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.
We investigate topological phase transitions in a family of deformed
Z3 toric-code wavefunctions prepared from a cluster state by local deformations and projective measurements. Their norms map to the
Q=3 Potts model for single-parameter deformations and to a three-state Ashkin--Teller-like (AT
3) 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 (
e) anyons are confined or condensed. These phases are separated by critical structures with central charges
c=4/5,
8/5, and isolated
c=1 antiferromagnetic (AFM) endpoints. A normalized finite-distance
e-anyon pair-state norm provides a Fredenhagen--Marcu-type check of the confinement boundary, while the topological data of the quantum double
D(Z3) imply a topological entanglement entropy
γ=log3 throughout the gapped toric-code phase. Relative to the
Z2 case, the absence of sign-change folding leaves the AFM endpoints unfolded, and the extreme deformation reaches square ice with an emergent
U(1) one-form symmetry, Hilbert-space fragmentation, and exact scar configurations.
We study effectively one-dimensional systems that emerge at the edge of a two-dimensional topologically ordered state, or at the boundary between two topologically ordered states. We argue that anyons of the bulk are associated with emergent symmetries of the edge, which play a crucial role in the structure of its phase diagram. Using this symmetry principle, transitions between distinct gapped phases at the boundaries of Abelian states can be understood in terms of symmetry-breaking transitions or transitions between symmetry-protected topological phases. Yet more exotic phenomena occur when the bulk hosts non-Abelian anyons. To demonstrate these principles, we explore the phase diagrams of the edges of a single and a double layer of the toric code, as well as those of domain walls in a single- and double-layer Kitaev spin liquid (KSL). In the case of the KSL, we find that the presence of a non-Abelian anyon in the bulk enforces Kramers-Wannier self-duality as a symmetry of the effective boundary theory. These examples illustrate a number of surprising phenomena, such as spontaneous duality breaking, two-sector phase transitions, and unfreezing of marginal operators at a transition between different gapless phases.
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.
We analyze the effect of perturbations on a toric code-like Hamiltonian enriched with global symmetries. The ground states of this Hamiltonian are superpositions of loops decorated with cluster states, resulting in anyonic excitations that fractionalize a global on-site
Z2×Z2, inversion and time reversal symmetry. We use a modified version of the string order parameters proposed in [New Journal of Physics 21, 113016 (2019)], adapted to work beyond the framework of tensor networks, to capture the symmetry fractionalization pattern of the anyons. We show that this order parameter accurately captures the symmetry fractionalization in a direct Hamiltonian interpolation between the decorated and trivial toric codes, and in the presence of symmetry-preserving local magnetic fields and Ising interactions. We also observe how the condensation of an anyon that fractionalizes the symmetry forces the symmetry to spontaneously break at the phase transition. This notably changes the phase diagram of this Hamiltonian under parallel magnetic fields compared to the regular toric code since the condensation of charge and flux anyons drives the system to two distinct trivial phases, a symmetry broken phase and symmetric phase. Moreover, we provide a proof of spontaneous symmetry breaking from anyon condensation in the framework of projected entangled pair states.
The algebraic tools used to study topological phases of matter are not clearly suited to studying processes in which the bulk energy gap closes, such as phase transitions. We describe an elementary two edge thought experiment, which reveals the effect of an anyon condensation phase transition on the robust edge properties of a sample, bypassing a limitation of the algebraic description. In particular, the two edge construction allows some edge degrees of freedom to be tracked through the transition, despite the bulk gap closing. The two edge model demonstrates that bulk anyon condensation induces symmetry breaking in the edge model. Further, the construction recovers the expected result that the number of chiral current carrying modes at the edge cannot change through anyon condensation. We illustrate the construction through detailed analysis of anyon condensation transitions in an achiral phase, the toric code, and in chiral phases, the Kitaev spin liquids.