Entanglement Entropy in an Ising Chain with Alternating Field

Jie Ren, Shiqun Zhu

Haopeng Tian, Tianhao He, Xintian Wu

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.

Hai Tao Wang, Sam Young Cho

Non-local orders, entanglement entropy, and quantum fidelity are investigated in an infinite-size bond-alternating Ising chain with the Dzyaloshinskii-Moriya interaction by employing the infinite matrix product state representation with the infinite time evolving block decimation method. Directly computing two distinct types of finite string correlations for very large lattice distances, in contrast to an extrapolated extreme value for finite size chains, reveals two topologically ordered phases. As the bond alternation varies, a topological quantum phase transition with continuously variable critical exponents along the phase boundary occurs between the two Haldane phases for a Dzyaloshinskii-Moriya interaction stronger than the Ising interaction while a topological quantum crossover between them happens through an intermediate antiferromagentic phase demonstrated with the quantum fidelity for a Dzyaloshinskii-Moriya interaction weaker than the Ising interaction. The critical exponents of the order parameters and the central charges from the entanglement entropy quantify the universality classes of the phase transition points. Anisotropic Heisenberg types of spin chains with bond alternations are finally discussed to share the same criticality.

Geoffrey R. Grimmett, Tobias J. Osborne, Petra F. Scudo

Durrani, Ijaz

He, Tianhao, Xintian Wu

The entanglement entropies of $XY$ chains and quantum Ising chains (QICs) with fixed boundary conditions are studied here. Three kinds of boundary conditions (BCs) are considered: fixed up--up or down--down (the spins at both ends are aligned in the same direction), fixed up--down or down--up (the spins at the two ends are aligned in opposite directions), and fixed--free (the spin at one end is aligned, and the other end is free). It is shown that i) the entanglement entropy of an $XY$ chain with a fixed--free BC is the sum of those of QICs with a fixed--free BC and with a free--free BC; ii) the entanglement entropy of an $XY$ chain with a fixed up--up boundary condition is the sum of those of QICs with a fixed up--up BC and with a free--free BC; and iii) the entanglement entropy of an $XY$ chain with a fixed up--down BC is the sum of that of a QIC with a fixed up--up BC and that of the first excited state of a QIC with a free--free BC.