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arXiv:cond-mat/0701463 · Journal of Physics A Mathematical and Theoretical

Density matrices for finite segments of Heisenberg chains of arbitrary length

Jens Damerau, Frank Göhmann, Nils P Hasenclever, Andreas Klümper

Published Apr 11, 2007• 67 citations
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Abstract

We derive a multiple integral representing the ground-state density matrix of a segment of length m of the XXZ spin chain on L lattice sites, which depends on L only parametrically. This allows us to treat chains of arbitrary finite length. Specializing to the isotropic limit of the XXX chain we show for small m that the multiple integrals factorize. We conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double Cauchy type integral in the exponent. We demonstrate the efficiency of our formula by computing the next-to-nearest neighbour zz -correlation function for chain lengths ranging from two to macroscopic numbers.

DOI
10.1088/1751-8113/40/17/002

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We derive a multiple integral representing the ground-state density matrix of a segment of length m of the XXZ spin chain on L lattice sites, which depends on L only parametrically. This allows us to treat chains of arbitrary finite length. Specializing to the isotropic limit of the XXX chain we show for small m that the multiple integrals factorize. We conjecture that this property holds for arbitrary m and suggest an exponential formula for the density matrix which involves only a double Cauchy type integral in the exponent. We demonstrate the efficiency of our formula by computing the next-to-nearest neighbour zz -correlation function for chain lengths ranging from two to macroscopic numbers.
We consider the inhomogeneous generalization of the density matrix of a finite segment of length mmm of the antiferromagnetic Heisenberg chain. It is a function of the temperature TTT and the external magnetic field hhh, and further depends on mmm `spectral parameters' xij\\xi_jxij​. For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the infinite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary mmm and TTT but zero magnetic field.
We consider the inhomogeneous generalization of the density matrix of a finite segment of length m of the antiferromagnetic Heisenberg chain. It is a function of the temperature T and the external magnetic field h , and further depends on m ‘spectral parameters’ ξ j . For short segments of length 2 and 3 we decompose the known multiple integrals for the elements of the density matrix into finite sums over products of single integrals. This provides new numerically efficient expressions for the two-point functions of the infinite Heisenberg chain at short distances. It further leads us to conjecture an exponential formula for the density matrix involving only a double Cauchy-type integral in the exponent. We expect this formula to hold for arbitrary m and T but zero magnetic field.
We have analytically obtained all the density matrix elements up to six lattice sites for the spin-1/2 Heisenberg XXZ chain at Δ = 1/2. We use the multiple integral formula of the correlation function for the massless XXZ chain derived by Jimbo and Miwa. As for the spin–spin correlation functions, we have newly obtained the fourth- and fifth-neighbour transverse correlation functions. We have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution. Using these results the exact values of the entanglement entropy for the reduced density matrix up to six lattice sites have been obtained. We observe that our exact results agree quite well with the asymptotic formula predicted by the conformal field theory.