875 research outputs found
Renormalized transport properties of randomly gapped 2D Dirac fermions
We investigate the scaling properties of the recently acquired fermionic
non--linear --model which controls gapless diffusive modes in a
two--dimensional disordered system of Dirac electrons beyond charge neutrality.
The transport on large scales is governed by a novel renormalizable nonlocal
field theory. For zero mean random gap, it is characterized by the absence of a
dynamic gap generation and a scale invariant diffusion coefficient. The
function of the DC conductivity, computed for this model, is in perfect
agreement with numerical results obtained previously.Comment: Version published with minor change
Real Computational Universality: The Word Problem for a class of groups with infinite presentation
The word problem for discrete groups is well-known to be undecidable by a
Turing Machine; more precisely, it is reducible both to and from and thus
equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of the word problem
for a certain class of groups which are presented as quotient groups of a free
group and a normal subgroup. Most important, the free group will be generated
by an uncountable set of generators with index running over certain sets of
real numbers. This allows to include many mathematically important groups which
are not captured in the framework of the classical word problem.
Our contribution extends computational group theory from the discrete to the
Blum-Shub-Smale (BSS) model of real number computation. We believe this to be
an interesting step towards applying BSS theory, in addition to semi-algebraic
geometry, also to further areas of mathematics.
The main result establishes the word problem for such groups to be not only
semi-decidable (and thus reducible FROM) but also reducible TO the Halting
Problem for such machines. It thus provides the first non-trivial example of a
problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.
Quantum Hall effect induced by electron-phonon interaction
When phonons couple to fermions in 2D semimetals, the interaction may turn
the system into an insulator. There are several insulating phases in which the
time reversal and the sublattice symmetries are spontaneously broken. Examples
are many-body states commensurate to Haldane's staggered flux model or to
lattice models with periodically modulated strain. We find that the effective
field theories of these phases exhibit characteristic Chern-Simons terms, whose
coefficients are related to the topological invariants of the microscopic
model. This implies that the corresponding quantized Hall conductivities
characterize these insulating states.Comment: Accepted for publishing with Annals of Physics on April 30th, 202
Two-parameter scaling theory of transport near a spectral node
We investigate the finite-size scaling behavior of the conductivity in a
two-dimensional Dirac electron gas within a chiral sigma model. Based on the
fact that the conductivity is a function of system size times scattering rate,
we obtain a two-parameter scaling flow toward a finite fixed point. The latter
is the minimal conductivity of the infinite system. Depending on boundary
conditions, we also observe unstable fixed points with conductivities much
larger than the experimentally observed values, which may account for results
found in some numerical simulations. By including a spectral gap we extend our
scaling approach to describe a metal-insulator transition.Comment: 4.5 pages, 4 figures, published versio
Perturbative analysis of the conductivity in disordered monolayer and bilayer graphene
The DC conductivity of monolayer and bilayer graphene is studied
perturbatively for different types of disorder. In the case of monolayer, an
exact cancellation of logarithmic divergences occurs for all disorder types.
The total conductivity correction for a random vector potential is zero, while
for a random scalar potential and a random gap it acquires finite corrections.
We identify the diagrams which are responsible for these corrections and
extrapolate the finite contributions to higher orders which gives us general
expressions for the conductivity of weakly disordered monolayer graphene. In
the case of bilayer graphene, a cancellation of all contributions for all types
of disorder takes place. Thus, the minimal conductivity of bilayer graphene
turns out to be very robust against disorder.Comment: 4 pages, 2 figures + supplementary material. Final version as
published with PR
Emergent Chern-Simons excitations due to electron--phonon interaction
We address the problem of Dirac fermions interacting with longitudinal
phonons. A gap in the spectrum of fermions leads to the emergence of the
Chern--Simons excitations in the spectrum of phonons. We study the effect of
those excitations on observable quantities: the phonon dispersion, the phonon
spectral density, and the Hall conductivity.Comment: 9 pages, 4 figure
Spontaneous mass generation due to phonons in a two-dimensional Dirac fermion system
Fermions with one and two Dirac nodes are coupled to in-plane phonons to
study a spontaneous transition into the Hall insulating phase. At sufficiently
strong electron-phonon interaction a gap appears in the spectrum of fermions,
signaling a transition into a phase with spontaneously broken parity and
time-reversal symmetry. The structure of elementary excitations above the gap
in the corresponding phase reveals the presence of scale invariant parity
breaking terms which resemble Chern-Simons excitations. Evaluating the Kubo
formula for both models we find quantized Hall plateaux in each case, with
conductance of binodal model exactly twice as large as of the mononodal model
Effect of Coulomb interaction on the gap in monolayer and bilayer graphene
We study effects of a repulsive Coulomb interaction on the spectral gap in
monolayer and bilayer graphene in the vicinity of the charge neutrality point
by employing the functional renormalization-group technique. In both cases
Coulomb interaction supports the gap once it is open. For monolayer graphene we
correctly reproduce results obtained previously by several authors, e.g., an
apparent logarithmic divergence of the Fermi velocity and the gap as well as a
fixed point corresponding to a quantum phase transition at infinitely large
Coulomb interaction. On the other hand, we show that the gap introduces an
additional length scale at which renormalization flow of diverging quantities
saturates. An analogous analysis is also performed for bilayer graphene with
similar results. We find an additional fixed point in the gapless regime with
linear spectrum corresponding to the vanishing electronic band mass. This fixed
point is unstable with respect to gap fluctuations and can not be reached as
soon as the gap is opened. This preserves the quadratic scaling of the spectrum
and finite electronic band mass.Comment: 6 pages, 5 figures, final version to appear at PR
Conductivity of disordered 2d binodal Dirac electron gas: Effect of the internode scattering
We study the dc conductivity of a weakly disordered 2d Dirac electron gas
with two bands and two spectral nodes, employing a field theoretical version of
the Kubo--Greenwood conductivity formula. In this paper we are concerned with
the question how the internode scattering affects the conductivity. We use and
compare two established techniques for treating the disorder scattering: The
perturbation theory, there ladder and maximally crossed diagrams are summed up,
and the functional integral approach. Both turn out to be entirely equivalent.
For a large number of random potential configurations we have found only two
different conductivity scenarios. Both scenarios appear independently of
whether the disorder does or does not create the internode scattering. In
particular we do not confirm the conjecture that the internode scattering tends
to Anderson localization
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