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arXiv:0807.2436 · Physical Review A

Dislocation-mediated melting in superfluid vortex lattices

S. Andrew Gifford, Gordon Baym

Published Oct 10, 2008• 14 citations
arXivHTMLPDF

Abstract

We describe thermal melting of the two-dimensional vortex lattice in a rotating superfluid by generalizing the Halperin and Nelson theory of dislocation-mediated melting and derive a melting temperature proportional to the renormalized shear modulus of the vortex lattice. The rigid-body rotation of the superfluid attenuates the effects of lattice compression on the energy of dislocations and hence the melting temperature, while not affecting the shearing. Finally, we discuss dislocations and thermal melting in inhomogeneous rapidly rotating Bose-Einstein condensates; we delineate a phase diagram in the temperature--rotation rate plane, and infer that the thermal melting temperature should lie below the Bose-Einstein transition temperature.

DOI
10.1103/physreva.78.043607

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We describe thermal melting of the two-dimensional vortex lattice in a rotating superfluid by generalizing the Halperin and Nelson theory of dislocation-mediated melting and derive a melting temperature proportional to the renormalized shear modulus of the vortex lattice. The rigid-body rotation of the superfluid attenuates the effects of lattice compression on the energy of dislocations and hence the melting temperature, while not affecting the shearing. Finally, we discuss dislocations and thermal melting in inhomogeneous rapidly rotating Bose-Einstein condensates; we delineate a phase diagram in the temperature--rotation rate plane, and infer that the thermal melting temperature should lie below the Bose-Einstein transition temperature.
"We approach the static elasticity of a vortex lattice using the elastohydrodyamic approach. We derive the elastic Green's function for the vortex lattice, and connect it to the lattice distortion in an inhomogeneous condensate. We find that the longitudinal elastic response of a vortex lattice is screened by the effect of the ""global"" fluid velocity, but the shear response remains the same as conventional elastic systems. Applying the theory of dislocation-mediated melting to the vortex lattice, we find that the lattice melts at a temperature depending only on the shear modulus. Finally, we apply the theory of melting to trapped rapidly rotating BECs."
We investigate a shift in the critical temperature of rotating Bose-Einstein condensates mediated by the melting of the vortex lattice. Numerical simulations reveal that this temperature exhibits contrasting behavior depending on the system configuration: a negative shift occurs for fixed trap potentials due to the expansion of the condensate, while a positive shift is observed for fixed volumes, where vortex-lattice rigidity suppresses thermal fluctuations. We introduce a vortex-energy model that captures the role of vortex interactions, the positional energy of the vortex lattice, as well as the phase transition and how the vortex lattice disappears. The findings provide insights into the thermodynamic properties of rotating condensates and the dynamics of vortex-lattice melting, offering potential parallels with other quantum systems such as type-II superconductors.
We report the observation of the melting of a vortex lattice in a fast rotating quasi-two dimensional Bose gas, under the influence of thermal fluctuations. We image the vortex lattice after a time-of-flight expansion, for increasing rotation frequency at constant atom number and temperature. We detect the vortex positions and study the order of the lattice using the pair correlation function and the orientational correlation function. We evidence the melting transition by a change in the decay of orientational correlations, associated with a proliferation of dislocations. Our findings are consistent with the hexatic to liquid transition in the Kosterlitz, Thouless, Halperin, Nelson, and Young scenario for two-dimensional melting.
Abstract We present a classical field simulation study of the thermal melting of a two-dimensional (2D) vortex lattice in a rotating Bose gas, focusing on the role of finite-size effects on the melting temperature. This work constitutes a numerical continuation of the recent experimental investigation reported in (Sharma 2024 Phys. Rev. Lett. 133 143401), which addressed the thermal melting of a vortex lattice in a quasi-2D Bose gas. Using the stochastic projected Gross–Pitaevskii equation in a harmonic plus quartic trap, we simulate the finite-temperature equilibrium state and extract vortex configurations from density snapshots. Clear signatures of the two-step Kosterlitz–Thouless–Halperin–Nelson–Young melting scenario are identified. Our simulations enable a detailed characterization of the crystalline, hexatic, and liquid phases through correlation functions quantifying the translational and orientational order and through defect statistics. Finite-size effects are shown to play a crucial role at lower rotation frequencies, affecting the proliferation of lattice defects.
We present a classical field simulation study of the thermal melting of a two-dimensional vortex lattice in a rotating Bose gas, focusing on the role of finite-size effects on the melting temperature. This work constitutes a numerical continuation of the recent experimental investigation reported in [Physical Review Letters 133, 143401 (2024)], which addressed the thermal melting of a vortex lattice in a quasi-two-dimensional Bose gas. Using the stochastic projected Gross-Pitaevskii equation in a harmonic plus quartic trap, we simulate the finite-temperature equilibrium state and extract vortex configurations from density snapshots. Clear signatures of the two-step Kosterlitz--Thouless--Halperin--Nelson--Young melting scenario are identified. Our simulations enable a detailed characterization of the crystalline, hexatic, and liquid phases through correlation functions quantifying the translational and orientational order and through defect statistics. Finite-size effects are shown to play a crucial role at lower rotation frequencies, affecting the proliferation of lattice defects.
After delineating the physical regimes that vortex lattices encounter in rotating Bose-Einstein condensates as the rotation rate $\ensuremath{\Omega}$ increases, we derive the normal modes of the vortex lattice in two dimensions at zero temperature. Taking into account effects of the finite compressibility, we find an inertial mode of frequency $\ensuremath{\geqslant}2\ensuremath{\Omega}$, and a primarily transverse Tkachenko mode, whose frequency goes from being linear in the wave vector in the slowly rotating regime, where $\ensuremath{\Omega}$ is small compared with the lowest compressional mode frequency, to quadratic in the wave vector in the opposite limit. We calculate the correlation functions of vortex displacements and phase, density and superfluid velocities, and find that the zero-point excitations of the soft quadratic Tkachenko modes lead in a large system to a loss of long-range phase correlations, growing logarithmically with distance, and hence lead to a fragmented state at zero temperature. The vortex positional ordering is preserved at zero temperature, but the thermally excited Tkachenko modes cause the relative positional fluctuations to grow logarithmically with separation at finite temperature. The superfluid density, defined in terms of the transverse velocity autocorrelation function, vanishes at all temperatures. Finally we construct the long-wavelength single-particle Green's function in the rotating system and calculate the condensate depletion as a function of temperature.