Ising Model and Mean Field Theory

The Ising model provides the simplest, most profound mathematical description of a phase transition, the sudden change in a material's macroscopic state, such as when a metal becomes magnetic. While deceptively simple in its rules, it captures the essential physics of cooperative phenomena where the collective behavior of many interacting parts yields emergent order. Understanding its exact solution in one dimension and its mean-field approximation in higher dimensions is foundational to modern statistical mechanics, condensed matter physics, and even fields like neuroscience and machine learning, where similar models of interacting agents appear.

The Ising Model: Hamiltonian and Observables

The Ising model abstracts a magnetic material into a lattice of microscopic magnetic moments, or spins. Each spin, located at lattice site $i$, can point only "up" ($s_i=+1$) or "down" ($s_i=-1$). The key physics is encoded in the Hamiltonian, the function that gives the system's energy for any configuration of spins:

$$H=-J\sum _{⟨i,j⟩}{s}_i{s}_j-h\sum _i{s}_i.$$

Here, $J$ is the coupling constant. For $J>0$, the interaction is ferromagnetic: neighboring spins lower the system's energy by aligning. For $J<0$, it is antiferromagnetic, favoring alternating spins. The sum ${\sum }_{⟨i,j⟩}$ runs over all pairs of nearest-neighbor sites on the lattice. The term with $h$ represents an external magnetic field, which energetically favors spins aligning with it.

At a finite temperature $T$, thermal fluctuations compete with the aligning force of $J$. The central goal is to compute the partition function $Z={\sum }_{\{s\}}{e}^{-\beta H}$, where $\beta =1/(k_{B}T)$ and the sum is over all $2^N$ possible spin configurations. From $Z$, all thermodynamic observables follow. The most important are the average magnetization per spin $m=⟨s_i⟩$ and the magnetic susceptibility $\chi =\frac{\partial m}{\partial h}$, which measures how strongly the system magnetizes in response to a small field.

Exact Solution in One Dimension

The model is exactly solvable in one dimension—a linear chain of spins. This solution is instructive because it reveals a crucial truth: a one-dimensional Ising model with finite-range interactions does not exhibit a phase transition at any non-zero temperature.

The solution employs the transfer matrix method. For a chain of $N$ spins with periodic boundary conditions, the partition function can be written as the trace of a product of identical $2\times 2$ matrices. Each matrix $T$ has elements $T_{s_i,s_{i+1}}=\exp (\beta J{s}_i{s}_{i+1}+\beta h(s_i+s_{i+1})/2)$. The partition function becomes $Z=\text{Tr}(T^N)={\lambda }_{+}^N+{\lambda }_{-}^N$, where ${\lambda }_{+}$ and ${\lambda }_{-}$ are the eigenvalues of $T$, with ${\lambda }_{+}>{\lambda }_{-}$. In the thermodynamic limit ($N\to \infty$), the larger eigenvalue dominates completely: $Z\approx {\lambda }_{+}^N$.

From this, one can derive the zero-field magnetization $m(h=0)=0$ for all $T>0$. While spins influence their neighbors, a single broken bond anywhere in the chain can flip an entire domain, making long-range order impossible against thermal fluctuations. The susceptibility $\chi$ can be calculated and remains finite, confirming the absence of a critical point.

Mean-Field Theory: The Approximation of Self-Consistency

In two or three dimensions, exact solutions become intractable. Mean-field theory (MFT) is a powerful, albeit approximate, analytical method that becomes exact in the limit of infinite spatial dimension. Its core idea is to replace the complex interaction a spin feels from its specific neighbors with an average, or mean, field produced by all other spins.

We start from the interaction term for a specific spin $s_i$: $-J{s}_i{\sum }_{j\in \text{n.n. of }i}{s}_j$. In MFT, we approximate the neighboring spins by their thermal average $m=⟨s_j⟩$. Crucially, we assume this mean field is uniform across the lattice. The sum over $z$ nearest neighbors then gives $-J{s}_i(zm)$. The effective Hamiltonian for spin $s_i$ becomes that of a single spin in an effective magnetic field: $$H_i^{\text{MF}}=-(Jzm+h)s_i.$$ This is a monumental simplification, as spins are now independent in this effective field. The self-consistency condition arises because $m$ is both the source and the result of this field. For a spin in field $h_{\text{eff}}=Jzm+h$, its average magnetization is given by the Brillouin function. For Ising spins ($\pm 1$), it simplifies to: $$m=\tanh (\beta (Jzm+h)).$$ This is the central mean-field equation of state.

Spontaneous Magnetization and the Critical Point

Setting the external field $h=0$, the self-consistency equation becomes $m=\tanh (\beta Jzm)$. This always has the solution $m=0$ (the paramagnetic state). A non-zero solution $m\ne 0$, representing spontaneous magnetization, appears when the ferromagnetic interaction can overcome thermal disorder.

We find the critical temperature $T_c$ by asking when a non-zero solution first becomes possible. For small $m$, we expand the hyperbolic tangent: $\tanh (x)\approx x-x^3/3$. The equation becomes: $$m\approx (\beta Jz)m-\frac{1}{3}(\beta Jz{)}^3{m}^3.$$ The non-zero solution exists when the linear coefficient on the right exceeds 1. This yields the mean-field critical temperature: $$k_B{T}_c^{\text{MF}}=Jz.$$ Below $T_c$, the $m=0$ solution becomes unstable, and the system develops a spontaneous magnetization that grows as $m\sim (T_c-T{)}^{1/2}$ near the transition—a classic mean-field critical exponent.

The susceptibility $\chi$ is found by differentiating the equation of state with respect to $h$. Above $T_c$, and at $h=0$, one finds the Curie-Weiss law: $$\chi \sim \frac{1}{T-T_c},$$ which diverges as $T\to T_c^{+}$, signaling the system's extreme sensitivity to an external field at the critical point. Below $T_c$, the zero-field susceptibility behaves differently, reflecting the stability of the ordered phase.

Common Pitfalls

1. Confusing the mean-field critical temperature with the exact one: A major pitfall is taking $T_c^{\text{MF}}=Jz$ as the true physical critical temperature. For a 2D square lattice ($z=4$), MFT predicts $k_B{T}_c=4J$, but the famous exact solution by Onsager gives $k_B{T}_c\approx 2.269J$. MFT overestimates $T_c$ because it overestimates the strength of ordering by neglecting fluctuations, which are particularly strong in lower dimensions and act to destroy order more easily.

1. Applying MFT in one dimension: Using MFT on a 1D chain ($z=2$) incorrectly predicts a phase transition at $k_B{T}_c=2J$. The exact 1D solution proves no such transition occurs. This stark failure highlights that MFT is a high-dimensional approximation; it completely fails in 1D where fluctuations are dominant.

1. Misinterpreting the self-consistency condition: Students sometimes treat the mean field $Jzm$ as a fixed external parameter. The essence of MFT is the feedback loop: the field determines $m$, and $m$ determines the field. The correct solution is the fixed point where both relations hold simultaneously. Solving the transcendental equation $m=\tanh (\beta Jzm)$ graphically is an excellent way to visualize how solutions appear below $T_c$.

1. Ignoring the range of validity for expansions: The expansion of $\tanh (x)$ used to find $T_c$ and critical exponents is only valid for small $m$, i.e., very close to the critical point. Using this parabolic form far below $T_c$ will give incorrect results. Far from $T_c$, the full non-linear equation must be solved.

Summary

• The Ising model is a cornerstone lattice model of statistical physics, defined by spins with nearest-neighbor interactions, used to study ferromagnetism and phase transitions.
• An exact solution in one dimension via the transfer matrix method proves the absence of a phase transition at any finite temperature, highlighting the role of dimensionality.
• Mean-field theory approximates the interaction on a spin by an average field from its neighbors, leading to a self-consistency equation: $m=\tanh (\beta (Jzm+h))$.
• MFT predicts a phase transition at a critical temperature $k_B{T}_c=Jz$, with spontaneous magnetization ($m\ne 0$) for $T<T_c$ and a diverging susceptibility $\chi \sim 1/(T-T_c)$ near $T_c$.
• While qualitatively insightful, MFT is quantitatively inaccurate in low dimensions because it neglects correlations and fluctuations, leading to an overestimated $T_c$ and incorrect critical exponents.