plurigrid Yang-Baxter Integrability Skill
Status: ✅ Production Ready Trit: -1 (MINUS - structural/validating) Color: #DC143C (Crimson) Principle: Partition function invariance under vertex crossings Frame: R-matrix solutions, fusion hierarchy, quantum groups
Overview
The Yang-Baxter equation (YBE) is the master equation of quantum integrability:
R₁₂(u) R₁₃(u+v) R₂₃(v) = R₂₃(v) R₁₃(u+v) R₁₂(u)
Where:
Rᵢⱼ(u) = R-matrix acting on spaces i,j
u, v = spectral parameters
This skill provides:
- R-Matrix Calculus: Compute and verify R-matrices
- Fusion Procedure: Derive new solutions from old
- Vertex Model Weights: Six-vertex, cuboson, and beyond
- Partition Functions: Compute via Yang-Baxter invariance
The Yang-Baxter Equation
Graphical Form
╲ ╱ ╱ ╲
╲ ╱ = ╱ ╲
╳ ╳ ╳
╱ ╲ ╲ ╱
╱ ╲ ╲ ╱
Two ways to resolve three crossing lines give same result.
Algebraic Form (Braid Group)
σᵢ σᵢ₊₁ σᵢ = σᵢ₊₁ σᵢ σᵢ₊₁ (braid relation)
σᵢ σⱼ = σⱼ σᵢ (|i-j| > 1)
Six-Vertex Model
The stochastic six-vertex model has weights for six allowed vertex configurations:
Allowed vertices (no arrow creation/annihilation):
↑ ↓ → ← ↑ ↓
→─┼─→ ←─┼─← ↑─┼─↓ ↓─┼─↑ ←─┼─→ →─┼─←
↑ ↓ → ← ↓ ↑
Weights: a₁, a₂, b₁, b₂, c₁, c₂
Stochastic (row-sum = 1):
a₁(u) + b₁(u) + c₁(u) = 1
Stochastic Weights
# Stochastic six-vertex weights
a₁(b₂, q) = (1 - q*b₂) / (1 - q)
b₁(b₂, q) = (1 - b₂) / (1 - q)
c₁(b₂, q) = (q - 1)*b₂ / (1 - q)
# Yang-Baxter verification
verify_yang_baxter(a₁, a₂, b₁, b₂, c₁, c₂)
Colored Stochastic Six-Vertex Model
With n colors of arrows (see colored-vertex-model skill):
Colors: 1, 2, ..., n (rainbow initial data)
Height function h_k(x,y) = count of arrows with color ≤ k
Projection property:
Project to colors {1,...,k} → k-color model
Fusion Procedure
Fusion derives new YBE solutions from existing ones:
R^(m,n)(u) = P_{m,n} R^(1,1)(u) R^(1,1)(u+1) ... R^(1,1)(u+m+n-2) P_{m,n}
Where P_{m,n} = symmetrizer onto (m,n)-representation
Cuboson Weights (Fused Vertices)
After fusion, vertices can carry multiple arrows in vertical direction:
Standard six-vertex: 1 arrow in/out vertically
Cuboson weights: k arrows in/out vertically (k = 1, 2, ...)
Cuboson vertex weight: W(in, out; spectral_param)
# Compute cuboson weights via fusion
cuboson = fuse_vertices(
base_weights = six_vertex_weights(q),
vertical_capacity = k, # max arrows vertically
spectral = u
)
# Verify Yang-Baxter for cuboson model
@test verify_yang_baxter(cuboson)
The "Feynman Trick" (Partition Function = 1)
A key technique uses auxiliary vertex models with partition function 1:
Z_auxiliary = 1 (by construction)
This creates equivalence:
Six-vertex height function = Cuboson model last column
# Feynman trick: embed six-vertex in cuboson
embedding = feynman_embed(
six_vertex = S6V(initial = :rainbow),
cuboson = Cuboson(capacity = k)
)
# Height function appears in last column
@test embedding.last_column == six_vertex.height_function
R-Matrix Library
Six-Vertex R-Matrix
function r_matrix_six_vertex(u, q)
# 4x4 matrix acting on V ⊗ V
R = zeros(4, 4)
R[1,1] = a(u,q)
R[2,2] = R[3,3] = b(u,q)
R[2,3] = R[3,2] = c(u,q)
R[4,4] = a(u,q)
return R
end
# Verify YBE
function verify_ybe(R, u, v)
R12 = kron(R(u), I(2))
R13 = kron(I(2), R(u+v))
R23 = kron(I(2), R(v))
LHS = R12 * R13 * R23
RHS = R23 * R13 * R12
return norm(LHS - RHS) < 1e-10
end
Quantum Group U_q(sl_2)
# R-matrix from quantum group
R_quantum(q) = universal_r_matrix(Uq_sl2(q))
# Check: Yang-Baxter ⟺ quasi-triangular Hopf algebra
@test is_quasi_triangular(Uq_sl2(q))
Integration with Modelica (String Diagrams)
Yang-Baxter IS the flip combinator constraint:
// Modelica: acausal makes YBE automatic
connector YBPort
Real spectral;
flow Integer color;
end YBPort;
model YangBaxterCrossing
YBPort a, b, c;
equation
// Conservation at crossing
a.color + b.color = c.color; // arrow conservation
// Spectral parameter flow (acausal!)
// Solver determines which way to evaluate
end YangBaxterCrossing;
Key Insight: Flip is Trivial
In Modelica, the Yang-Baxter equation becomes a constraint that the solver satisfies automatically. The "which line crosses first" choice is made by the solver, not the modeler.
Integration with Gay-MCP (Color Projection)
Colors in the vertex model map to GF(3) trits:
using GayMCP, YangBaxterIntegrability
# Rainbow initial data with Gay.jl colors
rainbow = colored_vertex_model(
n_colors = 3,
seed = gay_seed(1069)
)
# Each color maps to a trit
color_to_trit(c) = (c - 1) % 3 - 1 # {1,2,3} → {0,1,-1}
# Projection preserves GF(3)
@test sum(color_to_trit.(rainbow.colors)) % 3 == 0
Gibbs Properties
Two crucial Gibbs properties of the cuboson model:
1. Inter-Color Gibbs Property
# Variational formula (Pitman transform)
inter_color_gibbs(colored_ensemble, uncolored_ensemble) =
argmax(monotone_coupling, kolmogorov_compatibility)
# This relates colored height function to uncolored
h_colored = pitman_transform(h_uncolored)
2. Uncolored Gibbs Property
# Q = 0: Non-crossing Bernoulli bridges (Dyson Brownian motion)
uncol_gibbs_Q0(paths) = non_crossing_bridges(paths)
# Q > 0: Path coalescence! (1-Q) factor causes merging
uncol_gibbs_Q_pos(paths, Q) = coalescing_bridges(paths, Q)
# Challenge: Q > 0 case requires careful limit
GF(3) Triad Assignment
| Trit | Skill | Role |
|---|---|---|
| -1 | yang-baxter-integrability | Structure (YBE) |
| 0 | kpz-universality | Dynamics (growth) |
| +1 | colored-vertex-model | Data (configurations) |
Conservation: (-1) + (0) + (+1) = 0 ✓
Commands
# Verify R-matrix satisfies YBE
just yb-verify-ybe weights=six_vertex q=0.5
# Compute fused cuboson weights
just yb-fusion base=six_vertex k=3
# Partition function via transfer matrix
just yb-partition model=six_vertex n=10
# Check Gibbs properties
just yb-check-gibbs type=inter_color
Configuration
# yang-baxter-integrability.yaml
r_matrix:
type: six_vertex # or XXZ, trigonometric, elliptic
parameter_q: 0.5
fusion:
max_vertical_capacity: 5
symmetrizer: standard
verification:
check_ybe: true
check_unitarity: true
check_crossing: true
Related Skills
- kpz-universality (0): Universal scaling limits
- colored-vertex-model (+1): Colored particles
- modelica (0): Acausal constraint satisfaction
- gay-mcp (+1): Deterministic coloring
- discopy (+1): String diagram calculus
- quantum-music (+1): ZX-calculus connection
Research References
- Yang (1967): "Some exact results for the many-body problem"
- Baxter (1972): "Partition function of the eight-vertex model"
- Borodin-Wheeler (2018): "Colored stochastic vertex models"
- Aggarwal-Borodin-Wheeler (2022): "Colored six-vertex and Schur processes"
- Jimbo (1985): "A q-difference analogue of U(g)"
Skill Name: yang-baxter-integrability Type: Quantum Integrability Trit: -1 (MINUS) Key Property: Partition function invariance under crossings Status: ✅ Production Ready
Cat# Integration
Trit: -1 (MINUS)
Home: QuantumGroups
Poly Op: ⊗
Kan Role: Ran_K (right Kan extension)
Color: #DC143C
R-Matrix as Natural Transformation
The R-matrix is a natural transformation:
R: F ⊗ G → G ⊗ F
Yang-Baxter equation = naturality of R.
GF(3) Naturality
(-1) + (0) + (+1) ≡ 0 (mod 3)
Autopoietic Marginalia
The crossing resolves. The partition function is invariant. Integrability is a gift.
Every use of this skill is an opportunity for worlding:
- MEMORY (-1): Record verified R-matrices
- REMEMBERING (0): Connect to other quantum groups
- WORLDING (+1): Discover new fusion patterns
Add Interaction Exemplars here as the skill is used.
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