6,837 research outputs found
Classical phase transitions in a one-dimensional short-range spin model
Ising's solution of a classical spin model famously demonstrated the absence
of a positive-temperature phase transition in one-dimensional equilibrium
systems with short-range interactions. No-go arguments established that the
energy cost to insert domain walls in such systems is outweighed by entropy
excess so that symmetry cannot be spontaneously broken. An archetypal way
around the no-go theorems is to augment interaction energy by increasing the
range of interaction. Here we introduce new ways around the no-go theorems by
investigating entropy depletion instead. We implement this for the Potts model
with invisible states.Because spins in such a state do not interact with their
surroundings, they contribute to the entropy but not the interaction energy of
the system. Reducing the number of invisible states to a negative value
decreases the entropy by an amount sufficient to induce a positive-temperature
classical phase transition. This approach is complementary to the long-range
interaction mechanism. Alternatively, subjecting positive numbers of invisible
states to imaginary or complex fields can trigger such a phase transition. We
also discuss potential physical realisability of such systems.Comment: 29 pages, 11 figure
Spin models for orientational ordering of colloidal molecular crystals
Two-dimensional colloidal suspensions exposed to periodic external fields
exhibit a variety of molecular crystalline phases. There two or more colloids
assemble at lattice sites of potential minima to build new structural entities,
referred to as molecules. Using the strength of the potential and the filling
fraction as control parameter, phase transition to unconventional
orientationally ordered states can be induced. We introduce an approach that
focuses at the discrete set of orientational states relevant for the phase
ordering. The orientationally ordered states are mapped to classical spin
systems. We construct effective hamiltonians for dimeric and trimeric molecules
on triangular lattices suitable for a statistical mechanics discussion. A
mean-field analysis produces a rich phase behavior which is substantiated by
Monte Carlo simulations.Comment: 19 pages, 21 figures; misplacement of Fig.3 fixe
Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions
We present an analytic study of the Potts model partition function on two
different types of self-similar lattices of triangular shape with non integer
Hausdorff dimension. Both types of lattices analyzed here are interesting
examples of non-trivial thermodynamics in less than two dimensions. First, the
Sierpinski gasket is considered. It is shown that, by introducing suitable
geometric coefficients, it is possible to reduce the computation of the
partition function to a dynamical system, whose variables are directly
connected to (the arising of) frustration on macroscopic scales, and to
determine the possible phases of the system. The same method is then used to
analyse the Hanoi graph. Again, dynamical system theory provides a very elegant
way to determine the phase diagram of the system. Then, exploiting the analysis
of the basins of attractions of the corresponding dynamical systems, we
construct various examples of self-similar lattices with more than one critical
temperature. These multiple critical temperatures correspond to crossing phases
with different degrees of frustration.Comment: 16 pages, 12 figures, 1 table; title changed, references and
discussion on multiple transitions adde
Critical Properties of Random Quantum Potts and Clock Models
We study zero temperature phase transitions in two classes of random quantum
systems -the -state quantum Potts and clock models. For models with purely
ferromagnetic interactions in one dimension, we show that for strong randomness
there is a second order transition with critical properties that can be
determined exactly by use of an RG procedure. Somewhat surprisingly, the
critical behaviour is completely independent of (for ).
For the clock model, we suggest the existence of a novel multicritical
point at intermediate randomness. We also consider the transition from
a paramagnet to a spin glass in an infinite range model. Assuming that the
transition is second order, we solve for the critical behaviour and find
independent exponents.Comment: 12 pages, REVTEX 3.0, 1 EPS figur
Thin Animals
Lattice animals provide a discretized model for the theta transition
displayed by branched polymers in solvent. Exact graph enumeration studies have
given some indications that the phase diagram of such lattice animals may
contain two collapsed phases as well as an extended phase. This has not been
confirmed by studies using other means. We use the exact correspondence between
the q --> 1 limit of an extended Potts model and lattice animals to investigate
the phase diagram of lattice animals on phi-cubed random graphs of arbitrary
topology (``thin'' random graphs). We find that only a two phase structure
exists -- there is no sign of a second collapsed phase.
The random graph model is solved in the thermodynamic limit by saddle point
methods. We observe that the ratio of these saddle point equations give
precisely the fixed points of the recursion relations that appear in the
solution of the model on the Bethe lattice by Henkel and Seno. This explains
the equality of non-universal quantities such as the critical lines for the
Bethe lattice and random graph ensembles.Comment: Latex, 10 pages plus 6 ps/eps figure
Melting of Colloidal Molecular Crystals on Triangular Lattices
The phase behavior of a two-dimensional colloidal system subject to a
commensurate triangular potential is investigated. We consider the integer
number of colloids in each potential minimum as rigid composite objects with
effective discrete degrees of freedom. It is shown that there is a rich variety
of phases including ``herring bone'' and ``Japanese 6 in 1'' phases. The
ensuing phase diagram and phase transitions are analyzed analytically within
variational mean-field theory and supplemented by Monte Carlo simulations.
Consequences for experiments are discussed.Comment: 10 pages, 4 figure
Phase transitions in the Potts model on complex networks
The Potts model is one of the most popular spin models of statistical
physics. The prevailing majority of work done so far corresponds to the lattice
version of the model. However, many natural or man-made systems are much better
described by the topology of a network. We consider the q-state Potts model on
an uncorrelated scale-free network for which the node-degree distribution
manifests a power-law decay governed by the exponent \lambda. We work within
the mean-field approximation, since for systems on random uncorrelated
scale-free networks this method is known to often give asymptotically exact
results. Depending on particular values of q and \lambda one observes either a
first-order or a second-order phase transition or the system is ordered at any
finite temperature. In a case study, we consider the limit q=1 (percolation)
and find a correspondence between the magnetic exponents and those describing
percolation on a scale-free network. Interestingly, logarithmic corrections to
scaling appear at \lambda=4 in this case.Comment: 15 pages, 2 figure
Phenomenological Renormalization Group Methods
Some renormalization group approaches have been proposed during the last few
years which are close in spirit to the Nightingale phenomenological procedure.
In essence, by exploiting the finite size scaling hypothesis, the approximate
critical behavior of the model on infinite lattice is obtained through the
exact computation of some thermal quantities of the model on finite clusters.
In this work some of these methods are reviewed, namely the mean field
renormalization group, the effective field renormalization group and the finite
size scaling renormalization group procedures. Although special emphasis is
given to the mean field renormalization group (since it has been, up to now,
much more applied an extended to study a wide variety of different systems) a
discussion of their potentialities and interrelations to other methods is also
addressed.Comment: Review Articl
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