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arXiv:cond-mat/0605102 · Physical Review Letters

Topological Defects and the Superfluid Transition of the s=1 Spinor Condensate in Two Dimensions

Subroto Mukerjee, Cenke Xu, Joel E. Moore

Published Sep 20, 2006• 102 citations
arXivHTMLPDF

Abstract

The s=1 spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We analyze the topological defects of the polar condensate, correcting previous studies, and show that the polar condensate in two dimensions is unstable at any finite temperature; instead, there is a nematic or paired superfluid phase with algebraic order in exp(2itheta), where theta is the superfluid phase, and no magnetic order. The Kosterlitz-Thouless transition out of this phase is driven by unbinding of half-vortices (the spin-disordered version of the combined spin and phase defects found by Zhou), and the anomalous universal 8T(c)/pi stiffness jump at the transition is confirmed in numerical simulations. The anomalous stiffness jump is a clear experimental signature of this phase and the corresponding phase transition.

DOI
10.1103/physrevlett.97.120406

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The s=1 spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We analyze the topological defects of the polar condensate, correcting previous studies, and show that the polar condensate in two dimensions is unstable at any finite temperature; instead, there is a nematic or paired superfluid phase with algebraic order in exp(2itheta), where theta is the superfluid phase, and no magnetic order. The Kosterlitz-Thouless transition out of this phase is driven by unbinding of half-vortices (the spin-disordered version of the combined spin and phase defects found by Zhou), and the anomalous universal 8T(c)/pi stiffness jump at the transition is confirmed in numerical simulations. The anomalous stiffness jump is a clear experimental signature of this phase and the corresponding phase transition.
We study the phase diagram of spin-one polar condensates in a two-dimensional optical lattice with magnetic anisotropy. We show that the topological binding of vorticity to nematic disclinations allows for a rich variety of phase transitions. These include Kosterlitz-Thouless-like transitions with a superfluid stiffness jump that can be experimentally tuned to take a continuous set of values, and a cascaded Kosterlitz-Thouless transition, characterized by two divergent length scales. For higher integer spin bosons SSS, the thermal phase transition out of the planar polar phase is strongly affected by the parity of SSS.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.
The s=1s=1s=1 spinor Bose condensate at zero temperature supports ferromagnetic and polar phases that combine magnetic and superfluid ordering. We investigate the formation of magnetic domains at finite temperature and magnetic field in two dimensions in an optical trap. We study the general ground state phase diagram of a spin-1 system and focus on a phase that has a magnetic Ising order parameter and numerically determine the nature of the finite-temperature superfluid and magnetic phase transitions. We then study three different dynamical models: model A, which has no conserved quantities, model F, which has a conserved second-sound mode, and the Gross-Pitaevskii (GP) equation, which has a conserved density and magnetization. We find the dynamic critical exponent to be the same for models A and F (z=2)(z=2)(z=2) but different for GP $(z\ensuremath{\approx}3)$. Externally imposed magnetization conservation in models A and F yields the value $z\ensuremath{\approx}3$, which demonstrates that the only conserved density relevant to domain formation is the magnetization density.
Spin-1 condensates in the polar (antiferromagnetic) phase in two dimensions are shown to undergo a transition of the Ising type, in addition to the expected Kosterlitz-Thouless (KT) transition of half-vortices, due to the quadratic Zeeman effect. We establish the phase diagram in terms of temperature and the strength of the Zeeman effect using Monte Carlo simulations. When the Zeeman effect is sufficiently strong, the Ising and KT transitions meet. For very strong Zeeman field the remaining transition is of the familiar integer KT type.
We numerically study the Berezinskii-Kosterlitz-Thouless (BKT) transition of a spin-1 spinor Bose gas under the quadratic Zeeman effect. A calculation of the mass and spin superfluid densities shows that (i) the BKT transition occurs only when vortices are classified by the integer group Z\mathbb{Z}Z, and Z2\mathbb{Z}_2Z2​ vortices do not contribute to the BKT transition, (ii) the two BKT transition temperatures for mass and spin superfluid densities are different for a positive quadratic Zeeman effect and equal for a negative quadratic Zeeman effect, and (iii) the universal relation of the superfluid densities at the BKT transition temperature is changed when multiple kinds of vortices contribute to the transition. We have further found that (iv) spin-singlet pairs in non-magnetic states show the quasi-off-diagonal-long-range order at the different temperature lower than the BKT transition temperature, giving the new universal relation of the superfluid density.
This paper develops and implements the stochastic projected Gross-Pitaevskii equation for spin-1 Bose gases, addressing key considerations for numerical simulations. As an application of the theory we explore equilibrium phases in a two-dimensional spin-1 gas, where quasi-long-range order emerges via a Berezinskii-Kosterlitz-Thouless transition. Our analysis includes the definition of superfluid densities for both mass and spin degrees of freedom in a manner suitable for implementation within a stochastic projected Gross-Pitaevskii equation simulation. We present a finite-temperature phase diagram for the ferromagnetic spin-1 Bose gas and identify three distinct superfluid phases: two exhibiting conventional Berezinskii-Kosterlitz-Thouless-like behavior and an additional phase that simultaneously supports independent mass and spin superflows. As temperature increases, the stability region of this additional phase shrinks. We provide a characterization of the phase transitions through consideration of the spin-component densities and the unbinding of multiple types of vortices. This work provides a foundation for further studies of nonequilibrium and finite-temperature phenomena in spinor Bose gases.