Ab initio low-energy dynamics of superfluid and solid ${}^4\text{H}\text{e}$

We have extracted information about real time dynamics of ${}^4\text{H}\text{e}$ systems from noisy imaginary-time correlation functions $f\left(\tau \right)$ computed via quantum Monte Carlo (QMC): production and falsification of model spectral functions $s\left(\omega \right)$ are obtained via a survival-to-compatibility with $f\left(\tau \right)$ evolutionary process, based on genetic algorithms. Statistical uncertainty in $f\left(\tau \right)$ is promoted to be an asset via a sampling of equivalent $f\left(\tau \right)$ within the noise, which give rise to independent evolutionary processes. In the case of pure superfluid ${}^4\text{H}\text{e}$ we have recovered from exact QMC simulations sharp quasiparticle excitations with spectral functions displaying also the multiphonon branch. As further applications, we have studied the impuriton branch of one ${}^3\text{H}\text{e}$ atom in liquid ${}^4\text{H}\text{e}$ and the vacancy-wave excitations in hcp solid ${}^4\text{H}\text{e}$ finding an unexpected rotonlike feature.

Figure 1

(Line) $S_{\text{GIFT}}\left(q,\omega \right)$ for $q=0.783{\text{Å}}^{-1}$ and $\rho =0.0218{\text{Å}}^{-3}$; (open circles) observed (Ref. 24) dynamic structure factor $S\left(q,\omega \right)$ in liquid ${}^4\text{H}\text{e}$ for $q=0.7{\text{Å}}^{-1}$ at saturated vapor pressure (SVP) and $T=1.3\text{K}$. Notice the logarithmic scale. Notice also the difference between the wave vector of $S_{\text{GIFT}}\left(q,\omega \right)$ and the one of the experimental available (Ref. 24) dynamic structure factor; the experimental single particle peak position is known to increase by about 0.8 K in moving from $q=0.7{\text{Å}}^{-1}$ to $q=0.783{\text{Å}}^{-1}$.

Figure 2

(a) and (b) $S_{\text{GIFT}}\left(q,\omega \right)$ at $q=1.755{\text{Å}}^{-1}$ and $\rho =0.0218{\text{Å}}^{-3}$; (a) single quasiparticle (qp) peak; (b) multiphonon (mp) contribution (notice change in scale). Lines corresponding to a $S_{\text{GIFT}}\left(q,\omega \right)$ obtained with a nonzero entropic prior $\left(\eta \ne 0\right)$ are also shown. (c) $\epsilon \left(q\right)$ extracted at $\rho =0.0218{\text{Å}}^{-3}$ from the position of the qp (circles) peaks and the positions of the maxima of the mp contribution (triangles) are shown. The error bars represent the 1/2–height widths. (d) $\epsilon \left(q\right)$ and mp contribution extracted at $\rho =0.0262{\text{Å}}^{-3}$. Lines in (c) and (d): experimental data (Refs. 25, 26); in the mp region in (c) the lower curve (dotted) represents the position of the maximum while the upper one (dashed) represents the 1/2–height width.

Figure 3

Upper panels: $S_{\text{GIFT}}\left(q,\omega \right)$ at $q=1.755{\text{Å}}^{-1}$ and $\rho =0.0218{\text{Å}}^{-3}$ extracted from noisy imaginary-time correlation functions with different level of accuracy; (left) single quasiparticle peak; (right) multiphonon contribution (notice change in scale). Lower panel: imaginary-time correlation functions $f\left(\tau \right)$ used in the GIFT reconstructions shown in the upper panel.

Figure 4

Evolution of the deviation (10) during the stochastic evolution of the genetic algorithm for the reconstruction plotted in Figs. 2a, 2b for $\eta =0$ averaged with respect to the sampled sets ${\mathcal{D}}^{\star }$. The dashed horizontal line represents the value $\delta =\frac{1}{l+1}{\sum }_{j=0}^l{\sigma }_{f_j}^2$.

Figure 5

(a) GIFT strength of the quasiparticle peak $Z\left(q\right)$ as function of $q$ at two densities and experimental data (Ref. 27). (b) GIFT Static density response function $\chi \left(q\right)$ at two densities and experimental data (Refs. 25, 28) Error bars of theoretical results are smaller than the symbol size.

Figure 6

Impurity ${}^3\text{H}\text{e}$ quasiparticle peak in superfluid ${}^4\text{H}\text{e}$ at SVP for several wave vectors; in the inset the extracted excitation energies are shown together with experimental data (Ref. 32).

Figure 7

Vacancy excitation spectrum in solid ${}^4\text{H}\text{e}$ extracted from the $S_{\text{GIFT}}\left(q,\omega \right)$ of the vacancy–vector position ${\vec{x}}_v$ at $\rho =0.0293{\text{Å}}^{-3}$ in a hcp lattice with $N=447$ particles along the principal symmetry directions: (a) $\Gamma \text{K}$, (b) $\Gamma \text{M}$, and (c) $\Gamma \text{A}$. Two different algorithms have been used to obtain ${\vec{x}}_v$: a coarse-grain algorithm (Ref. 36) and the Hungarian algorithm (Refs. 37, 38). Dotted lines represent the spectrum of a tight-binding model for the hcp lattice (Ref. 39) obtained imposing the values of the bandwidth along the $\Gamma \text{K}$ and $\Gamma \text{A}$ directions equal to the average between the values extracted from the two different algorithms. The arrow points out the vacancy-roton mode.

Figure 8

(Color online) Single peak reconstruction. Upper panel: two noisy imaginary-time correlation functions obtained via Eq. (B4) from $f^{\left(a\right)}\left(\tau \right)$ (open circles) and $f^{\left(b\right)}\left(\tau \right)$ (x symbols), which are the Laplace transforms of $s_a\left(\omega \right)$ (dotted line in the middle panel: $\mu =10$ and $\alpha =0.1$) and $s_b\left(\omega \right)$ (dotted line in the lower panel: $\mu =10$ and $\alpha =1$). The inset is a zoom on one imaginary-time instant. Middle panel: $s_a\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ (stairstep lines) using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$); green and blue lines represent MEM-like reconstructions with different values of the $\eta$ parameter in the fitness (see legend); greater the values for this parameter wider are the spectral functions. Lower panel: $s_b\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$).

Figure 9

(Color online) Double peak reconstruction for well-separated peaks. Upper panel: noisy imaginary-time correlation function obtained via Eq. (B4) from $f\left(\tau \right)$ (dotted line) which is the Laplace transform of $s\left(\omega \right)$ (dotted line in the lower panel, see text for parameters). Lower panel: $s\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ (stairstep lines) from $f\left(\tau \right)$ using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$); green and blue lines represent MEM-like reconstructions with different values of the $\eta$ parameter in the fitness (see legend).

Figure 10

Integral properties for double peak reconstruction. Upper panel: $I_0\left(\omega \right)$ from the exact $s\left(\omega \right)$ (dotted line) and $I_0\left(\omega \right)$ obtained with the reconstructed $S_{\text{GIFT}}\left(\omega \right)$ in Fig. 9 (case $\eta =0$). Lower panel: $I_{-1}\left(\omega \right)$ from the exact $s\left(\omega \right)$ (dotted line) and $I_{-1}\left(\omega \right)$ obtained with the reconstructed $S_{\text{GIFT}}\left(\omega \right)$ in Fig. 9.

Figure 11

Double peak reconstruction for overlapping peaks. Upper panel: two noisy imaginary-time correlation functions obtained via Eq. (B4) from $f^{\left(a\right)}\left(\tau \right)$ (open circles) and $f^{\left(b\right)}\left(\tau \right)$ (x symbols), which are the Laplace transforms of $s_a\left(\omega \right)$ (dotted line in the middle panel) and $s_b\left(\omega \right)$ (dotted line in the lower panel); see text for parameters. The inset is a zoom on one imaginary-time instant. Middle panel: $s_a\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$). Lower panel: $s_b\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$).

Figure 12

Multiple peak reconstruction. Upper panel: noisy imaginary-time correlation function obtained via Eq. (B4) from $f\left(\tau \right)$ which is the Laplace transform of $s\left(\omega \right)$ (dotted line in the lower panel). Lower panel: $s\left(\omega \right)$ (dotted line) and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ from $f\left(\tau \right)$ using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$).

Figure 13

Integral properties for multiple peak reconstruction. Upper panel: $I_0\left(\omega \right)$ from the exact (dotted line) and $I_0\left(\omega \right)$ obtained with the reconstructed $S_{\text{GIFT}}\left(\omega \right)$ in Fig. 12. Lower panel: $I_{-1}\left(\omega \right)$ from the exact (dotted line) and $I_{-1}\left(\omega \right)$ obtained with the reconstructed $S_{\text{GIFT}}\left(\omega \right)$ in Fig. 12.

Figure 14

Panel (a): $f_j$ values obtained with ${\sigma }_{{\epsilon }_j}=0.01$ and used by GIFT to reconstruct $s\left(\omega \right)$ as shown in panel (b). Panel (c): $f_j$ values obtained with ${\sigma }_{{\epsilon }_j}=0.05$ and used by GIFT to reconstruct $s\left(\omega \right)$ as shown in panel (d). Panel (b): exact $s\left(\omega \right)$ (dotted line) as in Fig. 12 and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ using the noisy observation of $f\left(\tau \right)$ in panel (a). Panel (d): exact $s\left(\omega \right)$ (dotted line) as in Fig. 12 and reconstructed $S_{\text{GIFT}}\left(\omega \right)$ using the noisy observation of $f\left(\tau \right)$ in panel (c).

Figure 15

$s\left(\omega \right)$ (dotted line) as in Fig. 12. Upper panel: reconstructed $S_{\text{GIFT}}\left(\omega \right)$ from the exact $f\left(\tau \right)$ (i.e., without noise) assumed to be known for $l=60$ number of points in imaginary time with $\delta \tau =1/160$, with $\Delta \omega =0.25$ and using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$). Middle panel: reconstructed $S_{\text{GIFT}}\left(\omega \right)$ from the exact $f\left(\tau \right)$ (i.e., without noise) assumed to be known for $l=240$ number of points in imaginary time with $\delta \tau =1/640$, with $\Delta \omega =0.25$ and using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$). Lower panel: reconstructed $S_{\text{GIFT}}\left(\omega \right)$ from the exact $f\left(\tau \right)$ (i.e., without noise) assumed to be known for $l=60$ number of points in imaginary time with $\delta \tau =1/160$, with $\Delta \omega =0.1$ and using in the fitness ${\Phi }_{{\mathcal{D}}^{\star }}$ only the first moment (i.e., ${\gamma }_n=0\forall n\ne 1$).