Re-examining the directional-ordering transition in the compass model with screw-periodic boundary conditions

We study the directional-ordering transition in the two-dimensional classical and quantum compass models on the square lattice by means of Monte Carlo simulations. An improved algorithm is presented which builds on the Wolff cluster algorithm in one-dimensional subspaces of the configuration space. This improvement allows us to study classical systems up to $L=512$. Based on the new algorithm we give evidence for the presence of strongly anomalous scaling for periodic boundary conditions which is much worse than anticipated before. We propose and study alternative boundary conditions for the compass model which do not make use of extended configuration spaces and show that they completely remove the problem with finite-size scaling. In the last part, we apply these boundary conditions to the quantum problem and present a considerably improved estimate for the critical temperature which should be of interest for future studies on the compass model. Our investigation identifies a strong one-dimensional magnetic ordering tendency with a large correlation length as the cause of the unusual scaling and moreover allows for a precise quantification of the anomalous length scale involved.Comment: 10 pages, 8 figures; version as publishe

Evidence of Unconventional Universality Class in a Two-Dimensional Dimerized Quantum Heisenberg Model

The two-dimensional $J$-$J^\prime$ dimerized quantum Heisenberg model is studied on the square lattice by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio \hbox{$\alpha=J^\prime/J$}. The critical point of the order-disorder quantum phase transition in the $J$-$J^\prime$ model is determined as \hbox{$\alpha_\mathrm{c}=2.5196(2)$} by finite-size scaling for up to approximately $10 000$ quantum spins. By comparing six dimerized models we show, contrary to the current belief, that the critical exponents of the $J$-$J^\prime$ model are not in agreement with the three-dimensional classical Heisenberg universality class. This lends support to the notion of nontrivial critical excitations at the quantum critical point.Comment: 4+ pages, 5 figures, version as publishe

Zero-temperature Monte Carlo study of the non-coplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice

We investigate the ground-state properties of the highly degenerate non-coplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice with Monte Carlo simulations. For that purpose, we introduce an Ising pseudospin representation of the ground states, and we use a simple Metropolis algorithm with local updates, as well as a powerful cluster algorithm. At sizes that can be sampled with local updates, the presence of long-range order is surprisingly combined with an algebraic decay of correlations and the complete disordering of the chirality. It is only thanks to the investigation of unusually large systems (containing $\sim 10^8$ spins) with cluster updates that the true asymptotic regime can be reached and that the system can be proven to consist of equivalent (i.e., equally ordered) sublattices. These large-scale simulations also demonstrate that the scalar chirality exhibits long-range order at zero temperature, implying that the system has to undergo a finite-temperature phase transition. Finally, we show that the average distance in the order parameter space, which has the structure of an infinite Cayley tree, remains remarkably small between any pair of points, even in the limit when the real space distance between them tends to infinity.Comment: 15 pages, 10 figure

Finite-Temperature N\'eel Ordering of Fluctuations in a Plaquette Orbital Model

We present a pseudospin model which should be experimentally accessible using solid-state devices and, being a variation on the compass model, adds to the toolbox for the protection of qubits in the area of quantum information. Using Monte Carlo methods, we find for both classical and quantum spins in two and three dimensions Ising-type Neel ordering of energy fluctuations at finite temperatures without magnetic order. We also readdress the controversy concerning the stability of the ordered state in the presence of quenched impurities and present numerical results which are at clear variance with earlier claims in the literature.Comment: 4 pages, 5 figures, revised and extended versio

Comprehensive quantum Monte Carlo study of the quantum critical points in planar dimerized/quadrumerized Heisenberg models

We study two planar square lattice Heisenberg models with explicit dimerization or quadrumerization of the couplings in the form of ladder and plaquette arrangements. We investigate the quantum critical points of those models by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio $\alpha = J^\prime/J$. The critical point of the order-disorder quantum phase transition in the ladder model is determined as $\alpha_\mathrm{c} = 1.9096(2)$ improving on previous studies. For the plaquette model we obtain $\alpha_\mathrm{c} = 1.8230(2)$ establishing a first benchmark for this model from quantum Monte Carlo simulations. Based on those values we give further convincing evidence that the models are in the three-dimensional (3D) classical Heisenberg universality class. The results of this contribution shall be useful as references for future investigations on planar Heisenberg models such as concerning the influence of non-magnetic impurities at the quantum critical point.Comment: 10+ pages, 7 figures, 4 table

Monte Carlo simulations of the directional-ordering transition in the two-dimensional classical and quantum compass model

A comprehensive study of the two-dimensional (2D) compass model on the square lattice is performed for classical and quantum spin degrees of freedom using Monte Carlo and quantum Monte Carlo methods. We employ state-of-the-art implementations using Metropolis, stochastic series expansion and parallel tempering techniques to obtain the critical ordering temperatures and critical exponents. In a pre-investigation we reconsider the classical compass model where we study and contrast the finite-size scaling behavior of ordinary periodic boundary conditions against annealed boundary conditions. It is shown that periodic boundary conditions suffer from extreme finite-size effects which might be caused by closed loop excitations on the torus. These excitations also appear to have severe effects on the Binder parameter. On this footing we report on a systematic Monte Carlo study of the quantum compass model. Our numerical results are at odds with recent literature on the subject which we trace back to neglecting the strong finite-size effects on periodic lattices. The critical temperatures are obtained as $T_\mathrm{c}=0.1464(2)J$ and $T_\mathrm{c}=0.055(1)J$ for the classical and quantum version, respectively, and our data support a transition in the 2D Ising universality class for both cases.Comment: 8 pages, 7 figures, differs slightly from published versio

Evidence of columnar order in the fully frustrated transverse field Ising model on the square lattice

Using extensive classical and quantum Monte Carlo simulations, we investigate the ground-state phase diagram of the fully frustrated transverse field Ising model on the square lattice. We show that pure columnar order develops in the low-field phase above a surprisingly large length scale, below which an effective U(1) symmetry is present. The same conclusion applies to the Quantum Dimer Model with purely kinetic energy, to which the model reduces in the zero-field limit, as well as to the stacked classical version of the model. By contrast, the 2D classical version of the model is shown to develop plaquette order. Semiclassical arguments show that the transition from plaquette to columnar order is a consequence of quantum fluctuations.Comment: 5 pages (including Supplemental Material), 5 figure