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arXiv:2203.04339 · Journal of Statistical Mechanics Theory and Experiment

Replicated entanglement negativity for disjoint intervals in the Ising conformal field theory

Gavin Rockwood

Published Aug 1, 2022• 3 citations
arXivHTMLPDF

Abstract

Abstract We calculate the N interval replicated negativity for the Ising conformal field theory using correlation functions of branch point twist fields. For some subset P ⊂ N , this is a calculation of Tr ( ρ A T P ) n where n is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression for Tr ( ρ A T P ) n . The results are expressed such that when P = Ø, which corresponds to calculating the Rényi entropy, the formulas aligns with known results for the N interval Rényi entropy.

DOI
10.1088/1742-5468/ac873f

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Abstract We calculate the N interval replicated negativity for the Ising conformal field theory using correlation functions of branch point twist fields. For some subset P ⊂ N , this is a calculation of Tr ( ρ A T P ) n where n is the replica index. This can be reformulated as a calculation of partition functions over superelliptic Riemann surfaces, and for the models in question, this partition function can be expressed in terms of the period matrix of this surface. We detail how to construct the period matrices for these surfaces, giving an analytic expression for Tr ( ρ A T P ) n . The results are expressed such that when P = Ø, which corresponds to calculating the Rényi entropy, the formulas aligns with known results for the N interval Rényi entropy.
Abstract We reconsider the computation of the entanglement entropy of two disjoint intervals in a (1+1) dimensional conformal field theory by conformal block expansion of the 4-point correlation function of twist fields. We show that accurate results may be obtained by taking into account several terms in the operator product expansion of twist fields and by iterating the Zamolodchikov recursion formula for each conformal block. We perform a detailed analysis for the Ising conformal field theory and for the free compactified boson. Each term in the conformal block expansion can be easily analytically continued and so this approach also provides a good approximation for the von Neumann entropy.
We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure E based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that E has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating E at each order n is always a torus topologically. By analytic continuation, the result of n=1/2 gives the value of E. Furthermore, the results of other values of n also yield meaningful conclusions: The n=1 result gives a general formula for the two-interval purity, which enables us to calculate the Rényi-2 N-partite information for N≤4 intervals; while the n=∞ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-1/2 XXZ chain, whose ground state is described by the Luttinger liquid.
Abstract We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point.
In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D conformal field theory (CFT) at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on nnn-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is a one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the noncompact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (10) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.
Abstract We study conformal twist field four-point functions on a ℤN orbifold. We examine in detail the case N = 3 and analyze theories obtained by replicated N-times a minimal model with central charge c < 1. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge Nc is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density matrix. Finally, we refine the bounds on the structure constants of unitary CFTs determined previously by the genus two modular bootstrap.
We propose a universal approach to compute the Renyi entropy with generically multiple disjoint intervals from the swapping operations. The proposal is grounded in our observation that there is a striking similarity between the replica trick utilized in quantum field theory and the process of swapping operations. Consequently, the Renyi entropy of multiple disjoint intervals can be obtained by evaluating the expectation values of the swapping operator. As an application, we use our proposal to study the Renyi entropy of a one-dimensional transverse-field Ising model with two, three and four disjoint intervals. In particular, as the system is at the critical point, our results of the Renyi entropy are consistent with the analytical results from the conformal field theory. Remarkably, our approach can go beyond the critical regime where the analytical methods from conformal field theory can not apply.