Exact physical quantities of the XYZ spin chain in the thermodynamic limit

TL;DR

This work derives the exact thermodynamic-limit quantities for the XYZ spin chain under periodic and twisted boundaries by recasting the transfer-matrix spectrum in terms of elliptic-polynomial zeros via an inhomogeneous $T$-$Q$ framework. It shows that the zeros organize into bulk strings and discrete modes whose densities determine the surface energy and elementary excitations, with pronounced parity dependencies tied to Neel order on a Mobius-like spin space. The analysis covers both real and pure-imaginary crossing parameters $\\eta$, identifying distinct zero-pattern regimes and providing explicit energy decompositions $E = e_r N + \sum_t E^{w}_r(w_t)$ with boundary-dependent shifts, corroborated by DMRG. The results illuminate how boundary conditions and system size parity influence low-energy properties and offer a path to finite-temperature properties via quantum transfer methods.

Abstract

The thermodynamic limits of the XYZ spin chain with periodic or twisted boundary conditions are studied. By using the technique of characterizing the eigenvalue of the transfer matrix by the $T-Q$ relation and by the zeros of the associated polynomial, we obtain the constraints of the Bethe roots and the zeros for the eigenvalues. With the help of structure of Bethe roots, we obtain the distribution patterns of zeros. Based on them, the physical quantities such as the surface energy and excitation energy are calculated. We find that both of them depend on the parity of sites number due to the topological long-range Neel order on the Mobius manifold in the spin space. We also check our results with those obtaining by the density matrix renormalization group. The method provided in this paper can be applied to study the thermodynamic properties at the thermal equilibrium state with finite temperature.

Figures (12)

• Figure 1: The exact numerical results of zeros $\{\bar{z}_j\}$ at the ground state (asterisks) and the first excited state (circles) with the system-size $N=10$. The boundary conditions are (a) $\beta=0$, (b) $\beta=x$, (c) $\beta=y$, and (d) $\beta=z$. Here $\tau=0.6i$ and $\eta=0.7$.
• Figure 2: The exact numerical results of zeros $\bar{z}_j$ at the ground state (asterisks) and the first excited state (circles) with the system-size $N=11$. The boundary conditions are (a) $\beta=0$, (b) $\beta=x$, (c) $\beta=y$, and (d) $\beta=z$. Here $\tau=0.6i$ and $\eta=0.7$.
• Figure 3: The patterns of zeros at the ground state (asterisks) and those at the first excited state (circles) for the boundary conditions (\ref{['bc']}) with (a) $\beta=0$, (b) $\beta=x$, (c) $\beta=y$ and (d) $\beta=z$. Here $\tau=0.6i$, $\eta=0.3$ and $N=10$.
• Figure 4: The patterns of zeros at the ground state (asterisks) and those at the first excited state (circles) for the boundary conditions (\ref{['bc']}) with (a) $\beta=0$, (b) $\beta=x$, (c) $\beta=y$ and (d) $\beta=z$. Here $\tau=0.6i$, $\eta=0.3$ and $N=11$.
• Figure 5: The surface energies $E^{\beta}_{rs}$ with $\beta=x, y, z$ versus the model parameter $\tau$ for $\eta=0.7$. The blue lines denote the analytic results obtained from Eq.(\ref{['Ersyz']}). The red dots, blue circles and black asterisks denote the DMRG results for $E^{y}_{rs}$, $E^{z}_{rs}$ and $E^{x}_{rs}$, respectively. (a) DMRG results with $N=150$, (b) DMRG results with $N=151$.