Numerical extraction of crosscap coefficients in microscopic models for ( 2 + 1 ) D conformal field theory

Conformal field theory (CFT) can be placed on disparate space-time manifolds to facilitate investigations of their properties. For ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mn>2</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:math> )-dimensional [ <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mo stretchy="false">(</c:mo> <c:mn>2</c:mn> <c:mo>+</c:mo> <c:mn>1</c:mn> <c:mo stretchy="false">)</c:mo> <c:mi mathvariant="normal">D</c:mi> </c:mrow> </c:math> ] theories, one useful choice is the real projective space <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mrow> <h:msup> <h:mi mathvariant="double-struck">RP</h:mi> <h:mn>3</h:mn> </h:msup> </h:mrow> </h:math> obtained by identifying antipodal points on the boundary sphere of a three-dimensional ball. One-point functions of scalar primary fields on this manifold generally do not vanish and encode the so-called crosscap coefficients. These coefficients also manifest on the sphere as the overlaps between certain crosscap states and CFT primary states. Taking the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mo stretchy="false">(</k:mo> <k:mn>2</k:mn> <k:mo>+</k:mo> <k:mn>1</k:mn> <k:mo stretchy="false">)</k:mo> <k:mi mathvariant="normal">D</k:mi> </k:mrow> </k:math> Ising CFT as a concrete example, we demonstrate that crosscap coefficients can be extracted from microscopic models. We construct crosscap states in both lattice models defined on polyhedrons and continuum models in Landau levels, where the degrees of freedom at antipodal points are entangled in Bell-type states. By computing their overlaps with the eigenstates of many-body Hamiltonians, we obtain results consistent with those from conformal bootstrap. Importantly, our approach directly reveals the absolute values of crosscap overlaps, whereas bootstrap calculations typically yield only their ratios. Furthermore, we investigate the finite-size scaling of these overlaps and their evolution under perturbations away from criticality.