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Physical review. B, Condensed matter

Zeros of the partition function for the two-dimensional Z5 -symmetric spin model

B. Bonnier, Y. Leroyer

Published Nov 1, 1991• 5 citations
Publisher

Abstract

We have numerically computed the exact distribution of zeros of the partition function for the ${\mathit{Z}}_{5}$-symmetric spin model on square lattices of linear size 3\ensuremath{\le}L\ensuremath{\le}7. At a point of the phase diagram close to the clock-model point, we observe two distinct lines of zeros, which converge towards two well-separated points of the real axis as L increases, thereby clearly signaling the existence of two phase transitions and confirming the emergence of the expected soft phase.

DOI
10.1103/physrevb.44.9700

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We have numerically computed the exact distribution of zeros of the partition function for the Z5{\mathit{Z}}_{5}Z5​-symmetric spin model on square lattices of linear size 3\ensuremath{\le}L\ensuremath{\le}7. At a point of the phase diagram close to the clock-model point, we observe two distinct lines of zeros, which converge towards two well-separated points of the real axis as L increases, thereby clearly signaling the existence of two phase transitions and confirming the emergence of the expected soft phase.
We study a statistical mechanics model of multiple-phase transitions called the ZQ-symmetric model on the square lattice for five possible spin directions (Q = 5). This study aims to investigate the existence of phase transition(s) and discuss the distribution of partition function zeros on the complex-temperature plane. Some cases of energy list χ are considered where they are written in an arbitrary arrangement of integer numbers (usually in decreasing value due to the angle of separation). The partition functions are computed using a transfer matrix approach, and their zeros are found numerically using the NewtonRaphson method. A Monte-Carlo Metropolis simulation is then performed to study the critical point of phase transition. The results are compared to the zeros distribution of the partition functions for the considered cases. Studying this generalised model is vital to mimic the system in the real world and to help experimental works use a more sustainable approach.
We have performed a Monte Carlo simulation of the Z5{\mathit{Z}}_{5}Z5​ model in the region of the phase diagram where the transition is expected to be first order. Analyzing our results by means of a method recently proposed by Lee and Kosterlitz, we confirm its sensitivity in discriminating between the weak first-order nature of the transition at the five-state Potts point and the second-order one near the Fateev-Zamolodchikov integrability point. In addition, we find the transition to be first order in the intermediate region, as already suggested by previous analysis.
We have investigated the dynamic critical behavior of the two-dimensional Z(5)-symmetric spin model by using short-time Monte Carlo (MC) simulations. We have obtained estimates of some critical points in its rich phase diagram and included, among the usual critical lines the study of first-order (weak) transition by looking into the order-disorder phase transition. In addition, we also investigated the soft-disorder phase transition by considering empiric methods. A study of the behavior of β/νz along the self-dual critical line has been performed and special attention has been devoted to the critical bifurcation point, or Fateev-Zamolodchikov (FZ) point. First, by using a refinement method and taking into account simulations out of equilibrium, we were able to localize parameters of this point. In a second part of our study, we turned our attention to the behavior of the model at the early stage of its time evolution in order to find the dynamic critical exponent z as well as the static critical exponents β and ν of the FZ point on square lattices. The values of the static critical exponents and parameters are in good agreement with the exact results, and the dynamic critical exponent z≈2.28 very close to the four-state Potts model (z≈2.29).
The loci of zeros of the partition functions of the following one dimensional Ising models are obtained, (i) Those of S = 1 / 2 and with the nearest (J) and the next nearest (J') neighbour interactions. The loci of zeros are classified in several patterns according to a combination of J and J'. In particular when J<0 and J'>0, the existence of two nearly concentric circular arcs whose heads approach points on the positive real axis of z corresponding to the critical field at zero temperature is confirmed. This suggests that the locus of the two and three dimensional Ising antiferromagnet (J<0, J'>0) will be closed two concentric circles below the Neel temperature. (ii) Those of S=1 and of S= 3 / 2 with the nearest neighbour (J) interaction. The locus shrinks to 25 points ( theta = 2 pi j/(2S+1), J=1,2,...,2S) on the unit circle at T = infinity . It extends along the unit circle for J>0 and along a nearly radial direction for J<0 when the temperature decreases.
We have performed multicanonical simulations to study the critical behavior of the two-dimensional Ising model with dipole interactions. This study concerns the thermodynamic phase transitions in the range of the interaction δ where the phase characterized by striped configurations of width h = 1 is observed. Controversial results obtained from local update algorithms have been reported for this region, including the claimed existence of a second-order phase transition line that becomes first order above a tricritical point located somewhere between δ = 0.85 and 1. Our analysis relies on the complex partition function zeros obtained with high statistics from multicanonical simulations. Finite size scaling relations for the leading partition function zeros yield critical exponents ν that are clearly consistent with a single second-order phase transition line, thus excluding such a tricritical point in that region of the phase diagram. This conclusion is further supported by analysis of the specific heat and susceptibility of the orientational order parameter.
There is a study on a square lattice that can predict the existence of multiple phase transitions on a complex plane. We extend the study on the different types of ZQ-symmetric model and different lattices in order to provide more evidence to the existence of multiple phase transitions. We focus on the ZQ-symmetric model with the nearest neighbour interaction on the six spin directions between molecular dipole, i.e. Q = 6 on a triangular lattice. Mainly, the model is defined on the triangular lattice graph with the nearest neighbour interaction. By using the transfer matrix approach, the partition functions are computed for increasing lattice sizes. The roots of polynomial partition function are also computed and plotted in the complex Argand plane. The specific heat equation is used for further comparison. The result supports the existence of the multiple phase transitions by the emergence of the multiple line curves in the locus of zeros distribution for specific type of energy level.