Exotic phases and associated phase transitions in low
dimensions have been a fascinating frontier and a driving force in
modern condensed matter physics since the 80s. Due to strong
correlation effect, they are beyond the description of mean-field
theory based on a single-particle picture and Landau's
symmetry-breaking theory of phase transitions. These new phases of
matter require new physical quantities to characterize them and new
languages to describe them. This thesis is devoted to the study on
exotic phases of correlated electrons in two spatial dimensions. We
present the following efforts in understanding two-dimensional
exotic phases: (1) Using Zn vertex algebra, we give a complete
classification and characterization of different one-component
fractional quantum Hall (FQH) states, including their ground state
properties and quasiparticles. (2) In terms of a non-unitary
transformation, we obtain the exact form of statistical
interactions between composite fermions in the lowest Landau level
(LLL) with v=1/(2m), m=1,2... By studying the pairing instability
of composite fermions we theoretically explains recently observed
FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin
liquids (SLs) on kagome lattice in Schwinger-fermion representation
using projective symmetry group (PSG). We propose one most
promising candidate for the numerically discovered SL state in
nearest-neighbor Heisenberg model on kagome lattice}. (4) By
analyzing different Z2 spin liquids on honeycomb lattice within PSG
classification, we find out the nature of the gapped SL phase in
honeycomb lattice Hubbard model, labeled sublattice pairing state
(SPS) in Schwinger-fermion representation. We also identify the
neighboring magnetic phase of SPS as a chiral-antiferromagnetic
(CAF) phase and analyze the continuous phase transition between SPS
and CAF phase. For the first time we identify a SL called 0-flux
state in Schwinger-boson representation with one (SPS) in
Schwinger-fermion representation by a duality transformation. (5)
We show that when certain non-collinear magnetic order coexists in
a singlet nodal superconductor, there will be Majorana bound states
in vortex cores/on the edges of the superconductor. This proposal
opens a window for discovering Majorana fermions in strongly
correlated electrons. (6) Motivated by recent numerical discovery
of fractionalized phases in topological flat bands, we construct
wavefunctions for spin-polarized fractional Chern insulators (FCI)
and time reversal symmetric fractional topological insulators (FTI)
by parton approach. We show that lattice symmetries give rise to
different FCI/FTI states even with the same filling fraction. For
the first time we construct FTI wavefunctions in the absence of
spin conservation which preserve all lattice symmetries. The
constructed wavefunctions also set up the framework for future
variational Monte Carlo simulations.
Exotic phases and associated phase transitions in low
dimensions have been a fascinating frontier and a driving force in
modern condensed matter physics since the 80s. Due to strong
correlation effect, they are beyond the description of mean-field
theory based on a single-particle picture and Landau's
symmetry-breaking theory of phase transitions. These new phases of
matter require new physical quantities to characterize them and new
languages to describe them. This thesis is devoted to the study on
exotic phases of correlated electrons in two spatial dimensions. We
present the following efforts in understanding two-dimensional
exotic phases: (1) Using Zn vertex algebra, we give a complete
classification and characterization of different one-component
fractional quantum Hall (FQH) states, including their ground state
properties and quasiparticles. (2) In terms of a non-unitary
transformation, we obtain the exact form of statistical
interactions between composite fermions in the lowest Landau level
(LLL) with v=1/(2m), m=1,2... By studying the pairing instability
of composite fermions we theoretically explains recently observed
FQHE in LLL with v=1/2,1/4. (3) We classify different Z2 spin
liquids (SLs) on kagome lattice in Schwinger-fermion representation
using projective symmetry group (PSG). We propose one most
promising candidate for the numerically discovered SL state in
nearest-neighbor Heisenberg model on kagome lattice}. (4) By
analyzing different Z2 spin liquids on honeycomb lattice within PSG
classification, we find out the nature of the gapped SL phase in
honeycomb lattice Hubbard model, labeled sublattice pairing state
(SPS) in Schwinger-fermion representation. We also identify the
neighboring magnetic phase of SPS as a chiral-antiferromagnetic
(CAF) phase and analyze the continuous phase transition between SPS
and CAF phase. For the first time we identify a SL called 0-flux
state in Schwinger-boson representation with one (SPS) in
Schwinger-fermion representation by a duality transformation. (5)
We show that when certain non-collinear magnetic order coexists in
a singlet nodal superconductor, there will be Majorana bound states
in vortex cores/on the edges of the superconductor. This proposal
opens a window for discovering Majorana fermions in strongly
correlated electrons. (6) Motivated by recent numerical discovery
of fractionalized phases in topological flat bands, we construct
wavefunctions for spin-polarized fractional Chern insulators (FCI)
and time reversal symmetric fractional topological insulators (FTI)
by parton approach. We show that lattice symmetries give rise to
different FCI/FTI states even with the same filling fraction. For
the first time we construct FTI wavefunctions in the absence of
spin conservation which preserve all lattice symmetries. The
constructed wavefunctions also set up the framework for future
variational Monte Carlo simulations.
The quantum nature of electrons emerges in various exotic ways when they are put together with quantum many-body interactions. In Mott insulating systems with strong electron correlation, frustration and strong quantum fluctuations promote localized electron spins to form nontrivial quantum paramagnetic phases such as quantum spin liquid and valence bond solid preventing the development of conventional magnetic orders. In this thesis, we study such exotic quantum phases in various different frustrated quantum magnets. First, encouraged by the spin liquid phenomenology observed in Ba3XSb2O9 (X=Cu, Ni), we investigate Z2 spin liquids on a three dimensional triangular lattice by employing the Schwinger boson mean-field theory with projective symmetry group analysis. It is found that a nematic spin liquid is energetically favorable on this lattice structure rather than symmetric spin liquids. Next, we study various valence bond solid phases that can appear in kagome lattice antiferromagnets by generalizing the recently proposed phenomenological Z2 gauge theory for the topological Z2 spin liquid state found in the nearest-neighbor kagome lattice Heisenberg model. This work provides a bridge between the topological spin liquid and valence bond solid phases previously proposed for kagome lattice antiferromagnets via vison condensation in the gauge theory. Then, we consider optical conductivity in Mott insulators as an experimental probe to detect quantum paramagnetic phases like valence bond solid. Applying this idea to deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with a 12-site valence bond solid state, we estimate the size of optical conductivity in this quantum magnet. After this, another quantum paramagnet BiCu2PO6 is carefully investigated to clarify its puzzling ground state nature and quasiparticle excitations. Utilizing recent high resolution neutron scattering data, we construct a microscopic spin model and also an effective theory for the low energy spin excitations in the compound. We find significant influences of magnetic anisotropies on the ground state and quasiparticle dynamics. Lastly, we investigate emergent quantum phases in the thin film geometries of the pyrochlore iridates, where a number of exotic quantum ground states are proposed to occur in bulk materials as a result of the interplay between electron correlation and strong spin-orbit coupling. The fate of these bulk phases, as well as the novel quantum states that may arise only in a thin film setting, are studied via a theoretical model that allows layer-dependent magnetic structures.
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.\nFor the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.\nFor the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory.
The concept of topology in condensed matter physics has led to the discovery of rich and exotic physics in recent years. Especially when strong correlations are included, phenomenons such as fractionalization and anyonic particle statistics can arise. In this thesis, we study several systems hosting topological phases of interacting fermions. In the first part, we consider one-dimensional systems of parafermions, which are generalizations of Majorana fermions, in the presence of a Z_N charge symmetry. We classify the symmetry-protected topological (SPT) phases that can occur in these systems using the projective representations of the symmetries and find a finite number of distinct phases depending on the prime factorization of N. The different phases exhibit characteristic degeneracies in their entanglement spectrum (ES). Apart from these SPT phases, we report the occurrence of parafermion condensate phases for certain values of N. When including an additional Z_N symmetry, we find a non-Abelian group structure under the addition of phases. In the second part of the thesis, we focus on two-dimensional lattice models of spinless fermions. First, we demonstrate the detection of a fractional Chern insulator (FCI) phase in the Haldane honeycomb model on an infinite cylinder by means of the density-matrix renormalization group (DMRG). We report the calculation of several quantities characterizing the topological order of the state, i.e., (i)~the Hall conductivity, (ii)~the spectral flow and level counting in the ES, (iii)~the topological entanglement entropy, and (iv)~the charge and topological spin of the quasiparticles. Since we have access to sufficiently large system sizes without band projection with DMRG, we are in addition able to investigate the transition from a metal to the FCI at small interactions which we find to be of first order. In a further study, we consider a time-reversal symmetric model on the honeycomb lattice where a Chern insulator (CI) induced by next-nearest neighbor interactions has been predicted by mean field theory. However, various subsequent studies challenged this picture and it was still unclear whether the CI would survive quantum fluctuations. We therefore map out the phase diagram of the model as a function of the interactions on an infinite cylinder with DMRG and find evidence for the absence of the CI phase. However, we report the detection of two novel charge-ordered phases and corroborate the existence of the remaining phases that had been predicted in mean field theory. Furthermore, we characterize the transitions between the various phases by studying the behavior of correlation length and entanglement entropy at the phase boundaries. Finally, we develop an improvement to the DMRG algorithm for fermionic lattice models on cylinders. By using a real space representation in the direction along the cylinder and a real space representation in the perpendicular direction, we are able to use the momentum around the cylinder as conserved quantity to reduce computational costs. We benchmark the method by studying the interacting Hofstadter model and report a considerable speedup in computation time and a severely reduced memory usage.
In this thesis, we investigate several topics pertaining to emergent collective quantum phenomena in the domain of correlated fermions, using the quantum Monte Carlo method. They display exotic low temperature phases as well as phase transitions which are beyond the Landau–Ginzburg theory. The interplay between three key points is crucial for us: fermion statistics, many body effects and topology. We highlight the following several achievements: 1. Successful modeling of continuum field theories with lattice Hamiltonians, 2. their sign-problem-free Monte Carlo simulations of these models, 3. and numerical results beyond mean field descriptions. First, we consider a model of Dirac fermions with a spin rotational invariant inter- action term that dynamically generates a quantum spin Hall insulator. Surprisingly, an s-wave superconducting phase emerges due to the condensation of topological de- fects of the spin Hall order parameter. When particle-hole symmetry is present, the phase transition between the topological insulator and the superconducting phase is an example of a deconfined quantum critical point(DQCP). Although its low energy effec- tive field theory is purely bosonic, the exact conservation law of the skyrmion number operator rules out the possibility of realizing this critical point in lattice boson models. This work is published in Ref. [1]. Second, we dope the dynamically generated quantum spin Hall insulator mentioned above. Hence it is described by a field theory without Lorentz invariance due to the lack of particle-hole symmetry. This sheds light on the extremely hot topic of twisted bilayergraphene: Why is superconductivity generated when the repulsive Coulomb interaction is much stronger than the electron-phonon coupling energy scale? In our case, Cooper pairs come from the topological skyrmion defects of the spin current order parameter, which are charged. Remarkably, the nature of the phase transition is highly non-mean-field-like: one is not allowed to simply view pairs of electrons as single bosons in a superfluid-Mott insulator transition, since the spin-current order parameter can not be ignored. Again, due to the aforementioned skyrmions, the two order parameters are intertwined: One phase transition occurs between the two symmetry breaking states. This work is summarized in Ref. [2]. Third, we investigate the 2 + 1 dimensional O(5) nonlinear sigma model with a topological Wess-Zumino-Witten term. Remarkably, we are able to perform Monte Carlo calculations with a UV cutoff given by the Dirac Landau level quantization. It is a successful example of simulating a continuous field theory without lattice regularization which leads to an additional symmetry breaking. The Dirac background and the five anti-commuting Dirac mass terms naturally introduce the picture of a non-trivial Berry phase contribution in the parameter space of the five component order parameter. Using the finite size scaling method given by the flux quantization, we find a stable critical phase in the low stiffness region of the sigma model. This is a candidate ground state of DQCP when the O(5) symmetry breaking terms are irrelevant at the critical point. Again, it has a bosonic low energy field theory which is seemingly unable to be realized in pure boson Hamiltonians. This work is summarized in Ref. [3].
Frustrated magnetism provides a rich platform for realizing exotic quantum phases in condensed matter systems. Among these, quantum spin liquids (QSLs) are a particularly intriguing class of states that exhibit no magnetic ordering and preserve all symmetries down to zero temperature. Just as conventional magnets can be thought of as spin crystals,QSLs represent their liquid analogs. This thesis explores two broad types of QSLs in two-dimensional magnets, with a focus on their dynamic properties and the nature of transitions from conventional magnetic states into QSL phases. QSLs are topological phases that host fractionalized excitations known as spinons. In these states, mobile spinons are coupled to an emergent gauge field, forming a highly entangled phase of matter. The first part of this thesis focuses on Dirac spin liquids (DSLs), a class of QSLs where spinons exhibit a gapless Dirac dispersion. To study the dynamicsof these spin liquids beyond leading-order mean-field theory, we develop a novel large-N fermionic limit of gapless spinons. In this limit, all gauge excitations are suppressed, and the leading-order corrections instead arise from order parameter fluctuations, which may condense and drive transitions into symmetry-broken phases from the parent DSL. Incorporating random phase approximation (RPA) corrections to the spin susceptibility, we find that sharp, spin-wave–like modes are recovered in the excitation spectrum. This methodology successfully captures characteristic features of the dynamics near phase transitions and in weakly symmetry-broken phases proximate to a DSL. We apply this approach to microscopic models, including the J1-J2 XXZ model on the triangular lattice and the J1-J3 XXZ model on the honeycomb lattice. These models closely represent a wide range of frustrated materials, such as triangular and honeycomb cobaltates and triangular ytterbium compounds. Our theoretical predictions show excellent agreement with experimentally measured inelastic neutron scattering spectra from these materials. Gapless spinons can also sometimes become gapped through chiral symmetry breaking, leading to the formation of a chiral spin liquid (CSL). The CSL is an archetypal bosonic realization of a fractional quantum Hall (FQH) state at half-filling, and has been proposed as a time-reversal–breaking ground state in a variety of frustrated spin models. The second part of this thesis addresses the intriguing possibility of realizing a CSL through the "melting" of a chiral spin crystal—a conventional magnet with non-coplanar ordering. Such a transition offers a physically appealing analogue to the melting of a Wigner crystal into a fermionic FQH liquid. Focusing on the kagome lattice, where chiral interactions are knownto stabilize a CSL ground state, we study the effects of additional long-range Heisenberg interactions and demonstrate a phase transition into a non-coplanar octahedral magnetic order. To describe the universal class of transition between a CSL and a non-coplanar anti-ferromagnetic order, we then develop a novel Chern-Simons–matter field theory framework, constrained by symmetry principles and symmetry fractionalization patterns. This thesis thus provides new insights into the dynamical signatures and phase transitions of quantum spin liquids, offering a bridge between theoretical models and experimental observations in frustrated magnetic systems.
Materials exhibiting exotic quantum phenomena at a macroscopic level are of great
interest because of its intrinsic value to Physics, and also because of its potential
applications in technology. Strongly correlated electronic systems form interesting
examples of these. Exploring them calls for a good understanding of its physics,
and development of new tools that are applicable to their analysis. Robustness
of the fractional quantum Hall effect (FQHE) makes it an ideal stage to study
strongly correlated electron systems. The complexity of the FQHE problem stems
directly from the strong interactions between the constituent particles and not
from the specific nature of the material it is implemented in. This work attempts
to gain a theoretical description of some of the less understood aspects of FQHE.
FQHE in the lowest Landau level is well understood using the theory of weakly
interacting emergent particles called composite fermions. Composite fermions
formed in the second Landau level however appear to be strongly interacting.
Of particular interest is the FQHE at 2 + 1/2 filling fraction, where the many body
ground state is believed to be described by a paired state of composite fermions,
represented by the Pfaffian wavefunction. Based on predicted properties of its
excitations, there has been proposals for implementing topological quantum com-
putations in this system. However a good understanding of the nature of these
excitations is still lacking.
This thesis introduces a class of many body wavefunctions called “bipartite
composite fermion states” that provide a unified description of all excitations of the
paired state at 5/2. In the bipartite composite fermion (BCF) picture, constituent
composite fermions split into two partitions which are correlated to each other.
Within each partition, they fill the lowest composite fermion Landau level of its
partition.
Properties of the BCF state can now be related to the properties of the individual partitions, which are well understood in terms of the composite fermion
theory. In particular, we show that the excitations of BCF states are obtained by
creating excitations in the individual partitions. The paired state with odd number of composite fermions should have an unpaired composite fermion. The BCF
model predicts that this unpaired composite fermion behaves as an exciton in the
quasiparticle and quasihole are in separate partitions. We confirm this prediction
numerically in finite size systems. Generalizations of bipartite composite fermion
functions to other filling fractions especially 2 + 4/7 and 2 + 3/5 are also studied.
Idea of bipartite composite fermions can be generalized to the case of more
partitions. Specifically, we look at tripartite composite fermion states which are
similar to the bipartite states, except that these have three correlated partitions
instead of two. Tripartite states have a filling fraction of ν = 3/5, and its particle
hole conjugate is a candidate wavefunction for describing the ν = 2 + 2/5…