A general description of weak and strong tunneling fixed points is developed in the chiral-Luttinger-liquid model of quantum Hall edge states. Tunneling fixed points are a subset of ``termination'' fixed points, which describe boundary conditions on a multicomponent edge. The requirement of unitary time evolution for bounded edges gives a nontrivial consistency condition for possible low-energy boundary conditions. The effects of interactions and random hopping on fixed points are studied through a perturbative renormalization-group (RG) approach which generalizes the Giamarchi-Schulz RG for disordered Luttinger liquids to broken left-right symmetry and multiple modes. We apply our approach to a number of examples, such as tunneling between a quantum Hall edge and a superconductor and tunneling between two quantum Hall edges in the presence of interactions. Interactions are shown to induce a continuous renormalization of effective tunneling charge for the integrable case of tunneling between two Laughlin states.
The edge of the electronic fractional quantum Hall (FQH) system obeys the law of the chiral Luttinger liquid theory due to its intrinsic topological properties and the relation of bulk-edge correspondence. However, in a realistic experimental system, such as the usual Hall bar setup, the softening of the background confinement potential can induce the reconstruction of the edge spectrum, which breaks the chirality and universality of the FQH edge. The entanglement spectrum (ES) of the FQH ground state has the same counting structure as that of the energy spectrum indicating the topological characters of the quantum state. In this work, we report that the ES can also have an edge reconstruction while sweeping the area of the subsystem in real space cut. Moreover, we found that the critical area of the subsystem matches accurately with the intrinsic building block of the FQH liquids, namely the correlation hole of the FQH liquids. The above results seem to be universal after our study of a series of typical FQH states, such as two Laughlin states at $\ensuremath{\nu}=1/3$ and
1/5, and the Moore-Read state for $\ensuremath{\nu}=5/2$.
By numerical exact diagonalization techniques, we obtain the quantum phase diagram of the lattice fractional quantum Hall (FQH) systems in the presence of quenched disorder. By implementing an array of local potential traps representing the disorder, we show that the system undergoes a series of quantum phase transitions as the disorder and/or the interaction is tuned. As the strength of potential traps is increased, the FQH state turns into a compressible liquid and then into a topologically trivial insulator. We use the numerically calculated energy gap, quantum degeneracy, the Chern number, the entanglement spectrum, and the fidelity metric to identify various quantum phases. The connection to continuum FQH effects is also discussed.
Fractional quantum Hall (FQH) system at Landau level filling fraction $\ensuremath{\nu}=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=5{e}^{2}/2h$. Thermal Hall conductances of the Pf and APf states are quantized at ${\ensuremath{\kappa}}_{xy}=7/2$ and ${\ensuremath{\kappa}}_{xy}=3/2$, respectively, in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\ensuremath{\nu}=5/2$ FQH state is ${\ensuremath{\kappa}}_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a ${\ensuremath{\kappa}}_{xy}=5/2$ phase. In this paper, we do perturbative and nonperturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold ${\mathrm{\ensuremath{\Lambda}}}_{1}$, which is lowered to an emergent $U(1)\ifmmode\times\else\texttimes\fi{}U$(1) symmetry at energy scales between ${\mathrm{\ensuremath{\Lambda}}}_{1}$ and a higher value ${\mathrm{\ensuremath{\Lambda}}}_{2}$, and is finally lowered to the composite fermion parity symmetry
Z2F above ${\mathrm{\ensuremath{\Lambda}}}_{2}$. Based on the emergent symmetries, we propose a phase diagram of the disordered $\ensuremath{\nu}=5/2$ FQH system and show that a ${\ensuremath{\kappa}}_{xy}=5/2$ phase arises at disorder energy scales $\mathrm{\ensuremath{\Lambda}}>{\mathrm{\ensuremath{\Lambda}}}_{1}$. Furthermore, we show the gapped double-semion sector of
ND compact domain walls contributes nonlocal topological degeneracy ${2}^{{N}_{D}\ensuremath{-}1}$, causing a low-temperature peak in the heat capacity. We implement a nonperturbative method to bootstrap generic topological
1+1D domain walls (two-surface defects) applicable to any
2+1D non-Abelian topological order. We also identify potentially relevant spin topological quantum field theories (TQFTs) for various $\ensuremath{\nu}=5/2$ FQH states in terms of fermionic version of $U{(1)}_{\ifmmode\pm\else\textpm\fi{}8}$ Chern-Simons $\mathrm{theory}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$-class TQFTs.
In this thesis we investigate topological phases of matter that have a global, unbroken symmetry group -- also known as symmetry enriched topological (SET) phases. We address three questions about these phases: (1) how can we build exactly solvable models that realize them? (2) how can we determine if their edge theories can be gapped without breaking the symmetry? and (3) how do we understand the phenomenon of decoupled charge and neutral modes which occurs in certain fractional quantum Hall states? ,More specifically, we address the first question by constructing exactly solvable models for a wide class of symmetry enriched topological (SET) phases, which we call symmetry-enriched string nets. The construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group G, and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2 symmetry which exchanges the 'e' and 'm' type anyons. We further illustrate our construction with a number of additional examples. ,For the second question, we focus on the edge theories of 2D SET phases with Z2 symmetry. The central problem we seek to solve is to determine which edge theories can be gapped without breaking the symmetry. Previous attempts to answer this question in special cases relied on constructing perturbations of a particular type to gap the edge. This method proves the edge can be gapped when the appropriate perturbations can be found, but is inconclusive if they cannot be found. We build on this previous work by deriving a necessary and sufficient algebraic condition for when the edge can be gapped. Our results apply to Z2 symmetry protected topological phases as well as Abelian Z2 SET phases,Finally, in the fourth chapter, we describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode --- i.e. an edge mode that does not carry electric charge. These models consist of two counter-propagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the nu = 2/3 state is the most prominent example.
We describe solvable models that capture how impurity scattering in certain fractional quantum Hall edges can give rise to a neutral mode---i.e., an edge mode that does not carry electric charge. These models consist of two counterpropagating chiral Luttinger liquids together with a collection of discrete impurity scatterers. Our main result is an exact solution of these models in the limit of infinitely strong impurity scattering. From this solution, we explicitly derive the existence of a neutral mode and we determine all of its microscopic properties including its velocity. We also study the stability of the neutral mode and show that it survives at finite but sufficiently strong scattering. Our results are applicable to a family of Abelian fractional quantum Hall states of which the $\ensuremath{\nu}=2/3$ state is the most prominent example.