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arXiv:hep-th/0603064 · Journal of Statistical Mechanics Theory and Experiment

Factorization of multiple integrals representing the density matrix of a finite segment of the Heisenberg spin chain

Herman E Boos, Frank Göhmann, Andreas Klümper, Junji Suzuki

Published Apr 6, 2006• 49 citations
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Abstract

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.

DOI
10.1088/1742-5468/2006/04/p04001

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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 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 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 present a conjecture for the density matrix of a finite segment of the XXZ chain coupled to a heat bath and to a constant longitudinal magnetic field. It states that the inhomogeneous density matrix, conceived as a map which associates with every local operator its thermal expectation value, can be written as the trace of the exponential of an operator constructed from weighted traces of the elements of certain monodromy matrices related to and only two transcendental functions pertaining to the one-point function and the neighbour correlators, respectively. Our conjecture implies that all static correlation functions of the XXZ chain are polynomials in these two functions and their derivatives with coefficients of purely algebraic origin.
It was recently shown by Jimbo et al (2008 arXiv:0811.0439) that the correlation functions of a generalized XXZ chain associated with an inhomogeneous six-vertex model with a disorder parameter α and with arbitrary inhomogeneities on the horizontal lines factorize and can all be expressed in terms of only two functions ρ and ω. Here we approach the description of the same correlation functions and, in particular, of the function ω from a different direction. We start from a novel multiple integral representation for the density matrix of a finite chain segment of length m in the presence of a disorder field α. We explicitly factorize the integrals for m = 2. Based on this, we present an alternative description of the function ω in terms of the solutions of certain linear and nonlinear integral equations. We then prove directly that the two definitions of ω describe the same function. The definition in the work of Jimbo et al (2008 arXiv:0811.0439) was crucial for the proof of the factorization. The definition given here together with the known description of ρ in terms of the solutions of nonlinear integral equations is useful for performing, e.g., the Trotter limit in the finite temperature case, or for obtaining numerical results for the correlation functions at short distances. We also address the issue of the construction of an exponential form of the density matrix for finite α.