Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.Long-range interactions are relevant for a large variety of quantum systems in quantum optics and condensed matter physics. In particular, the control of quantum-optical platforms promises to gain deep insights into quantum-critical properties induced by the long-range nature of interactions. From a theoretical perspective, long-range interactions are notoriously complicated to treat. Here, we give an overview of recent advancements to investigate quantum magnets with long-range interactions focusing on two techniques based on Monte Carlo integration. First, the method of perturbative continuous unitary transformations where classical Monte Carlo integration is applied within the embedding scheme of white graphs. This linked-cluster expansion allows extracting high-order series expansions of energies and observables in the thermodynamic limit. Second, stochastic series expansion quantum Monte Carlo integration enables calculations on large finite systems. Finite-size scaling can then be used to determine the physical properties of the infinite system. In recent years, both techniques have been applied successfully to one- and two-dimensional quantum magnets involving long-range Ising, XY, and Heisenberg interactions on various bipartite and non-bipartite lattices. Here, we summarise the obtained quantum-critical properties including critical exponents for all these systems in a coherent way. Further, we review how long-range interactions are used to study quantum phase transitions above the upper critical dimension and the scaling techniques to extract these quantum critical properties from the numerical calculations.

arXiv:2403.00421EntropyMay 1, 202418 citations

Dynamic critical behavior of the one-dimensional XY model with a long-range interaction

Seong-Gyu Yang, Hye Jin Park, Beom Jun Kim

We numerically study dynamic critical behavior of the one-dimensional $XY$ model with a long-range interaction by using the Monte Carlo method and the resistively-shunted Josephson junction model. The two dynamic models exhibit the mean-field universality class in equilibrium as expected, but the dynamic critical behavior is shown to sensitively depend on details of numerical simulation. In more detail, the trial angle range in the Monte Carlo simulation is found to alter the value of the dynamic critical exponent, and the scaling of the Monte Carlo time unit by the acceptance ratio is shown to be useful to improve the estimation of the dynamic critical exponent. We compare the Monte Carlo result of the dynamic critical exponent with the result from the more realistic dynamic model of the resistively-shunted junction. We conclude that the small value of the trial angle range should be used to properly detect dynamic critical behavior.We numerically study dynamic critical behavior of the one-dimensional $XY$ model with a long-range interaction by using the Monte Carlo method and the resistively-shunted Josephson junction model. The two dynamic models exhibit the mean-field universality class in equilibrium as expected, but the dynamic critical behavior is shown to sensitively depend on details of numerical simulation. In more detail, the trial angle range in the Monte Carlo simulation is found to alter the value of the dynamic critical exponent, and the scaling of the Monte Carlo time unit by the acceptance ratio is shown to be useful to improve the estimation of the dynamic critical exponent. We compare the Monte Carlo result of the dynamic critical exponent with the result from the more realistic dynamic model of the resistively-shunted junction. We conclude that the small value of the trial angle range should be used to properly detect dynamic critical behavior.

Physical review. ESep 17, 20181 citation

Quantum phase transitions from competing short- and long-range interactions on a π -flux lattice

Xingchuan Zhu, Yiqun Huang, Huaiming Guo +1 more

Quantum phase transitions from the cluster-charge interaction, which is composed of competing short- and long-range interactions, are investigated on a $\ensuremath{\pi}$-flux lattice by using the mean-field theory and determinant quantum Monte Carlo (DQMC) simulations. Both methods identify a plaquette-dimer phase, which develops from a finite interaction strength. While its signature in DQMC is relatively weak, an obvious antiferromagnetic transition is revealed in the spin structure factor instead. The corresponding critical interaction and exponents are readily obtained by finite-size scalings, with the plaquette-dimer structure factor that can also be well scaled. These results suggest a possible deconfined quantum critical point between the plaquette-dimer and antiferromagnetic phases driven by the cluster-charge interaction on a $\ensuremath{\pi}$-flux lattice.Quantum phase transitions from the cluster-charge interaction, which is composed of competing short- and long-range interactions, are investigated on a $\ensuremath{\pi}$-flux lattice by using the mean-field theory and determinant quantum Monte Carlo (DQMC) simulations. Both methods identify a plaquette-dimer phase, which develops from a finite interaction strength. While its signature in DQMC is relatively weak, an obvious antiferromagnetic transition is revealed in the spin structure factor instead. The corresponding critical interaction and exponents are readily obtained by finite-size scalings, with the plaquette-dimer structure factor that can also be well scaled. These results suggest a possible deconfined quantum critical point between the plaquette-dimer and antiferromagnetic phases driven by the cluster-charge interaction on a $\ensuremath{\pi}$-flux lattice.

arXiv:2204.12147Physical review. B./Physical review. BAug 3, 20227 citations

Quantum statistical mechanics : a Monte Carlo study of clusters

Fung-kiu Chow

Jan 1, 20000 citations

We numerically study dynamic critical behavior of the one-dimensional

XY

model with a long-range interaction by using the Monte Carlo method and the resistively-shunted Josephson junction model. The two dynamic models exhibit the mean-field universality class in equilibrium as expected, but the dynamic critical behavior is shown to sensitively depend on details of numerical simulation. In more detail, the trial angle range in the Monte Carlo simulation is found to alter the value of the dynamic critical exponent, and the scaling of the Monte Carlo time unit by the acceptance ratio is shown to be useful to improve the estimation of the dynamic critical exponent. We compare the Monte Carlo result of the dynamic critical exponent with the result from the more realistic dynamic model of the resistively-shunted junction. We conclude that the small value of the trial angle range should be used to properly detect dynamic critical behavior.