1147
J. Phys. Chem. 1990, 94, 1147-1148
Pressure Dependence of the Superfluid 4He Transition in Two Dimensions D. S. Peters* and B. J. Alder Lawrence Livermore National Laboratory, P.0. Box 808, Livermore, California 94550 (Received: July 5, 1989)
It is experimentally known that the superfluid transition temperature in three-dimensional (3D)4He is reduced at higher densities. A similar effect is predicted in two dimensions (2D) based upon path integral quantum Monte Carlo simulations for small systems which have been scaled to the thermodynamic limit by using Kosterlitz-Thouless (KT) theory.
Thermodynamic behavior of a system at temperature P = l / k T is determined through the trace of the density matrix
TABLE I: Kosterlitz-Thouless Parameters and Infinite System Results for 2D ‘He at Three Densities
p(d,d’;p)= (dle-@Hld’) which can be rewritten in the equivalent form l...lp(d,dl;7)
P,
0.0432
p(dl,d2;7)... p ( d , , , d ’ ; 7 ) d d , d d 2 ... dd,,
where each density matrix in the integrand is now at a higher effective temperature, 7 = P / s . The advantage is that for, high enough temperatures (large s), excellent approximations to the exact density matrix are known. Quantum statistics can be introduced by the appropriate permutations, P, of the particles’ positions in the density matrix, so that, for bosons
Phs,,(d,d’;P)= (N)-~cP(d,Pd’;P) P
A technique for numerically calculating these Feynman path integrals for quantum many-body systems has recently been developed and applied to normal-density 3D and 2D 4He.293 Here, this algorithm is applied to high-density 2D 4He. The superfluid fraction, p , / p , vs temperature for a 16-particle simulation at density p = 0.0720 A-2 is shown in Figure 1. It is impractical at this time to treat much larger systems because of problems in sampling phase space. These small-system results must then be scaled to the thermodynamic limit by using finite-size Kosterlitz-Thouless (KT) theory4 as a guide. This technique was shown to accurately predict the number dependence for 9-, 16-, and 25-particle systems as well as the experimental transition temperature of the lower density 2D 4He system.j In contrast, the small systems results for 3D 4He2 do not (and should not) follow the scaling predicted by finite-size KT theory. In the KT theory superfluidity is destroyed by the unbinding of vortex pairs in the presence of other vortices which screen the interaction. The superfluid density, ps, is given in terms of K(1) = ii2p,(l)/m2kBT,with the concentration of vortices given in terms of their fugacity, y = exp(-p/kBr), or p, their chemical potential. A self-consistent solution is obtained by integration of the recursion relations
where 1 = In (r/dcore)represents the logarithmic interaction of vortices in two dimensions and d,,,, is the vortex core diameter. Extension of these arguments to finite systems, however tenuous, can be made by cutting off the intervortex distance r a t L/2, where ( I ) Angus, S.;de Reuck, K. M. International Thermodynamic Tables of the Fluid State Helium-#; Pergamon: Oxford, U . K . , 1977; p 263. (2) Pollock, E. L.; Ceperley, D. M. Phys. Reu. B 1987, 36, 8343. (3) Ceperley, D. M.; Pollock, E. L. Phys. Reu. B 1989, 39, 2084. (4) Kosterlitz, J. M.; Thouless, D. J. J . Phys. C, 1973, 6, 1181. Kotsubo, V.; Williams, G. A. Phys. Reu. B 1986, 33,6106. A good review on the subject of 2D superfluidity is given by: Minnhagen, P. Rev. Mod. Phys. 1987, 59,
1001.
0022-3654/90/2094-1147$02.50/0
‘4-=
0.0576 16.7 2.6 (2) 3.2 4.2 (4) 0.89 0.82 (2) 1.16 (2) 1.2
0.0720 14.9 2.0 (2) 2.6 5.0 (4) 1.2 0.78 (2) 1.33 (2) 1.5
“Numbers in parentheses represent the error in the last digit. L is the size of the periodic simulation box (see Table I). Finite-size effects lead to a smearing out of the phase transition and a high-temperature tail in the p s / p data shown in Figure 1. The dashed curve shown in Figure 1 is the least-squares fit to the data using finite-size KT theory. The best fit values for p and d,, are given in Table I, along with the same parameters for the other densities, p = 0.04323 and 0.0576 A-2. The core diameter is on the order of the interparticle spacing, 2a, as expected. The small size of the cores means that it is possible for vortex pairs to fit into the simulation box, while the large ratio p/kTF guarantees that the density of vortices is low, a necessary condition for this small screening theory to be valid. It is also interesting to note that as the correlations get stronger (at higher densities) the size of the cores gets correspondingly larger and so do their concentrations. The thermodynamic limit is then obtained by using the values of p and d,,,, as initial conditions and integrating the recursion relations out to L = -, giving the solid curve shown in Figure 1. In this limit, the superfluid fraction drops abruptly to zero at T,, as given by the universal jump criterion, p s / p = (2/7r)(m/h)*kTC, shown as the diagonal line in Figure 1. Above this line all vortices are bound in vortex, antivortex pairs and below this line, free vortices exist. The presence of these unpaired vortices causes the helicity modulus of the fluid to vanish, analogous to the vanishing of the rotational elasticity modulus in the 2D melting transition. The helicity modulus is a measure of the phase rigidity of the superfluid wave function and the associated long-range correlations of the phase which signify superfluidity. In a 3D fluid this order-disorder transition occurs as the spatial correlations go from nondecaying in the superfluid state to exponentially decaying in the normal state. In 2D the phase correlations decay algebraically below T, and also exponentially above T,. The values for T, at the three densities are given in Table I. The transition temperatures vs density for 4He films confined to two dimensions are plotted in Figure 2. The three numerical points, shown as open circles and a square, agree very well with the experimental data pointS shown as the filled circle. It is seen that at sufficiently high densities, near the solid phase boundary, the transition temperature is suppressed. Free 4He films, such as those existing on a substrate in contact with 4He gas, are not ( 5 ) Rudnick, I . Phys. Rev. Lett. 1978, 40, 1454.
0 1990 American Chemical Society
1148
Peters and Alder
The Journal of Physical Chemistry, Vol. 94, No. 3, 1990
t 0.8
normal
n
Y,
E3 E
0
4
0.6
-
0- k
/
/
IC,
0.4 -
E
/
c
0.2
I I
/
I I
superfluid
- / . /
I I I I
solid
I I
I I I I
I
0.0
0.0
0.5
1 .o
1.5
2.0
Temperature (K) Figure 1. Superfluid fraction as a function of temperature for 4He at Open circles are Monte Carlo path integral density p = 0.0720 results for a 16-particle system. Typical error bars are shown. The dashed curve is the best fit to this data using finite-size KosterlitzThouless theory. The solid curve represents the extrapolated infinite system results. The KT universal jump criterion is the diagonal line.
expected to exhibit this behavior. Rather, those films become more three dimensional in nature at high density, leading to a monotonic increase in Tc.5 In contrast, in bulk 3D 4He the gas phase interrupts the low-density part of the phase diagram, so that only a decrease in transition temperature with increased density is observed. It has also been suggested6 that a similar effect causes the observed reduction of the superconducting transition temperature at high dopant densities in the recently discovered high-temperature superconductors. Also given in Table I is the degeneracy parameter, A( Tc)/a, where the thermal deBroglie wavelength X = (h2/2mkBT)'/*,and the density of the particles n = l/aa2. On simple physical grounds it is expected that the superfluid transition should occur when h/a = 0(1), because that is when the exchange processes which produce superfluidity become prevalent. The standard Bose condensation criterion for the ideal 3D gas is equivalent to this result. Further, in the Kosterlitz-Thouless theory of the ideal 2D gas the temperature scale is set by T / J , J = nh2/m, so that X(T,)/a = constant falls out of this theory as well. The only difference between the 2D and 3D ideal Bose gas is that the constant is different. Particle interactions complicate the picture somewhat. The exchange processes will be restricted at higher densities by the presence of hard-core interactions between the particles, qualitatively explaining the reduction in the superfluid density. This effect is expected to be larger in 2D than in 3D because of the more restricted possible moves of the particles in 2D space that lead to exchange of position; that is, there are fewer paths available that avoid overlap in 2D. A direct comparison between 2D and 3D can be made by looking at the behavior of (6) Alder, B. J.; Peters, D. S. Europhys. Lett., to be published.
~ '0.02~ ~0.04' ' 0.06 " " 0.08 ' ~ '0.10~ ' ~ " ' Density
(A**-2)
Figure 2. Phase diagram for 2D 4He. The previous numerical result3 is shown as a square, and the present results are shown as open circles. Typical error bars are shown. The experimental value of T, = 0.75 K at p = 0.0460 (ref 5 ) is shown as a filled circle. p , / p for equivalent sized systems, N = 16 = 42 in 2D and N =
64 = 43 in 3D, at the same reduced density, r,/a = 0.61, where r , = 1.35 %.,the hard-core radius of 4He. The 2D system at that density reaches the condition p s / p = 0.5 at a temperature of 1.13 K, whereas the same condition is reached at 1.95 K in the 3D system, showing that a lower temperature is necessary in 2D to reach the same probability of cyclic exchanges. As a closing remark, we note that the decrease in the superfluid transition temperature at high densities can be understood in terms of an increase in the effective mass of the 4He atoms. Feynman showed,' using the path integral technique, that to a good degree of accuracy a system of hard-core bosons of mass m is equivalent to a system of free bosons with increased effective mass, m*. At low density this correction is of the form m* = m[l + (rb/a)2]. This suggests that the ideal gas criterion, X(Tc)/a = constant, could be valid for the interacting system as well if one redefines the deBroglie wavelength as A * = (f1~/2m*k~T)'/~. Thus the reduction in T, at high densities can be interpreted to be a result of the increased effective mass, m*. Although the exact form of Feynman's result does not hold at these high densities, we can infer a relative effective mass from the ratios of A( T,)/a at the different densities. This leads to the relative mass ratios m*/m shown in Table I. (relative to the mass at p = 0.0432 Acknowledgment. We thank David Ceperley and Roy Pollock for instruction in the use of their path integral program as well as many helpful discussions. This work was funded by the U S . Department of Energy and Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Registry No. He, 7440-59-7. (7) Feynman, R. P. Statisfical Mechanics; Benjamin/Cummings; Reading, MA, 1972; p 345.
