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arXiv:2201.03650 · Physical review. B./Physical review. B

Berezinskii-Kosterlitz-Thouless transitions in classical and quantum long-range systems

Guido Giachetti, Andrea Trombettoni, Stefano Ruffo, Nicolò Defenu

Published Jul 18, 2022• 28 citations
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Abstract

In the past decades, considerable efforts have been made to understand the critical features of both classical and quantum long-range (LR) interacting models. The case of the Berezinskii-Kosterlitz-Thouless (BKT) universality class, as in the two-dimensional (2D) classical $XY$ model, is considerably complicated by the presence, for short-range interactions, of a line of renormalization group fixed points. In this paper, we discuss a field-theoretical treatment of the 2D $XY$ model with LR couplings, and we compare it with results from the self-consistent harmonic approximation. These methods lead to a rich phase diagram, where both power law BKT scaling and spontaneous symmetry breaking appear for the same (intermediate) decay rates of LR interactions. We also discuss the Villain approximation for the 2D $XY$ model with power law couplings, providing hints that, in the LR regime, it fails to reproduce the correct critical behavior. The obtained results are then applied to the LR quantum $XXZ$ spin chain at zero temperature. We discuss the relation between the phase diagrams of the two models, and we give predictions about the scaling of the order parameter of the quantum chain close to the transition.

DOI
10.1103/physrevb.106.014106

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In the past decades, considerable efforts have been made to understand the critical features of both classical and quantum long-range (LR) interacting models. The case of the Berezinskii-Kosterlitz-Thouless (BKT) universality class, as in the two-dimensional (2D) classical XYXYXY model, is considerably complicated by the presence, for short-range interactions, of a line of renormalization group fixed points. In this paper, we discuss a field-theoretical treatment of the 2D XYXYXY model with LR couplings, and we compare it with results from the self-consistent harmonic approximation. These methods lead to a rich phase diagram, where both power law BKT scaling and spontaneous symmetry breaking appear for the same (intermediate) decay rates of LR interactions. We also discuss the Villain approximation for the 2D XYXYXY model with power law couplings, providing hints that, in the LR regime, it fails to reproduce the correct critical behavior. The obtained results are then applied to the LR quantum XXZXXZXXZ spin chain at zero temperature. We discuss the relation between the phase diagrams of the two models, and we give predictions about the scaling of the order parameter of the quantum chain close to the transition.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.
Abstract The crossover between short-range and long-range (LR) universal behaviors remains a central theme in the physics of LR interacting systems. The competition between LR coupling and the Berezinskii–Kosterlitz–Thouless mechanism makes the problem more subtle and less understood in the two-dimensional (2D) XY model, a cornerstone for investigating low-dimensional phenomena and their implications in quantum computation. We study the 2D XY model with algebraically decaying interaction ∼1/r 2+σ . Utilizing an advanced update strategy, we conduct LR Monte Carlo simulations of the model up to a linear size of L = 8192. Our results demonstrate continuous phase transitions into a ferromagnetic phase for σ < 2, which exhibit the simultaneous emergence of a long-ranged order and a power-law decaying correlation function due to the Goldstone mode. Furthermore, we find logarithmic scaling behaviors in the low-temperature phase at σ = 2. The observed scaling behaviors in the low-temperature phase for σ ≤ 2 agree with our theoretical analysis. Our findings request further theoretical understanding and can be of practical application in cutting-edge experiments like Rydberg atom arrays.
Abstract We derive the self-consistent harmonic approximation for the 2 D XY model with non-local interactions. The resulting equation for the variational couplings holds for any form of the spin-spin coupling as well as for any dimension. Our analysis is then specialized to power-law couplings decaying with the distance r as in order to investigate the robustness, at finite σ , of the Berezinskii-Kosterlitz-Thouless (BKT) transition, which occurs in the short-range limit . We propose an ansatz for the functional form of the variational couplings and show that for any the BKT mechanism occurs. The present investigation provides an upper bound for the critical threshold above which the traditional BKT transition persists in spite of the non-local nature of the couplings.
We discuss the critical behaviour of the 1-dimensional classical XY model with power law decay long range interactions by using Monte Carlo simulation. It has been predicted by renormalization group calculation that the critical behaviour changes with the power of interactions. From the simulation result of temperature dependence of the specific heat, we confirm that the system shows the critical behaviour of the mean field type and the non-trivial long range order. We also estimate the critical exponent η for the system with the inverse square long range interactions, and by using the phenomenological renormalization group plot, we confirm the 'Berezinskii-Kosterlitz-Thouless (BKT)-like' transition which has been predicted by the spin wave calculations.
We study the universal critical behavior of two-dimensional (2D) lattice bosonic gases at the Berezinskii-Kosterlitz-Thouless (BKT) transition, which separates the low-temperature superfluid phase from the high-temperature normal phase. For this purpose, we perform quantum Monte Carlo simulations of the hard-core Bose-Hubbard (BH) model at zero chemical potential. We determine the critical temperature by using a matching method that relates finite-size data for the BH model with corresponding data computed in the classical XYXYXY model. In this approach, the neglected scaling corrections decay as inverse powers of the lattice size LLL, and not as powers of 1/lnL1/\mathrm{ln}L1/lnL, as in more standard approaches, making the estimate of the critical temperature much more reliable. Then, we consider the BH model in the presence of a trapping harmonic potential, and we verify the universality of the trap-size dependence at the BKT critical point. This issue is relevant for experiments with quasi-2D trapped cold atoms.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is an archetypal example of a topological phase transition, which is driven by the proliferation of vortices. In this Letter, we analyze the persistence of the BKT transition in the XY model under the influence of long-range algebraically decaying interactions of the form ∼1/r^{2+σ}. The model hosts a magnetized low temperature phase for sufficiently small σ. Crucially, in the presence of long-range interactions, spin waves renormalize the interaction between vortices, which stabilizes the BKT transition. As a result, we find that there is no direct transition from the magnetized to the disordered phase and that the BKT transition persists for arbitrary long-range exponents, which is distinct from previous results. The key methodological advance is the inclusion of coupling between spin wave and topological excitations. We do this by developing Landau-Peierls-type arguments and using higher-order renormalization group calculations, obtaining compatible results. We emphasize that Landau-Peierls-type arguments are a powerful tool for analyzing continuous spin models. We discuss the relevance of our findings for current Rydberg atom experiments, and highlight the importance of long-range couplings for other types of topological defects.