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arXiv:1708.08822 · Journal of High Energy Physics

Thermal diffusivity and butterfly velocity in anisotropic Q-lattice models

Hyun-Sik Jeong, Yongjun Ahn, Dujin Ahn, Chao Niu, Wei-Jia Li, Keun-young Kim A

Published Jan 1, 2018• 53 citations
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Abstract

We study a relation between the thermal diffusivity (D T ) and two quantum chaotic properties, Lyapunov time (τ L ) and butterfly velocity (v B ) in strongly correlated systems by using a holographic method. Recently, it was shown that $$ {\mathbb{E}}_i:={D}_{T,i}/\left({v}_{{}^{B,i}}^2{\tau}_L\right)\left(i=x,y\right) $$ is universal in the sense that it is determined only by some scaling exponents of the IR metric in the low temperature limit regardless of the matter fields and ultraviolet data. Inspired by this observation, by analyzing the anisotropic IR scaling geometry carefully, we find the concrete expressions for $$ {\mathbb{E}}_i $$ in terms of the critical dynamical exponents z i in each direction, $$ {\mathbb{E}}_i={z}_i/2\left({z}_i-1\right) $$ . Furthermore, we find the lower bound of $$ {\mathbb{E}}_i $$ is always 1/2, which is not affected by anisotropy, contrary to the η/s case. However, there may be an upper bound determined by given fixed anisotropy.

DOI
10.1007/jhep01(2018)140

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We study a relation between the thermal diffusivity (D T ) and two quantum chaotic properties, Lyapunov time (τ L ) and butterfly velocity (v B ) in strongly correlated systems by using a holographic method. Recently, it was shown that Ei:=DT,i/(vB,i2τL)(i=x,y){\mathbb{E}}_i:={D}_{T,i}/\left({v}_{{}^{B,i}}^2{\tau}_L\right)\left(i=x,y\right)Ei​:=DT,i​/(vB,i2​τL​)(i=x,y) is universal in the sense that it is determined only by some scaling exponents of the IR metric in the low temperature limit regardless of the matter fields and ultraviolet data. Inspired by this observation, by analyzing the anisotropic IR scaling geometry carefully, we find the concrete expressions for Ei{\mathbb{E}}_iEi​ in terms of the critical dynamical exponents z i in each direction, Ei=zi/2(zi−1){\mathbb{E}}_i={z}_i/2\left({z}_i-1\right)Ei​=zi​/2(zi​−1) . Furthermore, we find the lower bound of Ei{\mathbb{E}}_iEi​ is always 1/2, which is not affected by anisotropy, contrary to the η/s case. However, there may be an upper bound determined by given fixed anisotropy.
When the Lyapunov exponent λL in a quantum chaotic system saturates the bound λL ≤ 2πk B T , it is proposed that this system has a holographic dual described by a gravity theory. In particular, the butterfly effect as a prominent phenomenon of chaos can ubiquitously exist in a black hole system characterized by a shockwave solution near the horizon. In this paper we propose that the butterfly velocity can be used to diagnose quantum phase transition (QPT) in holographic theories. We provide evidences for this proposal with an anisotropic holographic model exhibiting metal-insulator transitions (MIT), in which the derivatives of the butterfly velocity with respect to system parameters characterizes quantum critical points (QCP) with local extremes in zero temperature limit. We also point out that this proposal can be tested by experiments in the light of recent progress on the measurement of out-of-time-order correlation function (OTOC).
We study the thermal diffusivity D[subscript T] in models of metals without quasiparticle excitations (“strange metals”). The many-body quantum chaos and transport properties of such metals can be efficiently described by a holographic representation in a gravitational theory in an emergent curved spacetime with an additional spatial dimension. We find that at generic infrared fixed points D[subscript T] is always related to parameters characterizing many-body quantum chaos: the butterfly velocity v[subscript B] and Lyapunov time τ[subscript L] through D[subscript T]∼v[subscript B][superscript 2]τ[subscript L]. The relationship holds independently of the charge density, periodic potential strength, or magnetic field at the fixed point. The generality of this result follows from the observation that the thermal conductivity of strange metals depends only on the metric near the horizon of a black hole in the emergent spacetime and is otherwise insensitive to the profile of any matter fields.
We investigate the butterfly effect and charge diffusion near the quantum phase transition in holographic approach. We argue that their criticality is controlled by the holographic scaling geometry with deformations induced by a relevant operator at finite temperature. Specifically, in the quantum critical region controlled by a single fixed point, the butterfly velocity decreases when deviating from the critical point. While, in the non-critical region, the behavior of the butterfly velocity depends on the specific phase at low temperature. Moreover, in the holographic Berezinskii-Kosterlitz-Thouless transition, the universal behavior of the butterfly velocity is absent. Finally, the tendency of our holographic results matches with the numerical results of Bose-Hubbard model. A comparison between our result and that in the O(N ) nonlinear sigma model is also given.
Abstract We study an anisotropic holographic bottom-up model displaying a quantum phase transition (QPT) between a topologically trivial insulator and a non-trivial Weyl semimetal phase. We analyze the properties of quantum chaos in the quantum critical region. We do not find any universal property of the Butterfly velocity across the QPT. In particular it turns out to be either maximized or minimized at the quantum critical point depending on the direction of propagation. We observe that instead of the butterfly velocity, it is the dimensionless information screening length that is always maximized at a quantum critical point. We argue that the null-energy condition (NEC) is the underlying reason for the upper bound, which now is just a simple combination of the number of spatial dimensions and the anisotropic scaling parameter.
We study the chaotic dynamics in a classical many-body system of interacting spins on the kagome lattice. We characterize many-body chaos via the butterfly effect as captured by an appropriate out-of-time-ordered commutator. Due to the emergence of a spin-liquid phase, the chaotic dynamics extends all the way to zero temperature. We thus determine the full temperature dependence of two complementary aspects of the butterfly effect: the Lyapunov exponent, $\ensuremath{\mu}$, and the butterfly speed, vb{v}_{b}vb​, and study their interrelations with usual measures of spin dynamics such as the spin-diffusion constant, DDD, and spin-autocorrelation time, $\ensuremath{\tau}$. We find that they all exhibit power-law behavior at low temperature, consistent with scaling of the form $D\ensuremath{\sim}{v}_{b}^{2}/\ensuremath{\mu}$ and ${\ensuremath{\tau}}^{\ensuremath{-}1}\ensuremath{\sim}T$. The vanishing of $\ensuremath{\mu}\ensuremath{\sim}{T}^{0.48}$ is parametrically slower than that of the corresponding quantum bound, $\ensuremath{\mu}\ensuremath{\sim}T$, raising interesting questions regarding the semiclassical limit of such spin systems.
Recently, it has been proposed that the butterfly velocity — a speed at which quantum information propagates — may provide a fundamental bound on diffusion constants in dirty incoherent metals. We analytically compute the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength “hydrodynamic” disorder in a single spatial direction. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.