arXiv:1003.1110 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy of two disjoint blocks inXYchains

Maurizio Fagotti, Pasquale Calabrese

Published Apr 16, 2010• 194 citations

Abstract

We study the Rényi entanglement entropies of two disjoint intervals in XY chains. We exploit the exact solution of the model in terms of free Majorana fermions and we show how to construct the reduced density matrix in the spin variables by taking the Jordan–Wigner string between the two blocks properly into account. From this we can evaluate any Rényi entropy of finite integer order. We study in detail critical XX and Ising chains and we show that the asymptotic results for large blocks agree with recent conformal field theory predictions if corrections to the scaling are included in the analysis correctly. We also report results for the gapped phase and after a quantum quench.

DOI: 10.1088/1742-5468/2010/04/p04016

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arXiv:1003.1110 · Journal of Statistical Mechanics Theory and Experiment

Entanglement entropy of two disjoint blocks inXYchains

Maurizio Fagotti, Pasquale Calabrese

Published Apr 16, 2010• 194 citations

arXiv HTML PDF

Abstract

DOI: 10.1088/1742-5468/2010/04/p04016

Related papers

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The half-chain entanglement entropy (EE) of the one-dimensional $XX$ (isotropic $XY$ ) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed ( $++$ ) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the $XX$ chain and QICs: (1) the EE of the $XX$ chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the $XX$ chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the $XX$ chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the $XX$ chain with fixed-fixed ( $++$ ) BCs is the sum of the EE of a QIC with fixed-fixed ( $++$ ) BCs and that with free-free BCs; and (5) the EE of the $XX$ chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed ( $++$ ) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where $L$ is the length of the spin chain. We find that the fitted parameters for the leading term $lnL$ and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.The half-chain entanglement entropy (EE) of the one-dimensional $XX$ (isotropic $XY$) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed ($++$) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the $XX$ chain and QICs: (1) the EE of the $XX$ chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the $XX$ chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the $XX$ chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the $XX$ chain with fixed-fixed ($++$) BCs is the sum of the EE of a QIC with fixed-fixed ($++$) BCs and that with free-free BCs; and (5) the EE of the $XX$ chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed ($++$) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where $L$ is the length of the spin chain. We find that the fitted parameters for the leading term $lnL$ and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.

Physical review. B./Physical review. BMar 27, 20252 citations

A bstract We take the paradigm of interacting spin chains, the Heisenberg spin-

\frac{1}{2}

1 2 XXZ model, as a reference system and consider interacting models that are related to it by Jordan-Wigner transformations and restrictions to sub-chains. An example is the fermionic analogue of the gapless XXZ Hamiltonian, which, in a continuum scaling limit, is described by the massless Thirring model. We work out the Rényi- α entropies of disjoint blocks in the ground state and extract the universal scaling functions describing the Rényi- α tripartite information in the limit of infinite lengths. We consider also the von Neumann entropy, but only in the limit of large distance. We show how to use the entropies of spin blocks to unveil the spin structures of the underlying massless Thirring model. Finally, we speculate about the tripartite information after global quenches and conjecture its asymptotic behaviour in the limit of infinite time and small quench. The resulting conjecture for the “residual tripartite information”, which corresponds to the limit in which the intervals’ lengths are infinitely larger than their (large) distance, supports the claim of universality recently made studying noninteracting spin chains. Our mild assumptions imply that the residual tripartite information after a small quench of the anisotropy in the gapless phase of XXZ is equal to − log 2.

The half-chain entanglement entropy (EE) of the one-dimensional

XX

(isotropic

XY

) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed (

++

) and ($+\ensuremath{-}$), and fixed-free boundary conditions (BCs) are studied. The models are transformed into a fermion system with a Jordan-Wigner transformation. The correlation functions of these fermions are obtained numerically. With the correlation functions, the entanglement entropy is obtained. The numerical calculation is performed with quadruple precision. The numerical results agree with the exact relationship with an accuracy of ${10}^{\ensuremath{-}28}$ between the EEs of the

XX

chain and QICs: (1) the EE of the

XX

chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the

XX

chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the

XX

chain with fixed-free BCs is the sum of the EE of a QIC with fixed-free BCs and that with free-free BCs; (4) the EE of the

XX

chain with fixed-fixed (

++

) BCs is the sum of the EE of a QIC with fixed-fixed (

++

) BCs and that with free-free BCs; and (5) the EE of the

XX

chain with fixed-fixed ($+\ensuremath{-}$) BCs is the sum of the EE of a QIC with fixed-fixed (

++

) BCs and that of the first excited state of a QIC with free-free BCs. We fit the half-chain EE with finite-size scaling expansion of terms ${L}^{\ensuremath{-}n}$ and ${L}^{\ensuremath{-}n}lnL$, where

L

is the length of the spin chain. We find that the fitted parameters for the leading term

lnL

and the constant term are consistent with conformal field theory and that the exact solutions are highly accurate. The fitting accuracy is so high that the fitted central charge of the Ising model is $0.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}$ for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist ${L}^{\ensuremath{-}4}lnL$ and higher-order corrections for the fixed-fixed ($+\ensuremath{-}$) boundary conditions, whereas there exist ${L}^{\ensuremath{-}3}lnL$ and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.