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Physical review. B./Physical review. B

Universal ratio in random matrix theory and chaotic-to-integrable transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models

Fadi Sun, Yu Yi-Xiang, Jinwu Ye, Wei-Jia Liu

Published Dec 17, 2021• 6 citations
Publisher

Abstract

We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both $q=4$ SYK with interaction $J$ and $q=2$ SYK with an interaction $K$ in type-I or ${(q=2)}^{2}$ SYK with an interaction $\sqrt{K}$ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $N$ even and $N$ odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $1/N$ expansion in the large $N$ limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at $N=\ensuremath{\infty}$, $1/N$ expansions at a finite $N$, and the CIT characterized by the RMT at a finite $N$: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.

DOI
10.1103/physrevb.104.235133

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We investigate chaotic to integrable transition in two types of hybrid SYK models, which contain both q=4q=4q=4 SYK with interaction JJJ and q=2q=2q=2 SYK with an interaction KKK in type-I or (q=2)2{(q=2)}^{2}(q=2)2 SYK with an interaction K\sqrt{K}K​ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both NNN even and NNN odd case. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy levels. We introduce a new universal ratio, which is the ratio of the next-nearest-neighbor (NNN) energy level spacing to characterize the RMT. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study chaotic to integrable transitions (CIT) in both types of hybrid SYK models. Some preliminary results on possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem and its dual version in the quantum chaotic side are given. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the 1/N1/N1/N expansion in the large NNN limit at a suitable temperature range. We stress the crucial differences between the quantum phase transition (QPT) characterized by renormalization groups at N=\ensuremath∞N=\ensuremath{\infty}N=\ensuremath∞, 1/N1/N1/N expansions at a finite NNN, and the CIT characterized by the RMT at a finite NNN: the former focus on the ground state and its low-energy excitations (edge states in the Fock space), the latter on excited states (bulk states in the Fock space). We also discuss eigenstate thermalization hypothesis (ETH)'s power on a quantum chaotic bulk state and its inability to encode the edge states. Comments on some previously related works are given. Some future perspectives, especially the failure of the Zamoloddchikov's c-theorem in 1d CFT RG flow are outlined.
We investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or (q=2)2(q=2)^2(q=2)2 SYK with an interaction $ \sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy level. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. After defining the quantum analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of quantum chaos. While the stability of quantum chaos in all the other cases are much more robust than the KAM theorem in the integrable side. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial difference between the quantum phase transition (QPT) characterized by renormalization groups at $ N=\infty $, $ 1/N $ expansions at a finite $ N $, and the CIT characterized by the RMT at a finite $ N $.
We investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in type-I or (q=2)2(q=2)^2(q=2)2 SYK with an interaction $ \sqrt{K} $ in type-II. These models include hybrid Majorana fermion, complex fermion, and bosonic SYK. For the Majorana fermion case, we discuss both $ N $ even and $ N $ odd case. We introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the KAM theorem. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10-fold way by random matrix theory (RMT) and also work out the degeneracy of each energy level. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio, then use both ratios to study Chaotic to Integrable transitions (CIT) in both types of hybrid SYK models. After defining the quantum analog of the KAM theorem in the context of RMT, we show that the KAM theorem holds when $ J/K \sim \sqrt{N} e^{-N} $ and shrinks exponentially fast in the thermodynamic limit. In the GOE or GSE case in type-I, we find a lower bound of the dual form of the KAM theorem which states the stability of quantum chaos. While the stability of quantum chaos in all the other cases are much more robust than the KAM theorem in the integrable side. We explore some intrinsic connections between the two complementary approaches to quantum chaos: the RMT and the Lyapunov exponent by the $ 1/N $ expansion in the large $ N $ limit at a suitable temperature range. We stress the crucial difference between the quantum phase transition (QPT) characterized by renormalization groups at $ N=\infty $, $ 1/N $ expansions at a finite $ N $, and the CIT characterized by the RMT at a finite $ N $.
Here we investigate possible quantum analog of Kolmogorov-Arnold-Moser (KAM) theorem in two types of hybrid SYK models which contain both $ q=4 $ SYK with interaction $ J $ and $ q=2 $ SYK with an interaction $ K $ in Type I or $ (q=2)^2 $ SYK with an interaction $ \sqrt{K} $ in Type II . These models include hybrid Majorana fermion, complex fermion and bosonic SYK. We first introduce a new universal ratio which is the ratio of the next nearest neighbour (NNN) energy level spacing to characterize the random matrix behaviours. We make exact symmetry analysis on the possible symmetry class of both types of hybrid SYK in the 10 fold way and also work out the degeneracy of each energy levels. We perform exact diagonalization to evaluate both the known NN ratio and the new NNN ratio. In Type I, as $ K/J $ changes, there is always a chaotic to non-chaotic transition (CNCT) from the GUE to Poisson in all the hybrid fermionic SYK models, but not the hybrid bosonic SYK model. In Type II, there are always CNCT from the corresponding GOE, GUE or GSE dictated by the symmetry of the $ q=4 $ SYK to the Poisson dictated by $ ( q=2 )^2 $ SYK. When the double degeneracy at the $ q=4 $ ( or $ (q=2)^2 $ ) side is broken by the $ q=2 $ ( or $ q=4 $ ) perturbation in Type I ( or Type II), the new NNN ratio can be most effectively to quantify the stability of quantum chaos ( or the KAM ). We compare the stability of quantum chaos and KAM theorem near the integrability in all these hybrid SYK models. We also discuss some possible connections between CNCT characterized by the random matrix theory and the quantum phase transitions (QPT) characterized by renormalization groups. Quantum chaos in both types of hybrid SYK models are also contrasted with that in the $ U(1)/Z_2 $ Dicke model in quantum optics.
The random matrix theory (RMT) can be used to classify both topological phases of matter and quantum chaos. We develop a systematic and transformative RMT to classify the quantum chaos in the colored Sachdev-Ye-Kitaev (SYK) model first introduced by Gross and Rosenhaus. Here we focus on the two-colored case and the four-colored case with a balanced number of Majorana fermions NNN. By identifying the maximal symmetries, the independent parity conservation sectors, the minimum (irreducible) Hilbert space, and especially the relevant antiunitary and unitary operators, we show that the color degrees of freedom lead to novel quantum chaotic behaviors. When NNN is odd, different symmetry operators need to be constructed to make the classifications complete. The two-colored case only shows the threefold Wigner-Dyson way, and the four-colored case shows the tenfold generalized Wigner-Dyson way which may also have nontrivial edge exponents. We also study two- and four-colored hybrid SYK models, which display many salient quantum chaotic features hidden in the corresponding pure SYK models. These features motivate us to develop a systematic RMT to study the energy level statistics of two or four uncorrelated random matrix ensembles. The exact diagonalizations are performed to study both the bulk energy level statistics and the edge exponents and to find excellent agreement with our exact maximal symmetry classifications. Our complete and systematic methods can be easily extended to study the generic imbalanced cases. They may be transferred to the classifications of colored tensor models, quantum chromodynamics with pairings across different colors, quantum black holes, and interacting symmetry protected (or enriched) topological phases.
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.
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.