Phase transitions in coupled Ising chains and SO($N$)-symmetric spin chains

TL;DR

We address the quantum criticality of a $N$-fold copy of the Ising CFT in a $(1+1)$-dimensional field theory perturbed by competing relevant operators: the mass term $m$ and the $N$-spin product operator $\lambda_1$. We combine one-loop renormalization group analysis (near $N=16$ via an $\epsilon$-expansion) with large-scale matrix-product-state simulations to classify the transition as a function of $N$, and we connect the field theory to lattice realizations in coupled Ising chains and SO($N$)-symmetric spin chains. The main results show a continuous transition for $N=2$ (Ising universality, $c=1/2$) and $N=3$ (four-state Potts universality, $c=1$) but a first-order transition for all $N \ge 4$; this establishes a critical $N_c\in(3,16)$. These findings constrain direct SPT-to-trivial transitions in SO($N$) spin systems and refine conjectures about the existence and nature of such critical points.

Abstract

We investigate the nature of quantum phase transitions in a (1+1)-dimensional field theory composed of $N$ copies of the Ising conformal field theory interacting via competing relevant perturbations. The field theory governs the competition between a mass term and an interaction involving the product of $N$ order-parameter fields, which is realized, e.g. in coupled Ising chains, two-leg spin ladders, and SO($N$)-symmetric spin chains. By combining a perturbative renormalization group analysis and large-scale matrix-product state simulations, we systematically determine the nature of the phase transition as a function of $N$. For $N=2$ and $N=3$, we confirm that the transition is continuous, belonging to the Ising and four-state Potts universality classes, respectively. In contrast, for $N \ge 4$, our results provide compelling evidence that the transition becomes first order. We further apply these findings to specific lattice models with SO($N$) symmetry, including spin-$1/2$ and spin-$1$ two-leg ladders, that realize a direct transition between an SO($N$) symmetry-protected topological phase and a trivial phase. Our results refine a recent conjecture regarding the criticality of transitions between SPT phases.

Figures (15)

• Figure 1: Numerical results for the $N=2$ coupled Ising chains in Eq. \ref{['eq:NIsingHam']} with $K=0$ and $g=0.5$. (a) Magnetization $M$, (b) Correlation length $\xi$, and (c) half-system von Neumann entanglement entropy $S_\textrm{vN}$ are plotted as functions of $h$. (d) Plot of $S_\textrm{vN}$ vs the correlation length $\xi$ in the vicinity of the transition. The dashed line is a fitting function of the form $(c/6) \ln \xi +c_0$.
• Figure 2: Numerical results for the $N=2$ coupled Ising chains in Eq. \ref{['eq:NIsingHam']} with $K=0$ and $g=0.5$. Two-point connected correlation functions $\langle O_i O_j \rangle_c$ for (a) $O_i = \sigma^z_{i,1}$ and (b) $O_i = \sigma^z_{i,1} \sigma^z_{i+1,1}$, $\sigma^x_{i,1}$, and $\sigma^z_{i,1} \sigma^z_{i,2}$ are computed in the vicinity of the phase transitions. The dashed line is a fitting function of the power-law form $C |i-j|^{-2\Delta}$ in (a), whereas it is of the form $C|i-j|^{-2}$ shown as a reference in (b).
• Figure 3: Numerical results for the $N=3$ coupled Ising chains in Eq. \ref{['eq:NIsingHam']} with $K=0$ and $g=0.5$. (a) Magnetization $M$, (b) Correlation length $\xi$, and (c) von Neumann entanglement entropy are plotted as functions of $h$. (d) $S_\textrm{vN}$ are plotted against the correlation length $\xi$ at the vicinity of the transition. The dashed line is a fitting function of the form $(c/6) \ln \xi +c_0$.
• Figure 4: Numerical results for the $N=3$ coupled Ising chains \ref{['eq:NIsingHam']} with $K=0$ and $g=0.5$ at the transition $h=1.3403$. Two-point connected correlation functions $\langle O_i O_j \rangle_c$ for (a) $O_i = \sigma^z_{i,1}$, (b) $O_i = \sigma^z_{i,1}$, and (c) $O_i = \sigma^z_{i,1} \sigma^z_{i,2} \sigma^z_{i,3}$ are computed. The string correlation function $\langle O^\textrm{str}_{i,j} \rangle$ is also shown in (d). The blue dashed lines are fitting functions of the power-law form $C |i-j|^{-2\Delta}$, whereas the orange ones are those with exact exponents of the four-state Potts CFT shown as references.
• Figure 5: Numerical results for the $N=4$ coupled Ising chains in Eq. \ref{['eq:NIsingHam']} with $K=0$ and $g=0.5$. (a) Magnetization $M$, (b) Correlation length $\xi$, and (c) half-system von Neumann entanglement entropy $S_\textrm{vN}$ are plotted as functions of $h$. (d) Plot of $S_\textrm{vN}$ vs the correlation length $\xi$ in the vicinity of the transition.