44 research outputs found
KPZ modes in -dimensional directed polymers
We define a stochastic lattice model for a fluctuating directed polymer in
dimensions. This model can be alternatively interpreted as a
fluctuating random path in 2 dimensions, or a one-dimensional asymmetric simple
exclusion process with conserved species of particles. The deterministic
large dynamics of the directed polymer are shown to be given by a system of
coupled Kardar-Parisi-Zhang (KPZ) equations and diffusion equations. Using
non-linear fluctuating hydrodynamics and mode coupling theory we argue that
stationary fluctuations in any dimension can only be of KPZ type or
diffusive. The modes are pure in the sense that there are only subleading
couplings to other modes, thus excluding the occurrence of modified
KPZ-fluctuations or L\'evy-type fluctuations which are common for more than one
conservation law. The mode-coupling matrices are shown to satisfy the so-called
trilinear condition.Comment: 22 pages, 2 figure
The geometry of the double gyroid wire network: quantum and classical
Quantum wire networks have recently become of great interest. Here we deal
with a novel nano material structure of a Double Gyroid wire network. We use
methods of commutative and non-commutative geometry to describe this wire
network. Its non--commutative geometry is closely related to non-commutative
3-tori as we discuss in detail.Comment: pdflatex 9 Figures. Minor changes, some typos and formulation
Re-gauging groupoid, symmetries and degeneracies for graph Hamiltonians and applications to the Gyroid wire network
We study a class of graph Hamiltonians given by a type of quiver representation to which we can associate (non)-commutative geometries. By selecting gauging data, these geometries are realized by matrices through an explicit construction or a Kan extension. We describe the changes in gauge via the action of a re-gauging groupoid. It acts via matrices that give rise to a noncommutative 2-cocycle and hence to a groupoid extension (gerbe). We furthermore show that automorphisms of the underlying graph of the quiver can be lifted to extended symmetry groups of re-gaugings. In the commutative case, we deduce that the extended symmetries act via a projective representation. This yields isotypical decompositions and super-selection rules. We apply these results to the primitive cubic, diamond, gyroid and honeycomb wire networks using representation theory for projective groups and show that all the degeneracies in the spectra are consequences of these enhanced symmetries. This includes the Dirac points of the G(yroid) and the honeycomb systems
Local models and global constraints for degeneracies and band crossings
We study topological properties of families of Hamiltonians which may contain degenerate energy levels aka. band crossings. The primary tool are Chern classes, Berry phases and slicing by surfaces. To analyse the degenerate locus, we study local models. These give information about the Chern classes and Berry phases. We then give global constraints for the topological invariants. This is an hitherto relatively unexplored subject. The global constraints are more strict when incorporating symmetries such as time reversal symmetries. The results can also be used in the study of deformations. We furthermore use these constraints to analyse examples which include the Gyroid geometry, which exhibits Weyl points and triple crossings and the honeycomb geometry with its two Dirac points
Thermodynamics of the Complex su(3) Toda Theory
We present the first computation of the thermodynamic properties of the
complex su(3) Toda theory. This is possible thanks to a new string hypothesis,
which involves bound states that are non self-conjugate solutions of the Bethe
equations. Our method provides equivalently the solution of the su(3)
generalization of the XXZ chain. In the repulsive regime, we confirm that the
scattering theory proposed over the past few years - made only of solitons with
non diagonal S-matrices - is complete. But we show that unitarity does not
follow, contrary to early claims, eigenvalues of the monodromy matrix not being
pure phases. In the attractive regime, we find that the proposed minimal
solution of the bootstrap equations is actually far from being complete. We
discuss some simple values of the couplings, where, instead of the few
conjectured breathers, a very complex structure (involving E_6, or two E_8) of
bound states is necessary to close the bootstrap.Comment: 6 pages, 2 figures; some minor changes; accepted for publication in
Phys. Lett.
Asymmetric XXZ chain at the antiferromagnetic transition: Spectra and partition functions
The Bethe ansatz equation is solved to obtain analytically the leading
finite-size correction of the spectra of the asymmetric XXZ chain and the
accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary
at zero vertical field. The energy gaps scale with size as and
its amplitudes are obtained in terms of level-dependent scaling functions.
Exactly on the phase boundary, the amplitudes are proportional to a sum of
square-root of integers and an anomaly term. By summing over all low-lying
levels, the partition functions are obtained explicitly. Similar analysis is
performed also at the phase boundary of zero horizontal field in which case the
energy gaps scale as . The partition functions for this case are found
to be that of a nonrelativistic free fermion system. From symmetry of the
lattice model under rotation, several identities between the partition
functions are found. The scaling at zero vertical field is
interpreted as a feature arising from viewing the Pokrovsky-Talapov transition
with the space and time coordinates interchanged.Comment: Minor corrections only. 18 pages in RevTex, 2 PS figure
Finite size scaling for quantum criticality using the finite-element method
Finite size scaling for the Schr\"{o}dinger equation is a systematic approach
to calculate the quantum critical parameters for a given Hamiltonian. This
approach has been shown to give very accurate results for critical parameters
by using a systematic expansion with global basis-type functions. Recently, the
finite element method was shown to be a powerful numerical method for ab initio
electronic structure calculations with a variable real-space resolution. In
this work, we demonstrate how to obtain quantum critical parameters by
combining the finite element method (FEM) with finite size scaling (FSS) using
different ab initio approximations and exact formulations. The critical
parameters could be atomic nuclear charges, internuclear distances, electron
density, disorder, lattice structure, and external fields for stability of
atomic, molecular systems and quantum phase transitions of extended systems. To
illustrate the effectiveness of this approach we provide detailed calculations
of applying FEM to approximate solutions for the two-electron atom with varying
nuclear charge; these include Hartree-Fock, density functional theory under the
local density approximation, and an "exact"' formulation using FEM. We then use
the FSS approach to determine its critical nuclear charge for stability; here,
the size of the system is related to the number of elements used in the
calculations. Results prove to be in good agreement with previous Slater-basis
set calculations and demonstrate that it is possible to combine finite size
scaling with the finite-element method by using ab initio calculations to
obtain quantum critical parameters. The combined approach provides a promising
first-principles approach to describe quantum phase transitions for materials
and extended systems.Comment: 15 pages, 19 figures, revision based on suggestions by referee,
accepted in Phys. Rev.
Complete Exact Solution of Diffusion-Limited Coalescence, A + A -> A
Some models of diffusion-limited reaction processes in one dimension lend
themselves to exact analysis. The known approaches yield exact expressions for
a limited number of quantities of interest, such as the particle concentration,
or the distribution of distances between nearest particles. However, a full
characterization of a particle system is only provided by the infinite
hierarchy of multiple-point density correlation functions. We derive an exact
description of the full hierarchy of correlation functions for the
diffusion-limited irreversible coalescence process A + A -> A.Comment: 4 pages, 2 figures (postscript). Typeset with Revte