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Updated: Apr 27, 2026
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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
Published on: September 26, 2016
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Grassmannian representation of the two-dimensional monomer-dimer model.
Nicolas Allegra1, Jean-Yves Fortin1
1Institut Jean Lamour, CNRS/UMR 7198, Groupe de Physique Statistique, Université de Lorraine, BP 70239, F-54506 Vandœuvre-lès-Nancy Cedex, France.
Summary
We applied Grassmann algebra to monomer-dimer statistics on a 2D square lattice. The partition function was found using quadratic fermionic theory and pfaffians, extending Kasteleyn
Area of Science:
- Statistical mechanics
- Condensed matter physics
- Mathematical physics
Background:
- Monomer-dimer models are crucial for understanding statistical properties of systems.
- The Kasteleyn method provides exact solutions for pure dimer problems on lattices.
- Extending these solutions to include monomers presents a significant theoretical challenge.
Purpose of the Study:
- To develop a method for calculating the partition function of monomer-dimer systems on a 2D square lattice.
- To express the solution using Grassmann algebra and fermionic theories.
- To compare the results with existing theories for pure dimer systems and boundary/bulk monomer configurations.
Main Methods:
- Application of Grassmann algebra for fermionic representations.
- Utilizing quadratic fermionic theory to model the system.
- Expressing the partition function as a product of two pfaffians.
- Calculating correlation functions for boundary and bulk monomers.
Main Results:
- An exact expression for the partition function of n monomers on a 2D square lattice was derived.
- The solution is presented as a product of two pfaffians, linking it to Kasteleyn's work.
- Correlation functions were found to be consistent with previous discrete and continuous boundary results and numerical bulk evaluations.
Conclusions:
- Grassmann algebra provides an effective framework for solving monomer-dimer statistics problems.
- The pfaffian-based solution offers a generalized approach beyond pure dimer systems.
- The results validate and extend existing knowledge in statistical mechanics and lattice models.
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