Topological frustration (or topological mechanics) is the existence of classical zero modes that are robust to many but not all distortions of the Hamiltonian. It arises naturally from locality in systems whose interactions form a set of constraints such as in geometrically frustrated magnets and balls and springs metamaterials. For a magnet whose classical limit exhibits topological frustration, an important question is what happens to this topology when the degrees of freedom are quantized and whether such frustration could lead to exotic quantum phases of matter like a spin liquid. We answer these questions for a geometrically frustrated spin ladder model. It has the feature of having infinitely many conserved quantities that aid the solution. We find classical zero modes all get lifted by quantum fluctuations and the system is left with a unique rung singlet ground state -- a trivial quantum spin liquid. Moreover, we find low-energy eigenstates corresponding to known symmetry protected topological (SPT) ground states, and a special role of $SU(2)$ symmetry, that it demands the existence of extra dimensions of classical zero modes -- the phenomena we call symmetry-enriched topological frustration (SETF). These results suggest small violations of the conservation laws in the nearly SETF regime could lead to quantum scars. We further study a two-dimensional bilayer triangular lattice model and find a similar SETF phenomena which also leads to suppressed low-energy topological eigenstates in the quantum regime. These results suggest that in the absence of magnetic order, classical topological frustration manifests at finite spin as asymptotically low energy modes with support for exotic quantum phenomena.
Topological frustration (or topological mechanics) is the existence of classical zero modes that are robust to many but not all distortions of the Hamiltonian. It arises naturally from locality in systems whose interactions form a set of constraints such as in geometrically frustrated magnets and balls and springs metamaterials. For a magnet whose classical limit exhibits topological frustration, an important question is what happens to this topology when the degrees of freedom are quantized and whether such frustration could lead to exotic quantum phases of matter like a spin liquid. We answer these questions for a geometrically frustrated spin ladder model. It has the feature of having infinitely many conserved quantities that aid the solution. We find classical zero modes all get lifted by quantum fluctuations and the system is left with a unique rung singlet ground state -- a trivial quantum spin liquid. Moreover, we find low-energy eigenstates corresponding to known symmetry protected topological (SPT) ground states, and a special role of SU(2) symmetry, that it demands the existence of extra dimensions of classical zero modes -- the phenomena we call symmetry-enriched topological frustration (SETF). These results suggest small violations of the conservation laws in the nearly SETF regime could lead to quantum scars. We further study a two-dimensional bilayer triangular lattice model and find a similar SETF phenomena which also leads to suppressed low-energy topological eigenstates in the quantum regime. These results suggest that in the absence of magnetic order, classical topological frustration manifests at finite spin as asymptotically low energy modes with support for exotic quantum phenomena.
In spin lattices, frustration happens when there exists a competition between different terms of the Hamiltonian, which cannot be simultaneously minimized in all local physical bases. Frustration can destabilize the bipartite Neel order, which is the conventional groundstate of antiferromagnetic Heisenberg-type models in two dimensions. Hence, the focus of two-dimensional quantum magnetism studies has been directed toward elucidating the zero-temperature properties of Heisenberg-type models on the Archimedean lattices that exhibit geometrical frustration. This type of frustration is of great interest since it can lead to exotic forms of the quantum matter, such as topological spin-liquid (SL) and many-sublattice long-range-ordered phases. The spin-1/2 triangular Heisenberg model (THM) is a prototypical model with geometrical frustration, which has the highest coordination number of all Archimedean lattices. However, there exist two major, distinct difficulties for numerical methods to efficiently determine the groundstate properties of the THM: the immense dimension of the Hilbert space and the geometrical frustration phenomenon. In this thesis, to deal with such difficulties, we employ the state-of-the- art non-Abelian matrix product states (MPS) and density matrix renormalization group (DMRG) methods. We exploit the symmetries of the Hamiltonian to reduce the working dimension of the Hilbert space and, to lessen the effects of the geometrical frustration, we depend on the local entanglement (area-law) nature of the MPS as it scales with the size of a lattice subspace with reduced spatial dimensions. In such a way, we thoroughly examine the phase diagram of the THM with nearest and next-nearest neighbor (NN and NNN) exchange couplings on finite- and infinite-length cylinders of widths up to 12 sites. In addition, we develop new numerical tools to analyze topological and symmetry-broken phases in the context of the SU(2)-symmetric translation-invariant MPS algorithm. We establish that on infinite cylinders, the anyonic sectors of a topological order can be classified through the fractionalization of global symmetries and degeneracy patterns of the entanglement spectrum (ES). On the other hand, we detect symmetry-breaking phase transitions by a combination of the correlation lengths and second and fourth cumulants of the magnetic order parameters, even though symmetry implies that the order parameter itself is strictly zero. Furthermore, the appearance of Nambu-Goldstone modes in the excitation spectrum can be detected using “tower of states” (TOS) levels in the momentum-resolved ES. By applying the new tools to the THM, we discover a variety of exotic groundstates such as a Z2-gauge toric-code-type topological order (with vison and spinon excitations), a NNN Majumdar-Gosh, and a long-range 120◦-ordered state.
Frustration in many-body spin systems is a situation where the spins cannot orient themselves to fully minimize the interactions with their neighbors. It is caused, in general, by competing interactions or by the lattice structure, as in the antiferromagnets on non-bipartite lattices. Over the years frustrated systems have been studied substantially, since frustration can destabilize the antiferromagnetic order, lead to different phases of matter and induce various exotic properties. Frustration can be caused also by a choice of boundary conditions, i.e. the lattice topology. For instance, a ring with an odd number of sites is non-bipartite, which makes antiferromagnets in this setting frustrated. The subject of this thesis are zero-temperature one-dimensional quantum systems exactly in this setting. The major content is discussing how such a simple setting can influence the antiferromagnetic order, by studying spin-1/2 chains with discrete symmetries, whose breaking is related to the onset of the order. After reviewing the known results on topological frustration, dealing with non-thermodynamic quantities, we develop an approach for studying symmetry breaking, that is both suitable for exact analytical computations and meaningful for discussing the order also in a finite system. It consists of the realization that many spin-1/2 chains without external magnetic fields, such as the quantum XY chain, possess anticommuting global symmetries when the number of lattice sites is an odd number, implying the ground-state degeneracy already in a finite system. In this framework, we discover that topological frustration, despite being induced only by the choice of boundary conditions and even/odd system size choice, can affect the magnetization in the thermodynamic limit and system's quantum phase transitions. We find that topological frustration can destroy local order, create a site-dependent magnetization that varies in space with an incommensurate pattern, induce a first-order quantum phase transition that is not present without frustration and modify the nature of a second-order transition, by destroying local order at both sides of the transition and preserving only non-local string order parameters. All these results indicate the incompleteness of the approach to quantum many-body systems, based on the Ginzburg-Landau theory, that tries to capture the properties of the system by taking the system size to infinity before computing the observables and neglects the influence of the chain length as a relevant scale. We find that topological frustration can affect also non-equilibrium properties, by considering a local quantum quench setup. Namely, we study the Loschmidt echo and find it displays qualitatively and quantitatively different behavior for antiferromagnetic rings with an even and odd number of sites. Most of all, the differences become clearer for large system sizes, thus allowing to distinguish in a simple out-of-equilibrium experiment a system made by a certain, large, number of spins from the one with a single additional spin. The thesis also contains a mathematical part. In studying order parameters of models mappable to free fermions, Toeplitz determinants with symbols that possess a part proportional to the delta function arise. We derive asymptotic formulas for this type of determinants.
Magnetic frustrations are the fundamental cause of many interesting phenomena in materials that have been intensely studied since the middle of the 20th century. Magnetic systems with competing interactions, ranging from insulating antiferromagnets to arrays of superconducting qubits, exhibit nontrivial quantum e ects that arise from the need to satisfy a jagged free energy landscape. The search for and characterization of quantum spin liquids, nontrivial topological phases, and quantum phase transitions is conducted through investigations of such frustrated models. Within the large umbrella of frustrated magnetism I have studied some speci c models with the purpose of answering the following questions: (1) What are the interesting phases and less-interesting phases, and how do they fall in the phase diagram relative to one another? (2) What properties do the phases have, and how can we characterize what happens when one phase transitions to another? Armed with the answers to these questions, researchers can further explore these interesting phenomena and apply them towards new materials and devices. In this thesis, I rst develop a computational methodology for studying both classical and quantum spin models using various forms of Monte Carlo simulation. I use classical Monte Carlo to study highly-frustrated models of Ising spins both at nite temperatures and in the T = 0 limit, to map the phase diagram and determine the universality class of phase transitions. Using these methods, I show that in a dissipative chain of qubits, bath-induced qubit-qubit interactions lead to collective decoherence above a critical dissipation strength. This causes the qubits to lose all stored information and become fully correlated with one another, an e ect that one would need to mitigate in order to perform quantum computation. In another study, I show that in antiferromagnetic insulators with anisotropic interactions, frustration can cause an intermediate paramagnetic phase with oblong \droplets" of correlated spins. This phenomenon is quite general and the presence of these droplets can help explain experimental signatures that occur at unexpected temperatures in such materials. In addition to the classical simulations, I develop a variational Monte Carlo (VMC) technique to search for ground states in Heisenberg-type models, speci cally for spins S = 1. The case of S = 1 is more complicated than the case for S = 1=2 due to an additional local degree of freedom, which has interesting implications for the possible states that can be realized. Using semiclassical energy minimization as well as VMC, I show that spin S = 1 diamond-lattice antiferromagnets experience higher-order spinspin interactions that destroy previously theorized liquid-like spiral spin orderings. However, the presence of these higher-order interactions can also potentially allow other phases such as quantum spin liquids and topological paramagnets. Such results can provide an avenue for explaining anomalous phenomena like low-temperature paramagnetism in future experimental materials.
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.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.
Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the procedure of diagonalizing the Hamiltonian matrix are proposed and suggest a randomized distributed memory parallelization strategy. Benchmarks of computations on various supercomputers with system size up to 50 spin-1/2 particles have been performed. Results concerning the emergence of a chiral spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented. The stability and extent of this phase are discussed. In an extended Heisenberg model on the triangular lattice, we establish another chiral spin liquid phase. We discuss the special case of the Heisenberg J1-J2 model and present a scenario where the critical point of phase transition from the 120-degree Néel to a putative Z2 spin liquid is described by a Dirac spin liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of freedom on the triangular lattice with an additional ring-exchange term is discussed. We present our contribution to the project and the final results that suggest a series of chiral spin liquid phases in an extended parameter range. Finally, we present preliminary data from a Quantum Monte Carlo study of an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and multi-column representations. We establish the phase boundary between the Néel ordered phase and the disordered phases. The disordered phase in the four-column representation is expected to be a two-dimensional analog of the Haldane phase for the spin-1 Heisenberg chain.