We study the entanglement and Rényi entropies of two disjoint intervals in minimal models of conformal field theory. We use the conformal block expansion and fusion rules of twist fields to define a systematic expansion in the elliptic parameter of the trace of the n−th power of the reduced density matrix. Keeping only the first few terms we obtain an approximate expression that is easily analytically continued to n → 1, leading to an approximate formula for the entanglement entropy. These predictions are checked against some known exact results as well as against existing numerical data.
We first provide a formula to calculate the probability of occurrence of different configurations (formation probabilities) in a generic free fermion system. We then study the scaling of these probabilities with respect to the size in the case of critical transverse-field XY-chain in the σ z bases. In the case of the transverse field Ising model, we show that all the "crystal" configurations follow the formulas expected from conformal field theory (CFT). In the case of critical XX chain, we show that the only configurations that follow the formulas of the CFT are the ones which respect the filling factor of the system. By repeating all the calculations in the presence of open and periodic boundary conditions we find further support to our classification of different configurations. Using the developed technique, we also study Shannon information of a subregion in our system. In this respect we distinguish particular configurations that are more important in the study of the scaling limit of the Shannon information of the subsystem. Finally, we study the evolution of formation probabilities, Shannon information and Shannon mutual information after a quantum quench in free fermion system. In particular, for the intial state considered in this paper, we demonstrate that the Shannon information after quantum quench first increases with the time and then saturates at time, where l is the size of the subsystem.
We consider the Shannon mutual information of subsystems of critical quantum chains in their ground states. Our results indicate a universal leading behavior for large subsystem sizes. Moreover, as happens with the entanglement entropy, its finite-size behavior yields the conformal anomaly c of the underlying conformal field theory governing the long distance physics of the quantum chain. We studied analytically a chain of coupled harmonic oscillators and numerically the Q-state Potts models (Q = 2, 3 and 4), the XXZ quantum chain and the spin-1 Fateev-Zamolodchikov model. The Shannon mutual information is a quantity easily computed, and our results indicate that for relatively small lattice sizes its finite-size behavior already detects the universality class of quantum critical behavior.PACS numbers: 11.25. Hf, 03.67.Bg, 89.70.Cf, 75.10.Pq Entanglement measures have emerged nowadays as powerful tools for the study of quantum many body systems [1,2]. In one dimension, where most quantum critical systems have their long-distance physics ruled by a conformal field theory (CFT), the entanglement entropy has been proved the most important measure of entanglement. It allows one to identify the distinct universality classes of critical behaviors. Let us consider a periodic quantum chain with L sites, and partition the system into subsystems A and B of length ℓ and L − ℓ, respectively. The entanglement entropy is defined as the von Neumann entropy of the reduced density matrix ρ A of the partition A: S ℓ = −T r A ρ A ln ρ A . If the system is critical and in the ground state, in the regime where the subsystems are large compared with the lattice spacing, S ℓ is given by [3,4] where c is the central charge of the underlying CFT and γ s is a non-universal constant. A remarkable fact is that even in the case where the system is in a pure state formed by an excited state, the conformal anomaly dictates the overall behavior of the entanglement, similarly as in (1) [5]. It is worth mentioning that recently many interesting methods were proposed [6-8] to calculate the entanglement entropy and ultimately central charge, however, up to know they have not been implemented experimentally. A natural question concerns the possible existence of other measures of shared information that, similarly as the entanglement entropy, are also able to detect the several universality classes of critical behavior of quantum critical chains. In this Letter we present results that indicate that the Shannon mutual information of local observables is such a measure. The Shannon mutual information of the subsystems A and B, of sizes ℓ and L − ℓ is defined aswhere Sh(X ) = − x p x ln p x is the Shannon entropy of the subsystem X with probabilities p x of being in a configuration x. These probabilities, in the case where A is a subsystem of a quantum chain with wavefunction |Ψ A∪B = n,m c n,m |φ It is important to notice that the Shannon entropy and the Shannon mutual information are basis dependent quantities, reflecting the several kinds of obse...
-We study the ferromagnetic Ising model with long-range interactions in two dimensions. We first present results of a Monte Carlo study which shows that the long-range interactions dominate over the short-range ones in the intermediate regime of interaction range. Based on a renormalization group analysis, we propose a way of computing the influence of the long-range interactions as a dimensional change.Introduction. -In recent years there has been a lot of interest in the statistical physics of classical and quantum systems with long-range interactions, for a review see [1]. The role of quasi-stationary states and ergodicity breaking in long-range interacting systems was investigated in [2] and [3]. In [4] the approaching to equilibrium for long-range quantum systems was examined and there has been a lot of enthusiasm in investigating the entanglement entropy in long-range spin chains [5][6][7]. Very recently an experiment was conducted on a quantum system with tunable long-range interactions [8].In the present study we focus on the Ising model which is probably the most studied model in statistical mechanics, especially in the context of critical phenomena. Most of the studies about the Ising model are concentrated around the short-range case which is exactly solvable in one and two dimensions [9]. In three dimensions the problem was perturbatively studied using the ǫ-expansion technique [10] of the renormalization group (RG) combined with the Borel resummation of the perturbation series, see [11] and references therein. Most recently the problem was revisited by using conformal bootstrap technique [12]. Although now there are little unknown facts around shortrange Ising model the long-range Ising model is still the subject of many contradicting theoretical and numerical studies. We define the long-range Ising model as
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