Tensor network renormalization study on the crossover in classical Heisenberg and RP^{2} models in two dimensions

Area of Science:

• Condensed Matter Physics
• Quantum Field Theory

Background:

• The phase diagrams of the 2D Heisenberg and RP^{2} models are complex due to large correlation lengths at low temperatures.
• Previous studies have faced challenges in definitively characterizing critical points in these models.

Purpose of the Study:

• To determine the phase diagram and critical behavior of the 2D Heisenberg and RP^{2} models.
• To identify the relevant conformal field theories (CFTs) at zero temperature.
• To investigate the nature of phases at finite temperatures.

Main Methods:

• Tensor-Network Renormalization (TNR) was employed to analyze the models.
• Finite-size spectrum analysis of the transfer matrix was used to identify CFTs.
• Zero-temperature and finite-temperature properties were investigated.

Main Results:

• At zero temperature, the Heisenberg and ferromagnetic RP^{2} models exhibit a CFT with central charge c=2, consistent with Nambu-Goldstone modes.
• The antiferromagnetic Lebwohl-Lasher model shows a CFT with central charge c=4, aligning with SO(5) symmetry.
• At finite temperatures (T>0), spectral convergence is poor, suggesting a single disordered phase with an effective central charge c~1.9 for both models.

Conclusions:

• The zero-temperature behavior of the Heisenberg and ferromagnetic RP^{2} models is well-described by CFT with c=2.
• The antiferromagnetic Lebwohl-Lasher model's zero-temperature behavior is consistent with CFT with c=4 and SO(5) symmetry.
• At finite temperatures, both models likely possess a single disordered phase, with the ferromagnetic RP^{2} model showing a crossover related to Z_{2} vortex dissociation.