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arXiv:1812.11060 · Physical review. D/Physical review. D.

Impact of irrelevant deformations on thermodynamics and transport in holographic quantum critical states

Richard A. Davison, Simon A. Gentle, Blaise Goutéraux

Published Oct 29, 2019• 27 citations
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Abstract

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 ${v}_{s}$, and in this case the relation $D={v}_{s}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}$ holds exactly in the limit ${\ensuremath{\tau}}_{\mathrm{eq}}T\ensuremath{\gg}1$.

DOI
10.1103/physrevd.100.086020

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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/\ensuremathτeq\ensuremath≪T1/{\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\ll}T1/\ensuremathτeq​\ensuremath≪T. At very late times, transport is diffusion-dominated, with a diffusivity that can be written simply in terms of \ensuremathτeq{\ensuremath{\tau}}_{\mathrm{eq}}\ensuremathτeq​ and the butterfly velocity, D\ensuremath∼vB2\ensuremathτeqD\ensuremath{\sim}{v}_{B}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}D\ensuremath∼vB2​\ensuremathτeq​. 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=vs2\ensuremathτeqD={v}_{s}^{2}{\ensuremath{\tau}}_{\mathrm{eq}}D=vs2​\ensuremathτeq​ holds exactly in the limit \ensuremathτeqT\ensuremath≫1{\ensuremath{\tau}}_{\mathrm{eq}}T\ensuremath{\gg}1\ensuremathτeq​T\ensuremath≫1.
The dissipative dynamics of strongly interacting systems are often characterized by the timescale set by the inverse temperature ${\ensuremath{\tau}}_{P}\ensuremath{\sim}\ensuremath{\hbar}/({k}_{B}T)$. We show that near a class of strongly interacting quantum critical points that arise in the infrared limit of translationally invariant holographic theories, there is a collective excitation (a quasinormal mode of the dual black hole spacetime) whose lifetime ${\ensuremath{\tau}}_{\mathrm{eq}}$ is parametrically longer than ${\ensuremath{\tau}}_{P}$: ${\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\gg}{T}^{\ensuremath{-}1}$. The lifetime is enhanced due to its dependence on a dangerously irrelevant coupling that breaks the particle-hole symmetry and the invariance under Lorentz boosts of the quantum critical point. The thermal diffusivity (in units of the butterfly velocity) is anomalously large near the quantum critical point and is governed by ${\ensuremath{\tau}}_{\mathrm{eq}}$ rather than ${\ensuremath{\tau}}_{P}$. We conjecture that 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 $T{\ensuremath{\tau}}_{\mathrm{eq}}\ensuremath{\gg}1$. While scale invariance is broken, a generalized scaling theory still holds provided that the dependence of observables on the dangerously irrelevant coupling is incorporated. Our work further underlines the connection between dangerously irrelevant deformations and slow equilibration.
Hydrodynamics provides a universal description of interacting quantum field theories at sufficiently long times and wavelengths, but breaks down at scales dependent on microscopic details of the theory.In the vicinity of a quantum critical point, it is expected that some aspects of the dynamics are universal and dictated by properties of the critical point.We use gauge-gravity duality to investigate the breakdown of diffusive hydrodynamics in two low-temperature states dual to black holes with AdS 2 horizons, which exhibit quantum critical dynamics with an emergent scaling symmetry in time.We find that the breakdown is characterized by a collision between the diffusive pole of the retarded Green's function with a pole associated to the AdS 2 region of the geometry, such that the local equilibration time is set by infrared properties of the theory.The absolute values of the frequency and wave vector at the collision (ω eq and k eq ) provide a natural characterization of all the low-temperature diffusivities D of the states via D ¼ ω eq =k 2 eq , where ω eq ¼ 2πΔT is set by the temperature T and the scaling dimension Δ of an operator of the infrared quantum critical theory.We confirm that these relations are also satisfied in a Sachdev-Ye-Kitaev chain model in the limit of strong interactions.Our work paves the way toward a deeper understanding of transport in quantum critical phases.
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.
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.
Quantum phase transitions occur strictly at zero temperature and are accessed by varying a physical parameter such as a magnetic field or chemical potential. While these transitions themselves are experimentally inaccessible, the existence of a zero temperature quantum critical point has remarkable effects on the dynamical properties of interacting quantum many-body systems at finite temperatures. In this thesis, we study dynamics in four strongly interacting many-body systems near quantum critical points in the context of NMR relaxation, quantum quenches, many-body quantum chaos, and charge diffusion. In chapter 1, we begin by revisiting the issue of NMR relaxation in the quantum disordered phase of Ising spin chains. We find that the imaginary part of the spin susceptibility at low frequences is dominated by rare quantum processes which leads to an NMR relaxation rate that scales as 1/T1∼exp⁡(−2ΔT)1/T_{1}\sim \exp(-\frac{2\Delta}{T})1/T1​∼exp(−T2Δ​) near the quantum critical point rather than the previously conjectured 1/T1∼exp⁡(−ΔT)1/T_{1}\sim \exp(-\frac{\Delta}{T})1/T1​∼exp(−TΔ​). These results resolve certain discrepancies between the energy scales measured with different experimental probes in the quantum disordered paramagnetic phase of the Ising chain system CoNb2O6\text{CoNb}_{2}\text{O}_{6}CoNb2​O6​. In chapter 2, we study the non-equilibrium dynamics of Sachdev-Ye-Kitaev (SYK) models with qqq body interactions after a quantum quench, considering a quench protocol which drives the system from an initial state with long-lived quasiparticle excitations to a final state lacking any long-lived quasiparticle excitations. For q=4q=4q=4, we find that the timescale for the system to locally thermalize τeq∼ℏkBT\tau_{eq}\sim\frac{\hbar}{k_{B}T}τeq​∼kB​Tℏ​ and for q→∞q\rightarrow \inftyq→∞ we find that the system thermalizes instantaneously. In chapter 3, we study the real time dynamics of quantum electrodynamics in 2+1 dimensions with NfN_{f}Nf​ flavors of fermions. We focus on thermalization and the onset of many-body quantum chaos and compute the quantum Lyapunov exponent in a 1Nf\frac{1}{N_{f}}Nf​1​ expansion. We briefly comment on chaos, locality, and gauge invariance. Finally, in Chapter 4 we consider whether the butterfly velocity - a speed at which quantum information propagates - may provide a fundamental bound on diffusion constants in dirty incoherent metals by computing the charge diffusion constant and the butterfly velocity in charge-neutral holographic matter with long wavelength ``hydrodynamic" disorder. In this limit, we find that the butterfly velocity does not set a sharp lower bound for the charge diffusion constant.