Crosscap states and duality of Ising field theory in two dimensions

We propose two distinct crosscap states for the two-dimensional (2D) Ising field theory. These two crosscap states, identifying Ising spins or dual spins (domain walls) at antipodal points, are shown to be related via the Kramers-Wannier duality transformation. We derive their Majorana free field representations and extend bosonization techniques to calculate correlation functions of the 2D Ising conformal field theory (CFT) with different crosscap boundaries. Away from criticality, we develop a conformal perturbation theory to calculate the Klein bottle entropy (norm-square of the crosscap overlap) as a universal scaling function [Phys. Rev. Lett. 130, 151602 (2023)]. For the Ising field theory, our analytical results support the conjectured monotonicity of the Klein bottle entropy under relevant perturbations. The formalism provides a general framework for studying perturbed 2D CFTs on non-orientable manifolds.

💡 Research Summary

The paper introduces and thoroughly investigates two distinct cross‑cap states in the two‑dimensional Ising field theory, each corresponding to a different identification of degrees of freedom at antipodal points on a non‑orientable surface. The first state, denoted |C⁺⟩, identifies the original Ising spins σ at opposite points, while the second, |C⁻⟩, identifies the dual spins (disorder operators µ, i.e., domain walls) at antipodal locations. On the lattice, these are constructed as |C⁺latt⟩ = ∏{j=1}^{N/2}(1+σ^x_jσ^x_{j+N/2})|↑↑…↑⟩ and |C⁻_latt⟩ = U_KW|C⁺_latt⟩, where U_KW is the unitary Kramers‑Wannier duality operator. The two states are exact duals: U_KW|C⁺_latt⟩ = |C⁻_latt⟩ and vice‑versa, and |C⁻_latt⟩ can be written as an equal‑weight superposition of |C⁺_latt⟩ and a modified cross‑cap state |C′_latt⟩.

Using the Jordan‑Wigner transformation followed by a Bogoliubov rotation, the critical Ising chain is diagonalized into free Majorana fermions in the Neveu‑Schwarz (NS) sector. The authors compute analytically the overlaps ⟨ψ_{k₁…k_M}|C⁺_latt⟩ for all fermionic eigenstates. Remarkably, these overlaps are universal constants (√2/2, (√2±1)/2) depending only on the parity of M, with no finite‑size corrections. The overlap for |C⁻_latt⟩ acquires an extra factor (−1)^M, reflecting the duality transformation.

Passing to the continuum limit, the low‑energy modes become the left‑ and right‑moving Majorana fields b_n and \bar b_n of the 2D Ising CFT. The lattice overlaps translate into exact expressions for the continuum cross‑cap states: |C⁺⟩ = (√2+√2)/2 |1⟩⟩_C + (√2−√2)/2 |ε⟩⟩_C, |C⁻⟩ = (1/√2)(|C₁⟩+|C_ε⟩), where |C₁⟩ and |C_ε⟩ are the standard Pradisi‑Sagnotti‑Stanev (PSS) cross‑cap states. Thus |C⁺⟩ coincides with the usual PSS cross‑cap, while |C⁻⟩ is a symmetric superposition of the PSS state and its ε‑twisted partner, exactly matching the lattice relation (5)–(6).

The authors then study correlation functions on a semi‑infinite cylinder with a cross‑cap boundary, which under the conformal map w = e^{2πz/L} become correlation functions on the real projective plane RP². By extending bosonization to include cross‑cap boundaries, they express Ising correlators in terms of those of a Z₂‑orbifolded compact boson. For example, the two‑point function of spin fields reads ⟨σ(w₁)σ(w₂)⟩{C^±} = (√2±√2)/2 G±(η) |w₁−w₂|^{-1/4}, with η the cross‑cap cross‑ratio and G_±(η) given by elementary hypergeometric functions. The +‑sign reproduces known results for the standard PSS cross‑cap, while the –‑sign corresponds to the new superposed state.

Moving away from criticality, the paper treats the Ising field theory perturbed by the relevant thermal (ε) and magnetic (σ) operators: H = H₀ − g₁∫ε − g₂∫σ. A conformal perturbation theory is developed for the overlap ⟨ψ₀(s)|C⟩, where s = g L^{2−2h} is the dimensionless coupling (h is the scaling dimension of the perturbing primary). The overlap is factorized as Z(s) exp