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arXiv:2407.10395

R-matrix for the XX spin chain at the special imaginary value of staggered magnetic field

P. N. Bibikov

Published Jul 15, 2024• 0 citations
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Abstract

It is pointed, that the $16\times16$ Hamiltonian density matrix, corresponding to XX spin chain in a staggered magnetic field, satisfies the Reshetikhin condition as well, as the two higher ones. All of them are necessary for the existence of the corresponding $R$-matrix. At the special imaginary value of the staggered field, the $R$-matrix is obtained explicitly. It is shown that the corresponding Hamiltonian has purely imaginary spectrum.

DOI
10.48550/arxiv.2407.10395

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It is pointed, that the 16×1616\times1616×16 Hamiltonian density matrix, corresponding to XX spin chain in a staggered magnetic field, satisfies the Reshetikhin condition as well, as the two higher ones. All of them are necessary for the existence of the corresponding RRR-matrix. At the special imaginary value of the staggered field, the RRR-matrix is obtained explicitly. It is shown that the corresponding Hamiltonian has purely imaginary spectrum.
We suggest the elliptic 16×1616\times1616×16 R-matrix for the XX spin chain in a staggered magnetic field. This integrable model is a special case of the more general one, studied long ago within the alternative approach by V. M. Kontorovich and V. M. Tsukernik. The Yang-Baxter equation is verified with the full rigorous only in the two special degenerate trigonometric cases. For the general one, studying the first terms of the Taylor expansions for the RRR-matrix entries, we give the strong evidence for implementation of the Yang-Baxter equation.
Using the density-matrix-renormalization-group technique, we calculate numerically the low-energy excitation spectrum and magnetization curve of the spin-1 antiferromagnetic chain in a staggered magnetic field, which is expected to describe the physics of the ${R}_{2}{\mathrm{BaNiO}}_{5} (R\ensuremath{\ne}Y)$ family below the N\'eel temperature of the magnetic rare-earth (R) sublattice. These results are valid in the entire range of the staggered field, and agree with those given by the nonlinear $\ensuremath{\sigma}$ model study for small fields, but differ from the latter for large fields. They are consistent with the available experimental data. The correlation functions for this model are also calculated. The transverse correlations display the anticipated exponential decay with shorter correlation length, while the longitudinal correlations show explicitly the induced staggered magnetization.
The Reshetikhin condition for the general Hamiltonian density matrix of the S=1S=1S=1 axially symmetric spin chain is completely solved. 16 new integrable models and corresponding RRR-matrices are presented.
We study the phase diagram of the spin-1/2 XXZ Heisenberg chain in a staggered magnetic field. An exact solution is obtained along the phase boundary, where the exchange anisotropy and the staggered-field strength satisfy an exact analytical relation, yielding a highly degenerate manifold of ground states with off-diagonal long-range order. We demonstrate that these exact ground states lie on the boundary between the ferromagnetic phase and a gapless Luttinger liquid phase. Upon further decreasing the anisotropy, the gapless phase eventually gives way to a gapped quantum-disordered phase. Large-scale density-matrix renormalization group calculations confirm this phase structure and indicate that the transition across the exact boundary is of first order. Our results provide an exact benchmark for understanding quantum phase transitions driven by staggered magnetic fields in one-dimensional quantum spin systems.