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The master equation describing non-equilibrium one-dimensional problems like diffusion limited reactions or critical dynamics of classical spin systems can be written as a Schrödinger equation in which the wave function is the probability distribution and the Hamiltonian is that of a quantum chain with nearest neighbor interactions. Since many one-dimensional quantum chains are integrable, this opens a new field of applications. At the same time physical intuition and probabilistic methods bring new insight into the understanding of the properties of quantum chains. A simple example is the asymmetric diffusion of several species of particles which leads naturally to Hecke algebras and qdeformed quantum groups. Many other examples are given. Several relevant technical aspects like critical exponents, correlation functions and finite-size scaling are also discussed in detail.

In a quantum critical chain, the scaling regime of the energy and momentum of the ground state and low lying excitations are described by conformal field theory (CFT). The same holds true for the von Neumann and Rényi entropies of the ground state, which display a universal logarithmic behaviour depending on the central charge. In this letter we generalize this result to those excited states of the chain that correspond to primary fields in CFT. It is shown that the n-th Rényi entropy is related to a 2n-point correlator of primary fields. We verify this statement for the critical XX and XXZ chains. This result uncovers a new link between quantum information theory and CFT.Entanglement is one of the central concepts in quantum physics since Schroedinger used the term in an answer to the Einstein-Podolsky-Rosen article in 1935. A particularly active line of research is concerned with the role played by entanglement in the physics of many-body systems [1]. One is typically interested in the amount of entanglement between two spatial partitions, say A and B, of a many-body system in its ground state. For a pure ground state the amount of entanglement is usually quantified with the entanglement entropy, or the von Neumann entropy of the reduced density matrix ρ A :n , the entanglement entropy being lim n→1 S n . One of the most important results in this topic is the celebrated area law [2-4], which, roughly speaking, states that ground states of gapped many-body systems with short-range interactions have an entanglement entropy proportional to the area of the hypersurface separating both partitions. The area law restricts the fraction of the Hilbert space accessible to ground states of local Hamiltonians in an essential way, allowing for their efficient numerical simulation [4].Violations of the area law occur in gapless (critical) systems. In one dimension most of critical systems, as well as being gapless, are also conformal invariant. The attention to the entanglement properties on these systems came after the seminal result of Holzhey, Larsen and Wilczek [5], who showed that the leading behavior of the ground state entropies S gs n is proportional to the central charge of the underlying conformal field theory (CFT) governing the long-distance physics of the discrete quantum chain. If ℓ and N are the lengths of the partition A and of the total system, both measured in lattice spacing units, then the Rényi entropy of the ground state, with periodic boundary conditions, is [5-7]where c is the central charge of the CFT and γ n is a nonuniversal constant. In a critical model, the finite-size scaling of the energy of excitations is given by the scaling dimension of the corresponding conformal operators [8]. This fact suggests that also the entanglement entropy could be related to properties of these operators. Entanglement of excited states has been considered previously. In [9] it was shown that the negativity of the excited states in the XXZ critical model shows a universal scaling. In [10] it was shown that a violat...

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The entanglement in a quantum system that possess an internal symmetry, characterized by the S z -magnetization or U (1)-charge, is distributed among different sectors. The aim of this letter is to gain a deeper understanding of the contribution to the entanglement entropy in each of those sectors for the ground state of conformal invariant critical one dimensional systems. We find surprisingly that the entanglement entropy is equally distributed among the different magnetization sectors. Its value is given by the standard area law violating logarithmic term, that depends on the central charge c, minus a double logarithmic correction related to the zero temperature susceptibility. This result provides a new method to estimate simultaneously the central charge c and the critical exponents of U (1)-symmetric quantum chains. The method is numerically simple and gives precise results for the spin-1 2 quantum XXZ chain. We also compute the probability distribution of the magnetization in contiguous sublattices.