, Volume 10, Issue 3, pp 325-341
Date: 28 Sep 2010

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

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Abstract

We consider 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 expression of Rényi entropy in terms of modular functions. The eigenvalues λ n form exact geometric sequence. For example, for strong magnetic field λ n = C exp(−π τ 0 n), here τ 0 > 0 and C > 0 depend on anisotropy and magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but degeneracy g n increases sub-exponentially as eigenvalues diminish: \({g_{n}\sim \exp{(\pi \sqrt{n/3})}}\) . For weak magnetic field expressions are similar.

F. Franchini was supported in part by PRIN Grant 2007JHLPEZ. Alexander Its was supported by NSF Grant DMS-0701768. Vladimir Korepin was supported by NSF Grant DMS 0905744.