Spectrum of the density matrix of a large block of spins of the XY model in one dimension

2010/02/15 by Fabio Franchini, F. Franchini, A. R. Its +3 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Opinion Dynamics and Social Influence #Quantum Information and Cryptography #Quantum many-body systems #cond-mat.stat-mech #cond-mat.str-el #hep-th #math-ph #math.MP #quant-ph

paper · pdf · doi:10.1007/s11128-010-0197-7

published as Quantum Information Processing 10: 325-341 (2011) · 13 pages, 5 figures

arxiv created 2010/02/15 · openalex publication_date 2010/09/27 · arxiv updated 2012/01/31 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/29

Abstract

We consider the reduced density matrix of a large block of consecutive spins in the ground states of the XY spin chain on an infinite lattice. We derive the spectrum of the density matrix using the expression of the Renyi entropy in terms of modular functions. The eigenvalues λ_n form an exact geometric sequence. For example, for strong magnetic field λ_n = C exp(-πτ₀ n), here τ₀>0 and C > 0 depend on the anisotropy and the magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but the degeneracy g_n increases sub-exponentially as eigenvalues diminish: g_n ∼ exp(π√(n/3)). For weak magnetic field expressions are similar.

Citations

• Area laws for the entanglement entropy - a review
• Two soluble models of an antiferromagnetic chain

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