Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Sustainability Science and Management

ZEROS OF PARTITION FUNCTION FOR Z5-SYMMETRIC MODEL ON SQUARE LATTICE-PARTICULAR CASE

NIK AIZAD NIK HASMI, Siti Fatimah Zakaria, Fatimah Abdul Razak, Universiti Kebangsaan Malaysia

Published Jun 30, 2024• 0 citations
Publisher

Abstract

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.

DOI
10.46754/jssm.2024.06.001

Related papers

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

B. Bonnier, Y. Leroyer

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.

Physical review. B, Condensed matter•Nov 1, 1991•5 citations

The Z_6-symmetric model partition function on triangular lattice

Nor Sakinah Mohd Manshur, Siti Fatimah Zakaria, Nasir Ganikhodjaev

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.

Malaysian Journal of Fundamental and Applied Sciences•Jun 15, 2020•0 citations

Analytic Properties of Potts and Ising Model Partition Functions and the Relationship between Analytic Properties and Phase Transitions in Equilibrium Statistical Mechanics

Siti Fatimah Zakaria

The Ising and $Q$-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena $-$ how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of $Q$-state Potts models to a more general case called the $Z_Q$-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case $Q=5,6,$ and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent $\\alpha$.

•Nov 1, 2016•3 citations

On the partition function for the three-dimensional Ising model

Yogendra P. Valani

Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.

•Aug 1, 2011•2 citations

7 A NEW LOOK AT THE 2D ISING MODEL FROM EXACT PARTITION FUNCTION ZEROS FOR LARGE LATTICE SIZES

Nelson A. Alves, J. R. Drugowich de Felício, Ulrich H. E. Hansmann

A general numerical method is presented to locate the partition function zeros in the complex β plane for large lattice sizes. We apply this method to the 2D Ising model and results are reported for square lattice sizes up to L = 64. We also propose an alternative method to evaluate corrections to scaling which relies only on the leading zeros. This method is illustrated with our data.

•Aug 18, 2016•7 citations

Estimating the partition function zeros by using the Wang-Landau Monte Carlo algorithm

Seung‐Yeon Kim

The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).

Journal of the Korean Physical Society•Mar 1, 2017•4 citations
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 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.
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.
The Ising and QQQ-state Potts models are statistical mechanical models of spins interaction on crystal lattices. \nWe study the partition functions on a range of lattices, particularly two- and three-dimensional cases. The study aims to investigate cooperative phenomena −-− how higher level structure is affected by the detailed activity of a very large number of lower level structures. \nWe investigate the analytic properties of the partition functions and their relationship to physical observables in equilibrium near phase transition. \nOur study is focussed on describing the partition function and the distribution of zeros of the partition function in the complex-temperature plane close to phase transitions. \nHere we first consider the solved case of the Ising model on square lattice as a benchmark for checking our method of computation and analysis. \nThe partition function is computed using a transfer matrix approach and the zeros are found numerically by Newton-Raphson method. \nWe extend the study of QQQ-state Potts models to a more general case called the ZQZ_QZQ​-symmetric model. \nWe evidence the existence of multiple phase transitions for this model in case Q=5,6,Q=5,6,Q=5,6, and discuss the possible connection to different stages of disordered state. \nGiven sufficient and efficient coding and computing resources, we extend many previously studied cases to larger lattice sizes. \nOur analysis of zeros distribution close to phase transition point is based on a certain power law relation which leads to critical exponent of physical observable. \nWe evidence for example, that our method can be used to give numerical estimates of the specific heat critical exponent alpha\\alphaalpha.
Our aim is to investigate the critical behaviour of lattice spin models such as the three-dimensional Ising model in the thermodynamic limit. The exact partition functions (typically summed over the order of 1075 states) for finite simple cubic Ising lattices are computed using a transfer matrix approach. Q-state Potts model partition functions on two- and three-dimensional lattices are also computed and analysed. Our results are analysed as distributions of zeros of the partition function in the complex-temperature plane. We then look at sequences of such distributions for sequences of lattices approaching the thermodynamic limit. For a controlled comparison, we show how a sequence of zero distributions for finite 2d Ising lattices tends to Onsager’s thermodynamic solution. Via such comparisons, we find evidence to suggest, for example, a thermodynamic limit singular point in the behaviour of the specific heat of the 3d Ising model.
The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).