In this thesis, we consider two topics that involve strong spatial or temporal fluctuations in interacting quantum many-body systems. We first consider far-from equilibrium quantum hydrodynamics in light-driven helical edge states of a 2D topological insulator. We then consider the enhancement of superconducting and magnetic instabilities in fermion systems with random or structured inhomogeneity, driven by the nesting of fractal wave functions. In the first part, we study quantum quenches of helical edge liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized “hot spots” that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to an interaction-mediated frequency doubling that could be detected experimentally from emitted terahertz radiation. In the second part, we study the enhanced correlation effects in many-fermion systems induced by structured or random inhomogeneities. We consider the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions (“spectrum-wide quantum criticality,” SWQC) in the one-dimensional Aubry-Andr ́e and power-law random-banded matrix models (PRBMs) with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, whereas the pairing amplitude is maximized near this transition in both models. Although increasing inhomogeneity always depletes the superfluid stiffness, we find that superconductivity is still controlled by the fractal-enhanced amplitude near the Anderson transition. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconductor models, can robustly enhance superconductivity. We also study the interplay of magnetic ordering and the spectrum-wide quantum critical wave functions in PRBMs with repulsive Hubbard interaction. We employ self-consistent Hartree-Fock numerics and analytical field theory calculations using a Keldysh Finkel’stein nonlinear sigma model. We find that weak interactions delocalize the system, while strong interaction localizes the system. Ferromagnetism sets in near the Anderson transition, where the ferromagnetic spin susceptibility is also shown to peak. This indicates that ferromagnetic ordering is enhanced by the fractal wave functions. Using the sigma model, we calculate the quantum corrections to the conductance and spin susceptibility. We find that the analytical and numerical results are mutually consistent.
In this thesis, we consider two topics that involve strong spatial or temporal fluctuations in interacting quantum many-body systems. We first consider far-from equilibrium quantum hydrodynamics in light-driven helical edge states of a 2D topological insulator. We then consider the enhancement of superconducting and magnetic instabilities in fermion systems with random or structured inhomogeneity, driven by the nesting of fractal wave functions. In the first part, we study quantum quenches of helical edge liquids with spin-flip inelastic scattering. Counterpropagating charge packets in helical edges can be created by an ultrashort electric pulse applied across a 2D topological insulator. Localized “hot spots” that form due to scattering enable two types of strongly nonlinear wave dynamics. First, propagating packets develop self-focusing shock fronts. Second, colliding packets with opposite charge can exhibit near-perfect retroreflection, despite strong dissipation. This leads to an interaction-mediated frequency doubling that could be detected experimentally from emitted terahertz radiation. In the second part, we study the enhanced correlation effects in many-fermion systems induced by structured or random inhomogeneities. We consider the interplay of superconductivity and a wide spectrum of critical (multifractal) wave functions (“spectrum-wide quantum criticality,” SWQC) in the one-dimensional Aubry-Andr ́e and power-law random-banded matrix models (PRBMs) with attractive interactions, using self-consistent BCS theory. We find that SWQC survives the incorporation of attractive interactions at the Anderson localization transition, whereas the pairing amplitude is maximized near this transition in both models. Although increasing inhomogeneity always depletes the superfluid stiffness, we find that superconductivity is still controlled by the fractal-enhanced amplitude near the Anderson transition. Our results suggest that SWQC, recently discovered in two-dimensional topological surface-state and nodal superconductor models, can robustly enhance superconductivity. We also study the interplay of magnetic ordering and the spectrum-wide quantum critical wave functions in PRBMs with repulsive Hubbard interaction. We employ self-consistent Hartree-Fock numerics and analytical field theory calculations using a Keldysh Finkel’stein nonlinear sigma model. We find that weak interactions delocalize the system, while strong interaction localizes the system. Ferromagnetism sets in near the Anderson transition, where the ferromagnetic spin susceptibility is also shown to peak. This indicates that ferromagnetic ordering is enhanced by the fractal wave functions. Using the sigma model, we calculate the quantum corrections to the conductance and spin susceptibility. We find that the analytical and numerical results are mutually consistent.
The shape of electron wave functions determines key properties of quantum materials. Very recently, it was found that disorder can turn all wave functions into fractals (``spectrum-wide quantum criticality,'' SWQC) in models of topological and strongly correlated superconductors. The implications of SWQC for superconductivity itself remain largely unexplored. Here, the authors show that superconductivity can be strongly enhanced near the SWQC Anderson transition in two one-dimensional models. The study highlights a new potential mechanism for achieving higher transition temperatures via random or structured inhomogeneity.
We study superfluidity in strongly correlated systems by focusing three features as follows. First, interplay between superfluidity and other order parameters near quantum critical points is studied. The onset positions of the order parameters with or without superconductivity determines how they interact with each other. For the spin density wave (SDW) transition, we find that the quantum critical point shifts by order |∆|, where ∆ is pairing amplitude, so that the region of SDW order is smaller in the superconductor than in the metal. This shift is larger than the |∆|2 shift predicted by theories of competing orders which ignore Fermi surface effects. Second, we study a doped antiferromagnetic Mott insulator motivated by copper based high temperature superconductors. Based on spin and charge fractionalization, the algebraic charge liquid phase whose bosonic spinons have energy gap but fermionic holons form Fermi surfaces and its superconductivity are studied. We show fluctuations of local SDW order orientation lead to d-wave superconductivity, and its quantum critical pairing mechanisms are investigated. Also, a theory of bound states between spinons and holons is studied. We show that the fractionalization induces a doubling electron-like quasiparticles, which naturally induces the fractionalized Fermi liquid. The bound state theory describes exotic properties of the underdoped Cuprates such as “nodal-antinodal dichotomy” at a mean field level description. At last, we study superfluidity of fermions in optical lattices near the Feshbach resonance. Due to divergence of scattering length at the resonance, universality is observed and characteristics of superfluidity has been studied. We extend such universal properties by considering superfluid-insulator transition with an even integer number of fermions per unit cell. The resulting insulator interpolates between a band insulator of fermions and a Mott insulator of fermion pairs.
In this thesis, we consider two different topics: the probes of out-of-equilibrium topological matter and many-body-localization (MBL) in higher dimensions. The first part focuses on the two-dimensional (2D) topological Floquet p+ip superfluid induced by an instantaneous quench of the interaction strength. We investigate radio-frequency (rf) spectroscopy, metal-to-superconductor tunneling, and angle-resolved photoemission spectroscopy (ARPES) as probes of this out-of-equilibrium quantum system in both the cold-atomic and solid-state realizations. We notice a single avoided crossing in the Floquet band structure and a population inversion in the occupation of these bands. In addition, the rf spectrum is well captured by a quasiequilibrium approximation and shows a robust gap. We also compute the rf signal when only the lower Floquet band is occupied as well as the tunneling signal. In both cases, the spectra show wildly different behaviors compared with the rf signal from the post-quench state, and the gap disappears for strong quenches. We attribute this to the fact that the distribution function influences the rf but not the tunneling signal. Finally, we look into the ARPES spectrum which exhibits a clear series of Floquet copies. The second part is devoted to interacting fermions in the presence of quenched disorder. We derive a finite-temperature Keldysh response theory for such a system in the delocalized phase, in the form of the Finkel'stein nonlinear sigma model (FNLσM). Applicable to any symmetry class with at least a U(1) symmetry, our formulation automatically incorporates the correct infrared cutoffs for quantum corrections to transport, and provides a framework to study the real and virtual scattering processes. In particular, we are able to rederive from the FNLσM the Altshuler-Aronov-Khmelnitsky equations for dephasing. We then propose a strategy to approach the 2D ergodic-to-MBL transition from the ergodic side. Such a transition is explored as a dephasing catastrophe in an isolated system with short-range interactions. In this case, the self-generated heat bath responsible for dephasing exhibits diffusive (non-Markovian) density fluctuations. We recast the dephasing of quantum interference corrections as a self-interacting polymer loop, and study its critical behavior using renormalization group approach. In dimensions d=4-ε, we identify a nontrivial fixed point corresponding to a finite temperature where the dephasing time diverges. This fixed point can be associated with a toy version of 2D ergodic-to-MBL transition, provided that it survives to ε=2.
Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important. The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder. The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.
We consider three topics, all concerning the confluence of dynamics, disorder, and interactions in low-dimensional quantum many-body systems of interest for the potential manipulation of quantum matter. Particle fractionalization is believed to orchestrate the physics of many strongly correlated systems, yet its direct experimental detection remains a challenge. We propose a measurement for an ultracold matter system, in which correlations in initially decoupled 1D chains are imprinted via quantum quench upon two-dimensional Dirac fermions. Coupling fractionalized initial states launches relativistic "fractionalization waves'' along the chains, while coupling noninteracting chains induces perpendicular dispersion. These could be easily distinguished in an ultracold gas experiment. As a potential window on transitions out of the ergodic phase of weakly disordered, quasi-one-dimensional fermion systems, we study dephasing due to a diffusive noise bath generated by inelastic scattering due to short-ranged interactions. We calculate the dephasing of weak localization perturbatively through second order in the bath coupling and discover a failure of the self-consistent Born approximation. We also consider a many-channel quantum wire where short-ranged, spin-exchange interactions coexist with screened Coulomb interactions. We calculate the dephasing rate, treating the short-ranged interaction perturbatively and the Coulomb interaction exactly, giving the first controlled calculation of quasi-1D dephasing due to diffusive noise. Our results are relevant to the search for precursors to the many-body localization transition. They also link dephasing to the enhancement of the spin-triplet interaction near a metal-insulator transition, providing for the amplification of dephasing at low temperatures in spin SU(2)-symmetric quantum wires. The physics of massless, 2D Dirac fermions with a spatially modulated Dirac cone could have important applications in low-dimensional superconductivity and other Dirac materials. This is mathematically identical to a theory of massless fermions on a certain class of curved spacetime manifolds. We study these manifolds, which are dominated by curvature singularities, and find that isotropic and nematic fluctuations have profound yet different effects on the spacetime geometry. We present analytical results on geodesic geometry, give some exact solutions for some highly-symmetric toy models, and speculate about the role of induced one-dimensionality for such bound orbits in the physics of 2D dirty d-wave superconductivity.
This thesis presents studies of the role of disorder in non-equilibrium quantum systems. The quantum states relevant to dynamics in these systems are very different from the ground state of the Hamiltonian. Two distinct systems are studied, (i) periodically driven Hamiltonians in two dimensions, and (ii) electrons in a one-dimensional lattice with power-law decaying hopping amplitudes. In the first system, the novel phases that are induced from the interplay of periodic driving, topology and disorder are studied. In the second system, the Anderson transition in all the eigenstates of the Hamiltonian are studied, as a function of the power-law exponent of the hopping amplitude. In periodically driven systems the study focuses on the effect of disorder in the nature of the topology of the steady states. First, we investigate the robustness to disorder of Floquet topological insulators (FTIs) occurring in semiconductor quantum wells. Such FTIs are generated by resonantly driving a transition between the valence and conduction band. We show that when disorder is added, the topological nature of such FTIs persists as long as there is a gap at the resonant quasienergy. For strong enough disorder, this gap closes and all the states become localized as the system undergoes a transition to a trivial insulator. Interestingly, the effects of disorder are not necessarily adverse, disorder can also induce a transition from a trivial to a topological system, thereby establishing a Floquet Topological Anderson Insulator (FTAI). Such a state would be a dynamical realization of the topological Anderson insulator. We identify the conditions on the driving field necessary for observing such a transition. We realize such a disorder induced topological Floquet spectrum in the driven honeycomb lattice and quantum well models. Finally, we show that two-dimensional periodically driven quantum systems with spatial disorder admit a unique topological phase, which we call the anomalous Floquet-Anderson insulator (AFAI). The AFAI is characterized by a quasienergy spectrum featuring chiral edge modes coexisting with a fully localized bulk. Such a spectrum is impossible for a time-independent, local Hamiltonian. These unique characteristics of the AFAI give rise to a new topologically protected nonequilibrium transport phenomenon: quantized, yet nonadiabatic, charge pumping. We identify the topological invariants that distinguish the AFAI from a trivial, fully localized phase, and show that the two phases are separated by a phase transition. The thesis also present the study of disordered systems using Wegner's Flow equations. The Flow Equation Method was proposed as a technique for studying excited states in an interacting system in one dimension. We apply this method to a one-dimensional tight binding problem with power-law decaying hoppings. This model presents a transition as a function of the exponent of the decay. It is shown that the the entire phase diagram, i.e. the delocalized, critical and localized phases in these systems can be studied using this technique. Based on this technique, we develop a strong-bond renormalization group that procedure where we solve the Flow Equations iteratively. This renormalization group approach provides a new framework to study the transition in this system.