Energy Probability Distribution Zeros: A Route to Study Phase Transitions

Abstract

In the study of phase transitions a very few models are accessible to exact solution. In the most cases analytical simplifications have to be done or some numerical technique has to be used to get insight about their critical properties. Numerically, the most common approaches are those based in Monte Carlo simulations together finite size scaling analysis. The use of Monte Carlo techniques requires the estimate of quantities like the specific heat or susceptibilities in a wide range of temperature or the construction of the density of states in large intervals of energy. Although many of these techniques are well developed they may be very time consuming when the system size becomes large enough. It should be suitable to have a method that could surpass those difficulties. In this work we present an iterative method to study the critical behavior of a system based on the partial knowledge of the complex Fisher zeros set of the partition function. The method is general with advantages over most conventional techniques since it does not need to identify any order parameter \emph{a priori}. The critical temperature and exponents can be obtained with great precision even in the most unamenable cases like the two dimensional $XY$ model. To test the method and to show how it works we applied it to some selected models where the transitions are well known: The 2D Ising, Potts and XY models and to a homopolymer system. Our choice cover systems with first order, continuous and Berezinskii-Kosterlitz-Thouless transitions as well as the homopolymer that has two pseudo-transitions. The strategy can easily be adapted to any model, classical or quantum, once we are able to build the corresponding energy probability distribution.

9 pages, 11 figures, 5 tables

References in corpus (3)

• More is the Same; Phase Transitions and Mean Field Theories
• Surface effects in the crystallization process of elastic flexible polymers
• Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: A comparison for elastic flexible polymers

Cited by in corpus (15)

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• The fully frustrated XY model revisited: A new universality class
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• Tensor network calculation of the logarithmic correction exponent in the XY model
• Universal energy and magnetisation distributions in the Blume-Capel and Baxter-Wu models
• Two-dimensional dilute Baxter-Wu model: Transition order and universality
• Intermediate phase and pseudo phase transition in an artificial spin ice model
• Phase Diagram of Flexible Polymers with Quenched Disordered Charged Monomers
• Pushing the limits of EPD zeros method
• Finding the Dominant Zero of the Energy Probability Distribution
• Connecting the Unstable Region of the Entropy to the Pattern of the Fisher's Zeros Map
• Using the Energy probability distribution zeros to obtain the critical properties of the two-dimensional anisotropic Heisenberg model
• A Microcanonical Inflection Point Analysis via Parametric Curves and its Relation to the Zeros of the Partition Function
• Determination of melting temperature of hexagonal ice using Lee-Yang phase transition theory
• Padé Approximation and Partition Function Zeros