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arXiv:1203.4908 · Physical Review Letters

Hawking Radiation and Nonequilibrium Quantum Critical Current Noise

Julian Sonner, Andrew G. Green

Published Aug 27, 2012• 72 citations
arXivHTMLPDF

Abstract

The dynamical scaling of quantum critical systems in thermal equilibrium may be inherited in the driven steady state, leading to universal out-of-equilibrium behavior. This attractive notion has been demonstrated in just a few cases. We demonstrate how holography-a mapping between the quantum critical system and a gravity dual-provides an illuminating perspective and new results. Nontrivial out-of-equilibrium universality is particularly apparent in current noise, which is dual to Hawking radiation in the gravitational system. We calculate this in a two-dimensional system driven by a strong in-plane electric field and deduce a universal scaling function interpolating between previously established equilibrium and far-from-equilibrium current noise. Since this applies at all fields, out-of-equilibrium experiments no longer require very high fields for comparison with theory.

DOI
10.1103/physrevlett.109.091601

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The dynamical scaling of quantum critical systems in thermal equilibrium may be inherited in the driven steady state, leading to universal out-of-equilibrium behavior. This attractive notion has been demonstrated in just a few cases. We demonstrate how holography-a mapping between the quantum critical system and a gravity dual-provides an illuminating perspective and new results. Nontrivial out-of-equilibrium universality is particularly apparent in current noise, which is dual to Hawking radiation in the gravitational system. We calculate this in a two-dimensional system driven by a strong in-plane electric field and deduce a universal scaling function interpolating between previously established equilibrium and far-from-equilibrium current noise. Since this applies at all fields, out-of-equilibrium experiments no longer require very high fields for comparison with theory.
A system driven in the vicinity of its critical point by varying a relevant field in an arbitrary function of time is a generic system that possesses a long relaxation time compared with the driving time scale and thus represents a large class of nonequilibrium systems. For such a manifestly nonlinear nonequilibrium strongly fluctuating system, we show that there exists universal nonequilibrium critical behavior that is incredibly well described by its equilibrium critical properties. A dynamic renormalization-group theory is developed to account for the behavior. The weak driving may give rise to several time scales depending on its form and thus rich nonequilibrium phenomena of various regimes and their crossovers, negative susceptibilities, as well as a violation of fluctuation-dissipation theorem and hysteresis. An initial condition that can be in either equilibrium or nonequilibrium but has longer correlations than the driving scales also results in a unique regime and complicates the situation. The implication of the results on measurement is also discussed. The theory may shed light on the study of other nonequilibrium systems and even nonlinear science.
Quantum criticality provides an important route to revealing universal nonequilibrium behavior. A canonical example of a critical point is the Bose-Hubbard model, which we study under the application of an electric field. A Boltzmann transport formalism and $\ensuremath{\epsilon}$ expansion are used to obtain the nonequilibrium conductivity and current noise. This approach allows us to explicitly identify how a universal nonequilibrium steady state is maintained, by identifying the rate-limiting step in balancing Joule heating and dissipation to a heat bath. It also reveals that the nonequilibrium distribution function is very far from a thermal distribution.
We investigate bias-driven non-equilibrium quantum phase transitions in a paradigmatic quantum-transport setup: an interacting quantum dot coupled to non-interacting metallic leads. Using the Random Phase Approximation, which is exact in the limit of a large number of dot levels, we map out the zero-temperature non-equilibrium phase diagram as a function of interaction strength and applied bias. We focus our analysis on the behavior of the charge susceptibility and the current noise in the vicinity of the transition. Remarkably, despite the intrinsically non-equilibrium nature of the steady state, critical charge fluctuations admit an effective-temperature description, Teff(T,V)T_{\text{eff}}(T,V)Teff​(T,V), that collapses the steady-state behavior onto its equilibrium form. In sharp contrast, current fluctuations exhibit genuinely non-equilibrium features: the fluctuation-dissipation ratio becomes negative in the ordered phase, corresponding to a negative effective temperature for the current degrees of freedom. These results establish current noise as a sensitive probe of critical fluctuations at non-equilibrium quantum phase transitions and open new directions for exploring voltage-driven critical phenomena in quantum transport systems.
We investigate the dynamical properties of low-dimensional systems, driven by external noise sources. Specifically we consider a resistively shunted Josephson junction and a one-dimensional quantum liquid in a commensurate lattice potential, subject to 1/f1/f1/f noise. In absence of nonlinear coupling, we have shown previously that these systems establish a nonequilibrium critical steady state [Dalla Torre, Demler, Giamarchi, and Altman, Nat. Phys. 6, 806 (2010)]. Here, we use this state as the basis for a controlled renormalization group analysis using the Keldysh path integral formulation to treat the nonlinearities: the Josephson coupling and the commensurate lattice. The analysis to first order in the coupling constant indicates transitions between superconducting and localized regimes that are smoothly connected to the respective equilibrium transitions. However, at second order, the back action of the mode coupling on the critical state leads to renormalization of dissipation and emergence of an effective temperature. In the Josephson junction, the temperature is parametrically small allowing to observe a universal crossover between the superconducting and insulating regimes. The I-V characteristics of the junction displays algebraic behavior controlled by the underlying critical state over a wide range. In the noisy one-dimensional liquid, the generated dissipation and effective temperature are not small as in the junction. We find a crossover between a quasilocalized regime dominated by dissipation and another dominated by temperature. However, since in the thermal regime the thermalization rate is parametrically small, signatures of the nonequilibrium critical state may be seen in transient dynamics.
Universality and scaling laws are hallmarks of equilibrium phase transitions and critical phenomena. However, extending these concepts to non-equilibrium systems is an outstanding challenge. Despite recent progress in the study of dynamical phases, the universality classes and scaling laws for non-equilibrium phenomena are far less understood than those in equilibrium. In this work, using a trapped-ion quantum simulator with single-spin resolution, we investigate the non-equilibrium nature of critical fluctuations following a quantum quench to the critical point. We probe the scaling of spin fluctuations after a series of quenches to the critical Hamiltonian of a long-range Ising model. With systems of up to 50 spins, we show that the amplitude and timescale of the post-quench fluctuations scale with system size with distinct universal critical exponents, depending on the quench protocol. While a generic quench can lead to thermal critical behavior, we find that a second quench from one critical state to another (i.e. a double quench) results in a new universal non-equilibrium behavior, identified by a set of critical exponents distinct from their equilibrium counterparts. Our results demonstrate the ability of quantum simulators to explore universal scaling beyond equilibrium.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.