Quantum Phases and Transitions in Spin Chains with Non-Invertible Symmetries

TL;DR

This work constructs two exactly solvable1D spin chains with non-invertible generalized symmetries: an S3-symmetric model and its dual Rep(S3)-symmetric counterpart obtained by gauging a Z2 subgroup. It shows that their phase diagrams host four gapped symmetry-breaking patterns and, along self-dual lines, intrinsically non-invertible KW self-duality, with enhanced symmetry described by the 2+1d JK4 × JK4 topological order (SymTO). The authors develop explicit lattice dualities via gauging subgroups, identify order/disorder patch operators to diagnose SSB, and provide a SymTO-centered framework that classifies gapped and gapless states, including incommensurate gapless phases with anomalous U(1) symmetries arising from lattice translation. The results illustrate Morita-equivalent fusion-category symmetries yielding identical symmetric spectra and reveal a rich structure of domain walls and condensations linking distinct SymTOs, thereby advancing lattice realizations of fusion-category symmetries and their phase structure. The work further connects to holographic symmetry descriptions and modular-bootstrap analyses, and outlines future directions for implementing gauging by algebra objects and exploring broader fusion-category landscapes.

Abstract

Generalized symmetries often appear in the form of emergent symmetries in low energy effective descriptions of quantum many-body systems. Non-invertible symmetries are a particularly exotic class of generalized symmetries, in that they are implemented by transformations that do not form a group. Such symmetries appear in large families of gapless states of quantum matter and constrain their low-energy dynamics. To provide a UV-complete description of such symmetries, it is useful to construct lattice models that respect these symmetries exactly. In this paper, we discuss two families of one-dimensional lattice Hamiltonians with finite on-site Hilbert spaces: one with (invertible) $S^{\,}_3$ symmetry and the other with non-invertible $\mathsf{Rep}(S^{\,}_3)$ symmetry. Our models are largely analytically tractable and demonstrate all possible spontaneous symmetry breaking patterns of these symmetries. Moreover, we use numerical techniques to study the nature of continuous phase transitions between the different symmetry-breaking gapped phases associated with both symmetries. Both models have self-dual lines, where the models are enriched by so-called intrinsically non-invertible symmetries generated by Kramers-Wannier-like duality transformations. We provide explicit lattice operators that generate these non-invertible self-duality symmetries. We show that the enhanced symmetry at the self-dual lines is described by a 2+1D symmetry-topological-order (SymTO) of type $\mathrm{JK}^{\,}_4\boxtimes \overline{\mathrm{JK}}^{\,}_4$. The condensable algebras of the SymTO determine the allowed gapped and gapless states of the self-dual $S^{\,}_3$-symmetric and $\mathsf{Rep}(S^{\,}_3)$-symmetric models.

Figures (16)

• Figure 1: Schematic of the Hamiltonian \ref{['eq:def Ham gen S3']} showing the couplings between qutrit (depicted by a tripartitioned disk) and qubit (depicted by a bipartitioned disk) degrees of freedom. Single-body terms $J^{\,}_2,J^{\,}_4$ are suppressed.
• Figure 2: Phase diagram of Hamiltonian \ref{['eq:Ham gen S3 reparameterized']} based on analytical arguments for $\theta\approx 0$. The ground state degeneracy (GSD) for each of the SSB phases are labeled. The vertical and horizontal critical lines correspond to the Potts (6,5) and Ising (4,3) minimal model CFTs, respectively. They intersect at a multi-critical point belonging to the 3-state Potts $\boxtimes$ Ising universality class.
• Figure 3: Numerical phase diagram obtained from the TEFR algorithm showing GSD (a) and central charge (b) as a function of $J^{\,}_1$ and $J_3$. The effective system size for these plots is $L=128$. Figs. (c) and (d) show the central charges computed from bipartite entanglement entropy scaling in the ground state obtained from DMRG, as discussed in the main text. We plot the central charges along a vertical ($J^{\,}_1=0.5$) and a horizontal ($J_3=0.5$) slice of the phase diagram shown in Fig. (b), for a chain of $L=100$ sites. We fix $\theta=0.1$, with $J^{\,}_2=1-J^{\,}_1$, $J_4=1-J_3$ in all of these plots.
• Figure 4: Numerical phase diagram obtained from the TEFR algorithm with fixed $\theta=0.7\approx\frac{2\pi}{9}$, showing GSD (Fig. (a)), and central charge (Fig. (b)) as a function of $J^{\,}_1$ and $J_3$, setting $J^{\,}_2=1-J^{\,}_1$ and $J_4=1-J_3$ everywhere. The nominal GSD in this gapless region is much larger than that shown in Fig. (a); we have capped the maximum allowed values to 10 so that these plots may be easily compared with the ones in Fig. \ref{['fig:s3pd-numerics']}. In these figures, the effective system size is $L=128$. Fig. (c) shows the absolute value of the Fourier transform of ground state expectation value $\langle\hat{\sigma}^{z}_{i}\rangle$ for $J^{\,}_{1}=0.5$ and $J^{\,}_{1}=0.52$ with fixed $J^{\,}_{3}=0.5$ and $L=101$ sites.
• Figure 5: Numerical phase diagram from TEFR algorithm, with fixed $J_{\perp}$ and $\theta=0$, showing GSD (left) and central charge(right) as heatmaps, in the $J^{\,}_1, J^{\,}_{3}$ plane (with $J^{\,}_2=1-J^{\,}_1$ and $J_4=1-J^{\,}_{3}$ everywhere). The effective system size is $L=128$. (a) For positive $J_{\perp}$, we find the multicritical point widens into a critical line between the $S_3$ symmetric and $S_3$ SSB phases. (b) For negative $J_{\perp}$, we find the multicritical point widens into a critical line between the phases which spontaneously break $S_3$ down to $\mathbb{Z}_3$ and $\mathbb{Z}_2$, instead.