08.13.08 · stat-mech / exactly-solved-models

The Ising model on the Bethe lattice

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Anchor (Master): Baxter Ch. 4 §4.5–4.8; Bethe 1935 Proc. R. Soc. A 150:552; Eggarter 1974 Phys. Rev. B 9:2989

Intuition Beginner

The square-lattice Ising model is hard because every spin sits inside a maze of loops: you can walk away from a spin along the bonds and come back to where you started. Those loops are exactly what makes the two-dimensional model resist an elementary solution. The Bethe lattice removes them. It is a network in which every site has the same number of neighbours, say $q$ of them, but there are no closed loops at all. Walk away from any spin and you never return. The picture is a branching tree: a central spin with $q$ branches, each branch splitting into $q - 1$ new branches, and so on forever.

Why bother with a model on a tree that no real crystal looks like? Because the no-loop structure lets you solve the Ising model exactly, and the answer behaves like the mean-field guess that physicists reach for first. On the tree, a single spin feels an effective pull from each of its branches, and because the branches never reconnect, the pull from one branch can be computed once and reused. That single idea — summing up a whole branch into one number and feeding it to the next spin — turns an impossible-looking sum over all spin patterns into a short repeated calculation.

The payoff is a clean exactly-solved model that shows a genuine phase transition for $q \geq 3$ , unlike the one-dimensional chain. The temperature at which order sets in, and the way the magnetisation grows just below it, come out in closed form. The tree is the bridge between the crude mean-field approximation and the genuinely hard loop-bearing lattices.

Visual Beginner

A schematic of a Bethe lattice with coordination number $q = 3$ : a central spin connected to three branches, each branch site connected to two further sites, and those to two more, fanning outward without ever closing a loop. Arrows on each branch indicate the effective field that the branch transmits inward toward the centre.

The picture captures the one structural fact that makes the model solvable: because no two branches ever reconnect, the spins on different branches of the central site are statistically independent once the central spin is fixed. Each branch can be replaced by a single effective field acting on the centre, and that field is the same for every branch by symmetry.

Worked example Beginner

Take the smallest interesting case: coordination number $q = 3$ , and find the temperature where order first appears.

Step 1. Each spin has $q = 3$ neighbours. Define the convenient combination $K = J / (k_{B} T)$ , a single dimensionless number that measures how strong the coupling is compared with the temperature. Large $K$ means cold and strongly coupled; small $K$ means hot.

Step 2. The exact condition for the ordered phase to begin on a Bethe lattice is that $(q - 1) tanh K = 1$ . With $q = 3$ this reads $2 tanh K = 1$ , so $tanh K = 1/2$ .

Step 3. Solve for $K$ . The number whose hyperbolic tangent is $1/2$ is $K_{c} = \frac{1}{2} ln 3 \approx 0.5493$ , using the identity that $tanh K = 1/2$ rearranges to $e^{2 K} = 3$ .

Step 4. Translate back to temperature. Since $K = J / (k_{B} T)$ , the critical temperature is $k_{B} T_{c} = J / K_{c} = 2 J / ln 3 \approx 1.820 J$ .

Step 5. Compare with the chain. For a one-dimensional chain every spin has $q = 2$ neighbours, so $(q - 1) tanh K = tanh K$ , which only reaches $1$ as $K \to \infty$ , that is at zero temperature. The chain never orders at a positive temperature. The moment $q$ reaches $3$ , a finite-temperature transition appears.

What this tells us: adding a single extra branch per site — going from a chain to a three-branched tree — is enough to create a genuine phase transition, and the tree hands you the exact transition temperature in one line.

Check your understanding Beginner

Formal definition Intermediate+

The Cayley tree of coordination number $q$ and depth $n$ is the connected acyclic graph built by starting from a central site, attaching $q$ neighbours, and attaching $q - 1$ further neighbours to each newly added site, repeated $n$ times. The Bethe lattice is the bulk (deep interior) of the Cayley tree in the limit $n \to \infty$ : the regular graph in which every site has exactly $q$ neighbours and no cycles occur. The distinction is essential and is taken up in the key theorem: on the Cayley tree a finite fraction of the sites lies on the outermost shell, so surface effects do not vanish, whereas the Bethe lattice refers to the central region where every site is locally identical.

The Ising model on this graph assigns a spin $σ_{i} \in {+ 1, - 1}$ to each site $i$ and energy $$ \mathcal{H}(\sigma) = -J \sum_{\langle i j \rangle} \sigma_i \sigma_j - H \sum_i \sigma_i, $$ where $⟨ ij ⟩$ runs over bonds, $J > 0$ is the ferromagnetic coupling, and $H$ is an external field. Write $K = β J$ and $h = β H$ with $β = 1/ (k_{B} T)$ . The partition function is $Z = \sum_{σ} e^{- β H (σ)}$ .

Fix the central spin $σ_{0}$ and consider one of its $q$ branches. Let $g_{n} (σ_{0})$ denote the sum of Boltzmann weights over all spin configurations on that branch out to depth $n$ , with the root of the branch (the first site below $σ_{0}$ ) summed over and $σ_{0}$ held fixed. Because the branches are disjoint and meet only at $σ_{0}$ , the partition function factorises: $$ Z = \sum_{\sigma_0 = \pm 1} e^{h \sigma_0} , \big[g_n(\sigma_0)\big]^{q}. $$ The single ratio $$ x_n = \frac{g_n(+1)}{g_n(-1)} $$ carries all the information the branch transmits inward, and the recursion that propagates it is the analytic heart of the model.

Counterexamples to common slips

The Bethe lattice is not the whole Cayley tree. The free energy per site of the entire Cayley tree is dominated by its boundary and does not equal the bulk free energy; treating the two as identical produces a wrong (boundary-contaminated) thermodynamic limit. Eggarter 1974 made the distinction precise.
The recursion uses $q - 1$ , not $q$ , in the iterated step. A site one level below the centre has one bond pointing back toward the centre and $q - 1$ bonds continuing outward; using $q$ double-counts the inward bond.
The transition exists only for $q \geq 3$ . The case $q = 2$ is the linear chain, whose recursion map has no symmetry-breaking bifurcation at finite temperature.

Key theorem with proof Intermediate+

Theorem (recursion, critical point, and mean-field exponent). On the Bethe lattice of coordination number $q$ , define the cavity field $x_{n} = g_{n} (+ 1) / g_{n} (- 1)$ . In zero external field this ratio obeys the recursion $$ x_{n+1} = \left( \frac{e^{K} x_n + e^{-K}}{e^{-K} x_n + e^{K}} \right)^{q-1}. $$ The paramagnetic fixed point $x^{⋆} = 1$ loses stability, and a ferromagnetic fixed point $x^{⋆} \neq = 1$ appears, precisely when $$ (q-1)\tanh K = 1. $$ Just below the critical temperature the spontaneous magnetisation vanishes as $m \sim (T_{c} - T)^{1/2}$ , the mean-field value of the exponent $β$ .

Proof. Consider a branch rooted at a site $s$ one level below the central spin. The weight of that branch with $s = \pm 1$ summed over everything beyond satisfies, by attaching one bond from the parent to $s$ and then $q - 1$ sub-branches below $s$ , $$ g_{n+1}(\sigma_0) = \sum_{s = \pm 1} e^{K \sigma_0 s} , \big[g_n(s)\big]^{q-1}. $$ Forming the ratio and writing $x_{n} = g_{n} (+ 1) / g_{n} (- 1)$ , $$ x_{n+1} = \frac{e^{K} , [g_n(+1)]^{q-1} + e^{-K} , [g_n(-1)]^{q-1}}{e^{-K} , [g_n(+1)]^{q-1} + e^{K} , [g_n(-1)]^{q-1}} = \left( \frac{e^{K} x_n + e^{-K}}{e^{-K} x_n + e^{K}} \right)^{q-1}, $$ after dividing numerator and denominator by $[g_{n} (- 1)]^{q - 1}$ . This is the stated recursion. Denote the map $x \mapsto F (x)$ .

The disordered (paramagnetic) state corresponds to the fixed point $x^{⋆} = 1$ , where the two orientations of every branch are equally weighted; substituting $x = 1$ gives $F (1) = 1$ , so it is a fixed point at all temperatures. Its stability is governed by $F^{'} (1)$ . Differentiating, $$ F'(x) = (q-1)\left( \frac{e^K x + e^{-K}}{e^{-K} x + e^K}\right)^{q-2} \cdot \frac{e^{K}(e^{-K}x + e^K) - e^{-K}(e^K x + e^{-K})}{(e^{-K} x + e^K)^2}. $$ At $x = 1$ the first factor is $1$ , and the numerator of the second fraction is $e^{2 K} - e^{- 2 K} = 2 sinh 2 K$ while the denominator is $(e^{- K} + e^{K})^{2} = (2 cosh K)^{2} = 2 (1 + cosh 2 K)$ . Hence $$ F'(1) = (q-1)\frac{2\sinh 2K}{2(1+\cosh 2K)} = (q-1)\tanh K, $$ using $sinh 2 K / (1 + cosh 2 K) = tanh K$ . The fixed point $x^{⋆} = 1$ is stable while $F^{'} (1) < 1$ and unstable once $F^{'} (1) > 1$ . The marginal case $F^{'} (1) = 1$ is the critical point, giving $(q - 1) tanh K_{c} = 1$ .

For the exponent, set $x = 1 + ϵ$ with $ϵ$ small and expand $F$ to third order. The even structure of the map (the symmetry $x \leftrightarrow 1/ x$ inherited from $σ \leftrightarrow - σ$ ) forces the expansion of the symmetric deviation to take the form $ϵ_{n + 1} = a ϵ_{n} - b ϵ_{n}^{3} + \dots$ with $a = F^{'} (1) = (q - 1) tanh K$ and $b > 0$ in a neighbourhood of $K_{c}$ . A nonzero fixed deviation satisfies $ϵ^{2} = (a - 1) / b$ . Near criticality $a - 1 = F^{'} (1) - 1$ is linear in $K - K_{c}$ , hence linear in $T_{c} - T$ , so $ϵ \sim (T_{c} - T)^{1/2}$ . The magnetisation is an odd analytic function of $ϵ$ vanishing with it, so $m \sim ϵ \sim (T_{c} - T)^{1/2}$ , the mean-field exponent $β = 1/2$ . $□$

Bridge. This recursion builds toward every exactly-solved tree and cavity calculation in statistical mechanics, and the foundational reason it closes in one line is the absence of loops: each branch is statistically independent of the others once the spin where they meet is fixed, so a whole branch collapses to the single number $x_{n}$ . This is exactly the structure that reappears under a different name in the mean-field model 08.02.01, where the self-consistency equation $m = tanh (β q J m + h)$ plays the role the fixed-point equation plays here; putting these together, the Bethe-lattice fixed point generalises the mean-field self-consistency by keeping the local fluctuations of a single bond exact while still discarding loop correlations. The central insight is that the loop-free geometry is what makes a local effective field globally consistent, and this same cavity logic appears again in the corner-transfer-matrix recursion of 08.13.04, whose spectrum is likewise generated by an iterated local map. The bridge is the recognition that an exactly-solvable model need not be low-dimensional: the Bethe lattice is effectively infinite-dimensional, which is why its exponents coincide with mean-field theory rather than with the loop-bearing planar Ising model of 08.03.01.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Show that the recursion map $F (x) = (\frac{e ^{K} x + e ^{- K}}{e ^{- K} x + e ^{K}})^{q - 1}$ has the symmetry $F (1/ x) = 1/ F (x)$ , and explain what physical symmetry this encodes.

Hint

Substitute $x \to 1/ x$ and multiply numerator and denominator by $x$ . The ratio $x = g (+) / g (-)$ flips under a global spin flip.

Answer

Replacing $x$ by $1/ x$ : the inner fraction becomes $(e^{K} / x + e^{- K}) / (e^{- K} / x + e^{K})$ . Multiply top and bottom by $x$ to get $(e^{K} + e^{- K} x) / (e^{- K} + e^{K} x)$ , which is the reciprocal of the original inner fraction. Raising to the power $q - 1$ preserves the reciprocal, so $F (1/ x) = 1/ F (x)$ . Physically this is the $σ \to - σ$ symmetry of the zero-field Ising model: flipping every spin exchanges $g (+ 1)$ and $g (- 1)$ , hence inverts $x$ . The symmetric fixed point $x = 1$ is the paramagnet, and the broken-symmetry fixed points come in reciprocal pairs $x^{⋆}, 1/ x^{⋆}$ . Rubric: full credit for the algebra plus the spin-flip interpretation.

Exercise 6 (medium, short-answer).

Explain why the susceptibility $χ = \partial m / \partial H$ diverges as $T \to T_{c}^{+}$ on the Bethe lattice, and identify the exponent $γ$ .

Hint

Linearise the field-driven recursion about $x = 1$ . A small field $h$ produces a response amplified by the geometric series $\sum_{n} [F^{'} (1)]^{n}$ , which diverges as $F^{'} (1) \to 1$ .

Answer

A small field perturbs the cavity ratio by $δ x_{0} \propto h$ at the boundary, and each step inward multiplies the accumulated response by $F^{'} (1) = (q - 1) tanh K$ while adding a fresh field term. Summing the geometric series, the central response is $\propto 1/ [1 - F^{'} (1)]$ . Since $F^{'} (1) - 1$ is linear in $T - T_{c}$ near criticality, $χ \propto 1/ (T - T_{c})$ , so $γ = 1$ , the mean-field value. The divergence is the loss of stability of the paramagnetic fixed point. Rubric: full credit for $γ = 1$ with the geometric-series argument.

Exercise 7 (hard, short-answer).

Show that the free energy per site of the entire Cayley tree of depth $n$ does not converge to the bulk Bethe-lattice value, and explain the role of the boundary.

Hint

Count the sites. The number of sites in the outermost shell of a depth- $n$ Cayley tree with $q = 3$ is comparable to the total number of sites. A finite fraction of all sites lives on the surface.

Answer

The number of sites at shell $k$ grows like $q (q - 1)^{k - 1}$ , so the total through depth $n$ is $1 + q \frac{( q - 1 ) ^{n} - 1}{q - 2}$ , dominated by the last shell: the outermost shell holds a fraction $\to (q - 2) / (q - 1)$ of all sites as $n \to \infty$ . Because surface sites have fewer bonds and a different local environment, their contribution to the extensive free energy does not become negligible, so the whole-tree free energy per site is a boundary-weighted average that differs from the bulk value. The Bethe-lattice free energy is defined as the contribution of the deep interior, obtained either by Eggarter's interior-site bookkeeping or by computing the change in $ln Z$ on adding one bulk site. Rubric: full credit for the shell count plus the boundary-fraction argument.

Exercise 8 (hard, short-answer).

Identify the Bethe-Peierls approximation for a loop-bearing lattice of coordination number $q$ with the exact Bethe-lattice solution, and state why the approximation improves on simple mean field.

Hint

Bethe-Peierls treats one central spin and its $q$ neighbours exactly, replacing the rest of the lattice by an effective field on the neighbours. This is exactly the central cluster of the Cayley tree.

Answer

The Bethe-Peierls approximation treats a cluster of one central spin and its $q$ nearest neighbours with the exact pair Boltzmann weight $e^{K σ_{0} σ_{j}}$ , and represents the rest of the lattice by a self-consistent effective field $h_{eff}$ acting on each neighbour. The self-consistency that the central magnetisation equals the neighbour magnetisation reproduces exactly the Bethe-lattice fixed-point equation $(q - 1) tanh K_{c} = 1$ for the transition. It improves on simple (Curie-Weiss) mean field because the central bond is treated exactly rather than averaged, so nearest-neighbour correlations $⟨ σ_{0} σ_{j} ⟩$ are nonzero above $T_{c}$ , and the predicted $T_{c}$ is lower (closer to the true value) than the mean-field $k_{B} T_{c}^{MF} = q J$ . Rubric: full credit for identifying the central cluster plus the exact-bond improvement.

Advanced results Master

Proposition (exact bulk free energy). Let $κ = ln x^{⋆}$ be the logarithm of the stable cavity ratio and let the per-site free energy of the Bethe lattice be defined by the interior increment $- β f = lim_{n \to \infty} [ln Z_{n} - ln Z_{n - 1}]$ restricted to bulk addition of one site with its $q - 1$ outgoing branches. Then $$ -\beta f = \ln!\Big( e^{K} \cosh!\big[(q-1)\tfrac{\kappa}{q-1} + h\big] + e^{-K}\cosh!\big[\cdots\big]\Big) - \tfrac{q}{2}\ln!\big(e^{K} x^\star{}^{-1} + e^{-K}\big) + \text{(bond correction)}, $$ where the bracketed corrections subtract the over-counted bonds. In zero field the construction reduces to a closed expression in $K$ , $q$ , and the fixed point $x^{⋆} (K)$ , and is analytic in $T$ except at $T_{c}$ , where the second derivative (the specific heat) has a finite jump — a mean-field discontinuity, not the logarithmic divergence of the two-dimensional Ising model.

The cleanest route to the bulk free energy avoids the boundary entirely. Following Eggarter 1974, add a single site to the deep interior together with the bonds and sub-trees required to keep every site $q$ -coordinated; the change in $ln Z$ is the bulk free energy per site, with the surface contribution algebraically removed. The result is exact and matches the recursion-derived critical point.

Proposition (specific-heat jump and absent latent heat). The transition on the Bethe lattice for $q \geq 3$ is continuous (second order): the magnetisation rises from zero as $(T_{c} - T)^{1/2}$ , there is no latent heat, and the specific heat $c (T)$ has a finite discontinuity at $T_{c}$ with $c (T_{c}^{-}) > c (T_{c}^{+})$ . The discontinuity is the Bethe-lattice analogue of the mean-field cusp; the absence of a divergence reflects the effectively infinite dimensionality of the tree, which exceeds the upper critical dimension $d_{c} = 4$ of the Ising universality class.

Proposition (Eggarter boundary dominance). On the full Cayley tree, a finite fraction of the sites lies on the outermost shell for every $q \geq 3$ ; consequently the magnetisation of the whole tree, averaged over all sites, can remain nonzero down to $T = 0$ behaviour driven by boundary fields, and the whole-tree thermodynamic functions differ from the bulk Bethe-lattice functions. This is the precise sense in which "Cayley tree" and "Bethe lattice" name different objects: the first is boundary-dominated, the second is its bulk.

The exponents obtained — $β = 1/2$ , $γ = 1$ , $α = 0$ (jump), $δ = 3$ — are exactly the mean-field exponents, and they satisfy the Rushbrooke and Widom scaling relations with the mean-field gap exponent. The Bethe lattice is therefore the canonical exactly-solved realisation of mean-field critical behaviour: it has genuine short-range interactions and a finite coordination number, yet its loop-free geometry makes fluctuations subcritical and pins the exponents to their classical values. This contrasts sharply with the planar Ising model of 08.03.01, whose loops produce the nonclassical $β = 1/8$ and a logarithmically divergent specific heat.

Synthesis. The Bethe-lattice Ising model is the foundational reason mean-field theory is exact in high effective dimension, and the central insight is that loop-free geometry decouples the branches so that a single cavity recursion captures the entire model. Putting these together, the recursion $x_{n + 1} = F (x_{n})$ , the bifurcation at $(q - 1) tanh K_{c} = 1$ , and the order parameter $m = (x^{⋆ q} - 1) / (x^{⋆ q} + 1)$ form one self-contained exact solution whose every critical exponent is mean-field; this is exactly the classical-exponent regime that the Wilson-Fisher analysis of 08.04.02 recovers above the upper critical dimension. The fixed-point bifurcation here generalises the Curie-Weiss self-consistency of 08.02.01 by keeping each bond exact, and the cavity construction is dual to the message-passing recursions of inference on trees, where the same fixed-point equation reappears as belief propagation. The bridge is the recognition that the tree is effectively infinite-dimensional: its exact solubility and its mean-field exponents are two faces of the same loop-free structure, and the corner-transfer-matrix recursion of 08.13.04 is the loop-bearing refinement that recovers the nonclassical exponents the tree cannot see.

Full proof set Master

Proposition (exact critical coupling), proof. The cavity recursion $x_{n + 1} = F (x_{n})$ with $F (x) = ((e^{K} x + e^{- K}) / (e^{- K} x + e^{K}))^{q - 1}$ has fixed point $x^{⋆} = 1$ for all $K$ , since $F (1) = ((e^{K} + e^{- K}) / (e^{- K} + e^{K}))^{q - 1} = 1$ . Linearising about $x = 1$ , the multiplier is $$ F'(1) = (q-1) \cdot \left.\frac{d}{dx}\frac{e^K x + e^{-K}}{e^{-K}x + e^K}\right|_{x=1} = (q-1)\frac{e^{2K} - e^{-2K}}{(e^{-K}+e^K)^2} = (q-1)\tanh K, $$ where the last equality uses $e^{2 K} - e^{- 2 K} = 2 sinh 2 K$ and $(e^{- K} + e^{K})^{2} = 2 (1 + cosh 2 K)$ together with $sinh 2 K = 2 sinh K cosh K$ and $1 + cosh 2 K = 2 cosh^{2} K$ , so the ratio is $tanh K$ . The paramagnetic fixed point is attracting for $∣ F^{'} (1) ∣ < 1$ and repelling for $F^{'} (1) > 1$ ; since $F^{'} (1)$ increases monotonically from $0$ (at $K = 0$ ) toward $q - 1$ (as $K \to \infty$ ), it crosses $1$ at a unique $K_{c}$ for $q \geq 3$ , determined by $(q - 1) tanh K_{c} = 1$ . For $q = 2$ the supremum of $F^{'} (1)$ is $1$ , reached only as $K \to \infty$ , so no finite-temperature transition occurs. $□$

Proposition (mean-field order-parameter exponent), proof. Write $u = ln x$ so that the map becomes odd in $u$ by the reciprocal symmetry $F (1/ x) = 1/ F (x)$ established in Exercise 3, i.e. $G (u) := ln F (e^{u})$ satisfies $G (- u) = - G (u)$ . An odd smooth map has Taylor expansion $G (u) = a u + c u^{3} + O (u^{5})$ with $a = G^{'} (0) = F^{'} (1) = (q - 1) tanh K$ . A nonzero fixed point of $G$ satisfies $u = a u + c u^{3}$ , hence $u^{2} = (1 - a) / c$ . Near $K_{c}$ , $c < 0$ (so that a real branch exists on the ordered side where $a > 1$ ), and $1 - a = 1 - (q - 1) tanh K$ is a smooth function vanishing linearly at $K_{c}$ with nonzero slope, so $1 - a \propto (K - K_{c}) \propto (T_{c} - T)$ to leading order. Therefore $u^{⋆} \propto (T_{c} - T)^{1/2}$ . The magnetisation $m = (x^{⋆ q} - 1) / (x^{⋆ q} + 1) = tanh (\frac{q}{2} u^{⋆})$ is analytic and odd in $u^{⋆}$ , so $m \propto u^{⋆} \propto (T_{c} - T)^{1/2}$ , giving $β = 1/2$ . $□$

Proposition (susceptibility exponent), proof. In a small uniform field $h$ , linearise the inhomogeneous recursion $x_{n + 1} = F (x_{n}) + h \partial_{h} (\dots)$ about $x = 1$ in the paramagnetic phase. The deviation $δ_{n} = x_{n} - 1$ obeys $δ_{n + 1} = F^{'} (1) δ_{n} + b h + O (δ^{2}, h^{2})$ with a nonzero constant $b$ from the explicit field dependence. Iterating from a fixed boundary value and summing the geometric series in $F^{'} (1) < 1$ gives the steady central deviation $δ^{⋆} = b h / (1 - F^{'} (1))$ . The magnetisation is $m \propto δ^{⋆}$ to linear order, so $χ = \partial m / \partial H \propto 1/ (1 - F^{'} (1))$ . Since $1 - F^{'} (1) = 1 - (q - 1) tanh K \propto (T - T_{c})$ near criticality, $χ \propto (T - T_{c})^{- 1}$ , giving $γ = 1$ . The same computation on the ordered side gives $γ^{'} = 1$ , and at $T_{c}$ the response saturates as $m \propto H^{1/3}$ , i.e. $δ = 3$ , since the cubic term in $G$ then controls the balance $a = 1$ . $□$

Proposition (boundary dominance of the Cayley tree), proof. Label shells $k = 0, 1, \dots, n$ from the centre. Shell $0$ has one site; shell $k \geq 1$ has $N_{k} = q (q - 1)^{k - 1}$ sites. The total is $N = 1 + q \sum_{k = 1}^{n} (q - 1)^{k - 1} = 1 + q \frac{( q - 1 ) ^{n} - 1}{q - 2}$ for $q \geq 3$ . The outer shell has $N_{n} = q (q - 1)^{n - 1}$ , and $$ \frac{N_n}{N} \xrightarrow{n\to\infty} \frac{q(q-1)^{n-1}}{q(q-1)^n/(q-2)} = \frac{q-2}{q-1} > 0. $$ A nonzero fraction of sites is therefore on the boundary for every $q \geq 3$ . Because boundary sites carry one bond and lower coordination than interior sites, the per-site free energy of the whole tree is a fixed convex combination of bulk and surface contributions with strictly positive surface weight, hence differs from the pure-bulk value computed by the interior increment. This is the Eggarter distinction between the Cayley tree and the Bethe lattice. $□$

Connections Master

Mean-field model 08.02.01. The Bethe-lattice fixed-point equation reduces to the Curie-Weiss self-consistency $m = tanh (β q J m)$ in the limit of large coordination number, where $tanh K \approx K$ and the exact treatment of a single bond becomes negligible. The Bethe lattice is the systematic correction to mean field that keeps nearest-neighbour correlations exact, and the Bethe-Peierls cluster approximation for loop-bearing lattices is exactly this tree solution applied as an approximation.
The Ising model 08.01.02. The same spin Hamiltonian $H = - J \sum_{⟨ ij ⟩} σ_{i} σ_{j} - H \sum_{i} σ_{i}$ is placed on a different graph; the loop-free geometry is the sole change, and it converts the intractable lattice sum into an exactly solvable recursion. The Bethe lattice is the cleanest demonstration that the critical behaviour of the Ising model depends on the graph topology, not only on the local couplings.
The Onsager solution 08.03.01. The planar Ising model has loops and therefore nonclassical exponents ( $β = 1/8$ , logarithmic specific heat); the Bethe lattice has none and therefore mean-field exponents ( $β = 1/2$ , specific-heat jump). The contrast isolates loops as the source of nonclassical critical fluctuations, and the two solutions together bound the behaviour of intermediate finite-dimensional lattices.
The corner transfer matrix 08.13.04. Both the cavity recursion of the Bethe lattice and the corner-transfer-matrix recursion of the planar models build the thermodynamic limit by an iterated local map whose fixed point or spectrum encodes the order parameter. The corner transfer matrix is the loop-bearing refinement that recovers the exponents the tree's mean-field recursion cannot capture.
Wilson-Fisher fixed point 08.04.02. The Bethe-lattice exponents are exactly the classical exponents that the renormalisation-group analysis predicts above the upper critical dimension $d_{c} = 4$ . The tree, being effectively infinite-dimensional, sits firmly in the mean-field regime, and its exact solution is a finite-coordination check on the high-dimensional limit of the Wilson-Fisher picture.

Historical & philosophical context Master

Hans Bethe introduced the cluster approximation that bears his name in his 1935 paper on the statistical theory of superlattices (Proc. R. Soc. Lond. A 150:552) ^{[Bethe 1935]}, treating order-disorder transitions in binary alloys by solving a central atom together with its shell of nearest neighbours exactly and closing the system with a self-consistent effective field. Rudolf Peierls reformulated and extended the method to unequal concentrations the following year (Proc. R. Soc. Lond. A 154:207) ^{[Peierls 1936]}, and the combined construction became the Bethe-Peierls approximation. The recognition that the approximation is in fact the exact solution of the Ising model on a Cayley tree came later, as the tree was understood to be the geometry on which the cluster self-consistency closes without error. Baxter's Ch. 4 presents the recursion in its mature form, as the warm-up exactly-solved model before the genuinely two-dimensional chapters.

The subtle point that the Cayley tree and the Bethe lattice are different objects was made precise by Eggarter in 1974 (Phys. Rev. B 9:2989) ^{[Eggarter 1974]}, who showed that the boundary of a Cayley tree holds a finite fraction of its sites and therefore contaminates the naive whole-tree thermodynamic limit; the Bethe-lattice free energy must be defined by the bulk interior. The model occupies a permanent place in the theory of phase transitions as the exactly-solved realisation of mean-field critical behaviour with genuine short-range interactions, and the cavity recursion it introduced was rediscovered as belief propagation in the theory of inference on graphs and as the cavity method in the statistical mechanics of disordered systems and combinatorial optimisation.

Bibliography Master

@book{BaxterExactlySolved,
  author    = {Baxter, Rodney J.},
  title     = {Exactly Solved Models in Statistical Mechanics},
  publisher = {Academic Press},
  address   = {London},
  year      = {1982}
}

@article{Bethe1935Superlattices,
  author  = {Bethe, Hans A.},
  title   = {Statistical theory of superlattices},
  journal = {Proc. R. Soc. Lond. A},
  volume  = {150},
  number  = {871},
  year    = {1935},
  pages   = {552--575}
}

@article{Peierls1936Superlattices,
  author  = {Peierls, Rudolf},
  title   = {Statistical theory of superlattices with unequal concentrations of the components},
  journal = {Proc. R. Soc. Lond. A},
  volume  = {154},
  number  = {881},
  year    = {1936},
  pages   = {207--222}
}

@article{Eggarter1974,
  author  = {Eggarter, T. P.},
  title   = {Cayley trees, the {Ising} problem, and the thermodynamic limit},
  journal = {Phys. Rev. B},
  volume  = {9},
  number  = {7},
  year    = {1974},
  pages   = {2989--2992}
}

@article{MullerHartmann1974,
  author  = {M{\"u}ller-Hartmann, E. and Zittartz, J.},
  title   = {New type of phase transition},
  journal = {Phys. Rev. Lett.},
  volume  = {33},
  number  = {15},
  year    = {1974},
  pages   = {893--897}
}

@book{MezardMontanari2009,
  author    = {M{\'e}zard, Marc and Montanari, Andrea},
  title     = {Information, Physics, and Computation},
  publisher = {Oxford University Press},
  year      = {2009}
}

Prerequisites

08.01.02
08.02.01

Tier anchors

beginner: Baxter Ch. 4 §4.1 the Cayley tree; one spin pulling on its branches
intermediate: Baxter Ch. 4 §4.2–4.4 the recursion relation, magnetisation and free energy of the deep interior
master: Baxter Ch. 4 §4.5–4.8; Bethe 1935 Proc. R. Soc. A 150:552; Eggarter 1974 Phys. Rev. B 9:2989

References

raw/pdfs/kardar-1-3-laws-entropy-potentials.pdf · Kardar statistical physics notes (Ising, mean-field and transfer-matrix background)
quantum-well · statistical physics index
Baxter — Exactly Solved Models in Statistical Mechanics · Ch. 4 (the Ising model on the Bethe lattice; recursion relation; Bethe–Peierls approximation)
Bethe — Statistical theory of superlattices · Proc. R. Soc. Lond. A 150:552 (1935), the Bethe–Peierls cluster approximation
Peierls — Statistical theory of superlattices with unequal concentrations of the components · Proc. R. Soc. Lond. A 154:207 (1936), the pair-cluster reformulation
Eggarter — Cayley trees, the Ising problem, and the thermodynamic limit · Phys. Rev. B 9:2989 (1974), boundary-dominance of the Cayley tree and the interior Bethe-lattice free energy

Estimated time

beginner: 16m
intermediate: 38m
master: 70m