In the quantum Hall effect, a planar electron gas exposed to a perpendicular magnetic field condenses into a macroscopic quantum mechanical liquid characterized by a topological integer or fractional quantization of the transverse conductance. For the fractionally quantized liquid, the low-energy elementary excitations carry fractional charge and obey fractional statistics. At low energies, the degrees of freedom in the quantum Hall states are purely topological. The states are classified into universality classes according to their topological order, and for each universality class, there is a low-energy effective quantum field theory, which determines the properties of the associated quasiparticles. In a finite system with a boundary, the two-dimensional effective theory reduces into a one-dimensional edge theory, characteristic of each universality class. The thesis investigates few-electron quantum Hall droplets and the associated quasiparticles by means of numerical exact diagonalization of the Hamiltonian matrix in a high magnetic field approximation, which constitutes a Landau level projection. The inter-electron interaction depends on the Landau level as well as the details of the experimental sample. In the results, we compare the states obtained with realistic effective interactions to the theoretical models concerning the universality classes of the Laughlin states and the Pfaffian state, by studying overlaps of the state vectors, phase structure of the wave functions, as well as the quasiparticle and edge excitations.
The subject of this work is the application of conformal field theory in the fractional quantum Hall effect. The fractional quantum Hall effect is observed in a two-dimensional electron gas in a strong perpendicular magnetic field and at low temperature. The electron system forms a strongly correlated quantum liquid and exhibits the exact quantization of the Hall conductivity in to a fractional value vet/h, where v is the fractional filling fraction of Landau-levels in the system. In addition, the elementary excitations of the electron fluid have fractional charge and fractional statistics of anyons. First, we review the basic of two-dimensional conformal field theory and show how certain conformal quantum field theories can be solved non-perturbatively by using the representation theory of conformal symmetries. The low-energy effective field theory in the bulk of a fractional quantum Hall state is known to be a 2+1 dimensional topological field theory. The states of the topological quantum field theory are directly connected to the holomorphic correlation functions of an equivalent two-dimensional conformal field theory. It is known that the gapless excitations at the boundary of the quantum Hall state are described by the same conformal field theory. By using this connection, the so-called Laughlin and Moore-Read states are derived in detail using conformal field theory. The fractional charge and statistics of the quasiparticle excitations in these states are obtained from the conformal field theory description. Furthermore, the non-Abelian statistics of the Moore-Read states are verified explicitly from the quasiparticle wave functions. We argue how the different fractional quantum Hall phases of matter are described by the quantum numbers obtained using conformal field theory. Lastly, we study model interactions that have the Laughlin v = 1/3 and the Moore-Read v = 1/2 states as exact ground states. We compute the matrix elements of these interactions and diagonalize the Hamiltonian matrix in the disk geometry for both states. The low-energy edge excitations of these states are studied in the finite geometry and we verify that they are in accordance with the conformal field theory on the boundary. By creating a localized density disturbance in the system, we create the e/4 quasiparticle of the Moore-Read state, and by studying the properties of the edge theory it is seen that quasiparticle has the same properties as the non-Abelian quasihole in the conformal field theory. The obtained results verify the correspondence between the bulk state and the boundary conformal field theory at the edge in the fractional quantum Hall states in question.
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.
Topological quantum phases of matter are often characterized by the presence of fractionalized quasiparticles, which exhibit non-trivial braiding statistics or carry fractional quantum numbers, or of protected gapless surface states. In this thesis, we study topological phases in two and three spatial dimensions, from the perspective of searching for new exotic quantum phases and of characterizing their experimental signatures.\nWe first study topological defects in fermionic paired superfluids and discover that in the presence of a multiply quantized vortex, such a state hosts unpaired fermions in the BCS regime. We predict that these unpaired fermions will result in an experimentally measurable deviation of the system's angular momentum from its value in the BEC regime. Focusing on two-dimensions, we then study superconductors coupled to dynamically fluctuating electromagnetism, and establish a universal framework for studying the low-energy physics of such systems. We derive topological field theories describing all spin-singlet superconductors which naturally capture the interplay of symmetry and topology in these gapped states.\nThe remainder of this thesis is focused on a novel class of long-range entangled states of matter, known as “fracton” phases, in both two- and three-dimensions. These phases exhibit an intriguing phenomenology, most surprising of which is the presence of excitations which are immobile in isolation. Studying the non-equilibrium dynamics of gapped fracton phases, we discover that they naturally exhibit glassy quantum dynamics in the absence of quenched disorder, and hence may have potential technological applications as robust quantum memories. Finally, we conclude with a description of fracton phases as higher-rank tensor gauge theories and discuss the emergent phases of matter in which a finite density of fractons or their bound states may exist.
Topological phases of matter describe systems beyond the paradigm of spontaneous symmetry breaking, giving rise to unconventional phenomena with possible applications in topological quantum computing. Almost 40 years after their discovery, topological phases remain among the most active research fields of condensed matter physics. As evidenced by the recent discovery of higher order topological insulators (HOTIs), novel physics can emerge even in non-interacting systems. Theoretical investigations of strongly correlated systems are extremely difficult due to the exponential growth of the quantum-mechanical Hilbert space and frequently require the application of approximative methods such as model wave functions. Tensor network states (TNS) are a class of variational wave functions which allow an efficient encoding of relevant quantum many body states and which form the basis for very successful numerical algorithms. Projected entangled pair states (PEPS), a class of TNS in two and higher dimensions, permit the exact representation of many interacting topological phases with time reversal symmetry. However, the description of experimentally relevant chiral topological phases like the integer and fractional quantum Hall effects is much more subtle using the TNS framework. The first part of this thesis focuses on the description of chiral topological phases using PEPS. Firstly, we study a previously proposed chiral PEPS that was conjectured to possess anyonic excitations. By a careful analysis of its symmetries we are able to align important entanglement observables more closely with the expected universal behavior. This highlights that efficient PEPS can possess characteristic properties of chiral topological phases. Secondly, we focus on the issue that all known chiral PEPS have algebraic bulk correlation functions and therefore cannot be the ground states of gapped local Hamiltonians. Since this problem arises already for non-interacting chiral topological phases, we focus on two examples of such systems and show that their ground states can be represented exactly by efficient PEPS in a hybrid lattice with one momentum-space direction. After an inverse Fourier transform, the PEPS with only real-space coordinates requires an exponentially growing number of parameters for an exact representation. This provides a concrete illustration of the impossibility to encode gapped chiral phases exactly with efficient PEPS. Finally, we analyze a model wave function for a three-dimensional (3D) HOTI with strong intrinsic correlations using large-scale variational Monte Carlo simulations. This wave function is obtained by projection of two copies of a non-interacting HOTI with chiral hinge states. We characterize the gapless hinge states of the interacting system and show that they are of the same nature as the edge states of the
1/2 Laughlin state. Surprisingly, the gapped surfaces host a two-dimensional (2D) phase whose topological entanglement entropy is half of that of the
1/2 Laughlin state. Such a value cannot be obtained by any of the known 2D topological orders, showing a clear departure not only from the Laughlin
1/2 physics but from conventional 2D topological order. This demonstrates that 3D topological phases can host rich phenomena that are not yet fully understood.
We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low-energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the ground-state manifold. We extract the modular transformation matrices
S (
U) which contain the information of mutual (self) statistics, quantum dimensions, and the fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.