Spin Structure and Critical Nucleation Frequency of Fractionalized Vortices in 2D Topologically Ordered Superfluids of Cold Atoms
We have studied spin structures of fluctuation-driven fractionalized vortices and topological spin order in 2D nematic superfluids of cold sodium atoms. Our Monte Carlo simulations suggest a softened -spin disclination structure in a half-quantum vortex when spin correlations are short ranged; in addition, calculations indicate that a unique non-local topological spin order emerges simultaneously as cold atoms become a superfluid below a critical temperature. We have also estimated fluctuation-dependent critical frequencies for half-quantum vortex nucleation in rotating optical traps and discussed probing these excitations in experiments.
Quantum number fractionalization has been one of the most fundamental and exciting concepts studied in modern many-body physics and topological field theoriesSu80 (); Tsui82 (); Laughlin83 (); Jackiw76 (). During the past few years, low dimensional fractionalized quantum states have further been proposed to be promising candidates for carrying out fault tolerant quantum computationKitaev03 () and their realizations in optical lattices were exploredDuan03 (); Buchler05 (); Micheli06 (). A closely related topic in which there has also been a growing interest is vortex fractionalization in cold gases (see for instance Ref.Demler02 (); Mukerjee06 (); Semenoff07 (); Thouless98 ()). Especially, motivated by experiments on low dimensional cold gasesHadzibabic06 (), Mukerjee et al studied 2D superfluids of cold atoms and analyzed the role played by fractionalized vortices in phase transitionsMukerjee06 (). However, spin structures of those half-quantum vortices induced by thermal fluctuations and potential topological orderWen04 () haven’t been thoroughly explored and remain to be understood. In this Letter we illustrate spin structures of fractionalized vortices; in addition, we also show that 2D quantum gases with short ranged spin correlations can have a topological spin order. We further estimate critical nucleation frequencies of fractionalized vortices in optical traps which can potentially be studied in experimentsStenger98 ().
Our simulations illustrate that in a fundamental vortex carrying one-half of circulation quantum ( is the Planck constant and is the mass of atoms), there exists a softened spin disclination (i.e. a disclination in the absence of spin stiffness) even when local spin moments are strongly fluctuating at finite temperatures. The topological winding number associated with softened spin disclinations is conserved as far as the phase rigidity remains finite. This effectively leads to a non-local topological spin order. Such a nonlocal order is absent in a conventional condensate of atoms or pairs of atoms. We have further studied creation of these excitations in rotating superfluids and obtained fluctuation-dependent critical frequencies for half-quantum vortex(HQV) nucleation. HQVs in traps can be probed by measuring a precession of eigenaxes of surface quadrupole modes.
Here is the lattice site index and are the nearest neighbor sites. is the chemical potential and is the one-particle hopping amplitude. Two coupling constants are ; , are effective s-wave scattering lengths, is the localized Wannier function for atoms in a periodical potential. Operators , create hyperfine spin-one atoms in , and states respectively. The spin and number operators are defined as , and . Spin correlations are mainly induced by interaction . Here we consider antiferromagnetic spin-dependent interactions such as in sodium atoms where . Minimization of this antiferromagnetic spin-dependent interaction requires that the order parameter be a real vector up to a global phase, i.e. where is a unit director on a two-sphere, represents a phase director and is the number of atoms per site. All low energy degrees of freedom are characterized by configurations where and vary slowly in space and timeDemler02 (). Low lying collective modes include spin-wave excitations with energy dispersion , and phase-wave excitations with , (here is the lattice constant). In one-dimensions, low energy quantum fluctuations destroy spin order leading to quantum spin disordered superfluidsZhou01 (). In two-dimensions, the amplitude of quantum spin fluctuations is of order of and is negligible in shallow lattices as is order-of-magnitude bigger than . At finite temperatures, spin correlations are mainly driven by long wave length thermal fluctuations, analogous to quantum cases. This aspect was also paid attention to previously and normal-superfluid transitions were investigatedMukerjee06 ().
We therefore study the following Hamiltonian which effectively captures long wave length thermal fluctuations
here states at each site are specified by two unit directors: a nematic director, and a phase director, . is the effective coupling between two neighboring sites and depends on , the number of atoms per site. The model is invariant under the following local Ising gauge transformation: , , and . In the following, we present results of our simulations on 2D superfluids, especially spin structures, energetics of HQVs and nucleation of HQVs in rotating optical traps using the effective Hamiltonian in Eq.2.
HQV and underlying topological spin order Around a HQV, both phase and nematic directors rotate slowly by ; in polar coordinates , a HQV in condensates is represented by , with Thouless98 (); Zhou01 (). The question here is whether, when nematic directors are not ordered, a spin disclination is still present in a HQV. To fully take into account thermal fluctuations, we carry out Monte Carlo simulations on a square lattice of sites and study spatial correlations between a HQV and a -spin disclination, and topological order.
We first identify critical temperatures of the normal-superfluid phase transition by calculating correlations and the phase rigidity. The gauge-invariant quadrupole-quadrupole correlation functions we have studied are
Here , ; , . In simulations, we have studied these correlation functions and found that the phase correlation length for becomes divergent at a temperature which is identified as a critical temperature . We also calculate the phase rigidity or the renormalized phase coupling
here is a small phase difference applied across the opposite boundaries of the lattice and is the corresponding free energy. We indeed find that it approaches zero at while at takes a bare value . Meanwhile, the spin correlation function remains to be short ranged across . By extrapolating our data to lower temperatures, we find that the spin correlation length diverges only at (see Fig.1a). Our simulations for correlation lengths are in agreement with previous results in Ref.Mukerjee06 (); they are also consistent with the continuum limit of the model in Eq.2 which is equivalent to an model and an nonlinear-sigma model.
In order to keep track of the winding of nematic directors in a wildly fluctuating back ground, we introduce the following gauge invariant -rotation checking operator, which is essentially a product of sign-checking operators
Here the product is carried out along a closed square-shape path centered at the origin of a lattice. can be either or ; and () is when encloses a -spin disclination (HQV). The gauge invariant circulation of supercurrent velocity (in units of ) is defined as ; this quantity is equal to one in a HQV.
In our simulations, we investigate the winding number averaged over configurations where phase directors rotate by around the boundary of the lattice and the center plaquette. At temperatures above the normal-superfluid transition temperature , both winding numbers and circulation are averaged to zero within our numerical accuracy(see Fig.1). And our choice of boundary conditions does not lead to a vortex or disclination configuration in the absence of phase rigidity. Below , the circulation is averaged to one indicating that the boundary conditions effectively project out HQV configurations. Meanwhile, we observe loop-perimeter dependent which can be attributed to the background fluctuations of HQV or disclination pairs. The loop-perimeter dependence of here is almost identical to that for uniform boundary conditions, i.e. the back ground value. After normalizing in terms of background winding numbers , we find both and approach (see Fig.1). We thus demonstrate that a softened disclination is spatially correlated with a HQV. At the temperatures we carry out these simulations the spin correlation length is sufficiently short compared to the size of the lattice. At further lower temperatures, the spin correlation length becomes longer than the lattice size and fluctuations of pairs of disclination-anti-disclination are strongly suppressed; are equal to for almost all loops, which corresponds to a mean field result.
Results in Fig.1 indicate that there exists a softened spin disclination in a HQV. This is a distinct feature in our systems and there exist no such additional magnetic structures in HQVs in conventional molecular condensates of atom pairs discussed perviouslyRomans04 (). Thus, -disclinations like HQVs have logarithmically divergent energies and are fully suppressed in ground states. Our results also illustrate that although the average local spin quadrupole moments vanish because of strong fluctuations, an overall -rotation of nematic directors in disclinations is still conserved because of a coupling to the superfluid component. This coupling between a HQV and disclination can also be attributedSong08 () to a coupling between Higgs matter and discrete gauge fieldsFradkin79 (). Furthermore, the absence of unbound -disclinations in superfluids indicates a topological order, similar to the one introduced previously for an isotropic phase of liquid crystalToner93 (). Consequently, once a conventional phase order appears below a critical temperature, a topological spin order simultaneously emerges while spin correlations remain short ranged.
The emergent topological order can be further verified by examining the average of product-operator over all configurations (with open boundaries). Above the normal-superfluid transition temperature , we again find that both are averaged to zero within our numerical accuracy implying proliferation of unbound HQVs or disclinations. Below , we study the loop-perimeter dependence of average winding numbers and find that both and are linear functions of loop-perimeter analogous to the Wilson-loop-product of deconfining gauge fieldsWilson74 (); if there were unbound disclinations, one should expect that is proportional to, instead of the loop-perimeter, the loop-area which represents the number of unbound disclinations enclosed by the loop.
Critical frequency for HQV nucleation Let us now turn to the nucleation of those excitations in rotating trapsMadison00 (); Haljan01 (); AboShaeer01 (); Fetter01 (); Dalfovo01 (); Tsubota02 (); Isoshima02 (). To understand the critical frequency for nucleation, we study the free energy of a vortex, in a rotating frame, as a function of the distance from the axis of a cylindrical optical trap (the axis is along the -direction),
Here is the free energy of a HQV located at distance from the trap axis in the absence of rotation, is the angular momentum of the vortex state and is the rotating frequency.
In a lattice without a trapping potential, is approximately equal to , with leading contributions from phase winding and spin twisting; here are renormalized phase and spin coupling respectively and is the size of system. For an integer-quantum vortex (IQV), is equal to . The ratio between and depends on the ratio or spin fluctuations; in the limit approaches infinity, the ratio changes discontinuously from at where to at finite low temperatures in where vanishes. In simulations of a finite trap (see below), because of a finite size effect we find that this ratio varies from to smoothly as temperatures increase from to .
To study nucleation of half-quantum vortices in an optical trap, we assume a nearly harmonic trapping potential , with being the trap frequency. The average number of particles per site has a Thomas-Fermi profile; , here is the number density at the center and is the Thomas-Fermi radius. Furthermore, the optical lattice potential along the axial direction is sufficiently deep so that atoms are confined in a two-dimensional plane; the in-plane lattice potential depth is set to be ( is the photon recoil energy).
For the trap and lattice geometry described above, we calculate parameters in Eq.1 and obtain nK, nK and nK. For and trap frequency , we also find that and where is the harmonic oscillator length. The coupling in Eq.2 depends on the distance from the center of trap and at the center, the coupling is about nK. In non-rotating or slowly rotating traps, the free energy maximum is located at the center and there should be no vortices in the trap. As frequencies are increased, a local energy minimum appears at the center and becomes degenerate with the no-vortex state at a thermodynamic critical frequency (which is about at ); however because of a large energy barrier separating the two degenerate states as shown in Fig.2, vortices are still prohibited from entering the trap.
Further speeding up rotations results in an energetically lower and spatially narrower barrier. Within the range of temperatures studied, thermal activation turns out to be insignificant within an experimental time scale () because of low attempt frequencies. So only when the spatial width of barrier becomes comparable to a hydrodynamic breakdown lengthFeder00 (), the barrier can no longer be felted and vortices start to penetrate into the trap. We use this criterion to numerically determine the dynamical critical frequency for vortex nucleation ; for IQVs, the calculated is a flat function of (see Fig.2b) which is qualitatively consistent with earlier estimatesSimula02 ().
For HQVs, depends on the renormalized spin coupling and therefore the amplitude of spin fluctuations. Because of this, varies from about at where and due to a finite size effect (see the inset of Fig.2), to about at temperatures close to where and . In other words, this unique temperature dependence can be considered to be an indicator of fluctuation-driven vortex fractionalization. It is worth remarking that in the thermodynamic limit where approaches zero at any finite temperatures, approaches as mentioned before. Consequently, (about ) for HQVs is about one-half of the critical frequency for IQVs (about for the finite trap studied here). Also note that the zero temperature estimate of is close to the previously obtained value of critical frequencies of HQVs in Bose-Einstein condensatesIsoshima02 ().
The interaction between two HQVs with the same vorticity at a separation distance contains two parts. One, is from interactions between two supercurrent velocity fields which is logarithmic as a function of ; and the other, is from interaction between two spin twisting fields accompanying HQVs. For a disclination-anti disclination pair, in the dilute limit one finds that resulting in a cancellation of long range interactions. The resultant short-range repulsions lead to square vortex lattices found in numerical simulationsAndrew08 (). For fluctuation-driven fractionalized vortices, is almost zero and the overall interactions are always logarithmically repulsive. HQVs nucleated in a rotating trap should therefore form a usual triangular vortex lattice.
Individual vortex lines can be probed either by studying a precession of eigenaxes of surface quadrupole mode in rotating superfluidsChevy00 (). In the later approach, one studies the angular momentum carried per particle in a HQV state. When a HQV is nucleated in the trap, superfluids are no longer irrotational and the angular momentum per particle is rather than per particle for an integer vortex state. When a surface quadrupole oscillation across a rotating superfluid is excited, larger axes of quadrupole oscillation start to precess just as in the case of integer vortices. However, the precession rate is only one half of the value for an integer vortex state which can be studied in experiments.
In conclusion, 2D superfluids of sodium atoms have a non-local topological spin order. In rotating traps, fluctuation-driven fractionalized vortices can nucleate at a critical frequency which is about half of that for integer vortices. Observation of these exotic excitations could substantially improve our understanding of topological order and fractionalization. We thank J. Zhang and Z. C. Gu for contributions at an early stage of the project. This work is in part supported by the office of the Dean of Science, UBC, NSERC (Canada), Canadian Institute for Advanced Research, and the A. P. Sloan foundation.
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