This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20163号 / 理博第4248号 / 新制||理||1611(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 高橋 義朗, 准教授 柳瀬 陽一 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

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The purpose of the present thesis is to investigate the effects of interactions in interferometers in the integer quantum Hall regime. The behaviour of electrons in condensed matter systems is mostly determined by the repulsive Coulomb interaction. However, under special circumstances the Coulomb interaction can be effectively attractive, giving rise to electron-pairing in unconventional superconductors and specifically designed mesoscopic setups. In particular, electron interactions can play a crucial role in quantum Hall systems, leading for instance to the emergence of quasi-particles with fractional charge and anyonic statistics. Quantum Hall Fabry-Pérot interferometers (FPI) have attracted increasing attention due to their ability to probe such exotic physics. In addition, such interferometers are affected by electron interactions themselves in unexpected ways. For example, experimental evidence for electron-pairing in a quantum Hall FPI was recently found. Interactions play a crucial role in quantum Hall FPI, and a deeper investigation of their effects is necessary in order to reach a better understanding of these systems. We start by analysing the magnetic field and gate voltage dependence of the longitudinal resistance in an electronic FPI, taking into account the interactions between an outer-most interfering edge mode, an inner non-interfering edge mode and the bulk. For weak bulk-edge coupling and sufficiently strong inter-edge interaction, we obtain that the interferometer operates in the Aharonov-Bohm regime with a flux periodicity halved with respect to the usual expectation, as recently observed. We do not find evidence for a connection between a reduced flux period and electron pairing, though. Then, we compute electron shot noise of an integer quantum Hall FPI in the limit of strong backscattering in the presence of a inter-edge repulsive interaction. We find that the Fano factor for strong edge-edge coupling is considerably enhanced with respect to the Fano factor of a non-interacting interferometer, indicating a significant correlation between the tunnelling of subsequent electrons. In particular, we find a two-fold enhanced Fano factor for some parameters, indicative of electron pairing as found experimentally. We interpret this result in terms of a dynamical attraction between electrons taking place in the interfering edge via the exchange of neutral inter-edge plasmons. We argue that our results for interferometers in the strong backscattering limit are related to an enhancement of shot noise observed experimentally in more open devices.

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We reserve the first part of this thesis to a brief (and by far incomplete, but hopefully self-contained) introduction to the vast subject of quantum Hall physics. We dedicate the first chapter to a discursive broad introduction. The second one is instead used to introduce the integer and fractional quantum Hall effects, with an eye to the synthetic quantum matter platforms for their realization. In the third chapter we present famous Laughlin's wavefunction and discuss its basic features, such as the gapless edge modes and the gapped quasiparticle excitations in the bulk. We close this introductory part with a fourth chapter which presents a brief overview on the chiral Luttinger liquid theory. In the second part of this thesis we instead proceed to present our original results. In the fifth chapter we numerically study the linear and non-linear dynamics of the chiral gapless edge modes of fractional quantum Hall Laughlin droplets -- both fermionic and bosonic -- when confined by anharmonic trapping potentials with model short range interactions; anharmonic traps allow us to study the physics beyond Wen's low-energy/long-wavelength chiral Luttinger liquid paradigm in a regime which we believe is important for synthetic quantum matter systems; indeed, even though very successful, corrections to Wen's theory are expected to occur at higher excitation energies/shorter wavelengths. Theoretical works pointed to a modified hydrodynamic description of the edge modes, with a quadratic correction to Wen's linear dispersion $\omega_k=vk$ of linear waves; even though further works based on conformal field theory techniques casted some doubt on the validity of the theoretical description, the consequences of the modified dispersion are very intriguing. For example, in conjunction with non-linearities in the dynamics, it allowed for the presence of fractionally quantized solitons propagating ballistically along the edge. The strongly correlated nature of fractional quantum Hall liquids poses technical challenges to the theoretical description of its dynamics beyond the chiral Luttinger liquid model; for this reason we developed a numerical approach which allowed us to follow the dynamics of macroscopic fractional quantum Hall clouds, focusing on the neutral edge modes that are excited by applying an external weak time-dependent potential to an incompressible fractional quantum Hall cloud prepared in a Laughlin ground state. By analysing the dynamic structure factor of the edge modes and the semi-classical dynamics we show that the edge density evolves according to a Korteweg-de Vries equation; building on this insight, we quantize the model obtaining an effective chiral Luttinger liquid-like Hamiltonian, with two additional terms, which we believe captures the essential low-energy physics of the edge beyond Wen's highly successful theory. We then move forward by studying -- even though only partially -- some of the physics of this effective model and analyse some of its consequences. In the sixth chapter we look at the spin properties of bulk abelian fractional quantum Hall quasiparticles, which are closely related to their anyonic statistics due to a generalized spin-statistics relation - which we prove on a planar geometry exploiting the fact that when the gauge-invariant generator of rotations is projected onto a Landau level, it fractionalizes among the quasiparticles and the edge. We then show that the spin of Jain's composite fermion quasielectron satisfies the spin-statistics relation and is in agreement with the theory of anyons, so that it is a good anti-anyon for the Laughlin's quasihole. On the other hand, even though we find that the Laughlin’s quasielectron satisfies the spin-statistics relation, it carries the wrong spin to be the anti-anyon of Laughlin’s quasihole. Leveraging on this observation, we show how Laughlin's quasielectron is a non-local object which affects the system's edge and thus affecting the fractionalization of the spin. Finally, in the seventh chapter we draw our conclusions.

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.

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<p>Topological materials have been under intense investigation for more than 30 years and have experienced astonishing growth  over the last decade. The two-dimensional electron gas has long served as a model system for the exploration of topological physics, supporting a diverse array of strongly correlated emergent phenomena. Indeed, some of the most stunning topological phases in condensed matter systems are the integer and fractional quantum Hall states forming in two-dimensional electron gases.</p> <p>It was realized early on that electron localization in the bulk has an important role in attaining topological phases, where the sample bulk is well described by randomly localized electrons, known as the Anderson insulator. However, a different type of topological phases forms when charge carriers order in the bulk. Such a charge ordering can only occur in the presence of strong electron-electron interactions and low disorder. Localization of this kind is of a collective nature and differs fundamentally from the single particle physics of the Anderson insulator. The nature of charge ordering, however, is more nuanced than first thought. Indeed, in high Landau levels, Hartree-Fock theories predict the proliferation of numerous exotic bulk insulators, where in the limit of no disorder electrons cluster together and form a hexagonal lattice. Initial observations of these highly correlated insulating phases were limited to low disorder two-dimensional electron gases confined to GaAs/AlGaAs heterostructures. However, recent discoveries of charge ordering in two-dimensional electron gases confined to graphene highlight the universality of this phenomena, irrespective of host material. While progress has been made in understanding the collective insulators residing within integer quantum Hall plateaus, many aspects remain unresolved. In this Dissertation, I discuss the transport properties and energetics of collective insulators forming near integer quantum Hall plateaus in the latest generation of very low disorder two-dimensional electron gases.</p> <p>In chapter  1  I briefly introduce recent developments in our current understanding of the integer quantum Hall effect, where the topological phase is described by both a topological invariant as well as a local order parameter related to the Landau symmetry breaking paradigm. Next, I introduce the basic principles of two-dimensional electron gases confined to semiconductor heterostructures and provide a short summary of recent technological breakthroughs in molecular beam epitaxial growth protocols. The chapter concludes with an introduction to the essential physics of both the integer and the fractional quantum Hall effect.</p> <p>Chapter  2  contains a brief review of the existing literature on the collective insulators forming in sufficiently low disorder two-dimensional electron gases. The primary focus of chapter  2  is on the unique magnetotransport patterns seen at various Landau level filling factors, which support the collective insulator  interpretation. Throughout this chapter I tend to lean on theoretical models that describe these collective phases through the lens of the Hartree-Fock theory; however, it is important to note that both density matrix renormalization group theories and direct diagonalization of small electron systems reach similar conclusions.</p> <p>In chapter  3  I present our data displaying the hallmark transport signatures of a collective insulator residing within the flanks of the nu = 1 integer quantum Hall plateau. Our sample belongs to the latest generation of low disorder 2DEGs confined to GaAs/AlGaAs. I provide a detailed analysis of its development in both temperature and filling factor. The distinct transport signatures we observe strongly overlap in filling factor with prior microwave resonance, surface acoustic wave, compressibility, and tunneling measurements, all of which point to the formation of a collective insulator known as the integer quantum Hall Wigner solid. One puzzling aspect, however, is that while the latter measurements exhibit the integer quantum Hall Wigner solid in older generation samples, transport signatures of this phase appear to be present only in the newest and highest mobility samples. By using distinct features in the magnetoresistance, I propose a stability diagram of the integer quantum Hall Wigner solid in nu −T phase space. Analysis of magnetoresistance profiles at fixed filling factors display sharp peaks within the region of integer quantum Hall Wigner solid phase. It is believed that these sharp peaks are a shared property of collective insulators forming in low disorder two-dimensional electron gases and signal the onset of the electron solid formation. Additional analysis of the magnetoresistance profiles suggests activated transport behavior with a gap energy comparable to that of the plateau center. Lastly, I present large signal measurements of the nu = 1 integer quantum Hall Wigner solid. The data displays strong non-linear behavior in the current-voltage characteristics consistent with the depinning and sliding conduction. However, similar threshold conduction is also seen in the current-voltage characteristics near the center of the integer quantum Hall plateau, where the bulk is an Anderson insulator. Much to our surprise, trends in the threshold current are monotonic in filling factor.</p> <p>In chapter  4  I report on the recent emergence of a newly observed collective insulator residing within the nu = 2 integer quantum Hall plateau and centered at filling factor nu = 1.79. Based on the range of filling factors which stabilizes this collective insulator, we find it distinct from the aforementioned integer quantum Hall Wigner solid. Indeed, the transport behavior is eerily reminiscent to the reentrant insulating phase seen at low filling factors between 1/5 < nu < 2/9. Hence, we term this collective insulator the reentrant integer quantum Hall Wigner solid. Evoking concepts of particle-hole symmetry, we find the reentrant integer quantum Hall Wigner solid to be one member of the larger family of Wigner solids, which is intimately linked through this fundamental symmetry of the system.</p> <p>Lastly in chapter  5 , I explore the energetics of the collective insulators which develop in the N = 2 and N = 3 Landau level, specifically the two- and three-electron bubble phase. We extract the onset temperatures of these exotic bulk insulators from the sharp peaks in the magnetoresistance at fixed filling factor. We compare our measured onset energies with the cohesive energies found from numerical calculations. We find the onset temperatures for the both two- and three-electron bubble phase show an approximately linear trend in filling factor within a single Landau level. In addition, we observe that the three-electron bubble phase has a larger onset temperature than the two-electron bubble phase, a result which is inconsistent with some numerical energy calculations. Thus, our measurements of bubble phase energetics call attention to the importance of the short-range Coulomb interaction in the formation of multi-electron bubble phases and is expected to serve as guide towards the refinement of existing theoretical models.</p>

In this dissertation, we use exact diagonalization to study a few strongly correlated systems, ranging from the Fermi-Hubbard model to the fractional quantum Hall effect (FQHE). The discussion starts with an overview of strongly correlated systems and what is meant by strongly correlated. Then, we extend cluster perturbation theory (CPT), an economic method for computing the momentum and energy resolved Green's function for Hubbard models to higher order correlation functions, specifically the spin susceptibility. We benchmark our results for the one-dimensional Fermi-Hubbard model at half-filling. In addition we study the FQHE at fillings $nu = 5/2$ for fermions and $nu = 1/2$ for bosons. For the $nu = 5/2$ system we investigate a two-body model that effectively captures the three-body model that generates the Moore-Read Pfaffian state. The Moore-Read Pfaffian wave function pairs composite fermions and is believed to cause the FQHE at $nu = 5/2$. For the $nu = 1/2$ system we estimate the entropy needed to observe Laughlin correlations with cold atoms via an ansatz partition function. We find entropies achieved with conventional cooling techniques are adequate. / Doctor of Philosophy / Strongly correlated quantum many-body physics is a rich field that hosts a variety of exotic phenomena. By quantum many-body we mean physics that is concerned with the behavior of interacting particles, such as electrons, where the quantum behavior cannot be ignored. By strongly correlated, we mean when the interactions between particles are sufficiently strong such that they cannot be treated as a small perturbation. In contrast to weakly correlated systems, strongly correlated systems are much more difficult to solve. That is because methods that reduce the many-body problem to a single independent body problem do not work well. In this dissertation we use exact diagonalization, a method to computationally solve quantum many-body systems, to study two strongly correlated systems: the Hubbard model and the fractional quantum Hall effect.The Hubbard model captures the physics of many interesting materials and is the standard toy model. Originally developed with magnetic properties in mind, it has been extended to study superconductivity, topological phases, cold atoms, and much more. The fractional quantum Hall effect is a novel phase of matter that hosts exotic excitations, some of which may have applications to quantum computing.