Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Ising 100: review of solutions

Oğuz Alp Ağırbaş, Anıl Ata, Eren Demirci, Ilmar Gahramanov, Tuğba Hırlı, R. Semih Kanber, Ahmet Berk Kavruk, Mustafa Mullahasanoğlu, Zehra Özcan, Cansu Özdemir, Irmak Özgüç, Sinan Ulaş Öztürk, Uveys Turhan, Ali Mert T. Yetkin, Yunus Emre Yıldırım, Reyhan Yumuşak

TL;DR

Ising 100 surveys the exact solutions of the two-dimensional Ising model through four complementary paradigms: combinatorial determinant methods, Pfaffian/dimer techniques, star-triangle correlation approaches, and fermionic Grassmann formulations. It elucidates how each method yields exact partition functions and correlation functions across diverse lattices (rectangular, triangular, honeycomb, Kagome) and boundary conditions (free, cylindrical, toroidal). The work highlights unifying structures such as determinant/Pfaffian representations, elliptic-integral correlations, and local transformation rules that underpin exact results. Collectively, these approaches reinforce the profound connections between statistical mechanics, graph theory, and integrable systems, offering a consolidated reference for exact 2D Ising solutions and their mathematical underpinnings.

Abstract

We present several known solutions to the two-dimensional Ising model. This review originated from the ``Ising 100'' seminar series held at Boğaziçi University, Istanbul, in 2024.

Ising 100: review of solutions

TL;DR

Ising 100 surveys the exact solutions of the two-dimensional Ising model through four complementary paradigms: combinatorial determinant methods, Pfaffian/dimer techniques, star-triangle correlation approaches, and fermionic Grassmann formulations. It elucidates how each method yields exact partition functions and correlation functions across diverse lattices (rectangular, triangular, honeycomb, Kagome) and boundary conditions (free, cylindrical, toroidal). The work highlights unifying structures such as determinant/Pfaffian representations, elliptic-integral correlations, and local transformation rules that underpin exact results. Collectively, these approaches reinforce the profound connections between statistical mechanics, graph theory, and integrable systems, offering a consolidated reference for exact 2D Ising solutions and their mathematical underpinnings.

Abstract

We present several known solutions to the two-dimensional Ising model. This review originated from the ``Ising 100'' seminar series held at Boğaziçi University, Istanbul, in 2024.

Paper Structure

This paper contains 85 sections, 912 equations, 64 figures, 1 table.

Table of Contents

  1. Introduction
  2. Combinatorial Approach (Eren Demirci)
  3. Introduction
  4. The Method
  5. Construction of the Determinant
  6. Deriving the Determinant
  7. Formulating the Determinant
  8. Obtaining the Exact Partition Function
  9. Obtaining the Exact Partition Function of Triangular Model
  10. Kasteleyn's Dimer Method and Pfaffian Technique (Ahmet Berk Kavruk and Irmak Özgüç)
  11. Graphs, Perfect Matchings and Hafnians
  12. The Hafnian
  13. Pfaffian, Transition Cycles and All That
  14. On Lattices with Free Boundary Conditions
  15. The Pfaffian
  16. ...and 70 more sections

Figures (64)

  • Figure 1: Hyperbolic triangle whose edges are $2H'$, $2H^*$ and $\gamma$
  • Figure 2: The illustration of $n\times m$ model.
  • Figure 3: The left cycle will be referred to as $(v)$ and the right one as $(f)$.
  • Figure 4: $\Omega$ for this one is 0.
  • Figure 5: $\Omega$ for this one is $2\pi$.
  • ...and 59 more figures