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Many-body structure of quantum vortices in thin 4He films
26June1995 PHYSICS
LETTERS
A
Physics Letters A 202 (1995) 317-323
Many-body structure of quantum vortices in thin 4He films M. Saarela a, F.V. Kusmartsev
a9b
aDepartment of Theoretical Physics, Uniuersiry of Ouly SF-90570 Oulu, Finland b L.D. Landau Institate for Theoretical Physics, GSP-I, Kosygina 2, V-334, Moscow I1 7940, Russian Federation Received 7 February 1995; accepted for publication 6 April 1995 Communicated by V.M. Agranovich
Abstract We propose a microscopic theory of quantum vortices in two-dimensional 4He, applicable to thin superfluid 4He films. The theory describes the vortex core and reproduces the classical logarithmic behavior at large distances. The single vortex is treated as a quasiparticle. We calculate the energy needed to create a single vortex, the vortex inertial mass, the microscopic interaction between vortices and the binding energy of the vortex-antivortex pair as a function of density at zero temperature. At 4He densities less than 0.037 A-’ the binding energy of the vortex-antivortex pair becomes negative, indicating an instability for the creation of a novel state.
It was Vadim Berezinskii [l] who first recognized that two-dimensional (2D) condensed matter systems may have a phase transition, associated with the creation of vortices. Kosterlitz and Thouless [2,3] developed those ideas by introducing an intuitive Hamiltonian and derived the scaling equations. In thin superfluid 4He films it is assumed that at the critical temperature of the Berezinskli-Kosterlitz-Thouless (BKT) phase transition macroscopic amounts of unpaired vortices are created, and that destroys the superfluidity [4]. With recent possibilities of making thin films and building up new quantum devices it has become very important to understand the role of these topological excitations in 2D fluids. The microscopic structure of a vortex, its interaction with the many-particle background and with other vortices is a fascinating many-body problem [5-71. Traditionally a single vortex has been inserted into the superfluid as an external potential and the response of the system is shown as the inhomogeneity of the density profile [8-lo]. At the center of the 2D vortex the superfluid density is zero and far away it equals the uniform bulk density. More recently the inertial muss of the vortex has received considerable attention and various estimates of its value have been made [ll-141. This inspired us to present a many-body calculation of quantum vortices in the Boson fluid. The calculations are done in purely 2D 4He superfluid at zero temperature. However, a very good resemblance of that system in found in thin 4He films on smooth substrates like the first superfluid layers on top of two solid 4He layers on graphite or on top of the solid hydrogen on glass [15-171. We start from the many-particle Hamiltonian with a given pair interaction between superfluid particles and include vortices through the quantized phase in the wave function [B]. The kinetic energy part of the many-particle Hamiltonian transforms the phase into an effective interaction between the vortex core and the superfluid particles. That is responsible of the repulsion of the Elsevier Science B.V. SD1 0375-9601(95)00319-3
318
M. Saarela, F. V. Kusmartser
/Physics
Letters A 202 (1995) 317-323
material away from the core region. We show that in 2D “He the volume of the expelled superfluid is of the order of the volume occupied by one 4He atom. The vortex is then a microscopic object with non-zero kinetic energy and its inertial mass must be taken into account. This is in contrast to the treatment of a vortex line in full three dimensions, where the mass becomes infinite [9]. We set the vortex mass equal to the mass of the expelled superfluid [ll]. Its value is calculated through a self-consistent many-body procedure and is proportional to the compressibility. This means that the mass becomes infinite when the superfluid becomes soft against density oscillations. In the limit of a large vortex radius our result reproduces the estimate obtained by Duan and Leggett [13]. The mass together with the effective interaction defines the effective Hamiltonian for the vortex core motion. The modulus of the many-body wave function includes correlations between the vortex core and the superfluid 4He particles [8]. These correlation functions are determined by minimizing the energy required to create a vortex excitation, i.e. we minimize the chemical potential of the vortex quasiparticle in the superfluid. Without an external source of rotation, vortices in superfluid films must only be created in pairs with different signs of vorticities. In classical hydrodynamics the vortex motion is described by two coordinates x and y chosen as conjugate variables, which conserves the vorticity [18]. Vortex-antivortex pairs move perpendicular to the line joining them and their separation distance is proportional to their center of mass (CM) momentum [7]. The “electrostatic forces” arising due to different vorticities are in equilibrium with the “electrodynamic forces” arising due to their motion. For the quantum vortices which we are discussing here the distance between the vortex and antivortex is determined by the balance between the kinetic energy of the single vortices and their attractive interaction. Thus the pair can be created even at zero CM-momentum. The direct part of the interaction between the vortex and antivortex is determined by the phase with two centers of circulation. It vanishes at zero separation distance and diverges logarithmically at large distances. The interaction contains also the induced part which is due to modification of the correlations between the superfluid particles in the region of the created pair, and that enhances the effective attraction. At non-zero CM-momenta the probability distribution of the pair will be polarized in such a way that the separation distance will increase in the perpendicular direction of the motion due to the coupling between the linear CM-motion and rotational motion [19]. The probability distribution is evaluated by minimizing the energy required to create the pair. We assume that the vortex can be described by adding a singular, quantized phase into the many-body wave function,
The coordinates ri, . . . , r,,, refer to the superfluid particles and rv to the vortex core. The gradient of the phase 4(r,, rj), called the vector potential, is related to the superfluid velocity field, A(r,, rj) = KVj+(r,, rj) = (m,/h>u,(r,, rj>. We have denoted with ma the mass of the superfluid particle. The condition that the wave function !Z$ must be single ualued requires that the circulation around the point r, using any closed path should be an integer multiple of 2~, (i.e. $aA(r,., r) * dr = 27r~). The integer K is the conserued uorticity quantum number of the circulation. The modulus of the wave function describes the response of the superfluid to the vortex. We separate the vortex-background correlations from the pure background correlations by using the Jastrow-Feenberg type exponential factors [20],
(rv>
rj)
+
C UVBB(rV1rk) vB( r,, rN). rj,
j
(2)
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
The ground state wave function Hamiltonian [ 151, HB=-x---
i-1 2m,
Pa
is determined
by minimizing
the expectation
319
value of the microscopic
4He
i.j= 1 i
The energy of the system with one infinite muss vortex is the expectation to the wave function qV [8,9],
value of the Hamiltonian
with respect
(4) where the phase has been transformed into an interaction-like term I. The connection to the classical Hamiltonian is found by inserting the superfluid velocity field in place of the vector potential. Since the vortex in 2D 4He occupies only a microscopic volume of the order of one atom size its kinetic energy must be added into the effective Hamiltonian,
The energy of the vortex excitation the mass m,, @‘I H, IV’> pv=
(TPyl!P)
The unknown correlation Euler equations, 6E.L” 6P(ry,
is then equal to the chemical
(‘-& IH, -
w, functions
= SU”as(r”,
of a quasiparticle
W’,>
IW
are determined
rl, r2) =O.
impurity
carrying
(6)
.
SPV rr)
potential
from the variational
ansatz leading
to the set of coupled
(7)
From these Euler equations we obtain the vortex-background pair distribution function gvB(r) and its Fourier transform, the vortex-background structure function SvB(k> = pBjdr ~‘*~“[g”~(r) - 11. Here, pB is the average superfluid density. Details of the method are described in Refs. [15,21,22]. The volume of the “hole” created by the vortex is equal to the value of the structure function at the origin, S”‘(O + ), and the vortex mass is the 4He mass times this volume, m, = -mBSvB(O + ), which is calculated iteratively. We study properties of the vortex excitations in the liquid phase where the superfluid order is possible and present results for the 2D 4He densities ranging from 0.037 A-” to 0.065 A-‘. In experiments with thin films the whole liquid density range of the 2D 4He can be reached because the growth of the film on a substrate can be accurately controlled. For the interaction between 4He particles we use the Aziz potential [23]. Our result for the equation of state agrees well with the Monte Carlo simulations [24]. In the low density limit the liquid phase kas the spinodal instability where the speed of sound becomes zezo. In our model that occurs at about 0.031 Am2. In the high density limit the liquid solidifies at about 0.070 Am2 [15].
-getting Eq. (4) we have written the kinetic energy term in the following form (‘P” 1V2 lly,> = (tL2/2m,X(P IiV- A + 2iA .V - A* + V* IT) + h.c.]. The first term on the right hand side disappears with the choice of the Landau gauge. The second term, which is the usual linear term in the vector potential in the case of magnetic field, also disappears, because our wave function ly is the wave function for a Boson system at zero momentum, and it is a real quantity.
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
320
I
“0
2
4
6 r (11
8
10
12
Fig. 1. The pair distribution functions between the single vortex and the background ‘He particle with vorticity K = 1 (solid curves) and the pair distribution functions for the pure 4He (dashed curves) at densities listed in Table 1. The first peak increases with the density for both sets of curves.
In 2D superfluid the vortex-background
ii2
interaction term is (see Eq. (5))
h*
V’B(‘“,‘,)‘2mIA(r,,r,)12=-
2mB
B
K* 1 TV -‘j
1”
For the vector potential A(r,, r,> we have used the Landau gauge and the quantization condition V, X r,) = 27r~&r, - ri). The solutions for the vortex-background pair distribution functions from the Euler equations (7) for a single vortex with vorticity K = 1 are shown in Fig. 1 for different densities. The pair distribution function starts quadratically from zero and at large distances approaches unity. The increase of the vorticity moves the background particles further away from the center of the vortex. The values of the vortex yass calculated from the structure function are given in Table 1. In the region of the saturation density 0.043 Ae2 the vortex mass is about twice the 4He mass. It increases rapidly when the density is decreased and diverges at the spinodal point. The increasing density squeezes the vortex into a smaller volume and at the solidification density its mass is only half the helium mass. The chemical potential of a single vortex in 2D diverges logarithmically with the size of the system due to the effective interaction of Eq. (8). That is a slow divergence which does not cause any problems in the numerical work. In Table 1 we give the full chemical potential and its “core” part, CL’,where the interaction term of Eq. (8) has been subtracted as a function of
Ah”,
Table 1 The vortex chemical potential, chemical
F”, its “core” part, yc, described in the text and the vortex mass as a function of density. We also give the
potential of the vortex-antivortex
pair, pva, and the root mean square radius, fi
pB (A- ‘)
F” (K)
CL’(K)
m, (amu)
cLy=00
fi
0.037 0.040 0.045 0.050 0.055 0.060 0.065
4.63 5.58 6.90 8.17 9.43 10.7 12.0
1.27 1.32 1.43 1.58 1.77 1.99 2.24
24.3 14.2 8.18 5.61 4.21 3.24 2.73
0.35 1.71 3.50 5.28 7.12 9.04 11.0
2.04 2.07 2.16 2.27 2.39 2.52 2.65
(A)
321
IU. Saarela, F. V. Kusmartsev / Physics Letters A 202 (1995) 317-323
density. For the size of the system the radius R = 36 A has been used. These results are quite insensitive to the precise value of the vortex mass. When the vortex-antivortex pair is created the logarithmic divergence of the chemical potential with the size of the system cancels off. The many-body wave function of the pair contains two phase factors corresponding to two centers of circulation with opposite vorticities.
The sum of these phases transforms into the direct part of the interaction, which can be divided into three terms: vortex and antivortex interactions with the background and the vortex-antivortex interaction mediated by one background particle, raY
gdA(C, B
rj) +VvaB(rv,
fj)-VvB('a,
rj)12=VVB(rv9
fa,
Tj),
with
vvaB(fvr r,, r,) = -K2-
fi* (r,-rj)
-(rv--r,)
mg 1r, - rj 1* 1rv - rj 1*
(11)
’
The centers of the vortex and antivortex are labeled with subindices v and a, respectively. The variational problem is to minimize the energy of the pair added into the superfluid with respect to the vortex-antivortex correlations. The problem is complicated by the fact that the vortex-4He and the 4He-4He correlations will also be modified due to the presence of two vortices. Using the hypemetted chain approximation the vortex-antivortex correlations can be related to the radial distribution function gva(rV, ra). The chemical potential for the pair can then be written into the form
+gvatrv, cJWC(~“~r,>+ -
h2
2mB
/
pBd*r, gvaB(rV7 r,, rl)IA(r,,
ra, rl)l’
I ,
(14
where R is the volume of integration. The induced potential, wzJrV, r,), in the second line enters due to the modification of the medium [25] *. In momentum space it can be expressed in terms of the background and vortex-background structure functions, 6.$(k)
= --
h2k2 TB(k) -
4m,
i P(k)
2 i(
29B(k) ”
We also approximate the triplet distribution function functions [20]. This is a well justified approximation
+ 1 . i
(13)
gvaB(rV, r,, r,> with the product of the pair distribution here because the vortex-antivortex pair is tightly bound.
* The kinetic energy of 4He atoms, exactly speaking the sum of nodal diagrams, induces an additional short-range attraction between vortices given in Eq. (13). In the long wave length limit this is the phonon induced interaction introduced by Bardeen, Baym and Pines [29] between two 3He impurities in 4He.
M. Saarela, F. V. Kusmartseo / Physics Letters A 202 (1995) 317-323
322
16 12 8 4 0 -4 0
2
4
6
8
10
12
14
r CA, Fig. 2. The vortex-antivortex interaction at densities full interaction and the upper line is its direct part.
The expression
Vy”( r)
=
pE = 0.045 A-’
for the effective uortex-antirrortex
p”/
d'r,
g’“(
rv, r,)
g’“(
(solid curves) and 0.065 k’
interaction
r,, r,)VVaB(
rv, ra,
(dashed curves). The lower line is the
can then be written in the form,
rl) + wyi,( r),
(14)
The notation r = 1rv - ra 1has been used. In the limit of a large intervortex distance we find the logarithmic term. V”“(r) + 2rrK2pB(h2/mg)ln(r/r,), owhere rC is a fictitious core radius of a single vortex 3. The values of rC range from 4 A at pB = 0.037 A-* to 0.8 A at pB = 0.065 A-’ when fitted to the value of the full potential at 18 A. In Fig. 2 we show the full effective vortex-antiyortex potential and separately, its direct part - the integral term in Eq. (14) - at densities 0.045 AP2 and 0.065 A-*. The short range attraction caused by the induced potential is clearly seen, which at r = 0 is between - 3 and - 4 K. The minimization of the chemical potential of Eq. (12) with respect to g”“(r) leads to the Schrijdinger-like equation with the interaction (14). We solve it for the lowest eigensiate and calculate from that the vortex-antivortex chemical potential. The results are given in Table 1. We find that at densities less than 0.037 A-’ the chemical potential for the pair creation becomes negative. This signals a new kind of instability in the system different from the liquid-solid phase transition and the spinodal instability. Here macroscopic amounts of vortex-antivortex pairs are spontaneously created. The vzrtex-antivortex chemical potential depends strongly on the density and near the solidification density 0.065 ,A-’ it is 11 K. The root mean square radius is much less density dependent ranging between 2.04 A and 2.65 A as shown also in Table 1. That is because the smaller the density the heavier the mass. Such a small separation distance implies that the nearest neighbor 4He atoms play the main role in the structure of the pair core. The pair has many bound states corresponding to the different principle quantum numbers and angular momenta. At a large separation distance the pair energy depends logarithmically on the principle quantum number. The existence of the bound pair clarifies the microscopic picture of the BKT phase transition. The transition may go through intermediate stages associated with different quantum bound states and the appropriate scaling equations must be modified. Note that the interaction between two 4He atoms adjacent to the vortex-antivortex pair increases effectively, and they may form a bound state,similar to the Anderson resonance valence bond [26] and in the films the
3 In order to make the comparison present formalism.
with quasiclassical
theory of vortices
we introduce
the core radius
rC which is not needed in the
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
323
resonance valence bond liquid may be created. Since the interaction between the vortex-antivortex pairs is long-ranged, the liquid of vortex-antivortex pairs may crystallize [27]. Thus, we propose two novel states: liquid and vortex-antivortex crystal, which may exist in films of 4He. Our resonance valence bond liquid state recalls the flux phase state, studied by Kalmeyer and Laughlin [28] in the hard-core Boson representation of the Heisenberg Hamiltonian. We call for experimentalists to look for the suggested phases. After the submission of this work we became aware of the work by Chen et al. [30], where, probably, the vortex-antivortex crystal we proposed has been observed, and also of the work by Zhang [31], where, independently from us the idea of the vortex-antivortex crystal has been presented, although Zhang used different arguments than we use. We thank Sandy Fetter, Alpo Kallio, Pekka Pietilainen, Lev Pitaevski and Grisha Volovik for many valuable discussions. We thank the Academy of Finland for financial support, and one of us (FVK) thanks also the International Centre of Theoretical Physics for partial support.
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
V. Berezinskil, Sov. Phys. JETP 32 (1971) 493. J.M. Kosterlitz and D.J. Thouless, .I. Phys. C 6 (1973) 1181. J.M. Kosterlitz and D.J. Thouless, Progr. Low Temp. Phys. VIIb (1978) 373. G. Agnolet, D.F. McQueeney and J.D. Reppy, Phys. Rev. B 39 (19891, 8934, and references therein. R.J. Donnelly, Quantized vortices in helium II (Cambridge Univ. Press, Cambridge, 1991). E.M. Lifshitz and L.P. Pitaevskii. in: Landau and Lifshitz Course of Theoretical Physics, Vol. 9, Statistical physics, Part 2 (Pergamon, Oxford, 1981) p. 111. A.L. Fetter. in: The physics of liquid and solid helium, Vol. I, eds. K.H. Bennemann and J.B. Ketterson (Wiley, New York, 1976) p. 207. R.P. Feynman, in: Progress in low temperature physics, Vol. I, ed. C.J. Gorter (North-Holland, Amsterdam, 1955) Ch. 2. G.V. Chester, R. Metz and L. Reatto, Phys. Rev. 175 (1968) 275. F. Dalfovo, Phys. Rev. B 46 (1992) 5482. C.M. Muirhead, W. Vinen and R. Donnelly, Philos. Trans. R. Sot. A 311 (1984) 433. E. Simanek, Phys. Lett. A 154 (1991) 309. J.-M. Duan and A.J. Leggett, Phys. Rev. Lett. 68 (1992) 1216. Q. Niu, P. Ao and D.J. Thouless,Phys. Rev. Len. 72 (1994) 1706. B.E. Clements, J.L. Epstein, E. Krotscheck and M. Saarela, Phys. Rev. B 48 (1993) 7450. B.E. Clements, E. Krotscheck and H.J. Lauter, Phys. Rev. Lctt. 70 (1993) 1287. B.E. Clements et al., Phys. Rev. B 50 (1994) 6958. A.L. Fetter, Phys. Rev. 162 (1967) 143. M. Saarela, B.E. Clements, E. Krotscheck and F.V. Kusmartsev, in: Condensed matter theories, Vol. 10, eds. M. deLano and M. Casas (Nova, New York, 1995), in press. E. Feenberg, Theory of quantum liquids (Academic Press, New York, 1969). M. Saarela and E. Krotscheck,J. Low Temp. Phys. 90 (1993) 41.5. E. Krotscheck and M. Saarela, Phys. Rep. 232 (1993) 1. R.A. Aziz et al., J. Chem. Phys. 70 (1979) 4330. P.A. Whitlock, G.V. Chester and M.H. Kalos, Phys. Rev. B 38 (1988) 2418. J.C. Owen, Phys. Rev. Lett. 47 (1981) 586. F.V. Kusmartsev, Phys. Rev. B 43 (1991) 6132. K. Moulopoulos and N.W. Ashcroft, Phys. Rev. Lett. 69 (1992) 2555. V. Kalmeyer and R.B. Laughlin, Phys. Rev. Len. 59 (1987) 2095. J. Bardeen, G. Baym and D. Pines, Phys. Rev. 156 (1967) 207. M.T. Chen, J.M. Roesler and J.M. Mochel, J. Low Temp. Phys. 89 (1992) 125. S.C. Zhang, Phys. Rev. Lett. 71 (1993) 2142.
LETTERS
A
Physics Letters A 202 (1995) 317-323
Many-body structure of quantum vortices in thin 4He films M. Saarela a, F.V. Kusmartsev
a9b
aDepartment of Theoretical Physics, Uniuersiry of Ouly SF-90570 Oulu, Finland b L.D. Landau Institate for Theoretical Physics, GSP-I, Kosygina 2, V-334, Moscow I1 7940, Russian Federation Received 7 February 1995; accepted for publication 6 April 1995 Communicated by V.M. Agranovich
Abstract We propose a microscopic theory of quantum vortices in two-dimensional 4He, applicable to thin superfluid 4He films. The theory describes the vortex core and reproduces the classical logarithmic behavior at large distances. The single vortex is treated as a quasiparticle. We calculate the energy needed to create a single vortex, the vortex inertial mass, the microscopic interaction between vortices and the binding energy of the vortex-antivortex pair as a function of density at zero temperature. At 4He densities less than 0.037 A-’ the binding energy of the vortex-antivortex pair becomes negative, indicating an instability for the creation of a novel state.
It was Vadim Berezinskii [l] who first recognized that two-dimensional (2D) condensed matter systems may have a phase transition, associated with the creation of vortices. Kosterlitz and Thouless [2,3] developed those ideas by introducing an intuitive Hamiltonian and derived the scaling equations. In thin superfluid 4He films it is assumed that at the critical temperature of the Berezinskli-Kosterlitz-Thouless (BKT) phase transition macroscopic amounts of unpaired vortices are created, and that destroys the superfluidity [4]. With recent possibilities of making thin films and building up new quantum devices it has become very important to understand the role of these topological excitations in 2D fluids. The microscopic structure of a vortex, its interaction with the many-particle background and with other vortices is a fascinating many-body problem [5-71. Traditionally a single vortex has been inserted into the superfluid as an external potential and the response of the system is shown as the inhomogeneity of the density profile [8-lo]. At the center of the 2D vortex the superfluid density is zero and far away it equals the uniform bulk density. More recently the inertial muss of the vortex has received considerable attention and various estimates of its value have been made [ll-141. This inspired us to present a many-body calculation of quantum vortices in the Boson fluid. The calculations are done in purely 2D 4He superfluid at zero temperature. However, a very good resemblance of that system in found in thin 4He films on smooth substrates like the first superfluid layers on top of two solid 4He layers on graphite or on top of the solid hydrogen on glass [15-171. We start from the many-particle Hamiltonian with a given pair interaction between superfluid particles and include vortices through the quantized phase in the wave function [B]. The kinetic energy part of the many-particle Hamiltonian transforms the phase into an effective interaction between the vortex core and the superfluid particles. That is responsible of the repulsion of the Elsevier Science B.V. SD1 0375-9601(95)00319-3
318
M. Saarela, F. V. Kusmartser
/Physics
Letters A 202 (1995) 317-323
material away from the core region. We show that in 2D “He the volume of the expelled superfluid is of the order of the volume occupied by one 4He atom. The vortex is then a microscopic object with non-zero kinetic energy and its inertial mass must be taken into account. This is in contrast to the treatment of a vortex line in full three dimensions, where the mass becomes infinite [9]. We set the vortex mass equal to the mass of the expelled superfluid [ll]. Its value is calculated through a self-consistent many-body procedure and is proportional to the compressibility. This means that the mass becomes infinite when the superfluid becomes soft against density oscillations. In the limit of a large vortex radius our result reproduces the estimate obtained by Duan and Leggett [13]. The mass together with the effective interaction defines the effective Hamiltonian for the vortex core motion. The modulus of the many-body wave function includes correlations between the vortex core and the superfluid 4He particles [8]. These correlation functions are determined by minimizing the energy required to create a vortex excitation, i.e. we minimize the chemical potential of the vortex quasiparticle in the superfluid. Without an external source of rotation, vortices in superfluid films must only be created in pairs with different signs of vorticities. In classical hydrodynamics the vortex motion is described by two coordinates x and y chosen as conjugate variables, which conserves the vorticity [18]. Vortex-antivortex pairs move perpendicular to the line joining them and their separation distance is proportional to their center of mass (CM) momentum [7]. The “electrostatic forces” arising due to different vorticities are in equilibrium with the “electrodynamic forces” arising due to their motion. For the quantum vortices which we are discussing here the distance between the vortex and antivortex is determined by the balance between the kinetic energy of the single vortices and their attractive interaction. Thus the pair can be created even at zero CM-momentum. The direct part of the interaction between the vortex and antivortex is determined by the phase with two centers of circulation. It vanishes at zero separation distance and diverges logarithmically at large distances. The interaction contains also the induced part which is due to modification of the correlations between the superfluid particles in the region of the created pair, and that enhances the effective attraction. At non-zero CM-momenta the probability distribution of the pair will be polarized in such a way that the separation distance will increase in the perpendicular direction of the motion due to the coupling between the linear CM-motion and rotational motion [19]. The probability distribution is evaluated by minimizing the energy required to create the pair. We assume that the vortex can be described by adding a singular, quantized phase into the many-body wave function,
The coordinates ri, . . . , r,,, refer to the superfluid particles and rv to the vortex core. The gradient of the phase 4(r,, rj), called the vector potential, is related to the superfluid velocity field, A(r,, rj) = KVj+(r,, rj) = (m,/h>u,(r,, rj>. We have denoted with ma the mass of the superfluid particle. The condition that the wave function !Z$ must be single ualued requires that the circulation around the point r, using any closed path should be an integer multiple of 2~, (i.e. $aA(r,., r) * dr = 27r~). The integer K is the conserued uorticity quantum number of the circulation. The modulus of the wave function describes the response of the superfluid to the vortex. We separate the vortex-background correlations from the pure background correlations by using the Jastrow-Feenberg type exponential factors [20],
(rv>
rj)
+
C UVBB(rV1rk) vB( r,, rN). rj,
j
(2)
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
The ground state wave function Hamiltonian [ 151, HB=-x---
i-1 2m,
Pa
is determined
by minimizing
the expectation
319
value of the microscopic
4He
i.j= 1 i
The energy of the system with one infinite muss vortex is the expectation to the wave function qV [8,9],
value of the Hamiltonian
with respect
(4) where the phase has been transformed into an interaction-like term I. The connection to the classical Hamiltonian is found by inserting the superfluid velocity field in place of the vector potential. Since the vortex in 2D 4He occupies only a microscopic volume of the order of one atom size its kinetic energy must be added into the effective Hamiltonian,
The energy of the vortex excitation the mass m,, @‘I H, IV’> pv=
(TPyl!P)
The unknown correlation Euler equations, 6E.L” 6P(ry,
is then equal to the chemical
(‘-& IH, -
w, functions
= SU”as(r”,
of a quasiparticle
W’,>
IW
are determined
rl, r2) =O.
impurity
carrying
(6)
.
SPV rr)
potential
from the variational
ansatz leading
to the set of coupled
(7)
From these Euler equations we obtain the vortex-background pair distribution function gvB(r) and its Fourier transform, the vortex-background structure function SvB(k> = pBjdr ~‘*~“[g”~(r) - 11. Here, pB is the average superfluid density. Details of the method are described in Refs. [15,21,22]. The volume of the “hole” created by the vortex is equal to the value of the structure function at the origin, S”‘(O + ), and the vortex mass is the 4He mass times this volume, m, = -mBSvB(O + ), which is calculated iteratively. We study properties of the vortex excitations in the liquid phase where the superfluid order is possible and present results for the 2D 4He densities ranging from 0.037 A-” to 0.065 A-‘. In experiments with thin films the whole liquid density range of the 2D 4He can be reached because the growth of the film on a substrate can be accurately controlled. For the interaction between 4He particles we use the Aziz potential [23]. Our result for the equation of state agrees well with the Monte Carlo simulations [24]. In the low density limit the liquid phase kas the spinodal instability where the speed of sound becomes zezo. In our model that occurs at about 0.031 Am2. In the high density limit the liquid solidifies at about 0.070 Am2 [15].
-getting Eq. (4) we have written the kinetic energy term in the following form (‘P” 1V2 lly,> = (tL2/2m,X(P IiV- A + 2iA .V - A* + V* IT) + h.c.]. The first term on the right hand side disappears with the choice of the Landau gauge. The second term, which is the usual linear term in the vector potential in the case of magnetic field, also disappears, because our wave function ly is the wave function for a Boson system at zero momentum, and it is a real quantity.
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
320
I
“0
2
4
6 r (11
8
10
12
Fig. 1. The pair distribution functions between the single vortex and the background ‘He particle with vorticity K = 1 (solid curves) and the pair distribution functions for the pure 4He (dashed curves) at densities listed in Table 1. The first peak increases with the density for both sets of curves.
In 2D superfluid the vortex-background
ii2
interaction term is (see Eq. (5))
h*
V’B(‘“,‘,)‘2mIA(r,,r,)12=-
2mB
B
K* 1 TV -‘j
1”
For the vector potential A(r,, r,> we have used the Landau gauge and the quantization condition V, X r,) = 27r~&r, - ri). The solutions for the vortex-background pair distribution functions from the Euler equations (7) for a single vortex with vorticity K = 1 are shown in Fig. 1 for different densities. The pair distribution function starts quadratically from zero and at large distances approaches unity. The increase of the vorticity moves the background particles further away from the center of the vortex. The values of the vortex yass calculated from the structure function are given in Table 1. In the region of the saturation density 0.043 Ae2 the vortex mass is about twice the 4He mass. It increases rapidly when the density is decreased and diverges at the spinodal point. The increasing density squeezes the vortex into a smaller volume and at the solidification density its mass is only half the helium mass. The chemical potential of a single vortex in 2D diverges logarithmically with the size of the system due to the effective interaction of Eq. (8). That is a slow divergence which does not cause any problems in the numerical work. In Table 1 we give the full chemical potential and its “core” part, CL’,where the interaction term of Eq. (8) has been subtracted as a function of
Ah”,
Table 1 The vortex chemical potential, chemical
F”, its “core” part, yc, described in the text and the vortex mass as a function of density. We also give the
potential of the vortex-antivortex
pair, pva, and the root mean square radius, fi
pB (A- ‘)
F” (K)
CL’(K)
m, (amu)
cLy=00
fi
0.037 0.040 0.045 0.050 0.055 0.060 0.065
4.63 5.58 6.90 8.17 9.43 10.7 12.0
1.27 1.32 1.43 1.58 1.77 1.99 2.24
24.3 14.2 8.18 5.61 4.21 3.24 2.73
0.35 1.71 3.50 5.28 7.12 9.04 11.0
2.04 2.07 2.16 2.27 2.39 2.52 2.65
(A)
321
IU. Saarela, F. V. Kusmartsev / Physics Letters A 202 (1995) 317-323
density. For the size of the system the radius R = 36 A has been used. These results are quite insensitive to the precise value of the vortex mass. When the vortex-antivortex pair is created the logarithmic divergence of the chemical potential with the size of the system cancels off. The many-body wave function of the pair contains two phase factors corresponding to two centers of circulation with opposite vorticities.
The sum of these phases transforms into the direct part of the interaction, which can be divided into three terms: vortex and antivortex interactions with the background and the vortex-antivortex interaction mediated by one background particle, raY
gdA(C, B
rj) +VvaB(rv,
fj)-VvB('a,
rj)12=VVB(rv9
fa,
Tj),
with
vvaB(fvr r,, r,) = -K2-
fi* (r,-rj)
-(rv--r,)
mg 1r, - rj 1* 1rv - rj 1*
(11)
’
The centers of the vortex and antivortex are labeled with subindices v and a, respectively. The variational problem is to minimize the energy of the pair added into the superfluid with respect to the vortex-antivortex correlations. The problem is complicated by the fact that the vortex-4He and the 4He-4He correlations will also be modified due to the presence of two vortices. Using the hypemetted chain approximation the vortex-antivortex correlations can be related to the radial distribution function gva(rV, ra). The chemical potential for the pair can then be written into the form
+gvatrv, cJWC(~“~r,>+ -
h2
2mB
/
pBd*r, gvaB(rV7 r,, rl)IA(r,,
ra, rl)l’
I ,
(14
where R is the volume of integration. The induced potential, wzJrV, r,), in the second line enters due to the modification of the medium [25] *. In momentum space it can be expressed in terms of the background and vortex-background structure functions, 6.$(k)
= --
h2k2 TB(k) -
4m,
i P(k)
2 i(
29B(k) ”
We also approximate the triplet distribution function functions [20]. This is a well justified approximation
+ 1 . i
(13)
gvaB(rV, r,, r,> with the product of the pair distribution here because the vortex-antivortex pair is tightly bound.
* The kinetic energy of 4He atoms, exactly speaking the sum of nodal diagrams, induces an additional short-range attraction between vortices given in Eq. (13). In the long wave length limit this is the phonon induced interaction introduced by Bardeen, Baym and Pines [29] between two 3He impurities in 4He.
M. Saarela, F. V. Kusmartseo / Physics Letters A 202 (1995) 317-323
322
16 12 8 4 0 -4 0
2
4
6
8
10
12
14
r CA, Fig. 2. The vortex-antivortex interaction at densities full interaction and the upper line is its direct part.
The expression
Vy”( r)
=
pE = 0.045 A-’
for the effective uortex-antirrortex
p”/
d'r,
g’“(
rv, r,)
g’“(
(solid curves) and 0.065 k’
interaction
r,, r,)VVaB(
rv, ra,
(dashed curves). The lower line is the
can then be written in the form,
rl) + wyi,( r),
(14)
The notation r = 1rv - ra 1has been used. In the limit of a large intervortex distance we find the logarithmic term. V”“(r) + 2rrK2pB(h2/mg)ln(r/r,), owhere rC is a fictitious core radius of a single vortex 3. The values of rC range from 4 A at pB = 0.037 A-* to 0.8 A at pB = 0.065 A-’ when fitted to the value of the full potential at 18 A. In Fig. 2 we show the full effective vortex-antiyortex potential and separately, its direct part - the integral term in Eq. (14) - at densities 0.045 AP2 and 0.065 A-*. The short range attraction caused by the induced potential is clearly seen, which at r = 0 is between - 3 and - 4 K. The minimization of the chemical potential of Eq. (12) with respect to g”“(r) leads to the Schrijdinger-like equation with the interaction (14). We solve it for the lowest eigensiate and calculate from that the vortex-antivortex chemical potential. The results are given in Table 1. We find that at densities less than 0.037 A-’ the chemical potential for the pair creation becomes negative. This signals a new kind of instability in the system different from the liquid-solid phase transition and the spinodal instability. Here macroscopic amounts of vortex-antivortex pairs are spontaneously created. The vzrtex-antivortex chemical potential depends strongly on the density and near the solidification density 0.065 ,A-’ it is 11 K. The root mean square radius is much less density dependent ranging between 2.04 A and 2.65 A as shown also in Table 1. That is because the smaller the density the heavier the mass. Such a small separation distance implies that the nearest neighbor 4He atoms play the main role in the structure of the pair core. The pair has many bound states corresponding to the different principle quantum numbers and angular momenta. At a large separation distance the pair energy depends logarithmically on the principle quantum number. The existence of the bound pair clarifies the microscopic picture of the BKT phase transition. The transition may go through intermediate stages associated with different quantum bound states and the appropriate scaling equations must be modified. Note that the interaction between two 4He atoms adjacent to the vortex-antivortex pair increases effectively, and they may form a bound state,similar to the Anderson resonance valence bond [26] and in the films the
3 In order to make the comparison present formalism.
with quasiclassical
theory of vortices
we introduce
the core radius
rC which is not needed in the
M. Saarela, F. V. Kusmartseu / Physics Letters A 202 (1995) 317-323
323
resonance valence bond liquid may be created. Since the interaction between the vortex-antivortex pairs is long-ranged, the liquid of vortex-antivortex pairs may crystallize [27]. Thus, we propose two novel states: liquid and vortex-antivortex crystal, which may exist in films of 4He. Our resonance valence bond liquid state recalls the flux phase state, studied by Kalmeyer and Laughlin [28] in the hard-core Boson representation of the Heisenberg Hamiltonian. We call for experimentalists to look for the suggested phases. After the submission of this work we became aware of the work by Chen et al. [30], where, probably, the vortex-antivortex crystal we proposed has been observed, and also of the work by Zhang [31], where, independently from us the idea of the vortex-antivortex crystal has been presented, although Zhang used different arguments than we use. We thank Sandy Fetter, Alpo Kallio, Pekka Pietilainen, Lev Pitaevski and Grisha Volovik for many valuable discussions. We thank the Academy of Finland for financial support, and one of us (FVK) thanks also the International Centre of Theoretical Physics for partial support.
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