Correlation functions at the topological quantum phase transition in the S=1 XXZ chain with single-ion anisotropy

TL;DR

This work analyzes the quantum critical state at the transition between the Haldane and large-$D$ phases in the $S=1$ XXZ chain with single-ion anisotropy, showing it is governed by a Gaussian ($c=1$) theory. Using bosonization, the authors derive the asymptotic forms of spin, dimer, and squared-spin correlations, revealing that uniform components of some operators decay algebraically while staggered components are gapped, and that the transverse spin correlations decay algebraically only in the staggered sector. They verify these predictions with DMRG on open chains, extracting the TLL parameter $K_+$ and correlation lengths, and they demonstrate how weak bond alternation reintroduces algebraic components in previously exponential terms. The results provide a detailed, quantitatively validated picture of Gaussian criticality in this model, with implications for related 1D systems such as spin ladders and multi-mode chains. The methodology combines a rigorous field-theoretical framework with high-precision numerics to map correlation functions across the transition and under perturbations like bond alternation. The study advances understanding of topological-to-disordered transitions in integer-spin chains and offers practical benchmarks for related low-dimensional quantum systems.

Abstract

We study the one-dimensional S=1 XXZ spin model with single-ion anisotropy. It is known that at the transition points between the Haldane and large-D phases, the model exhibits a quantum criticality described by the Gaussian theory, i.e., a conformal field theory with the central charge c=1. Using the bosonization approach, we investigate various correlation functions at the phase transition and derive their asymptotic forms. This allows us to clarify their peculiar behavior: the longitudinal (transverse) two-point spin correlation function has components that decay algebraically only in the uniform (staggered) sector. These theoretical predictions are verified by the numerical calculations using the density-matrix renormalization group method. The effect of weak bond alternation on the critical ground state at the phase transition is also discussed. It is shown that the bond alternation induces the missing power-law components in the correlation functions.

Figures (12)

• Figure 1: One-point function of (a) $\langle \mathcal{O}_{\rm d}^z(j)\rangle$, (b) $\langle \mathcal{O}_{\rm d}^{xy}(j)\rangle$, and (c) $\langle (S^z_j)^2 \rangle$ for $\Delta=1.50$ and $L=256$. Open and solid circles respectively represent the DMRG data and their fits to the analytic forms Eqs. (\ref{['eq:form_dimeroperator']}) and (\ref{['eq:form_Sz2operator']}).
• Figure 2: Leading uniform component of the one-point functions: (a) $\langle \mathcal{O}_{\rm d}^z(j)\rangle_{c1}$, (b) $\langle \mathcal{O}_{\rm d}^{xy}(j)\rangle_{c1}$, and (c) $\langle (S^z_j)^2 \rangle_{c1}$ for $\Delta=1.50$. Circles (blue) and squares (purple) represent the data for $L=256$ and $128$, respectively.
• Figure 3: Exponentially-decaying component of one-point functions; (a) $\langle \mathcal{O}_{\rm d}^z(j)\rangle_{\rm exp}$, (b) $\langle \mathcal{O}_{\rm d}^{xy}(j)\rangle_{\rm exp}$, and (c) $\langle (S^z_j)^2\rangle_{\rm exp}$, for $\Delta=1.50$. The absolute values of the data near the open boundary, $j \lesssim 12$, are plotted as a function of the center position of the operators, i.e., $j+\frac{1}{2}$ for the dimer operators and $j$ for the squared-spin operator. Circles (blue) and squares (purple) represent the data for $L=256$ and $128$, respectively. The solid line shows the result of fitting to the exponentially-decaying form.
• Figure 4: Transverse two-spin correlation function $\langle S^+_j S^-_k \rangle$ for $\Delta=1.50$ and $L=256$. The absolute values are plotted as a function of $|j-k|$. Open and solid circles respectively represent the DMRG data and their fits to the analytic form Eq. (\ref{['eq:form_S+S-cor']}).
• Figure 5: Longitudinal two-spin correlation function $\langle S^z_j S^z_k \rangle$ for $\Delta=1.50$ and $L=256$. The absolute values are plotted as a function of $|j-k|$. The open circles represent the DMRG data. The dashed curve represents the analytic form Eq. (\ref{['eq:form_SzSzcor']}) with $K_+$ obtained from the fitting of $\langle \mathcal{O}^z_{\rm d}(j) \rangle$.