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arXiv:1609.07832 · Journal of High Energy Physics

Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

Yingfei Gu, Xiao-Liang Qi, Douglas Stanford

Published May 1, 2017• 430 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep05(2017)125

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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 an MBL transition. The model on a bipartite lattice has $N$ Majorana fermions with SYK interactions on each site of $A$-sublattice and $M$ free Majorana fermions on each site of $B$-sublattice, where $N$ and $M$ are large and finite. For $r$$\equiv$$M/N\! $$r_c$) distributions. Note that no subdiffusive phase intervenes between diffusive and MBL phases. Moreover, the critical exponent $\nu$=0, violating the Harris criterion. Our higher-dimensional SYK model may provide a promising arena to explore exotic MBL transitions.

•Mar 6, 2017•10 citations
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 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.
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 describes Majorana fermions with random interaction, which displays many interesting properties such as non-Fermi liquid behavior, quantum chaos, emergent conformal symmetry and holographic duality. Here we consider a SYK model or a chain of SYK models with N Majorana fermion modes coupled to another SYK model with N 2 Majorana fermion modes, in which the latter has many more degrees of freedom and plays the role as a thermal bath. For a single SYK model coupled to the thermal bath, we show that although the Lyapunov exponent is still proportional to temperature, it monotonically decreases from 2π/β (β = 1/(k B T), T is temperature) to zero as the coupling strength to the thermal bath increases. For a chain of SYK models, when they are uniformly coupled to the thermal bath, we show that the butterfly velocity displays a crossover from a T \sqrt{T} T​ -dependence at relatively high temperature to a linear T-dependence at low temperature, with the crossover temperature also controlled by the coupling strength to the thermal bath. If only the end of the SYK chain is coupled to the thermal bath, the model can introduce a spatial dependence of both the Lyapunov exponent and the butterfly velocity. Our models provide canonical examples for the study of thermalization within chaotic models.
We compute the transport and chaos properties of lattices of quantum Sachdev-Ye-Kitaev islands coupled by single fermion hopping, and with the islands coupled to a large number of local, low energy phonons. We find two distinct regimes of linear-in-temperature (T)(T)(T) resistivity, and describe the crossover between them. When the electron-phonon coupling is weak, we obtain the ``incoherent metal'' regime, where there is near-maximal chaos with front propagation at a butterfly velocity vB{v}_{B}vB​, and the associated diffusivity ${D}_{\mathrm{chaos}}={v}_{B}^{2}/(2\ensuremath{\pi}T)$ closely tracks the energy diffusivity. On the other hand, when the electron-phonon coupling is strong, and the linear resistivity is largely due to near-elastic scattering of electrons off nearly free phonons, we find that the chaos is far from maximal and spreads diffusively. We also describe the crossovers to low TTT regimes where the electronic quasiparticles are well defined.
We construct a new higher dimensional SYK model with complex fermions on bipartite lattices. As an extension of the original zero-dimensional SYK model, we focus on the one-dimension case, and similar Hamiltonian can be obtained in higher dimensions. This model has a conserved U(1) fermion number Q and a conjugate chemical potential μ. We evaluate the thermal and charge diffusion constants via large q expansion at low temperature limit. The results show that the diffusivity depends on the ratio of free Majorana fermions to Majorana fermions with SYK interactions. The transport properties and the butterfly velocity are accordingly calculated at low temperature. The specific heat and the thermal conductivity are proportional to the temperature. The electrical resistivity also has a linear temperature dependence term.
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.