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arXiv:0704.0850 · Journal of Physics A Mathematical and Theoretical

Density matrix elements and entanglement entropy for the spin-1/2 XXZ chain at Δ = 1/2

Jun Sato, Masahiro Shiroishi

Published Jul 12, 2007• 26 citations
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Abstract

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.

DOI
10.1088/1751-8113/40/30/009

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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.
We have obtained all the density matrix elements on six lattice sites for the spin-1/2 Heisenberg chain via the algebraic method based on the quantum Knizhnik–Zamolodchikov equations. Several interesting correlation functions, such as chiral correlation functions, dimer–dimer correlation functions, etc, have been analytically evaluated. Furthermore, we have calculated all the eigenvalues of the density matrix and analyse the eigenvalue distribution. As a result, the exact von Neumann entropy for the reduced density matrix on six lattice sites has been obtained.
We have obtained all the density matrix elements on six lattice sites for the spin-1/2 Heisenberg chain via the algebraic method based on the quantum Knizhnik-Zamolodchikov equations. Several interesting correlation functions, such as chiral correlation functions, dimer-dimer correlation functions, etc... have been analytically evaluated. Furthermore we have calculated all the eigenvalues of the density matrix and analyze the eigenvalue-distribution. As a result the exact von Neumann entropy for the reduced density matrix on six lattice sites has been obtained.
The exact expression for the density matrix of the kink ground state of the ferromagnetic XXZ chain is obtained. Utilizing this, we exactly calculate various correlation functions such as the longitudinal and transverse spin-spin correlation functions, and the ferromagnetic and antiferromagnetic string formation probabilities. The asymptotic behaviors of these correlation functions are also analyzed. As a consequence, we find that the spin-spin correlation functions decay exponentially for large distances, while the string formation probabilities exhibit Gaussian decay for large strings. We also evaluate the entanglement entropy, which shows interesting behaviors due to the lack of the translational invariance of the state.
We review the recent progresses of the exact calculation of the correlation functions for the spin-1/2 Heisenberg XXX chain as well as the anisotropic XXZ chain. The spin-1/2 Heisenberg chain is exactly solvable by Bethe ansatz. Nevertheless the exact calculation of the correlation functions is still a very difficult problem in general. However, the recent theoretical developments have enabled us to derive the explicit analytical forms of some correlation functions in short range. For example, polynomial representations in terms of the one-dimensional integrals are obtained for the correlation functions of the XXZ chain up to four lattice sites. For the special case of the XXX chain, the correlation functions are further calculated up to five lattice sites.
Abstract We study the relative entanglement entropy (EE) among various primary excited states in two critical spin chains: the S = 1 / 2 XXZ chain and the transverse field Ising chain at criticality. For the S = 1 / 2 XXZ chain, which corresponds to c = 1 free boson conformal field theory (CFT), we numerically calculate the relative EE by exact diagonalization and find a perfect agreement with the predictions by the CFT. For the transverse field Ising chain at criticality, which corresponds to the c = 1 / 2 Ising CFT, we analytically relate its relative EE to that of the S = 1 / 2 XXZ chain and confirm the relation numerically. We also calculate the ‘sandwiched’ Rényi relative EE and again the numerical results agree well with the analytical predictions. Our results are the first direct confirmation of the CFT predictions on the relative EE of the primary excited states in critical spin chains.
We carry out a systematic study of the exact block entanglement in the XXZ spin chain at Δ = −1/2. We present the first analytic expressions for reduced density matrices for n spins in a chain of length L (for n ≤6 and arbitrary but odd L ) for a truly interacting model. The entanglement entropy and the moments of the reduced density matrix and its spectrum are then easily derived. We explicitly construct the 'entanglement Hamiltonian' as the logarithm of this matrix. Exploiting the degeneracy of the ground state, we find the scaling behavior of the entanglement of the zero-temperature mixed state.