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arXiv:1707.02843 · Journal of High Energy Physics

Holographic butterfly effect and diffusion in quantum critical region

Yi Ling, Zhuo-Yu Xian

Published Sep 1, 2017• 28 citations
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Abstract

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.

DOI
10.1007/jhep09(2017)003

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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.
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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 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 study thermodynamic and transport observables of quantum critical states that arise in the infrared limit of holographic renormalization group flows. Although these observables are expected to exhibit quantum critical scaling, there are a number of cases in which their frequency and temperature dependences are in apparent contradiction with scaling theories. We study two different classes of examples and show in both cases that the apparent breakdown of scaling is a consequence of the dependence of observables on an irrelevant deformation of the quantum critical state. By assigning scaling dimensions to the near-horizon observables, we formulate improved scaling theories that are completely consistent with all explicit holographic results once the dependence on the dangerously irrelevant coupling is properly accounted for. In addition to governing thermodynamic and transport phenomena in these states, we show that the dangerously irrelevant coupling also controls late-time equilibration, which occurs at a rate parametrically slower than the temperature $1/{\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\ll}T$. At very late times, transport is diffusion-dominated, with a diffusivity that can be written simply in terms of ${\ensuremath{\tau}}_{\mathrm{eq}}$ and the butterfly velocity, $D\ensuremath{\sim}{v}_{B}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}$. We conjecture that in such cases there exists a long-lived, propagating collective mode with velocity vs{v}_{s}vs​, and in this case the relation $D={v}_{s}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}$ holds exactly in the limit ${\ensuremath{\tau}}_{\mathrm{eq}}T\ensuremath{\gg}1$.
Abstract The region near a critical point is studied using holographic models of second-order phase transitions. In a previous paper, we argued that the quantum circuit complexity of the vacuum (C 0) is the largest at the critical point. When deforming away from the critical point by a term the complexity C ( τ ) has a piece non-analytic in τ, namely C 0 − C ( τ ) ∼ | τ − τ c | ν ( d − 1 ) + a n a l y t i c . Here, as usual, ν = 1 d − Δ and ξ is the correlation length ξ ∼ | τ − τ c | − ν and there are possible logarithmic corrections to this expression. That was derived using numerical results for the Bose–Hubbard model and general scaling considerations. In this paper, we show that the same is valid in the case of holographic complexity providing evidence that the results are universal, and at the same time providing evidence for holographic computations of complexity.