(PHD, 2021)

Abstract: Several investigations are presented around the general topic of the ground state and low-energy behavior of models for many-body quantum physics in one dimension (1d). We develop a novel numerical method for the ground and low-energy sectors of local Hamiltonians in 1d which is based on proofs from quantum information theory. This method, the rigorous renormalization group (RRG), enjoys the benefits of explicit global information from the Hamiltonian in its local step, allowing it to avoid spurious convergence in systems with challenging energy landscapes. We apply RRG to the random XYZ spin chain in an unbiased numerical study evaluating infinite-randomness fixed point physics and continuously varying critical exponents in the ground state, finding evidence for both. In a related effective model with correlations preventing the exact solution of the strong-disorder renormalization group equations, we use the framework of random walks to rigorously establish continuously varying critical exponents. We also perform detailed studies of deconfined quantum critical points (DQCP) in 1d, providing strong evidence for phase transitions which display similar phenomenology to the canonical examples in 2d. A family of DQCP phase transitions in 1d is exhibited which appears to controlled by complex fixed points corresponding to a walking scenario for renormalization group flows.

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(PHD, 2019)

Abstract: Quantum chaos and the eigenstate thermalization hypothesis are based on the assumption of the validity of random matrix theory description on the spectrum and eigenstates. They provide the foundation and descriptions for the typical dynamics and thermalization in generic closed quantum systems. In this thesis, we investigate situations where the systems show atypical dynamics or anomalous thermalization, conflicting with the usual expectations from quantum chaos and eigenstate thermalization hypothesis. We first examine weak thermalization in a nonintegrable spin chain. The system shows long-lived strong oscillations and relaxes to the thermal equilibrium weakly. We identify the dynamics describable by quasiparticles and recognize the oscillation frequency to be the quasiparticle mass gap. We also estimate the damping time for the oscillations. Next, we study prethermalization, a phenomenon where a system relaxes to an intermediate almost-equilibrium stage before reaching the true thermal equilibrium. We study a nonintegrable spin chain in the strong coupling limit, where an almost-conserved quantity emerges and gives rise to the prethermalization. We also study a newly proposed diagnostic for quantum chaos: out-of-time-ordered correlators. Contrasting to the chaotic systems, we inspect their behaviors in various noninteracting integrable models. Finally, we dig into the quantum many-body scar states in the PXP model which describes a Rydberg atom chain. These special states do not satisfy the random matrix theory description nor the eigenstate thermalization hypothesis, therefore defying quantum chaos.

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(PHD, 2015)

Abstract:

The topological phases of matter have been a major part of condensed matter physics research since the discovery of the quantum Hall effect in the 1980s. Recently, much of this research has focused on the study of systems of free fermions, such as the integer quantum Hall effect, quantum spin Hall effect, and topological insulator. Though these free fermion systems can play host to a variety of interesting phenomena, the physics of interacting topological phases is even richer. Unfortunately, there is a shortage of theoretical tools that can be used to approach interacting problems. In this thesis I will discuss progress in using two different numerical techniques to study topological phases.

Recently much research in topological phases has focused on phases made up of bosons. Unlike fermions, free bosons form a condensate and so interactions are vital if the bosons are to realize a topological phase. Since these phases are difficult to study, much of our understanding comes from exactly solvable models, such as Kitaev’s toric code, as well as Levin-Wen and Walker-Wang models. We may want to study systems for which such exactly solvable models are not available. In this thesis I present a series of models which are not solvable exactly, but which can be studied in sign-free Monte Carlo simulations. The models work by binding charges to point topological defects. They can be used to realize bosonic interacting versions of the quantum Hall effect in 2D and topological insulator in 3D. Effective field theories of “integer” (non-fractionalized) versions of these phases were available in the literature, but our models also allow for the construction of fractional phases. We can measure a number of properties of the bulk and surface of these phases.

Few interacting topological phases have been realized experimentally, but there is one very important exception: the fractional quantum Hall effect (FQHE). Though the fractional quantum Hall effect we discovered over 30 years ago, it can still produce novel phenomena. Of much recent interest is the existence of non-Abelian anyons in FQHE systems. Though it is possible to construct wave functions that realize such particles, whether these wavefunctions are the ground state is a difficult quantitative question that must be answered numerically. In this thesis I describe progress using a density-matrix renormalization group algorithm to study a bilayer system thought to host non-Abelian anyons. We find phase diagrams in terms of experimentally relevant parameters, and also find evidence for a non-Abelian phase known as the “interlayer Pfaffian”.

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(PHD, 2012)

Abstract:

The recent experimental realizations of spin-1/2 gapless quantum spin liquids in two-dimensional triangular lattice organic compounds EtMe_{3}Sb[Pd(dmit)_{2}]_{2} and κ-(ET)_{2}Cu_{2}(CN)_{3} have stimulated the investigation of the gapless spin liquid theories. The models in dimensions greater than one (D>1) usually involve multispin interactions, such as ring exchange interactions, that are difficult to study, while effective gauge theory descriptions are not well-controlled to give reliable physics information. Driven by the need for a systematic and controlled analysis of such phase, such models on ladders are seriously studied. This thesis first focuses on such ladder models. We propose that the gapless spin liquid phase can be accessed from a two-band interacting electron model by metal-Mott insulator phase transition. We use Bosonization analysis and weak-coupling Renormalization Group to further study the gapless spin liquid state in the presence of Zeeman magnetic fields or orbital magnetic fields. Several new exotic gapless spin liquids with dominant spin nematic correlations are predicted. In such a ladder spin liquid, we also consider the impurity effects. We conclude that the local energy textures and oscillating spin susceptibilities around the impurities are nontrivial and can be observed in the experiments. We then shift our focus to another theoretical candidate, an SU(2)-invariant spin liquid with Majorana excitations, which can also qualitatively explain the experimental phenomenology. We construct an exactly solvable Kitaev-type model realizing the long-wavelength Majorana spin liquid state and study its properties. We find that the state has equal power-law spin and spin-nematic correlations and behaves nontrivially in the presence of Zeeman magnetic fields. Finally, we realize such Majorana spin liquid states on a two-leg ladder and further explore their stability. We conclude the states can be stable against short-range interactions and gauge field fluctuations.

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(PHD, 2011)

Abstract: The strong interest in strongly correlated systems in condensed matter physics has continued unabated for the past few decades. In recent years, the number of novel, exotic quantum phases found in theoretical studies has seen a phenomenal rise. Among those interesting quantum states are bose liquids and spin liquids, where strong quantum fluctuations have prevented the systems from developing a long range order. Our work in this thesis seeks to further the understanding of frustrated systems. In the study of a hard-core boson model with ring-only exchange interactions on a square lattice, we obtain concrete numerical realization of the unconventional Exciton Bose Liquid (EBL) phase, which possesses interesting properties such as a “Bose surface” which resembles the Fermi surface in a metal, as well as unusual thermodynamic properties such as a T Log T dependence for specific heat. An equally important result from this work is the demonstration that the widely used Gutzwiller projection on slave-particle wave functions may generally fail to capture the correct long wavelength physics in the respective systems. For the Heisenberg antiferromagnet on the kagome lattice, which is a promising candidate for realizing a spin-disordered ground state, our variational study shows that the projected Schwinger boson wave function is energetically better than the Dirac spin liquid wave function when a small antiferromagnetic second-neighbor spin coupling is added to the nearest-neighbor model. We also study the anisotropic triangular Heisenberg antiferromagnetic in magnetic field, and find simple, yet accurate wave functions for various regions of the surprisingly rich phase diagram, thus providing insights into the energetics of the competing phases in this interesting model. Finally, our work also highlights permanent-type wave functions as potentially useful constructions in variational studies of systems with short-ranged correlations, e.g., a Mott insulator and a gapped spin liquid.

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