I define quantum loop models whose degrees of freedom are Ising spins on the square lattice as in the toric code, but where the excitations should have non-abelian statistics. The inner product is topological, allowing a direct implementation of the anyonic fusion matrix on the lattice. It also makes deconfined anyons possible for a variety of values of the weight per loop
d in the ground state. For d=\sqrt{2}, a gapped non-abelian topological phase can occur with only four-spin interactions.
Condensation of quantum loops naturally leads to topological phases with Abelian excitations. Here, I propose that non-Abelian topological phases can arise from merging two (or several) identical Abelian quantum loop condensates. I define merging through a symmetrization operation, which makes the two loop condensates indistinguishable, inducing the possibility of a topological degeneracy in the space of quasiparticles. To illustrate the construction, the case of two identical toric-code quantum loop gases is considered. A spin-1 model for the two-dimensional square lattice is presented for which the resulting merged state is the exact unique ground state. This Hamiltonian involves four-body interactions between spins located at the same plaquette or vertex, which are quadratic in the spin-1 operators. Vertex and plaquette interaction terms are not mutually commuting. The model displays gapped excitations that result from merging anyons in both toric-code copies. These excitations exhibit non-Abelian braiding properties.
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.
String-net condensation can give rise to non-Abelian anyons whereas loop condensation usually gives rise to Abelian anyons. It has been proposed that generalized quantum loop gases with non-orthogonal inner products can produce non-Abelian anyons. We detail an exact mapping between the string-net and the generalized loop models and explain how the non-orthogonal products arise. We also introduce a loop model of double-stranded nets where quantum loops with an orthogonal inner product and local interactions supports non-Abelian Fibonacci anyons. Finally we emphasize the origin of the sign problem in such systems and its consequences on the complexity of their ground state wave functions.
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⟩, where
∣TC⟩ is a fixed-point toric code wavefunction. Although it has a confined regime for
β>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) 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.
Quantum phases of matter can be classified into two kinds: gapped or gapless, depending on whether the equilibrium ground state is separated from excited states by a finite energy window or not in the thermodynamic limit. Topology has played an important role in the understanding of both gapped and gapless phases of matter. This thesis uncovers several robust physical properties of both kinds of systems which are largely unexplored before. \\ For gapped systems, we study topological orders, which are characterized by the existence of anyonic quasiparticles with fractional braiding statistics. In the first part, we introduce a family of Abelian quantum Hall insulators, at filling factors
ν=p/2q for bosonic systems and
ν=p/(p+2q) for fermionic systems, with
p and
q being two coprime integers. These states are termed the \textit{nondiagonal} quantum Hall states, which are constructed in a coupled wire model that exhibits an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for anyons. We unveil a nontrivial interplay between charge symmetry and translation symmetry in this system, and establish the nondiagonal quantum Hall states as symmetry-enriched topological orders. In the second part, we introduce a related model of interacting electrons, but in the absence of a strong quantizing magnetic field, known as the \textit{toric code insulator}. It is constructed from a two-dimensional array of strongly coupled one-dimensional topological superconductors, which feature electron fractionalization and possesses anyonic excitations described by the
Z2 topological order. The motion of anyons is again constrained, and different types of anyons can be related by the translation symmetry. Nondiagonal quantum Hall states and the toric code insulator are convincing examples that itinerant electronic systems can realize interesting symmetry-enriched topological orders.\\ For gapless quantum matters we focus on a particularly familiar class, the metals, but we uncover their topological aspects which were much less appreciated. Every metal is characterized by a Fermi surface of gapless excitations, and the Fermi surface surrounds a manifold in the momentum space, known as the Fermi sea. The ground state of a non-interacting metal, or a Fermi gas, consists of all single-electron states within the Fermi sea. The shape of the Fermi sea has a natural topological characterization in terms of the Euler characteristic
χF, and a fundamental question that we address in this thesis is: \textbf{does
χF imply any robust physical properties in metals, and how can they be measured?} In the third part of the thesis,
χF is connected to physically measurable quantities, such as equal-time density-density correlation functions, and multipartite entanglement measures, such as the mutual information. Particularly, for three dimensions, we establish that the 4-partite mutual information is a robust probe of the Fermi sea topology even in the presence of Fermi-liquid interactions. In the last part, we focus on two dimensions, and propose an experimentally accessible platform to probe
χF for any two-dimensional electron gas (2DEG) that respects time-reversal symmetry. The proposed experimental setup is a Josephson
π-junction, and we predict that the electrical transport in the direction \textit{along} the junction exhibits a current-rectification effect that is controlled by the underlying Fermi sea topology. Due to the relation to Andreev state transport, we term this effect \textit{topological Andreev rectification}. The size of rectification is quantified by a \textit{nonlocal rectified conductance}, which is established to attain an integer quantized value that equals
χF (in unit of
e2/h). Extensive numerical simulations are performed to support our theoretical proposal, and material platforms such as InAs heterostructures and graphene are identified as promising candidates to demonstrate this effect.