Simulation of dynamical properties of normal and superfluid helium

Abstract

The formation of a superfluid when ⁴He is cooled below the characteristic lambda transition temperature is accompanied by intricate quantum mechanical phenomena, including the emergence of a Bose condensate. A combination of path integral and semiclassical techniques is used to calculate the single-particle velocity autocorrelation function across the normal-to-superfluid transition. We find that the inclusion of particle exchange alters qualitatively the shape of the correlation function below the characteristic transition temperature but has no noticeable effect on the dynamics in the normal phase. The incoherent structure factor extracted from the velocity autocorrelation function is in very good agreement with neutron scattering data, reproducing the width, height, frequency shift, and asymmetry of the curves, as well as the observed increase in peak height characteristic of the superfluid phase. Our simulation demonstrates that the peak enhancement observed in the neutron scattering experiments below the transition temperature arises exclusively from particle exchange, illuminating the role of Bose-statistical effects on the dynamics of the quantum liquid.

The remarkable properties of superfluids, in particular the observed frictionless flow and heat conduction without a temperature gradient, have fascinated scientists for several decades (1–5). Liquid ⁴He has for long served as the paradigm of superfluidity, and the unusual properties of this system around and below the characteristic lambda transition temperature have been the subject of persistent experimental and theoretical investigations. Superfluidity is intimately connected with Bose–Einstein condensation (6, 7) (BEC), but many questions still surround their relationship. A number of experimental properties of superfluid ⁴He are consistent with theoretical descriptions that assume the presence of BEC. However, direct observation of BEC in this system has been elusive, owing primarily to strong repulsive interactions associated with the closed-shell electronic structure of ⁴He.

Theoretical and computational studies have yielded a wealth of information regarding the momentum distribution of superfluid ⁴He, from which one can extract the fraction of particles in the zero momentum state (8, 9). These works, the majority of which are based on quantum mechanical simulations such as quantum Monte Carlo (10–14) and path integral Monte Carlo (15) methods, estimate the condensate fraction to be ≈7% at saturated vapor pressure (SVP) (12–15). Experimental studies of these properties are largely based on inelastic neutron scattering measurements (16). At small values of momentum transfer, the differential cross section obtained from these experiments displays characteristics associated with collective response (with a sharp peak that follows the well known phonon–maxon–roton dispersion curve) (3–5, 17, 18). On the other hand, when the momentum transferred to the fluid by the neutrons is very large, the observed result reduces to the incoherent scattering function. Although the latter is characterized by single-particle properties, it is also rich in information about the dynamical behavior of the Bose fluid. Careful analysis of recent neutron scattering experiments in the intermediate-to-high momentum regime supports the existence of a condensate density consistent with that obtained from numerical simulations of equilibrium properties (5, 8, 9, 19–21).

In this article, we present direct real-time simulation of dynamical properties in superfluid ⁴He employing a combined quantum-semiclassical approach. An earlier attempt to compute dynamical properties in superfluid ⁴He by using the maximum entropy analytic continuation method to invert imaginary time correlation functions was met with limited success in the superfluid phase, where the simulation results (reported at lower momentum transfer values) significantly overestimated the width of the dynamic structure factor (22). We calculate the single-particle velocity autocorrelation function in the temperature range 0.35–4.0 K (across the normal-to-superfluid transition) near SVP. The incoherent scattering function at intermediate values of the neutron momentum transfer is extracted from the correlation functions and compared with available experimental data. Comparison with results obtained in the absence of particle exchange (for a fictitious system obeying Boltzmann statistics) is also made. The simulation results reveal the distinct roles of zero-point motion and quantum statistical effects in the dynamics of the quantum liquid.

Methods and Computational Details

Numerical simulations of real-time properties in condensed phase systems exhibiting quantum mechanical effects are generally hindered by severe obstacles, which arise from the nonlocality of the Schrödinger equation along with the oscillatory nature of the quantum mechanical phase. Our approach combines a fully quantum mechanical path integral representation of the equilibrium density (23) with a semiclassical description of the dynamics (24–26) in a form that reduces the severity of the phase cancellation problem (27, 28). Dynamical information is obtained from time correlation functions, which have the general form

[1]

where is the (appropriately symmetrized) quantum statistical operator that describes the equilibrium density of the system, and Â, B̂ are operators that probe the desired properties. The time development of B̂ is obtained through a forward–backward semiclassical approximation (27, 28), which minimizes the oscillatory character of the integrand, in a phase space representation. The density operator is evaluated by using the path integral formulation of quantum statistical mechanics (23, 29). With these procedures, the time correlation function takes the form

[2]

Here, R and P are the 3n-dimensional vectors describing the position and momentum coordinates of the n-atom system, R(t), P(t) are the coordinates reached by a classical trajectory with initial phase space coordinates R₀, P₀, and represents permutations of atoms. The N auxiliary path integral variables R_k = (r_k⁽¹⁾, r_k⁽²⁾,..., r_k⁽ⁿ⁾) form the basis of the quantum-classical isomorphism: Each atom becomes equivalent to a chain or “necklace” of N classical “beads” r₁^(j), r₂^(j),..., r_N^(j) that account for the quantum dispersion of the particle (15, 29, 30). In the present context where real-time information is obtained, both ends of the path integral chain are linked to a special bead that designates the origin of a classical trajectory (see Fig. 1). The statistical weight of each path integral configuration is given by the complex-valued function Θ, and f_AB is a function of the path integral and trajectory values specific to the operators under consideration. Identical particle exchange effects (quantum statistics) are taken into account by allowing chains to “cross-link” (15, 31) (see Fig. 1). The method becomes an exact quantum mechanical treatment at zero time.

Fig. 1.

Schematic representation of the path integral necklace quantizing two ⁴He atoms. The exchange of one bead is indicated in Lower. The red bead represents the initial coordinate of a classical trajectory.

By using this combined quantum-semiclassical approach (32, 33), the single-particle velocity autocorrelation function for liquid ⁴He

[3]

(where m is the atomic mass, n is the number of ⁴He atoms in the simulation box, and p̂_j is the momentum operator of the jth atom) is calculated from the properties of classical trajectories emanating from an appropriate phase space density. We employ the pair-product density matrix as a high temperature form at T = 20 K and T = 10 K in a position (15) and a mixed coordinate–phase space (33) representation, respectively. Dynamical information is obtained from the time-dependence of this classical trajectory, averaged by Monte Carlo methods with respect to the coordinates of all of the beads and permutation spaces. Numerical tests indicate that the combined quantum-semiclassical method yields sufficiently accurate approximations for the velocity autocorrelation function in the incoherent regime, which in the case of ⁴He is observed at moderately large values of the momentum transfer.

Within a Gaussian approximation, the velocity autocorrelation function uniquely determines the self-intermediate scattering function through the relation (34)

[4]

In this_– expression, Q is the neutron momentum transfer and ω_R = hQ²/2m is the atom recoil energy. Eq. 4 is obtained from a second-order cumulant expansion and is valid at small to intermediate values of Q. Fourier transformation of Eq. 4 yields the incoherent structure factor.

The simulation used a cubic box containing n = 40 ⁴He atoms, interacting with a potential given by Aziz et al. (35), with periodic boundary conditions. The dynamical properties presented in the next section are converged with respect to system size. The equations of motion for the classical trajectories were solved by using the velocity Verlet algorithm with a time step equal to 2.5 fs. Within the temperature range studied, converged results were obtained with up to 54 path integral beads. All calculations were performed on a PC cluster configured with 16 nodes.

Results and Discussion

A representative snapshot of the path integral chains in the superfluid regime is shown in Fig. 2. One can observe closed loops, indicating the quantum dispersion of individual ⁴He atoms, as well as long loops winding across the unit cell; the latter arise from particle exchange and are responsible for superfluid behavior. The transition between normal and superfluid phases is steep, as one concludes from the step-like shape of the energy–temperature curve (15). Slightly above the lambda transition temperature, exchange operations are infrequent. On the other hand, slightly below the transition temperature, once chains consisting of two or three atoms are formed, they quickly incorporate many other atoms, filling the simulation cell with long loops. These qualitative differences are responsible for the dramatic change in the properties of ⁴He across the normal-to-superfluid transition observed in the simulation.

Fig. 2.

Snapshot of the path integral representation of the system in the superfluid regime at T = 0.9 K. Twenty path integral beads are used for each ⁴He atom. Chains shown in orange form closed loops indicating atoms that are not exchanged. Long linked chains, corresponding to exchanged atoms, are shown in light blue. One can see long exchange paths winding across the unit cell. Red balls indicate a bead from which classical trajectories are launched. Image beads that are outside the unit cell are displayed in gray.

Fig. 3 shows the single-particle velocity autocorrelation function at the temperatures T = 1.18, 2.5, and 4.0 K at SVP. The zero-time results of our simulation reproduce the well known equilibrium properties of the system obtained from path integral Monte Carlo calculations (15). All calculated functions exhibit a large imaginary part in this temperature range, affirming the significance of quantum mechanical effects in this system, primarily zero-point motion. To clarify the effects of quantum statistics, we performed additional calculations in which we omitted particle exchange. In Fig. 3b, the velocity autocorrelation function for the system of boson particles at T = 1.18 K is compared with the correlation function for a fictitious system of interacting ⁴He atoms that obey Boltzmann statistics. As seen in the initial value of the correlation function, quantum statistical effects decrease the kinetic energy appreciably, owing primarily to the existence of a zero-momentum state that characterizes the Bose condensate. This kinetic energy depression has been observed in equilibrium path integral Monte Carlo calculations. More importantly, quantum statistics alter the shape of the correlation function. It is seen that inclusion of particle exchange effects leads to a faster decay of the correlation function and to the appearance of a prominent negative part. These features can be attributed to additional attraction between ⁴He atoms originating from Bose statistics.

Fig. 3.

Single-particle velocity autocorrelation function of liquid ⁴He at various temperatures. (a) Simulation results with full inclusion of Bose exchange at T = 4.0 K, ρ = 0.01932 Å^–3(red), T = 2.5 K, ρ = 0.02179 Å^–3(green), and T = 1.18 K, ρ = 0.02186 Å^–3(blue). Real and imaginary parts are displayed as solid and dashed lines, respectively. (b) Comparison of the real part of the velocity autocorrelation function (solid lines) against that of a fictitious system where particle exchange has been neglected (dashed lines) at T = 1.18 K, ρ = 0.02186 Å^–3(blue) and T = 4.0 K, ρ = 0.01932 Å^–3(red). The results of the two calculations are indistinguishable at T = 4.0 K. Results in the time domain where quantum coherence may not be negligible are shown in gray.

Exchange effects were found to be negligible at the two higher temperatures considered (T = 2.5 and 4.0 K), both of which are higher than the superfluid transition temperature T_λ = 2.17 K. In this normal regime, which is described well by Boltzmann statistics, thermal excitation is minimal and the large zero-point motion of interacting ⁴He atoms (≈16 K per atom) makes the dominant contribution to the kinetic energy of the system. For this reason, the initial value of the three correlation functions in Fig. 3 that are characterized by Boltzmann statistics exhibits a weak temperature dependence.

Incoherent scattering experiments have proven very useful for measuring single-particle properties, with particular emphasis on extracting the condensate fraction. In the limit of large momentum transfer, Q → ∞ [the “impulse approximation” (36)], the scattering function is simply related to the momentum distribution function of the atom struck by the neutron (provided that interactions between atoms are not infinite anywhere). In this limit, the existence of a Bose condensate manifests itself as a distinct sharp peak in the spectrum. Deviations from this limit at finite values of Q, known as final state effects, broaden the incoherent structure factor owing to the interaction of the surrounding ⁴He atoms. Thus, comparison with available experimental results at moderate Q values requires knowledge not only of static properties (such as the momentum distribution) but also of the system's dynamical response.

The single-particle velocity autocorrelation function obtained from our simulation is used to calculate the self-intermediate scattering function (34) according to Eq. 4. The resulting incoherent structure factor is shown in Fig. 4 and compared with that reported from neutron scattering experiments at various values of the neutron momentum transfer Q. It has been found that the incoherent limit is adequately reached for Q > 12 Å^–1 (5). At smaller values of Q, quantum coherence effects are likely to become important. Such effects in dynamics of ⁴He atoms are not captured by our simulation method and are deemed responsible for the differences between simulation results and neutron scattering experiments observed in Fig. 4a for Q = 8 Å^–1. The use of the Gaussian approximation to extract the incoherent structure factor introduces small errors at large Q values, which account for some of the discrepancies observed in Fig. 4 b and c.

Fig. 4.

The incoherent structure factor for three values of the momentum transfer at various temperatures. Simulation results are shown as solid lines, and experimental data are shown as circles. The recoil energy, ω_R, is shown as a dotted line. (a) Q = 8.0 Å^–1. Simulation results are shown at T = 1.43 K, ρ = 0.02186 Å^–3(blue) and T = 2.5 K, ρ = 0.02179 Å^–3(red). Experimental results (19) are shown at T = 1.42 K (blue) and T = 2.5 K (red) at SVP. (b) Q = 23.0 Å^–1. Simulation results are shown at T = 0.35 K, ρ = 0.02210 Å^–3(blue) and T = 3.33 K, ρ = 0.02210 Å^–3(red). Experimental results (20) are shown at T = 0.35 K, ρ = 0.02210 Å^–3(blue) and T = 3.5 K, ρ = 0.02210 Å^–3(red). (c) Q = 28.5 Å^–1. Simulation results are shown at T = 1.18 K, ρ = 0.02186 Å^–3(blue) and T = 4.0 K, ρ = 0.01932 Å^–3(red). Experimental results (21) are shown at T = 1.3 K (blue) and T = 3.5 K (red) at SVP.

When exchange effects are negligible and Boltzmann statistics are operative, the simulation results obtained at different temperatures appear similar, centered about the recoil energy, ω_R. A small shift of the peak in the negative direction with respect to the recoil energy can also be observed. The asymmetry of the peak is attributed to the imaginary component of the velocity autocorrelation function, which is a manifestation of large quantum mechanical features in the dynamics of the system. In the normal fluid, such effects are associated with zero-point motion. Particle exchange also contributes to the imaginary part of the correlation function in the superfluid regime, leading to a more pronounced asymmetry in the incoherent structure factor. These effects are more clearly observed at the smaller Q values, for which the scattering function is determined by the dynamical response of the system over a larger time interval. A pronounced peak around the center of the simulated structure factor is seen in all cases where quantum statistical effects play a significant role. These subtle features of the simulation results are in line with experimental observations (19–21). As seen in Fig. 5, the enhancement of the peak height is absent from the simulations where particle exchange effects were omitted. This feature is attributed to the effect of Bose statistics on the short time values of the velocity autocorrelation function. The association of the observed peak enhancement with Bose exchange effects is in line with theoretical models that employ the computed momentum distribution (and thus, implicitly, the existence of a BEC) to fit the neutron scattering data (5, 8, 9, 14, 19–21, 37). We note that none of the available final state effects theories can correctly account for the small shift of the peak observed in the experiments.

Fig. 5.

The incoherent structure factor obtained from the single-particle velocity autocorrelation function within a Gaussian approximation at T = 1.18 K, ρ = 0.02186 Å^–3 at the neutron momentum transfer Q = 28.5 Å^–1(solid line) in comparison with the same function obtained by omitting particle exchange effects (dashed line).

Apart from small discrepancies between simulation and experimental results, our calculations are successful at reproducing the overall shape of the incoherent structure factor and various features that characterize the superfluid phase, in particular the asymmetry and the condensate peak. This is particularly gratifying, because the interaction potential between ⁴He atoms is the only input required in our calculations. At the smallest Q values considered, errors in the simulation results are deemed to arise from the long time tail of the velocity autocorrelation function, where quantum coherence effects probably become important. At large Q values, the primary source of error is the truncation of the cumulant expansion of the intermediate scattering function at the second order, which allows its evaluation from the two-time correlation function obtained from the simulation.

There is increasing interest in the dynamical properties of systems that are characterized by prominent quantum statistical effects, such as the spectroscopy of impurity atoms or molecules embedded in superfluid ⁴He nanodroplets (38–40), the manifestation of superfluidity and BEC in confined geometries such as on surfaces and in porous media (41), and the recently reported observation of “supersolidity” (42). The combined quantum-semiclassical methodology used in the calculations presented in this article will help deepen the microscopic understanding of the intriguing dynamical behaviors of such systems.