Numerical tensor-network results locate a critical point at theta=pi in the 2D CP(1) model, consistent with Haldane's conjecture.
A Strong-Coupling Analysis of the Lattice $CP^{N-1}$ Models in the Presence of a $\theta$ Term
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abstract
A $\theta$ term, which couples to topological charge, is added to the lattice $CP^{N-1}$ model. The strong-coupling character expansion is developed. The series for the free energy and mass gap are respectively computed to tenth order and fourth order. Several features of the strong-coupling analysis emerge. One is the loss of superconfinement. Another is that in the intermediate coupling constant region, there are indications of a transition to a deconfining phase when $\theta$ is sufficiently large. The transition is like the one which has been observed in Monte Carlo simulations of a similar lattice $CP^{N-1}$ action.
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Phase structure analysis of CP(1) model with $\theta$ term by tensor renormalization group
Numerical tensor-network results locate a critical point at theta=pi in the 2D CP(1) model, consistent with Haldane's conjecture.