08.13.05 · stat-mech / exactly-solved-models

The hard-hexagon model (Baxter 1980)

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Anchor (Master): Baxter Ch. 14 §14.5–14.8; Baxter 1980 J. Phys. A 13:L61; Andrews–Baxter–Forrester 1984 J. Stat. Phys. 35:193

Intuition Beginner

Picture a triangular grid of sites, and a supply of identical pieces shaped to cover one site each. You drop pieces onto sites at random, but with one rule: no two pieces may sit on neighbouring sites. Each piece keeps its six nearest neighbours empty. Because the forbidden region around an occupied site looks like a hexagon, this is the hard-hexagon model: a lattice gas of particles that exclude one another out to nearest-neighbour distance on the triangular lattice.

A single number controls how crowded the lattice gets: the fugacity $z$ , which rewards each placed particle. When $z$ is small, particles are rare and scattered, and the lattice looks the same on average everywhere. When $z$ is large, the particles want to pack as tightly as the exclusion rule allows. The densest packing fills one of three interleaved sub-grids, leaving the other two empty, so the lattice spontaneously picks a favoured sub-grid and the symmetry between the three breaks.

Why does anyone care about a packing puzzle? Because Baxter solved it exactly in 1980, and the answer turned out to be a famous identity from number theory in disguise. The density at which the lattice orders, and the order parameter that measures the broken symmetry, are written in the same products that Rogers and Ramanujan studied a century earlier. A counting problem about hexagons and a counting problem about partitions of integers are the same problem.

Visual Beginner

The figure shows a patch of the triangular lattice partitioned into three interleaved sub-lattices, drawn in three shades. Particles (filled discs) sit only on the sites of one favoured sub-lattice; every site of the other two sub-lattices stays empty, and no two filled discs are nearest neighbours. A faint hexagon is drawn around one particle to mark the six neighbour sites it forbids.

Reading the picture: in the disordered phase at low fugacity the three shades are occupied equally on average and the picture would look uniform. As the fugacity is raised past its critical value, one shade wins and the lattice locks into the pattern shown. The order parameter measures how strongly one sub-lattice is preferred over the other two.

Worked example Beginner

Take the smallest interesting piece of the triangular lattice: a single up-triangle of three mutually adjacent sites, labelled $1$ , $2$ , $3$ . Because all three are nearest neighbours, the hard-hexagon rule forbids any two of them from being occupied at the same time. Count the allowed configurations and weight each by the fugacity $z$ for every particle present.

The empty configuration has no particles, weight $1$ . There are three configurations with a single particle (on site $1$ , or $2$ , or $3$ ), each of weight $z$ . No configuration with two or three particles is allowed, because any pair among the three sites is adjacent. So the partition function of this tiny system is $Z_{△} = 1 + 3 z .$

The average number of particles on the triangle is the fugacity-weighted count: $⟨ N ⟩ = \frac{0 \cdot 1 + 1 \cdot ( 3 z )}{1 + 3 z} = \frac{3 z}{1 + 3 z} .$ At $z = 1$ this gives $⟨ N ⟩ = 3/4$ , an average of three-quarters of one particle shared across three sites, so a density of $1/4$ per site.

What this tells us: even the three-site system already shows the exclusion at work — the doubly and triply occupied terms are absent, which is the whole content of "hard". The full lattice does the same bookkeeping over the entire triangular grid, where the competition between the fugacity reward and the exclusion penalty produces a genuine ordering transition.

Check your understanding Beginner

Formal definition Intermediate+

Let $T$ be the two-dimensional triangular lattice with site set $V$ and nearest-neighbour edge set $E$ . A hard-hexagon configuration is an occupation map $σ : V \to {0, 1}$ , with $σ_{i} = 1$ meaning site $i$ is occupied, subject to the hard-core constraint $σ_{i} σ_{j} = 0$ for every edge ${i, j} \in E$ : no two adjacent sites are simultaneously occupied. The grand partition function at fugacity (activity) $z > 0$ is $$ \Xi(z) = \sum_{\sigma} z^{,\sum_i \sigma_i} \prod_{{i,j}\in E} (1 - \sigma_i \sigma_j), $$ the product enforcing the exclusion by killing every configuration with an adjacent occupied pair. The density is $ρ = lim_{∣ V ∣ \to \infty} ∣ V ∣^{- 1} \sum_{i} ⟨ σ_{i} ⟩$ , and the free energy per site is $κ (z) = lim_{∣ V ∣ \to \infty} ∣ V ∣^{- 1} lo g Ξ (z)$ .

The triangular lattice decomposes into three sub-lattices $A, B, C$ , each a triangular lattice of one-third the density, such that no two sites of a single sub-lattice are adjacent. Write $ρ_{A}, ρ_{B}, ρ_{C}$ for the sub-lattice densities $⟨ σ_{i} ⟩$ on each. The order parameter is $$ R = \rho_A - \rho_B \quad (\text{with the convention } \rho_B = \rho_C \text{ in an } A\text{-ordered state}), $$ which vanishes in the disordered phase ( $ρ_{A} = ρ_{B} = ρ_{C}$ ) and is positive in the ordered phase, where sub-lattice $A$ is preferentially occupied.

Baxter's solution embeds the hard-hexagon model into a one-parameter family of commuting transfer matrices by assigning the model a set of Boltzmann weights that satisfy the star–triangle relation of 08.13.01. The exclusion product is recovered as the $z$ -dependent limit of weights parametrised by Jacobi elliptic functions of a spectral variable $u$ and nome $q$ ; the physical fugacity $z$ is recovered from the elliptic modulus, and the corner transfer matrix of 08.13.04 computes the order parameter as a ratio of two infinite products in $q$ .

Counterexamples to common slips

The hard-hexagon model is not the hard-square model. On the square lattice, nearest-neighbour exclusion (hard squares) is not known to be exactly solvable — Baxter's star–triangle integrability is special to the triangular geometry, where the three-site face supports the star–triangle relation. The hard-square model has been studied numerically and by series expansion but has no closed-form free energy.
The exclusion is nearest-neighbour only. A particle forbids its six neighbours but does not forbid the next ring of sites; two particles two steps apart are allowed. Strengthening the exclusion to second neighbours changes the model and destroys the Rogers–Ramanujan structure.
The order parameter $R$ is a difference of sub-lattice densities, not the total density $ρ$ . The total density is analytic across the transition in the variable that matters for ordering; it is the sub-lattice imbalance that develops a singularity, with exponent $β = 1/9$ .

Key derivation Intermediate+

Theorem (Baxter 1980; hard-hexagon order parameter). Let $z$ exceed the critical fugacity $z_{c} = (11 + 55) /2$ , so the model is in the ordered phase. Introduce the nome $q \in (0, 1)$ through Baxter's elliptic parametrisation of the fugacity. The corner-transfer-matrix order parameter is $$ R(q) ;=; \prod_{n=1}^{\infty} \frac{(1 - q^{5n-4})(1 - q^{5n-1})}{(1 - q^{5n-3})(1 - q^{5n-2})}, $$ and the two competing sub-lattice densities are the products $ρ_{A}, ρ_{B}$ built from the same family. The fugacity is recovered from $q$ by $z = - q^{- 1} \prod_{n \geq 1} [(1 - q^{5 n - 4}) (1 - q^{5 n - 1}) / ((1 - q^{5 n - 3}) (1 - q^{5 n - 2}))]^{5}$ -type relations, and as $q \to 1^{-}$ the order parameter $R$ vanishes, marking the critical point.

Derivation. The route runs through the corner transfer matrix of 08.13.04. Baxter first finds Boltzmann weights for the hard-hexagon faces that satisfy the star–triangle relation of 08.13.01. The triangular-lattice partition function is then written in the corner geometry as $Ξ = tr (A B C D)$ , a trace of four corner transfer matrices, each summing the weights of one angular quadrant with its boundary occupations held fixed.

Because the weights solve the star–triangle relation, each corner transfer matrix has the exponential form $A (u) = exp (- u H_{CTM})$ with a spectral-parameter-linear exponent, exactly as established for the general solvable model in 08.13.04. The generator $H_{CTM}$ has an equally spaced spectrum: its eigenvalues form an arithmetic progression $ϵ (m_{1} + m_{2} + \dots)$ over occupation numbers, so the diagonalised corner trace becomes a sum of products of geometric factors $x = e^{- ϵ}$ , with $x$ a power of the nome $q$ .

Carrying out the corner trace, the order parameter is the ratio of two single-variable infinite products. After the diagonalisation, the sub-lattice density on $A$ is computed as a graded sum over the corner spectrum, and the imbalance $R = ρ_{A} - ρ_{B}$ collapses to $$ R(q) = \frac{\prod_{n\ge 1}(1 - q^{5n-4})(1 - q^{5n-1})}{\prod_{n\ge 1}(1 - q^{5n-3})(1 - q^{5n-2})}. $$ The exponents $5 n - 4, 5 n - 1, 5 n - 3, 5 n - 2$ — residues $1, 4$ over $3, 2$ modulo $5$ — are the signature of the Rogers–Ramanujan identities: the very products that Rogers (1894) and later Ramanujan proved equal to the sum-side $q$ -series. The appearance of modulus $5$ is forced by the three-state structure of the model combined with the corner spectrum, and it fixes the universality class.

The critical fugacity follows by tracking where $R$ vanishes. As $q \to 1$ the elliptic functions degenerate and the product $R (q) \to 0$ ; transcribing the limit back through Baxter's fugacity relation gives the algebraic value $z_{c} = (11 + 55) /2 \approx 11.09$ . Below $z_{c}$ the model is disordered and $R = 0$ ; above it, $R > 0$ with the leading singularity $R \sim (z - z_{c})^{β}$ , $β = 1/9$ . $□$

Bridge. This derivation builds toward the central identification of the whole exactly-solved-models programme: an order parameter is a corner-transfer-matrix $q$ -series, and a $q$ -series with the right modular signature is a number-theoretic identity. The foundational reason the hard-hexagon density is computable is exactly the geometric spectrum of 08.13.04 — the same equally spaced corner spectrum that delivered the Ising magnetisation $M = (1 - k^{- 2})^{1/8}$ here delivers a Rogers–Ramanujan product, and this is exactly the structure that appears again in 08.13.03 for the eight-vertex staggered polarisation. The central insight is that the star–triangle relation of 08.13.01 is what forces the corner spectrum to be arithmetic, and the arithmetic spectrum is what makes the trace resum into an infinite product. Putting these together, the hard-hexagon solution identifies the physics of sub-lattice ordering with the combinatorics of integer partitions into parts of fixed residue mod $5$ : the bridge is the Rogers–Ramanujan identity, read in one direction by the statistical mechanic and in the other by the number theorist.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

State the first Rogers–Ramanujan identity in product-and-sum form, and identify which residues mod $5$ appear in its product side. Relate them to the hard-hexagon order-parameter exponents.

Hint

The first identity has product side $\prod_{n \geq 0} (1 - q^{5 n + 1})^{- 1} (1 - q^{5 n + 4})^{- 1}$ and sum side $\sum_{n \geq 0} q^{n^{2}} / (q; q)_{n}$ .

Answer

The first Rogers–Ramanujan identity is $n = 0 \sum \infty \frac{q ^{n^{2}}}{( q ; q ) _{n}} = n = 0 \prod \infty \frac{1}{( 1 - q ^{5 n + 1} ) ( 1 - q ^{5 n + 4} )},$ where $(q; q)_{n} = \prod_{k = 1}^{n} (1 - q^{k})$ . The product side uses residues $1$ and $4$ modulo $5$ . The hard-hexagon order parameter $R (q)$ is built from the same residue classes ( $1, 4$ versus $2, 3$ modulo $5$ ): the numerator carries $5 n - 4 \equiv 1$ and $5 n - 1 \equiv 4$ , the denominator $5 n - 3 \equiv 2$ and $5 n - 2 \equiv 3$ . The modulus- $5$ partition structure of Rogers–Ramanujan is exactly the structure of the hard-hexagon corner-transfer-matrix series. Rubric: full credit for the identity plus the residue identification.

Exercise 5 (medium, short-answer).

Explain why the row-to-row transfer matrix of 08.03.02 does not give the hard-hexagon order parameter as directly as the corner transfer matrix does.

Hint

The order parameter is a local quantity at one site; the corner construction places that site at the meeting of the four quadrants.

Answer

The row transfer matrix builds the partition sum one horizontal strip at a time, so its largest eigenvalue controls the bulk free energy but a single central spin's expectation is buried in the full eigenvector and is awkward to extract. The corner transfer matrix slices the lattice into four quadrants meeting at one central site, putting the site whose occupation defines the order parameter at the apex of all four corners. The order parameter then becomes a graded trace of the diagonalised corner operators, which for star–triangle weights has an arithmetic spectrum and resums to a closed-form $q$ -product. Rubric: full credit for identifying the central-site geometry and the geometric spectrum as the reason.

Exercise 6 (hard, short-answer).

The hard-hexagon transition is continuous, in the universality class with central charge $c = 4/5$ and density exponent $β = 1/9$ . Explain, in terms of the symmetry that breaks, why this class differs from the Ising class ( $c = 1/2$ , $β = 1/8$ ) of the corner-transfer-matrix Ising magnetisation in 08.13.04.

Hint

The hard-hexagon ordered phase chooses one of three sub-lattices; count the broken-symmetry states and compare to the two-state Ising case.

Answer

The Ising transition breaks a two-state ( $Z_{2}$ ) symmetry between up- and down-magnetised phases, placing it in the $c = 1/2$ Ising class with $β = 1/8$ . The hard-hexagon ordered phase chooses one of three interleaved sub-lattices, so it breaks a three-state ( $Z_{3}$ ) symmetry; the relevant fixed point is the three-state Potts / $c = 4/5$ minimal model, whose order-parameter exponent is $β = 1/9$ . The different number of degenerate ordered states (two versus three) changes the universality class, even though both order parameters come out of the same corner-transfer-matrix machinery. The modulus- $5$ Rogers–Ramanujan products encode the $c = 4/5$ structure just as the Ising elliptic series encodes $c = 1/2$ . Rubric: full credit for tying the $Z_{3}$ three-sub-lattice symmetry to the distinct universality class.

Exercise 7 (hard, short-answer).

Sketch how the Andrews–Baxter–Forrester (1984) restricted solid-on-solid (RSOS) models generalise the hard-hexagon order parameter into a hierarchy of Rogers–Ramanujan-type identities.

Hint

The RSOS models have heights restricted to $1, \dots, L$ ; hard hexagons is the case $L = 4$ in regime III, and varying $L$ produces the Andrews–Gordon identities.

Answer

Andrews, Baxter, and Forrester introduced a family of restricted solid-on-solid (RSOS) models on the square lattice, with integer heights constrained to ${1, \dots, L}$ and adjacent heights differing by one. Solving these by the corner transfer matrix produces order-parameter $q$ -series that are exactly the Andrews–Gordon generalisations of the Rogers–Ramanujan identities, indexed by $L$ and by the regime (I–IV). Hard hexagons sits at $L = 4$ in regime III, where the two Rogers–Ramanujan identities (moduli $5$ ) appear; larger $L$ gives moduli $2 L + 1$ and the full Andrews–Gordon hierarchy. The corner-transfer-matrix calculation thus became a machine for producing — and conjecturally proving — partition identities, turning Baxter's lattice method into a tool in additive number theory. Rubric: full credit for the height-restriction picture plus the $L = 4$ /modulus- $5$ placement of hard hexagons in the hierarchy.

Advanced results Master

Theorem (Baxter 1980; exact free energy and density). The hard-hexagon free energy per site $κ (z)$ and the density $ρ (z)$ are given in closed form by infinite products in the nome $q$ , with the fugacity $z$ recovered from $q$ through Baxter's elliptic relation. The model has a single critical point at $z_{c} = (11 + 55) /2$ , separating a disordered low-fugacity phase ( $z < z_{c}$ , $ρ_{A} = ρ_{B} = ρ_{C}$ , $R = 0$ ) from an ordered high-fugacity phase ( $z > z_{c}$ , $ρ_{A} > ρ_{B} = ρ_{C}$ , $R > 0$ ).

The free energy is analytic except at $z_{c}$ , where the density has an infinite-slope inflection and the order parameter rises from zero. The critical density is $ρ_{c} = (5 - 5) /10 \approx 0.276$ .

Theorem (Rogers–Ramanujan form of the order parameter). In the ordered regime the order parameter is the ratio of Rogers–Ramanujan products $$ R(q) = \prod_{n=1}^{\infty}\frac{(1-q^{5n-4})(1-q^{5n-1})}{(1-q^{5n-3})(1-q^{5n-2})}, $$ and by the two Rogers–Ramanujan identities each product equals a sum-side $q$ -series $$ G(q) = \sum_{n\ge0}\frac{q^{n^2}}{(q;q)n} = \prod{n\ge0}\frac{1}{(1-q^{5n+1})(1-q^{5n+4})},\qquad H(q) = \sum_{n\ge0}\frac{q^{n^2+n}}{(q;q)n} = \prod{n\ge0}\frac{1}{(1-q^{5n+2})(1-q^{5n+3})}. $$ The order-parameter and sub-lattice-density series are rational in $G$ and $H$ , so the hard-hexagon ordering is governed entirely by the Rogers–Ramanujan functions.

Theorem (critical exponents and the $c = 4/5$ universality class). The hard-hexagon transition is continuous, with density-imbalance exponent $β = 1/9$ and correlation-length exponent $ν = 5/6$ . These are the exponents of the three-state Potts universality class, the $c = 4/5$ unitary minimal model of conformal field theory, consistent with the $Z_{3}$ sub-lattice symmetry that breaks at the transition.

The exponent $β = 1/9$ is not a $Z_{2}$ -Ising value; it reflects the three-fold degeneracy of the ordered state. The continuous transition with these exponents was a sharp confirmation that exactly solvable lattice models reproduce the conformal-field-theory predictions for their universality classes.

Theorem (Andrews–Baxter–Forrester hierarchy). The hard-hexagon model is the $L = 4$ , regime-III member of the RSOS family of Andrews, Baxter, and Forrester (1984). Solving the RSOS models by the corner transfer matrix yields order-parameter series equal to the Andrews–Gordon identities, an infinite family of Rogers–Ramanujan-type identities of modulus $2 L + 1$ , with hard hexagons supplying the modulus- $5$ base case.

This embedding turned Baxter's corner-transfer-matrix method into a source of partition identities: the physical requirement that the corner trace converge to a sub-lattice density forced the $q$ -series into Rogers–Ramanujan–Andrews–Gordon form, and the resulting identities were later given independent combinatorial (Bailey-chain) proofs.

Synthesis. The hard-hexagon model is the foundational reason the exactly-solved-models programme touches additive number theory. The central insight is that the corner transfer matrix of 08.13.04, whose arithmetic spectrum is forced by the star–triangle relation of 08.13.01, produces an order parameter that is exactly a Rogers–Ramanujan product, and this is exactly the same mechanism that gave the Ising magnetisation $M = (1 - k^{- 2})^{1/8}$ and the eight-vertex staggered polarisation of 08.13.03. Putting these together, the three headline corner-transfer-matrix order parameters — Ising, eight-vertex, hard-hexagon — are one calculation specialised to three models, and the hard-hexagon case identifies the physics of $Z_{3}$ sub-lattice ordering with the combinatorics of integer partitions into parts of fixed residue modulo $5$ . The bridge is the Rogers–Ramanujan identity: a statement about lattice-gas densities on the physics side and a statement about partition-generating functions on the number-theory side, the same product read two ways. The Andrews–Baxter–Forrester hierarchy generalises this single identification into an infinite tower, so that the corner-transfer-matrix method becomes a machine that manufactures Rogers–Ramanujan-type identities indexed by the height restriction $L$ , and the $c = 4/5$ universality class of the transition ties the whole structure to the three-state Potts conformal field theory.

Full proof set Master

Proposition (densest packing and maximum density). The maximum density of the hard-hexagon model is $ρ_{m a x} = 1/3$ , attained exactly by full occupation of one of the three sub-lattices.

Proof. Partition $V$ into the three sub-lattices $A, B, C$ , each a triangular lattice whose own sites are pairwise non-adjacent (each sub-lattice has lattice spacing $3$ times the original, so no two of its sites share an edge of $T$ ). Fully occupying one sub-lattice, say $A$ , places a particle on every third site with no two adjacent, so it is an allowed configuration of density $1/3$ . For the upper bound, consider any allowed configuration and any elementary up-triangle ${i, j, k}$ of three mutually adjacent sites. The exclusion forbids more than one of $i, j, k$ from being occupied, so each up-triangle carries at most one particle. The up-triangles tile $V$ with one triangle per site on average (each site belongs to two up-triangles, and there are $∣ V ∣$ up-triangles for $∣ V ∣$ sites in the thermodynamic limit), giving $\sum_{i} σ_{i} \leq ∣ V ∣/3$ . Hence $ρ \leq 1/3$ , with equality realised by the sub-lattice packing. $□$

Proposition (critical fugacity as an algebraic number). The critical fugacity satisfies $z_{c}^{2} - 11 z_{c} - 1 = 0$ with positive root $z_{c} = (11 + 55) /2$ .

Proof. At criticality the nome reaches the boundary value $q \to 1^{-}$ where the order parameter $R (q) \to 0$ . Baxter's elliptic relation between the fugacity and the nome, expanded about $q = 1$ via the modular ( $q \to \tilde{q}$ ) transformation of the relevant theta constants, gives $z$ as an algebraic function of the modulus that degenerates to a root of a quadratic at the critical modulus. Tracking the degeneration, $z_{c}$ is the larger root of $x^{2} - 11 x - 1 = 0$ . The discriminant is $121 + 4 = 125 = 25 \cdot 5$ , so $125 = 55$ and the roots are $(11 \pm 55) /2$ . The physical (positive, $> 1$ ) root is $z_{c} = (11 + 55) /2 \approx 11.09$ ; the conjugate root $(11 - 55) /2 \approx - 0.09$ is unphysical as a fugacity. That $z_{c}$ is a quadratic irrational in $5$ — rather than a transcendental — is the fingerprint of the modulus- $5$ Rogers–Ramanujan structure. $□$

Proposition (order parameter and Rogers–Ramanujan functions). The order parameter $R (q)$ is expressible through the Rogers–Ramanujan functions $G (q)$ and $H (q)$ , and consequently the sub-lattice imbalance is controlled entirely by the modulus- $5$ partition functions.

Proof. Write the Euler products $(q; q)_{\infty} = \prod_{n \geq 1} (1 - q^{n})$ . Grouping the factors of $R (q)$ by residue class modulo $5$ , $$ R(q) = \frac{\prod_{n\ge1}(1-q^{5n-4}),\prod_{n\ge1}(1-q^{5n-1})}{\prod_{n\ge1}(1-q^{5n-3}),\prod_{n\ge1}(1-q^{5n-2})}. $$ By the product side of the two Rogers–Ramanujan identities, $\prod_{n \geq 0} (1 - q^{5 n + 1})^{- 1} (1 - q^{5 n + 4})^{- 1} = G (q)$ and $\prod_{n \geq 0} (1 - q^{5 n + 2})^{- 1} (1 - q^{5 n + 3})^{- 1} = H (q)$ . The numerator of $R$ is the reciprocal of $G (q)$ (residues $1, 4$ ) and the denominator is the reciprocal of $H (q)$ (residues $2, 3$ ), so $$ R(q) = \frac{H(q)}{G(q)}. $$ Both $G$ and $H$ equal their sum-side $q$ -series $\sum_{n} q^{n^{2}} / (q; q)_{n}$ and $\sum_{n} q^{n^{2} + n} / (q; q)_{n}$ by the Rogers–Ramanujan identities, which were first proved by Rogers (1894) and rediscovered by Ramanujan; the equality $R = H / G$ makes the order parameter the Rogers–Ramanujan continued fraction's defining ratio, since the continued fraction $1 + q / (1 + q^{2} / (1 + \dots))$ equals $H (q) / G (q)$ up to a factor $q^{1/5}$ . The sub-lattice ordering is therefore a function of the two modulus- $5$ partition generating functions alone. $□$

Theorem (free energy and density), stated without full proof — see Baxter 1980 J. Phys. A 13 and Baxter 1982 Ch. 14 ^{[Baxter 1980]}. The closed forms for $κ (z)$ and $ρ (z)$ as products in $q$ , together with the modular relation $z = z (q)$ , require the full elliptic-function apparatus of Baxter Ch. 14 — the theta-constant identities of 06.06.05, the inversion relation, and the corner-transfer-matrix diagonalisation — and are stated here with the order-parameter computation carrying the structural content.

Connections Master

The hard-hexagon order parameter is computed by the corner transfer matrix of 08.13.04: the partition function factorises as $tr (A B C D)$ , the corner operators have the exponential form with an arithmetic spectrum forced by integrability, and the graded corner trace resums to the Rogers–Ramanujan product $R = H / G$ . This unit is the second headline application of that method, after the Ising magnetisation; the hard-hexagon case is the one where the corner $q$ -series turns out to be a named number-theoretic object.

The integrability that underlies the solution is the star–triangle / Yang–Baxter relation of 08.13.01, read for the three-site faces of the triangular lattice. The same relation that placed the eight-vertex and six-vertex models in commuting families places the hard-hexagon weights in a commuting family, and it is this commutation that forces the corner-transfer-matrix spectrum to be equally spaced — the property that makes the trace resum to a product at all.

The row-to-row transfer matrix of 08.03.02 is the operator whose largest eigenvalue gives the hard-hexagon free energy; the corner transfer matrix is its angular companion that reaches the order parameter the row matrix cannot. The two operators commute as members of the same star–triangle family, and the spectral parameter that drives the row solution reappears as the angular variable of the corner construction.

The elliptic parametrisation of the fugacity and the modular degeneration at criticality use the Jacobi theta constants of 06.06.05: the nome $q$ of the order-parameter products is the elliptic nome, the critical limit $q \to 1$ is the modular ( $τ \to 0$ ) degeneration, and the algebraic critical fugacity $z_{c} = (11 + 55) /2$ is the value the modulus- $5$ theta-constant ratios reach at that limit.

Historical & philosophical context Master

Rodney Baxter announced the exact solution of the hard-hexagon model in a 1980 letter to the Journal of Physics A, titled "Hard hexagons: exact solution," where he reported the free energy, the density, the critical fugacity $z_{c} = (11 + 55) /2$ , and the order parameter as an infinite product, solved by the corner-transfer-matrix method he had developed for the eight-vertex model ^{[Baxter 1980]}. The hard-hexagon lattice gas had been studied numerically for years as a model of adsorbed monolayers — helium or noble gases on a graphite substrate, where the substrate's triangular symmetry and the atoms' mutual repulsion realise the nearest-neighbour exclusion — and series expansions had estimated its critical point, but no closed form was known until Baxter found that the model's Boltzmann weights satisfy the star–triangle relation.

The appearance of the Rogers–Ramanujan products was the surprise. The identities had been proved by Leonard James Rogers in an 1894 memoir in the Proceedings of the London Math Society, then rediscovered and made famous by Srinivasa Ramanujan around 1913 ^{[Rogers 1894]}; for decades they had lived in additive number theory, with combinatorial interpretations as partition-counting statements (partitions into parts differing by at least two, versus partitions into parts congruent to $1, 4$ modulo $5$ ). Baxter's calculation produced exactly these products as the sub-lattice densities of a physical lattice gas. George Andrews, Baxter, and Peter Forrester then showed in 1984 that hard hexagons is the base case of an infinite RSOS hierarchy whose order parameters are the Andrews–Gordon generalisations of the Rogers–Ramanujan identities, indexed by a height restriction $L$ , with modulus $2 L + 1$ ^{[Andrews-Baxter-Forrester 1984]}. The corner-transfer-matrix method thereby became a manufacturing process for partition identities, and the physical convergence of the corner trace gave a route to identities whose only prior proofs had been purely combinatorial.

Bibliography Master

@article{Baxter1980hardhexagons,
  author  = {Baxter, Rodney J.},
  title   = {Hard Hexagons: Exact Solution},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {13},
  number  = {3},
  year    = {1980},
  pages   = {L61--L70}
}

@book{Baxter1982,
  author    = {Baxter, Rodney J.},
  title     = {Exactly Solved Models in Statistical Mechanics},
  publisher = {Academic Press},
  address   = {London},
  year      = {1982}
}

@article{ABF1984,
  author  = {Andrews, George E. and Baxter, Rodney J. and Forrester, Peter J.},
  title   = {Eight-Vertex SOS Model and Generalized Rogers--Ramanujan-Type Identities},
  journal = {Journal of Statistical Physics},
  volume  = {35},
  year    = {1984},
  pages   = {193--266}
}

@article{Rogers1894,
  author  = {Rogers, Leonard J.},
  title   = {Second Memoir on the Expansion of Certain Infinite Products},
  journal = {Proceedings of the London Mathematical Society},
  volume  = {25},
  year    = {1894},
  pages   = {318--343}
}

@book{AndrewsPartitions,
  author    = {Andrews, George E.},
  title     = {The Theory of Partitions},
  publisher = {Addison-Wesley},
  series    = {Encyclopedia of Mathematics and its Applications},
  volume    = {2},
  year      = {1976}
}

@article{Baxter1982hh,
  author  = {Baxter, Rodney J. and Pearce, Paul A.},
  title   = {Hard Hexagons: Interfacial Tension and Correlation Length},
  journal = {Journal of Physics A: Mathematical and General},
  volume  = {15},
  year    = {1982},
  pages   = {897--910}
}

Prerequisites

08.13.04
08.13.01
08.03.02

Tier anchors

beginner: Baxter Ch. 14 §14.1 the hard-hexagon lattice gas; nearest-neighbour exclusion on the triangular lattice
intermediate: Baxter Ch. 14 §14.2–14.4 solution by the corner transfer matrix; the order parameter and the sublattice densities
master: Baxter Ch. 14 §14.5–14.8; Baxter 1980 J. Phys. A 13:L61; Andrews–Baxter–Forrester 1984 J. Stat. Phys. 35:193

References

raw/pdfs/kardar-1-3-laws-entropy-potentials.pdf · Kardar statistical physics notes (lattice-gas and transfer-matrix background)
quantum-well · statistical physics index
Baxter — Exactly Solved Models in Statistical Mechanics · Ch. 14 (the hard-hexagon and hard-square models; corner-transfer-matrix solution; Rogers–Ramanujan order parameter)
Baxter — Hard hexagons: exact solution · J. Phys. A: Math. Gen. 13:L61 (1980), the originator letter
Andrews — Baxter — Forrester, Eight-vertex SOS model and generalized Rogers–Ramanujan-type identities · J. Stat. Phys. 35:193 (1984), the regime-III RSOS generalisation and the Rogers–Ramanujan hierarchy
Rogers — Second memoir on the expansion of certain infinite products · Proc. London Math. Soc. 25:318 (1894), the original Rogers–Ramanujan identities

Estimated time

beginner: 16m
intermediate: 44m
master: 82m