Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1501.04311 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy and negativity of disjoint intervals in CFT: some numerical extrapolations

Cristiano De Nobili, Andrea Coser, Erik Tonni C

Published Jun 10, 2015• 97 citations
arXivHTMLPDF

Abstract

The entanglement entropy and the logarithmic negativity can be computed in quantum field theory through a method based on the replica limit. Performing these analytic continuations in some cases is beyond our current knowledge, even for simple models. We employ a numerical method based on rational interpolations to extrapolate the entanglement entropy of two disjoint intervals for the conformal field theories given by the free compact boson and the Ising model. The case of three disjoint intervals is studied for the Ising model and the non compact free massless boson. For the latter model, the logarithmic negativity of two disjoint intervals has been also considered. Some of our findings have been checked against existing numerical results obtained from the corresponding lattice models.

DOI
10.1088/1742-5468/2015/06/p06021

Related papers

Entanglement entropy of two disjoint intervals in conformal field theory: II

Pasquale Calabrese, John Cardy, Erik Tonni C

We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.

arXiv:1011.5482•Journal of Statistical Mechanics Theory and Experiment•Jan 20, 2011•318 citations

Entanglement entropy of two disjoint intervals in conformal field theory

Pasquale Calabrese, John Cardy, Erik Tonni C

We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr ρ A n for any integer n is calculated as the four-point function of twist fields of a particular type and the final result is expressed in a compact form in terms of the Riemann–Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.

arXiv:0905.2069•Journal of Statistical Mechanics Theory and Experiment•Nov 4, 2009•362 citations

Entanglement entropy of two disjoint intervals inc= 1 theories

Vincenzo Alba, Luca Tagliacozzo, Pasquale Calabrese

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 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 to the leading scaling behavior.

arXiv:1103.3166•Journal of Statistical Mechanics Theory and Experiment•Jun 15, 2011•85 citations

Entanglement negativity in extended systems: a field theoretical approach

Pasquale Calabrese, John Cardy, Erik Tonni C

We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.

arXiv:1210.5359•Journal of Statistical Mechanics Theory and Experiment•Feb 1, 2013•311 citations

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

Gavin Rockwood

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.

arXiv:2203.04339•Journal of Statistical Mechanics Theory and Experiment•Aug 1, 2022•3 citations

Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks

Paola Ruggiero, Erik Tonni C, Pasquale Calabrese

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.

arXiv:1805.05975•Journal of Statistical Mechanics Theory and Experiment•Nov 1, 2018•66 citations
The entanglement entropy and the logarithmic negativity can be computed in quantum field theory through a method based on the replica limit. Performing these analytic continuations in some cases is beyond our current knowledge, even for simple models. We employ a numerical method based on rational interpolations to extrapolate the entanglement entropy of two disjoint intervals for the conformal field theories given by the free compact boson and the Ising model. The case of three disjoint intervals is studied for the Ising model and the non compact free massless boson. For the latter model, the logarithmic negativity of two disjoint intervals has been also considered. Some of our findings have been checked against existing numerical results obtained from the corresponding lattice models.
We continue the study of the entanglement entropy of two disjoint intervals in conformal field theories that we started in J. Stat. Mech. (2009) P11001. We compute Trρ_A^n for any integer n for the Ising universality class and the final result is expressed as a sum of Riemann-Siegel theta functions. These predictions are checked against existing numerical data. We provide a systematic method that gives the full asymptotic expansion of the scaling function for small four-point ratio (i.e. short intervals). These formulas are compared with the direct expansion of the full results for free compactified boson and Ising model. We finally provide the analytic continuation of the first term in this expansion in a completely analytic form.
We study the entanglement of two disjoint intervals in the conformal field theory of the Luttinger liquid (free compactified boson). Tr ρ A n for any integer n is calculated as the four-point function of twist fields of a particular type and the final result is expressed in a compact form in terms of the Riemann–Siegel theta functions. In the decompactification limit we provide the analytic continuation valid for all model parameters and from this we extract the entanglement entropy. These predictions are checked against existing numerical data.
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 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 to the leading scaling behavior.
We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.
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.