on Ising models

Overview

This post serves as a study note for the Ising model, a fundamental paradigm in statistical mechanics for understanding phase transitions and collective behavior in discrete systems. We cover the Hamiltonian formulation, symmetry breaking in the thermodynamic limit, Kramers-Wannier duality, and the transfer matrix method.

🏷️ Introduction

The Ising model was originally proposed to explain ferromagnetism. It consists of discrete variables called spins ${\sigma }_i\in \{+1,-1\}$ placed on a lattice. The system’s energy is defined by the Hamiltonian:

$$H(\sigma )=-J\sum _{⟨i,j⟩}{\sigma }_i{\sigma }_j-h\sum _i{\sigma }_i$$

where interact strength $J$ determines ferromagnetic ($J>0$) or anti-ferromagnetic ($J<0$) behavior, and $h$ is an external magnetic field. The probability of a configuration $\sigma$ is given by the Boltzmann distribution:

$$P(\sigma )=\frac{1}{Z}{e}^{-\beta H(\sigma )},\beta =\frac{1}{k_{B}T}$$

where $Z={\sum }_{\sigma }{e}^{-\beta H(\sigma )}$ is the Partition Function.

🏷️ Phase Transitions and Symmetry Breaking

A phase transition occurs when a system changes its collective state abruptly. In the Ising model, this manifests as Spontaneous Symmetry Breaking (SSB).

🏷️ The Mathematical Perspective: Non-commutativity of Limits

For any finite system ($N<\infty$) at $h=0$, the Gibbs measure is unique and symmetric, meaning $⟨M⟩=0$ for all $T>0$. Symmetry breaking only emerges in the thermodynamic limit ($N\to \infty$). The defining signature of SSB is the non-commutativity of limits for the magnetization $M$:

$$M^{\ast }=\underset{h\to 0^{+}}{\lim }\left(\underset{N\to \infty }{\lim }⟨M{⟩}_{N,h}\right)\ne \underset{N\to \infty }{\lim }\left(\underset{h\to 0}{\lim }⟨M{⟩}_{N,h}\right)=0$$

Below $T_c$, the inner limit produces a non-zero value even as $h\to 0^{+}$. This implies that the free energy density $f(\beta ,h)$ is non-analytic, and the magnetization $M=-\partial f/\partial h$ is discontinuous.

🏷️ Peierls Argument: The Energy-Entropy Tension

To prove that $M^{\ast }>0$ exists in 2D, we analyze the stability of an ordered state against the formation of “islands” of reversed spins. Consider a domain boundary of length $L$. The energy cost scales as $\Delta E=2J\cdot L$, while the entropy gain for a closed loop of length $L$ is approximately $\Delta S=L{k}_B\ln 3$. The resulting free energy balance is:

$$\Delta F=\Delta E-T\Delta S=L(2J-k_{B}T\ln 3)$$

For small $T$, $\Delta F>0$ for all $L$, meaning large-scale fluctuations are suppressed and the system remains ordered.

🏷️ Kramers-Wannier Duality

Kramers and Wannier discovered a deep symmetry relating the high-temperature and low-temperature regimes of the 2D Ising model.

🏷️ Primal and Dual Lattices

The duality maps a square lattice to a dual square lattice where nodes sit at the centers of the primal faces.

• At Low $T$, the system is described by the alignment of spins.
• At High $T$, the system is better described by the configuration of domain walls.

🏷️ The Duality Relation

The partition function $Z$ at inverse temperature $\beta$ is related to the partition function $Z^{\ast }$ at a dual temperature ${\beta }^{\ast }$ via:

$$\sinh (2\beta J)\sinh (2{\beta }^{\ast }J)=1$$

If a unique phase transition exists, it occurs at the self-dual point where $\beta ={\beta }^{\ast }={\beta }_c$, yielding:

$$k_B{T}_c=\frac{2J}{\ln (1+\sqrt{2})}\approx 2.269J$$

🏷️ Transfer Matrix Technique (1D Case)

For the 1D Ising model with $N$ spins and periodic boundary conditions, the partition function for $h=0$ is:

$$Z=\sum _{\sigma }\prod _{i=1}^N{e}^{\beta J{\sigma }_i{\sigma }_{i+1}}$$

We define the Transfer Matrix $V$:

$$V=\left(\begin{array}{cc}{e}^{\beta J}& e^{-\beta J}\\ e^{-\beta J}& e^{\beta J}\end{array}\right)$$

The partition function simplifies to the trace of the $N$-th power of $V$:

$$Z=\mathrm{Tr}(V^N)={\lambda }_1^N+{\lambda }_2^N$$

where ${\lambda }_1=2\cosh (\beta J)$ and ${\lambda }_2=2\sinh (\beta J)$. In the thermodynamic limit:

$$\frac{1}{N}\ln Z\to \ln (2\cosh (\beta J))$$

Because the free energy is analytic for all $T>0$, no phase transition exists in 1D.

📊 Numerical Verification

Since the state space is $2^{N^2}$, exact computation is impossible for large $N$. Instead, we use Markov Chain Monte Carlo (MCMC) algorithms like Metropolis-Hastings (see content/codes/2023 Spring/ising_2d_simulation.py).

The simulation reveals the sharp transition at $T_c\approx 2.269J/k_B$:

• $T<T_c$: Large domains of aligned spins dominate. The system has long-range order.
• $T\approx T_c$: Fractal-like clusters of all scales appear. The correlation length $\xi$ diverges.
• $T>T_c$: Thermal noise destroys order, resulting in a disordered state.

(Above: Magnetization curve and lattice configurations at different temperatures. Note the spontaneous symmetry breaking as $T$ drops below $T_c$.)

🏷️ Metropolis Algorithm

• Duality: The 2D Ising model satisfies Kramers-Wannier Duality, allowing the exact calculation of $T_c$.
• Mean Field Theory: An approximation $m=\tanh (\beta (zJm+h))$ that overestimates $T_c$ but provides qualitative insight.

📚 References