[Paper Review] Entanglement Entropy for Disjoint Subsystems in XX Spin Chain

Bai-Qi Jin, V. E. Korepin|arXiv (Cornell University)|Apr 6, 2011

Quantum many-body systems1 references3 citations

TL;DR

This paper presents an exact determinant formulation for the entanglement entropy of multiple disjoint intervals in the ground state of the XX spin chain using free fermion techniques. By generalizing the fermionic mapping to non-contiguous subsystems, it derives a block-Toeplitz determinant representation, enabling analytical treatment of Rényi and von Neumann entropies. The key result shows that mutual information between two equal-length, equally separated intervals vanishes in the infinite-size limit, even in the critical phase.

ABSTRACT

Fisher-Hartwig formula has been successful applied to describe the von Neumann and Rényi entropies of a block of spins in the ground state of XX spin chain. It was based on a determinant representation. In this paper, we generalize the free fermion method to obtain an exact formulation for the entropy of any finite subsystem in XX spin chain. Based on this, we derive a determinant representation of the entropy of multiple disjoint intervals in the ground state of $XX$ model.

Motivation & Objective

• To extend the free fermion method beyond contiguous subsystems to derive exact entanglement entropy for multiple disjoint intervals in the XX spin chain.
• To establish a determinant representation of the reduced density matrix for non-contiguous subsystems using Majorana fermion operators and Wick's theorem.
• To analyze the mutual information between two disjoint intervals in the critical XX chain and determine its asymptotic behavior.
• To provide a framework for analytical treatment of entanglement entropy in extended quantum spin systems with non-adjacent subsystems.

Proposed method

• Map the XX spin chain to a system of free Majorana fermions using the Jordan-Wigner transformation and define two sets of Majorana operators: one for the full chain and one restricted to subsystem A.
• Use Wick's theorem to compute all two-point correlation functions ⟨GS|~cₘ~cₙ|GS⟩ for the restricted Majorana operators on subsystem A, which depend on the positions of sites in A and B.
• Construct a block matrix A representing the correlation structure of the reduced density matrix, with blocks corresponding to intervals in A and off-diagonal blocks encoding correlations across gaps in B.
• Express the entanglement entropy via the eigenvalues of the correlation matrix A, using the formula S = -∑ᵢ λᵢ log λᵢ + (1 - λᵢ) log(1 - λᵢ) for von Neumann entropy.
• Derive a determinant representation of the entropy as a block-Toeplitz-like matrix Tₘₙ, with generators depending on the Fourier symbol g(θ) of the XX model.
• Apply the Fisher-Hartwig formula to analyze the asymptotic behavior of the determinant, particularly for mutual information between two disjoint intervals.

Experimental results

Research questions

• RQ1 Can the free fermion method be generalized to compute entanglement entropy for non-contiguous, multiple disjoint intervals in the XX spin chain?
• RQ2 What is the structure of the reduced density matrix for a subsystem composed of disjoint intervals in the ground state of the XX model?
• RQ3 How does the mutual information between two disjoint intervals scale in the thermodynamic limit, especially in the critical phase?
• RQ4 Can a determinant representation be derived for the entanglement entropy of multiple disjoint intervals to enable analytical analysis?
• RQ5 Does the mutual information between two separated intervals vanish in the infinite-size limit, even when the system is at a quantum critical point?

Key findings

• An exact determinant formulation for the entanglement entropy of any finite, non-contiguous subsystem in the XX spin chain is derived using free fermion techniques and correlation matrices.
• The reduced density matrix of a disjoint subsystem is expressed via a block-Toeplitz-like correlation matrix, with eigenvalues determining the von Neumann and Rényi entropies.
• For two disjoint intervals of length m separated by distance m in the critical XX chain, the mutual information I_{A₁:A₂} is computed numerically for large m.
• The mutual information decreases with increasing m and approaches zero in the limit m → ∞, indicating asymptotic independence of the intervals.
• This result confirms that even in the critical phase, long-range entanglement between disjoint intervals vanishes when both interval length and separation scale to infinity.
• The paper establishes a general framework for the analytical study of entanglement entropy in extended quantum spin systems with non-adjacent subsystems.