The role of initial conditions in the ageing of the long-range spherical model

The kinetics of the long-range spherical model evolving from various initial states is studied. In particular, the large-time auto-correlation and -response functions are obtained, for classes of long-range correlated initial states, and for magnetized initial states. The ageing exponents can depend on certain qualitative features of initial states. We explicitly find the conditions for the system to cross over from ageing classes that depend on initial conditions to those that do not.Comment: 15 pages; corrected some typo

Entanglement entropy of random quantum critical points in one dimension

For quantum critical spin chains without disorder, it is known that the entanglement of a segment of N>>1 spins with the remainder is logarithmic in N with a prefactor fixed by the central charge of the associated conformal field theory. We show that for a class of strongly random quantum spin chains, the same logarithmic scaling holds for mean entanglement at criticality and defines a critical entropy equivalent to central charge in the pure case. This effective central charge is obtained for Heisenberg, XX, and quantum Ising chains using an analytic real-space renormalization group approach believed to be asymptotically exact. For these random chains, the effective universal central charge is characteristic of a universality class and is consistent with a c-theorem.Comment: 4 pages, 3 figure

The Ubiquitous 'c': from the Stefan-Boltzmann Law to Quantum Information

I discuss various aspects of the role of the conformal anomaly number c in 2- and 1+1-dimensional critical behaviour: its appearance as the analogue of Stefan's constant, its fundamental role in conformal field theory, in the classification of 2d universality classes, and as a measure of quantum entanglement, among other topics.Comment: 8 pages, 2 figures. Boltzmann Medal Lecture, Statphys24, Cairns 2010. v3: minor revision

Correlations in an expanding gas of hard-core bosons

We consider a longitudinal expansion of a one-dimensional gas of hard-core bosons suddenly released from a trap. We show that the broken translational invariance in the initial state of the system is encoded in correlations between the bosonic occupation numbers in the momentum space. The correlations are protected by the integrability and exhibit no relaxation during the expansion

Bi-partite entanglement entropy in integrable models with backscattering

In this paper we generalise the main result of a recent work by J. L. Cardy and the present authors concerning the bi-partite entanglement entropy between a connected region and its complement. There the expression of the leading order correction to saturation in the large distance regime was obtained for integrable quantum field theories possessing diagonal scattering matrices. It was observed to depend only on the mass spectrum of the model and not on the specific structure of the diagonal scattering matrix. Here we extend that result to integrable models with backscattering (i.e. with non-diagonal scattering matrices). We use again the replica method, which connects the entanglement entropy to partition functions on Riemann surfaces with two branch points. Our main conclusion is that the mentioned infrared correction takes exactly the same form for theories with and without backscattering. In order to give further support to this result, we provide a detailed analysis in the sine-Gordon model in the coupling regime in which no bound states (breathers) occur. As a consequence, we obtain the leading correction to the sine-Gordon partition function on a Riemann surface in the large distance regime. Observations are made concerning the limit of large number of sheets.Comment: 22 pages, 2 figure

Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization-group approximation

The critical thermodynamics of the two-dimensional N-vector cubic and MN models is studied within the field-theoretical renormalization-group (RG) approach. The beta functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various N are investigated. For N=2 it is found that the continuous line of fixed points running from the XY fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both beta functions closer to each another. For the cubic model with N\geq 3, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under N>2 is an artifact of the perturbative analysis. For the quenched dilute O(M) models ($MN$ models with N=0) the results are compatible with a stable pure fixed point for M\geq1. For the MN model with M,N\geq2 all the non-perturbative results are reproduced. In addition a new stable fixed point is found for moderate values of M and N.Comment: 26 pages, 3 figure

Entanglement Dynamics after a Quench in Ising Field Theory: A Branch Point Twist Field Approach

We extend the branch point twist field approach for the calculation of entanglement entropies to time-dependent problems in 1+1-dimensional massive quantum field theories. We focus on the simplest example: a mass quench in the Ising field theory from initial mass m0 to final mass m. The main analytical results are obtained from a perturbative expansion of the twist field one-point function in the post-quench quasi-particle basis. The expected linear growth of the Rényi entropies at large times mt ≫ 1 emerges from a perturbative calculation at second order. We also show that the Rényi and von Neumann entropies, in infinite volume, contain subleading oscillatory contributions of frequency 2m and amplitude proportional to (mt)−3/2. The oscillatory terms are correctly predicted by an alternative perturbation series, in the pre-quench quasi-particle basis, which we also discuss. A comparison to lattice numerical calculations carried out on an Ising chain in the scaling limit shows very good agreement with the quantum field theory predictions. We also find evidence of clustering of twist field correlators which implies that the entanglement entropies are proportional to the number of subsystem boundary points

Bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories

This paper is a review of the main results obtained in a series of papers involving the present authors and their collaborator J L Cardy over the last 2 years. In our work, we have developed and applied a new approach for the computation of the bi-partite entanglement entropy in massive (1+1)-dimensional quantum field theories. In most of our work we have also considered these theories to be integrable. Our approach combines two main ingredients: the 'replica trick' and form factors for integrable models and more generally for massive quantum field theory. Our basic idea for combining fruitfully these two ingredients is that of the branch-point twist field. By the replica trick, we obtained an alternative way of expressing the entanglement entropy as a function of the correlation functions of branch-point twist fields. On the other hand, a generalization of the form factor program has allowed us to study, and in integrable cases to obtain exact expressions for, form factors of such twist fields. By the usual decomposition of correlation functions in an infinite series involving form factors, we obtained exact results for the infrared behaviours of the bi-partite entanglement entropy, and studied both its infrared and ultraviolet behaviours for different kinds of models: with and without boundaries and backscattering, at and out of integrability

Universal parity effects in the entanglement entropy of XX chains with open boundary conditions

We consider the Renyi entanglement entropies in the one-dimensional XX spin-chains with open boundary conditions in the presence of a magnetic field. In the case of a semi-infinite system and a block starting from the boundary, we derive rigorously the asymptotic behavior for large block sizes on the basis of a recent mathematical theorem for the determinant of Toeplitz plus Hankel matrices. We conjecture a generalized Fisher-Hartwig form for the corrections to the asymptotic behavior of this determinant that allows the exact characterization of the corrections to the scaling at order o(1/l) for any n. By combining these results with conformal field theory arguments, we derive exact expressions also in finite chains with open boundary conditions and in the case when the block is detached from the boundary.Comment: 24 pages, 9 figure

Entanglement entropy of two disjoint intervals in c=1 theories

We study the scaling of the Renyi entanglement entropy of two disjoint blocks of critical lattice models described by conformal field theories with central charge c=1. We provide the analytic conformal field theory result for the second order Renyi entropy for a free boson compactified on an orbifold describing the scaling limit of the Ashkin-Teller (AT) model on the self-dual line. We have checked this prediction in cluster Monte Carlo simulations of the classical two dimensional AT model. We have also performed extensive numerical simulations of the anisotropic Heisenberg quantum spin-chain with tree-tensor network techniques that allowed to obtain the reduced density matrices of disjoint blocks of the spin-chain and to check the correctness of the predictions for Renyi and entanglement entropies from conformal field theory. In order to match these predictions, we have extrapolated the numerical results by properly taking into account the corrections induced by the finite length of the blocks to the leading scaling behavior.Comment: 37 pages, 23 figure