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.
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.
The most general description of a quantum many-body system is given by a wave- function that lives in a Hilbert space with dimension exponential in the number of particles. This makes it extremely hard to study strongly correlated phenomena like the fractional quantum Hall effect and high-temperature superconductivity. Whenever interactions are sufficiently local and temperature is low, the system does not explore the full Hilbert space, but its ground state resides in the small corner of Hilbert space described by the area law. Containing little entanglement, the states can then be expressed as tensor networks, a family of wavefunctions with a polynomial number of parameters. On the one hand, tensor networks can be used as a variational manifold in nu- merical computations. On the other hand, they allow building model wavefunctions much like locality allows writing down physically realistic Hamiltonians. Besides allowing for an analytical treatment, these models grant access both to the physical and the entanglement degrees of freedom. This is particularly useful in classifying phases of matter. A large number of phases can be explained in terms of Landau’s symmetry-breaking paradigm. This framework, however, is not complete, as exemplified by the existence of phases with intrinsic topological order in two dimensions. It was a major conceptual advance when tensor networks could explain (non-chiral) topological phases as those where the symmetry resides in the entanglement degrees of freedom. The symmetries corresponding to those topological phases act as discrete, finite groups on the virtual degrees of freedom. The purpose of this Thesis is to generalize this program to include other symmetries. We investigate a class of tensor networks with continuous symmetries and find that they cannot describe gapped physics with a unique ground state. The abelian case is found to describe a non-Lorentz invariant phase transition point into a topologically ordered phase. The physics of the non- abelian case is that of a plaquette state that spontaneously breaks the translation symmetry of the lattice. The non-abelian PEPS arises as the ground state of a local parent Hamiltonian whose ground state manifold is completely characterized by the tensor network. In both cases, we find two types of corrections to the entanglement entropy: first there is a correction that is logarithmic in the size of the boundary and independent of the shape. A further correction depends only on the shape of the partition, imposing further restrictions on regions that are suffciently thin. Finally, we investigate symmetries that mix the virtual with the physical degrees of freedom and are furthermore anisotropic. Their physics is described by subsystem symmetry protected topological order. In particular, we focus on the entanglement entropy in the cluster phase and show that there is a universal constant correction to the entropy throughout the phase. This is important in the program of establishing the entanglement entropy as a detection mechanism for topologically ordered phases. We put forward a numerical algorithm to compute the correction and use it to discover a novel phase of matter in which the cluster phase is embedded.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
Tensor Networks are a tool from quantum information theory to describe many-body
quantum systems. It has been shown that states arising from Tensor Networks in 1D (Matrix
Product States - MPS) accurately describe ground states of local, gapped Hamiltonians.
The same project for higher dimensional systems is currently the subject of great efforts.
However, quantum many-body systems in dimension 2 or greater can exhibit much richer
behaviour than their one-dimensional counterparts. One of the hallmarks of this is the
scarcely understood notion of topological order. It has been shown recently, that Projected
Entangled-Pair States (PEPS - Tensor Networks in 2D) can describe chiral topological
states, that lie in the ground state manifold of a certain class of gapless Hamiltonians. In
this work, we try to perturb the Hamiltonian into a gapped chiral phase. We will then
study, whether this gapped, chiral phase can be described in terms of PEPS or how the
PEPS-construction has to be modified to accommodate for this kind of behaviour.
This thesis is divided into two mainly independent parts: In the first part, we derive a criterion to determine when a translationally invariant Matrix Product State (MPS) has long range localizable entanglement, which indicates that the corresponding state has some kind of non-local hidden order. We give examples fulfilling this criterion and eventually use it to obtain all such MPS with bond dimension 2 and 3. In the second part, we show that Projected Entangled Pair States (PEPS) in two spatial dimensions can describe chiral topological states by explicitly constructing a family of such states with a non-trivial Chern number. We demonstrate that such free fermionic PEPS must necessarily be non-injective and have gapless parent Hamiltonians. Moreover, we provide numerical evidence that they can nevertheless approximate well the physical properties of Chern insulators with local Hamiltonians at arbitrary temperatures. We also construct long range, topological Hamiltonians with a flat energy spectrum for which those PEPS are unique ground states. As for non-chiral topological PEPS, the non-trivial, topological properties can be traced down to the existence of a symmetry on the virtual level of the PEPS tensor that is used to build the state. We use the special properties of PEPS to build the boundary theory and show how the symmetry results in the appearance of chiral modes, a ground state degeneracy of the parent Hamiltonian on the torus and a universal correction to the area law for the zero Rényi entropy. Finally, we show that PEPS can also describe chiral topologically ordered phases. For that, we construct a simple PEPS for spin-1/2 particles in a two-dimensional lattice. We reveal a symmetry of the PEPS tensor that gives rise to the global topological character. We also extract characteristic quantities of the edge Conformal Field Theory using the bulk-boundary correspondence.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
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.