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arXiv:1703.02051 · Physical Review Letters

Solvable Sachdev-Ye-Kitaev Models in Higher Dimensions: From Diffusion to Many-Body Localization

Shao-Kai Jian, Hong Yao

Published Nov 15, 2017• 119 citations
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Abstract

Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits a MBL transition. The model on a bipartite lattice has N Majorana fermions with SYK interactions on each site of the A sublattice and M free Majorana fermions on each site of the B sublattice, where N and M are large and finite. For r≡M/N<r_{c}=1, it describes a diffusive metal exhibiting maximal chaos. Remarkably, its diffusive constant D vanishes [D∝(r_{c}-r)^{1/2}] as r→r_{c}, implying a dynamical transition to a MBL phase. It is further supported by numerical calculations of level statistics which changes from Wigner-Dyson (r<r_{c}) to Poisson (r>r_{c}) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent ν=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.

DOI
10.1103/physrevlett.119.206602

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Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits a MBL transition. The model on a bipartite lattice has N Majorana fermions with SYK interactions on each site of the A sublattice and M free Majorana fermions on each site of the B sublattice, where N and M are large and finite. For r≡M/N<r_{c}=1, it describes a diffusive metal exhibiting maximal chaos. Remarkably, its diffusive constant D vanishes [D∝(r_{c}-r)^{1/2}] as r→r_{c}, implying a dynamical transition to a MBL phase. It is further supported by numerical calculations of level statistics which changes from Wigner-Dyson (r<r_{c}) to Poisson (r>r_{c}) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent ν=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.
Many aspects of many-body localization (MBL) transitions remain elusive so far. Here, we propose a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model and show that it exhibits an MBL transition. The model on a bipartite lattice has NNN Majorana fermions with SYK interactions on each site of AAA-sublattice and MMM free Majorana fermions on each site of BBB-sublattice, where NNN and MMM are large and finite. For rrr≡\equiv≡$M/N\! $rcr_crc​) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent ν\nuν=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.
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.
Many key features of the higher-dimensional Sachdev-Ye-Kitaev (SYK) model at finite NNN remain unknown. Here we study the SYK chain consisting of $N\phantom{\rule{4pt}{0ex}}(N\ensuremath{\ge}2)$ fermions per site with random interactions and hoppings between neighboring sites. In the limit of vanishing SYK interactions, from both supersymmetric field theory analysis and numerical calculations we find that the random hopping model exhibits Anderson localization at finite NNN, irrespective of the parity of NNN. Moreover, the localization length scales linearly with NNN, implying no Anderson localization only at $N=\ensuremath{\infty}$. For finite SYK interaction JJJ, from the exact diagonalization we show that there is a dynamic phase transition between many-body localization and thermal diffusion as JJJ exceeds a critical value Jc{J}_{c}Jc​. In addition, we find that the critical value Jc{J}_{c}Jc​ decreases with the increase of NNN, qualitatively consistent with the analytical result of ${J}_{c}/t\ensuremath{\propto}\frac{1}{{N}^{5/2}logN}$ derived from the weakly interacting limit.
The SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
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.
The Sachdev-Ye-Kitaev (SYK) model is a concrete model for a non-Fermi liquid with maximally chaotic behavior in (0+1)(0+1)(0+1) dimensions. In order to gain some insights into real materials in higher dimensions where fermions could hop between different sites, here we consider coupling a SYK lattice by constant hopping. We call this the dispersive SYK model. Focusing on (1+1)(1+1)(1+1)-dimensional homogeneous hopping, by either tuning the temperature or the relative strength of the random interaction (hopping) and constant hopping, we find a crossover between a dispersive metal to an incoherent metal, where the dynamic exponent zzz changes from 1 to $\ensuremath{\infty}$. We study the crossover by calculating the spectral function, charge density correlator, and the Lyapunov exponent. We further find the Lyapunov exponent becomes larger when the chemical potential is tuned to approach a van Hove singularity because of the large density of states near the Fermi surface. The effect of the topological nontrivial bands is also discussed.