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arXiv:2108.08060 · Journal of High Energy Physics

Root patterns and energy spectra of quantum integrable systems without U(1) symmetry: the antiperiodic XXZ spin chain

Xiong Le, Yi Qiao, Junpeng Cao, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Nov 1, 2021• 9 citations
arXivHTMLPDF

Abstract

Abstract Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without U(1) symmetry, their spectra are usually given by inhomogeneous T − Q relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic XXZ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.

DOI
10.1007/jhep11(2021)044

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Abstract Finding out root patterns of quantum integrable models is an important step to study their physical properties in the thermodynamic limit. Especially for models without U(1) symmetry, their spectra are usually given by inhomogeneous T − Q relations and the Bethe root patterns are still unclear. In this paper with the antiperiodic XXZ spin chain as an example, an analytic method to derive both the Bethe root patterns and the transfer-matrix root patterns in the thermodynamic limit is proposed. Based on them the ground state energy and elementary excitations in the gapped regime are derived. The present method provides an universal procedure to compute physical properties of quantum integrable models in the thermodynamic limit.
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A novel Bethe ansatz scheme is proposed to calculate physical properties of quantum integrable systems without U(1)U(1)U(1) symmetry. As an example, the antiperiodic XXZ spin chain, a typical correlated many-body system embedded in a topological manifold, is examined. Conserved ``momentum'' and ``charge'' operators are constructed despite the absence of translational invariance and U(1)U(1)U(1) symmetry. The ground state energy and elementary excitations are derived exactly. It is found that two intrinsic fractional (one half) zero modes accounting for the double degeneracy exist in the eigenstates. The elementary excitations show quite a different picture from that of a periodic chain. This method can be applied to other quantum integrable models either with or without U(1)U(1)U(1) symmetry.
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We study the thermodynamic limit of the antiperiodic XXZ spin chain with the anisotropic parameter η=πi3. We parameterize eigenvalues of the transfer matrix by their zero points instead of Bethe roots. We obtain patterns of the distribution of zero points. Based on them, we calculate the ground state energy and the elementary excitations in the thermodynamic limit. We also obtain the two-body scattering matrix of elementary excitations. Two types of elementary excitations and three types of scattering processes are discussed in detailed.
Rational QQQ-systems for integrable spin chains without U(1)U(1)U(1) symmetry
The Q Q -system is an efficient method for finding complete physical solutions of Bethe Ansatz equations, but so far its application has been confined to systems possessing U(1) U ( 1 ) symmetry. We extend the rational Q Q -system framework to integrable spin chains without U(1) U ( 1 ) symmetry, exemplified by the closed XXZ model with anti-diagonal twists and the open XXZ model with non-diagonal boundary fields. We demonstrate that the Q Q -system can be derived by combining TQ T Q -relation with fusion relations of higher-spin transfer matrices. This yields QQ Q Q -relations analogous to the U(1) U ( 1 ) symmetric case but incorporating additional inhomogeneous terms. We present numerical solutions that are validated against exact diagonalization, confirming that it generates all and exclusively physical solutions.