The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.
The concept of non-trivial electronic topology in condensed matter physics is crucial for understanding the electronic spectrum of solids and for tailoring various thermodynamic and transport properties, useful for real-world applications. Research on topological insulators and superconductors primarily utilizes the electronic band theory and symmetry classes of gapped quantum systems, with the existence of spectral or mobility gaps ensured even in the presence of impurities and defects. However, structural disorder, naturally manifested in materials such as glasses and amorphous solids, can significantly impact the critical properties of phase transitions in topological quantum matters. This dissertation systematically studies topological phase transitions in non-crystalline free-fermion systems via tight-binding quantum Hamiltonians on 2D random geometries. Leveraging various theoretical and experimentally accessible measures, we demonstrate the resilience of topological phases in non-crystalline systems against structural randomness, described by classical percolation theory, and characterize their unique critical behavior through finite-size scaling. We also extend the notion of amorphous topological states to non-integer dimensional random systems. We reveal that these 2D-embedded random fractals can admit well-defined thermodynamic topological phases with unconventional critical properties. We further study quantum dynamics in these random topological systems. By employing a family of discrete-time topological quantum walks on random percolation lattices, we address wavefunction propagation, superdiffusive quantum speedup, and localization transition by identifying the critical probabilities and timescales. Lastly, we introduce topological lattices with adjustable spatial correlation to investigate Landau levels and quantum oscillations in a low magnetic field regime.
Condensed matter physics studies how many quantum particles, when brought together, conspire to give rise to collective behaviours that are far richer than those of the individual particles themselves. This is the physics of emergence: the idea that a large system can have properties that are not obvious from its microscopic building blocks alone. The work presented in this thesis explores how to characterise the properties of such many-particle states from three interconnected points of view: topology, quantum transport, and quantum information. The common idea behind these themes is that the properties of quantum matter may be accessed directly from quantum states themselves, without necessarily referring to an underlying Hamiltonian, circuit, or protocol used to generate them. Symmetry-protected topological phases of matter arise from the impossibility of continuously deforming certain quantum states into others whilst preserving their symmetries. In fermionic quantum matter, the phases resulting from such impossibility are those of topological insulators and superconductors. While having an insulating bulk, these materials possess metallic boundary states robust to perturbations preserving the protecting symmetries of their phase. Mathematically, the topological nature of these materials is characterised by the calculation of topological invariants, integer or half-integer numbers distinguishing between different topological phases. These invariants have been traditionally formulated for crystalline quantum matter, where one may use momentum space to perform mathematical calculations. This thesis focuses instead on characterising the topological phases of noncrystalline matter, which includes disorder and amorphicity. We develop a series of mathematical tools, known as local topological markers, which reformulate traditional topological invariants in real space and are therefore applicable even in the absence of crystallinity. Local topological markers are formulated in terms of the one-particle density matrix, a correlation function that only requires the knowledge of the quantum state in question to be calculated. Thus, the framework for topological characterisation we discuss, which also extends to certain higher-order topological phases, is purely state-based: it does not need to specify any parent Hamiltonian nor protocol used to prepare the quantum state to be characterised. Contrary to the classical picture of electricity where electrons behave as solid spheres drifting through a wire, the electronic transport through a nanostructure is shaped by quantum effects like tunnelling and interference. Quantum transport studies how these effects determine the conducting properties of mesoscopic samples---those that are neither microscopic nor fully macroscopic, but lie in an intermediate regime. In this thesis, we use quantum transport to study topological insulator nanowires, where the interplay between topology, deviations from crystallinity, and symmetry constraints shape electronic transmission. The characteristic feature of topological nanowires is the appearance of a perfectly transmitted mode when a certain magnetic flux is threaded through the wire's cross section. Varying the magnetic flux interpolates from perfect transmission to situations where electronic interference is enhanced, suppressing transmission. This is seen in the behaviour of conductance, which oscillates as a function of magnetic flux in what is known as Aharanov-Bohm oscillations. The mechanism originating the perfectly transmitted mode in a topological nanowire relies on time-reversal symmetry appearing at specific values of the magnetic flux. In amorphous nanowires, where time-reversal symmetry may be broken at any magnetic flux, the perfectly transmitted mode loses its protection, and the characteristic conductance oscillations of a topological nanowire may be lost. We study how the perfectly transmitted mode may be protected even when time-reversal symmetry is broken, and relate its loss to a topological phase transition driven by amorphicity. Entanglement is a distinctive feature of condensed matter systems: the possibility that quantum particles may be correlated in a way that has no classical analogue. Although inherently quantum-mechanical, entanglement alone is not a sufficient ingredient for quantum computation to outperform classical computers. The reason is that there exists a special class of quantum states, stabiliser states, that may be highly entangled whilst admitting an efficient classical representation---calculations involving stabiliser states are tractable in modern-day computers. The deviation of a quantum state from the set of stabiliser states goes by the name of quantum nonstabiliserness or magic, and, in the same way as entanglement, it is not a guarantee of quantum complexity on its own. In this thesis, we develop a method to characterise quantum states that combines entanglement and nonstabiliserness. To do so, we use the framework of the information lattice, whichsystematically decomposes the correlations in a quantum state in terms of their scale---the spatial extent of correlations. Through the information lattice, we distinguish different contributions to the nonstabiliserness content of a quantum state, which may have either a single-particle origin or come from intrinsic many-particle correlations. This framework, which is purely state-based, provides a first step to a fully scale-resolved description of nonstabiliserness and contributes to the characterisation of quantum states in terms of their complexity.
Clearly, in nature, but also in technological applications, complex systems built in an entirely ordered and regular fashion are the exception rather than the rule. In this thesis we explore how critical phenomena are influenced by quenched spatial randomness. Specifically, we consider physical systems undergoing a continuous phase transition in the presence of topological disorder, where the underlying structure, on which the system evolves, is given by a non-regular, discrete lattice. We therefore endeavour to achieve a thorough understanding of the interplay between collective dynamics and quenched randomness. According to the intriguing concept of universality, certain laws emerge from collectively behaving many-body systems at criticality, almost regardless of the precise microscopic realization of interactions in those systems. As a consequence, vastly different phenomena show striking similarities at their respective phase transitions. In this dissertation we pursue the question of whether the universal properties of critical phenomena are preserved when the system is subjected to topological perturbations. For this purpose, we perform numerical simulations of several prototypical systems of statistical physics which show a continuous phase transition. In particular, the equilibrium spin-1/2 Ising model and its generalizations represent -- among other applications -- fairly natural approaches to model magnetism in solids, whereas the non-equilibrium contact process serves as a toy model for percolation in porous media and epidemic spreading. Finally, the Manna sandpile model is strongly related to the concept of self-organized criticality, where a complex dynamic system reaches a critical state without fine-tuning of external variables. Our results reveal that the prevailing understanding of the influence of topological randomness on critical phenomena is insufficient. In particular, by considering very specific and newly developed lattice structures, we are able to show that -- contrary to the popular opinion -- spatial correlations in the number of interacting neighbours are not a key measure for predicting whether disorder ultimately alters the behaviour of a given critical system.
Quantum states of matter at zero temperature are called quantum phases, which are characterized by their symmetries and topology. If two quantum phases cannot be related by symmetry-preserving continuous transformations, they are defined to be topologically distinct. The zero-temperature transitions among quantum phases are quantum phase transitions. Quantum phase transitions that happen among topologically distinct phases are called topological quantum phase transitions. The study of topological quantum phases and topological quantum phase transitions is a central topic in condensed matter physics.My research on quantum phases has been focused on the theoretical study of topologically nontrivial quantum phases in the crystals governed by non-interacting Hamiltonians, including topological insulator phases, topological semimetal phases, topological superconductor phases, and so on. First, we proposed a sufficient criterion that can efficiently determine whether a three-dimensional crystal is a topological semimetal or not, based on the compatibility of different expressions for a special topological invariant, the quantized bulk average value of the effective axion field. Second, we predicted the existence of various topological insulator phases and topological semimetal phases in half-Heusler materials. Third, we proposed the magnetic-resonance-induced current as a feasible experimental probe of a special kind of topological insulators, called the axion insulators. Fourth, we proposed a new pairing mixing mechanism, the singlet-quintet mixing, for superconductors with spin-3/2 fermions, demonstrated the topological superconductor phase induced by it, and studied various properties of it, including spin-susceptibility, upper critical field, stability against the disorder, and surface Majorana flat bands.Besides the topological quantum phases, I have also worked on quantum phase transitions. First, we constructed the first theoretical model for the emergent supersymmetry at a discontinuous quantum phase transition and proposed to realize it on the surface of a topological superconductor. Second, we proposed the discontinuous change of piezoelectricity as a probe of two-dimensional topological quantum phase transitions between insulating phases, through a systematic study of all the relevant gap closing cases. In this dissertation, I first briefly introduce the basic concepts in the field of topological quantum phases and topological quantum phase transitions with a focus on the crystals governed by noninteracting Hamiltonians. Then, I review in detail all my first-authored research works mentioned above. My other research works are briefly mentioned.
From the alignment of magnets to the melting of ice, the transition between different phases of matter underpins our exploitation of materials. Both a quantum and a classical phase can undergo an instability into another state. In this thesis, we study the stability of matter in both contexts: topological states and crystalline solids. We start with the stability of fractional quantum Hall states on a lattice, known as fractional Chern insulators. We investigate, using exact diagonalization, fractional Chern insulators in higher Chern bands of the Harper-Hofstadter model, and examine the robustness of their many-body energy gap in the effective continuum limit. We report evidence of stable states in this regime; comment on two cases associated with a bosonic integer quantum Hall effect; and find a modulation of the correlation function in higher Chern bands. We next examine the stability of molecules using variational and diffusion Monte Carlo. By incorporating the matrix of force constants directly into the algorithms, we find that we are able to improve the efficiency and accuracy of atomic relaxation and eigenfrequency calculation. We test the performance on a diverse selection of case studies, with varying symmetries and mass distributions, and show that the proposed formalism outperforms existing restricted Hartree-Fock and density functional theory methods. Finally, we analyze the stability of three-dimensional crystals. We note that for repulsive Coulomb crystals of point nuclei, cubic systems have a zero matrix of force constants at second order. We investigate this by constructing an analytical model in the tight-binding approximation, and present a phase diagram of the most stable crystal structures, as we tune core and valence orbital radii. We reconcile our results with calculations in the nearly free electron regime, as well as current research in condensed matter and plasma physics.
This thesis focuses on the effects of strong electron-electron interactions on topological phase transitions.In very general terms, topological phase transitions are typically associated with the formation of gapless states at the boundary between two gapped phases.The opening of a gap is usually associated with a spontaneous symmetry breaking that can obliterate the topological nature of the system.In this work we highlight a completely different destiny for a topological transition in the presence of interaction and we develop a theoretical framework to address this physics identifying the physical origin for the unconventional scenario.In particular, we show that strong electronic correlation can lead to a gap opening at the topological transition without any symmetry breaking.This turns the topological transition discontinuous, thereby giving rise to a quantum critical point connecting the continuous region with the discontinuous one.This unconventional scenario establishes a surprising connection between a topological transition and quantum criticality.By solving a Bernevig-Hughes-Zhang model supplemented with a local symmetric interaction, we demonstrate that the crucial role of orbital polarization naturally emerges, already at the mean-field level.This local parameter effectively captures the essence of the topological transition, complementing the established evaluation of topological invariants and enabling a Landau-like description.Within this framework, we analyze the interplay between band topology and the dynamical effects of electronic correlations, achieved by developing a non-perturbative method for calculating the effective free energy, the Variational Gaussian Approximation, which is compared with both a simple static mean-field, which is not able to capture any critical behavior for the topological transition and with the accurate Dynamical Mean-Field Theory which provides an important benchmark for the accuracy of the approach.We show indeed that dynamical quantum fluctuations can lead to gap opening without any symmetry breaking.The growth of these quantum effects as the interaction is increased turns indeed the topological transition from continuous to discontinuous.This change of behavior is associated with the presence of massive Dirac fermion at the transition point showing a Gross-Neveu critical behaviour near the quantum critical endpoint.We identify the gap opening as a condensed matter analog of the Coleman-Weinberg mechanism of mass generation.The theory presented offers a broad scope for extension in various directions, ranging from changing the topological model and modifying the interactions, to incorporating non-local effects, or even applying it to entirely trivial strongly correlated models.
Topological phases are quantum phases of matter that appear close to absolute zero temperature. In contrast to commonly known condensed phases of matter, such as a magnet or a superconductor, distinct topological phases may have the same symmetry. Instead, these phases are distinguished by the topology of the electronic wavefunction. Already the single-particle excitations described by band theory in an insulator or a metal or by Bogoliubov-de Gennes theory in a superconductor may have novel anomalous properties associated to their topology. This topological band theory is the subject of this thesis. We aim towards identifying the anomalous boundary and defect states associated to the topology of the band structure in crystalline insulators and superconductors. Furthermore, we determine criteria to easily identify the topological phases from the band structure of a given insulator or superconductor. In Chapter 2 we show that crystalline topological insulators and superconductors may host anomalous gapless or in-gap excitations on corners of a two-dimensional crystal or hinges of a three-dimensional crystal. These topological phases are called "second-order" according to the codimension "2" of their anomalous boundary states. We discuss the precise conditions for their appearance and when they can be linked to the bulk topology. Our discussion includes mirror, twofold-rotation and inversion as crystalline symmetries. In Chapter 3 we discuss the appearance of anomalous gapless or ingap states at defects in a crystalline insulator or superconductor. In particular, we link the existence of anomalous states at disclinations -- lattice defects that violate a rotation symmetry only locally -- to second-order topological crystalline phases. As a side product, we identify the possible contributions of other topological phases to the disclination anomaly. The results of this chapter allow to determine precisely how the disclination anomaly is linked to the bulk topology. In Chapter 4 we present how to construct "symmetry-based indicators" for topological superconductors that admit a description within the Bogoliubov-de Gennes framework. Symmetry-based indicators are necessary criteria for a given Bogoliubov-de Gennes Hamiltonian to realize a topological crystalline phase with anomalous boundary exitations. These criteria are formulated in terms of data from a small set of high-symmetry momenta only. Our approach guarantees to extract the maximal information on the anomalous boundary exitations that can be extracted from a point-wise evaluation of the Bogoliubov-de Gennes Hamiltonian. In the limit of weak superconducting pairing, our criteria can be formulated in terms of data from the normal-state band structure alone.