Symmetry-deformed toric codes and the quantum dimer model

TL;DR

Symmetry-deformed toric codes study how dropping Gauss-law star terms from Kitaev’s toric code generates global and subsystem symmetries, yielding models such as the $U(1)$TC and $XY$TC. The authors map out the resulting phase structure, showing that acquiring global symmetry typically destroys emergent gauge structure and topological order, while subsystem symmetries produce subextensive ground-state degeneracies organized by Wilson loops; they also analyze the extreme deformation to the quantum dimer model (QDM). Using solvable limits and iDMRG numerics, they show a plaquette valence-bond solid ground state in the $U(1)$TC, a robust $2^{L+1}$-fold degeneracy in the $XY$TC, and a hierarchy of models connected to TFIM and QDM through term dropping. In the QDM limit, an emergent SO(2) symmetry appears near a quasi-critical region with a gapless mode (central charge $c\approx 1$), and sublattice modulation stabilizes pVBS, suggesting a rich interplay between symmetry, quantum order-by-disorder, and topological concepts. Overall, the work illuminates a broader landscape of symmetry-deformed quantum orders beyond traditional gauging/ungauging, with potential extensions to larger gauge groups such as $\mathbb{Z}_4$.

Abstract

Motivated by the recent introduction of a $U(1)$-symmetric toric code model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the toric code (TC) Hamiltonian. This "term-dropping" protocol introduces global symmetries that go beyond the alternative framework of "ungauging" topological order in symmetry-deformed models and gives rise to models such as the $U(1)$TC or $XY$TC. These models inherit (emergent) subsystem symmetries (from the original 1-form symmetry of the TC) that can give rise to (subextensive) ground-state degeneracies, which can still be organized by the eigenvalues of Wilson loop operators. However, we demonstrate that these models do not support topological or fracton order (as has been conjectured in the literature) due to the loss of (emergent) gauge symmetry. An extreme deformation of the TC is the quantum dimer model (QDM), which we discuss along the family of symmetry-deformed models from the perspective of subsystem symmetries, sublattice modulation, and quantum order-by-disorder mechanisms resulting in rich phase diagrams. For the QDM, this allows us to identify an emergent SO(2) symmetry for what appears to be a gapless ground state (by numerical standards) that is unstable to the formation of a plaquette valence bond solid upon sublattice modulation.

Figures (7)

• Figure 1: Schematic model overview for the term-dropping protocol. Shown are instances of 8-term, 6-term, 4-term, and 2-term models that arise from restricting the 16 individual operators (and their Hermitian conjugates) of the toric code's star terms shown in the upper left (orange shaded) panel. The plaquette term of the original toric code can be similarly represented (see the upper right, pink-shaded panel) where we distinguish terms enforcing $B_p=+1$$(B_p=-1)$ plaquette eigenvalues corresponding to an even (odd) gauge theory. For each model instance, we illustrate a representative example of a ground-state configuration in the exactly solvable limit of maximal sublattice modulation $(J_{s_2} = 0)$. The black diagonal lines emanating from the plaquettes serve as a visual aid to identify the locations of fluxes, as explained in detail in Fig. \ref{['fig:TGSD']} and in the main text. For the quantum dimer model (QDM), the gray plaquettes indicate those that do not gain energy from the star term (see the discussion in Sec. \ref{['sec:QDM']}).
• Figure 2: Phase diagram of symmetry-deformed toric codes. (a) Phase diagram interpolating the $U(1)$TC and $XY$TC models to the TC model using parameters $\lambda$ and $\lambda'$, respectively, with a sublattice modulation of the star terms, $J_{s_2} = 1 - J_{s_1}$. The red/blue lines indicate first/second order transitions. The label "pVBS" stands for plaquette valence bond solid, while "SSSB" stands for spontaneous subsystem symmetry breaking phase. For the solvable limit of the $U(1)$TC the subsystem symmetry is emergent, indicated by "SSSB*". (b) Energy difference between the full-line configuration and the pVBS configuration, $\Delta E = E_{\text{full-line}} - E_{\text{pVBS}}$. (c) Schematic illustration of the quantum order-by-disorder mechanism induced in 4th-order perturbation theory. Energy costs are localized on the plaquettes marked in red. (d) The inverse correlation length $1/\xi$, which scales with the gap, along horizontal cuts in the upper half, indicating continuous phase transitions (with a gap closing) for $\lambda < 0.851$ and first-order transitions (with a finite gap) for $\lambda > 0.851$. (e) Scaling of the correlation length $\xi$ with bond dimension $\chi$ at phase boundaries.
• Figure 3: Graphical representation of ground-state configurations in the exactly solvable limit $J_{s_2} = 0$. (a) The sites, on which the star terms act, are split into two sublattices, $s_1$ and $s_2$. Ground-state configurations can be constructed by assigning to each $s_1$ site a plaquette valence bond favored by the corresponding $A_s$ term, e.g., $\ket{ }$, $\ket{ }$, or $\ket{ }$ in the case of the $U(1)$TC. $B_p = -1$ plaquettes (indicated by a "$+$") appear at the endpoints of (anti-)diagonal lines (see text for details). (b) Ground-state configurations related by the $Z_2^{\text{sub}}$ symmetry can be sequentially generated by repeatedly applying the operator $\prod_{i \in C_{\text{diag}}} \sigma^x_i$, where $C_{\text{diag}}$ denotes the contour along the diagonal direction. (c) Examples of the subextensive ground-state manifold of the $U(1)$TC, characterized by different compactification angles and classified by the eigenvalues of the operators $W_x$ and $W_y$.
• Figure 4: Equivalence between TFIM and symmetry-deformed TC models. (a) Schematics of a spin configuration for symmetry-deformed TC/TFIM. (b) Energy spectra of two variants of the TFIM-type two-term symmetry-deformed TC model with uniform $B_p = +1$ and $B_p = -1$ constraints and the TFIM , obtained via ED for a system of size $N = 32$. While all three models share the same low-energy spectra (indicated by the blue lines), the symmetry-deformed TC models also exhibit additional states that are dynamically disconnected from the TFIM subspace.
• Figure 5: Equivalence between QDM and symmetry-deformed TC models. (a) Schematics of a spin/dimer configuration for symmetry-deformed TC/QDM. (b) Energy spectra of the QDM and two variants of the QDM-type two-term symmetry-deformed TC model with uniform $B_p = +1$ and $B_p = -1$ constraints, obtained via ED for a system of size $N = 32$. While all three models share the same low-energy spectra (indicated by the blue lines), the symmetry-deformed TC models also exhibit additional states that are dynamically disconnected from the QDM subspace.