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Register a new userThe Quantum Compass Model on the Square and Simple Cubic Lattices
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We use high-temperature series expansions to obtain thermodynamic properties of the quantum compass model, and to investigate the phase transition on the square and simple cubic lattices. On the square lattice we obtain evidence for a phase transition, consistent with recent Monte Carlo results. On the simple cubic lattice the same procedure provides no sign of a transition, and we conjecture that there is no finite temperature transition in this case.
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By means of Monte Carlo simulations, we study long-range site percolation on square and simple cubic lattices with various combinations of nearest neighbors, up to the eighth neighbors for the square lattice and the ninth neighbors for the simple cubic lattice. We find precise thresholds for 23 systems using a single-cluster growth algorithm. Site percolation on lattices with compact neighborhoods can be mapped to problems of lattice percolation of extended shapes, such as disks and spheres, and the thresholds can be related to the continuum thresholds $eta_c$ for objects of those shapes. This mapping implies $zp_{c} sim 4 eta_c = 4.51235$ in 2D and $zp_{c} sim 8 eta_c = 2.73512$ in 3D for large $z$ for circular and spherical neighborhoods respectively, where $z$ is the coordination number. Fitting our data to the form $p_c = c/(z+b)$ we find good agreement with $c = 2^d eta_c$; the constant $b$ represents a finite-$z$ correction term. We also study power-law fits of the thresholds.
We study the quantum criticality at finite temperature for three two-dimensional (2D) $JQ_3$ models using the first principle nonperturbative quantum Monte Carlo calculations (QMC). In particular, the associated universal quantities are obtained and their inverse temperature dependence are investigated. The considered models are known to have quantum phase transitions from the Neel order to the valence bond solid. In addition, these transitions are shown to be of second order for two of the studied models, with the remaining one being of first order. Interestingly, we find that the outcomes obtained in our investigation are consistent with the mentioned scenarios regarding the nature of the phase transitions of the three investigated models. Moreover, when the temperature dependence of the studied universal quantities is considered, a substantial difference between the two models possessing second order phase transitions and the remaining model is observed. Remarkably, by using the associated data from both the models that may have continuous transitions, good data collapses are obtained for a number of the considered universal quantities. The findings presented here not only provide numerical evidence to support the results established in the literature regarding the nature of the phase transitions of these $JQ_3$ models, but also can be employed as certain promising criterions to distinguish second order phase transitions from first order ones for the exotic criticalities of the $JQ$-type models. Finally, based on a comparison between the results calculated here and the corresponding theoretical predictions, we conclude that a more detailed analytic calculation is required in order to fully catch the numerical outcomes determined in our investigation.
A new quantum spin model with frustration, the `Union Jack model on the square lattice, is analyzed using spin-wave theory. For small values of the frustrating coupling $alpha$, the system is N{ e}el ordered as usual, while for large $alpha$ the frustration is found to induce a canted phase. The possibility of an intermediate spin-liquid phase is discussed.
Quantum phase transition in the one-dimensional period-two and uniform quantum compass model are studied by using the pseudo-spin transformation method and the trace map method. The exact solutions are presented, the fidelity, the nearest-neighbor pseudo-spin entanglement, spin and pseudo-spin correlation functions are then calculated. At the critical point, the fidelity and its susceptibility change substantially, the gap of pseudo-spin concurrence is observed, which scales as $1/N$ (N is system size). The spin correlation functions show smooth behavior around the critical point. In the period-two chain, the pseudo-spin correlation functions exhibit a oscillating behavior, which is absent in the unform chain. The divergent correlation length at the critical point is demonstrated in the general trend for both cases.
We study bond percolation on the simple cubic (SC) lattice with various combinations of first, second, third, and fourth nearest-neighbors by Monte Carlo simulation. Using a single-cluster growth algorithm, we find precise values of the bond thresholds. Correlations between percolation thresholds and lattice properties are discussed, and our results show that the percolation thresholds of these and other three-dimensional lattices decrease monotonically with the coordination number $z$ quite accurately according to a power law $p_{c} sim z^{-a}$, with exponent $a = 1.111$. However, for large $z$, the threshold must approach the Bethe lattice result $p_c = 1/(z-1)$. Fitting our data and data for lattices with additional nearest neighbors, we find $p_c(z-1)=1+1.224 z^{-1/2}$.
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