Exactly solvable models of topologically ordered phases with non-abelian anyons typically require complicated many-body interactions which do not naturally appear in nature. This motivates the "inverse problem" of quantum many-body physics: given microscopic systems with experimentally realistic two-body interactions, how to design a Hamiltonian that realizes a desired topological phase? Here we solve this problem on a platform motivated by Rydberg atoms, where elementary two-level systems couple via simple blockade interactions. Within this framework, we construct Hamiltonians that realize topological orders described by non-abelian quantum double models. We analytically prove the existence of topological order in the ground state, and present efficient schemes to prepare these states. We also introduce protocols for the controlled adiabatic braiding of anyonic excitations to probe their non-abelian statistics. Our construction is generic and applies to quantum doubles
D(G) for arbitrary finite groups
G. We illustrate braiding for the simplest non-abelian quantum double
D(S3).
We present a numerical study of a blockade Hamiltonian with a local
Z2 symmetry. For such models, the existence of a topologically ordered ground state has been proven rigorously for weak Rabi driving [T. F. Maier et al., PRX Quantum 6, 030340 (2025)]. However, the phase diagram beyond weak driving is still an open question. We perform large-scale infinite-size density matrix renormalization group (iDMRG) simulations on a cylinder and map out the full phase diagram of the model. We find a single continuous phase transition between the topologically ordered phase and a trivial paramagnetic phase, and confirm numerically the stability of the topologically ordered phase up to Rabi drives comparable to the characteristic energy scale of the system. Furthermore, we compute the excitation gaps and show that the charge gap remains open across the phase transition and charges become confined in the trivial phase. On the other hand, the gap in the symmetric sector closes at the transition, consistent with the condensation of flux excitations.
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.
We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary.Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter.The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a Z 4 toric code, where the condensation is implemented at the level of the ground states by two-body measurements.We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps its ground-state subspace to that of a DS string-net model.We show that the construction of the DS stabilizer Hamiltonian generalizes to all twisted quantum doubles (TQDs) with Abelian anyons.This yields a Pauli stabilizer code on composite-dimensional qudits for each such TQD, implying that the classification of topological Pauli stabilizer codes extends well beyond stacks of toric codes-in fact, exhausting all Abelian anyon theories that admit a gapped boundary.We also demonstrate that symmetry-protected topological phases of matter characterized by type-I and type-II cocycles can be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form symmetries of the TQD stabilizer models.
Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.
We develop and investigate several quantum many-body spin models of use for topological quantum information processing and storage. These models fall into two categories: those that are designed to be more realistic than alternative models with similar phenomenology, and those that are designed to have richer phenomenology than related models. In the first category, we present a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the perturbative low-energy limits of entirely two-body Hamiltonians. This construction reproduces the target models' behavior using only couplings which are natural in terms of the original Hamiltonians. As an extension of this work, we construct parent Hamiltonians involving only local 2-body interactions for a broad class of Projected Entangled Pair States (PEPS). We define a perturbative Hamiltonian with a finite order low energy effective Hamiltonian that is a gapped, frustration-free parent Hamiltonian for an encoded version of a desired PEPS. For topologically ordered PEPS, the ground space of the low energy effective Hamiltonian is shown to be in the same phase as the desired state to all orders of perturbation theory. We then move on to define models that generalize the phenomenology of several well-known systems. We first define generalized cluster states based on finite group algebras, and investigate properties of these states including their PEPS representations, global symmetries, relationship to the Kitaev quantum double models, and possible applications. Finally, we propose a generalization of the color codes based on finite groups. For non-Abelian groups, the resulting model supports non-Abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.