Deformed LDPC codes with spontaneously broken non-invertible duality symmetries

TL;DR

This work develops symmetry‑preserving deformations of LDPC‑coded spin systems in a transverse field and identifies a unique frustration‑free point where a trivial gapped phase and a nontrivial code/TO phase coexist, with a spontaneous breaking of non‑invertible duality symmetry if present. The authors introduce a graph‑theoretic coarse‑graining and martingale method to prove a finite gap in the thermodynamic limit for these non‑commuting, frustration‑free Hamiltonians defined on arbitrary Tanner graphs. The results extend to quantum CSS codes and yield concrete examples (e.g., Toric Code and X‑Cube) across 1+1, 2+1, and 3+1 dimensions, illustrating coexisting phases and symmetry breaking under dualities. Collectively, the paper provides a general lattice framework for realizing and certifying gapped phases with spontaneously broken non‑invertible dualities, and outlines pathways to tricritical behavior and Z_N generalizations with broad implications for fracton and topological order physics.

Abstract

Low-density parity check (LDPC) codes are a well known class of Pauli stabiliser Hamiltonians that furnish fixed-point realisations of nontrivial gapped phases such as symmetry breaking and topologically ordered (including fracton) phases. In this work, we propose symmetry-preserving deformations of these models, in the presence of a transverse field, and identify special points along the deformations with interesting features: (i) the special point is frustration-free, (ii) its ground states include a product state and the code space of the underlying code, and (iii) it remains gapped in the thermodynamic (infinite volume) limit. So the special point realises a first-order transition between (or the coexistence of) the trivial gapped phase and the nontrivial gapped phase associated with the code. In addition, if the original model has a non-invertible duality symmetry, then so does the deformed model. In this case, the duality symmetry is spontaneously broken at the special point, consistent with the associated anomaly. A key step in proving the gap is a coarse-graining/blocking procedure on the Tanner graph of the code that allows us to apply the martingale method successfully. Our model, therefore, provides the first application of the martingale method to a frustration-free model, that is not commuting projector, defined on an arbitrary Tanner graph. We also discuss several familiar examples on Euclidean spatial lattice. Of particular interest is the 2+1d transverse field Ising model: while there is no non-invertible duality symmetry in this case, our results, together with known numerical results, suggest the existence of a tricritical point in the phase diagram.

Figures (3)

• Figure 1: The bipartite graphs associated with (a) the 1+1d Ising model, (b) the 2+1d Ising model, and (c) the 2+1d plaquette Ising model. In all cases, the solid (black) vertices represent the physical qubits in $V$ and the hollow (white) vertices represent the interactions in $\widehat{V}$. In (a) and (c), the red arrow $\rho$ represents a reversing automorphism, given by a "half-translation", that generates the non-invertible duality symmetry. The thin grey lines in (c) represent the underlying square lattice. Since the 2+1d Ising model and the 2+1d Toric Code are related via gauging, they have the same bipartite graph, except for exchanging the solid and hollow vertices (and a "half-translation" in the (1,1) direction).
• Figure 2: The bipartite graphs associated with (a) the 3+1d Toric Code and (b) the X-Cube model. In both cases, the solid (black) vertices represent the physical qubits in $V$ and the hollow (white) vertices represent the interactions in $\widehat{V}$. In (a), the red arrow $\rho$ represents a reversing automorphism, given by a "half-translation" in the $(1,1,1)$ direction, that generates the non-invertible Wegner duality symmetry. The thick grey lines represent the underlying cubic lattice.
• Figure 3: The possible phase diagram of the deformed model on a Euclidean lattice. The dashed (solid) line represents a first (second/higher) order transition between the trivial phase and the spontaneously symmetry broken (SSB) or topologically ordered (TO) phase (the latter includes fracton order). TP denotes the transition point between the two phases at $\lambda = 0$, FF denotes the frustration-free point at $\lambda = 1$, TCP denotes a tricritical point, and B denotes a bifurcation point. The transition line shown here is just a representative, and depending on the model, it could bend the other way. In particular, when there is a non-invertible duality symmetry, the entire transition line is vertical along the self-dual line $h=J$.

Theorems & Definitions (7)

• Claim 4.8
• Theorem B.1
• proof : Proof of Theorem \ref{['thm:goodcover']}
• Theorem C.35: MinrankZeroforcing
• Theorem C.37: GENTNER2018203
• Proposition C.42
• proof