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arXiv:1702.08462 · SciPost Physics

Energy diffusion and the butterfly effect in inhomogeneous Sachdev-Ye-Kitaev chains

Yingfei Gu, Andrew Lucas, Xiao-Liang Qi

Published May 28, 2017• 95 citations
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Abstract

We compute the energy diffusion constant D D , Lyapunov time \tau_{\textsc{l}} and butterfly velocity v_{\textsc{b}} in an inhomogeneous chain of coupled Majorana Sachdev-Ye-Kitaev (SYK) models in the large N N and strong coupling limit. We find D\le v_{\textsc{b}}^2 \tau_{\textsc{l}} from a combination of analytical and numerical approaches. Our example necessitates the sharpening of postulated transport bounds based on quantum chaos.

DOI
10.21468/scipostphys.2.3.018

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We compute the energy diffusion constant D D , Lyapunov time \tau_{\textsc{l}} and butterfly velocity v_{\textsc{b}} in an inhomogeneous chain of coupled Majorana Sachdev-Ye-Kitaev (SYK) models in the large N N and strong coupling limit. We find D\le v_{\textsc{b}}^2 \tau_{\textsc{l}} from a combination of analytical and numerical approaches. Our example necessitates the sharpening of postulated transport bounds based on quantum chaos.
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.
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 study the nonequilibrium dynamics of a one-dimensional complex Sachdev-Ye-Kitaev chain by directly solving for the steady state Green's functions in terms of small perturbations around their equilibrium values. The model exhibits non-Fermi-liquid behavior without quasiparticles and features diffusive propagation of both energy and charge. We explore the thermoelectric transport properties of this system by imposing uniform temperature and chemical potential gradients. We then expand the conserved charges and their associated currents to leading order in these gradients, which we can compute numerically and analytically for different parameter regimes. This allows us to extract the full temperature and chemical potential dependence of the transport coefficients. In particular, we uncover that the diffusivity matrix takes on a simple form in various limits and leads to simplified Einstein relations. At low temperatures, we also recover a previously known result for the Wiedemann-Franz ratio. Furthermore, we establish a relationship between diffusion and quantum chaos by showing that the diffusivity eigenvalues are upper bounded by the chaos propagation rate at all temperatures. Our work showcases an important example of an analytically tractable calculation of transport properties in a strongly interacting quantum system and reveals a more general purpose method for addressing strongly coupled transport.
We develop a general framework for studying the equilibrium and non-equilibrium properties of arbitrary networks of Sachdev-Ye-Kitaev clusters coupled to thermal baths. We proceed to apply this technique to the problem of energy transport, which is known to be diffusive due to the strange metal behavior of these models. We use the external baths to impose a temperature gradient in the system and study the emerging non-equilibrium steady state using the Schwinger-Keldysh formalism. We consider two different configurations for the baths, implementing either a boundary or bulk driving, and show that the latter leads to a significantly faster convergence to the steady state. This setup allows us to compute both the temperature and frequency dependence of the diffusion constant. At low temperatures, our results agree perfectly with the previously known values for diffusivity in the conformal limit. We also establish a relationship between energy transport and quantum chaos by showing that the diffusion constant is upper bounded by the chaos propagation rate at all temperatures. Moreover, we find a simple analytical form for the non-equilibrium Green's functions in the linear response regime and use it to derive exact closed-form expressions for the diffusion constant in various limits. We mostly focus on uniform one-dimensional chains, but we also discuss higher-dimensional generalizations.
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.
We study chaos in a classical limit of the Sachdev-Ye-Kitaev (SYK) model obtained in a suitably defined large-SSS limit. The low-temperature Lyapunov exponent is found to depend linearly on temperature, with a slope that is parametrically different than in the quantum case: It is proportional to N/SN/SN/S. The classical dynamics can be understood as the rotation of an NNN-dimensional body with a random inertia tensor, corresponding to the random couplings of the SYK Hamiltonian. This allows us to find an extensive number of fixed points, corresponding to the body's principal axes of rotation. The thermodynamics is mapped to the ppp-spin model with p=2p=2p=2, which exhibits a spin glass phase at low temperature whose presence does not preclude the existence of chaos.