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arXiv:2607.28740

Gapped Parent Hamiltonians for the Strongly Deformed Toric Code

Nandagopal Manoj, Zack Weinstein, Jason Alicea

cond-mat.str-elcond-mat.stat-mechquant-phSubmitted Jul 30, 2026• Announced Aug 3, 2026• 0 citations
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Abstract

Local non-unitary deformations of topologically ordered wavefunctions can drive transitions into peculiar states that challenge modern perspectives on gapped quantum matter. The strongly deformed toric code offers a curious case, hosting $m$ anyon condensation alongside perimeter-law scaling of Wilson loops charged under an exact 1-form symmetry---properties that typically do not coexist in gapped ground states. Nevertheless, we rigorously construct local gapped parent Hamiltonians for these strongly deformed toric code states. The Hamiltonians we construct are not strictly finite-range, but contain sums of Wilson loop operators whose coefficients decay exponentially in their diameter. If one adopts standard locality bounds used to define gapped phases---which allow for such exponentially decaying terms---our construction shows that these states realize a trivial gapped phase. Within this locality class, we demonstrate that perimeter-law scaling of Wilson loops does not imply a spontaneously broken 1-form symmetry, and from a dual perspective, that long-range ferromagnetic order and perimeter-law disorder parameter correlations can coexist in a 2D gapped ground state. We evade a recent no-go theorem [Sahay et al., arXiv:2503.01977] by relaxing its assumptions in a manner that we quantify as benign in the thermodynamic limit. More broadly, our results highlight that stronger notions of locality are necessary for prohibiting these counterintuitive properties within a gapped phase.

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7 + 25 pages, 1 + 2 figures

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Local non-unitary deformations of topologically ordered wavefunctions can drive transitions into peculiar states that challenge modern perspectives on gapped quantum matter. The strongly deformed toric code offers a curious case, hosting mmm anyon condensation alongside perimeter-law scaling of Wilson loops charged under an exact 1-form symmetry---properties that typically do not coexist in gapped ground states. Nevertheless, we rigorously construct local gapped parent Hamiltonians for these strongly deformed toric code states. The Hamiltonians we construct are not strictly finite-range, but contain sums of Wilson loop operators whose coefficients decay exponentially in their diameter. If one adopts standard locality bounds used to define gapped phases---which allow for such exponentially decaying terms---our construction shows that these states realize a trivial gapped phase. Within this locality class, we demonstrate that perimeter-law scaling of Wilson loops does not imply a spontaneously broken 1-form symmetry, and from a dual perspective, that long-range ferromagnetic order and perimeter-law disorder parameter correlations can coexist in a 2D gapped ground state. We evade a recent no-go theorem [Sahay et al., arXiv:2503.01977] by relaxing its assumptions in a manner that we quantify as benign in the thermodynamic limit. More broadly, our results highlight that stronger notions of locality are necessary for prohibiting these counterintuitive properties within a gapped phase.
It is well known that an exponentially localized Hamiltonian must be gapless if its ground state has algebraic correlations. We show that even certain exponentially decaying correlations can imply gaplessness. This is exemplified by the deformed toric code ∝exp⁡(β∑ℓZℓ)∣TC⟩\propto \exp(β\sum_{\ell} Z_{\ell}) |\mathsf{TC}\rangle∝exp(β∑ℓ​Zℓ​)∣TC⟩, where ∣TC⟩|\mathsf{TC}\rangle∣TC⟩ is a fixed-point toric code wavefunction. Although it has a confined regime for β>βcβ> β_cβ>βc​, recent work has drawn attention to its perimeter law loop correlations. Here, we show that these unusual loop correlations -- namely, perimeter law coexisting with a 1-form symmetry whose disorder operator has long-range order -- imply that any local parent Hamiltonian must either be gapless or have a degeneracy scaling with system size. Moreover, we construct a variational low-energy state for arbitrary local frustration-free Hamiltonians, upper bounding the finite-size gap by O(1/L3)O(1/L^3)O(1/L3) on periodic boundary conditions. Strikingly, these variational states look like loop waves -- non-quasiparticle analogs of spin waves -- generated from the ground state by non-local loop operators. Our findings have implications for identifying the subset of Hilbert space to which gapped ground states belong, and the techniques have wide applicability. For instance, a corollary of our first result is that Glauber dynamics for the ordered phase of the two-dimensional classical Ising model on the torus must have a gapless Markov transition matrix, with our second result bounding its gap.
We examine the condensation and confinement mechanisms exhibited by a deformed toric code model proposed in [Castelnovo and Chamon, Phys. Rev. B 77, 054433 (2008)]. The model describes both sides of a phase transition from a topological phase to a trivial phase. Our findings reveal an unconventional confinement mechanism that governs the behavior of the toric code excitations within the trivial phase. Specifically, the confined magnetic charge can still be displaced without any energy cost, albeit only via the application of non-unitary operators that reduce the norm of the state. This peculiar phenomenon can be attributed to a previously known feature of the model: It maintains the non-trivial ground state degeneracy of the toric code throughout the transition. We describe how this degeneracy arises in both phases in terms of spontaneous symmetry breaking of a generalized (1-form) symmetry and explain why such symmetry breaking is compatible with the trivial phase. The present study implies the existence of subtle considerations that must be addressed in the context of recently posited connections between topological phases and broken higher-form symmetries.
Topological phase transition of deformed Z3{\mathbb Z}_3Z3​ toric code
We investigate topological phase transitions in a family of deformed Z3\mathbb Z_3Z3​ toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the Q=3Q=3Q=3 Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT3_33​) construction with two independent four-spin couplings in the general case. Projected entangled-pair-state (PEPS) and variational uniform matrix-product-state (VUMPS) calculations identify the toric-code (TC) phase and phases in which electric (eee) anyons are confined or condensed. These phases are separated by critical structures with central charges c=4/5c=4/5c=4/5, 8/58/58/5, and isolated c=1c=1c=1 antiferromagnetic (AFM) endpoints. A normalized finite-distance eee-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double D(Z3)D(\mathbb Z_3)D(Z3​) imply a topological entanglement entropy γ=log⁡3γ=\log3γ=log3 throughout the gapped toric-code phase. Relative to the Z2\mathbb Z_2Z2​ case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent U(1)U(1)U(1) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.
Motivated by the recent introduction of a U ( 1 ) -symmetric toric code (TC) model, we investigate symmetry-based deformations of topological order by systematically deconstructing the Gauss-law-enforcing star terms of the 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 X Y 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 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.
In the tensor network representation, a deformed Z2Z_{2}Z2​ topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate of pairs of logarithmically confined electric charges and magnetic fluxes, and the scaling behavior of various anyon correlations can be exactly derived, revealing the effective interaction between anyons and their condensation. Deformations away from the self-duality drive the Coulomb state into either the gapped Higgs phase or confining phase.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.