We propose a new method to numerically calculate transition points that belongs to 2D Ising universality class for quantum spin models.Generally, near the multicritical point, in conventional methods, a finite size correction becomes very large.To suppress the effect of the multicritical point, we use a z-axis twisted boundary condition and a y-axis twisted boundary condition.We apply our method to an S = 1 2 bond-alternating XXZ model.The multicritical point of this model has a BKT transition, where the correlation length diverges singularly.However, with our method, the convergence of calculation is highly improved, thus we can calculate the transition point even near the multicritical point.
A point where several critical lines intersect is called a multicritical point.Near such a point, multiple critical phenomena interfere and a finite-size correction becomes large.Therefore, conventional methods cannot be applied near the multicritical point.We propose a new method to numerically calculate a transition point for a quantum spin chain near the multicritical point.We treat a bond-alternating (BA) XXZ model for S = 1/2, 1, 3/2.In a ground-state phase diagram of this model, a 2D Gaussian universality transition line bifurcates into two 2D Ising universality transition lines, which make a multicritical point.At this point, Berezinskii-Kosterlitz-Thouless transition occurs, where the correlation length diverges singularly.To deal with a 2D Ising universality, we review a transverse field 1D quantum Ising (TFI) model, corresponding to the classical 2D Ising model with transfer matrix.A relation between a disorder phase and an order phase is known as the Kramers-Wannier duality transformation.We find a proper duality transformation that is exact in the finite-size TFI model with a periodic and anti-periodic boundary condition.This shows that the energies for the two boundary conditions are crossing at the transition point.A free fermion field theory is derived from the critical Ising model by taking a continuum limit.From a conformal field theory, we verify that the energy-crossing is realized in the 2D Ising universality class, not only in the TFI model.We apply our method to the BA XXZ model.The energy-crossing happens in boundary conditions that are twisted around a z-axis and y-axis.In an anisotropic limit, two energies in a finite-size are crossing at the transition point since the BA XXZ model is identical to the TFI model.on the other hand, at the multicritical point, the finite-size correction vanishes by an isotropy and a twist translation symmetry.Therefore, near the multicritical point, our method has a smaller finite-size correction.In this paper, we also review the 2D Gaussian universality.By a bosonization, the BA XXZ model can be transformed to a phase Hamiltonian composed of boson operators.The degenerate energies for a twisted boundary condition are split by a perturbation around a Gaussian fixed point.By our method, we numerically calculate a transition point of the BA XXZ model for S = 1/2, 1, 3/2.As expected, the finite-size correction of numerical results becomes very small near the multicritical point.i
The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions
J1<0 (nearest neighbor, ferromagnetic) and
J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a
Z4 symmetry and is stabilized at low temperatures for
g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The
J1-
J2 model has a line of very weak first-order phase transitions in the whole region
1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from
g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point
g∗ in the
J1-
J2 model. The continuous transitions near
g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of
g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.
Quantum entanglement at critical points is often marked by universal characteristics. Here, the entanglement entropy is calculated at the quantum multicritical point of the random transverse-field Ising model (RTIM). We use an efficient implementation of the strong disorder renormalization group method in two and three dimensions for two types of disorder. For cubic subsystems we find a universal logarithmic corner contribution to the area law $bln(\ensuremath{\ell})$ that is independent of the form of disorder. Our results agree qualitatively with those at the quantum critical points of the RTIM, but with new
b prefactors due to having both geometric and quantum fluctuations at play. By studying the vicinity of the multicritical point, we demonstrate that the corner contribution serves as an ``entanglement susceptibility,'' a useful tool to locate the phase transition and to measure the correlation length critical exponents.
Two nonperturbative methods by means of the transfer-matrix finite-size-scaling and Monte Carlo simulations are used to investigate the spin-1 Ashkin-Teller model (ATM). We have obtained rich phase diagrams with first- and second-order phase transitions with several multicritical points of higher order. Also this model exhibits a partially ordered phase
PO2 which does not exist in the spin-1/2 Ashkin-Teller model (ATM). Finally, the critical behavior of this model is discussed.
The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-para-magnetic phase boundary is probed, with the following estimates for the associated critical indices: $\ensuremath{\nu}=1.49(2)$, $\ensuremath{\gamma}=2.71(4)$, ${\ensuremath{\eta}}_{1}=0.183(3)$, which are distinctly different from the percolation values.