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.
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 extend beyond the Euler scales the hydrodynamic theory for quantum
and classical integrable models developed in recent years, accounting
for diffusive dynamics and local entropy production. We ...review how the
diffusive scale can be reached via a gradient expansion of the
expectation values of the conserved fields and how the coefficients of
the expansion can be computed via integrated steady-state two-point
correlation functions, emphasising that
{\mathcal PT}
T
-symmetry
can fully fix the inherent ambiguity in the definition of conserved
fields at the diffusive scale. We develop a form factor expansion to
compute such correlation functions and we show that, while the dynamics
at the Euler scale is completely determined by the density of single
quasiparticle excitations on top of the local steady state, diffusion is
due to scattering processes among quasiparticles, which are only present
in truly interacting systems. We then show that only two-quasiparticle
scattering processes contribute to the diffusive dynamics. Finally we
employ the theory to compute the exact spin diffusion constant of a
gapped XXZ spin
-1/2
−
1
/
2
chain at finite temperature and half-filling, where we show that spin
transport is purely diffusive.
The quantum symmetric simple exclusion process (Q-SSEP) is a model for quantum stochastic dynamics of fermions hopping along the edges of a graph with Brownian noisy amplitudes and driven ...out-of-equilibrium by injection-extraction processes at a few vertices. We present a solution for the invariant probability measure of the one dimensional Q-SSEP in the infinite size limit by constructing the steady correlation functions of the system density matrix and quantum expectation values. These correlation functions code for a rich structure of fluctuating quantum correlations and coherences. Although our construction does not rely on the standard techniques from the theory of integrable systems, it is based on a remarkable interplay between the permutation groups and polynomials. We incidentally point out a possible combinatorial interpretation of the Q-SSEP correlation functions via a surprising connexion with geometric combinatorics and the associahedron polytopes.
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We introduce the asymmetric extension of the quantum symmetric simple exclusion process which is a stochastic model of fermions on a lattice hopping with random amplitudes. In this setting, we ...analytically show that the time-integrated current of fermions defines a height field that exhibits quantum nonlinear stochastic Kardar-Parisi-Zhang dynamics. Similarly to classical simple exclusion processes, we further introduce the discrete Cole-Hopf (or Gärtner) transform of the height field that satisfies a quantum version of the stochastic heat equation. Finally, we investigate the limit of the height field theory in the continuum under the celebrated Kardar-Parisi-Zhang scaling and the regime of almost-commuting quantum noise.
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A
bstract
The S-matrix of the well-studied sinh-Gordon model possesses a remarkable strong/weak coupling duality
b →
1
/b
. Since there is no understanding nor evidence for such a duality based on ...the quantum action of the model, it should be questioned whether the properties of the model for
b >
1 are simply obtained by analytic continuation of the weak coupling regime 0
< b <
1. In this article we assert that the answer is no, and we develop a concrete and specific proposal for the properties when
b >
1. Namely, we propose that in this region one needs to introduce a background charge
Q
∞
=
b
+ 1
/b −
2 which differs from the Liouville background charge by the shift of
−
2. We propose that in this regime the model has non-trivial massless renormalization group flows between two different conformal field theories. This is in contrast to the weak coupling regime which is a theory of a single massive particle. Evidence for our proposal comes from higher order beta functions. We show how our proposal correctly reproduces the freezing transitions in the multi-fractal exponents of a Dirac fermion in 2 + 1 dimensions in a random magnetic field, which provides a strong check since such transitions have several detailed features. We also point out a connection between a semi-classical version of this transition and the so-called Manning condensation phenomena in polyelectrolyte physics.
Quantum coherences characterize the ability of particles to quantum mechanically interfere within some given distances. In the context of noisy many-body quantum systems, these coherences can ...fluctuate. A simple toy model to study such fluctuations in an out-of-equilibrium setting is the open quantum symmetric simple exclusion process (Q-SSEP), which describes spinless fermions in one dimension hopping to neighboring sites with random amplitudes coupled between two reservoirs. Here, we show that the dynamics of fluctuations of coherences in Q-SSEP have a natural interpretation as free cumulants, a concept from free probability theory. Based on this insight, we provide heuristic arguments as to why we expect free probability theory to be an appropriate framework to describe coherent fluctuations in generic mesoscopic systems where the noise emerges from a coarse-grained description. In the case of Q-SSEP, we show how the link to free probability theory can be used to derive the time evolution of connected fluctuations of coherences as well as a simple steady-state solution.
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We consider two quantum Ising chains initially prepared at thermal equilibrium but with different temperatures and coupled at a given time through one of their end points. In the long-time limit the ...system reaches a nonequilibrium steady state. We discuss properties of this nonequilibrium steady state and characterize the convergence to the steady regime. We compute the mean energy flux through the chain and show that the heat transport is ballistic. We derive also the large-deviation function for the quantum and thermal fluctuations of this energy transfer.
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This review provides an introduction to two dimensional growth processes. Although it covers a variety of processes such as diffusion limited aggregation, it is mostly devoted to a detailed ...presentation of stochastic Schramm–Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of various examples of 2D growth processes and their connections with statistical mechanics. SLE is then introduced and Schramm's argument mapping conformally invariant interfaces to SLE is explained. A substantial part of the review is devoted to reveal the deep connections between statistical mechanics and processes, and more specifically to the present context, between 2D critical systems and SLE. Some of the remarkable properties of SLE are explained, together with the tools for computing with it. This review has been written with the aim of filling the gap between the mathematical and the physical literature on the subject.
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