Transfer Matrix Method: Exact Solution of the 1D Ising Model and Onsager's Legacy

Intuition Beginner

Imagine a chain of arrows, each pointing up or down, connected to their neighbours by springs. Each arrow "wants" to align with its neighbour (if the coupling is ferromagnetic) and also wants to point in the direction of an external magnetic field. How do you compute the probability of any arrangement of arrows?

The brute-force approach would be to list every possible configuration and add up their Boltzmann weights 11.04.01. For a chain of $N$ spins, there are $2^N$ configurations. Even for a modest chain of 60 spins, that is about $10^{18}$ terms, far too many to enumerate. The transfer matrix method sidesteps this exponential explosion by recognising that the problem has a local structure: each spin only interacts with its nearest neighbours. This locality lets you rewrite the partition function as a matrix problem.

The key idea is to build a small matrix $T$ (the transfer matrix) that encodes the energy cost of one pair of adjacent spins. For the 1D Ising model, each spin takes two values ($+1$ or $-1$), so $T$ is a $2\times 2$ matrix with four entries. The partition function of the entire chain then equals the trace (sum of diagonal elements) of $T$ raised to the $N$-th power:

$$Z=\mathrm{Tr}(T^N).$$

Why is this useful? Because raising a matrix to a power is governed by its eigenvalues. If $T$ has eigenvalues ${\lambda }_1$ and ${\lambda }_2$, then $T^N$ has eigenvalues ${\lambda }_1^N$ and ${\lambda }_2^N$, and:

$$Z={\lambda }_1^N+{\lambda }_2^N.$$

In the thermodynamic limit ($N\to \infty$), the larger eigenvalue ${\lambda }_1$ completely dominates: $Z\approx {\lambda }_1^N$. The free energy per spin is then $f=-kT\ln {\lambda }_1$. Everything about the thermodynamics -- the free energy, the magnetisation, the specific heat, the correlation between spins at different sites -- follows from this single eigenvalue.

The transfer matrix method is one of the most powerful exact techniques in statistical mechanics. It gives the complete solution of the 1D Ising model with full generality (arbitrary coupling, external field, boundary conditions). It also extends to two dimensions, where it leads to Onsager's celebrated exact solution of the 2D Ising model, one of the landmark achievements of 20th-century physics.

Visual Beginner

The 1D Ising chain consists of $N$ spins $s_1,s_2,\dots ,s_N$ on a ring, each taking the value $+1$ (up) or $-1$ (down). Each bond $(s_i,s_{i+1})$ contributes an energy $-J{s}_i{s}_{i+1}$ to the total.

The transfer matrix $T$ is a $2\times 2$ lookup table: the entry $T_{s_i,s_{i+1}}$ is the Boltzmann weight for the pair $(s_i,s_{i+1})$. The four entries are:

• $T_{++}$ (both up): $e^{\beta J}$
• $T_{--}$ (both down): $e^{\beta J}$
• $T_{+-}$ (up then down): $e^{-\beta J}$
• $T_{-+}$ (down then up): $e^{-\beta J}$

The partition function is obtained by "multiplying through" the chain: each bond contributes one factor of $T$, and connecting the chain into a ring gives the trace.

Worked example Beginner

A 1D Ising chain has $N=3$ spins on a ring (periodic boundary conditions: spin 3 connects back to spin 1), coupling $J=1$, and temperature $kT=1$ (so $\beta =1$). There is no external field.

Step 1: Build the transfer matrix. Each pair $(s_i,s_{i+1})$ contributes a weight $e^{\beta J{s}_i{s}_{i+1}}$. The matrix is:

$$T=\left(\begin{array}{cc}{e}^{\beta J}& e^{-\beta J}\\ e^{-\beta J}& e^{\beta J}\end{array}\right)=\left(\begin{array}{cc}e& e^{-1}\\ e^{-1}& e\end{array}\right).$$

Step 2: Compute eigenvalues. This symmetric matrix has eigenvalues ${\lambda }_1=e+e^{-1}=2\cosh (1)\approx 3.086$ and ${\lambda }_2=e-e^{-1}=2\sinh (1)\approx 2.350$.

Step 3: Compute the partition function. $Z={\lambda }_1^3+{\lambda }_2^3\approx 29.40+12.98=42.38$.

Verification by brute force. There are $2^3=8$ configurations. The energy of each is $E=-s_1{s}_2-s_2{s}_3-s_3{s}_1$ (three bonds on the ring). The all-up and all-down configurations have energy $-3$ (weight $e^3$ each). The remaining six configurations have exactly one frustrated bond and energy $+1$ (weight $e^{-1}$ each). Sum: $2e^3+6e^{-1}\approx 40.17+2.21=42.38$. The transfer matrix result matches exactly.

Check your understanding Beginner

Formal definition Intermediate+

The 1D Ising Hamiltonian and the transfer matrix

The 1D Ising model with periodic boundary conditions has Hamiltonian:

$$H=-J\sum _{i=1}^N{s}_i{s}_{i+1}-h\sum _{i=1}^N{s}_i,$$

where $s_{N+1}\equiv s_1$. The partition function is:

$$Z_N=\sum _{s_1=\pm 1}\cdots \sum _{s_N=\pm 1}\exp \left(\beta J\sum _{i=1}^N{s}_i{s}_{i+1}+\beta h\sum _{i=1}^N{s}_i\right).$$

The Boltzmann weight factorises over bonds when the magnetic field term is split evenly between the two spins at each end of a bond. Define the transfer matrix element:

$$⟨s|T|s^{\prime }⟩=\exp \left(\beta Js{s}^{\prime }+\frac{\beta h}{2}(s+s^{\prime })\right),$$

where $s,s^{\prime }\in \{-1,+1\}$. Then:

$$Z_N=\sum _{s_1,\dots ,s_N}⟨s_1|T|s_2⟩⟨s_2|T|s_3⟩\cdots ⟨s_N|T|s_1⟩=\mathrm{Tr}(T^N).$$

Explicitly:

$$T=\left(\begin{array}{cc}{e}^{\beta J+\beta h}& e^{-\beta J}\\ e^{-\beta J}& e^{\beta J-\beta h}\end{array}\right).$$

Diagonalisation and the partition function

The eigenvalues of $T$ are:

$${\lambda }_{1,2}=e^{\beta J}\cosh (\beta h)\pm \sqrt{e^{2\beta J}{\sinh }^2(\beta h)+e^{-2\beta J}}.$$

The partition function is:

$$Z_N={\lambda }_1^N+{\lambda }_2^N.$$

Since ${\lambda }_1>{\lambda }_2>0$ for all finite $\beta$ and $h$, the thermodynamic limit selects the largest eigenvalue:

$$Z_N={\lambda }_1^N\left(1+{\left(\frac{{\lambda }_2}{{\lambda }_1}\right)}^N\right)\stackrel{N\to \infty }{\to }{\lambda }_1^N.$$

Free energy in the thermodynamic limit

The free energy per spin is:

$$f=-\underset{N\to \infty }{\lim }\frac{1}{N}kT\ln Z_N=-kT\ln {\lambda }_1.$$

For zero field ($h=0$):

$${\lambda }_1=e^{\beta J}+e^{-\beta J}=2\cosh (\beta J),{\lambda }_2=e^{\beta J}-e^{-\beta J}=2\sinh (\beta J).$$

The exact free energy per spin is:

$$f=-kT\ln (2\cosh \beta J).$$

This function is analytic in $T$ for all $T>0$: no singularity, no phase transition.

Magnetisation and susceptibility

The magnetisation per spin follows from $m=-\partial f/\partial h$. For the zero-field case, $m=0$ at all $T>0$ by the symmetry $h\to -h$. The zero-field susceptibility is:

$$\chi =\frac{\partial m}{\partial h}{|}_{h=0}=\beta e^{2\beta J}.$$

At high temperature ($\beta J\ll 1$): $\chi \approx \beta$ (Curie law, independent spin response). At low temperature ($\beta J\gg 1$): $\chi$ grows exponentially, signalling the onset of long-range order as $T\to 0$.

Spin-spin correlation function

The two-point correlation function is:

$$⟨s_i{s}_j⟩=\frac{{\sum }_{\{s\}}{s}_i{s}_j{e}^{-\beta H}}{Z_N}.$$

Using the transfer matrix spectral decomposition, this evaluates to:

$$⟨s_0{s}_r⟩=(\tanh \beta J{)}^r,r=|i-j|.$$

Defining the correlation length $\xi$ via $⟨s_0{s}_r⟩\sim e^{-r/\xi }$:

$${\xi }^{-1}=-\ln \tanh (\beta J).$$

At high temperature: $\tanh (\beta J)\approx \beta J\ll 1$, so $\xi \approx -1/\ln (\beta J)\approx 1$ (short-range correlations). At low temperature: $\tanh (\beta J)\to 1$, so $\xi \to \infty$ (correlation length diverges). The divergence occurs only at $T=0$, confirming the absence of a finite-temperature phase transition.

Specific heat

The internal energy per spin at $h=0$ is:

$$u=\frac{\partial (\beta f)}{\partial \beta }=-J\tanh (\beta J).$$

The specific heat per spin is:

$$\frac{c}{k}=(\beta J{)}^2{\mathrm{sech}}^2(\beta J).$$

The specific heat has a broad peak near $kT\approx 0.83J$ (a Schottky-type anomaly reflecting the two-level character of individual bonds) but remains finite at all temperatures. There is no divergence, providing a third signature that no finite-$T$ phase transition occurs in 1D.

Absence of a finite-temperature phase transition

The 1D Ising model with nearest-neighbour interactions has no phase transition at any $T>0$. Three independent lines of evidence converge on this conclusion:

1. Correlation function. $⟨s_0{s}_r⟩=(\tanh \beta J{)}^r$ decays exponentially for all $T>0$. Long-range order never develops.

2. Free energy. $f=-kT\ln (2\cosh \beta J)$ is analytic in $T$ for all $T>0$, with no singularity.

3. Peierls argument. Creating a domain wall (a boundary between an "up" region and a "down" region) costs energy $2J$ but gains entropy $k\ln N$ (the wall can be placed at $N$ positions). The free energy cost is $\Delta F=2J-kT\ln N$, which is negative for sufficiently large $N$ at any $T>0$. Domain walls are always favourable in 1D, destroying long-range order.

The phase transition occurs only at $T=0$, where $\xi \to \infty$ and correlations become long-range. This result, first established by Ising (1925), motivated the question of whether higher-dimensional Ising models could exhibit finite-temperature transitions, a question answered definitively by Onsager (1944) for the 2D case.

Key derivation: Transfer matrix diagonalisation Intermediate+

Proposition. The partition function of the 1D Ising model on a ring of $N$ spins with coupling $J$ and field $h$ is $Z_N={\lambda }_1^N+{\lambda }_2^N$ where ${\lambda }_{1,2}=e^{\beta J}\cosh (\beta h)\pm \sqrt{e^{2\beta J}{\sinh }^2(\beta h)+e^{-2\beta J}}$. The free energy per spin in the thermodynamic limit is $f=-kT\ln {\lambda }_1$.

Proof. The trace and determinant of $T$ are:

$$\mathrm{tr}T=2e^{\beta J}\cosh (\beta h),\det T=e^{2\beta J+2\beta h}\cdot e^{\beta J-\beta h}-e^{-2\beta J}\cdot e^{-2\beta J}=e^{2\beta J}-e^{-2\beta J}=2\sinh (2\beta J).$$

The eigenvalues satisfy the characteristic equation ${\lambda }^2-(\mathrm{tr}T)\lambda +\det T=0$:

$${\lambda }_{1,2}=\frac{\mathrm{tr}T\pm \sqrt{(\mathrm{tr}T{)}^2-4\det T}}{2}.$$

Expanding the discriminant:

$$(\mathrm{tr}T{)}^2-4\det T=4e^{2\beta J}{\cosh }^2(\beta h)-8\sinh (2\beta J).$$

Using ${\cosh }^2(\beta h)=1+{\sinh }^2(\beta h)$ and $2\sinh (2\beta J)=e^{2\beta J}-e^{-2\beta J}$:

$$4e^{2\beta J}(1+{\sinh }^2\beta h)-4(e^{2\beta J}-e^{-2\beta J})=4e^{2\beta J}{\sinh }^2(\beta h)+4e^{-2\beta J}.$$

Hence:

$${\lambda }_{1,2}=e^{\beta J}\cosh (\beta h)\pm \sqrt{e^{2\beta J}{\sinh }^2(\beta h)+e^{-2\beta J}}.$$

The square root exceeds $e^{-\beta J}$, so ${\lambda }_1>e^{\beta J}\cosh (\beta h)\ge e^{\beta J}>0$ and ${\lambda }_2>0$ (since ${\lambda }_1{\lambda }_2=\det T>0$). With ${\lambda }_1>{\lambda }_2$, the thermodynamic limit gives $f=-kT\ln {\lambda }_1$. $\square$

Bridge. The transfer matrix method reduces the thermodynamics of the 1D Ising model to a linear-algebra problem: compute the eigenvalues of a $2\times 2$ matrix. This structure persists in higher dimensions, where the transfer matrix becomes a $2^L\times 2^L$ matrix (with $L$ the system size in the transverse direction). The 2D Ising model on an $L\times N$ lattice has a $2^L\times 2^L$ transfer matrix, and Onsager's solution comes from diagonalising this matrix in the limit $L\to \infty$ using spinor methods.

Exercises Intermediate+

Lean formalization Intermediate+

The transfer matrix solution of the 1D Ising model requires: (1) the Ising Hamiltonian on a periodic chain as a formal lattice structure; (2) the transfer matrix as a $2\times 2$ real matrix parametrised by $\beta$, $J$, $h$; (3) the identification $Z=\mathrm{Tr}(T^N)$ via the Gibbs measure; (4) eigenvalue computation and the thermodynamic limit $f=-kT\ln {\lambda }_1$; (5) the spin-spin correlation via the spectral decomposition. Mathlib has the linear algebra (eigenvalues, trace, matrix powers) but not the statistical-mechanical framework. The gap is well-defined and could be bridged by a formal Gibbs-measure construction for lattice systems.

Advanced results Master

Onsager's solution of the 2D Ising model

On September 1, 1944, Lars Onsager published the exact free energy of the square-lattice Ising model with zero external field. The result had been announced orally at a meeting of the New York Academy of Sciences in 1942 and took two more years to appear in print. The free energy per spin is:

$$f=-kT\left[\frac{\ln 2}{2}+\frac{1}{2\pi }{\int }_0^{\pi }d{\omega }_1{\int }_0^{\pi }d{\omega }_2\ln \left({\cosh }^2(2\beta J)-\sinh (2\beta J)(\cos {\omega }_1+\cos {\omega }_2)\right)\right].$$

The critical temperature is determined by the condition that the argument of the logarithm first vanishes, which occurs when $\sinh (2{\beta }_{c}J)=1$:

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

At $T_c$, the free energy has a logarithmic singularity in the specific heat:

$$c\sim -\frac{2}{\pi }{\left(\frac{2J}{k{T}_c}\right)}^2\ln |T-T_c|+\text{regular terms},$$

corresponding to the critical exponent $\alpha =0$ (logarithmic divergence rather than power law).

The significance of Onsager's result can hardly be overstated. It was the first exact solution of a model exhibiting a genuine phase transition in two dimensions. It demonstrated that:

• Mean-field theory gives incorrect critical exponents in $d=2$. The exact exponents differ from the Landau/mean-field predictions in every measurable quantity.
• The specific heat diverges logarithmically, a weaker singularity than the power-law divergence predicted by mean-field theory ($\alpha =0$ jump).
• The free energy is non-analytic at $T_c$, confirming that phase transitions correspond to genuine singularities in thermodynamic functions rather than mathematical artefacts.

On the method. Onsager used the transfer matrix in the row-to-row direction. For an $L\times N$ lattice with periodic boundary conditions in both directions, the transfer matrix is a $2^L\times 2^L$ matrix that propagates one row of $L$ spins to the next. Onsager diagonalised this matrix using what he described only as "a certain group of transformations" -- in modern language, a sequence of rotations through the Clifford algebra that maps the transfer matrix to a product of independent two-level systems. Kaufman (1949) later simplified the derivation using spinor analysis, making the connection to free fermions explicit and accessible.

Spontaneous magnetisation: Yang's result

Onsager's 1944 paper gave the free energy but not the spontaneous magnetisation. At a conference in 1949, Onsager announced the result on the blackboard during a discussion following a talk by Gregory Wannier, writing only the formula without proof:

$$M={\left[1-\frac{1}{{\sinh }^4(2\beta J)}\right]}^{1/8},T<T_c.$$

C.N. Yang published the first complete derivation in 1952 ("The Spontaneous Magnetization of a Two-Dimensional Ising Model," Phys. Rev. 85, 808--816). Yang's proof uses the transfer matrix and the asymptotic analysis of Toeplitz determinants (Szego's limit theorem). The critical exponent is $\beta =1/8$: near $T_c$, expanding $\sinh (2\beta J)\approx 1+const.\times (T_c-T)$ gives $M\sim (T_c-T{)}^{1/8}$. This value, dramatically different from the mean-field prediction $\beta =1/2$, demonstrated that fluctuations in two dimensions modify the critical behaviour in an essential way.

Critical exponents from the exact 2D Ising solution

The exact critical exponents of the 2D Ising model on a square lattice are:

Exponent	Observable	Exact value	Mean-field
$\alpha$	Specific heat	0 (log divergence)	0 (finite jump)
$\beta$	Magnetisation	1/8	1/2
$\gamma$	Susceptibility	7/4	1
$\delta$	Critical isotherm	15	3
$\nu$	Correlation length	1	1/2
$\eta$	Correlation function at $T_c$	1/4	0

These exponents satisfy all four scaling relations exactly. The Rushbrooke relation: $\alpha +2\beta +\gamma =0+1/4+7/4=2$. The Widom relation: $\gamma =\beta (\delta -1)=(1/8)(14)=7/4$. The Fisher relation: $\gamma =(2-\eta )\nu =(7/4)(1)=7/4$. The Josephson (hyperscaling) relation: $d\nu =2\cdot 1=2=2-\alpha$. The fact that hyperscaling is satisfied confirms that $d=2$ is below the upper critical dimension $d_c=4$, where fluctuations dominate and modify the critical exponents.

The exponent $\delta =15$ is particularly striking. At $T=T_c$, the magnetisation satisfies $M\sim h^{1/15}$ for small $h$, an extremely flat response compared to the mean-field prediction $h^{1/3}$.

Combinatorial methods: Kasteleyn cities and Pfaffians

An alternative route to the 2D Ising partition function, developed independently by Kasteleyn, Fisher, and Temperley, uses a combinatorial (graph-theoretic) approach. The partition function is mapped to a counting problem: the number of ways to arrange "dimers" (dominoes covering two adjacent sites) on a related lattice. This dimer model has an exact solution via the Pfaffian of a certain antisymmetric matrix (the Kasteleyn matrix).

A Pfaffian is a polynomial expression in the entries of an antisymmetric matrix $A$, related to the determinant by $\mathrm{Pf}(A{)}^2=\det (A)$. For a planar graph with $n$ vertices, the dimer partition function equals $|\mathrm{Pf}(A)|$, where $A$ is the signed adjacency matrix constructed according to Kasteleyn's rules (orient the edges so that every face has an odd number of clockwise-oriented edges). The Pfaffian can be computed in polynomial time, making this approach efficient as well as exact.

The Kasteleyn approach also illuminates the connection between the 2D Ising model and free fermions. The transfer matrix of the 2D Ising model can be written as a product of fermionic operators via the Jordan-Wigner transformation, and the Pfaffian solution makes this connection explicit. This fermionic representation is the starting point for the conformal field theory description of the 2D Ising critical point.

Quantum-to-classical correspondence: Trotter-Suzuki decomposition

There is a deep connection between classical statistical mechanics in $d$ dimensions and quantum mechanics in $d-1$ dimensions. The 1D classical Ising model maps to a 0D quantum system (a single quantum spin-1/2), and the 2D classical Ising model maps to a 1D quantum system (the transverse-field Ising chain).

The mapping works as follows. Write the transfer matrix of the 1D classical Ising model in the form:

$$T=e^{K_0}{e}^{h_x{\sigma }^x+h_z{\sigma }^z},$$

where $K_0$, $h_x$, $h_z$ depend on $\beta J$ and $\beta h$, and ${\sigma }^x$, ${\sigma }^z$ are Pauli matrices. The partition function $Z=\mathrm{Tr}(T^N)$ is then the partition function of a single quantum spin-1/2 evolving through $N$ imaginary-time steps at inverse temperature proportional to $N$. The eigenvalues of $T$ are the energy levels of the effective quantum Hamiltonian $H_{\mathrm{eff}}=-h_x{\sigma }^x-h_z{\sigma }^z$ (up to an additive constant).

In two dimensions, the transfer matrix of the classical Ising model on an $L\times N$ lattice maps to the Hamiltonian of a 1D quantum Ising chain of $L$ spins in a transverse field:

$$H_{\mathrm{quantum}}=-\sum _{i=1}^L\left[J_q{\sigma }_i^z{\sigma }_{i+1}^z+g{\sigma }_i^x\right],$$

where $J_q$ and $g$ are related to $\beta J$ of the classical model. The quantum phase transition at $g=J_q$ corresponds to the classical thermal phase transition at $T=T_c$. The Trotter-Suzuki decomposition (Trotter 1959, Suzuki 1976) provides the formal basis: a $d$-dimensional classical partition function can be written as the imaginary-time path integral of a $(d-1)$-dimensional quantum system.

This correspondence means that exact solutions of classical models yield exact results for quantum models, and vice versa. The 2D classical Ising model and the 1D quantum transverse-field Ising chain share the same critical exponents, a manifestation of the quantum-classical mapping that has become central to condensed matter physics and quantum information theory.

Full proof set Master

Proposition. The free energy of the 2D Ising model on a square lattice with zero field has a logarithmic singularity at $T_c=2J/(k\ln (1+\sqrt{2}))$, giving the critical exponent $\alpha =0$.

Proof (sketch). The transfer matrix in the row direction is a $2^L\times 2^L$ matrix $T_L$ acting on the space of row spin configurations. Using the Jordan-Wigner transformation (Kaufman 1949), the $2^L$-dimensional spin space is mapped to a Fock space of $L$ fermionic modes. The transfer matrix factorises into a product over independent two-level systems labelled by momenta $q_j=\pi j/(L+1)$ for $j=1,\dots ,L$:

$$ϵ(q)=2\cosh (2\beta J)\pm 2\sqrt{{\sinh }^2(2\beta J)+{\sin }^{2}q}.$$

The larger eigenvalue at each $q$ is ${ϵ}_{+}(q)=2\cosh (2\beta J)+2\sqrt{{\sinh }^2(2\beta J)+{\sin }^{2}q}$. The free energy per spin in the thermodynamic limit is:

$$f=-kT\left[\frac{\ln 2}{2}+\frac{1}{2\pi }{\int }_0^{\pi }dq\ln \left(\cosh (2\beta J)+\sqrt{{\sinh }^2(2\beta J)+{\sin }^{2}q}\right)\right].$$

After integration over $q$ this reduces to the Onsager form. Differentiating twice with respect to $T$ gives the specific heat. The integral has a singularity when the argument of the square root has a double zero at $q=0$, which occurs when $\sinh (2\beta J)=1$, i.e., at ${\beta }_c=\ln (1+\sqrt{2})/(2J)$. Near ${\beta }_c$, the singularity is logarithmic:

$$c\sim -\frac{2}{\pi }{\left(\frac{2J}{k{T}_c}\right)}^2\ln |T-T_c|+\text{regular terms}.$$

The logarithmic divergence corresponds to the critical exponent $\alpha =0$. $\square$

Proposition (Yang 1952). The spontaneous magnetisation of the 2D Ising model on a square lattice is $M=[1-{\sinh }^{-4}(2\beta J){]}^{1/8}$ for $T<T_c$.

Proof (sketch). Yang computed the spin-spin correlation function $⟨{\sigma }_{0,0}{\sigma }_{N,N}⟩$ using the transfer matrix and Toeplitz determinant methods. The correlation function is expressed as a determinant of an $N\times N$ Toeplitz matrix whose symbol has a discontinuity at $T_c$. The limit $M^2={\lim }_{N\to \infty }⟨{\sigma }_{0,0}{\sigma }_{N,N}⟩$ is evaluated using Szego's strong limit theorem for Toeplitz determinants, modified to handle the discontinuous symbol (the Fisher-Hartwig conjecture, in modern terminology). The result $M=[1-{\kappa }^2{]}^{1/8}$ where $\kappa =1/{\sinh }^2(2\beta J)$ follows from the analysis of the Wiener-Hopf factorisation of the symbol. The critical exponent $\beta =1/8$ follows: near $T_c$, $\sinh (2\beta J)\approx 1+const.\times (T_c-T)$, so $\kappa \approx 1-const.\times (T_c-T)$ and $M\sim (T_c-T{)}^{1/8}$. $\square$

Connections Master

• 11.06.01 The Ising model was introduced in 11.06.01 as the simplest lattice model exhibiting phase transitions. This unit provides the exact solution for the 1D case (Ising 1925) and the Onsager solution for the 2D case, establishing the baseline against which approximate methods are measured.
• 11.06.02 Mean-field theory gives critical exponents ($\beta =1/2$, $\gamma =1$) that differ from the exact 2D Ising values ($\beta =1/8$, $\gamma =7/4$). The discrepancy quantifies the role of fluctuations in low dimensions.
• 11.06.03 Landau theory predicts $\beta =1/2$ for the order parameter; the exact 2D Ising result $\beta =1/8$ demonstrates the failure of the Landau approach in $d=2<d_c=4$.
• 11.06.04 The Ginzburg criterion identifies $d_c=4$ as the upper critical dimension below which fluctuations modify the critical exponents. The exact 2D Ising exponents provide a concrete worked example in $d=2<d_c$.
• 11.07.01 The renormalization group must reproduce the exact 2D Ising exponents as a consistency check. The Wilson-Fisher fixed point in $d=2$ gives the exact Onsager exponents, providing the most stringent test of the RG framework.

Historical and philosophical context Master

Ernst Ising (1900--1998) solved the one-dimensional version of the model that bears his name in his 1924 doctoral dissertation under Wilhelm Lenz at the University of Hamburg. Ising's 1925 paper ("Beitrag zur Theorie des Ferromagnetismus," Z. Phys. 31, 253--258) showed that the 1D model has no phase transition at any finite temperature. Ising himself incorrectly extrapolated this result to higher dimensions, concluding that the model could not describe ferromagnetism in any dimension. This pessimistic conclusion was shared by many physicists through the late 1920s and early 1930s.

Rudolf Peierls gave a heuristic argument in 1936 that the 2D Ising model does have a phase transition, by showing that domain walls cost more free energy than they gain in two dimensions (unlike in one dimension). This reversed Ising's conclusion and motivated the search for an exact solution.

Lars Onsager (1903--1976), already famous for his reciprocal relations in irreversible thermodynamics (which would earn him the Nobel Prize in Chemistry in 1968), solved the 2D Ising model in 1944. His paper ("Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition," Phys. Rev. 65, 117--149) is one of the most remarkable calculations in theoretical physics. Densely written in idiosyncratic notation, it left many readers unable to follow the details. Onsager never provided a pedagogical exposition of his methods.

Bruria Kaufman (1918--1970), a mathematician and physicist who had been Onsager's student at Columbia, simplified the derivation in 1949 using spinor analysis (Clifford algebra). Her approach ("Crystal Statistics. II. Partition Function Evaluated by Spinor Analysis," Phys. Rev. 76, 1232--1243) replaced Onsager's opaque transformations with the transparent language of anticommuting operators, making the connection to free fermions explicit. Kaufman's work is a landmark in its own right and established the spinor approach as the standard tool for transfer-matrix diagonalisation.

The spontaneous magnetisation was not included in Onsager's 1944 paper. At a conference at Cornell in 1949, Onsager announced the result on the blackboard during the discussion following a talk by Wannier, writing $M\propto (1-k^2{)}^{1/8}$ with $k=1/\sinh (2K)$ and $K=\beta J$. He published only an abstract, never a full derivation. The first published proof is due to C.N. Yang in 1952, who used Toeplitz determinant methods. Yang later described it as the longest calculation of his career.

The exact 2D Ising solution has had profound consequences far beyond statistical mechanics. The conformal field theory of the 2D Ising critical point (Belavin, Polyakov, and Zamolodchikov, 1984) is one of the simplest and most deeply understood conformal field theories, with connections to string theory and the AdS/CFT correspondence. The identification of the transfer matrix with a quantum Hamiltonian (via the Trotter-Suzuki decomposition) established the quantum-classical correspondence that is now central to quantum information theory and condensed matter physics. The exact critical exponents serve as the benchmark for every approximate method in statistical physics, from mean-field theory to the renormalization group.

Bibliography Master

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}

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}

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}

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