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

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

Critical behavior at the interface between two systems belonging to different universality classes

We consider the critical behavior at an interface which separates two semi-infinite subsystems belonging to different universality classes, thus having different set of critical exponents, but having a common transition temperature. We solve this problem analytically in the frame of mean-field theory, which is then generalized using phenomenological scaling considerations. A large variety of interface critical behavior is obtained which is checked numerically on the example of two-dimensional q-state Potts models with q=2 to 4. Weak interface couplings are generally irrelevant, resulting in the same critical behavior at the interface as for a free surface. With strong interface couplings, the interface remains ordered at the bulk transition temperature. More interesting is the intermediate situation, the special interface transition, when the critical behavior at the interface involves new critical exponents, which however can be expressed in terms of the bulk and surface exponents of the two subsystems. We discuss also the smooth or discontinuous nature of the order parameter profile.Comment: 16 pages, 9 figures, published version, minor changes, some references adde

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

Voter Model with Time dependent Flip-rates

We introduce time variation in the flip-rates of the Voter Model. This type of generalisation is relevant to models of ageing in language change, allowing the representation of changes in speakers' learning rates over their lifetime and may be applied to any other similar model in which interaction rates at the microscopic level change with time. The mean time taken to reach consensus varies in a nontrivial way with the rate of change of the flip-rates, varying between bounds given by the mean consensus times for static homogeneous (the original Voter Model) and static heterogeneous flip-rates. By considering the mean time between interactions for each agent, we derive excellent estimates of the mean consensus times and exit probabilities for any time scale of flip-rate variation. The scaling of consensus times with population size on complex networks is correctly predicted, and is as would be expected for the ordinary voter model. Heterogeneity in the initial distribution of opinions has a strong effect, considerably reducing the mean time to consensus, while increasing the probability of survival of the opinion which initially occupies the most slowly changing agents. The mean times to reach consensus for different states are very different. An opinion originally held by the fastest changing agents has a smaller chance to succeed, and takes much longer to do so than an evenly distributed opinion.Comment: 16 pages, 6 figure

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

A possible combinatorial point for XYZ-spin chain

We formulate and discuss a number of conjectures on the ground state vectors of the XYZ-spin chains of odd length with periodic boundary conditions and a special choice of the Hamiltonian parameters. In particular, arguments for the validity of a sum rule for the components, which describes in a sense the degree of antiferromagneticity of the chain, are given.Comment: AMSLaTeX, 15 page

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