Two-point functions in boundary loop models

Using techniques of conformal bootstrap, we propose analytical expressions for a large class of two-point functions of bulk fields in critical loop models defined on the upper-half plane. Our results include the two-point connectivities in the Fortuin–Kasteleyn random cluster model with both free and wired boundary conditions. We link the continuum expressions to lattice quantities by computing universal ratios of amplitudes for the two-point connectivities, and find excellent agreement with transfer-matrix numerics.

💡 Research Summary

In this paper the authors develop a conformal‑bootstrap framework for computing bulk two‑point correlation functions in critical loop models defined on the upper half‑plane. Starting from the Q‑state Potts model, they use the Fortuin–Kasteleyn (FK) expansion to rewrite the partition function in terms of non‑intersecting loops, assigning a weight √Q (or √Q₁ for clusters touching the boundary) to each loop. Via the Coulomb‑gas mapping they relate the loop weight to the Coulomb‑gas coupling β, obtaining the standard relations Q = 4 cos²(πβ²) and central charge c = 1 − 6(β − β⁻¹)².

The relevant CFT operators are expressed in Kac notation: degenerate fields V_d(1,N), bulk N‑leg (watermelon) fields V(N/2,0), spin fields V(0,½) and boundary leg operators v(N,1). Their conformal weights are Δ(r,s)=P(r,s)²−P(1,1)² with P(r,s)=½(rβ − sβ⁻¹).

Conformal bootstrap imposes crossing symmetry on the half‑plane two‑point function
⟨V_i(z₁)V_j(z₂)⟩ = G_{ij}(σ) |z₁−\bar z₂|^{−4Δ_i} …
where σ is the standard cross‑ratio. The function G_{ij}(σ) must be consistent under both the s‑channel limit (σ→0, bulk OPE) and the t‑channel limit (σ→1, boundary OPE). By assuming diagonal boundary conditions that forbid open loop ends on the boundary, the authors restrict the exchanged spectra to S(s) = {Δ(1,N)} and S(t) = {Δ(N,1)}.

Using the semi‑analytical bootstrap of Ribault (ref.