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arXiv:1306.3192 · Journal of Statistical Mechanics Theory and Experiment

Energy flow and fluctuations in non-equilibrium conformal field theory on star graphs

Benjamin Doyon, Marianne Hoogeveen, Denis Bernard

Published Mar 11, 2014• 22 citations
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Abstract

We consider non-equilibrium quantum steady states in conformal field theory (CFT) on star-graph configurations, with a particular, simple connection condition at the vertex of the graph. These steady states occur after a large time as a result of initially thermalizing the legs of the graph at different temperatures, and carry energy flows. Using purely Virasoro algebraic calculations we evaluate the exact scaled cumulant generating function for these flows. We show that this function satisfies a generalization of the usual non-equilibrium fluctuation relations. This extends results by two of the authors to the case of more than two legs. It also provides an alternative derivation centered on Virasoro algebra operators rather than local fields, hence an alternative regularization scheme, thus confirming the validity and universality of the scaled cumulant generating function. Our derivation shows how the usual Virasoro algebra leads, in large volumes, to a continuous-index Virasoro algebra for which we develop diagrammatic principles, which may be of interest in other non-equilibrium contexts as well. Finally, our results shed light on the Poisson-process interpretation of the long-time energy transfer in CFT.

DOI
10.1088/1742-5468/2014/03/p03002

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We consider non-equilibrium quantum steady states in conformal field theory (CFT) on star-graph configurations, with a particular, simple connection condition at the vertex of the graph. These steady states occur after a large time as a result of initially thermalizing the legs of the graph at different temperatures, and carry energy flows. Using purely Virasoro algebraic calculations we evaluate the exact scaled cumulant generating function for these flows. We show that this function satisfies a generalization of the usual non-equilibrium fluctuation relations. This extends results by two of the authors to the case of more than two legs. It also provides an alternative derivation centered on Virasoro algebra operators rather than local fields, hence an alternative regularization scheme, thus confirming the validity and universality of the scaled cumulant generating function. Our derivation shows how the usual Virasoro algebra leads, in large volumes, to a continuous-index Virasoro algebra for which we develop diagrammatic principles, which may be of interest in other non-equilibrium contexts as well. Finally, our results shed light on the Poisson-process interpretation of the long-time energy transfer in CFT.
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures Tl, Tr, and waiting for a long time. We evaluate the current J(Tl, Tr) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov’s method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium c-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the “additivity ” property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl, Tr) is not of the form f(Tl)−f(Tr).
We evaluate the exact energy current and scaled cumulant generating function (related to the large-deviation function) in non-equilibrium steady states with energy flow, in any integrable model of relativistic quantum field theory (IQFT) with diagonal scattering. Our derivations are based on various recent results of D. Bernard and B. Doyon. The steady states are built by connecting homogeneously two infinite halves of the system thermalized at different temperatures TlT_lTl​, TrT_rTr​, and waiting for a long time. We evaluate the current J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) using the exact QFT density matrix describing these non-equilibrium steady states and using Al.B. Zamolodchikov's method of the thermodynamic Bethe ansatz (TBA). The scaled cumulant generating function is obtained from the extended fluctuation relations which hold in integrable models. We verify our formula in particular by showing that the conformal field theory (CFT) result is obtained in the high-temperature limit. We analyze numerically our non-equilibrium steady-state TBA equations for three models: the sinh-Gordon model, the roaming trajectories model, and the sine-Gordon model at a particular reflectionless point. Based on the numerics, we conjecture that an infinite family of non-equilibrium ccc-functions, associated to the scaled cumulants, can be defined, which we interpret physically. We study the full scaled distribution function and find that it can be described by a set of independent Poisson processes. Finally, we show that the "additivity" property of the current, which is known to hold in CFT and was proposed to hold more generally, does not hold in general IQFT, that is J(Tl,Tr)J(T_l,T_r)J(Tl​,Tr​) is not of the form f(Tl)−f(Tr)f(T_l)-f(T_r)f(Tl​)−f(Tr​).
We study the energy current and its fluctuations in quantum gapless 1d systems far from equilibrium modeled by conformal field theory, where two separated halves are prepared at distinct temperatures and glued together at a point contact. We prove that these systems converge towards steady states, and give a general description of such non-equilibrium steady states in terms of quantum field theory data. We compute the large deviation function, also called the full counting statistics, of energy transfer through the contact. These are universal and satisfy fluctuation relations. We provide a simple representation of these quantum fluctuations in terms of classical Poisson processes whose intensities are proportional to Boltzmann weights.
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.
We construct non-equilibrium steady states in the Klein–Gordon theory in arbitrary space dimension d following a local quench. We consider the approach where two independently thermalized semi-infinite systems, with temperatures and , are connected along a -dimensional hypersurface. A current-carrying steady state, described by thermally distributed modes with temperatures and for left and right-moving modes, respectively, emerges at late times. The non-equilibrium density matrix is the exponential of a non-local conserved charge. We obtain exact results for the average energy current and the complete distribution of energy current fluctuations. The latter shows that the long-time energy transfer can be described by a continuum of independent Poisson processes, for which we provide the exact weights. We further describe the full time evolution of local observables following the quench. Averages of generic local observables, including the stress-energy tensor, approach the steady state with a power-law in time, where the exponent depends on the initial conditions at the connection hypersurface. We describe boundary conditions and special operators for which the steady state is reached instantaneously on the connection hypersurface. A semiclassical analysis of freely propagating modes yields the average energy current at large distances and late times. We conclude by comparing and contrasting our findings with results for interacting theories and provide an estimate for the timescale governing the crossover to hydrodynamics. As a modification of our Klein–Gordon analysis we also include exact results for free Dirac fermions.
In this communication, we present a simple derivation, from time-reversal symmetry, of fluctuation relations for steady-state large deviation functions in non-equilibrium quantum systems. We further show that a condition of pure transmission implies extended fluctuation relations, connecting large deviation functions to mean currents at shifted temperatures and chemical potentials. We illustrate these concepts in various examples, including the interacting resonant level model and conformal or integrable models.