We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A log ρ A corresponding to the reduced density matrix ρ A of a subsystem A. For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c, we re-derive the result S A ∼ (c/3) log ℓ of Holzhey et al. when A is a finite interval of length ℓ in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals (See note added). For such a system away from its critical point, when the correlation length ξ is large but finite, we show that S A ∼ A(c/6) log ξ, where A is the number of boundary points of A. These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite-size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free-field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.

We study the unitary time evolution of the entropy of entanglement of a one-dimensional system between the degrees of freedom in an interval of length ℓ and its complement, starting from a pure state which is not an eigenstate of the hamiltonian. We use path integral methods of quantum field theory as well as explicit computations for the transverse Ising spin chain. In both cases, there is a maximum speed v of propagation of signals. In general the entanglement entropy increases linearly with time t up to t = ℓ/2v, after which it saturates at a value proportional to ℓ, the coefficient depending on the initial state. This behavior may be understood as a consequence of causality.

We review the conformal field theory approach to entanglement entropy in 1+1 dimensions. We show how to apply these methods to the calculation of the entanglement entropy of a single interval, and the generalization to different situations such as finite size, systems with boundaries, and the case of several disjoint intervals. We discuss the behaviour away from the critical point and the spectrum of the reduced density matrix. Quantum quenches, as paradigms of non-equilibrium situations, are also considered.

This text provides a thoroughly modern graduate-level introduction to the theory of critical behaviour. Beginning with a brief review of phase transitions in simple systems and of mean field theory, the text then goes on to introduce the core ideas of the renormalization group. Following chapters cover phase diagrams, fixed points, cross-over behaviour, finite-size scaling, perturbative renormalization methods, low-dimensional systems, surface critical behaviour, random systems, percolation, polymer statistics, critical dynamics and conformal symmetry. The book closes with an appendix on Gaussian integration, a selected bibliography, and a detailed index. Many problems are included. The emphasis throughout is on providing an elementary and intuitive approach. In particular, the perturbative method introduced leads, among other applications, to a simple derivation of the epsilon expansion in which all the actual calculations (at least to lowest order) reduce to simple counting, avoiding the need for Feynman diagrams.

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We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system. PACS numbers: 73.43. Nq, 11.25.Hf. 64.60.Ht Suppose that an extended quantum system in d dimensions (for example a quantum spin system), is prepared at time t = 0 in a pure state |ψ 0 which is the ground state of some hamiltonian H 0 (or, more generally, in a thermal state at a temperature less than the gap m 0 to the first excited state.) For times t > 0 the system evolves unitarily according to the dynamics given by a different hamiltonian H, which may be related to H 0 by varying a parameter such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than m −1 0 . How do the correlation functions, expectation values of products of local observables, then evolve? The answer to this question would appear to depend in detail on the system under consideration. It was first addressed in the context of the quantum Ising-XY model in Refs.[1] (see also [2]). Until recently it was, however, largely an academic question, because the time scales over which most condensed matter systems can evolve coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are inescapable. However, with the development of experimental tools for studying the behavior of optical lattices of ultracold atoms, and quantum phase transitions in these systems [3], there has been renewed interest in this theoretical problem (see, for example, [4,5].)In this Letter we study such problems in general and argue that, if H is at or close to a quantum critical point (while H 0 is not), there is a large degree of universality in the behavior at sufficiently large distances and late times, despite the fact that correlations typically fall off exponentially, rather than the power laws characteristic of the ground state near a quantum critical point. Our arguments are based on the path integral approach and the well-known mapping of the quantum problem to a classical one in d+1 dimensions. The initial state plays the role of a boundary condition, and we are able to then use the renormalisation group (RG) theory of boundary critical behavior (see, e.g., [6]). From this point of view, particularly powerful analytic results are available for d = 1 and when the quantum critical point has dynamic exponent z = 1 (or, equivalently, a linear quasiparticle dispersion relation ω = v|...

%e show that for conforrnally invariant two-dimensional systems, the amplitude of the finite-size corrections to the free energy of an infinitely long strip of width L at criticality is linearly related to the conformal anomaly number c, for various boundary conditions. The result is confirmed by renormalization-group arguments and numerical calculations. It is also related to the magnitude of the Casimir effect in an interacting one-dimensional field theory, and to the low-temperature specific heat in quantum chains.