119279423
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 us 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 on the leading scaling behavior.
118322763
Entanglement negativity and conformal field theory: a Monte Carlo study
We investigate the behavior of the moments of the partially transposed reduced density matrix in critical quantum spin chains. Given subsystem A as the union of two blocks, this is the (matrix) transpose of ρA with respect to the degrees of freedom of one of the two blocks. This is also the main ingredient for constructing the logarithmic negativity. We provide a new numerical scheme for efficiently calculating all the moments of using classical Monte Carlo simulations. In particular we study several combinations of the moments which are scale invariant at a critical point. Their behavior is fully characterized in both the critical Ising and the anisotropic Heisenberg XXZ chains. For two adjacent blocks we find, in both models, full agreement with recent conformal field theory (CFT) calculations. For disjoint blocks, in the Ising chain finite size corrections are nonnegligible. We demonstrate that their exponent is the same as that governing the unusual scaling corrections of the mutual information between the two blocks. Monte Carlo data fully match the theoretical CFT prediction only in the asymptotic limit of infinite intervals. Oppositely, in the Heisenberg chain scaling corrections are smaller and, already at finite (moderately large) block sizes, Monte Carlo data are in excellent agreement with the asymptotic CFT result.
118782522
Non-conformal limit of AGT relation from the 1-point torus conformal block
Given a 4d N=2 SUSY gauge theory, one can construct the Seiberg-Witten prepotentional, which involves a sum over instantons. Integrals over instanton moduli spaces require regularisation. For UV-finite theories the AGT conjecture favours particular, Nekrasov's way of regularization. It implies that Nekrasov's partition function equals conformal blocks in 2d theories with W_{N_c} chiral algebra. For $N_c=2$ and one adjoint multiplet it coincides with a torus 1-point Virasoro conformal block. We check the AGT relation between conformal dimension and adjoint multiplet's mass in this case and investigate the limit of the conformal block, which corresponds to the large mass limit of the 4d theory e.i. the asymptotically free 4d N=2 supersymmetric Yang-Mills theory. Though technically more involved, the limit is the same as in the case of fundamental multiplets, and this provides one more non-trivial check of AGT conjecture.