Exploring the fractional quantum Hall effect with electron tunneling

In this talk I present a summary of recent work on tunnel junctions of a fractional quantum Hall fluid and an electron reservoir, a Fermi liquid. I consider first the case of a single point contact. This is a an exactly solvable problem from which much can be learned. I also discuss in some detail how these solvable junction problems can be used to understand many aspects of the recent electron tunneling experiments into edge states. I also give a detailed picture of the unusual behavior of these junctions in their strong coupling regime. A pedagogical introduction to the theories of edge states is also included.Comment: Invited talk at the XXXIVth Rencontres de Moriond, Condensed Matter Physics Meeting {\sl Quantum Physics at the Mesoscopic Scale}, Les Arcs, Haute Savoie, France, January 1999. 20 pages, 17 figure

Boson-fermion duality in a gravitational background

We study the $2+1$ dimensional boson-fermion duality in the presence of background curvature and electromagnetic fields. The main players are, on the one hand, a massive complex $|\phi|^4$ scalar field coupled to a $U(1)$ Maxwell-Chern-Simons gauge field at level $1$, representing a relativistic composite boson with one unit of attached flux, and on the other hand, a massive Dirac fermion. We show that, in a curved background and at the level of the partition function, the relativistic composite boson, in the infinite coupling limit, is dual to a short-range interacting Dirac fermion. The coupling to the gravitational spin connection arises naturally from the spin factors of the Wilson loop in the Chern-Simons theory. A non-minimal coupling to the scalar curvature is included on the bosonic side in order to obtain agreement between partition functions. Although an explicit Lagrangian expression for the fermionic interactions is not obtained, their short-range nature constrains them to be irrelevant, which protects the duality in its strong interpretation as an exact mapping at the IR fixed point between a Wilson-Fischer-Chern-Simons complex scalar and a free Dirac fermion. We also show that, even away from the IR, keeping the $|\phi|^4$ term is of key importance as it provides the short-range bosonic interactions necessary to prevent intersections of worldlines in the path integral, thus forbidding unknotting of knots and ensuring preservation of the worldline topologies.Comment: Final version published in Annals of Physic

Pair-Density-Wave Superconducting Order in Two-Leg Ladders

We show using bosonization methods that extended Hubbard-Heisenberg models on two types of two leg ladders (without flux and with flux $\pi$ per plaquette) have commensurate pair-density wave (PDW) phases. In the case of the conventional (flux-less) ladder the PDW arises when certain filling fractions for which commensurability conditions are met. For the flux $\pi$ ladder the PDW phase is generally present. The PDW phase is characterized by a finite spin gap and a superconducting order parameter with a finite (commensurate in this case) wave vector and power-law superconducting correlations. In this phase the uniform superconducting order parameter, the $2k_F$ charge-density-wave (CDW) order parameter and the spin-density- wave N\'eel order parameter exhibit short range (exponentially decaying) correlations. We discuss in detail the case in which the bonding band of the ladder is half filled for which the PDW phase appears even at weak coupling. The PDW phase is shown to be dual to a uniform superconducting (SC) phase with quasi long range order. By making use of bosonization and the renormalization group we determine the phase diagram of the spin-gapped regime and study the quantum phase transition. The phase boundary between PDW and the uniform SC ordered phases is found to be in the Ising universality class. We generalize the analysis to the case of other commensurate fillings of the bonding band, where we find higher order commensurate PDW states for which we determine the form of the effective bosonized field theory and discuss the phase diagram. We compare our results with recent findings in the Kondo-Heisenberg chain. We show that the formation of PDW order in the ladder embodies the notion of intertwined orders.Comment: 21 pages, 4 figures (one with two subfigures), revised text, 5 new references; total of 49 reference

Fermionic Chern-Simons Field Theory for the Fractional Hall Effect

We review the fermionic Chern-Simons field theory for the Fractional Quantum Hall Effect (FQHE). We show that in this field theoretic approach to the problem of interacting electrons moving in a plane in the presence of an external magnetic field, the FQHE states appear naturally as the semiclassical states of the theory. In this framework, the FQHE states are the unique ground states of a system of electrons on a fixed geometry. The excitation spectrum is fully gapped and these states can be viewed as infrared stable fixed points of the system. It is shown that the long distance, low energy properties of the system are described exactly by this theory. It is further shown that, in this limit, the actual ground state wave function has the Laughlin form. We also discuss the application of this theory to the problem of the FQHE in bilayers and in unpolarized single layer systems.Comment: To appear in ``Composite Fermions in the Quantum Hall Effect", edited by Olle Heinonen. 57 page

Dirac Composite Fermions and Emergent Reflection Symmetry about Even Denominator Filling Fractions

Motivated by the appearance of a `reflection symmetry' in transport experiments and the absence of statistical periodicity in relativistic quantum field theories, we propose a series of relativistic composite fermion theories for the compressible states appearing at filling fractions $\nu=1/2n$ in quantum Hall systems. These theories consist of electrically neutral Dirac fermions attached to $2n$ flux quanta via an emergent Chern-Simons gauge field. While not possessing an explicit particle-hole symmetry, these theories reproduce the known Jain sequence states proximate to $\nu=1/2n$, and we show that such states can be related by the observed reflection symmetry, at least at mean field level. We further argue that the lowest Landau level limit requires that the Dirac fermions be tuned to criticality, whether or not this symmetry extends to the compressible states themselves.Comment: 25 pages, minor revision

Scrambling in the Quantum Lifshitz Model

We study signatures of chaos in the quantum Lifshitz model through out-of-time ordered correlators (OTOC) of current operators. This model is a free scalar field theory with dynamical critical exponent $z=2$. It describes the quantum phase transition in 2D systems, such as quantum dimer models, between a phase with an uniform ground state to another one with a spontaneously translation invariance. At the lowest temperatures the chaotic dynamics are dominated by a marginally irrelevant operator which induces a temperature dependent stiffness term. The numerical computations of OTOC exhibit a non-zero Lyapunov exponent (LE) in a wide range of temperatures and interaction strengths. The LE (in units of temperature) is a weakly temperature-dependent function; it vanishes at weak interaction and saturates for strong interaction. The Butterfly velocity increases monotonically with interaction strength in the studied region while remaining smaller than the interaction-induced velocity/stiffness.Comment: 15 pages + appendices. 12 figure

Effective field theory for the bulk and edge states of quantum Hall states in unpolarized single layer and bilayer systems

We present an effective theory for the bulk Fractional Quantum Hall states in spin-polarized bilayer and spin-1/2 single layer two-dimensional electron gases (2DEG) in high magnetic fields consistent with the requirement of global gauge invariance on systems with periodic boundary conditions. We derive the theory for the edge states that follows naturally from this bulk theory. We find that the minimal effective theory contains two propagating edge modes that carry charge and energy, and two non-propagating topological modes responsible for the statistics of the excitations. We give a detailed description of the effective theory for the spin-singlet states, the symmetric bilayer states and for the $(m,m,m)$ states. We calculate explicitly, for a number of cases of interest, the operators that create the elementary excitations, their bound states, and the electron. We also discuss the scaling behavior of the tunneling conductances in different situations: internal tunneling, tunneling between identical edges and tunneling into a FQH state from a Fermi liquid.Comment: 27 pages; new subsection with summary of results and two tables. Misprints and errors of an an earlier version are corrected. In particular the tunneling exponents for the SU(2) states at 2/3 and 4/7 have been corrected; same with the electron operator for the 2/3 stat

Loop Models, Modular Invariance, and Three Dimensional Bosonization

We consider a family of quantum loop models in 2+1 spacetime dimensions with marginally long-ranged and statistical interactions mediated by a U$(1)$ gauge field, both purely in 2+1 dimensions and on a surface in a 3+1 dimensional bulk system. In the absence of fractional spin, these theories have been shown to be self-dual under particle-vortex duality and shifts of the statistical angle of the loops by $2\pi$, which form a subgroup of the modular group, PSL$(2,\mathbb{Z})$. We show that careful consideration of fractional spin in these theories completely breaks their statistical periodicity and describe how this occurs, resolving a disagreement with the conformal field theories they appear to approach at criticality. We show explicitly that incorporation of fractional spin leads to loop model dualities which parallel the recent web of 2+1 dimensional field theory dualities, providing a nontrivial check on its validity.Comment: 41 pages, including two appendice

Field Theory of Nematicity in the Spontaneous Quantum Anomalous Hall effect

We derive from a microscopic model the effective theory of nematic order in a system with a spontaneous quantum anomalous Hall effect in two dimensions. Starting with a model of two-component fermions (a spinor field) with a quadratic band crossing and short range four-fermion marginally relevant interactions we use a 1/N expansion and bosonization methods to derive the effective field theory for the hydrodynamic modes associated with the conserved currents and with the local fluctuations of the nematic order parameter. We focus on the vicinity of the quantum phase transition from the isotropic Mott Chern insulating phase to a phase in which time-reversal symmetry breaking coexists with nematic order, the nematic Chern insulator. The topological sector of the effective field theory is a BF/Chern-Simons gauge theory. We show that the nematic order parameter field couples with the Maxwell-type terms of the gauge fields as the space components of a locally fluctuating metric tensor. The nematic field has $z=2$ dynamic scaling exponent. The low energy dynamics of the nematic order parameter is found to be governed by a Berry phase term. By means of a detailed analysis of the coupling of the spinor field of the fermions to the changes of their local frames originating from long-wavelength lattice deformations we calculate the Hall viscosity of this system and show that in this system is not the same as the Berry phase term in the effective action of the nematic field, but both are related to the concept of torque Hall viscosity which we introduce here.Comment: mildly edited version, one new reference; version to be published in Physical Review B; 22 pages, 3 figures (two with 2 subfigures each), 90 reference

Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum

The entanglement entropy of a pure quantum state of a bipartite system $A \cup B$ is defined as the von Neumann entropy of the reduced density matrix obtained by tracing over one of the two parts. Critical ground states of local Hamiltonians in one dimension have entanglement that diverges logarithmically in the subsystem size, with a universal coefficient that for conformally invariant critical points is related to the central charge of the conformal field theory. We find the entanglement entropy for a standard class of $z=2$ quantum critical points in two spatial dimensions with scale invariant ground state wave functions: in addition to a nonuniversal ``area law'' contribution proportional to the size of the $AB$ boundary, there is generically a universal logarithmically divergent correction. This logarithmic term is completely determined by the geometry of the partition into subsystems and the central charge of the field theory that describes the equal-time correlations of the critical wavefunction.Comment: 4 pages, 1 figure, 28 reference