John Cardy

Operator Content of Two-Dimensional Conformally Invariant Theories

It is shown how conformal invariance relates many numerically accessible properties of the transfer matrix of a critical system in a finite-width infinitely long strip to bulk universal quantities. Conversely, general properties of the transfer matrix imply constraints on the allowed operator content of the theory. We show that unitary theories with a finite number of primary operators must have a conformal anomaly number c < 1, and therefore must fall into the classification of Friedan, Qiu and Shenker. For such theories, we derive sum rules which constrain the numbers of operators with given scaling dimensions.

Entanglement entropy and quantum field theory

We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy SA = −Tr ρAlogρA corresponding to the reduced density matrix ρA 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 of Holzhey et al when A is a finite interval of length 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. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where 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.

Boundary Conditions, Fusion Rules and the Verlinde Formula

Abstract Boundary operators is conformal field theory are considered as arising from the juxtaposition of different types of boundary conditions. From this point of view, the operator content of the theory in an annulus may be related to the fusion rules. By considering the partition function in such a geometry, we give a simple derivation of the Verlinde formula.

Entanglement entropy and conformal field theory

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.

Conformal Invariance and Surface Critical Behavior

Abstract Conformal invariance constrains the form of correlation functions near a free surface. In two dimensions, for a wide class of models, it completely determines the correlation functions at the critical point, and yields the exact values of the surface critical exponents. They are related to the bulk exponents in a non-trivial way. For the Q -state Potts model (0 ⩽ Q ⩽ 4) we find η = 2 (3v − 1) , and for the O( N ) model (−2 ⩽ N ⩽ 2), η = (2v − 1) (4v − 1) .

Evolution of entanglement entropy in one-dimensional systems

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 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 , after which it saturates at a value proportional to , the coefficient depending on the initial state. This behaviour may be understood as a consequence of causality.

Is There a c Theorem in Four-Dimensions?

Abstract The difficulties of extending Zamolodchikovs c -theorem to dimensions d ≠ 2 are discussed. It is shown that, for d even, the one-point function of the trace of the stress tensor on the sphere, S d , when suitably regularized, defines a c -function, which, at least to one loop order, is decreasing along RG trajectories and is stationary at RG fixed points, where it is proportional to the usual conformal anomaly. It is shown that the existence of such a c -function, if it satisfies these properties to all orders, is consistent with the expected behavior of QCD in four dimensions.

Time-dependence of correlation functions following a quantum quench

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.

Quantum quenches in extended systems

We study in general the time evolution of correlation functions in a extended quantum system after the quench of a parameter in the Hamiltonian. We show that correlation functions in d dimensions can be extracted using methods of boundary critical phenomena in d+1 dimensions. For d = 1 this allows us to use the powerful tools of conformal field theory in the case of critical evolution. Several results are obtained in generic dimension in the Gaussian (mean field) approximation. These predictions are checked against the real time evolution of some solvable models that allow us also to understand which features are valid beyond the critical evolution. All our findings may be explained in terms of a picture generally valid, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate with a finite speed through the system. Furthermore we show that the long time results can be interpreted in terms of a generalized Gibbs ensemble. We discuss some open questions and possible future developments.

Critical percolation in finite geometries

The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only under changes of scale, but also under mappings of the region which are conformal in the interior and continuous on the boundary. This is a larger invariance than that expected for generic critical systems. Specific predictions are presented for the crossing probability between opposite sides of a rectangle, and are compared with recent numerical work. The agreement is excellent.