On the Modular Operator of Mutli-component Regions in Chiral CFT

We introduce a new approach to find the Tomita–Takesaki modular flow for multi-component regions in general chiral conformal field theory. Our method is based on locality and analyticity of primary fields as well as the so-called Kubo–Martin–Schwinger (KMS) condition. These features can be used to transform the problem to a Riemann–Hilbert problem on a covering of the complex plane cut along the regions, which is equivalent to an integral equation for the matrix elements of the modular Hamiltonian. Examples are considered.

Resolving modular flow: a toolkit for free fermions

Modular flow is a symmetry of the algebra of observables associated to space-time regions. Being closely related to entanglement, it has played a key role in recent connections between information theory, QFT and gravity. However, little is known about its action beyond highly symmetric cases. The key idea of this work is to introduce a new formula for modular flows for free chiral fermions in 1 + 1 dimensions, working directly from the resolvent, a standard technique in complex analysis. We present novel results — not fixed by conformal symmetry — for disjoint regions on the plane, cylinder and torus. Depending on temperature and boundary conditions, these display different behaviour ranging from purely local to non-local in relation to the mixing of operators at spacelike separation. We find the modular two-point function, whose analytic structure is in precise agreement with the KMS condition that governs modular evolution. Our ready-to-use formulae may provide new ingredients to explore the connection between spacetime and entanglement.

High temperature convergence of the KMS boundary conditions: The Bose-Hubbard model on a finite graph

The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs equilibrium measures. In this paper, we prove that in a certain limiting regime of high temperature the classical KMS condition can be derived from the quantum condition in the simple case of the Bose–Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden–Thompson inequality, Bogoliubov inequality and semiclassical analysis.

Mathematical aspects of Bose–Einstein condensation in equilibrium and local equilibrium conditions

In the paper4 the notion of local KMS condition, introduced in Ref. 3 and extended in Ref. 2, was shown to open new possibilities in the study of the problem of Bose–Einstein Condensation (BEC). In this paper we analyze the general structure of states on the CCR algebra over a pre-Hilbert space that satisfies the local KMS condition with respect to a free Hamiltonian [Formula: see text] and to a given inverse temperature function [Formula: see text]. The replacement of Hilbert space by a pre-Hilbert space allows one to deal with test functions more singular than those usually considered in the theory of distributions (thus allowing e.g., fractal critical surfaces) and is equivalent (in the language of Weyl algebras) to consider a degenerate symplectic form. It is precisely this degeneracy that allows one to introduce in an intrinsic way the notions of [Formula: see text]-critical subspace (resp. [Formula: see text]-critical surface) and of states exhibiting BEC, independently of infinite volume limits and of boundary conditions. We prove that the covariance of any local KMS state is uniquely determined by the pair [Formula: see text], through a nonlinear extension of the Planck factor. For a large class of such states (including all known examples) the covariance splits into a sum of two mutually singular terms: one corresponding to a regular state, the other one with support on a critical surface (or more generally a critical subspace) uniquely determined by [Formula: see text] and [Formula: see text]. In particular we prove that, if such a state is gauge invariant Gaussian (quasi-free), then the [Formula: see text]-equilibrium condition uniquely determines the regular part of the state, while the singular part is arbitrary.

On three new principles in non-equilibrium statistical mechanics and Markov semigroups of weak coupling limit type

We introduce three new principles: the nonlinear Boltzmann–Gibbs prescription, the local KMS condition and the generalized detailed balance (GDB) condition. We prove the equivalence of the first two under general conditions and we discuss a master equation formulation of the third one.

On the range of the generator of a quantum Markov semigroup

We show that the commutant of the range of the infinitesimal generator of a norm-continuous quantum Markov semigroup on [Formula: see text], not consisting of identity maps, with a faithful normal invariant state is trivial whenever the fixed point algebra is atomic. As a consequence, two formulations of the irreversible [Formula: see text]-KMS condition proposed in Ref. 2 are equivalent for this class of quantum Markov semigroups.