In this dissertation, we investigate the edge spin excitations and reconstructions of quantum Hall (QH) liquids. Edge spin reconstructions reflect the interplay of confining potential, electron-electron interaction, and Zeeman splitting at the edges of QH liquids. The main method used in this dissertation is the exact diagonalization calculation for finite size systems. The QH liquids studied in this dissertation are classified into three kinds, based on their topological properties and statistics of quasiparticles (quasiholes). The first kind of QH liquids is spin-polarized QH liquids. The QH liquids with filling factors ν = 1 and 1/3 are studied in this kind. We find the low-energy excitations of the ν = 1 and 1/3 are bosonic edge spin waves. Instabilities of these ferromagnetic state with altering confinement strength result from the softening of these spin waves and formation of edge spin textures. A microscopic trial wave function is used to study the edge instabilities of larger systems with particle number up to 40, and the obtained results are consistent to the ones calculated by exact diagonalization. The second kind of QH liquids is spin-unpolarized QH liquids. The liquids with ν = 2, 2/5 and 2/3 are studied in this kind. When ν = 2, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial ν = 2 instability is toward the compact state with total spin 1. At ν = 2/5, the edge instabilities is also triggered by softening of the edge spin waves with smoother confinement. The situation of ν = 2/3 unpolarized QH liquids is a little different, because it has one backward-moving edge spin wave mode and one forward-moving magnetoplasmon mode. There is no mystery about what kind of instability would occur with stronger or smoother confinement. The exact diagonalization calculation of finite systems can reproduce the instability induced by edge spin wave at the stronger confinement side and gives an estimate about the critical value of confining potential. But instability due to edge magnetoplasmon mode can not be realized in small size systems because of the high velocity of this mode. The last kind is the non-Abelian QH liquid (at ν = 5/2). Its edge instability triggered by softening of edge magnetoplasmon excitations with smoother confinement. Three different confinements are compared in the study of integer QH liquids in finite size systems. And some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit. The effect of Zeeman splitting to the edge instabilities is also considered in this dissertation.
In this dissertation, we investigate the edge spin excitations and reconstructions of quantum Hall (QH) liquids. Edge spin reconstructions reflect the interplay of confining potential, electron-electron interaction, and Zeeman splitting at the edges of QH liquids. The main method used in this dissertation is the exact diagonalization calculation for finite size systems. The QH liquids studied in this dissertation are classified into three kinds, based on their topological properties and statistics of quasiparticles (quasiholes). The first kind of QH liquids is spin-polarized QH liquids. The QH liquids with filling factors ν = 1 and 1/3 are studied in this kind. We find the low-energy excitations of the ν = 1 and 1/3 are bosonic edge spin waves. Instabilities of these ferromagnetic state with altering confinement strength result from the softening of these spin waves and formation of edge spin textures. A microscopic trial wave function is used to study the edge instabilities of larger systems with particle number up to 40, and the obtained results are consistent to the ones calculated by exact diagonalization. The second kind of QH liquids is spin-unpolarized QH liquids. The liquids with ν = 2, 2/5 and 2/3 are studied in this kind. When ν = 2, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial ν = 2 instability is toward the compact state with total spin 1. At ν = 2/5, the edge instabilities is also triggered by softening of the edge spin waves with smoother confinement. The situation of ν = 2/3 unpolarized QH liquids is a little different, because it has one backward-moving edge spin wave mode and one forward-moving magnetoplasmon mode. There is no mystery about what kind of instability would occur with stronger or smoother confinement. The exact diagonalization calculation of finite systems can reproduce the instability induced by edge spin wave at the stronger confinement side and gives an estimate about the critical value of confining potential. But instability due to edge magnetoplasmon mode can not be realized in small size systems because of the high velocity of this mode. The last kind is the non-Abelian QH liquid (at ν = 5/2). Its edge instability triggered by softening of edge magnetoplasmon excitations with smoother confinement. Three different confinements are compared in the study of integer QH liquids in finite size systems. And some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit. The effect of Zeeman splitting to the edge instabilities is also considered in this dissertation.
We study the effect of electron-electron interaction on the charge and spin structures at the edge of integer quantum Hall liquids, under three different kinds of confining potentials. Our exact diagonalization calculation for small systems indicates that the low-energy excitations of the $\ensuremath{\nu}=1$ ferromagnetic state are bosonic edge spin waves. Instabilities of the ferromagnetic state with altering confinement strength result from the softening of these edge spin waves and formation of edge spin textures. In the $\ensuremath{\nu}\ensuremath{\lesssim}2$ regime, exact diagonalization on edge electron systems indicates that compact Hartree-Fock states with different total spin always become ground states in some regions of parameter space, and the ground states appearing between two compact states are their edge spin waves. The initial $\ensuremath{\nu}=2$ instability is toward the compact state with total spin 1. Larger systems are studied using a microscopic trial wave function, and some quantitative predictions on the edge instabilities for a certain type of confining potential are reached in the thermodynamic limit.
Quantum Hall systems have a one-body energy spectrum consisting of dispersion-less Landau levels. Electron-electron interactions thus dominate in partially filled Landau levels, which exhibit a myriad of strongly correlated phases such as quantum hall ferromagnets and fractional quantum Hall phases. We study two examples of these phenomena. In the first project, we explore the ground state of a system with an interface between two semi-infinite regions with fillings ν= 4 and ν= 3 respectively. The width of the interface can be controlled by varying the background potential, which provides an additional tuning parameter. For a certain range of interaction strengths, the ν= 4 bulk is unpolarized whereas the ν= 3 bulk is fully polarized. In the parameter space spanned by the interaction strength and width of the interface, we find two phases at the interface. Phase A has spin as a good quantum number, and the long-wavelength spin edge excitations are gapped. In phase B, spin rotation symmetry is spontaneously broken at the mean-field level. Using symmetry arguments we find the effective theory near the interface of phase B. This effective theory is known to have gapless long-wavelength spin excitations. In the second project, we study the ground state of a tunnel-decoupled double-layer graphene system when both layers are undoped. We find a simple Hamiltonian in the continuum limit from symmetries of the system. Using the Hartree-Fock approximation we find a state with inter-layer coherence with broken layer U(1) symmetry. This phase becomes magnetized in presence of a non-zero Zeeman field. A first-order phase transition can be driven from the ferromagnetic phase to the magnetized inter-layer coherent phase by increasing the Zeeman field. We predict the number of gapless modes in the bulk.
In a microscopic model of fractional quantum Hall liquids with electron-electron interactions and confinement, we calculate the edge Green's function via exact diagonalization. Our results for nu=1/3 and 2/3 suggest that, in the presence of Coulomb interaction, "external" parameters such as the sharpness of the edge and the strength of the edge confining potential, which can lead to edge reconstruction, may cause deviations from universality in the edge-tunneling I-V exponent. In particular, we do not find any direct dependence of this exponent on the range of the interaction potential as suggested by recent calculations in contradiction to the topological nature of the edge.
This dissertation is devoted to theoretical studies of strongly interacting one-dimensional and quasi one-dimensional electron systems. The properties of one-dimensional electron systems can be studied within the bosonization technique, which presents fermions as collective bosonic density excitations. The power of this approach is the ability to treat electron-electron interaction exactly in the low-energy limit. The approach predicts the failure of Fermi liquid and an absence of long-range order in one-dimensions. The low-energy description of one-dimensional interacting systems is called the Tomonaga-Luttinger liquid theory. For example, the edges of quantum Hall systems are one-dimensional and described by a chiral Tomonaga-Luttinger liquid. Another example is a quantum spin Hall system with helical edge states, which are also described by a Tomonaga-Luttinger liquid. In our first work, a study of magnetized edge states of quantum spin-Hall system is presented. A magnetic field dependent signature of such edges is obtained, which can be verified in a Coulomb drag experiment. The second part of the dissertation is devoted to quasi-one dimensional antiferromagnetic lattices. A spatially anisotropic lattice antiferromagnet can be viewed as an array of one dimensional spin chains coupled in a way to match the lattice symmetry. This allows to use the non-Abelian bosonization technique to describe the low-energy physics of spin chains and study the inter-chain interactions perturbatively. The work presented in the dissertation studies the effect of Dzyaloshinskii-Moriya interaction on the magnetic phase diagram of the spatially anisotropic kagome antiferromagnet. We predict a Dzyaloshinskii-Moriya interaction driven phase transition from Neel to Neel+dimer state. In the third work, a novel model of the fractional quantum Hall effect is given. Wave functions of two-dimensional electrons in strong and quantizing magnetic field are essentially one-dimensional. That invites one to use the one-dimensional phenomenological bosonization to describe the density fluctuations of the two-dimensional interacting electrons in magnetic field. Remarkably, the constructed trial bosonized fermion operator describing the electron states with a fixed Landau gauge momentum is effectively two-dimensional.
In the quantum Hall effect, a planar electron gas exposed to a perpendicular magnetic field condenses into a macroscopic quantum mechanical liquid characterized by a topological integer or fractional quantization of the transverse conductance. For the fractionally quantized liquid, the low-energy elementary excitations carry fractional charge and obey fractional statistics. At low energies, the degrees of freedom in the quantum Hall states are purely topological. The states are classified into universality classes according to their topological order, and for each universality class, there is a low-energy effective quantum field theory, which determines the properties of the associated quasiparticles. In a finite system with a boundary, the two-dimensional effective theory reduces into a one-dimensional edge theory, characteristic of each universality class. The thesis investigates few-electron quantum Hall droplets and the associated quasiparticles by means of numerical exact diagonalization of the Hamiltonian matrix in a high magnetic field approximation, which constitutes a Landau level projection. The inter-electron interaction depends on the Landau level as well as the details of the experimental sample. In the results, we compare the states obtained with realistic effective interactions to the theoretical models concerning the universality classes of the Laughlin states and the Pfaffian state, by studying overlaps of the state vectors, phase structure of the wave functions, as well as the quasiparticle and edge excitations.
Topological phases of matter are purely quantum mechanical and have no classical analogue. Most phases in nature can be classified and studied classically through the concept of symmetry breaking and its theoretical description, Landau-Ginzburg field theory. In contrast to the general wisdom of Landau-Ginzburg field theory, the topological phases share the same symmetry as a trivial insulator and still are different phases. Having gapped spectrum in bulk, they support a metallic edge excition robust against symmetry-respecting perturbation or an emergent fractional excitation in bulk. Following after fractional quantum Hall fluids, many topologically orderd phases, such as spin liquids and topological insulators, have been found and studied. Spin liquids are disordered phases of frustrated antiferromagnets and do not freeze and order even at the lowest temperature. They support an electrically neutral spin-1/2 excitation, which does not exist in a microscopic scale, with emergent dynamical gauge field in bulk and do not have an adiabatic path to a trivial paramagnet phase. The topological insulators are time-reversal symmetric band insulators which cannot evolve smoothly into a trivial insulator with the symmetry, and they have been intensely studied theoretically and experimetally for the last decade. Though the topological insulators are inherently non-interacting systems, they have exotic gapless edge and surface states which demonstrate many interesting quantum phenomena such as fractionalization, axionic electromagnetism, and half quantum Hall effect. In this thesis, we study various quantum phenomena of the topological phases, mainly of the topological insulator and its close relatives in which the physics of spin liquids has been merged into. As they are intrinsically quantum many-body states, the quantum field theory is an invaluable tool to explore the venue of the phases and will be used thorougly in this work. First, we present a topological field theoretic description of the topological insulators and translational symmetric Z2 spin liquid. We demostrate that abelian BF theory is an effective field theory for the two and three dimensional topological insulators. We show that the phenomenologies of BF theory are consistent with those of the topological insulators such as gapless edge and surface excitations and response functions to external gauge field. The same form of the BF theory is also applicable to study the certain classes of Z2 spin liquids with the gapless edge states. We closely examine the possible gapless edge theory of the BF theory and the translational symmetric spin liquids, and we show that the effective BF theories capture the bulk properties as well as the gapless edge spectrum of the spin liquid. Also we discuss the zero-temperature phase diagram of a thin film three dimensional topological insulators with two competing mass terms: time-reversal symmetric exciton condensation and time-reversal symmetry breaking Zeeman effect. There are two phases, quantum spin Hall phase and quantum anomalous Hall phase, and both phases support fractional excitations. We derive an effective topological field theory for the fractional excitations and examine the origin of the fractional excitations. Next we consider the relatives of the topological insulators. First we discuss the proximate symmetry broken phases of topological Mott insulator, a U(1) spin liquid with fermionic spinons in the topological insulator state. The topologically non-trivial band structure of the spinon generates the axion term for the gauge field, and we show that the axion term changes the nature of the confined phase of the spin liquid. Contrast to the conventional confined phase which has only a bond order, the confined phase of the spin liquid is in general a coexisting phase of the two different orders : a current order and a bond order. We then consider another relative of the topological insulator: a bosonic symmetry protected topological phase in two dimensional space, with both PSU(N) and time reversal symmetry. We develop an effective field theory for the phase, which is a SU(N) principal chiral model with a theta-term, and reveal the physics of the topological phase in detail.Finally we consider a topological semimetal, namely a Weyl semimetal. We first demonstrate that the Weyl semimetal can be realized in the magnetically doped topological band insulators. We explicitly derive the low energy theory of Weyl points from the general continuum Hamiltonian and tight-binding model of the topological insulators. Then we study superconducting instabilities of doped Weyl semimetals. We consider a minimal model for a Weyl semimetal and study the superconducting instabilities induced by a short-ranged attractive interaction. With the interaction, we find two competing states: a fully gapped finite-momentum pairing state and a nodal even-parity pairing state. We show that, in a mean field approximateion, the finite-momentum pairing state wins energetically against the usual even-parity paired state. We also show that exotic modes are localized at the full and half quantum vortices of the finite-momentum pairing state.