Benjamin Doyon

Form factors of branch-point twist fields in quantum integrable models and entanglement entropy

Abstract In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.

Energy flow in non-equilibrium conformal field theory

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.

Emergent hydrodynamics in integrable quantum systems out of equilibrium

Understanding the general principles underlying strongly interacting quantum states out of equilibrium is one of the most important tasks of current theoretical physics. With experiments accessing the intricate dynamics of many-body quantum systems, it is paramount to develop powerful methods that encode the emergent physics. Up to now, the strong dichotomy observed between integrable and nonintegrable evolutions made an overarching theory difficult to build, especially for transport phenomena where space-time profiles are drastically different. We present a novel framework for studying transport in integrable systems: hydrodynamics with infinitely many conservation laws. This bridges the conceptual gap between integrable and nonintegrable quantum dynamics, and gives powerful tools for accurate studies of space-time profiles. We apply it to the description of energy transport between heat baths, and provide a full description of the current-carrying nonequilibrium steady state and the transition regions in a family of models including the Lieb-Liniger model of interacting Bose gases, realized in experiments.

Non-Equilibrium Steady States in Conformal Field Theory

We present a construction of non-equilibrium steady states in one-dimensional quantum critical systems carrying energy and charge fluxes. This construction is based on a scattering approach within a real-time hamiltonian reservoir formulation. Using conformal field theory techniques, we prove convergence towards steady states at large time. We discuss in which circumstances these states describe the universal non-equilibrium regime at low temperatures. We compute the exact large deviation functions for both energy and charge transfers, which encode for the quantum and statistical fluctuations of these transfers at large time. They are universal, depending only on fundamental constants (

Entanglement entropy of non-unitary conformal field theory

{\hbar, k_B}

Non-equilibrium steady states in the Klein–Gordon theory

ħ,kB), on the central charge and on the external parameters such as the temperatures or the chemical potentials, and they satisfy fluctuation relations. A key point consists in relating the derivatives of these functions to the linear response functions but at complex shifted external parameters.

Far from equilibrium energy flow in quantum critical systems

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.

A hydrodynamic approach to non-equilibrium conformal field theories

Here we show that the Renyi entanglement entropy of a region of large size l in a one-dimensional critical model whose ground state breaks conformal invariance (such as in those described by non-unitary conformal field theories), behaves as ceff(n+1)/2n log(L), where ceff=c-24Delta > 0 is the effective central charge, c (which may be negative) is the central charge of the conformal field theory and Delta < 0 is the lowest holomorphic conformal dimension in the theory. We also obtain results for models with boundaries, and with a large but finite correlation length, and we show that if the lowest conformal eigenspace is logarithmic, then there is an additional term proportional to