In the last decade, artificially designed systems have emerged as a promising path to the realisation of topological phases of matter. A particular focus has been on chains of magnetic impurities placed on superconductors which realise mixed dimensional topological phases. To better understand these phases we provide a comprehensive exploration of their topological properties. We demonstrate the challenges of mixed dimensional systems and provide a proof of the conditions of validity of a classification method based on an effective lower-dimensional topological Hamiltonian. The applicability is illustrated through an analytically tractable model of a spiral magnetic interface embedded into a superconductor. We also show that this method provides correct results in regimes where we can demonstrate that other methods proposed for mixed-dimensional models fail. A limitation arises for models with multiple interfaces though, where we find that zeros of Green’s functions can act as false phase boundaries that make the classification by the topological Hamiltonian generally unreliable. Nonetheless, the latter remains applicable for a model of two spiral magnetic interfaces embedded into a superconductor, where in combination with a low-energy effective Hamiltonian, it can probe the local structure of the topological phase. Finally we extend the classification to models with time-periodic and quasi-periodic modulations. We exactly solve a model of a periodically precessing magnetic interface in a superconductor, demonstrating that the driving reduces the superconductor gap. Considering then quasi-periodically evolving magnetic interface, we propose a classification by an extension of the topological Hamiltonian, the topological quasi-energy operator. For a purely one-dimensional model, we provide evidence that quasi-periodic driving tends to destabilise a topological phase. But for a mixed dimensional model, we demonstrate the existence of an enhanced localising potential causing a larger regime of the topological phase.
In the last decade, artificially designed systems have emerged as a promising path to the realisation of topological phases of matter. A particular focus has been on chains of magnetic impurities placed on superconductors which realise mixed dimensional topological phases. To better understand these phases we provide a comprehensive exploration of their topological properties. We demonstrate the challenges of mixed dimensional systems and provide a proof of the conditions of validity of a classification method based on an effective lower-dimensional topological Hamiltonian. The applicability is illustrated through an analytically tractable model of a spiral magnetic interface embedded into a superconductor. We also show that this method provides correct results in regimes where we can demonstrate that other methods proposed for mixed-dimensional models fail. A limitation arises for models with multiple interfaces though, where we find that zeros of Green’s functions can act as false phase boundaries that make the classification by the topological Hamiltonian generally unreliable. Nonetheless, the latter remains applicable for a model of two spiral magnetic interfaces embedded into a superconductor, where in combination with a low-energy effective Hamiltonian, it can probe the local structure of the topological phase. Finally we extend the classification to models with time-periodic and quasi-periodic modulations. We exactly solve a model of a periodically precessing magnetic interface in a superconductor, demonstrating that the driving reduces the superconductor gap. Considering then quasi-periodically evolving magnetic interface, we propose a classification by an extension of the topological Hamiltonian, the topological quasi-energy operator. For a purely one-dimensional model, we provide evidence that quasi-periodic driving tends to destabilise a topological phase. But for a mixed dimensional model, we demonstrate the existence of an enhanced localising potential causing a larger regime of the topological phase.
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.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
Abstract The topological properties of periodically driven many-body systems often have no static analogs and defy a simple description based on the effective Hamiltonian. To explore the emergent edge modes in driven p-wave superconductors in two dimensions, we analysed a toy model of Kitaev chains (one-dimensional spinless p-wave superconductors with Majorana edge states) coupled by time-periodic hopping. We showed that with proper driving, the coupled Kitaev chains can turn into a fully gapped superconductor, which is analogous to the p x +ip y state but has two, rather than one, chiral edge modes. A different driving protocol turns it into a gapless superconductor with isolated point nodes and completely flat edge states at quasienergy ω=0 or π/T, with T as the driving period. The time evolution operator U(k x , k y , t) of the toy model is computed exactly to yield the phase bands. And the “topological singularities” of the phase bands are exhausted and compared to those of a periodically driven Hofstadter model, which features counter-propagating chiral edge modes. These examples demonstrate the unique edge states in driven superconducting systems and suggest driving as a potentially fruitful route to engineer new topological superconductors.
We study aspects of gapped phases of matter, focusing on their classification, including the group law under stacking, and their relation to topological quantum field theories (TQFT). In one spatial dimension, it is well-known that Matrix Product States (MPS) efficiently approximate ground states of gapped systems; by showing that these states arise naturally in 1 + 1-dimensional lattice TQFT, which in turn are closely related to continuum TQFT, we provide a concrete connection between ground states of lattice systems and TQFT in 1 + 1 dimensions. We generalize this to systems with symmetries and fermions, and obtain a classification and group law for the stacking of 1 + 1-dimensional symmetry-protected topological phases. Further, we study the effect of turning on/off interactions for the classification: the phase classification of a given symmetry class of Hamiltonians can be different depending on whether we allow interactions or not, and in low dimensions we provide some concrete formulas relating the phases under the non-interacting classification and those under the interacting classification. Lastly, we study the phases of the 2 + 1-dimensional topological superconductor, and show that for all 16 phases braiding statistics of vortices, which determine the underlying TQFT, can be obtained by stacking layers of the basic p + ip superconductor.
The recent discovery of topological insulators has led to a tremendous interest in the exploration of topological phases of matter which do not fit into Landau's symmetry breaking paradigm. Numerous exotic topological materials are theoretically predicted. Some of them have been experimentally reported, but many remain not. In this thesis, we explore topological phases of matter from three aspects: their classification, realization and application. We first review some basic classification theories, which provide us a "big picture" and lay the foundation for the rest of the thesis. We then move on to propose a systematic method based on quaternion algebra to construct toy tight-binding Hamiltonians for all the exotic phases in a recently developed periodic table for topological insulators and superconductors. We also introduce two peculiar families of topological phases that are beyond the table---the Hopf and four-dimensional topological insulators without time reversal symmetry. Prototypical Hamiltonians are constructed and their topological properties, such as robust edge states, are numerically studied. Motivated by rapid experimental progress in engineering spin-orbit coupling and artificial gauge field for cold atoms, we continue the thesis by proposing a feasible experimental scheme to realize a three-dimensional chiral topological insulator with cold fermionic atoms in an optical lattice. To unambiguously probe topological phases, we also bring forth systematic and generic methods to measure the characteristic topological invariants, for both free and strongly interacting systems. Moreover, we demonstrate that a kaleidoscope of knot and link structures is encoded in the spin texture of Hopf insulators and show how to observe different knots and links in cold atoms via time-of-flight images. The last part of the thesis is about the application of topological materials. After a demonstration of how to create, braid and detect Majorana fermions with cold atoms, we put forward a proposal to construct a self-test quantum random number generator by using Majorana fermions. Majorana random number generators are able to generate certifiable true random numbers with unconditional security. They offer a new perspective to the utilization of topological materials and may have vital applications in cryptography and related areas.
Topological phases of matter that depend for their existence on interactions are fundamentally interesting and potentially useful as platforms for future quantum computers. Despite the multitude of theoretical proposals, the only interaction-enabled topological phase experimentally observed is the fractional quantum Hall liquid. To help identify other systems that can give rise to such phases, we present in this work a detailed study of the effect of interactions on Majorana zero modes bound to vortices in a superconducting surface of a three-dimensional topological insulator. This system is of interest because, as was recently pointed out, it can be tuned into the regime of strong interactions. We start with a zero-dimensional system suggesting an experimental realization of the interaction-induced Z8 ground-state periodicity previously discussed by Fidkowski and Kitaev [Phys. Rev. B 81, 134509 (2010); Phys. Rev. B 83, 075103 (2011)] . We argue that the periodicity is experimentally observable using a tunnel probe. We then focus on interaction-enabled crystalline topological phases that can be built with the Majoranas in a vortex lattice in higher dimensions. In one dimension, we identify an interesting exactly solvable model which is related to a previously discussed one that exhibits an interaction-enabled topological phase. We study these models using analytical techniques, exact numerical diagonalization, and density matrix renormalization group. Our results confirm the existence of the interaction-enabled topological phase and clarify the nature of the quantum phase transition that leads to it. We finish with a discussion of models in dimensions 2 and 3 that produce similar interaction-enabled topological phases.