Persistence of the Berezinskii-Kosterlitz-Thouless transition with long-range couplings

TL;DR

The paper tackles whether the BKT transition survives in a two-dimensional long-range XY model with interactions decaying as $1/r^{2+\sigma}$. Using complementary Landau–Peierls-type arguments and renormalization-group analyses that include coupling between spin waves and vortices, it shows that long-range couplings renormalize vortex interactions in a way that preserves the BKT transition for all $\sigma$, while introducing a magnetized phase for sufficiently small $\sigma$. The key result is a renormalized condition $\beta_{\mathrm{eff},c}=2/\pi$ that shifts the BKT temperature upwards as $\sigma$ decreases, implying no direct transition from magnetized to disordered phases. This has experimental relevance for Rydberg-atom platforms and highlights the importance of long-range couplings for the behavior of topological defects in continuous-spin systems.

Abstract

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 $\sim 1/{r^{2+σ}}$. 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. We use both Landau-Peierls-type arguments and renormalization group calculations - including a coupling between spin wave and topological excitations - and obtain similar 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.

Figures (2)

• Figure 1: (a) Vortex-antivortex configuration (grey) overlayed with a spin wave configuration (blue). The vortices are separated at a distance of $r$ in units of the lattice spacing $a$. In the short-range XY model, spin waves and vortices do not interact with each other, which is no longer true in the presence of long-range interactions. (b) Effective potential of a vortex-antivortex pair. Increasing the temperature decreases the incline of the slope until vortices deconfine at the BKT transition (see inset figure, incline becomes zero). Long-range interactions strengthen the vortex-antivortex potential by increasing the slope. (c) Phase diagram of the long-range XY model, as obtained by the renormalization of vortices by long-range interactions. The long-range interactions stabilize vortices, so that the BKT transition appears at higher temperatures. As a result, the QLRO phase persists at intermediate temperatures for arbitrarily low values of $\sigma$.
• Figure 2: RG flow sections of the long-range XY model for $\sigma=1$. Note, the colors are chosen in accordance to the phase diagram in Figure \ref{['fig:figure1']}. (a) RG flow in the $T-y^2$-plane for fixed $g=1$. Because $g$ is kept fixed, also the effect of $g$ on the renormalization of $\beta$ is neglected. As can be seen, the BKT transition line persists up to the renormalized temperature $T_{{\mathrm{BKT}}}$ (see end matter for definition). At $T_{\mathrm{c, lr}}=2\pi(2-\sigma)$ long-range interactions become relevant, and from Eq. \ref{['eq:rgflow_rgequations']} we observe a divergence in the flow of $y^2$. We cannot make decisive statements what happens at this point, because both analytical and numerical treatments struggle with the divergence. The full analysis of the RG equations remains left open for future work. (b) RG flow in the $T-g$-plane for $y^2=0$. As can be seen, long-range interactions become relevant for $T<T_{\mathrm{c, lr}}$.