6,344 research outputs found

### Fisher Zeroes and Singular Behaviour of the Two Dimensional Potts Model in the Thermodynamic Limit

The duality transformation is applied to the Fisher zeroes near the
ferromagnetic critical point in the q>4 state two dimensional Potts model. A
requirement that the locus of the duals of the zeroes be identical to the dual
of the locus of zeroes in the thermodynamic limit (i) recovers the ratio of
specific heat to internal energy discontinuity at criticality and the
relationships between the discontinuities of higher cumulants and (ii)
identifies duality with complex conjugation. Conjecturing that all zeroes
governing ferromagnetic singular behaviour satisfy the latter requirement gives
the full locus of such Fisher zeroes to be a circle. This locus, together with
the density of zeroes is then shown to be sufficient to recover the singular
form of the thermodynamic functions in the thermodynamic limit.Comment: 10 pages, 0 figures, LaTeX. Paper expanded and 2 references added
clarifying duality relationships between discontinuities in higher cumulant

### Non-Universal Critical Behaviour of Two-Dimensional Ising Systems

Two conditions are derived for Ising models to show non-universal critical
behaviour, namely conditions concerning 1) logarithmic singularity of the
specific heat and 2) degeneracy of the ground state. These conditions are
satisfied with the eight-vertex model, the Ashkin-Teller model, some Ising
models with short- or long-range interactions and even Ising systems without
the translational or the rotational invariance.Comment: 17 page

### Partition Function Zeros of a Restricted Potts Model on Lattice Strips and Effects of Boundary Conditions

We calculate the partition function $Z(G,Q,v)$ of the $Q$-state Potts model
exactly for strips of the square and triangular lattices of various widths
$L_y$ and arbitrarily great lengths $L_x$, with a variety of boundary
conditions, and with $Q$ and $v$ restricted to satisfy conditions corresponding
to the ferromagnetic phase transition on the associated two-dimensional
lattices. From these calculations, in the limit $L_x \to \infty$, we determine
the continuous accumulation loci ${\cal B}$ of the partition function zeros in
the $v$ and $Q$ planes. Strips of the honeycomb lattice are also considered. We
discuss some general features of these loci.Comment: 12 pages, 12 figure

### Two-dimensional O(n) model in a staggered field

Nienhuis' truncated O(n) model gives rise to a model of self-avoiding loops
on the hexagonal lattice, each loop having a fugacity of n. We study such loops
subjected to a particular kind of staggered field w, which for n -> infinity
has the geometrical effect of breaking the three-phase coexistence, linked to
the three-colourability of the lattice faces. We show that at T = 0, for w > 1
the model flows to the ferromagnetic Potts model with q=n^2 states, with an
associated fragmentation of the target space of the Coulomb gas. For T>0, there
is a competition between T and w which gives rise to multicritical versions of
the dense and dilute loop universality classes. Via an exact mapping, and
numerical results, we establish that the latter two critical branches coincide
with those found earlier in the O(n) model on the triangular lattice. Using
transfer matrix studies, we have found the renormalisation group flows in the
full phase diagram in the (T,w) plane, with fixed n. Superposing three
copies of such hexagonal-lattice loop models with staggered fields produces a
variety of one or three-species fully-packed loop models on the triangular
lattice with certain geometrical constraints, possessing integer central
charges 0 <= c <= 6. In particular we show that Benjamini and Schramm's RGB
loops have fractal dimension D_f = 3/2.Comment: 40 pages, 17 figure

### 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

### The packing of two species of polygons on the square lattice

We decorate the square lattice with two species of polygons under the
constraint that every lattice edge is covered by only one polygon and every
vertex is visited by both types of polygons. We end up with a 24 vertex model
which is known in the literature as the fully packed double loop model. In the
particular case in which the fugacities of the polygons are the same, the model
admits an exact solution. The solution is obtained using coordinate Bethe
ansatz and provides a closed expression for the free energy. In particular we
find the free energy of the four colorings model and the double Hamiltonian
walk and recover the known entropy of the Ice model. When both fugacities are
set equal to two the model undergoes an infinite order phase transition.Comment: 21 pages, 4 figure

### Incommensurate structures studied by a modified Density Matrix Renormalization Group Method

A modified density matrix renormalization group (DMRG) method is introduced
and applied to classical two-dimensional models: the anisotropic triangular
nearest- neighbor Ising (ATNNI) model and the anisotropic triangular
next-nearest-neighbor Ising (ANNNI) model. Phase diagrams of both models have
complex structures and exhibit incommensurate phases. It was found that the
incommensurate phase completely separates the disordered phase from one of the
commensurate phases, i. e. the non-existence of the Lifshitz point in phase
diagrams of both models was confirmed.Comment: 14 pages, 14 figures included in text, LaTeX2e, submitted to PRB,
presented at MECO'24 1999 (Wittenberg, Germany

### A model for collisions in granular gases

We propose a model for collisions between particles of a granular material
and calculate the restitution coefficients for the normal and tangential motion
as functions of the impact velocity from considerations of dissipative
viscoelastic collisions. Existing models of impact with dissipation as well as
the classical Hertz impact theory are included in the present model as special
cases. We find that the type of collision (smooth, reflecting or sticky) is
determined by the impact velocity and by the surface properties of the
colliding grains. We observe a rather nontrivial dependence of the tangential
restitution coefficient on the impact velocity.Comment: 11 pages, 2 figure

### The free energy of the Potts model: from the continuous to the first-order transition region

We present a large $q$ expansion of the 2d $q$-states Potts model free
energies up to order 9 in $1/\sqrt{q}$. Its analysis leads us to an ansatz
which, in the first-order region, incorporates properties inferred from the
known critical regime at $q=4$, and predicts, for $q>4$, the $n^{\rm th}$
energy cumulant scales as the power $(3 n /2-2)$ of the correlation length. The
parameter-free energy distributions reproduce accurately, without reference to
any interface effect, the numerical data obtained in a simulation for $q=10$
with lattices of linear dimensions up to L=50. The pure phase specific heats
are predicted to be much larger, at $q\leq10$, than the values extracted from
current finite size scaling analysis of extrema. Implications for safe
numerical determinations of interface tensions are discussed.Comment: 11 pages, plain tex with 3 Postscript figures included Postscript
file available by anonymous ftp://amoco.saclay.cea.fr/pubs.spht/93-022.p

### The Chiral Potts Models Revisited

In honor of Onsager's ninetieth birthday, we like to review some exact
results obtained so far in the chiral Potts models and to translate these
results into language more transparent to physicists, so that experts in Monte
Carlo calculations, high and low temperature expansions, and various other
methods, can use them. We shall pay special attention to the interfacial
tension $\epsilon_r$ between the $k$ state and the $k-r$ state. By examining
the ground states, it is seen that the integrable line ends at a superwetting
point, on which the relation $\epsilon_r=r\epsilon_1$ is satisfied, so that it
is energetically neutral to have one interface or more. We present also some
partial results on the meaning of the integrable line for low temperatures
where it lives in the non-wet regime. We make Baxter's exact results more
explicit for the symmetric case. By performing a Bethe Ansatz calculation with
open boundary conditions we confirm a dilogarithm identity for the
low-temperature expansion which may be new. We propose a new model for
numerical studies. This model has only two variables and exhibits commensurate
and incommensurate phase transitions and wetting transitions near zero
temperature. It appears to be not integrable, except at one point, and at each
temperature there is a point, where it is almost identical with the integrable
chiral Potts model.Comment: J. Stat. Phys., LaTeX using psbox.tex and AMS fonts, 69 pages, 30
figure

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