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arXiv:1805.01470 · Journal of High Energy Physics

Conjecture on the butterfly velocity across a quantum phase transition

Matteo Baggioli, Bikash Padhi, Philip Phillips, Chandan Setty

Published Jul 1, 2018• 40 citations
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Abstract

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.

DOI
10.1007/jhep07(2018)049

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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.
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 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 In this paper, we make a systematical and in-depth exploration on the phase structure and the behaviors of butterfly velocity in an Einstein-Maxwell-dilaton-axions (EMDA) model. Depending on the model parameter, there are two kinds of mechanisms driving quantum phase transition (QPT) in this model. One is the infrared (IR) geometry to be renormalization group (RG) unstable, and the other is the strength of lattice deformation leading to some kind of bifurcating solution. We also find a novel QPT in the metal phases. The study on the behavior of the butterfly velocity crossing QPT indicates that the butterfly velocity or its first derivative exhibiting local extreme depends on the QPT mechanism. Further, the scaling behaviors of the butterfly velocity in the zero-temperature limit confirm that different phases are controlled by different IR geometries. Therefore, the butterfly velocity is a good probe to QPT and it also provides a possible way to study QPT beyond holography.
We propose the c-function as a new and accurate probe to detect the location of topological quantum critical points. As a direct application, we consider a holographic model which exhibits a topological quantum phase transition between a topologically trivial insulating phase and a gapless Weyl semimetal. The quantum critical point displays a strong Lifshitz-like anisotropy in the spatial directions, and the quantum phase transition does not follow the standard Landau paradigm. The c-function robustly shows a global feature at the quantum criticality and distinguishes, with great accuracy, the two separate zero temperature phases. Taking into account the relation of the c-function with the entanglement entropy, we conjecture that our proposal is a general feature of quantum phase transitions and that it is applicable beyond the holographic framework.
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}}_i Ei​ 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}}_i Ei​ 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.
We investigate the holographic entanglement entropy (HEE) of a strip geometry in four dimensional Q-lattice backgrounds, which exhibit metal-insulator transitions in the dual field theory. Remarkably, we find that the HEE always displays a peak in the vicinity of the quantum critical points. Our model provides the first direct evidence that the HEE can be used to characterize the quantum phase transition (QPT). We also conjecture that the maximization behavior of HEE at quantum critical points would be universal in general holographic models.