5,377 research outputs found
Quantum fluctuations and glassy behavior: The case of a quantum particle in a random potential
In this paper we expand our previous investigation of a quantum particle
subject to the action of a random potential plus a fixed harmonic potential at
a finite temperature T. In the classical limit the system reduces to a
well-known ``toy'' model for an interface in a random medium. It also applies
to a single quantum particle like an an electron subject to random
interactions, where the harmonic potential can be tuned to mimic the effect of
a finite box. Using the variational approximation, or alternatively, the limit
of large spatial dimensions, together with the use the replica method, and are
able to solve the model and obtain its phase diagram in the
plane, where is the particle's mass. The phase diagram is similar to that
of a quantum spin-glass in a transverse field, where the variable
plays the role of the transverse field. The glassy phase is characterized by
replica-symmetry-breaking. The quantum transition at zero temperature is also
discussed.Comment: revised version, 23 pages, revtex, 5 postscript figures in a separate
file figures.u
Quantum Monte Carlo simulations of a particle in a random potential
In this paper we carry out Quantum Monte Carlo simulations of a quantum
particle in a one-dimensional random potential (plus a fixed harmonic
potential) at a finite temperature. This is the simplest model of an interface
in a disordered medium and may also pertain to an electron in a dirty metal. We
compare with previous analytical results, and also derive an expression for the
sample to sample fluctuations of the mean square displacement from the origin
which is a measure of the glassiness of the system. This quantity as well as
the mean square displacement of the particle are measured in the simulation.
The similarity to the quantum spin glass in a transverse field is noted. The
effect of quantum fluctuations on the glassy behavior is discussed.Comment: 23 pages, 7 figures included as eps files, uses RevTeX. Accepted for
publication in J. of Physics A: Mathematical and Genera
Directed polymers on a Cayley tree with spatially correlated disorder
In this paper we consider directed walks on a tree with a fixed branching
ratio K at a finite temperature T. We consider the case where each site (or
link) is assigned a random energy uncorrelated in time, but correlated in the
transverse direction i.e. within the shell. In this paper we take the
transverse distance to be the hierarchical ultrametric distance, but other
possibilities are discussed. We compute the free energy for the case of
quenched disorder and show that there is a fundamental difference between the
case of short range spatial correlations of the disorder which behaves
similarly to the non-correlated case considered previously by Derrida and Spohn
and the case of long range correlations which has a totally different overlap
distribution which approaches a single delta function about q=1 for large L,
where L is the length of the walk. In the latter case the free energy is not
extensive in L for the intermediate and also relevant range of L values,
although in the true thermodynamic limit extensivity is restored. We identify a
crossover temperature which grows with L, and whenever T<T_c(L) the system is
always in the low temperature phase. Thus in the case of long-ranged
correlation as opposed to the short-ranged case a phase transition is absent.Comment: 23 pages, 1 figure, standard latex. J. Phys. A, accepted for
publicatio
Replica field theory for a polymer in random media
In this paper we revisit the problem of a (non self-avoiding) polymer chain
in a random medium which was previously investigated by Edwards and Muthukumar
(EM). As noticed by Cates and Ball (CB) there is a discrepancy between the
predictions of the replica calculation of EM and the expectation that in an
infinite medium the quenched and annealed results should coincide (for a chain
that is free to move) and a long polymer should always collapse. CB argued that
only in a finite volume one might see a ``localization transition'' (or
crossover) from a stretched to a collapsed chain in three spatial dimensions.
Here we carry out the replica calculation in the presence of an additional
confining harmonic potential that mimics the effect of a finite volume. Using a
variational scheme with five variational parameters we derive analytically for
d<4 the result R~(g |ln \mu|)^{-1/(4-d)} ~(g lnV)^{-1/(4-d)}, where R is the
radius of gyration, g is the strength of the disorder, \mu is the spring
constant associated with the confining potential and V is the associated
effective volume of the system. Thus the EM result is recovered with their
constant replaced by ln(V) as argued by CB. We see that in the strict infinite
volume limit the polymer always collapses, but for finite volume a transition
from a stretched to a collapsed form might be observed as a function of the
strength of the disorder. For d<2 and for large
V>V'~exp[g^(2/(2-d))L^((4-d)/(2-d))] the annealed results are recovered and
R~(Lg)^(1/(d-2)), where L is the length of the polymer. Hence the polymer also
collapses in the large L limit. The 1-step replica symmetry breaking solution
is crucial for obtaining the above results.Comment: Revtex, 32 page
Localization of a polymer in random media: Relation to the localization of a quantum particle
In this paper we consider in detail the connection between the problem of a
polymer in a random medium and that of a quantum particle in a random
potential. We are interested in a system of finite volume where the polymer is
known to be {\it localized} inside a low minimum of the potential. We show how
the end-to-end distance of a polymer which is free to move can be obtained from
the density of states of the quantum particle using extreme value statistics.
We give a physical interpretation to the recently discovered one-step
replica-symmetry-breaking solution for the polymer (Phys. Rev. E{\bf 61}, 1729
(2000)) in terms of the statistics of localized tail states. Numerical
solutions of the variational equations for chains of different length are
performed and compared with quenched averages computed directly by using the
eigenfunctions and eigenenergies of the Schr\"odinger equation for a particle
in a one-dimensional random potential. The quantities investigated are the
radius of gyration of a free gaussian chain, its mean square distance from the
origin and the end-to-end distance of a tethered chain. The probability
distribution for the position of the chain is also investigated. The glassiness
of the system is explained and is estimated from the variance of the measured
quantities.Comment: RevTex, 44 pages, 13 figure
About Boundary Terms in Higher Order Theories
It is shown that when in a higher order variational principle one fixes
fields at the boundary leaving the field derivatives unconstrained, then the
variational principle (in particular the solution space) is not invariant with
respect to the addition of boundary terms to the action, as it happens instead
when the correct procedure is applied. Examples are considered to show how
leaving derivatives of fields unconstrained affects the physical interpretation
of the model. This is justified in particularl by the need of clarifying the
issue for the purpose of applications to relativistic gravitational theories,
where a bit of confusion still exists. On the contrary, as it is well known for
variational principles of order k, if one fixes variables together with their
derivatives (up to order k-1) on the boundary then boundary terms leave
solution space invariant.Comment: 7 page
Differential systems associated with tableaux over Lie algebras
We give an account of the construction of exterior differential systems based
on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom
14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie
algebra is revisited and extended in the light of the formalism of the Spencer
cohomology; the question of involutiveness for the associated systems and their
prolongations is addressed; examples are discussed.Comment: 16 pages; to appear in: "Symmetries and Overdetermined Systems of
Partial Differential Equations" (M. Eastwood and W. Miller, Jr., eds.), IMA
Volumes in Mathematics and Its Applications, Springer-Verlag, New Yor
Counting Giant Gravitons in AdS_3
We quantize the set of all quarter BPS brane probe solutions in global AdS_3
\times S^3 \times T^4/K3 found in arxiv:0709.1168 [hep-th]. We show that,
generically, these solutions give rise to states in discrete representations of
the SL(2,R) WZW model on AdS_3. Our procedure provides us with a detailed
description of the low energy 1/4 and 1/2 BPS sectors of string theory on this
background. The 1/4 BPS partition function jumps as we move off the point in
moduli space where the bulk theta angle and NS-NS fields vanish. We show that
generic 1/2 BPS states are protected because they correspond to geodesics
rather than puffed up branes. By exactly quantizing the simplest of the probes
above, we verify our description of 1/4 BPS states and find agreement with the
known spectrum of 1/2 BPS states of the boundary theory. We also consider the
contribution of these probes to the elliptic genus and discuss puzzles, and
their possible resolutions, in reproducing the elliptic genus of the symmetric
product.Comment: 47 pages; (v2) references and minor clarifications adde
The universal behavior of one-dimensional, multi-species branching and annihilating random walks with exclusion
A directed percolation process with two symmetric particle species exhibiting
exclusion in one dimension is investigated numerically. It is shown that if the
species are coupled by branching (, ) a continuous phase
transition will appear at zero branching rate limit belonging to the same
universality class as that of the dynamical two-offspring (2-BARW2) model. This
class persists even if the branching is biased towards one of the species. If
the two systems are not coupled by branching but hard-core interaction is
allowed only the transition will occur at finite branching rate belonging to
the usual 1+1 dimensional directed percolation class.Comment: 3 pages, 3 figures include
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