We investigate the first-order phase transitions of the q-state Potts models with q = 5, 6, 7, and 8 on the two-dimensional square lattice, using Monte Carlo simulations. At the very weakly firstorder transition of the q = 5 system, the standard data-collapse procedure for the order parameter, carried out with results for a broad range of system sizes, works deceptively well and produces non-trivial critical exponents different from the trivial values expected for first-order transitions. However, we show a more systematic study on the 'pseudo-critical' exponents as a function of the system size signals first-order phase transitions. We also derive a novel scaling behavior of Binder ratio based on a phenomenological theory for first-order transitions, which can detect the weakly first-order transitions in much smaller lattices than the correlation lengths. The results overall show that proper care is indispensable to diagnose the nature of a phase transition with limited system sizes.arXiv:1801.02786v3 [cond-mat.stat-mech]

We develop the tensor renormalization group (TRG) algorithm for statistical systems with open boundaries, which allows us to investigate not only the bulk but also the boundary property, such as the surface magnetization. We demonstrate that the tensors representing the boundary in our algorithm exhibit the fixed point structures just as bulk tensors in previous TRG algorithms. At criticality, the scale-invariant boundary fixed point tensors have the information of the conformal tower, which is described by the underlying boundary conformal field theory. arXiv:1905.02351v2 [cond-mat.stat-mech]

We numerically obtain the conformal spectrum of several classical spin models on a twodimensional lattice with open boundaries, for each boundary fixed points obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theories (BCFTs) for the Ising, tri-critical Ising, and 3-state Potts models on the lattice, which allows us to confirm the validity of the BCFT analyses for the surface critical behaviors of those lattice models. arXiv:1911.09907v1 [cond-mat.stat-mech]

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