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.
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 authors consider here the behavior of quantum Hall edges away from the Luttinger liquid fixed point that describes the low energy, large system limit. Using the links between quantum Hall wave functions and conformal field theories, they construct effective Hamiltonians and constrain their forms by considering bulk symmetries. In examining the effect of bulk interactions on this edge, they find remarkable simplifications to these effective theories that allow a very accurate description of the low-energy physics of quantum Hall edges relatively far away from the Luttinger liquid fixed point, and that apply to small systems and higher energies.
The effects of impurity scattering on a general Abelian fractional quantum Hall (FQH) edge state are analyzed within the chiral-Luttinger-liquid model of low-energy edge dynamics. We find that some disordered edges can have several different phases characterized by different symmetries. The stable impurity edge phases are in general more symmetric than the original clean system and demonstrate the phenomenon of dynamical symmetry restoration at low energies and long length scales. The phase transitions between different disordered phases are characterized by broken symmetries and obey Landau's symmetry-breaking principle for continuous phase transitions. Phase diagrams for various edges are found using a system of coordinates for the interactions between modes in a quantum Hall edge. The temperature dependence of tunneling through a point contact is calculated and is found to be able to distinguish different impurity edge phases of the same FQH state.
We consider a series of problems regarding quantum Hall edges, focusing on both dynamics and the mathematical structure of edge states. We begin in Chapter 3 with a limiting case of the Laughlin state placed in a very steep confining potential, but which is weak compared to the interactions. We find that the eigenstates have a Jack polynomial structure and an energy spectrum which is extremely different from the well-known Luttinger liquid edge. In Chapter 5 we analyse the inner products of edge state wavefunctions, using an effective description given by a large-N expansion ansatz proposed by J. Dubail, N. Read and E. Rezayi, PRB 86, 245310 (2012). As noted by these authors, the terms in this ansatz can be constrained using symmetry, a procedure we perform to high orders. We then check the conjecture by calculating overlaps exactly for small system sizes and comparing the numerics with our high-order expansion to find excellent agreement. Finally, Chapter 6 considers the behaviour of quantum Hall edges close to the Luttinger liquid fixed point that occurs in the low energy, large system limit. We construct effective Hamiltonians using a local field theory description and then consider the effect of bulk symmetries on this edge. The symmetry analysis produces remarkable simplifications which allow for very accurate descriptions of the low-energy edge physics even relatively far away from the Luttinger liquid fixed point.
We investigate the homogeneous chiral edge theory of the filling ν=4/3 fractional quantum Hall state, which is parameterized by a Luttinger liquid velocity matrix and an electron tunneling amplitude (ignoring irrelevant terms). We identify two solvable cases: one case where the theory gives two free chiral boson modes, and the other case where the theory yields one free charge 2e3 chiral fermion and two free chiral Bogoliubov (Majorana) fermions. For generic parameters, the energy spectrum from our exact diagonalization shows Poisson level spacing statistics (LSS) in each conserved charge and momentum sector, indicating the existence of hidden conserved quantities and the possibility that the generic edge theory of the ν=4/3 fractional quantum Hall state is integrable. We further show that a global symmetry preserving irrelevant nonlinear kinetic term will lead to the transition of LSS from Poisson to Wigner-Dyson at high energies. This further supports the possibility that the model without irrelevant terms is integrable.
We study a quantum point contact in a quantum spin Hall insulator. It has recently been shown that the Luttinger liquid theory of such a structure maps to the theory of a weak link in a Luttinger liquid with spin with Luttinger liquid parameters ${g}_{\ensuremath{\rho}}=1/{g}_{\ensuremath{\sigma}}=g<1$. We show that for weak interactions, 1/2<g<1, the pinch-off of the point contact as a function of gate voltage is controlled by a novel quantum critical point, which is a realization of a nontrivial intermediate fixed point found previously in the Luttinger liquid model with spin. We predict that the dependence of the conductance on gate voltage near the pinch-off transition for different temperatures collapses onto a universal curve described by a crossover scaling function associated with that fixed point. We compute the conductance and critical exponents of the critical point as well as the universal scaling function in solvable limits, which include $g=1\ensuremath{-}ϵ$, g=1/2+ϵ, and g=1/3. These results, along with a general scaling analysis, provide an overall picture of the critical behavior as a function of g. In addition, we analyze the structure of the four-terminal conductance of the point contact in the weak tunneling and weak backscattering limits. We find that different components of the conductance can have different temperature dependences. In particular, we identify a skew conductance GXY, which we predict vanishes as ${T}^{\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\ge}2$. This behavior is a direct consequence of the unique edge state structure of the quantum spin Hall insulator. Finally, we show that for strong interactions, g<1/2, the presence of spin nonconserving spin-orbit interactions leads to a novel time-reversal-symmetry breaking insulating phase. In this phase, the transport is carried by spinless chargons and chargeless spinons. These lead to nontrivial correlations in the low frequency shot noise. Implications for experiments on HgCdTe quantum well structures will be discussed.
We study the four-terminal junction of spinless Luttinger liquid wires, which describes either a corner junction of two helical edge states of topological insulators or the tunneling from the spinful wire into the helical edge state. We use the fermionic representation and the scattering state formalism, in order to compute the renormalization group (RG) equations for the linear response conductances. We establish our approach by considering a junction between two possibly nonequivalent helical edge states and find an agreement with the earlier analysis of this situation. Tunneling from the tip of the spinful wire to the edge state is further analyzed which requires some modification of our formalism. In the latter case we demonstrate (i) the existence of both fixed lines and conventional fixed points of RG equations, and (ii) certain proportionality relations holding for conductances during renormalization. The scaling exponents and phase portraits are obtained in all cases.