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arXiv:1408.4586

Second cluster integral from the spectrum of an infinite XXZ spin chain

P. N. Bibikov

Published Aug 20, 2014• 2 citations
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Abstract

First and second terms of the low-temperature cluster expansion for the free energy density of a magnetically polarized $XXZ$ spin chain is obtained within the propagator approach suggested by E. W. Montroll and J. C. Ward. All the calculations employ only one- and two-magnon infinite-chain spectrums. In the $XXX$-point the result reproduces the well known S. Katsura's formula obtained 50 years ago by finite-chain calculations.

DOI
10.48550/arxiv.1408.4586

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First and second terms of the low-temperature cluster expansion for the free energy density of a magnetically polarized XXZXXZXXZ spin chain is obtained within the propagator approach suggested by E. W. Montroll and J. C. Ward. All the calculations employ only one- and two-magnon infinite-chain spectrums. In the XXXXXXXXX-point the result reproduces the well known S. Katsura's formula obtained 50 years ago by finite-chain calculations.
This thesis contains two results for the low temperature behavior of quantum spin systems. First, we present a lower bound for the spin-1 XXZ chain in finite volumes in terms of the gap of the two-site Hamiltonian. The estimate is derived by a method developed by Nachtergaele in (cond-mat/9410110) called the Martingale Method. Our bound relies on an assumption which we have, as yet, been unable to verify analytically in all cases. We present numerical evidence that strongly indicates our assumption is valid. The second result is a proof that the spin-1/2, d-dimensional XY model in the presence of an external magnetic field does not undergo a phase transition at low temperature, provided that the strength of the field is great enough. Using a contour expansion inspired by Kennedy, we show that the weights of contours satisfy a condition of Kotecky and Preiss which allows us to express the free energy of the system as a cluster expansion. As part of the setup we give a simple proof that the all-spin-up state is the unique ground state when the external magnetic field has strength at least 2d.
The propagator of the finite XXZ spin-12\tfrac{1}{2}21​ chain
We derive contour integral formulas for the real space propagator of the spin- \tfrac12 1 2 XXZ chain. The exact results are valid in any finite volume with periodic boundary conditions, and for any value of the anisotropy parameter. The integrals are on fixed contours, that are independent of the Bethe Ansatz solution of the model and the string hypothesis. The propagator is obtained by two different methods. First we compute it through the spectral sum of a deformed model, and as a by-product we also compute the propagator of the XXZ chain perturbed by a Dzyaloshinskii-Moriya interaction term. As a second way we also compute the propagator through a lattice path integral, which is evaluated exactly utilizing the so-called F F -basis in the mirror (or quantum) channel. The final expressions are similar to the Yudson representation of the infinite volume propagator, with the volume entering as a parameter. As an application of the propagator we compute the Loschmidt amplitude for the quantum quench from a domain wall state.
Using the Bethe ansatz method and the TBA equations for the higher spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with an anisotropy 0≤Δ<10 \le Δ<10≤Δ<1. In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies, which allows the exact evaluation of the infinite temperature spin dc conductivity L\mathcal{L}L. We find that L\mathcal{L}L is discontinuous at all rational numbers of the anisotropy parameter p0=π/cos⁡−1Δp_0=π/\cos^{-1}Δp0​=π/cos−1Δ in the region p0≥2p_0 \ge 2p0​≥2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ=1Δ=1Δ=1 point is exceptional. Close to this point, L\mathcal{L}L slowly increases in proportion to the first power of the magnetization. On the other hand L\mathcal{L}L is proportional to the second power of the magnetization when p0p_0p0​ approaches irrational numbers.