In this Letter I mainly consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. This system is described by the Hamiltonian H xxz = -∑ j = 1 N {σ j x σ j + 1 x + σ j y σ j + 1 y + Δ σ j z σ j + 1 z }. As it turns out, the ground state energy is proportional to the number of sites E = -3 N /2 for a special value of the asymmetry parameter Δ = -1/2. The trigonometric polynomial Q ( u ), the zeros of which are parameters of the ground state Bethe eigenvector, is explicitly constructed. This polynomial of degree n = ( N -1)/2 satisfies the Baxter T - Q equation. Using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function ⟨σ j z σ j + 1 z ⟩ = -1/2 + 3/2 N 2 . This correlation function is related to the average number of spin strings for the ground state ⟨ N string ⟩ = ¾( N -1/ N ). I would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered.In this Letter I mainly consider a finite XXZ spin chain with periodic boundary conditions and an odd number of sites. This system is described by the Hamiltonian H xxz = -∑ j = 1 N {σ j x σ j + 1 x + σ j y σ j + 1 y + Δ σ j z σ j + 1 z }. As it turns out, the ground state energy is proportional to the number of sites E = -3 N /2 for a special value of the asymmetry parameter Δ = -1/2. The trigonometric polynomial Q ( u ), the zeros of which are parameters of the ground state Bethe eigenvector, is explicitly constructed. This polynomial of degree n = ( N -1)/2 satisfies the Baxter T - Q equation. Using the second independent solution of this equation that corresponds to the same eigenvalue of the transfer matrix, it is possible to find a derivative of the ground state energy w.r.t. the asymmetry parameter. This derivative is closely connected with the correlation function ⟨σ j z σ j + 1 z ⟩ = -1/2 + 3/2 N 2 . This correlation function is related to the average number of spin strings for the ground state ⟨ N string ⟩ = ¾( N -1/ N ). I would like to stress that all the above simple formulae are not applicable to the case of an even number of sites which is usually considered.

arXiv:cond-mat/0012035Journal of Physics A Mathematical and GeneralMar 26, 2001108 citations

Infinite temperature spin dc conductivity of the spin-1/2 XXZ chain

Shinya Ae

Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz 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 the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 ⁡ Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz 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 the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 ⁡ Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.

Journal of Physics A Mathematical and TheoreticalSep 23, 20241 citation

Dynamical correlation functions of the XXZ spin- 1/2 chain

N Kitanine, Jean Michel Maillet, N. A. Slavnov +1 more

arXiv:hep-th/0407108Nuclear Physics BSep 17, 200583 citations

On the logarithmic bipartite fidelity of the open XXZ spin chain at $Δ=-1/2$ On the logarithmic bipartite fidelity of the open XXZ spin chain at $Δ=-1/2$

Christian Hagendorf, Gilles Parez

The open XXZ spin chain with the anisotropy \Delta=-\frac12 Δ = − 1 2 and a one-parameter family of diagonal boundary fields is studied at finite length. A determinant formula for an overlap involving the spin chain’s ground-state vectors for different lengths is found. The overlap allows one to obtain an exact finite-size formula for the ground state’s logarithmic bipartite fidelity. The leading terms of its asymptotic series for large chain lengths are evaluated. Their expressions confirm the predictions of conformal field theory for the fidelity.The open XXZ spin chain with the anisotropy \Delta=-\frac12 Δ = − 1 2 and a one-parameter family of diagonal boundary fields is studied at finite length. A determinant formula for an overlap involving the spin chain’s ground-state vectors for different lengths is found. The overlap allows one to obtain an exact finite-size formula for the ground state’s logarithmic bipartite fidelity. The leading terms of its asymptotic series for large chain lengths are evaluated. Their expressions confirm the predictions of conformal field theory for the fidelity.

arXiv:2111.15223SciPost PhysicsJun 24, 20222 citations

Spin Correlations in the S=1 XXZ Chain

Katsunori Kubo

Springer proceedings in physicsJan 1, 19920 citations

On the logarithmic bipartite fidelity of the open XXZ spin chain at

Δ=-1/2