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Physical review. B./Physical review. B

Finite-size scaling of half-chain entanglement entropy in the one-dimensional transverse field Ising model and the X X model

Haopeng Tian, Tianhao He, Xintian Wu

Published Mar 27, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.111.104437

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The half-chain entanglement entropy (EE) of the one-dimensional XXXXXX (isotropic XYXYXY) chain and quantum Ising chain (QIC) at the critical point are calculated numerically for large lattices. The periodic, free-free, fixed-fixed (++++++) and (+\ensuremath−+\ensuremath{-}+\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{10}^{\ensuremath{-}28}10\ensuremath−28 between the EEs of the XXXXXX chain and QICs: (1) the EE of the XXXXXX chain with periodic BCs is the sum of two QICs with periodic BCs; (2) the EE of the XXXXXX chain with free-free BCs is the sum of two QICs with free-free BCs; (3) the EE of the XXXXXX 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 XXXXXX 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 XXXXXX chain with fixed-fixed (+\ensuremath−+\ensuremath{-}+\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{L}^{\ensuremath{-}n}L\ensuremath−n and L\ensuremath−nlnL{L}^{\ensuremath{-}n}lnLL\ensuremath−nlnL, where LLL is the length of the spin chain. We find that the fitted parameters for the leading term lnLlnLlnL 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±\else\textpm\fi1\ifmmode×\else\texttimes\fi10\ensuremath−260.5\ifmmode\pm\else\textpm\fi{}1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}0.5\ifmmode±\else\textpm\fi1\ifmmode×\else\texttimes\fi10\ensuremath−26 for the periodic boundary condition. From the higher-order corrections in the fitted expansion, we find that there exist L\ensuremath−4lnL{L}^{\ensuremath{-}4}lnLL\ensuremath−4lnL and higher-order corrections for the fixed-fixed (+\ensuremath−+\ensuremath{-}+\ensuremath−) boundary conditions, whereas there exist L\ensuremath−3lnL{L}^{\ensuremath{-}3}lnLL\ensuremath−3lnL and higher-order corrections for the fixed-free boundary conditions. These types of corrections mean corrections of the central charge.
The exact relation between the entanglement entropies of the XYXYXY and quantum Ising chains with free and fixed boundary conditions
The entanglement entropies of XYXYXY 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 XYXYXY 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 XYXYXY 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 XYXYXY 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.
Abstract We study the relative entanglement entropy (EE) among various primary excited states in two critical spin chains: the S = 1 / 2 XXZ chain and the transverse field Ising chain at criticality. For the S = 1 / 2 XXZ chain, which corresponds to c = 1 free boson conformal field theory (CFT), we numerically calculate the relative EE by exact diagonalization and find a perfect agreement with the predictions by the CFT. For the transverse field Ising chain at criticality, which corresponds to the c = 1 / 2 Ising CFT, we analytically relate its relative EE to that of the S = 1 / 2 XXZ chain and confirm the relation numerically. We also calculate the ‘sandwiched’ Rényi relative EE and again the numerical results agree well with the analytical predictions. Our results are the first direct confirmation of the CFT predictions on the relative EE of the primary excited states in critical spin chains.
We calculate the entanglement and the universal boundary entropy (BE) in critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to the simple fact that the boundary field breaks the bulk symmetry or does not. We present a procedure to estimate the universal BE, based on the finite-size corrections of the entanglement entropy, which apply to ADBMF. To calculate the universal BE in the XXZ chain, we use the density matrix renormalization group. The transverse field XY chain with ADBMF after Jordan-Wigner transformation is not a quadratic free fermion Hamiltonian. We map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem integrable. The eigenstates of the transverse field XY chain can be obtained by proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.
We present a study of the scaling behavior of the R\'enyi entanglement entropy (REE) in SU(N)\mathrm{SU}(N)SU(N) spin chain Hamiltonians, in which all of the spins transform under the fundamental representation. These SU(N)\mathrm{SU}(N)SU(N) spin chains are known to be quantum critical and described by a well known Wess-Zumino-Witten (WZW) nonlinear sigma model in the continuum limit. Numerical results from our lattice Hamiltonian are obtained using stochastic series expansion quantum Monte Carlo for both closed and open boundary conditions. As expected for this 1D critical system, the REE shows a logarithmic dependence on the subsystem size with a prefactor given by the central charge of the SU(N)\mathrm{SU}(N)SU(N) WZW model. We study in detail the subleading oscillatory terms in the REE under both periodic and open boundaries. Each oscillatory term is associated with a WZW field and decays as a power law with an exponent proportional to the scaling dimension of the corresponding field. We find that the use of periodic boundaries (where oscillations are less prominent) allows for a better estimate of the central charge, while using open boundaries allows for a better estimate of the scaling dimensions. We also present numerical data on the thermal R\'enyi entropy which equally allows for extraction of the central charge.
The following thesis focuses on the scaling of entanglement entropy in lower dimensions and is divided into three main parts. Chapter 2 studies the thermal reduced density matrices in fermion and spin systems on ladders. Chapter 3 studies the many-body localization phase transition in a Rokhsar-Kivelson type wave function. Chapter 4 studies the subleading correction term of entanglement entropy in 2+1 dimensional scale invariant systems. Chapter 5 studies the bulk-boundary correspondence in 3 + 1 dimensional topological phases and its entanglement entropy. In chapter 2, we investigate the reduced density matrices for a model of free fermions on a two-leg ladder (gapped by the inter-chain tunneling operator) and in 1/2 spin systems on a ladder with a gapped ground state using exact solutions and several controlled approximations. We calculate the reduced density matrix and the entanglement entropy for a leg of the ladder (i.e. cut made between the chains). In the fermionic system we find the exact form of the reduced density matrix for one of the chains and determine the entanglement spectrum explicitly. Here we find that in the weak tunneling limit of the ladder the entanglement entropy of one chain of the gapped ladder has a simple and universal form dictated by conformal invariance. In the case of the spin system, we consider the strong coupling limit by using perturbation theory and get the reduced density matrix by the Schmidt decomposition. The entanglement entropies of a general gapped system of two coupled conformal field theories (in 1+1 dimensions) is discussed using the replica trick and scaling arguments. We show that 1) for a system with a bulk gap, the reduced density matrix has the form of a thermal density matrix, 2) the long-wavelength modes of one subsystem (a chain) of a gapped coupled system are always thermal, 3) the von Neumann entropy equals to the thermodynamic entropy of one chain, and 4) the bulk gap plays the role of effective temperature. In chapter 3, we construct a family of many-body wave functions to study the many-body localization phase transition. The wave functions have a Rokhsar-Kivelson form, in which the weight for the configurations are chosen from the Gibbs weights of a classical spin glass model, known as the Random Energy Model, multiplied by a random sign structure to represent a highly excited state. These wave functions show a phase transition into an MBL phase. In addition, we see three regimes of entanglement scaling with subsystem size: scaling with entanglement corresponding to an infinite temperature thermal phase, constant scaling, and a sub-extensive scaling between these limits. Near the phase transition point, the fluctuations of the Renyi entropies are non-Gaussian. We find that Renyi entropies with different Renyi index transition into the MBL phase at different points and have different scaling behavior, suggesting a multifractal behavior. In chapter 4, we study the universal scaling behavior of the entanglement entropy of critical theories in 2+1 dimensions. We specially consider two fermionic scale-invariant models, free massless Dirac fermions and a model of fermions with quadratic band touching, and numerically study the two-cylinder entanglement entropy of the models on the torus. We find that in both cases the entanglement entropy satisfies the area law and has the subleading term which is a scaling function of the aspect ratios of the cylindrical regions. We test the scaling of entanglement in both the free fermion models using three possible scaling functions for the subleading term derived from a) the quasi-one-dimensional conformal field theory, b) the bosonic quantum Lifshitz model, and c) the holographic AdS/CFT correspondence. For the later case we construct an analytic scaling function using holography, appropriate for critical theories with a gravitational dual description. We find that the subleading term in the fermionic models is well described, for a range of aspect ratios, by the scaling form derived from the quantum Lifshitz model as well as that derived using the AdS/CFT correspondence (in this case only for the Dirac model). For the case where the fermionic models are placed on a square torus we find the fit to the different scaling forms is in agreement to surprisingly high precision. In chapter 5, we discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus T3 parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under SL(3, Z) modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular S and T matrices computed from the boundary field theories with those computed in the bulk. We propose the three-loop braiding statistics can be studied by constructing the modular S and T matrices from an appropriate boundary field theory. We also study the EE for 3 + 1- dimensional topological phase with or without three-loop braiding phase.
We develop the finite-size scaling (FSS) theory at quantum transitions. We consider various boundary conditions, such as open and periodic boundary conditions, and characterize the corrections to the leading FSS behavior. Using renormalization-group (RG) theory, we generalize the classical scaling ansatz to describe FSS in the quantum case, classifying the different sources of scaling corrections. We identify nonanalytic corrections due to irrelevant (bulk and boundary) RG perturbations and analytic contributions due to regular backgrounds and analytic expansions of the nonlinear scaling fields. To check the general predictions, we consider the quantum XYXYXY chain in a transverse field. For this model exact or numerically accurate results can be obtained by exploiting its fermionic quadratic representation. We study the FSS of several observables, such as the free energy, the energy differences between low-energy levels, correlation functions of the order parameter, etc., confirming the general predictions in all cases. Moreover, we consider bipartite entanglement entropies, which are characterized by the presence of additional scaling corrections, as predicted by conformal field theory.