258 research outputs found

### 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

### 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

### 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

### 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

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