166 research outputs found
Surface Properties of Aperiodic Ising Quantum Chains
We consider Ising quantum chains with quenched aperiodic disorder of the
coupling constants given through general substitution rules. The critical
scaling behaviour of several bulk and surface quantities is obtained by exact
real space renormalization.Comment: 4 pages, RevTex, reference update
Finite-lattice expansion for Ising models on quasiperiodic tilings
Low-temperature series are calculated for the free energy, magnetisation,
susceptibility and field-derivatives of the susceptibility in the Ising model
on the quasiperiodic Penrose lattice. The series are computed to order 20 and
estimates of the critical exponents alpha, beta and gamma are obtained from
Pade approximants.Comment: 16 pages, REVTeX, 26 postscript figure
Aperiodic Ising Quantum Chains
Some years ago, Luck proposed a relevance criterion for the effect of
aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In
this article, we show how Luck's criterion can be derived within an exact
renormalisation scheme for Ising quantum chains with coupling constants
modulated according to substitution rules. Luck's conjectures for this case are
confirmed and refined. Among other outcomes, we give an exact formula for the
correlation length critical exponent for arbitrary two-letter substitution
sequences with marginal fluctuations of the coupling constants.Comment: 27 pages, LaTeX, 1 Postscript figure included, using epsf.sty and
amssymb.sty (one error corrected, some minor changes
Anderson Localization, Non-linearity and Stable Genetic Diversity
In many models of genotypic evolution, the vector of genotype populations
satisfies a system of linear ordinary differential equations. This system of
equations models a competition between differential replication rates (fitness)
and mutation. Mutation operates as a generalized diffusion process on genotype
space. In the large time asymptotics, the replication term tends to produce a
single dominant quasispecies, unless the mutation rate is too high, in which
case the populations of different genotypes becomes de-localized. We introduce
a more macroscopic picture of genotypic evolution wherein a random replication
term in the linear model displays features analogous to Anderson localization.
When coupled with non-linearities that limit the population of any given
genotype, we obtain a model whose large time asymptotics display stable
genotypic diversityComment: 25 pages, 8 Figure
Real Space Renormalization Group Study of the S=1/2 XXZ Chains with Fibonacci Exchange Modulation
Ground state properties of the S=1/2 antiferromagnetic XXZ chain with
Fibonacci exchange modulation are studied using the real space renormalization
group method for strong modulation. The quantum dynamical critical behavior
with a new universality class is predicted in the isotropic case. Combining our
results with the weak coupling renormalization group results by Vidal et al.,
the ground state phase diagram is obtained.Comment: 9 pages, 9 figure
Quasiperiodic Hubbard chains
Low energy properties of half-filled Fibonacci Hubbard models are studied by
weak coupling renormalization group and density matrix renormalization group
method. In the case of diagonal modulation, weak Coulomb repulsion is
irrelevant and the system behaves as a free Fibonacci chain, while for strong
Coulomb repulsion, the charge sector is a Mott insulator and the spin sector
behaves as a uniform Heisenberg antiferromagnetic chain. The off-diagonal
modulation always drives the charge sector to a Mott insulator and the spin
sector to a Fibonacci antiferromagnetic Heisenberg chain.Comment: 4 pages, 4 figures; Final version to appear in Phys. Rev. Let
Mutation, selection, and ancestry in branching models: a variational approach
We consider the evolution of populations under the joint action of mutation
and differential reproduction, or selection. The population is modelled as a
finite-type Markov branching process in continuous time, and the associated
genealogical tree is viewed both in the forward and the backward direction of
time. The stationary type distribution of the reversed process, the so-called
ancestral distribution, turns out as a key for the study of mutation-selection
balance. This balance can be expressed in the form of a variational principle
that quantifies the respective roles of reproduction and mutation for any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth rate, as
given by the difference between the current mean reproduction rate, and an
asymptotic decay rate related to the mutation process; this tradeoff is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence evolution with
mutation coupled to reproduction but independent across sites, and a fitness
function that is invariant under permutation of sites. Here, the variational
principle is worked out in detail and yields a simple, explicit result.Comment: 45 pages,8 figure
Bosonization of interacting fermions in arbitrary dimension beyond the Gaussian approximation
We use our recently developed functional bosonization approach to bosonize
interacting fermions in arbitrary dimension beyond the Gaussian
approximation. Even in the finite curvature of the energy dispersion at
the Fermi surface gives rise to interactions between the bosons. In higher
dimensions scattering processes describing momentum transfer between different
patches on the Fermi surface (around-the-corner processes) are an additional
source for corrections to the Gaussian approximation. We derive an explicit
expression for the leading correction to the bosonized Hamiltonian and the
irreducible self-energy of the bosonic propagator that takes the finite
curvature as well as around-the-corner processes into account. In the special
case that around-the-corner scattering is negligible, we show that the
self-energy correction to the Gaussian propagator is negligible if the
dimensionless quantities are
small compared with unity for all patches . Here is the cutoff
of the interaction in wave-vector space, is the Fermi wave-vector,
is the chemical potential, is the usual dimensionless Landau
interaction-parameter, and is the {\it{local}} density of
states associated with patch . We also show that the well known
cancellation between vertex- and self-energy corrections in one-dimensional
systems, which is responsible for the fact that the random-phase approximation
for the density-density correlation function is exact in , exists also in
, provided (1) the interaction cutoff is small compared with
, and (2) the energy dispersion is locally linearized at the Fermi the
Fermi surface. Finally, we suggest a new systematic method to calculate
corrections to the RPA, which is based on the perturbative calculation of the
irreducible bosonic self-energy arising from the non-Gaussian terms of the
bosonized Hamiltonian.Comment: The abstract has been rewritten. No major changes in the text
Random Tilings: Concepts and Examples
We introduce a concept for random tilings which, comprising the conventional
one, is also applicable to tiling ensembles without height representation. In
particular, we focus on the random tiling entropy as a function of the tile
densities. In this context, and under rather mild assumptions, we prove a
generalization of the first random tiling hypothesis which connects the maximum
of the entropy with the symmetry of the ensemble. Explicit examples are
obtained through the re-interpretation of several exactly solvable models. This
also leads to a counterexample to the analogue of the second random tiling
hypothesis about the form of the entropy function near its maximum.Comment: 32 pages, 42 eps-figures, Latex2e updated version, minor grammatical
change
Anomalous Diffusion in Aperiodic Environments
We study the Brownian motion of a classical particle in one-dimensional
inhomogeneous environments where the transition probabilities follow
quasiperiodic or aperiodic distributions. Exploiting an exact correspondence
with the transverse-field Ising model with inhomogeneous couplings we obtain
many new analytical results for the random walk problem. In the absence of
global bias the qualitative behavior of the diffusive motion of the particle
and the corresponding persistence probability strongly depend on the
fluctuation properties of the environment. In environments with bounded
fluctuations the particle shows normal diffusive motion and the diffusion
constant is simply related to the persistence probability. On the other hand in
a medium with unbounded fluctuations the diffusion is ultra-slow, the
displacement of the particle grows on logarithmic time scales. For the
borderline situation with marginal fluctuations both the diffusion exponent and
the persistence exponent are continuously varying functions of the
aperiodicity. Extensions of the results to disordered media and to higher
dimensions are also discussed.Comment: 11 pages, RevTe
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