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Non-Equilibrium Thermodynamics in Conformal Field Theory

Stephan Hollands, Roberto Longo

Published May 5, 2016• 4 citations
OpenAlex

Abstract

We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon.

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We present a model independent, operator algebraic approach to non-equilibrium quantum thermodynamics within the framework of two-dimensional Conformal Field Theory. Two infinite reservoirs in equilibrium at their own temperatures and chemical potentials are put in contact through a defect line, possibly by inserting a probe. As time evolves, the composite system then approaches a non-equilibrium steady state that we describe. In particular, we re-obtain recent formulas of Bernard and Doyon.
We present a construction of non-equilibrium steady states within conformal field theory. These states sustain energy flows between two quantum systems, initially prepared at different temperatures, whose dynamical properties are represented by two, possibly different, conformal field theories connected through an impurity. This construction relies on a real time formulation of conformal defect dynamics based on a field scattering picture parallelizing—but yet different from—the Euclidean formulation. We present the basic characteristics of this formulation and give an algebraic construction of the real time scattering maps that we illustrate in the case of -based conformal field theories.
Abstract We provide a pedagogical review of the main ideas and results in non-equilibrium conformal field theory and connected subjects. These concern the understanding of quantum transport and its statistics at and near critical points. Starting with phenomenological considerations, we explain the general framework, illustrated by the example of the Heisenberg quantum chain. We then introduce the main concepts underlying conformal field theory (CFT), the emergence of critical ballistic transport, and the CFT scattering construction of non-equilibrium steady states. Using this we review the theory for energy transport in homogeneous one-dimensional critical systems, including the complete description of its large deviations and the resulting (extended) fluctuation relations. We generalize some of these ideas to one-dimensional critical charge transport and to the presence of defects, as well as beyond one-dimensional criticality. We describe non-equilibrium transport in free-particle models, where connections are made with generalized Gibbs ensembles, and in higher-dimensional and non-integrable quantum field theories, where the use of the powerful hydrodynamic ideas for non-equilibrium steady states is explained. We finish with a list of open questions. The review does not assume any advanced prior knowledge of conformal field theory, large-deviation theory or hydrodynamics.
Abstract We study the non-equilibrium dynamics of conformal field theory (CFT) in 1+1 dimensions with a smooth position-dependent velocity v(x) explicitly breaking translation invariance. Such inhomogeneous CFT is argued to effectively describe 1+1-dimensional quantum many-body systems with certain inhomogeneities varying on mesoscopic scales. Both heat and charge transport are studied, where, for concreteness, we suppose that our CFT has a conserved U(1) current. Based on projective unitary representations of diffeomorphisms and smooth maps in Minkowskian CFT, we obtain a recipe for computing the exact non-equilibrium dynamics in inhomogeneous CFT when evolving from initial states defined by smooth inverse-temperature and chemical-potential profiles β(x)\beta (x)β(x) β ( x ) and μ(x)\mu (x)μ(x) μ ( x ) . Using this recipe, the following exact analytical results are obtained: (i) the full time evolution of densities and currents for heat and charge transport, (ii) correlation functions for components of the energy–momentum tensor and the U(1) current as well as for any primary field, and (iii) the thermal and electrical conductivities. The latter are computed by direct dynamical considerations and alternatively using a Green–Kubo formula. Both give the same explicit expressions for the conductivities, which reveal how inhomogeneous dynamics opens up the possibility for diffusion as well as implies a generalization of the Wiedemann–Franz law to finite times within CFT.
Non-equilibrium steady states (NESS) describe particularly simple and stationary non-equilibrium situations. A possibility to obtain such states is to consider the asymptotic evolution of two infinite heat baths brought into thermal contact. In this work we generalise corresponding results of Doyon~et.~al. (J.\ Phys.\ A 18 (2015) no.9) for free Klein-Gordon fields in several directions. Our analysis is carried out directly at the level of correlation functions and in the algebraic approach to QFT. We discuss non-trivial chemical potentials, condensates, inhomogeneous linear models and homogeneous interacting ones. We shall not consider a sharp contact at initial time, but a smooth transition region. As a consequence, the states we construct will be of Hadamard type, and thus sufficiently regular for the perturbative treatment of interacting models. Our analysis shows that perturbatively constructed interacting NESS display thermodynamic properties similar to the ones of the NESS in linear models. In particular, perturbation theory appears to be insufficient to describe full thermalisation in non-linear QFT. Notwithstanding, we find that the NESS for linear and interacting models is stable under small perturbations, which is one of the characteristic features of equilibrium states.
A possible thermodynamic evolution to equilibrium of a quantum statistical system is discussed. The description is entirely algebraic and it is done by means of complex coherent states built in the Universal Enveloping Algebra of the complex algebra https://www.w3.org/1998/Math/MathML" display="inline"> A 1 . Finite dimensional representations of https://www.w3.org/1998/Math/MathML" display="inline"> A 1 give fermion distributions while infinite dimensional ones describe bosons. The scheme is thermodynamic in the sense that fluctuations are not considered and this allows us to resort to an evolution operator that realizes, for t → ∞, a projector on the state of equilibrium. The shrinking of the Hilbert space is related to non unitary evolution that change the normalization of the state vectors. The theory is invariant under time reversal: vectors at t → ±∞ differ only in phases and both correspond to the equilibrium probability distribution. Equilibrium and non-equilibrium thermodynamic quantities can be calculated at microscopic level in a framework that is in between thermodynamic and statistical mechanics. Our approach is a physically founded example of non-linear quantum theory.
Abstract We develop a hydrodynamic approach to non-equilibrium conformal field theory. We study non-equilibrium steady states in the context of one-dimensional conformal field theory perturbed by the irrelevant operator. By direct quantum computation, we show, to first order in the coupling, that a relativistic hydrodynamic emerges, which is a simple modification of one-dimensional conformal fluids. We show that it describes the steady state and its approach, and we provide the main characteristics of the steady state, which lies between two shock waves. The velocities of these shocks are modified by the perturbation and equal the sound velocities of the asymptotic baths. Pushing this approach further, we are led to conjecture that the approach to the steady state is generically controlled by the power law t −1/2 , and that the widths of the shocks increase with time according to t 1/3 .