Logarithmic finite-size scaling correction to the leading Fisher zeros in the p-state clock model: A higher-order tensor renormalization group study

Partition Function Zeros of the Frustrated J1-J2 Ising Model on the Honeycomb Lattice

We study the zeros of the partition function in the complex temperature plane (Fisher zeros) and in the complex external field plane (Lee-Yang zeros) of a frustrated Ising model with competing nearest-neighbor (J1>0) and next-nearest-neighbor (J2<0) interactions on the honeycomb lattice. We consider the finite-size scaling (FSS) of the leading Fisher and Lee-Yang zeros as determined from a cumulant method and compare it to a traditional scaling analysis based on the logarithmic derivative of the magnetization ∂ln⟨|M|⟩/∂β and the magnetic susceptibility χ. While for this model both FSS approaches are subject to strong corrections to scaling induced by the frustration, their behavior is rather different, in particular as the ratio R=J2/J1 is varied. As a consequence, an analysis of the scaling of partition function zeros turns out to be a useful complement to a more traditional FSS analysis. For the cumulant method, we also study the convergence as a function of cumulant order, providing suggestions for practical implementations. The scaling of the zeros convincingly shows that the system remains in the Ising universality class for R as low as -0.22, where results from traditional FSS using the same simulation data are less conclusive. Hence, the approach provides a valuable additional tool for mapping out the phase diagram of models afflicted by strong corrections to scaling.

Monte Carlo study of duality and the Berezinskii-Kosterlitz-Thouless phase transitions of the two-dimensional q-state clock model in flow representations

The two-dimensional q-state clock model for q≥5 undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for q=5-9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are η_{1}=1/4 and η_{2}=4/q^{2} at the two BKT transitions. Further, the approximate self-dual points β_{sd}(L), defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size L and behave as β_{sd}≃q/2π asymptotically at the large-q limit. The exponent η at β_{sd} is consistent with 1/q within statistical error as long as q≥5. Based on this, we further conjecture that η(β_{sd})=1/q holds exactly and is universal for systems in the q-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.

Critical properties of the two-dimensional q-state clock model

We perform the state-of-the-art tensor network simulations directly in the thermodynamic limit to clarify the critical properties of the q-state clock model on the square lattice. We determine accurately the two phase transition temperatures through the singularity of the classical analog of the entanglement entropy, and provide extensive numerical evidences to show that both transitions are of the Berezinskii-Kosterlitz-Thouless (BKT) type for q≥5 and that the low-energy physics of this model is well described by the Z_{q}-deformed sine-Gordon theory. We also determine the characteristic conformal parameters, especially the compactification radius, that govern the critical properties of the intermediate BKT phase.

Finite-m scaling analysis of Berezinskii-Kosterlitz-Thouless phase transitions and entanglement spectrum for the six-state clock model

We investigate the Berezinskii-Kosterlitz-Thouless transitions for the square-lattice six-state clock model with the corner-transfer matrix renormalization group (CTMRG). Scaling analyses for effective correlation length, magnetization, and entanglement entropy with respect to the cutoff dimension m at the fixed point of the CTMRG provide transition temperatures consistent with a variety of recent numerical studies. We also reveal that the fixed-point spectrum of the corner-transfer matrix in the critical intermediate phase of the six-state clock model is characterized by the scaling dimension consistent with the c=1 boundary conformal field theory associated with the effective Z_{6} dual sine-Gordon model.