35 research outputs found
Interface mapping in two-dimensional random lattice models
We consider two disordered lattice models on the square lattice: on the
medial lattice the random field Ising model at T=0 and on the direct lattice
the random bond Potts model in the large-q limit at its transition point. The
interface properties of the two models are known to be related by a mapping
which is valid in the continuum approximation. Here we consider finite random
samples with the same form of disorder for both models and calculate the
respective equilibrium states exactly by combinatorial optimization algorithms.
We study the evolution of the interfaces with the strength of disorder and
analyse and compare the interfaces of the two models in finite lattices.Comment: 7 pages, 6 figure
Rounding of first-order phase transitions and optimal cooperation in scale-free networks
We consider the ferromagnetic large- state Potts model in complex evolving
networks, which is equivalent to an optimal cooperation problem, in which the
agents try to optimize the total sum of pair cooperation benefits and the
supports of independent projects. The agents are found to be typically of two
kinds: a fraction of (being the magnetization of the Potts model) belongs
to a large cooperating cluster, whereas the others are isolated one man's
projects. It is shown rigorously that the homogeneous model has a strongly
first-order phase transition, which turns to second-order for random
interactions (benefits), the properties of which are studied numerically on the
Barab\'asi-Albert network. The distribution of finite-size transition points is
characterized by a shift exponent, , and by a different
width exponent, , whereas the magnetization at the transition
point scales with the size of the network, , as: , with
.Comment: 8 pages, 6 figure
Density of critical clusters in strips of strongly disordered systems
We consider two models with disorder dominated critical points and study the
distribution of clusters which are confined in strips and touch one or both
boundaries. For the classical random bond Potts model in the large-q limit we
study optimal Fortuin-Kasteleyn clusters by combinatorial optimization
algorithm. For the random transverse-field Ising chain clusters are defined and
calculated through the strong disorder renormalization group method. The
numerically calculated density profiles close to the boundaries are shown to
follow scaling predictions. For the random bond Potts model we have obtained
accurate numerical estimates for the critical exponents and demonstrated that
the density profiles are well described by conformal formulae.Comment: 9 pages, 9 figure
Scale Invariance in disordered systems: the example of the Random Field Ising Model
We show by numerical simulations that the correlation function of the random
field Ising model (RFIM) in the critical region in three dimensions has very
strong fluctuations and that in a finite volume the correlation length is not
self-averaging. This is due to the formation of a bound state in the underlying
field theory. We argue that this non perturbative phenomenon is not particular
to the RFIM in 3-d. It is generic for disordered systems in two dimensions and
may also happen in other three dimensional disordered systems
Unexpected non-Wigner behavior in level-spacing distributions of next-nearest-neighbor coupled XXZ spin chains
The level-spacing distributions of XXZ spin chains with next-nearest-neighbor
couplings are studied under periodic boundary conditions. We confirm that
integrable XXZ spin chains mostly have the Poisson distribution as expected. On
the contrary, the level-spacing distributions of next-nearest-neighbor coupled
XXZ chains are given by non-Wigner distributions. It is against the
expectations, since the models are nonintegrable.Comment: 4 pages, 4 figures, to be published in Physical Review
Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model
We study birational transformations of the projective space originating from
lattice statistical mechanics, specifically from various chiral Potts models.
Associating these models to \emph{stable patterns} and \emph{signed-patterns},
we give general results which allow us to find \emph{all} chiral -state
spin-edge Potts models when the number of states is a prime or the square
of a prime, as well as several -dependent family of models. We also prove
the absence of monocolor stable signed-pattern with more than four states. This
demonstrates a conjecture about cyclic Hadamard matrices in a particular case.
The birational transformations associated to these lattice spin-edge models
show complexity reduction. In particular we recover a one-parameter family of
integrable transformations, for which we give a matrix representationComment: 22 pages 0 figure The paper has been reorganized, splitting the
results into two sections : results pertaining to Physics and results
pertaining to Mathematic
Disorder induced rounding of the phase transition in the large q-state Potts model
The phase transition in the q-state Potts model with homogeneous
ferromagnetic couplings is strongly first order for large q, while is rounded
in the presence of quenched disorder. Here we study this phenomenon on
different two-dimensional lattices by using the fact that the partition
function of the model is dominated by a single diagram of the high-temperature
expansion, which is calculated by an efficient combinatorial optimization
algorithm. For a given finite sample with discrete randomness the free energy
is a pice-wise linear function of the temperature, which is rounded after
averaging, however the discontinuity of the internal energy at the transition
point (i.e. the latent heat) stays finite even in the thermodynamic limit. For
a continuous disorder, instead, the latent heat vanishes. At the phase
transition point the dominant diagram percolates and the total magnetic moment
is related to the size of the percolating cluster. Its fractal dimension is
found d_f=(5+\sqrt{5})/4 and it is independent of the type of the lattice and
the form of disorder. We argue that the critical behavior is exclusively
determined by disorder and the corresponding fixed point is the isotropic
version of the so called infinite randomness fixed point, which is realized in
random quantum spin chains. From this mapping we conjecture the values of the
critical exponents as \beta=2-d_f, \beta_s=1/2 and \nu=1.Comment: 12 pages, 12 figures, version as publishe
Level statistics of XXZ spin chains with a random magnetic field
The level-spacing distribution of a spin 1/2 XXZ chain is numerically studied
under random magnetic field. We show explicitly how the level statistics
depends on the lattice size L, the anisotropy parameter , and the mean
amplitude of the random magnetic field h. In the energy spectrum, quantum
integrability competes with nonintegrability derived from the randomness, where
the XXZ interaction is modified by the parameter . When ,
the level-spacing distribution mostly shows Wigner-like behavior, while when
=0, Poisson-like behavior appears although the system is nonintegrable
due to randomness. Poisson-like behavior also appears for in the
large h limit. Furthermore, the level-spacing distribution depends on the
lattice size L, particularly when the random field is weak.Comment: 4 pages, 3 figures, to be published in Phys. Rev.
Ground state non-universality in the random field Ising model
Two attractive and often used ideas, namely universality and the concept of a
zero temperature fixed point, are violated in the infinite-range random-field
Ising model. In the ground state we show that the exponents can depend
continuously on the disorder and so are non-universal. However, we also show
that at finite temperature the thermal order parameter exponent one half is
restored so that temperature is a relevant variable. The broader implications
of these results are discussed.Comment: 4 pages 2 figures, corrected prefactors caused by a missing factor of
two in Eq. 2., added a paragraph in conclusions for clarit
Nonequilibrium dynamics of fully frustrated Ising models at T=0
We consider two fully frustrated Ising models: the antiferromagnetic
triangular model in a field of strength, , as well as the Villain
model on the square lattice. After a quench from a disordered initial state to
T=0 we study the nonequilibrium dynamics of both models by Monte Carlo
simulations. In a finite system of linear size, , we define and measure
sample dependent "first passage time", , which is the number of Monte
Carlo steps until the energy is relaxed to the ground-state value. The
distribution of , in particular its mean value, , is shown to
obey the scaling relation, , for both models.
Scaling of the autocorrelation function of the antiferromagnetic triangular
model is shown to involve logarithmic corrections, both at H=0 and at the
field-induced Kosterlitz-Thouless transition, however the autocorrelation
exponent is found to be dependent.Comment: 7 pages, 8 figure