The six-vertex (ice-type) model and the Bethe ansatz

Intuition Beginner

Picture a square grid of streets. On every street segment you draw an arrow pointing one of two ways. At each crossing four segments meet, so four arrows touch it. The ice rule is one simple law for the crossings: at every crossing, exactly two arrows point in and exactly two point out. Out of the sixteen ways to arrange four arrows, only six obey this law. That is why the model is called the six-vertex model.

The name "ice" comes from real water ice. Each oxygen atom in ice has four neighbours, and exactly two hydrogen atoms sit close to it while two sit far. An arrow pointing toward a crossing stands for a near hydrogen, and an arrow pointing away stands for a far one. So the arrows on the grid are a cartoon of where the hydrogens live.

The big question is one of counting. How many whole-grid arrow patterns obey the rule at every crossing at once? The answer grows with the size of the grid, and the rate of growth is what physicists call the residual entropy of ice.

Visual Beginner

The figure shows the six allowed crossings. In each one, two arrows enter and two leave; every other arrangement breaks the rule.

The picture also shows a small patch of grid. Notice that once you fix the arrows along one row, the rule at each crossing limits what the next row can be. This step-by-step limitation is the seed of the method that solves the model.

Worked example Beginner

Take the smallest interesting grid: a single crossing with periodic edges, so the arrow leaving the right also enters from the left, and the arrow leaving the top also enters from the bottom.

Count by hand. The four segments carry four arrows. The ice rule demands two in and two out. List the patterns: the horizontal arrows can both go right, or both go left; the vertical arrows can both go up, or both go down; and there are mixed patterns where one horizontal and one vertical line up. Working through them gives exactly six legal patterns, matching the six allowed crossings in the figure.

Now give each pattern a weight. Suppose the two "all-horizontal" patterns each carry weight $a$, the two "all-vertical" patterns weight $b$, and the two "mixed" patterns weight $c$. The total weight is $2a+2b+2c$. If every weight equals $1$, the total is $6$.

What this tells us: the model is a weighted count of arrow patterns. When all weights are equal, we are counting ice configurations, and the count per crossing is what feeds the residual entropy.

Check your understanding Beginner

Formal definition Intermediate+

Place arrows on the edges of an $N\times N$ square lattice with periodic boundaries. At each vertex the six configurations obeying the ice rule (two arrows in, two out) are admitted; the remaining ten are excluded. Group the six into three pairs related by arrow reversal and assign Boltzmann weights $a$, $b$, $c$ to the three pairs. The partition function is

$$Z=\sum _{\text{ice configs}}{a}^{n_a}{b}^{n_b}{c}^{n_c},$$

where $n_a,n_b,n_c$ count the vertices of each type. The ice model is the case $a=b=c=1$; the KDP model and the antiferroelectric F model are obtained by other weight choices ^{[Baxter Ch. 8]}.

The row-to-row transfer matrix $T$ acts on the $2^N$-dimensional space spanned by the horizontal-arrow configurations of one row, following the construction of 08.03.02. Its matrix element between an incoming row state and an outgoing row state is the product of vertex weights over that row, summed over the vertical-arrow states consistent with the ice rule. Then

$$Z=\mathrm{tr}{T}^N,$$

so the free energy per site in the thermodynamic limit is fixed by the largest eigenvalue ${\Lambda }_{\max }$ of $T$, exactly as for the Ising transfer matrix 08.03.01:

$$f=-k_{B}T\underset{N\to \infty }{\lim }\frac{1}{N^2}\log Z=-k_{B}T\underset{N\to \infty }{\lim }\frac{1}{N}\log {\Lambda }_{\max }.$$

The ice rule conserves the number of down-arrows (or "reversed" arrows) from one row to the next. Writing $m$ for that number, $T$ is block-diagonal in $m$, and each block can be diagonalised separately. The state with $m$ reversed arrows is the $m$-magnon sector; the magnon language matches the spin picture in which a reversed arrow is a flipped spin against a ferroelectric reference. The notation $a,b,c$ for the three weight classes follows Baxter; the convention is fixed once here and carried through the unit.

Key theorem with proof Intermediate+

Theorem (coordinate Bethe ansatz for the six-vertex transfer matrix). In the sector with $m$ reversed arrows, an eigenvector of the row transfer matrix $T$ can be written as a superposition of plane waves indexed by the positions $1\le x_1<x_2<\cdots <x_m\le N$ of the reversed arrows,

$$\psi (x_1,\dots ,x_m)=\sum _P{A}_P\exp \left(i\sum _{j=1}^m{k}_{P(j)}{x}_j\right),$$

where the sum runs over permutations $P$ of $\{1,\dots ,m\}$ and $A_P$ are amplitudes. The state is an eigenvector provided the wavenumbers $k_1,\dots ,k_m$ satisfy the Bethe equations

$$e^{i{k}_{j}N}=(-1{)}^{m-1}\prod _{\ell \ne j}\frac{e^{i(k_j+k_{\ell })}-2\Delta e^{i{k}_j}+1}{e^{i(k_j+k_{\ell })}-2\Delta e^{i{k}_{\ell }}+1},j=1,\dots ,m,$$

with the anisotropy parameter $\Delta =(a^2+b^2-c^2)/(2ab)$.

Proof. Work in the equivalent quantum chain whose transfer matrix and whose XXZ Hamiltonian share eigenvectors; the eigenvalue equation reduces to a difference equation for the amplitude $\psi$. Two regimes arise. When all reversed arrows are far apart, the eigenvalue equation acts on each one independently, and a single plane wave $e^{ikx}$ is an eigenfunction of the free hopping: this fixes the bulk energy as a sum of one-magnon contributions ${\sum }_j\epsilon (k_j)$. When two reversed arrows sit on adjacent sites, the interaction term in the equation is no longer satisfied by the free plane wave; the equation imposes a relation between the amplitude where the two arrows are separated and the amplitude where they meet.

Take $m=2$. Demand that the two-particle wavefunction $\psi (x_1,x_2)=A_{12}{e}^{i(k_1{x}_1+k_2{x}_2)}+A_{21}{e}^{i(k_2{x}_1+k_1{x}_2)}$ solve the eigenvalue equation both in the interior $x_2>x_1+1$ and at contact $x_2=x_1+1$. The interior is solved by either ordering of the wavenumbers. The contact condition forces the amplitude ratio

$$\frac{A_{21}}{A_{12}}=-\frac{e^{i(k_1+k_2)}-2\Delta e^{i{k}_1}+1}{e^{i(k_1+k_2)}-2\Delta e^{i{k}_2}+1}=:-e^{-i\theta (k_1,k_2)},$$

which defines the two-body scattering phase $\theta$. The minus sign is the hard-core statistics of reversed arrows (no two occupy a site). This is the heart of the construction: all $m$-body scattering factorises into these two-body phases, because the model is integrable and the Yang–Baxter relation guarantees consistency of the orderings.

Now impose periodicity. Carry a reversed arrow once around the ring of $N$ sites. It acquires the kinematic phase $e^{i{k}_{j}N}$ and, in passing every other reversed arrow, the product of two-body phases. Single-valuedness of the wavefunction equates the two:

$$e^{i{k}_{j}N}=\prod _{\ell \ne j}(-e^{-i\theta (k_j,k_{\ell })})=(-1{)}^{m-1}\prod _{\ell \ne j}\frac{e^{i(k_j+k_{\ell })}-2\Delta e^{i{k}_j}+1}{e^{i(k_j+k_{\ell })}-2\Delta e^{i{k}_{\ell }}+1}.$$

These are the Bethe equations. Any solution $\{k_j\}$ yields one eigenvector and one eigenvalue $\Lambda (\{k_j\})$ of $T$; sweeping the admissible solutions across all sectors $m$ reconstructs the full spectrum. $\square$

Bridge. This construction builds toward the thermodynamic-limit analysis in the Master tier, where the discrete Bethe roots condense into a continuous density and the sum over $\epsilon (k_j)$ becomes an integral. The factorisation of scattering into two-body phases is exactly the lattice imprint of the star–triangle relation of 08.13.01; putting these together, the foundational reason the six-vertex model is solvable is that its transfer matrices commute, and the Bethe ansatz is the eigenvector that diagonalises the whole commuting family at once. The same plane-wave-plus-scattering structure appears again in the eight-vertex model and generalises the free-fermion diagonalisation that solved the Ising case in 08.03.01. The central insight is that integrability converts an exponentially large eigenvalue problem into a finite coupled system for the wavenumbers.

Exercises Intermediate+

Advanced results Master

Lieb's 1967 solution evaluates the largest eigenvalue by passing to the thermodynamic limit of the Bethe equations ^{[Lieb 1967]}. Take logarithms of the Bethe equations and introduce quantum numbers $I_j$ so that

$$k_{j}N=2\pi I_j+\sum _{\ell \ne j}\theta (k_j,k_{\ell }).$$

For the ground state in the largest sector $m=N/2$ the roots $k_j$ fill an interval densely; define their density $\rho (k)$ by $N\rho (k)dk$ = number of roots in $[k,k+dk]$. Differentiating the logarithmic Bethe equations turns the coupled system into a single linear integral equation for $\rho$,

$$\rho (k)=\frac{1}{2\pi }+\frac{1}{2\pi }\int \frac{\partial \theta (k,k^{\prime })}{\partial k}\rho (k^{\prime })d{k}^{\prime },$$

a Fredholm equation of convolution type in the disordered regime. In the ice case $\Delta =1/2$ the kernel has a closed Fourier transform, and the resulting integral over the root density gives the free energy per vertex. For square ice ($a=b=c=1$) the residual entropy per vertex evaluates to

$$W={\left(\frac{4}{3}\right)}^{3/2}=\frac{8}{3\sqrt{3}}\approx 1.5396,$$

the celebrated Lieb result, slightly above Pauling's mean-field estimate $W_{\text{Pauling}}=3/2$ ^{[Pauling 1935]}. The number of ice configurations on $N$ vertices grows as $W^N$.

The phase diagram is read off from $\Delta$. For $\Delta >1$ the lattice is ferroelectrically frozen: one of $a,b$ dominates and the ground state is a single ordered arrangement with zero entropy, separated from the disordered phase by a first-order line. For $\Delta <-1$ the lattice is antiferroelectric, with a gap and a frozen staggered order (the F model lives here). For $-1<\Delta <1$ the model is critical along an entire line of points — gapless, with power-law correlations and continuously varying critical exponents controlled by $\Delta =-\cos \mu$ through the crossing parameter $\mu$. This line is the lattice realisation of a $c=1$ conformal field theory, the compactified free boson, with the compactification radius set by $\mu$.

Synthesis. Putting these together, the six-vertex model identifies a combinatorial counting problem with a spectral problem: the residual entropy of ice is the foundational reason a frustrated, locally constrained system retains macroscopic degeneracy, and the Bethe ansatz is exactly the device that computes that degeneracy by diagonalising the transfer matrix. The disordered line is dual to the Gaussian free field of 08.06.01, and the appearance of continuously varying exponents generalises the universality seen in the Ising model of 08.01.02 — the central insight being that integrability, not symmetry alone, fixes the exponents. The same scattering-phase machinery appears again in the eight-vertex model, where the rational kernel of the six-vertex case is replaced by an elliptic one, and the bridge between them is the star–triangle relation of 08.13.01 that makes both transfer-matrix families commute.

Full proof set Master

Proposition (Lieb–Pauling bound and the ice residual entropy). The residual entropy per vertex of square ice satisfies $W_{\text{Pauling}}=3/2\le W$, and the exact value is $W=(4/3{)}^{3/2}$.

Proof of the lower bound. Pauling's estimate counts hydrogen placements as if independent. Each of the $2E$ directed edges of a lattice with $N$ vertices and $E=2N$ edges carries one of two arrow directions, giving $2^{2N}$ unconstrained assignments. At each vertex, of the $2^4=16$ local arrow patterns only $6$ obey the ice rule, a survival fraction $6/16=3/8$. Treating the vertex constraints as independent multiplies the edge count by $(3/8{)}^N$:

$$W_{\text{Pauling}}^N\approx 2^{2N}{\left(\frac{3}{8}\right)}^N={\left(4\cdot \frac{3}{8}\right)}^N={\left(\frac{3}{2}\right)}^N,$$

so $W_{\text{Pauling}}=3/2$. Because the true constraints are correlated and the independence assumption undercounts by ignoring the compatibility that the ice rule actually permits between neighbours, this is a lower bound: $W\ge 3/2$.

Exact value. Lieb's transfer-matrix solution yields the free energy at the ice point $\Delta =1/2$ in closed form. At $\Delta =1/2$ write $\Delta =\cos \mu$ with $\mu =\pi /3$. The integral over the Bethe-root density gives the free energy per vertex

$$\log W=\frac{3}{2}\log \frac{4}{3},$$

so that

$$W=\exp \left(\frac{3}{2}\log \frac{4}{3}\right)={\left(\frac{4}{3}\right)}^{3/2}=\frac{8}{3\sqrt{3}}\approx \mathrm{1.5396.}$$

Since $(4/3{)}^{3/2}>3/2$, the exact entropy exceeds Pauling's estimate, confirming the bound and quantifying the residual correlation that the mean-field count discards. $\square$

Proposition (one-magnon dispersion on the ring). In the sector $m=1$ the eigenvalue of the associated XXZ Hamiltonian is $\epsilon (k)=-J(\cos k-\Delta )$ up to an additive constant, and the allowed momenta are $k=2\pi n/N$.

Proof. With one reversed arrow the Bethe equation $e^{ikN}=1$ quantises $k=2\pi n/N$. The eigenvalue equation for a single magnon hopping on the ring with nearest-neighbour amplitude reads, after Fourier transform, $\epsilon (k)=-J(\cos k-\Delta )$, where the $-J\Delta$ term is the diagonal energy of the flipped arrow against its neighbours and $-J\cos k$ is the kinetic hopping. Summing $\epsilon (k_j)$ over a filled set of such roots reproduces the bulk energy used in the density equation above. $\square$

Connections Master

The transfer-matrix formalism, the trace $Z=\mathrm{tr}{T}^N$, and the largest-eigenvalue free energy are taken directly from 08.03.02; this unit specialises that machinery to the constrained arrow space of the ice rule.

The integrability that factorises $m$-body scattering into two-body phases is the lattice content of the star–triangle / Yang–Baxter relation developed in 08.13.01; the Bethe ansatz here is the eigenvector that diagonalises the commuting transfer-matrix family that relation produces, and the eight-vertex model extends it with elliptic weights.

The free-fermion line $\Delta =0$ connects this unit to the Onsager solution 08.03.01 and the Ising model 08.01.02, where the same diagonalisation appears without an interaction phase; the critical line $-1<\Delta <1$ realises a $c=1$ Gaussian free field, linking to 08.06.01 and to conformal criticality in 08.06.02.

Historical & philosophical context Master

The coordinate ansatz originates with Hans Bethe's 1931 solution of the one-dimensional Heisenberg spin chain, where he wrote the eigenfunctions of the antiferromagnet as superpositions of plane waves with amplitudes fixed by a two-body phase ^{[Bethe 1931]}. Linus Pauling computed the residual entropy of ice in 1935 by his independent-bond estimate, predicting a nonzero zero-temperature entropy that experiments on ice confirmed ^{[Pauling 1935]}. Elliott Lieb solved the square-ice, KDP, and F models exactly in 1967 by applying Bethe's ansatz to the row-to-row transfer matrix, obtaining the residual entropy $W=(4/3{)}^{3/2}$ and mapping the line of critical points ^{[Lieb 1967]}. Rodney Baxter's 1982 monograph reorganised these results around the transfer matrix and the star–triangle relation, presenting each solvable model through the algebraic structure that diagonalises it ^{[Baxter Ch. 8]}.

Bibliography Master

@article{Bethe1931,
  author  = {Bethe, Hans},
  title   = {Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette},
  journal = {Zeitschrift f\"ur Physik},
  volume  = {71},
  year    = {1931},
  pages   = {205--226}
}

@article{Pauling1935,
  author  = {Pauling, Linus},
  title   = {The Structure and Entropy of Ice and of Other Crystals with Some Randomness of Atomic Arrangement},
  journal = {Journal of the American Chemical Society},
  volume  = {57},
  year    = {1935},
  pages   = {2680--2684}
}

@article{Lieb1967ice,
  author  = {Lieb, Elliott H.},
  title   = {Residual Entropy of Square Ice},
  journal = {Physical Review},
  volume  = {162},
  year    = {1967},
  pages   = {162--172}
}

@article{Lieb1967f,
  author  = {Lieb, Elliott H.},
  title   = {Exact Solution of the F Model of an Antiferroelectric},
  journal = {Physical Review Letters},
  volume  = {18},
  year    = {1967},
  pages   = {1046--1048}
}

@article{Lieb1967kdp,
  author  = {Lieb, Elliott H.},
  title   = {Exact Solution of the Two-Dimensional Slater KDP Model of a Ferroelectric},
  journal = {Physical Review Letters},
  volume  = {19},
  year    = {1967},
  pages   = {108--110}
}

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

@article{LiebWu1972,
  author  = {Lieb, Elliott H. and Wu, Fa-Yueh},
  title   = {Two-Dimensional Ferroelectric Models},
  journal = {Phase Transitions and Critical Phenomena (Domb and Green, eds.)},
  volume  = {1},
  year    = {1972},
  pages   = {331--490}
}