Transfer Matrix Method for the 2D Ising Model

Hello! Welcome to the next lesson in our course on the statistical mechanics of phase transitions.

In our previous two lessons, we thoroughly explored the one-dimensional Ising model. We saw how the transfer matrix method allowed for an exact solution, and from this solution, we derived a crucial result: the 1D Ising model with short-range interactions does not exhibit a phase transition at any finite temperature ($T>0$). The core reason was the analyticity of the free energy, guaranteed by the properties of the transfer matrix's eigenvalues.

Today, we take a major step up in complexity and physical relevance by moving to two dimensions. As you intuited in the last lesson's exercise, dimensionality is key. The 2D Ising model does have a finite-temperature phase transition, a discovery that marked a watershed moment in statistical physics. While the full solution is famously complex, our goal today is to build the essential machinery for it. We will generalize the transfer matrix method from a 1D chain to a 2D lattice. This process will unveil a profound and beautiful connection between 2D classical statistical mechanics and 1D quantum mechanics, a theme that resonates deeply with modern condensed matter and quantum field theory.

This lesson will cover our learning outcome: to outline the transfer matrix formalism for the two-dimensional Ising model.

Setting up the 2D Ising Model

Let's consider a 2D square lattice with $M$ rows and $N$ columns. Each site $(i,j)$ is occupied by a spin ${\sigma }_{i,j}=\pm 1$. The spins interact with their nearest neighbors. We'll allow for anisotropy, meaning the horizontal interaction coupling $J_x$ can differ from the vertical one $J_y$.

The Hamiltonian for the system is:

$$H=-J_x\sum _{i=1}^N\sum _{j=1}^M{\sigma }_{i,j}{\sigma }_{i+1,j}-J_y\sum _{i=1}^N\sum _{j=1}^M{\sigma }_{i,j}{\sigma }_{i,j+1}$$

We assume periodic boundary conditions, so ${\sigma }_{N+1,j}={\sigma }_{1,j}$ and ${\sigma }_{i,M+1}={\sigma }_{i,j}$. As before, we'll use the dimensionless couplings $K_x=\beta J_x$ and $K_y=\beta J_y$. The partition function is the sum over all $2^{MN}$ spin configurations:

$$Z=\sum _{\{\sigma \}}{e}^{-\beta H}=\sum _{\{\sigma \}}\exp \left(K_x\sum _{i,j}{\sigma }_{i,j}{\sigma }_{i+1,j}+K_y\sum _{i,j}{\sigma }_{i,j}{\sigma }_{i,j+1}\right)$$

Generalizing the Transfer Matrix

In the 1D case, the transfer matrix $T$ "transferred" the calculation from one spin to the next. For our 2D lattice, we can think of the vertical direction as a kind of "time" axis. The transfer matrix will evolve the system from one entire row to the next.

Let's define the state of a single row $j$ as the configuration of its $N$ spins: $|{\varphi }_j⟩=|{\sigma }_{1,j},{\sigma }_{2,j},\dots ,{\sigma }_{N,j}⟩$. Since each of the $N$ spins can be up or down, there are $2^N$ possible configurations for a row. Our state vectors $|{\varphi }_j⟩$ live in a $2^N$-dimensional Hilbert space.

The transfer matrix $T$ will be a $2^N\times 2^N$ matrix whose elements $⟨{\varphi }_{j+1}|T|{\varphi }_j⟩$ encapsulate the total Boltzmann weight for the interactions within row $j$ and the interactions between row $j$ and row $j+1$.

If we can construct such a matrix, the partition function, which involves summing over all $M$ rows, becomes a trace over the product of $M$ transfer matrices:

$$Z=\sum _{{\varphi }_1,\dots ,{\varphi }_M}⟨{\varphi }_1|T|{\varphi }_M⟩\dots ⟨{\varphi }_3|T|{\varphi }_2⟩⟨{\varphi }_2|T|{\varphi }_1⟩=\text{Tr}(T^M)$$

This is the same beautiful simplification we achieved in 1D. In the thermodynamic limit ($M\to \infty$), the free energy per row will be dominated by the largest eigenvalue ${\Lambda }_{\text{max}}$ of $T$:

$$F=-k_{B}T\ln Z\approx -k_{B}T\ln ({\Lambda }_{\text{max}}^M)=-M{k}_{B}T\ln {\Lambda }_{\text{max}}$$

The problem of calculating the partition function is thus reduced to finding the largest eigenvalue of the $2^N\times 2^N$ transfer matrix $T$.

Constructing the 2D Transfer Matrix

How do we build this $2^N\times 2^N$ matrix? The key is to split the Boltzmann weight into two parts: one for the horizontal (intra-row) interactions and one for the vertical (inter-row) interactions. We will represent each part as an operator acting on the $2^N$-dimensional space of row configurations.

This approach, pioneered by Onsager and Kaufman, elegantly maps the problem onto the language of quantum mechanics.

Let's summarize and formalize the construction outlined in the notes. We define two operators, $V_1$ and $V_2$, that act on the $2^N$-dimensional space of row configurations.

1. The Intra-Row Operator ($V_1$)

This operator accounts for the interactions within a row. The corresponding part of the Boltzmann factor for a row in state $|\varphi ⟩=|{\sigma }_1,\dots ,{\sigma }_N⟩$ is $\exp (K_x{\sum }_i{\sigma }_i{\sigma }_{i+1})$.

This operator is diagonal in the basis of spin configurations $|\varphi ⟩$. To express it in quantum mechanical language, we can associate the spin values ${\sigma }_i=\pm 1$ with the eigenvalues of the Pauli ${\sigma }_i^z$ operator acting at site $i$ of our 1D quantum chain. The operator $V_1$ can then be written as:

$$V_1=\exp \left(K_x\sum _{i=1}^N{\sigma }_i^z{\sigma }_{i+1}^z\right)$$

Here, ${\sigma }_i^z$ is understood as the tensor product $\mathbf{I}\otimes \cdots \otimes {\sigma }^z\otimes \cdots \otimes \mathbf{I}$, where ${\sigma }^z$ is at the $i$-th position.

2. The Inter-Row Operator ($V_2$)

This operator connects row $j$ (state $|\varphi ⟩$) to row $j+1$ (state $|{\varphi }^{\prime }⟩$). The Boltzmann factor is $\exp (K_y{\sum }_i{\sigma }_i^{\prime }{\sigma }_i)$. This operator is not diagonal. Its role is to "flip" spins between rows, much like a quantum evolution operator.

As shown in the resource, this interaction can be cleverly rewritten. For each site $i$, the term $e^{K_y{\sigma }_i^{\prime }{\sigma }_i}$ is an element of a $2\times 2$ matrix connecting the spin at site $i$ in one row to the next. This matrix is:

$$\left(\begin{array}{cc}{e}^{K_y}& e^{-K_y}\\ e^{-K_y}& e^{K_y}\end{array}\right)=e^{K_y}\mathbf{I}+e^{-K_y}{\sigma }^x$$

Using the Kramers-Wannier duality relation $\tanh (K_y^{\ast })=e^{-2K_y}$, this matrix can be expressed (up to a constant prefactor) as $\exp (K_y^{\ast }{\sigma }^x)$. The operator $V_2$ that acts on the entire row is the product of these operators for each site:

$$V_2\propto \prod _{i=1}^N\exp (K_y^{\ast }{\sigma }_i^x)=\exp \left(K_y^{\ast }\sum _{i=1}^N{\sigma }_i^x\right)$$

(The prefactor, $(2\sinh 2K_y{)}^{N/2}$, can be carried along, but for understanding the operator structure, we can absorb it into the definition of the transfer matrix.)

3. The Full Transfer Matrix $T$

The total transfer matrix that evolves the system from one row to the next is the product of these two operators:

$$T=V_2^{1/2}{V}_1{V}_2^{1/2}$$

A symmetric form like this is often preferred as it leads to a Hermitian operator, which has real eigenvalues. However, since the trace is cyclically invariant, $\text{Tr}((V_1{V}_2{)}^M)=\text{Tr}((V_2^{1/2}{V}_1{V}_2^{1/2}{)}^M)$, so we can also work with the simpler, non-symmetric form $T^{\prime }=V_1{V}_2$.

Using the expressions we found (and dropping the * on $K_y$ for simplicity, as it's just a reparameterization), the transfer matrix is the product of two non-commuting operators:

$$T\approx \exp \left(K_x\sum _{i=1}^N{\sigma }_i^z{\sigma }_{i+1}^z\right)\exp \left(K_y^{\ast }\sum _{i=1}^N{\sigma }_i^x\right)$$

The Connection to Quantum Mechanics

This final expression is profound. The problem of finding the partition function of a 2D classical statistical system is equivalent to finding the eigenvalues of $T$. Notice the structure of $T$: it's composed of two exponentials of operators that do not commute, since $[{\sigma }^x,{\sigma }^z]\ne 0$. This is precisely the structure of a quantum mechanical evolution operator in imaginary time, $e^{-\beta H}$, where $H$ itself contains non-commuting parts.

In a specific limit (strong coupling vertically, $K_y\to \infty$, and weak coupling horizontally, $K_x\to 0$), the transfer matrix $T$ can be shown to be equivalent to the evolution operator $e^{-\Delta \tau H_Q}$ of a 1D quantum spin chain described by the transverse-field Ising Hamiltonian:

$$H_Q=-J\sum _i{\sigma }_i^z{\sigma }_{i+1}^z-h\sum _i{\sigma }_i^x$$

Here, the horizontal coupling $J_x$ plays the role of the quantum interaction strength $J$, and the vertical coupling $K_y$ maps to the transverse magnetic field $h$.

This mapping means that the phase transition in the 2D classical model corresponds to a quantum phase transition in the ground state of the 1D quantum model, which occurs at $T=0$ as the parameter $h/J$ is varied. This equivalence is a powerful tool, allowing insights from quantum field theory to be applied to classical critical phenomena, and vice versa.

Test your understanding!

In our construction, the transfer was from row to row (vertically). What would the operators $V_1$ (intra-"row") and $V_2$ (inter-"row") look like if we decided to build the transfer matrix to transfer from column to column (horizontally)?

Show answer

If we transfer from column to column, the roles of $x$ and $y$ are swapped.

• A "state" would be the configuration of a column of $M$ spins. The Hilbert space would be $2^M$-dimensional.
• The intra-"row" operator, which now corresponds to the interactions within a column, would be $V_1=\exp \left(K_y{\sum }_{j=1}^M{\sigma }_j^z{\sigma }_{j+1}^z\right)$.
• The inter-"row" operator, which now corresponds to interactions between columns, would be $V_2\approx \exp \left(K_x^{\ast }{\sum }_{j=1}^M{\sigma }_j^x\right)$.

The structure is identical, just with $N\leftrightarrow M$ and $K_x\leftrightarrow K_y$. This highlights the underlying symmetry of the lattice.

Conclusion

Today we have constructed the essential theoretical tool for analyzing the 2D Ising model: the transfer matrix.

Key Takeaways:

• The transfer matrix formalism can be extended to 2D by treating an entire row of $N$ spins as the fundamental state. This leads to a $2^N\times 2^N$ matrix $T$.
• The partition function is once again given by $Z=\text{Tr}(T^M)$, and the free energy is determined by the largest eigenvalue of $T$.
• The transfer matrix $T$ can be constructed as the product of two operators: $V_1$, representing interactions within a row, and $V_2$, representing interactions between rows.
• By expressing these operators using Pauli matrices, we revealed that the 2D classical Ising model is mathematically equivalent to a 1D quantum Ising model in a transverse field. A thermal phase transition in the classical system maps to a quantum phase transition in the 1D quantum system.

This formalism is the starting point for the exact solution. Finding the eigenvalues of this $2^N\times 2^N$ matrix is a formidable task, but one that was famously accomplished by Lars Onsager in 1944.

Preview of the Next Lesson:
We will not go through the complex derivation of the eigenvalues ourselves. Instead, we will state Onsager's celebrated exact solution for the free energy of the 2D Ising model. We will analyze this solution to see precisely how and where the non-analyticity signifying a phase transition appears, and we will finally be able to calculate the critical temperature.