Collective and single-particle excitations in two-dimensional dipolar Bose gases

The Berezinskii-Kosterlitz-Thouless transition in two-dimensional dipolar systems has been studied recently by path integral Monte Carlo simulations [A. Filinov et al., Phys. Rev. Lett. 105, 070401 (2010)]. Here, we complement this analysis and study temperature-coupling strength dependence of the density (particle-hole) and single-particle (SP) excitation spectra both in superfluid and normal phases. The dynamic structure factor, $S(q,\omega )$, of the longitudinal excitations is rigorously reconstructed with full information on damping. The SP spectral function, $A(q,\omega )$, is worked out from the one-particle Matsubara Green's function. A stochastic optimization method is applied for reconstruction from imaginary times. In the superfluid regime sharp energy resonances are observed in both the density and SP excitations. The involved hybridization of both spectra is discussed. In contrast, in the normal phase, when there is no coupling, the density modes, beyond acoustic phonons, are significantly damped. Our results generalize previous zero-temperature analyses based on variational many-body wave functions [F. Mazzanti et al., Phys. Rev. Lett. 102, 110405 (2009); D. Hufnagl et al., Phys. Rev. Lett. 107, 065303 (2011)], where the underlying physics of the excitation spectrum and the role of the condensate has not been addressed.

Figure 1

Matsubara Green's function $\langle \Psi ({\mathbf{r}}_2,t_2){\Psi }^{+}({\mathbf{r}}_1,t_1)\rangle$ (in units of $a^2$) for a set of $D$ values. Simulation parameters ($D,\mu ,V$) are specified in Table . The temperature ($T=1.0$) is below to the BKT transition. The superfluid fraction is ${\rho }_s/\rho >0.80$. Due to spatial homogeneity we plot the dependence on the relative spatial distance $|{\mathbf{r}}_2-{\mathbf{r}}_1|$ and relative imaginary time $\tau =t_2-t_1$.

Figure 2

The same as in Fig. 1 but for the density-density correlation function $\langle \rho ({\mathbf{r}}_2,t_2)\rho ({\mathbf{r}}_1,t_1)\rangle$ (in units $a^4$).

Figure 3

(a) Fourier transform of the one-particle Green's function $G_1$ (16): PIMC data (symbols) and the SO fit (solid lines). Vertical dashed error bars demonstrates the statistical error (shown only for two curves). Colors (and numbers) correspond to the chosen $q$ vectors shown on the left panel. (b) Relative deviation (20) after optimization. Simulation parameters: $D=12.5$ and $T=1.25$ (see Table ). Temperature is close to $T_c$. Superfluid fraction ${\rho }_s/\rho =0.38\left(2\right)$.

Figure 4

The same as in Fig. 3 for the density correlation function $G_2$ (18). Simulation parameters: $D=12.5$ and $T=1.0$. Superfluid fraction ${\rho }_s/\rho =0.81\left(2\right)$. $G_2(q,\tau )$ is evaluated only for $\tau \in [0,\beta /2]$ due to the symmetry with respect to the midpoint $\tau =\beta /2$ [see Eq. (29)].

Figure 5

Temperature dependence of the momentum distribution $n_q$ (a)–(c) and one-particle density matrix $G_1(r,\beta )$ (d)–(f) in the log scale for dipole coupling $D=0.1,1.75,12.5$. In the superfluid phase, the decay of $G_1(r,\beta )$ is characterized by (1) a fast short-range decay to a condensate fraction $n_0\left(T\right)$; (2) off-diagonal quasi-long-range order which depends on temperature and the superfluid density. Simulation parameters are $V=165$, chemical potential $\mu \left(D\right)=4.7,32.85,163.53$, the average particle number $\langle N\rangle \approx 164$. The used wave vectors are $qa=2\pi n(a/L),(n=0,1,...)$. The solid lines are obtained by interpolation (see text). The system is partially superfluid for $T\le 1.67$ ($D=0.1,1.75$) and $T\le 1.0$ ($D=12.5$).

Figure 6

Long-wavelength asymptotics of the momentum distribution $n_q$ in the superfluid phase, $T_{1\left(2\right)}=1.0\left(0.714\right)$ vs $D$ (log-log scale). The dashed lines are the prediction by Eq. (48) for $T_{1\left(2\right)}$. Simulation parameters: $V_{1\left(2\right)}=165\left(576\right)$, $T_{1\left(2\right)}=1.0\left(0.714\right)$.

Figure 7

Rescaled dynamic structure factor $S(q,\omega )/S\left(q\right)$ at dipolar coupling $D=0.1,0.5,1.75,7.5,12.5$ and three temperatures $T$. The system is superfluid (normal gas) at $T=1$ ($T=2,3.3$). The discrete wave-numbers, $q=2\pi n/L(n=0,1,...)$, are induced by the PBCs. The dashed line (guide to the eye) is the upper bound ${\omega }_{\chi }$ for the lower ($L$) excitation branch derived from the sum rules (see Sec. B). The upper ($H$) branch lies above.

Figure 8

Spectral density $\left|A\right(q,\omega \left)\right|$ at $D=0.1,0.5,1.75,12.5$ and three temperatures. The system is superfluid at $T=1$. The dashed line is the upper bound ${\omega }_{\chi }$ for the ${\omega }_L$ branch in $S(q,\omega )$. The spectral densities are not smoothed and demonstrate the raw data. The frequency resolution compared to Fig. 7 is lowered from $w_{\text{min}}=0.5$ to $w_{\text{min}}=1.5$ (see Sec. 6 for details). Hence, the sharpest features have the smallest possible width equal $w_{\text{min}}$.

Figure 9

Dispersion relations for $D=0.1$. Positions of the peaks of $S(q,\omega )$ (a)–(c) and $A(q,\omega )$ (d)–(f) and their half-widths shown as the error-bars. The full spectral densities $S(q,\omega )$ and $A(q,\omega )$ are shown in Figs. 7 and 8. Temperature $T=1$ is below the critical temperature $T_c$ and $T=2.0,3.3$ are above it (see Table ). The simulation parameters are specified in Table . The solid (black) line ${\omega }_{\chi }\left(q\right)$ and the dashed (blue) line ${\omega }_F\left(q\right)$ are the upper bounds for the lowest excitation branch derived from the compressibility and first-frequency sum rules (see Appendix B). The dotted (brown) lines are the free-particle ($q^2/2m$) and isothermal sound ($cq$) dispersions ($c$ is taken from Table ). The solid gray line is the solution of Eqs. (52) and (54) for the high-energy branch ${\omega }_H\left(q\right)$ plotted for the $q$ vectors where the spectral weight $S_H\left(q\right)$ is positive. The red symbols denote some additional features in the spectrum which are hardly seen in Figs. 7 and 8.

Figure 10

Dispersion relations $S(q,\omega )$ (a)–(c) and $A(q,\omega )$ (d)–(f) for $D=1.75$. The half-widths of the peaks are shown as the error bars. See Fig. 9 for further details.

Figure 11

Dispersion relations $S(q,\omega )$ (a)–(c) and $A(q,\omega )$ (d)–(f) for $D=12.5$. The half-widths of the peaks are shown by error bars. See Fig. 9 for further details. In (d)–(f) the black color denotes the uppermost branch which is omitted from the discussion in the text.

Figure 12

(a)–(f) Reconstructed spectral density $S\left(\omega \right)$ (solid red line) compared with the reference $S^0\left(\omega \right)$ (steplike black curve) consisting of two energy resonances [(d)–(f) on the logarithmic scale]. Three columns group the results for three cases of the relative statistical error $\delta G\left({\tau }_i\right)/G^0\left({\tau }_i\right)$ shown by a solid (black) line in (l) and (m). The first column (a),(d),(g),(j) is the reconstruction from the error-free [$\delta G\left(\tau \right)=0$] correlation function $G^0\left(\tau \right)$ shown by dashed (black) lines in (g)–(i). The bold symbols with error bars specify the data [$G\left({\tau }_i\right)\pm \delta G\left({\tau }_i\right),i=0–50$] used in the error-biased reconstruction (second and third columns). Different solutions $G_n\left(\tau \right)$ obtained by the SO are shown by solid (red) lines but are practically indistinguishable on the present scale. Each solution corresponds to a spectral density $S_n$ used in the ensemble average (25). Only part, $\tau \in [0.40,0.50]$, of the full imaginary time interval $[0,\beta =1]$ is shown. (j)–(l) Solid (gray) lines show the relative deviations $|1-G_n\left({\tau }_i\right)/G^0\left({\tau }_i\right)|$ from the exact correlation function. Only 25 from $n\sim 200–400$ samples are shown. Note, that the deviations (panels k-l) are within the statistical error $\delta G\left({\tau }_i\right)/G\left({\tau }_i\right)$ [solid (black) line]. Horizontal line ${10}^{-3}$ is guide for the eye. Three lower panels show the relative error of the ensemble average frequency moments (A3, A4, A5), that is, $\delta \langle {\omega }^n\rangle =|1-{\langle {\omega }^n\rangle }_{S\left(\omega \right)}/{\langle {\omega }^n\rangle }_{S^0\left(\omega \right)}|$, where $S\left(\omega \right)$ is shown in panels (a)–(c) and denoted on the $x$ axis as (a), (b), and (c). See text for further details and used parameters.

Figure 13

(a)–(f) Reconstructed spectral density $S\left(\omega \right)$ (solid red lines) compared with $S^0\left(\omega \right)$ (steplike black curves). (d)–(f) The same on the logarithmic scale. For further details, see text and caption of Fig. 12. For the relative errors $\delta \langle {\omega }^n\rangle$, see text.

Figure 14

(a)–(f) Reconstructed spectral density $S\left(\omega \right)$ (solid red lines) compared with $S^0\left(\omega \right)$ (steplike black curves). (d)–(f) The same on the logarithmic scale. For further details, see text and caption of Fig. 12. The plots of correlation functions $G_n\left({\tau }_i\right)$ are omitted. For the relative errors $\delta \langle {\omega }^n\rangle$, see text.

Figure 15

(a)–(f) Reconstructed spectral density $S\left(\omega \right)$ (solid red lines) compared with $S^0\left(\omega \right)$ (steplike black curves). (d)–(f) The same on the logarithmic scale. For further details, see text and caption of Fig. 12.