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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.

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Author notes

  1. F. Franchini

    Present address: SISSA, Via Beirut 2-4, 34151, Trieste, TS, Italy

Authors and Affiliations

  1. The Abdus Salam ICTP, Strada Costiera 11, 34014, Trieste, TS, Italy

    F. Franchini

  2. Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, Indianapolis, IN, 46202-3216, USA

    A. R. Its

  3. C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY, 11794-3840, USA

    V. E. Korepin

  4. Mathematics Department, State University of New York at Stony Brook, Stony Brook, NY, 11794-3840, USA

    L. A. Takhtajan

Authors
  1. F. Franchini
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  2. A. R. Its
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  3. V. E. Korepin
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  4. L. A. Takhtajan
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Corresponding author

Correspondence to V. E. Korepin.

Additional information

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.

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Franchini, F., Its, A.R., Korepin, V.E. et al. Spectrum of the density matrix of a large block of spins of the XY model in one dimension. Quantum Inf Process 10, 325–341 (2011). https://doi.org/10.1007/s11128-010-0197-7

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  • DOI: https://doi.org/10.1007/s11128-010-0197-7

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