glass theory. We construct the solutions which correspond to the replica-symmetric case and to one-step replica symmetry breaking. We compute the regularizized free energy on these solutions in the limit of zero replicas. It is observed that there are nondiagonal solutions with the regularized free energy lower than that of the standard diagonal conformal solution.
Abstract We investigate the nonperturbative effects in the Sadchev–Ye–Kitaev (SYK) model at large NNN . The replica-nondiagonal solutions are considered in the SYK model with an integer number of replicas M>1M > 1M>1 . An important property of such solutions is the fact that they realize spontaneous breaking of the time translation group on MMM replicas of the thermal circle. It is demonstrated in this study that this spontaneous symmetry breaking can be described in terms of the Bogolyubov quasi-averages, if we consider the system of nonlocally interacting replicas of the SYK model. The inclusion of the interaction of the replicas in this system transforms the subleading replica-nondiagonal saddle points from a pure SYK to thermodynamic phases, thereby generating a nontrivial phase structure. We investigate this phase structure exactly analytically in the SYK2 model and numerically in the SYK4 model.
We study the saddle points of the SYK model, formulated in terms of the replica bilocal fields, beyond the replica-diagonal assumption. We find a family of replica-nondiagonal saddle points in the IR limit, where the saddle point equations are separable. We use the Parisi ansatz to find the replicanondiagonal solutions and take the replica limit. The free energy on these solutions is computed, and we find that some of the replica-nondiagonal saddles have lower free energy than the replica-diagonal saddle point.
In this note we study the SYK model with one time point, recently considered by Saad, Shenker, Stanford, and Yao. Working in a collective field description, they derived a remarkable identity: the square of the partition function with fixed couplings is well approximated by a ``wormhole'' saddle plus a ``pair of linked half-wormholes'' saddle. It explains factorization of decoupled systems. Here, we derive an explicit formula for the half-wormhole contribution. It is expressed through a hyperpfaffian of the tensor of SYK couplings. We then develop a perturbative expansion around the half-wormhole saddle. This expansion truncates at a finite order and gives the exact answer. The last term in the perturbative expansion turns out to coincide with the wormhole contribution. In this sense the wormhole saddle in this model does not need to be added separately, but instead can be viewed as a large fluctuation around the linked half-wormholes.
Abstract One of the most promising routes to non-fermi liquids and strange metals has been through SYK models (Sachdev 2010 Phys. Rev. Lett. 105 151602) which necessarily involve large flavor degrees of freedom and interactions with imposed disorder. We introduce an interacting model of nonlocal spinless fermions in which large flavor and effective disorder emerge spontaneously in the extreme nonlocal limit. This model may be thought of as an expansion in a dimensionless ‘locality scale,’ α, with the limit α → 0 recovering a conventional local action. For finite α, the resulting interacting nonlocal action exhibits a large number of low-energy degrees of freedom—proportional to α—and the structure factor mediating the local interactions between these fermions is effectively random for large α. In one dimension, with finite α, we show that interactions are marginal to the one loop level (as they are in the conventional local case), preserving a gapless phase. At large α the interaction strength becomes comparable to the bandwidth and we analyze this large flavor limit in a manner similar to the diagrammatic approach to SYK. As α is increased we argue that the gapless phase established by conventional RG possibly crosses over to a gapless SYK phase, although the ‘melon’ diagrams are not exclusively dominant. We speculate on how this model might arise physically from an S-matrix connecting pure excited states—rather than the vacuum—to access finite temperature interacting fermions.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.