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arXiv:2003.07089 · Physical review. B./Physical review. B

Exact ground state and elementary excitations of a topological spin chain

Yi Qiao, Pei Sun, Junpeng Cao, Wen‐Li Yang, Kangjie Shi, Yupeng Wang

Published Aug 7, 2020• 20 citations
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Abstract

A novel Bethe ansatz scheme is proposed to calculate physical properties of quantum integrable systems without $U(1)$ 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 $U(1)$ 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 $U(1)$ symmetry.

DOI
10.1103/physrevb.102.085115

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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.
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.
A novel Bethe ansatz scheme is proposed to investigate the exact physical properties of an integrable anisotropic quantum spin chain with competing interactions among the nearest, next nearest neighbor and chiral three spin couplings, where the boundary condition is the twisted one. The eigenvalue of the transfer matrix is characterized by its zero roots instead of the traditional Bethe roots. Based on the exact solution, the conserved momentum and charge operators of this U(1)-symmetry broken system are obtained. The ground state energy and density of rapidities are calculated. It is found that there exist three kinds of elementary excitations and the corresponding dispersion relations are obtained, which gives a different picture from that with periodic boundary condition.
A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.
An analytic method is proposed to compute the surface energy and elementary excitations of the XXZ spin chain with generic nondiagonal boundary fields. For the gapped case, in some boundary parameter regimes the contributions of the two boundary fields to the surface energy are nonadditive. Such a correlation effect between the two boundaries also depends on the parity of the site number NNN even in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. For the gapless case, contributions of the two boundary fields to the surface energy are additive due to the absence of long-range correlation in the bulk. Although the U(1)U(1)U(1) symmetry of the system is broken, exact spinonlike excitations, which obviously do not carry spin-12\frac{1}{2}21​, are observed. The present method provides a universal procedure to deal with quantum integrable systems either with or without U(1)U(1)U(1) symmetry.
By using a set of gauge transformations, the exact solutions of the XXZ spin chain with unparallel boundary magnetic fields are derived in the framework of the algebraic Bethe ansatz. In the easy-plane case, we show the elementary excitations are some kind of spinons without definite spin because the U(1) symmetry is broken, while in the easy-axis ferromagnetic case a spiral state is realized in the ground state. The correlation functions, the spin torque as well as the spin voltage for the later case are also derived.
The exact elementary excitations in a typical U(1)U(1)U(1) symmetry broken quantum integrable system, that is the twisted ${J}_{1}\ensuremath{-}{J}_{2}$ spin chain with nearest-neighbor, next nearest neighbor, and chiral three spin interactions, are studied. The main technique is that we quantify the energy spectrum of the system by the zero roots of the transfer matrix instead of the traditional Bethe roots. From the numerical calculation and singularity analysis, we obtain the patterns of zero roots. Based on them, we analytically obtain the ground state energy and the elementary excitations in the thermodynamic limit. We find that the system also exhibits the nearly degenerate states in the regime of $\ensuremath{\eta}\ensuremath{\in}\mathbb{R}$, where the nearest-neighbor couplings among the zzz direction are ferromagnetic. More careful study shows that the competing of interactions can induce the gapless low-lying excitations and quantum phase transition in the antiferromagnetic regime with $\ensuremath{\eta}\ensuremath{\in}\mathbb{R}+i\ensuremath{\pi}$.