Quantum Information Processing

, Volume 10, Issue 3, pp 325–341

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

  • F. Franchini
  • A. R. Its
  • V. E. Korepin
  • L. A. Takhtajan

DOI: 10.1007/s11128-010-0197-7

Cite this article as:
Franchini, F., Its, A.R., Korepin, V.E. et al. Quantum Inf Process (2011) 10: 325. doi:10.1007/s11128-010-0197-7


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 λn form exact geometric sequence. For example, for strong magnetic field λn = C exp(−πτ0n), here τ0 > 0 and C > 0 depend on anisotropy and magnetic field. Different eigenvalues are degenerated differently. The largest eigenvalue is unique, but degeneracy gn increases sub-exponentially as eigenvalues diminish: \({g_{n}\sim \exp{(\pi \sqrt{n/3})}}\). For weak magnetic field expressions are similar.


Entanglement Entropy Entanglement spectrum Reduced density matrix XY model Partitions theory 

Mathematics Subject Classification (2000)

Primary 81P40 82B23 82B20 Secondary 05A17 94A17 

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • F. Franchini
    • 1
    • 2
  • A. R. Its
    • 3
  • V. E. Korepin
    • 4
  • L. A. Takhtajan
    • 5
  1. 1.The Abdus Salam ICTPTriesteItaly
  2. 2.SISSATriesteItaly
  3. 3.Department of Mathematical SciencesIndiana University-Purdue University IndianapolisIndianapolisUSA
  4. 4.C.N. Yang Institute for Theoretical PhysicsState University of New York at Stony BrookStony BrookUSA
  5. 5.Mathematics DepartmentState University of New York at Stony BrookStony BrookUSA