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Simulating the Berezinskii-Kosterlitz-Thouless transition with the complex Langevin algorithm

Philipp Heinen and Thomas Gasenzer
Phys. Rev. A 108, 053311 – Published 13 November 2023
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  • INTRODUCTION
  • COMPLEX LANGEVIN APPROACH TO A BOSE GAS
  • RESULTS OF THE CL SIMULATIONS
  • SUMMARY AND OUTLOOK
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Numerical simulations of the full quantum properties of interacting many-body systems by means of field-theoretic Monte Carlo techniques are often limited due to a sign problem. Here we simulate properties of a dilute two-dimensional Bose gas in the vicinity of the Berezinskii-Kosterlitz-Thouless (BKT) transition by means of the complex Langevin (CL) algorithm, thereby extending our previous CL study of the three-dimensional Bose gas to the lower-dimensional case. The purpose of the paper is twofold. On the one hand, it adds to benchmarking of the CL method and thus contributes to further exploring the range of applicability of the method. With the respective results, the universality of the equation of state is recovered, as well as the long-wave-length power-law dependence of the single-particle momentum spectrum below the BKT transition. Analysis of the rotational part of the current density corroborates vortex unbinding in crossing the transition. Beyond these measures of consistency we compute quantum corrections to the critical density and chemical potential in the weakly coupled regime. Our results show a shift of these quantities to lower values as compared to those obtained from classical field theory. It points in the opposite direction as compared to the shift of the critical density found by means of the path-integral Monte Carlo method at larger values of the coupling. Our simulations widen the perspective for precision comparisons with experiment.

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    • Received 19 June 2023
    • Accepted 19 October 2023

    DOI:https://doi.org/10.1103/PhysRevA.108.053311

    ©2023 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Bose-Einstein condensates
    1. Techniques
    Nonperturbative methodsQuantum Monte CarloStochastic differential equations
    Atomic, Molecular & Optical

    Authors & Affiliations

    Philipp Heinen1 and Thomas Gasenzer1,2

    • 1Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
    • 2Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

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    Issue

    Vol. 108, Iss. 5 — November 2023

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    • Figure 1
      Figure 1

      Critical density ρc (upper panel) and chemical potential μc (lower) for various coupling strengths mg as a function of the dimensionless parameter x=ln−2(LmgT). Solid lines represent a fit of (32) and (34) to the data points. Extrapolating them for x→0 we can recover the critical density and chemical potential in the infinite-volume limit. Fits of the higher-order formula (33) are shown as dashed lines, which hardly deviates from the leading-order expression. Where no error bars are seen, they are shorter than the width of the data points.

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    • Figure 2
      Figure 2

      Critical density ρc (upper panel) and chemical potential μc (lower) in the infinite-volume limit as functions of the coupling mg computed with the complex Langevin simulation of the full quantum model (blue points), within a regime of couplings for which runaway processes are absent. The results are compared to the predictions (18) with constant ζρ=380 and (19) with ζμ=13.2 as obtained from the classical simulation of [29] (black lines). The critical quantities appear to be shifted downwards in the full quantum simulation by about 4% (2%). The blue dashed lines represent (18) and (19) with constant ζρ=260±12 and (19) with ζμ=12.3±0.1, as obtained from fits to our data.

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    • Figure 3
      Figure 3

      Equation of state, relating density ρ, and chemical potential μ for three values of the coupling mg. The curves connecting the data points are obtained by spline interpolation. Error bars are too small to be visible. The inset of the upper panel shows the densities as functions of μ/T. In the main part of the upper panel, their arguments are rescaled with the coupling mg. The critical chemical potentials are shown as dotted vertical lines in the respective color. In the lower panel, the resulting curves are shifted by the critical chemical potentials and densities; cf. Eq. (20). The curves collapse for all three couplings considered, giving evidence of the universality of the 2D Bose gas near the BKT transition. For the critical chemical potentials and densities ρc and μc we took the result from Sec. 3a for mg=0.1, whereas ρc and μc for mg=0.025 and mg=0.2 are obtained from the differences μc(mg=0.025)−μc(mg=0.1) and μc(mg=0.2)−μc(mg=0.1) as well as ρc(mg=0.025)−ρc(mg=0.1) and ρc(mg=0.2)−ρc(mg=0.1), which by means of a least-squares fit were determined such that all three curves collapse. The black lines in the lower panel represent the mean field (MF) approximations (22) and (23), valid for X=(μ−μc)/(mgT)≫1 and X≪−1, respectively. The dashed lines show the MF behavior for ζρ=380 and ζμ=13.2 of [29], the solid lines for the values (36) and (37).

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    • Figure 4
      Figure 4

      Single-particle momentum spectra f(k)=f(|k|), Eq. (13), for a coupling mg=0.1 and two different chemical potentials: μ/(mgT)=2, slightly below the transition (upper panel), and μ/(mgT)=4, far below the transition (lower panel). The momentum is expressed in units of the thermal momentum kT=1/λT. Whereas slightly below the transition the spectrum approximately shows a k−7/4 power law in the infrared, below the thermal wave number kT, consistent with the near-critical scaling predicted by BKT theory, we find a MF Rayleigh-Jeans ∼k−2 fall off farther below the transition, which forms part of the Bogoliubov distribution (dashed line). The condensate fractions f(k=0)/Ntot are 61% and 82%, respectively, i.e., due to the finite extent of our system we find a macroscopic occupation of the condensate mode.

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    • Figure 5
      Figure 5

      Power-law exponent α as a function of μ/(mgT), as obtained from a least-squares fit of a linear function to lnf(k) vs lnk, obtained from our simulations for mg=0.1. The critical chemical potential is marked by a dotted black line. Going from close to the transition to far below the transition, α decreases from ≃1/4 to zero, in accordance with the prediction from BKT theory.

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    • Figure 6
      Figure 6

      Snapshots of the position-space density ρR≡φR2+χR2 at ϑ=4×103as−1 for mg=0.1 and four different chemical potentials (a) μ/(mgT)=0.8, (b) 1.2, (c) 1.6, and (d) 2.0 (upper panel). The temperature is kept fixed while the chemical potential is varied, such that the healing length ξh≡1/2mμ in units of the lattice spacing varies between ξh=3.16as and ξh=2as. The MF densities λT2ρ¯≡λT2μ/g are λT2ρ¯=5.02,7.54,10.05,12.56. As one can see, the position-space densities are completely dominated by fluctuations such that it is difficult to infer information about the topological phase transition from them. The lower panel shows the position-space phase ϕR≡arg(φR+iχR) for the same parameters, which indicates phase ordering across the transition.

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    • Figure 7
      Figure 7

      Mean integrated rotational part of the current squared, ρv,freeξh2, as defined in (40), as a function of μ/(mgT), for mg=0.1. It can be considered as a measure for the mean number of unbound vortices appearing in the system; see the discussion in the main text. Error bars are too small to be visible.

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    • Figure 8
      Figure 8

      Snapshots of the real part of ρv,free(x)ξh2, as defined in (27) and Appendix pp3 for mg=0.1 and the same four different chemical potentials (a) μ/(mgT)=0.8, (b) 1.2, (c) 1.6, and (d) 2.0, corresponding to the data shown in Fig. 6. For better visibility, fluctuations above the healing momentum kh=2mμ are filtered out by a low-pass filter. Note that by virtue of the complexification prescription of the CL algorithm, ρv,free(x)ξh2 will be in general complex in single realizations and its real part can become negative, while in the long-time average, ρv,free(x)ξh2 will come out real and positive.

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    • Figure 9
      Figure 9

      Determination of the BKT transition point by the secant method for mg=0.1 and L/as=128. Starting from two initial guesses for the critical chemical potential, μ0/(mgT)=1.633μ1/(mgT)=1.477, subsequent guesses are chosen as the intersection of the secant through the previous two data points and the μ/T axis (upper panel). The convergence is extremely fast. The lower panel shows the convergence of the superfluid densities as in Langevin time ϑ, with error bands obtained from the variance of several statistically independent runs.

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    • Figure 10
      Figure 10

      Critical density ρc as a function of the coupling mg computed by employing the Nelson criterion for the superfluid fraction (SFF), as described in Sec. 3a (blue points), in comparison to the results from the rescaling of the equation of state (green points). Since the latter yields only differences of critical densities, we chose the critical density at mg=0.1 to be equal to the result from the superfluid fraction method. The black line represents Eq. (18) with constant ζρ=380 obtained from the classical simulation of [29].

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    • Figure 11
      Figure 11

      Critical density ρc as a function of the coupling mg computed by Langevin simulations of the classical field theory, i.e., setting Nτ=1 (green points) in comparison with the full quantum simulations (blue points) and the result from the classical field theory simulation of [29]. The critical densities from the classical Langevin simulation are substantially higher than those from the full quantum simulation, and they result as higher than those from the classical field theory simulation [29].

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    • Figure 12
      Figure 12

      Comparison of the momentum spectrum from the full quantum simulation (blue points) and the classical simulation (green points) for mg=0.1 in the critical region, where the latter has been corrected in the UV by subtracting a Rayleigh-Jeans distribution and adding a Bose-Einstein distribution at zero chemical potential. Note that the zero mode has been shifted to a finite k value in order to make it visible on the double-logarithmic scale. The chemical potential μ of the quantum simulation is matched to the one of the classical simulation, μclass, according to μ=μclass−2gΔρ, Eq. (F2). Apart from small deviations at very high momenta, the spectra agree well over almost the entire momentum range. However, deviations are visible in the strongly correlated IR-modes, with the quantum system being farther in the superfluid phase than the classical one. The corresponding density of the quantum system results only by approximately 2% larger than the corrected density of the classical system.

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