A note on Lee-Yang zeros in the negative half-plane

Area of Science:

• Statistical Mechanics
• Phase Transitions
• Lattice Models

Background:

• Lee-Yang zeros of the grand-canonical partition function are crucial for understanding phase transitions and critical phenomena in statistical systems.
• The location of these zeros in the complex fugacity plane provides insights into the system's thermodynamic properties and stability.
• The monomer-dimer system on a lattice is a well-studied model exhibiting interesting phase behavior.

Purpose of the Study:

• To derive lower bounds on the inverse compressibility for systems whose Lee-Yang zeros are located in the left half of the complex fugacity plane.
• To investigate the virial expansion of pressure in terms of density for systems with Lee-Yang zeros on the negative real axis, such as the monomer-dimer system.
• To explore the relationship between the positivity of virial coefficients and the location of Lee-Yang zeros.

Main Methods:

• Analytical derivation of lower bounds for inverse compressibility based on the properties of Lee-Yang zeros.
• Analysis of the virial expansion of pressure for specific lattice models.
• Explicit calculation of the partition function for the monomer-dimer system on two rows.

Main Results:

• Established lower bounds on the inverse compressibility for systems with Lee-Yang zeros in the left complex fugacity plane, including those on the negative real axis.
• Demonstrated that there is no direct correlation between the positivity of virial coefficients and the location of Lee-Yang zeros in the negative real axis.
• Provided examples illustrating cases where Lee-Yang zeros are negative, virial coefficients are positive, or both conditions are met.

Conclusions:

• The study provides theoretical bounds on a key thermodynamic quantity, inverse compressibility, linked to Lee-Yang zero locations.
• The research clarifies the complex relationship between Lee-Yang zeros and virial coefficients, showing they are not directly coupled.
• An explicit calculation for the monomer-dimer system suggests a finite number of negative virial coefficients, offering specific insights into lattice models.