Finite-temperature phases are traditionally studied through spontaneous symmetry breaking, but symmetry-breaking fields can remove their distinction, as in the Ising model. This motivates the search for stable nontrivial thermal phases without symmetry protection. For classical $\mathbb Z_2$ lattice gauge models with spins on $m$-dimensional cells in a fixed spatial dimension $d$, we prove joint analyticity of the thermodynamic free-energy density in temperature and perturbation strength at sufficiently low positive temperatures under sufficiently weak local perturbations, whenever $1\le m\le d-2$. Perturbations need not preserve internal or gauge symmetries and may include noncommuting quantum interactions. A separate high-temperature expansion yields an analytic region reaching infinite temperature at any fixed finite perturbation strength. These results establish thermodynamic stability and support the conjecture that low- and high-temperature regimes remain separated by a thermal transition without symmetry protection.
Thermodynamic Stability of Classical Z2 Lattice Gauge Models
Finite-temperature phases are traditionally studied through spontaneous symmetry breaking, but symmetry-breaking fields can remove their distinction, as in the Ising model. This motivates the search for stable nontrivial thermal phases without symmetry protection. For classical Z2 lattice gauge models with spins on m-dimensional cells in a fixed spatial dimension d, we prove joint analyticity of the thermodynamic free-energy density in temperature and perturbation strength at sufficiently low positive temperatures under sufficiently weak local perturbations, whenever 1≤m≤d−2. Perturbations need not preserve internal or gauge symmetries and may include noncommuting quantum interactions. A separate high-temperature expansion yields an analytic region reaching infinite temperature at any fixed finite perturbation strength. These results establish thermodynamic stability and support the conjecture that low- and high-temperature regimes remain separated by a thermal transition without symmetry protection.
Local-Channel Stability of Classical Z2 Lattice Gauge Models
We establish sufficient conditions for two-way local quantum-channel stability in classical Z2 lattice gauge models with spins on m-dimensional cells of a d-dimensional lattice. Each interaction multiplies the spins on the boundary of an (m+1)-cell. This family, defined on the full spin Hilbert space without a Gauss-law constraint, includes the Ising model at m=0 and the magnetic sector of the toric code at m=1. The low-temperature proof combines two ingredients: locally indistinguishable flat sectors and extended magnetic defects of dimension d−m−1. For 1≤m≤d−2, sufficiently weak local classical perturbations at sufficiently low positive temperature preserve equivalence to the canonical flat ensemble. No internal or gauge symmetry is required of the perturbation. For weak noncommuting perturbations, we prove the same conclusion in a smaller sufficient window by reducing the diagonal probabilities to a weak classical interaction and recovering the coherences with a local channel. The conversion channels use product ancillary inputs, have polylogarithmic causal radius, and achieve any prescribed inverse-power whole-system trace-norm error. Sufficiently high-temperature states are instead two-way equivalent to a product state. Independent finite-dimensional onsite systems can be included with weak local couplings. An optimized covariance between two distant neighborhoods of parallel m-tori is nonincreasing under one-way local conversion and invariant under two-way conversion. Its values Fm=1 and 0 distinguish the proved low- and high-temperature domains without a protecting symmetry. These sufficient domains do not determine the full phase diagram.
Topologically ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors. We connect the two notions of stability by constructing classical statistical mechanics models for decoding general Calderbank-Shor-Steane codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized Z 2 lattice-gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the imaginary-time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative X or Z magnetic field. Motivated by error-correction considerations, we define general order parameters for all such generalized Z 2 lattice-gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the low-density parity-check condition and with a sufficiently large code distance, we prove the existence of a low-temperature ordered phase of the corresponding lattice-gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence for stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and timelike defects in the lattice-gauge theory. A high free-energy cost of spacelike defects corresponds to a successful “memory experiment” and suppresses the energy splitting among the ground states, while a high free-energy cost of timelike defects corresponds to a successful “stability experiment” and points to a nonzero gap to local excitations.
In the field of frustrated magnetism, Kitaev models provide prototypical examples of exactly solvable quantum spin liquids, in which the spin degrees of freedom fractionalize into Majorana fermions coupled to an emergent Z2 gauge field. While the ground states of these models are already well understood as exhibiting different kinds of Majorana (semi)metals, it has still been an open task to achieve a detailed understanding of the thermodynamics, and, namely, the low-temperature ordering of the gauge field into different Z2 flux configurations. In this thesis, we investigate the thermodynamics of Kitaev systems in two and three spatial dimensions with sign-problem-free quantum Monte Carlo simulations. In a first study of elementary 3D Kitaev models, we verify that the ground state Z2 flux sectors of these systems are entirely determined by their elementary plaquette length – a result which shows the validity of Lieb’s theorem for lattice geometries which lack the geometric requirements for its proof. We closely investigate the low-temperature phase transition associated with gauge-ordering, which is a particular realization of an inverted Ising phase transition, as it occurs in general lattice Z2 gauge theories. Our results corroborate the understanding of this transition as separating different regimes in terms of vison-loop excitations, by showing a clear correlation between the critical temperature and the vison gap. We also introduce the concept of “gauge frustration”, which, for a particular 3D Kitaev model, leads to the suppression of the phase transition and a more complex physical behavior at low temperatures. A second quantum Monte Carlo study focuses on a generalized 2D Kitaev model on a five-coordinated lattice system. Here, the gauge ordering is accompanied by the spontaneous breaking of time-reversal symmetry, a scenario which not only allows for a variety of topological ground states, but also for the occurrence of a phase transition in two spatial dimensions. We show that this model exhibits such a transition at a particularly high temperature scale. Moreover, the system possesses a number of phases where the Z2 fluxes are only partially ordered.
Deconfined Thermal Phase Transitions with Z2 Gauge Structures
Fathoming deconfined phases is one of the key issues in modern condensed matter. Striking many-body effects including massive quantum entanglement and coherence may be realized as manifested in quantum spin liquids and topological orders. Here, we demonstrate that deconfined phases even host exotic thermal phase transitions, dubbed deconfined thermal transitions. Constructing a Z2 lattice gauge model with strong interactions between Z2 gauge fluxes, we prove the existence of a thermal phase transition between deconfined and confined phases in two spatial dimensions in sharp contrast to its absence in the Wegner model. Incorporating deconfined fermions, it is shown that gapless excitations from Fermi surfaces endow line-tension to Z2 gauge fluxes at zero temperature, and we argue that a deconfined thermal transition with deconfined fermions may be interpreted as a hidden order transition with thermal gap-opening in Fermi surfaces. Moreover, it is shown that symmetry breaking transitions in deconfined phases may be unconventional. Global Z2 and U(1) symmetry breaking transitions in deconfined phases may be in the same universality class, which is impossible under the conventional Landau-Ginzburg-Wilson paradigm. Characteristic signatures of the transitions in experiments and candidate strongly correlated systems such as Kitaev materials are also discussed.
Slow mixing and emergent one-form symmetries in three-dimensional Z2 gauge theory
Symmetry-breaking order at low temperatures is often accompanied by slow relaxation dynamics, due to diverging free-energy barriers arising from interfaces between different ordered states. Here, we extend this correspondence to classical topological order, where the ordered states are locally indistinguishable, so there is no notion of interfaces between them. We study the relaxation dynamics of the three-dimensional (3D) classical Z2 lattice gauge theory (LGT) as a canonical example. We prove a lower bound on the mixing time in the deconfined phase, tmix=exp[Ω(L)], where L is the linear system size. This bound applies even in the presence of perturbations that explicitly break the one-form symmetry between different long-lived states. This perturbation destroys the energy barriers between ordered states, but we show that entropic effects nevertheless lead to diverging free-energy barriers at nonzero temperature. Our proof establishes the LGT as a robust finite-temperature classical memory. We further prove that entropic effects lead to an emergent one-form symmetry, via a notion that we make precise. We argue that the exponential mixing time follows from universal properties of the deconfined phase, and numerically corroborate this expectation by exploring mixing time scales at the Higgs and confinement transitions out of the deconfined phase. These transitions are found to exhibit markedly different dynamic scaling, even though both have the static critical exponents of the 3D Ising model. We expect this novel entropic mechanism for memory and emergent symmetry to also bring insight into self-correcting quantum memories.
Although condensed matter systems usually do not have higher-form symmetries, we show that, unlike 0-form symmetry, higher-form symmetries can emerge as exact symmetries at low energies and long distances. In particular, emergent higher-form symmetries at zero temperature are robust to arbitrary local UV perturbations in the thermodynamic limit. This result is true for both invertible and noninvertible higher-form symmetries. Therefore emergent higher-form symmetries are exact emergent symmetries: they are not UV symmetries but constrain low-energy dynamics as if they were. Since phases of matter are defined in the thermodynamic limit, this implies that a UV theory without higher-form symmetries can have phases characterized by exact emergent higher-form symmetries. We demonstrate this in three lattice models, the quantum clock model and emergent ZN and U(1)p-gauge theory, finding regions of parameter space with exact emergent (anomalous) higher-form symmetries. Furthermore, we perform a generalized Landau analysis of a 2+1D lattice model that gives rise to Z2 gauge theory. Using exact emergent 1-form symmetries accompanied by their own energy/length scales, we show that the transition between the deconfined and Higgs/confined phases is continuous and equivalent to the spontaneous symmetry-breaking transition of a Z2 symmetry, even though the lattice model has no symmetry. Also, we show that this transition line must always contain two parts separated by multi-critical points or other phase transitions. We discuss the physical consequences of exact emergent higher-form symmetries and contrast them to emergent 0-form symmetries. Lastly, we show that emergent 1-form symmetries are no longer exact at finite temperatures, but emergent p-form symmetries with $p\ensuremath{\ge}2$ are.