2000 character limit reached
A Pedestrian's Way to Baxter's Bethe Ansatz for the Periodic XYZ Chain
Published 30 Nov 2023 in cond-mat.stat-mech, math-ph, math.MP, and quant-ph | (2312.00161v2)
Abstract: A chiral coordinate Bethe ansatz method is developed to study the periodic XYZ chain. We construct a set of chiral vectors with fixed number of kinks. All vectors are factorized and have simple structures. Under roots of unity conditions, the Hilbert space has an invariant subspace and our vectors form a basis of this subspace. We propose a Bethe ansatz solely based on the action of the Hamiltonian on the chiral vectors, avoiding the use of transfer matrix techniques. This allows to parameterize the expansion coefficients and derive the homogeneous Bethe ansatz equations whose solutions give the exact energies and eigenstates. Our analytic results agree with earlier approaches, notably by Baxter, and are supported by numerical calculations.
- R. J. Baxter, Eight-vertex model in lattice statistics, Phys. Rev. Lett. 26, 832 (1971a).
- R. Baxter, One-dimensional anisotropic heisenberg chain, Phys. Rev. Lett. 26, 834 (1971b).
- R. J. Baxter, Partition function of the eight-vertex lattice model, Ann. Phys. 70, 193 (1972a).
- R. J. Baxter, One-dimensional anisotropic heisenberg chain, Ann. Phys. 70, 323 (1972b).
- R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. i. some fundamental eigenvectors, Ann. Phys. 76, 1 (1973a).
- R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. ii. equivalence to a generalized ice-type lattice model, Ann. Phys. 76, 25 (1973b).
- R. Baxter, Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. iii. eigenvectors of the transfer matrix and hamiltonian, Ann. Phys. 76, 48 (1973c).
- R. J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982).
- R. J. Baxter, Completeness of the Bethe ansatz for the six and eight-vertex models, J. Stat. Phys. 108, 1 (2002).
- L. A. Takhtadzhan and L. D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, Rush. Math. Surveys 34, 11 (1979).
- G. Felder and A. Varchenko, Algebraic bethe ansatz for the elliptic quantum group eτ,η(sl2)subscript𝑒𝜏𝜂𝑠subscript𝑙2e_{\tau,\eta}(sl_{2})italic_e start_POSTSUBSCRIPT italic_τ , italic_η end_POSTSUBSCRIPT ( italic_s italic_l start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ), Nucl. Phys. B 480, 485–503 ((1996).
- T. Deguchi, Construction of some missing eigenvectors of the xyz spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix, J. Phys. A: Math. Gen. 35, 879 (2002).
- K. Fabricius and B. M. McCoy, New developments in the eight vertex model. journal of statistical physics, J. Stat. Phys. 111, 323–337 (2003).
- K. Fabricius and B. M. McCoy, New developments in the eight vertex model ii. chains of odd length, J. Stat. Phys. 120, 37–70 (2005).
- K. Fabricius, A new q-matrix in the eight-vertex model, J. Phys. A: Math. Theor. 40, 4075 (2007).
- K. Fabricius and B. M. McCoy, An elliptic current operator for the eight-vertex model, J. Phys. A: Math. Gen. 39, 14869 (2006).
- K. Fabricius and B. M. McCoy, New q matrices and their functional equations for the eight vertex model at elliptic roots of unity, J. Stat. Phys. 134, 643–668 (2009).
- W.-L. Yang and Y.-Z. Zhang, T–Q relation and exact solution for the XYZ chain with general non-diagonal boundary terms, Nucl. Phys. B 744, 312 (2006).
- X. Zhang, A. Klümper, and V. Popkov, Phantom Bethe roots in the integrable open spin-1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG XXZ chain, Phys. Rev. B 103, 115435 (2021a).
- X. Zhang, A. Klümper, and V. Popkov, Chiral coordinate Bethe ansatz for phantom eigenstates in the open XXZ spin-1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG chain, Phys. Rev. B 104, 195409 (2021b).
- V. Popkov, X. Zhang, and T. Prosen, Boundary-driven XYZ chain: Inhomogeneous triangular matrix product ansatz, Phys. Rev. B 105, L220302 (2022).
- X. Zhang, A. Klümper, and V. Popkov, Invariant subspaces and elliptic spin-helix states in the integrable open spin-1212\frac{1}{2}divide start_ARG 1 end_ARG start_ARG 2 end_ARG xyz chain, Phys. Rev. B 106, 075406 (2022).
- E. T. Whittaker and G. N. Watson, A course of modern analysis (Cambridge University Press, 1950).
- C. Hagendorf and P. Fendley, The eight-vertex model and lattice supersymmetry, JOURNAL OF STATISTICAL PHYSICS 146, 1122 (2012).
- V. Popkov, X. Zhang, and A. Klümper, Phantom Bethe excitations and spin helix eigenstates in integrable periodic and open spin chains, Phys. Rev. B 104, L081410 (2021).
- T. Deguchi, K. Fabricius, and B. M. McCoy, The sl2 loop algebra symmetry of the six-vertex model at roots of unity, J. Stat. Phys. 102, 701 (2001).
- K. Fabricius and B. M. McCoy, Bethe’s equation is incomplete for the xxz model at roots of unity, J. Stat. Phys. 103, 647 (2001).
- V. Popkov, M. Žnidarič, and X. Zhang, Universality in relaxation of spin helices under the XXZ𝑋𝑋𝑍XXZitalic_X italic_X italic_Z- spin chain dynamics, Phys. Rev. B 107, 235408 (2023a).
- D. Braak and N. Andrei, On the spectrum of the xxz-chain at roots of unity, J. Stats. Phys. 105, 677 (2001).
- V. Popkov, X. Zhang, and A. Klümper, Chiral bases for qubits and their applications to integrable spin chains, arXiv:2303.14056 (2023b).
- G. M. Schütz, A reverse duality for the asep with open boundaries, Journal of Physics A: Mathematical and Theoretical 56, 274001 (2023).
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