Ising Model for Ferro- and Antiferromagnetism

Hello! Welcome to the first lesson of our new module on the Ising Model.

In the previous module, we concluded our study of the Van der Waals model, a powerful phenomenological theory that gave us deep insights into the liquid-gas phase transition, including the nature of metastable and unstable states. However, the VdW model is a "top-down" theory; it starts from a modified equation of state rather than from first principles.

Today, we shift gears and begin our exploration of the Ising model, which is arguably the most important microscopic model in the study of phase transitions. Our goal for this lesson is to formulate the Ising Hamiltonian for ferromagnetic and antiferromagnetic interactions on a lattice. This model, while seemingly simple, is rich enough to exhibit a phase transition and will serve as our primary theoretical laboratory for understanding critical phenomena. Mastering its formulation is the essential first step before we can attempt to solve it.

The Microscopic Picture: A Lattice of Spins

The Van der Waals model described a fluid in terms of continuous macroscopic variables like pressure and volume. The Ising model takes a different, "bottom-up" approach. It represents a physical system as a discrete lattice of interacting components.

Imagine a crystal lattice, a regular array of sites in 1, 2, or 3 dimensions. At each site, labeled by an index $i$, we place a classical "spin" variable, ${\sigma }_i$. This spin is a highly simplified representation of a magnetic moment and can only point in one of two directions: "up" or "down". We represent these two states with the values:

• ${\sigma }_i=+1$ (spin up)
• ${\sigma }_i=-1$ (spin down)

A complete specification of the state of all $N$ spins on the lattice, $\{{\sigma }_1,{\sigma }_2,\dots ,{\sigma }_N\}$, is called a configuration. The core task of the model is to assign an energy to every possible configuration.

While we use the language of "spins" and magnetism, this model is incredibly versatile. The two states could equally represent:

• The presence or absence of an atom on a lattice site (lattice gas model).
• Two types of atoms (A or B) in a binary alloy.
• The firing or non-firing state of a neuron.

Given your background in condensed matter physics, you'll recognize this as a drastic simplification of real magnetic systems. For instance, the Heisenberg model, which you may have encountered, uses quantum vector spin operators ${\mathbf{S}}_i$. The Ising model simplifies this to a single scalar component, which can be thought of as a system with very strong uniaxial anisotropy, forcing the spins to align only along one axis (e.g., the z-axis).

The Ising Hamiltonian

The energy of a given spin configuration is defined by the Hamiltonian, $\mathcal{H}$. For the Ising model, this Hamiltonian is composed of two main contributions: the interaction energy between the spins themselves and the interaction energy of the spins with an external magnetic field.

We will adopt the following standard form for the Ising Hamiltonian:
$\mathcal{H}=-J{\sum }_{⟨i,j⟩}{\sigma }_i{\sigma }_j-B{\sum }_i{\sigma }_i$

Let's break this down term by term.

1. The Interaction Term: $-J{\sum }_{⟨i,j⟩}{\sigma }_i{\sigma }_j$

   • ${\sigma }_i{\sigma }_j$: This product captures the interaction between two spins at sites $i$ and $j$. If the spins are parallel ($+1\cdot +1$ or $-1\cdot -1$), the product is $+1$. If they are anti-parallel ($+1\cdot -1$), the product is $-1$.
   • $J$: This is the coupling constant or exchange energy. It sets the strength and nature of the interaction. Its units are energy.
   • ${\sum }_{⟨i,j⟩}$: This notation signifies that we are summing over all pairs of nearest neighbors on the lattice. We only count each pair once. For example, on a 1D chain, we sum over pairs $(1,2),(2,3),(3,4),\dots$. On a 2D square lattice, each interior spin has four neighbors. This nearest-neighbor assumption is a simplification but captures the essential physics that interactions are typically short-ranged.

2. The External Field Term: $-B{\sum }_i{\sigma }_i$

   • $B$: This represents the strength of an external magnetic field. For simplicity, we've absorbed physical constants like the magnetic moment $\mu$ into this term, so it also has units of energy.
   • ${\sum }_i{\sigma }_i$: This is the sum over all individual spins. This term describes how each spin couples to the external field. The total magnetization of a configuration is $M={\sum }_i{\sigma }_i$. So this energy term can also be written as $-BM$. A configuration with more spins aligned with the field (larger $M$) will have a lower energy, as expected.

Now we have a complete recipe: for any arrangement of up and down spins on our lattice, we can calculate its total energy $\mathcal{H}$.

Ferromagnetism vs. Antiferromagnetism

The sign of the coupling constant $J$ is crucial, as it determines the preferred alignment of neighboring spins and thus the fundamental nature of the ordered state at low temperatures.

Ferromagnetic Case: $J>0$

If $J$ is positive, the interaction energy for a pair of neighbors is $-J{\sigma }_i{\sigma }_j$. To minimize this energy, we need the product ${\sigma }_i{\sigma }_j$ to be as large and positive as possible, which means ${\sigma }_i{\sigma }_j=+1$. This occurs when the spins are parallel (${\sigma }_i={\sigma }_j$).

• Physical Meaning: A positive $J$ favors the alignment of neighboring spins.
• Low Temperature Behavior: At low temperatures, where the system seeks its lowest energy state, thermal fluctuations are small. The system will tend to order into a ferromagnetic state, where all spins point in the same direction (either all up or all down).

Antiferromagnetic Case: $J<0$

If $J$ is negative, it's convenient to write $J=-|J|$. The interaction energy is then $-(-|J|){\sigma }_i{\sigma }_j=+|J|{\sigma }_i{\sigma }_j$. To minimize this energy, we need the product ${\sigma }_i{\sigma }_j$ to be as large and negative as possible, which means ${\sigma }_i{\sigma }_j=-1$. This occurs when the spins are anti-parallel (${\sigma }_i=-{\sigma }_j$).

• Physical Meaning: A negative $J$ favors the anti-alignment of neighboring spins.
• Low Temperature Behavior: At low temperatures, the system will try to arrange its spins in a staggered, alternating pattern to form an antiferromagnetic state.

Note that for an antiferromagnet, it's not always possible to satisfy all interactions simultaneously. For example, on a 2D triangular lattice, if two neighbors are anti-aligned, the third spin cannot be anti-aligned with both. This leads to a phenomenon called frustration, a topic of great interest in modern condensed matter physics. For the simple lattices we'll consider initially (like square lattices), this is not an issue.

Conclusion

In this lesson, we have laid the foundation for our study of the Ising model. We have moved from the phenomenological world of the Van der Waals equation to a concrete, microscopic model of interacting entities on a lattice.

Key Takeaways:

• The Ising model represents a system as a lattice of classical spins ${\sigma }_i$ that can take values $+1,-1$.
• The total energy of a configuration is given by the Hamiltonian: $\mathcal{H}=-J{\sum }_{⟨i,j⟩}{\sigma }_i{\sigma }_j-B{\sum }_i{\sigma }_i$.
• The term with coupling constant $J$ represents the nearest-neighbor spin-spin interaction energy.
• The term with field $B$ represents the interaction energy with an external magnetic field.
• Ferromagnetism ($J>0$) favors parallel alignment of neighboring spins.
• Antiferromagnetism ($J<0$) favors anti-parallel alignment of neighboring spins.

Preview of the Next Lesson:
Now that we have the Hamiltonian, which defines the energy of any microstate, the central task in statistical mechanics is to compute the partition function: $Z={\sum }_{\text{configs}}{e}^{-\beta \mathcal{H}}$. The sum runs over all $2^N$ possible spin configurations, which is computationally impossible for large systems. However, in the next lesson, we will see that for the one-dimensional case, this sum can be performed exactly. We will introduce the powerful transfer matrix method to solve the 1D Ising model and find a definitive answer to the question: does a phase transition occur in one dimension?