08.13.01 · stat-mech / exactly-solved-models

The Yang–Baxter equation and the star–triangle relation

shipped3 tiersLean: none

Anchor (Master): Baxter Ch. 9 §9.6–9.8; Yang 1967; Baxter 1972 Ann. Phys. 70:193

Intuition Beginner

Imagine a factory line that builds a long lattice one row at a time. A machine reads the current row of states and stamps out the next row, with each local pattern carrying a weight. Doing this row after row builds the whole lattice, and the total weight of every finished lattice is the quantity physicists most want to compute. The stamping machine is the transfer matrix: feed it a row, it returns the next.

Now suppose you own two slightly different stamping machines, tuned to two different settings. Usually the order in which you apply machines matters, the way putting on socks then shoes differs from shoes then socks. The deep question is: when do these two machines give the same result no matter which one you run first? When they do, the machines commute, and the lattice becomes special. It can be solved exactly.

The condition that makes two such machines commute is a single local rule about how three weights fit together at a crossing. That rule is the star–triangle relation, and its modern face is the Yang–Baxter equation.

Visual Beginner

The figure shows the heart of the rule. On the left, three lines cross so that one line passes on the left of a meeting point of the other two; on the right, the same three lines cross so that line passes on the right. The star–triangle relation says these two pictures carry the same total weight when you sum over the internal states.

Reading it as machines: sliding one line past a crossing of the other two does not change the answer. That single sliding move, repeated, is what lets you reorder a whole stack of stamping machines, which is why the lattice becomes exactly solvable.

Worked example Beginner

Take the simplest version: three lines, each carrying an arrow that can point one of two ways, meeting at a point. A weight is attached to each small crossing depending on the two arrows that meet there. Call the three pairwise crossing-weights $R$ , $S$ , $T$ .

In the left picture the lines cross in the order: first the lower two, then the outer line with each. In the right picture they cross in the opposite order. To compare them, fix the three arrows coming in from outside and the three going out, then add up the weight over every choice of the arrows on the internal segments.

Suppose every weight is just $1$ when the two arrows agree and $0$ when they disagree. Then in both pictures the only surviving term forces all three lines to carry the same arrow straight through, and each picture sums to exactly $1$ . The two sides match. With richer weights the matching becomes a real constraint on $R$ , $S$ , $T$ — and that constraint is the star–triangle relation.

What this tells us: the relation is a balance condition on local crossing-weights. When it holds, reordering crossings is free, and a global reordering of transfer matrices follows.

Check your understanding Beginner

Formal definition Intermediate+

Fix a finite-dimensional vector space $V$ , the local state space sitting on each edge of the lattice. The basic object is the $R$ -matrix, a linear operator $R (u) \in End (V \otimes V)$ depending on a complex spectral parameter $u$ (equivalently a rapidity). For each ordered pair $(i, j)$ of factors in a tensor product $V^{\otimes n}$ , write $R_{ij} (u)$ for the operator that acts as $R (u)$ on factors $i$ and $j$ and as the identity elsewhere.

The Yang–Baxter equation is the identity in $End (V^{\otimes 3})$

R_{12} (u) R_{13} (u + v) R_{23} (v) = R_{23} (v) R_{13} (u + v) R_{12} (u),

required to hold for all values of the spectral parameters $u$ , $v$ . The additive form of the argument, $u + v$ on the middle factor, is the convention for difference models, where $R$ depends only on differences of rapidities; this convention is fixed once here and carried through the unit.

In the lattice setting the same data appears as a vertex weight. Assign to each edge of the square lattice a state in $V$ , and to each vertex the Boltzmann weight $w (β α γ δ)$ depending on the four adjoining edge states $α, β, γ, δ$ (left, top, right, bottom). The components of the $R$ -matrix are exactly these vertex weights,

R (u)_{α β}^{γ δ} = w_{u} (δ α γ β),

with the spectral parameter setting the anisotropy of the weights. The star–triangle relation is the component form of the Yang–Baxter equation: for three rapidities it equates the weighted sum over the internal edge of a "star" (three weights meeting at a point) with the weighted sum over a "triangle" (the same three lines re-routed). Onsager's name star–triangle records its origin in the $Y$ – $Δ$ transformation of the triangular Ising lattice; Baxter's reading is that the same equation, parametrised by a spectral variable, controls every solvable vertex model ^{[Baxter Ch. 9]}.

The row-to-row transfer matrix built from these weights is

T (u) = tr_{a} (R_{a, 1} (u) R_{a, 2} (u) \dots R_{a, N} (u)),

an operator on the physical space $V^{\otimes N}$ obtained by multiplying $R$ -matrices along a row and tracing out the auxiliary space $V_{a}$ that threads the row (the monodromy matrix is the product before the trace). This is the construction of 08.03.02 cast in $R$ -matrix language. The spectral parameter $u$ labels a one-parameter family of transfer matrices on the same physical space.

Key theorem with proof Intermediate+

Theorem (Yang–Baxter implies commuting transfer matrices). Let $R (u)$ satisfy the Yang–Baxter equation and let $T (u) = tr_{a} (\prod_{k = 1}^{N} R_{a, k} (u))$ be the associated row-to-row transfer matrix on $V^{\otimes N}$ . If $R (u)$ is invertible for generic $u$ , then

[T (u), T (u^{'})] = 0 for all u, u^{'} .

Proof. Introduce two auxiliary spaces, $V_{a}$ carrying $u$ and $V_{b}$ carrying $u^{'}$ , alongside the $N$ physical spaces. The product $T (u) T (u^{'})$ is the partial trace over both auxiliary spaces of the operator

(k = 1 \prod N R_{a, k} (u)) (k = 1 \prod N R_{b, k} (u^{'})) = k = 1 \prod N R_{a, k} (u) R_{b, k} (u^{'}),

the last equality holding because operators on disjoint physical factors commute, so the two products interleave site by site. Write $M_{a} (u) = \prod_{k} R_{a, k} (u)$ and $M_{b} (u^{'}) = \prod_{k} R_{b, k} (u^{'})$ for the two monodromy matrices, so $T (u) T (u^{'}) = tr_{ab} (M_{a} (u) M_{b} (u^{'}))$ .

The Yang–Baxter equation, with the two auxiliary spaces playing the roles of factors $1$ and $2$ and a physical site playing factor $3$ , gives the intertwining (RTT) relation

R_{ab} (u - u^{'}) M_{a} (u) M_{b} (u^{'}) = M_{b} (u^{'}) M_{a} (u) R_{ab} (u - u^{'}),

where $R_{ab}$ acts on the two auxiliary spaces only. This is proved by induction on $N$ : for $N = 1$ it is the Yang–Baxter equation itself, written as $R_{ab} R_{a 1} (u) R_{b 1} (u^{'}) = R_{b 1} (u^{'}) R_{a 1} (u) R_{ab}$ ; the inductive step pushes $R_{ab}$ through one more site $R_{a, k} R_{b, k}$ using the same three-factor identity and leaves the already-treated sites untouched, since they act on different physical spaces.

Because $R_{ab} (u - u^{'})$ is invertible for generic $u - u^{'}$ , multiply the intertwining relation on the left by $R_{ab}^{- 1}$ :

M_{a} (u) M_{b} (u^{'}) = R_{ab}^{- 1} M_{b} (u^{'}) M_{a} (u) R_{ab} .

Take the trace over both auxiliary spaces. The trace is cyclic on the auxiliary factors, so the conjugating $R_{ab}^{- 1} (\dots) R_{ab}$ drops out:

T (u) T (u^{'}) = tr_{ab} (M_{a} (u) M_{b} (u^{'})) = tr_{ab} (M_{b} (u^{'}) M_{a} (u)) = T (u^{'}) T (u) .

By continuity the identity extends from generic $u - u^{'}$ to all values, including coincident spectral parameters. Hence the family ${T (u)}$ commutes. $□$

Bridge. This theorem builds toward the eigenvalue analysis of every solvable model in this chapter: a commuting family is simultaneously diagonalisable, so a single $u$ -independent eigenvector basis diagonalises $T (u)$ for all $u$ at once, and expanding $lo g T (u)$ in $u$ produces an infinite tower of mutually commuting conserved charges — the algebraic meaning of integrability that appears again in 05.02.03. The same factorised-scattering content reappears in 08.13.02, where the two-body phase that fixes the Bethe-ansatz amplitudes is exactly a matrix element of $R (u)$ and the Bethe equations are the consistency of carrying one rapidity around the ring. Putting these together, the star–triangle relation is not one technique among many but the structural origin of exact solvability, and the spectral parameter is the device that turns a single algebraic identity into a continuous commuting family. The same intertwining relation organises the eight-vertex model, whose elliptic $R$ -matrix is the apex solution of the very equation proved here.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Explain why a check-matrix $\overset{ˇ}{R} (u) = P R (u)$ (with $P$ the permutation) satisfies the braided Yang–Baxter equation $\overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} = \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23}$ when $R$ satisfies the unbraided form, and what the spectral parameters become.

Hint

Conjugating by $P$ swaps the labels $13 \leftrightarrow$ adjacent pairs; the braided form uses only adjacent factors.

Answer

Multiplying each $R_{ij}$ by the permutation converts the relation among factors ${12, 13, 23}$ into a relation among the adjacent pairs ${12, 23}$ only, because $P$ moves the non-adjacent $13$ action to an adjacent slot. The spectral parameters carried by the three operators are $u$ , $u + v$ , $v$ , distributed across the two adjacent positions. In the constant ( $u$ -independent) limit $\overset{ˇ}{R}$ becomes a representation of the braid group.

Rubric: full credit for the $P$ -conjugation argument and naming the braid-group limit.

Exercise 6 (hard, short-answer).

Carry out the inductive step in the proof of the intertwining relation: assuming $R_{ab} M_{a}^{(k)} M_{b}^{(k)} = M_{b}^{(k)} M_{a}^{(k)} R_{ab}$ for the monodromy over the first $k$ sites, prove it for $k + 1$ sites.

Hint

$M^{(k + 1)} = M^{(k)} R_{\cdot, k + 1}$ ; push $R_{ab}$ across the new site using the three-factor Yang–Baxter equation on ${a, b, k + 1}$ .

Answer

Write $M_{a}^{(k + 1)} = M_{a}^{(k)} R_{a, k + 1} (u)$ and similarly for $b$ . Then $R_{ab} M_{a}^{(k + 1)} M_{b}^{(k + 1)} = R_{ab} M_{a}^{(k)} R_{a, k + 1} M_{b}^{(k)} R_{b, k + 1}$ . Since $R_{a, k + 1}$ and $M_{b}^{(k)}$ act on disjoint physical spaces they commute, giving $R_{ab} M_{a}^{(k)} M_{b}^{(k)} R_{a, k + 1} R_{b, k + 1}$ . Apply the inductive hypothesis to the first three factors: this equals $M_{b}^{(k)} M_{a}^{(k)} R_{ab} R_{a, k + 1} R_{b, k + 1}$ . The Yang–Baxter equation on ${a, b, k + 1}$ rewrites $R_{ab} R_{a, k + 1} R_{b, k + 1} = R_{b, k + 1} R_{a, k + 1} R_{ab}$ . Commuting $R_{a, k + 1}$ back past $M_{b}^{(k)}$ assembles $M_{b}^{(k + 1)} M_{a}^{(k + 1)} R_{ab}$ , completing the step.

Rubric: full credit for the disjoint-space commutations plus the single use of Yang–Baxter on the new site.

Exercise 7 (hard, short-answer).

The free-fermion / six-vertex $R$ -matrix on $V = C^{2}$ has the form $R (u) = a (u) (E_{11} \otimes E_{11} + E_{22} \otimes E_{22}) + b (u) (E_{11} \otimes E_{22} + E_{22} \otimes E_{11}) + c (u) (E_{12} \otimes E_{21} + E_{21} \otimes E_{12})$ , with weights $a, b, c$ . State the single scalar constraint among $a, b, c$ at three rapidities that the star–triangle relation reduces to, and name the invariant it fixes.

Hint

The surviving component equation is a relation among the anisotropies $Δ = (a^{2} + b^{2} - c^{2}) / (2 ab)$ at the three rapidities.

Answer

The component (star–triangle) equations all reduce to the requirement that the combination $Δ (u) = \frac{a ( u ) ^{2} + b ( u ) ^{2} - c ( u ) ^{2}}{2 a ( u ) b ( u )}$ be independent of the spectral parameter $u$ , i.e. $Δ (u) = Δ (u^{'}) = Δ (u^{''})$ . The invariant fixed is the anisotropy $Δ$ , which labels the difference-model line on which the whole commuting family lives; varying $u$ at fixed $Δ$ sweeps the family.

Rubric: full credit for " $Δ$ constant along the family" and identifying $Δ$ as the invariant.

Advanced results Master

The intertwining relation proved above carries more information than commutation alone. Writing the monodromy matrix in the auxiliary space $V_{a} = C^{2}$ as a $2 \times 2$ matrix of operators on the physical space,

M_{a} (u) = (A (u) C (u) B (u) D (u)), T (u) = A (u) + D (u),

the single relation $R_{ab} (u - u^{'}) M_{a} (u) M_{b} (u^{'}) = M_{b} (u^{'}) M_{a} (u) R_{ab} (u - u^{'})$ encodes sixteen scalar commutation relations among $A, B, C, D$ at shifted spectral parameters. Among them is $[A (u) + D (u), A (u^{'}) + D (u^{'})] = 0$ , recovering the theorem, and the relations $[B (u), B (u^{'})] = 0$ together with $A (u) B (u^{'}) = f (u, u^{'}) B (u^{'}) A (u) + g (u, u^{'}) B (u) A (u^{'})$ . These are the engine of the algebraic Bethe ansatz: $B (u)$ creates excitations over a reference state, and the structure constants $f, g$ — themselves ratios of $R$ -matrix weights — reproduce the Bethe equations of 08.13.02 without ever solving a difference equation in position space. The factorised scattering of Yang's many-body problem ^{[Yang 1967]} is the statement that the $S$ -matrix of $n$ particles is an ordered product of two-body $S$ -matrices, consistent precisely because those two-body factors obey the Yang–Baxter equation; McGuire's earlier three-body analysis ^{[McGuire 1964]} is the special case $n = 3$ that first exposed the consistency condition.

Two structural specialisations sit on either side of the spectral-parameter axis. Setting the spectral parameter to a fixed value (or taking a rational, trigonometric, or elliptic limit in which $R$ becomes $u$ -independent) reduces the Yang–Baxter equation to the braid relation. Conjugating by the permutation, $\overset{ˇ}{R} = P R$ , the unbraided equation becomes $\overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} = \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23}$ , which is precisely the defining relation of the Artin braid group $B_{n}$ acting on $V^{\otimes n}$ . Each solution of the constant Yang–Baxter equation therefore yields a representation of every braid group, and through the Markov trace a link invariant; the trigonometric six-vertex $R$ -matrix produces the Jones polynomial. On the other side, the spectral-parameter-dependent solutions are organised by quantum groups: the $R$ -matrix is the image, in a tensor product of representations, of the universal $R$ -matrix of a quasitriangular Hopf algebra $U_{q} (g)$ , and the Yang–Baxter equation is the image of the abstract quantum-Yang–Baxter identity satisfied by that universal element ^{[Baxter Ch. 9]}.

The classification of solutions is graded by the analytic dependence on $u$ . Rational $R$ -matrices (Yangian symmetry) govern the isotropic Heisenberg chain and Yang's delta-interaction gas; trigonometric $R$ -matrices (quantum affine $U_{q} (sl_{2})$ ) govern the six-vertex model and the anisotropic XXZ chain; elliptic $R$ -matrices (Sklyanin algebra) govern the eight-vertex model and the XYZ chain. The elliptic case is the most general and the apex: Baxter's solution of the eight-vertex model is the parametrisation of $a, b, c, d$ by Jacobi theta functions of the spectral parameter such that the star–triangle relation becomes a theta-function addition theorem, linking this machinery laterally to 06.06.05.

A notation crosswalk fixes the conventions inherited from Baxter. The dimensionless coupling is $K = J / k_{B} T$ (Baxter's convention), in contrast to the $β J$ used in some shipped Ising units; the spectral parameter is written $u$ here (Baxter and the integrable-systems literature also use $λ$ or, multiplicatively, $x = e^{u}$ ); the elliptic modulus and nome are $k$ and $q$ respectively, to be matched against the theta-function conventions of 06.06.05; and the eight-vertex weights $a, b, c, d$ refer to the four arrow-conserving-mod-2 vertex classes. These pins hold across 08.13.02 and the eight-vertex unit.

Synthesis. Putting these together, the Yang–Baxter equation identifies three subjects that look unrelated from a distance: lattice statistical mechanics, where it is the star–triangle relation that makes transfer matrices commute and the free energy computable; low-dimensional topology, where its constant limit is the braid relation and its Markov trace is a knot invariant; and the representation theory of quantum groups, where it is the image of a universal $R$ -matrix. The spectral parameter is the thread tying the first to the third, and the commuting family it generates is the structural origin of integrability that reappears in 05.02.03 as commuting Hamiltonians in involution. The same factorised-scattering content appears again in 08.13.02, whose two-body Bethe phase is a matrix element of $R (u)$ , and the elliptic solution of the equation is the apex eight-vertex model whose non-universal critical exponents sharpen the universality picture of 08.05.01. The central identification is that solvability is a property of a single local equation, not of any individual model: a model is exactly solved when its Boltzmann weights solve the star–triangle relation.

Full proof set Master

Proposition (the braid limit). Let $R (u)$ be a difference-form solution of the Yang–Baxter equation that admits a limit $R_{\infty} = lim_{u \to u_{*}} R (u)$ at some value $u_{*}$ (rational, trigonometric, or elliptic degeneration), and set $\overset{ˇ}{R} = P R_{\infty}$ . Then $\overset{ˇ}{R}$ satisfies the braid relation $\overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} = \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23}$ , so $n$ copies generate a representation of the Artin braid group $B_{n}$ on $V^{\otimes n}$ .

Proof. In the difference form the three arguments $u, u + v, v$ collapse to a common value in the degeneration limit, so the spectral-parameter dependence drops and the Yang–Baxter equation becomes the constant identity $R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}$ . Multiply through by permutation operators using the intertwining identities $P_{12} R_{13} = R_{23} P_{12}$ and $P_{23} R_{13} = R_{12} P_{23}$ , which hold because conjugation by a permutation relabels the factors an $R$ -matrix acts on. Inserting $\overset{ˇ}{R}_{i, i + 1} = P_{i, i + 1} R_{i, i + 1}$ and pushing every permutation to the right, the non-adjacent operators $R_{13}$ are converted to adjacent ones, and the constant Yang–Baxter equation rearranges exactly into $\overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} = \overset{ˇ}{R}_{23} \overset{ˇ}{R}_{12} \overset{ˇ}{R}_{23}$ . Because the $\overset{ˇ}{R}_{i, i + 1}$ for $∣ i - j ∣ \geq 2$ commute (disjoint factors) and adjacent ones satisfy the braid relation, the assignment $σ_{i} \mapsto \overset{ˇ}{R}_{i, i + 1}$ extends to a homomorphism from $B_{n}$ . $□$

Proposition (uniqueness of the conserved tower). Let ${T (u)}$ be the commuting family of the theorem with $T (u)$ analytic and invertible near $u_{0}$ , and suppose $H$ is any operator commuting with every $T (u)$ . Then $H$ is a function of the conserved charges $Q_{n} = \partial_{u}^{n} lo g T (u)_{u_{0}}$ .

Proof. Since the $T (u)$ commute and are simultaneously diagonalisable, decompose the physical space into joint eigenspaces; on each, $T (u)$ acts by a scalar eigenvalue function $Λ (u)$ , analytic in $u$ . The charges act by $\partial_{u}^{n} lo g Λ (u) ∣_{u_{0}}$ , and an operator commuting with all $T (u)$ preserves each joint eigenspace, hence acts by a scalar there. Two joint eigenspaces carry distinct eigenvalue functions $Λ (u)$ (else they merge), so the family of Taylor coefficients ${\partial_{u}^{n} lo g Λ (u) ∣_{u_{0}}}_{n}$ separates the eigenspaces; therefore the scalar by which $H$ acts is determined by those coefficients, i.e. $H = F ({Q_{n}})$ for some function $F$ . $□$

The eigenvalue functions $Λ (u)$ themselves are constrained by a functional equation — Baxter's $T Q$ relation — whose derivation belongs to the eight-vertex unit; here it suffices that analyticity plus the commuting-family structure already forces the spectrum into one analytic family per joint eigenspace.

Connections Master

The transfer matrix, the trace formula $Z = tr T^{N}$ , and the largest-eigenvalue free energy are imported from 08.03.02; this unit supplies the structural reason a family of such matrices can be diagonalised simultaneously, which that unit could not provide.

Integrability as the existence of commuting conserved quantities is the classical-mechanics statement of 05.02.03; the present theorem is its lattice incarnation, with the spectral-parameter expansion of $lo g T (u)$ producing the tower of commuting charges in place of Poisson-commuting integrals of motion. The translation between commuting Hamiltonians in involution and commuting transfer matrices is the bridge between the two chapters.

The six-vertex model and coordinate Bethe ansatz of 08.13.02 are the first application: the two-body scattering phase that fixes its plane-wave amplitudes is a matrix element of $R (u)$ , the Bethe equations are the consistency of the intertwining relation around the ring, and the algebraic Bethe ansatz reconstructs the same eigenvectors from the operators $B (u)$ of the monodromy matrix. The eight-vertex model extends the construction to an elliptic $R$ -matrix, parametrised by the theta functions of 06.06.05, and its non-universal critical exponents are the integrable counterexample that refines the universality discussion of 08.05.01.

The constant limit links to low-dimensional topology: the braid-group representations built from $\overset{ˇ}{R}$ yield, via the Markov trace, polynomial invariants of knots and links, with the six-vertex $R$ -matrix producing the Jones polynomial. The spectral-parameter-dependent theory links forward to quantum groups, where $R (u)$ is the image of the universal $R$ -matrix of $U_{q} (g)$ .

Historical & philosophical context Master

The equation entered physics from two directions. On the lattice side, the star–triangle relation is the spectral-parametrised descendant of Onsager's $Y$ – $Δ$ transformation, used in his 1944 solution of the two-dimensional Ising model and in the duality analysis of triangular and honeycomb lattices. Rodney Baxter recognised in 1971–1972 that the condition for two transfer matrices to commute is exactly a star–triangle relation on their Boltzmann weights, and that commutation supplies the infinite family of conserved quantities behind exact solvability; this is the organising insight of his solution of the eight-vertex model and of his 1982 monograph, which presents each solvable model through the algebra that diagonalises it ^{[Baxter 1972]}.

On the quantum many-body side, J. B. McGuire showed in 1964 that the $N$ -body problem with delta-function interactions in one dimension has factorised, reflectionless scattering, with the three-body $S$ -matrix consistently decomposing into two-body factors ^{[McGuire 1964]}. Chen Ning Yang sharpened this in 1967 into the consistency condition on the two-body scattering matrix that now bears his name jointly with Baxter's, solving the repulsive delta-interaction gas by what became the nested Bethe ansatz ^{[Yang 1967]}. The two strands — Baxter's commuting transfer matrices and Yang's factorised scattering — were recognised as the same equation in the 1970s. Ludwig Faddeev, Evgeny Sklyanin and Leon Takhtajan abstracted it after 1979 into the quantum inverse scattering method, and Vladimir Drinfeld and Michio Jimbo found in 1985 that its spectral-parameter solutions are governed by quantum groups, the quasitriangular Hopf algebras whose universal $R$ -matrix solves the equation at the abstract level.

Bibliography Master

@article{McGuire1964,
  author  = {McGuire, J. B.},
  title   = {Study of Exactly Soluble One-Dimensional N-Body Problems},
  journal = {Journal of Mathematical Physics},
  volume  = {5},
  year    = {1964},
  pages   = {622--636}
}

@article{Yang1967,
  author  = {Yang, Chen Ning},
  title   = {Some Exact Results for the Many-Body Problem in One Dimension with Repulsive Delta-Function Interaction},
  journal = {Physical Review Letters},
  volume  = {19},
  year    = {1967},
  pages   = {1312--1315}
}

@article{Baxter1971eightvertex,
  author  = {Baxter, Rodney J.},
  title   = {Eight-Vertex Model in Lattice Statistics},
  journal = {Physical Review Letters},
  volume  = {26},
  year    = {1971},
  pages   = {832--833}
}

@article{Baxter1972,
  author  = {Baxter, Rodney J.},
  title   = {Partition Function of the Eight-Vertex Lattice Model},
  journal = {Annals of Physics},
  volume  = {70},
  year    = {1972},
  pages   = {193--228}
}

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

@article{FaddeevSklyaninTakhtajan1979,
  author  = {Faddeev, L. D. and Sklyanin, E. K. and Takhtajan, L. A.},
  title   = {The Quantum Inverse Problem Method. I},
  journal = {Theoretical and Mathematical Physics},
  volume  = {40},
  year    = {1979},
  pages   = {688--706}
}

@article{Drinfeld1985,
  author  = {Drinfeld, V. G.},
  title   = {Hopf Algebras and the Quantum Yang--Baxter Equation},
  journal = {Soviet Mathematics Doklady},
  volume  = {32},
  year    = {1985},
  pages   = {254--258}
}

@article{Jimbo1985,
  author  = {Jimbo, Michio},
  title   = {A $q$-Difference Analogue of $U(\mathfrak{g})$ and the Yang--Baxter Equation},
  journal = {Letters in Mathematical Physics},
  volume  = {10},
  year    = {1985},
  pages   = {63--69}
}

Prerequisites

08.03.02
05.02.03

Used in

08.13.02

Tier anchors

beginner: Baxter Ch. 9 §9.1–9.2 commuting-transfer-matrix picture; two cameras on the same lattice
intermediate: Baxter Ch. 9 §9.6 star–triangle relation; Yang 1967 factorised scattering
master: Baxter Ch. 9 §9.6–9.8; Yang 1967; Baxter 1972 Ann. Phys. 70:193

References

raw/pdfs/kardar-1-3-laws-entropy-potentials.pdf · Kardar statistical physics notes (transfer-matrix and lattice-model background)
quantum-well · statistical physics index
Baxter — Exactly Solved Models in Statistical Mechanics · Ch. 9 (commuting transfer matrices and the star–triangle relation)
Baxter — Partition function of the eight-vertex lattice model · Ann. Phys. 70:193 (1972), commuting-transfer-matrix / star–triangle originator
Yang — Some exact results for the many-body problem in one dimension with repulsive delta-function interaction · Phys. Rev. Lett. 19:1312 (1967), factorised scattering / consistency condition
McGuire — Study of exactly soluble one-dimensional N-body problems · J. Math. Phys. 5:622 (1964), three-body factorisation predecessor

Estimated time

beginner: 18m
intermediate: 48m
master: 85m