Quantized vortices in two dimensional solid He
Diagonal and off-diagonal properties of 2D solid He systems doped with a quantized vortex have been investigated via the Shadow Path Integral Ground State method using 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 the resulting Hamiltonian can be treated exactly with Quantum Monte Carlo methods. The vortex core is found to sit in an interstitial site and a very weak relaxation of the lattice positions away from the vortex core position has been observed. Also other properties like Bragg peaks in the static structure factor or the behavior of vacancies are very little affected by the presence of 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 tail of the density matrix. Within the assumed Onsager Feynman phase, we find that a quantized vortex cannot auto-sustain itself unless a condensate is already present like when dislocations are present. It remains to be investigated if backflow can change this conclusion.
Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy
Quantized vortices are one of the most genuine manifestation of the presence of superfluidity in many body quantum systems, but from the microscopic point of view no complete understanding of them has been reached yet. Recently quantized vortices have been related to the supersolidity issue ; in fact, arguments in favor of a vortex phase in low temperature solid He, preceding the supersolid transition, are appeared in literature [2, 3, 4]. With respect to this possible connection one of the fundamental questions to be answered is: what does a quantum 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 some approximations or assumptions [5-10]. In fact, the wave function has to be an eigenstate of the angular momentum, so it needs a phase. Following the well established routine for the ground state, once chosen a variational ansatz for the wave function, one could be tempted to correct it by means of exact zero temperature Quantum Monte Carlo (QMC) techniques. Unfortunately this is actually not possible because of the sign problem that affects QMC methods. The most followed recipe is to improve the variational description via QMC, but releasing the exactness of the methods in favor of approximations that allow to avoid the sign problem, like for example fixed phase  or fixed nodes .
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 as , where is a many-body phase, is the modulus of the wave function and are the coordinates of the particles. describes a quantum state of the system if it is a solution of the time independent Schrödinger equation: from it is possible to obtain two coupled differential equations for and for . The fixed phase approximation consists in assuming the functional form of as given and to solve the equation
for . Solving (Quantized vortices in two dimensional solid He) is equivalent to solve the original time independent Schrödinger equation for the -particle with an extra potential term .
The simplest choice for the phase is the well known Onsager-Feynman (OF) phase : (where is the angular polar coordinate of the -th particle). is an eigenstate of the component of the angular momentum operator with eigenvalue , being the quantum of circulation. This choice for gives rise to the standard OF flow field: in fact the extra-potential in (Quantized vortices in two dimensional solid He) reads
where is the radial polar coordinate of the -th particle.
In order to sustain a quantized vortex, the system should display a macroscopic phase coherence, and at 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], but if the vortex turns out to be able to induce a BEC it could be a self-sustaining excitation. On the other hand, it is largely accepted that defects are able to induce BEC , and then a defected crystal 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 2D crystal without imposing boundary constraints . Moreover the 2D system allows to reach large distances keeping the number of particle in the simulation at a tractable level, and this is a desirable feature when interested in off-diagonal properties of the system.
We face the task of solving (Quantized vortices in two dimensional solid He) with the extra-potential given by (Quantized vortices in two dimensional solid He) when with the SPIGS method [17, 18], which allows to obtain the lowest eigenstate of a given Hamiltonian by projecting in imaginary time a SWF  taken as trial wave function. The SPIGS method is unbiased by the choice of the trial wave function and the only inputs are the interparticle potential and the approximation for the imaginary time propagator . As He-He interatomic potential we have considered the HFDHE2 Aziz potential  and we have employed the pair-Suzuki approximation  for the imaginary time propagator with time step K. One difficulty with (Quantized vortices in two dimensional solid He) is that the potential is long range so that either one puts the system in a bucket [6, 7] or one should consider a vortex lattice . Such complications can be avoided by multiplying in (Quantized vortices in two dimensional solid He) by a smoothing function
( being the side of the simulation box) so that standard periodic boundary conditions can be applied. With this choice, the extra-potential is equivalent to the OF one only for , the provided is no more an exact eigenstate of but it is close to it in the interesting region of the vortex core if is large enough. Here we have used Å. We have performed simulations at Å in a nearly squared box designed to house a perfect triangular crystal with lattice sites, and a crystal with 10 vacancies (); such vacancies in the initial configuration transform themselves in dislocations .