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arXiv:2109.13263 · Physical review. B./Physical review. B

Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d

Yichen Hu, Biao Lian

Published Mar 8, 2022• 8 citations
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Abstract

We construct and study a chiral Sachdev-Ye (SY) model consisting of $N$ chiral $\mathrm{SU}{(M)}_{1}$ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the $0+1\mathrm{d}$ quantum chaotic SY spin model into $1+1\mathrm{d}$ chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of $2+1\mathrm{d}$ gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers $N$ and $M$, which is integrable and decomposes into a chiral $\mathrm{SU}{(M)}_{N}$ WZW model and its coset with a different speed of light. When $N=M=2$, the model maps to a free Majorana fermion model. The second limit is the large $N$ and $M$ limit with random interactions, which is solvable to the leading $\frac{1}{NM}$ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.

DOI
10.1103/physrevb.105.125109

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We construct and study a chiral Sachdev-Ye (SY) model consisting of NNN chiral SU(M)1\mathrm{SU}{(M)}_{1}SU(M)1​ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the 0+1d0+1\mathrm{d}0+1d quantum chaotic SY spin model into 1+1d1+1\mathrm{d}1+1d chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of 2+1d2+1\mathrm{d}2+1d gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers NNN and MMM, which is integrable and decomposes into a chiral SU(M)N\mathrm{SU}{(M)}_{N}SU(M)N​ WZW model and its coset with a different speed of light. When N=M=2N=M=2N=M=2, the model maps to a free Majorana fermion model. The second limit is the large NNN and MMM limit with random interactions, which is solvable to the leading 1NM\frac{1}{NM}NM1​ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound 2\ensuremathπ/\ensuremathβ2\ensuremath{\pi}/\ensuremath{\beta}2\ensuremathπ/\ensuremathβ at temperature \ensuremathβ\ensuremath−1{\ensuremath{\beta}}^{\ensuremath{-}1}\ensuremathβ\ensuremath−1.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=\infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=\infty $ at any finite $ M $, nor at $ M=\infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.
We reinterpret the chiral U(N)U(N)U(N) Wess-Zumino-Witten (WZW) model at level k>1k>1k>1 in (1+1)(1+1)(1+1) dimensions as an interacting chiral metal in two space dimensions. In this reinterpretation, spatial translations along one of the spatial dimensions in the two-dimensional chiral metal arise from a generator of the U(N)U(N)U(N) symmetry of the WZW model. The WZW model at k=1k=1k=1 is equivalent to Balents and Fisher's free chiral metal [L. Balents and M. P. A. Fisher, Phys. Rev. Lett. 76, 2782 (1996)]. Here, the U(N)U(N)U(N) symmetry corresponds to the IR symmetry of a chiral Fermi gas with (half of a) Fermi surface, with NNN equal to the number of points on the Fermi surface. We argue that exactly solvable interacting generalizations occur for levels k>1k>1k>1. Importantly, these interacting chiral metals maintain the U(N)U(N)U(N) symmetry of the free system. We calculate two-point correlation functions of the single-particle fermion operator, the U(1)U(1)U(1) number density, and current operators in these theories for general kkk. We find that interactions (k>1k>1k>1) produce 1/N1/N1/N corrections to scaling of the single-particle fermion operator as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and renormalize the amplitudes of the density and current two-point functions. This construction illustrates the ersatz Fermi liquid proposal of Else et al. [Phys. Rev. X 11, 021005 (2021)].
The Sachdev-Ye-Kitaev model is a (0 + 1)-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with N Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large N limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a “butterfly velocity” describing the propagation of chaos in space. We mainly present results for a (1 + 1)-dimensional example, and discuss the general case near the end.
Quantum chaos is one of the distinctive features of the Sachdev-Ye-Kitaev (SYK) model, N Majorana fermions in 0+1 dimensions with infinite-range two-body interactions, which is attracting a lot of interest as a toy model for holography. Here we show analytically and numerically that a generalized SYK model with an additional one-body infinite-range random interaction, which is a relevant perturbation in the infrared, is still quantum chaotic and retains most of its holographic features for a fixed value of the perturbation and sufficiently high temperature. However, a chaotic-integrable transition, characterized by the vanishing of the Lyapunov exponent and spectral correlations given by Poisson statistics, occurs at a temperature that depends on the strength of the perturbation. We speculate about the gravity dual of this transition.
Many-body chaos has emerged as a powerful framework for understanding thermalization in strongly interacting quantum systems. While recent analytic advances have sharpened our intuition for many-body chaos in certain large N theories, it has proven challenging to develop precise numerical tools capable of exploring this phenomenon in generic Hamiltonians. To this end, we utilize massively parallel, matrix-free Krylov subspace methods to calculate dynamical correlators in the Sachdev-Ye-Kitaev model for up to N=60 Majorana fermions. We begin by showing that numerical results for two-point correlation functions agree at high temperatures with dynamical mean field solutions, while at low temperatures finite-size corrections are quantitatively reproduced by the exactly solvable dynamics of near extremal black holes. Motivated by these results, we develop a novel finite-size rescaling procedure for analyzing the growth of out-of-time-order correlators. Our procedure accurately determines the Lyapunov exponent, λ, across a wide range in temperatures, including in the regime where λ approaches the universal bound, λ=2π/β.