Quantized vortices in two dimensional solid 4He
aa r X i v : . [ c ond - m a t . o t h e r] N ov Quantized vortices in two dimensional solid He M Rossi, D E Galli, P Salvestrini and L Reatto
Dipartimento di Fisica, Universit`a degli Studi di Milano, via Celoria 16, 20133 Milano, ItalyE-mail: [email protected]
Abstract.
Diagonal and off-diagonal properties of 2D solid He systems doped with aquantized vortex have been investigated via the Shadow Path Integral Ground State methodusing the fixed-phase approach. The chosen approximate phase induces the standard Onsager-Feynman flow field. In this approximation the vortex acts as a static external potential and theresulting Hamiltonian can be treated exactly with Quantum Monte Carlo methods. The vortexcore is found to sit in an interstitial site and a very weak relaxation of the lattice positionsaway from the vortex core position has been observed. Also other properties like Bragg peaksin the static structure factor or the behavior of vacancies are very little affected by the presenceof the vortex. We have computed also the one-body density matrix in perfect and defected He crystals finding that the vortex has no sensible effect on the off-diagonal long range tailof the density matrix. Within the assumed Onsager Feynman phase, we find that a quantizedvortex cannot auto-sustain itself unless a condensate is already present like when dislocationsare present. It remains to be investigated if backflow can change this conclusion.
Quantized vortices are one of the most genuine manifestation of the presence of superfluidityin many body quantum systems, but from the microscopic point of view no completeunderstanding of them has been reached yet. Recently quantized vortices have been relatedto the supersolidity issue [1]; in fact, arguments in favor of a vortex phase in low temperaturesolid He, preceding the supersolid transition, are appeared in literature [2, 3, 4]. With respectto this possible connection one of the fundamental questions to be answered is: what does aquantum vortex look like in solid Helium from a microscopic point of view?Dealing with vortices is a really hard task for microscopic methods, and it calls for someapproximations or assumptions [5-10]. In fact, the wave function has to be an eigenstate of theangular momentum, so it needs a phase. Following the well established routine for the groundstate, once chosen a variational ansatz for the wave function, one could be tempted to correct itby means of exact zero temperature Quantum Monte Carlo (QMC) techniques. Unfortunatelythis is actually not possible because of the sign problem that affects QMC methods. The mostfollowed recipe is to improve the variational description via QMC, but releasing the exactnessof the methods in favor of approximations that allow to avoid the sign problem, like for examplefixed phase [6] or fixed nodes [10].We study here the properties of a single vortex in solid He via the Shadow-PIGS (SPIGS)method with fixed phase approximation. The many-body wave function can be written asΨ( R ) = e i Ω( R ) Ψ ( R ), where Ω( R ) is a many-body phase, Ψ ( R ) is the modulus of the wavefunction and R = ( ~r , ~r , . . . , ~r N ) are the coordinates of the N particles. Ψ( R ) describes aquantum state of the system if it is a solution of the time independent Schr¨odinger equation:from ˆ H Ψ( R ) = E Ψ( R ) it is possible to obtain two coupled differential equations for Ω( R ) andfor Ψ ( R ). The fixed phase approximation consists in assuming the functional form of Ω( R ) asiven and to solve the equation " − ¯ h m N X i =1 ∇ i + V ( R ) + ¯ h m N X i =1 ( ~ ∇ i Ω) Ψ ( R ) = E Ψ ( R ) . (1)for Ψ ( R ). Solving (1) is equivalent to solve the original time independent Schr¨odinger equationfor the N -particle with an extra potential term V e = ¯ h m P Ni =1 ( ~ ∇ i Ω) .The simplest choice for the phase is the well known Onsager-Feynman (OF) phase [11]:Ω( R ) = l P Ni =1 θ i (where θ i is the angular polar coordinate of the i -th particle). Ψ( R ) is aneigenstate of the z component of the angular momentum operator ˆ L z with eigenvalue ¯ hN l ,being l = 1 , , . . . the quantum of circulation. This choice for Ω( R ) gives rise to the standardOF flow field: in fact the extra-potential in (1) reads V e = l ¯ h m N X i =1 r i (2)where r i is the radial polar coordinate of the i -th particle.In order to sustain a quantized vortex, the system should display a macroscopic phasecoherence, and at T = 0 K this means that solid He should house a Bose-Einstein condensate(BEC). It is known from QMC results that no BEC is present in the perfect crystal [12-15], butif the vortex turns out to be able to induce a BEC it could be a self-sustaining excitation. On theother hand, it is largely accepted that defects are able to induce BEC [16], and then a defectedcrystal can safely sustain a quantized vortex. Here we report on the study of a two dimensional(2D) He crystal with and without dislocations. In fact, dislocations can be included in the 2Dcrystal without imposing boundary constraints [16]. Moreover the 2D system allows to reachlarge distances keeping the number of particle in the simulation at a tractable level, and this isa desirable feature when interested in off-diagonal properties of the system.We face the task of solving (1) with the extra-potential given by (2) when l = 1 with theSPIGS method [17, 18], which allows to obtain the lowest eigenstate of a given Hamiltonianby projecting in imaginary time a SWF [19] taken as trial wave function. The SPIGS methodis unbiased by the choice of the trial wave function and the only inputs are the interparticlepotential and the approximation for the imaginary time propagator [20]. As He-He interatomicpotential we have considered the HFDHE2 Aziz potential [21] and we have employed the pair-Suzuki approximation [20] for the imaginary time propagator with time step δτ = 1 /
120 K − .One difficulty with (2) is that the potential is long range so that either one puts the system ina bucket [6, 7] or one should consider a vortex lattice [8]. Such complications can be avoided bymultiplying 1 /r in (2) by a smoothing function χ ( r ) = r < ∆e − ( r − ∆) / ( r − L/ ∆ ≤ r ≤ L/ r > L/ L being the side of the simulation box) so that standard periodic boundary conditions canbe applied. With this choice, the extra-potential is equivalent to the OF one only for r < ∆,the provided Ψ is no more an exact eigenstate of ˆ L z but it is close to it in the interestingregion of the vortex core if ∆ is large enough. Here we have used ∆ = 30˚A. We have performedsimulations at ρ = 0 . − in a nearly squared box designed to house a perfect triangularcrystal with M = 572 lattice sites, and a crystal with 10 vacancies ( N = 562); such vacanciesin the initial configuration transform themselves in dislocations [16].In Fig. 1 we report our results for the integrated vortex energies ε v ( r ) = ( E v ( r ) − E ( r )) /N ( E v ( r ) and E ( r ) are, respectively, the energy of the particles that lie inside the disk of radius r
10 20 30 40r ( A ° )051015 ε v ( K ) Figure 1.
Integrated vortex energy ε v with error bars as a function of thedistance from the core in perfect 2D solid He at ρ = 0 . − . ° )00.0010.0020.003 ρ d ( A ° − ) Figure 2.
Comparison between the radialdefect distribution ρ d ( r ) as a function ofthe distance from the origin (vortex corewhen vortex is present) for a 2D solid Heat ρ = 0 . − with (filled symbols)and without (open symbols) a vortexin the perfect (circles) and in defected(squares) crystal.in the system with and without vortex) as a function of the distance from the core in the perfectcrystal. The center of mass of the system is not fixed, and we find that, independently on thestarting configuration of the crystal, the vortex core sits in an interstitial site. We also find avery small relaxation of the surrounding lattice around the vortex core.In order to study the effects of the vortex on the crystal properties we have monitored thestatic structure factor, the pair distribution function and the radial defect distribution ρ d ( r )(i.e. the distribution of the particle whose coordination is different from 6 as a function of thedistance from the origin where the vortex core is located). In Fig. 2 we plot our results for ρ d ( r )both for the perfect and for the defected crystal. We find that in the defected case the radialdefect distribution is about an order of magnitude larger than in the perfect one; however, theresults of the system with and without vortex are very close each other so that we conclude thatthe OF vortex does not affect in a sensible way the disorder which behaves as in the systemwithout vortex. We come at a similar conclusion for the crystalline structure, since both thestatic structure factor and the pair distribution function show no appreciable differences for thesystem with and without the vortex.We have computed also the one-body density matrix ρ for the perfect and for the defectedcrystal with and without the vortex in order to investigate the vortex effects on the off-diagonalproperties. Our results are reported in Fig. 3. ρ for the crystal with vortex are indistinguishablewithin the error bars from the ones obtained without vortex. We conclude that the OF vortexis not able to induce a Bose-Einstein condensation (BEC) in the perfect crystal, or to increasethe already present condensate fraction in the defected one.
10 20 30 40 50r−r’ ( A ° )1e−101e−081e−060.00010.011 ρ (r − r ’ ) Figure 3.
One body density matrix ρ in 2D solid He without (open symbols)and with (filled symbols) a vortex for theperfect crystal (circles) and for the crystalwith dislocations (squares).Since no BEC is present in the perfect crystal and the OF vortex is not able to induce it byitself, we can conclude that perfect 2D solid He can not sustain vortices of the OF type. Thusthe OF wave function is a possible representation of a vortex in solid Helium only when BECis already present, like in a defected crystal. Preliminary results in three dimensional solid Heseem to confirm such conclusions for the OF vortex.In the liquid phase, the OF phase [11] has been improved with the inclusion of back-flow(BF) correlations [6, 8]. The effect of backflow increases [8] at higher densities and it mightwell become dominant in the solid phase. Computations along this line are in progress. Sincethe BF terms acts mainly near the vortex core, we might expect that BF will not modify muchthe off-diagonal properties in 2D, where the vortex core is a point defect, while it could becomerelevant in 3D where the vortex core is an extended defect.This work was supported by Regione Lombardia and CILEA Consortium througha LISA Initiative (Laboratory for Interdisciplinary Advanced Simulation) 2010 grant[link:http://lisa.cilea.it].
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