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Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture

B. -Q. Jin, V. E. Korepin

TL;DR

This work analyzes entanglement entropies of a block of L spins in the XX model with a transverse field on an infinite chain at zero temperature. By mapping to Majorana and fermionic modes and expressing the density matrix via a Toeplitz determinant, the authors derive a closed-form entropy in terms of ν_i eigenvalues, then apply Fisher-Hartwig theory to obtain the leading large-L behavior S(ρ_A) ∼ (1/3) ln L with a field-dependent constant. They also introduce a universal scaling variable 𝓛 = 2L sqrt{1−(h/2)^2} to describe crossovers and provide Rényi entropy expressions with analogous scaling components. The results quantify how the magnetic field modulates subsystem entanglement and establish a precise link between quantum entanglement and Toeplitz/Fisher-Hartwig determinants.

Abstract

We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnetic field interacting with spins is very simple: the magnetic field effectively reduce the size of the subsystem. We also calculate entropy of a subsystem of a small size. We also evaluated Renyi and Tsallis entropies of the subsystem. We represented the entropy in terms of a Toeplitz determinant and calculated the asymptotic analytically.

Quantum Spin Chain, Toeplitz Determinants and Fisher-Hartwig Conjecture

TL;DR

This work analyzes entanglement entropies of a block of L spins in the XX model with a transverse field on an infinite chain at zero temperature. By mapping to Majorana and fermionic modes and expressing the density matrix via a Toeplitz determinant, the authors derive a closed-form entropy in terms of ν_i eigenvalues, then apply Fisher-Hartwig theory to obtain the leading large-L behavior S(ρ_A) ∼ (1/3) ln L with a field-dependent constant. They also introduce a universal scaling variable 𝓛 = 2L sqrt{1−(h/2)^2} to describe crossovers and provide Rényi entropy expressions with analogous scaling components. The results quantify how the magnetic field modulates subsystem entanglement and establish a precise link between quantum entanglement and Toeplitz/Fisher-Hartwig determinants.

Abstract

We consider one-dimensional quantum spin chain, which is called XX model, XX0 model or isotropic XY model in a transverse magnetic field. We study the model on the infinite lattice at zero temperature. We are interested in the entropy of a subsystem [a block of L neighboring spins]. It describes entanglement of the block with the rest of the ground state. G. Vidal, J.I. Latorre, E. Rico, and A. Kitaev showed that for large blocks the entropy scales logarithmically. We prove the logarithmic formula for the leading term and calculate the next term. We discovered that the dependence on the magnetic field interacting with spins is very simple: the magnetic field effectively reduce the size of the subsystem. We also calculate entropy of a subsystem of a small size. We also evaluated Renyi and Tsallis entropies of the subsystem. We represented the entropy in terms of a Toeplitz determinant and calculated the asymptotic analytically.

Paper Structure

This paper contains 6 sections, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Density Matrix of Sub-system A
  3. Closed Form for The Entropy
  4. Toeplitz Matrix and Fisher-Hartwig Conjecture
  5. Asymptotic Form of The Entropy
  6. Summary

Figures (1)

  • Figure 1: The contour $c(\epsilon,\delta)$. Bold lines $(-\infty, -1-\epsilon)$ and $(1+\epsilon,\infty)$ are the cuts of integrand $e(1+\epsilon,\lambda)$. Zeros of $D_{\mathrm{L}}(\lambda)$ (Eq. \ref{['exd']}) are located on bold line $(-1, 1)$ and this line becomes the cut of $\mathrm{d} \log D_{\mathrm{L}}(\lambda)$ for $L\to \infty$ (Eq. \ref{['apd']}). The arrow is the direction of the route of integral we take and $\mathrm{R}$ is the radius of circles.