[Paper Review] Berezinskii-Kosterlitz-Thouless Phase Transitions with Long-Range Couplings

Guido Giachetti, Nicolò Defenu+2 more|arXiv (Cornell University)|Jan 1, 2021

Quantum many-body systems89 references44 citations

TL;DR

This paper investigates Berezinskii-Kosterlitz-Thouless (BKT) phase transitions in two-dimensional XY models with long-range power-law couplings ∼r⁻²⁻σ. It reveals a novel phase diagram for 7/4 < σ < 2, featuring a finite-temperature quasi-ordered phase between a low-T symmetry-broken phase and a high-T disordered phase, with Tc exhibiting unique universal behavior distinct from short-range models. The findings are relevant for current AMO quantum systems with tunable long-range interactions.

ABSTRACT

The Berezinskii-Kostelitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry-breaking, where a quasi-ordered 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 $T_{ m BKT}$. In this letter, we consider the effect of long-range decaying couplings $\sim r^{-2-\sigma}$ on this phenomenon. After pointing out the relevance of this non trivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features -- for $7/4<\sigma<2$ -- a quasi ordered phase in a finite temperature range $T_c < T < T_{ m BKT}$, which occurs between a symmetry broken phase for $T<T_c$ and a disordered phase for $T>T_{ m BKT}$. 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.

Motivation & Objective

• To understand the fate of the BKT topological phase transition when long-range interactions (J ∼ r⁻²⁻σ) are introduced in two-dimensional planar rotator systems.
• To resolve the open question of whether long-range interactions modify the universal critical behavior of the BKT transition, especially in the regime where the Hohenberg-Mermin-Wagner theorem no longer applies.
• To identify and characterize a new phase diagram featuring a finite-temperature quasi-ordered phase between symmetry-broken and disordered phases for 7/4 < σ < 2.
• To establish the universal nature of the critical temperature Tc in the long-range regime, contrasting it with the short-range XY model’s behavior.
• To provide a theoretical framework applicable to current experimental platforms such as ultracold atoms, trapped ions, and Rydberg gases with tunable long-range interactions.

Proposed method

• The study employs a classical O(2) spin model on a 2D lattice with long-range couplings J|i−j| ∼ g/|i−j|²⁺σ, where σ > 0 ensures thermodynamic additivity.
• The analysis combines analytical field-theory arguments, spin-wave theory, and renormalization group (RG) insights to assess the stability of the quasi-ordered phase.
• The authors use the Sak’s criterion for long-range criticality, adapted to the 2D XY model, to determine the crossover between short- and long-range critical behavior.
• They examine the infrared finiteness of spin-wave fluctuations via the integral ∫d²q/qσ, showing that for σ < 2, fluctuations remain infrared finite, enabling spontaneous magnetization.
• The phase diagram is constructed by combining known results: for σ < 7/4, the system exhibits a symmetry-broken phase at low T; for σ > 2, the BKT transition persists; and for 7/4 < σ < 2, a new quasi-ordered phase emerges.
• The critical temperature Tc is analyzed for its universal features, showing deviations from the short-range XY model’s behavior due to the interplay between long-range interactions and topological defects.

Experimental results

Research questions

• RQ1 How does the inclusion of long-range couplings ∼r⁻²⁻σ modify the BKT phase transition in two-dimensional XY models?
• RQ2 Does a finite-temperature quasi-ordered phase exist between the symmetry-broken and disordered phases for intermediate values of σ?
• RQ3 What is the nature of the critical temperature Tc in the long-range regime, and how does it differ from the short-range XY model?
• RQ4 To what extent do long-range interactions alter the universality class of the BKT transition, particularly in the presence of topological defects?
• RQ5 Can the predicted phase diagram and critical behavior be experimentally observed in current 2D atomic, molecular, and optical quantum systems?

Key findings

• For 7/4 < σ < 2, the system exhibits a finite-temperature quasi-ordered phase between a low-temperature symmetry-broken phase and a high-temperature disordered phase, characterized by power-law correlations.
• The critical temperature Tc for the onset of long-range order displays unique universal features that differ significantly from those of the short-range XY model.
• For σ < 7/4, the system remains in a symmetry-broken phase at low temperatures, with no intermediate quasi-ordered phase, consistent with mean-field critical exponents.
• For σ > 2, the BKT transition persists, and the system remains in the topological universality class, with the critical behavior governed by the standard BKT mechanism.
• The spin-wave theory shows that the infrared divergence of fluctuations is suppressed for σ < 2, allowing spontaneous magnetization and the emergence of a finite-temperature quasi-ordered phase.
• The results are robust and universal, suggesting that the predicted phase diagram can be probed in current experimental platforms such as ultracold atoms, trapped ions, and Rydberg gases with tunable long-range interactions.