Extend tensor renormalization methods to generalized XY models with p ≥ 3 fractional-vortex interactions

Develop and apply higher-order tensor renormalization group (HOTRG) techniques, as used for the q=2 generalized XY model on the square lattice, to generalized XY models whose Hamiltonians include interactions of the form cos(p(θ_i − θ_j)) for integers p ≥ 3 that support fractional vortices.

Background

This work analyzes the two-dimensional generalized XY (gXY) model with q=2 using higher-order tensor renormalization group (HOTRG) methods, mapping out Berezinskii-Kosterlitz-Thouless (BKT), half-BKT, and Ising transition lines and their meeting region in the T−Δ plane.

The authors note that for models with interactions cos(p(θ_i − θ_j)) with p ≥ 3, Monte Carlo studies suggest richer phase structures. However, a corresponding tensor-network treatment has not yet been carried out, motivating the open problem of extending and applying HOTRG to these fractional-vortex models.

References

It remains an open problem to apply and extend the methods used in this work to models involving fractional vortices, corresponding to terms like \cos(p\theta_i - p\theta_j) with p \ge 3 in the Hamiltonian.

— Phase diagram of generalized XY model using tensor renormalization group  (2404.17504 - Samlodia et al., 2024) in Section 4 (Conclusions)

It remains an open problem to apply and extend the methods used in this work to models involving fractional vortices, corresponding to terms like \cos(q \theta_i - q \theta_j) with q \ge 3 in the Hamiltonian. It is believed that these models have a more complicated phase diagram, as studied in Ref. using Monte Carlo methods.

— Numerical Investigations of Phase Transitions in Lattice Field Theories  (2609.20536 - Longia, 17 Sep 2026) in Chapter 3, Section "Conclusions" (Section \ref{sec:Conclusions})