Speaker
Description
While the concept of the renormalization group (RG) is a highly powerful framework for the phase transitions, applying it analytically to specific systems is generally difficult. In recent years, the Tensor Renormalization Group (TRG), which utilizes Tensor Networks (TNs), has been developed as a method for performing RG calculations numerically with high precision. TRG can be easily executed even in regimes where conventional numerical techniques, such as Monte Carlo methods, encounter severe challenges. However, it suffers from the drawback that a systematic error evaluation method has not yet been established.
For numerical analyses using TNs, gauge-invariant partition function ratios have been proposed as a tool to evaluate errors via finite-size scaling. These ratios exhibit distinct values depending on the RG fixed point, and these values at a stable fixed point are believed to correspond to the number of thermodynamic states in that phase. Because partition function ratios have primarily been applied to spin systems, applying them to models described by different degrees of freedom is necessary to achieve a deeper understanding of their significance. Therefore, in this presentation, we focus on a class of systems known as classical loop O($n$) models, whose essential degrees of freedom can be considered as graphs on a lattice. The partition function of this model is expressed as a sum of weights over loop configurations defined on the lattice.
In this presentation, we investigate the cubic loop O($n$) model on square lattice, a variant of the classical loop O($n$) model that is closely related to the Ising model. For this system, we perform real-space renormalization using bond-weighted TRG—an improved variant of TRG—and compute the gauge-invariant partition function ratios. We reveal that as the system passes through two critical points, the calculated ratios change like $1 \to 2n \to n$. Furthermore, we demonstrate that these values do not necessarily align with the conventional interpretation of "the number of thermodynamical states."