Correlation effects and collective excitations in bosonic bilayers: Role of quantum statistics, superfluidity, and the dimerization transition

A two-component, two-dimensional (2D) dipolar bosonic system in the bilayer geometry is considered. By performing quantum Monte Carlo simulations in a wide range of layer spacings we analyze in detail the pair correlation functions, the static response function, and the kinetic and interaction energies. By reducing the layer spacing we observe a transition from weakly to strongly bound dimer states. The transition is accompanied by the onset of short-range correlations, suppression of the superfluid response, and rotonization of the excitation spectrum. A dispersion law and a dynamic structure factor for the in-phase (symmetric) and out-of-phase (antisymmetric) collective modes during the dimerization is studied in detail with the stochastic reconstruction method and the method of moments. The antisymmetric mode spectrum is most strongly influenced by suppression of the inlayer superfluidity (specified by the superfluid fraction ${\gamma }_s={\rho }_s/\rho$). In a pure superfluid (normal fluid) phase, only an acoustic [optical (gapped)] mode is recovered. In a partially superfluid phase, both are present simultaneously, and the dispersion splits into two branches corresponding to a normal and a superfluid component. The spectral weight of the acoustic mode scales linearly with ${\gamma }_s$. This weight transfers to the optical branch when ${\gamma }_s$ is reduced due to formation of dimer states. In summary, we demonstrate how the interlayer dimerization in dipolar bilayers can be uniquely identified by static and dynamic properties.

Figure 1

(a),(d) Intra- and interlayer PDFs $g_{\alpha \beta }\left(r\right)$. (b),(e) Static structure factor $S_{\alpha \beta }\left(q\right)$. (c),(f) Density response function ${\chi }_{\alpha \beta }\left(q\right)\equiv \left|Re{\chi }_{\alpha \beta }(q,\omega =0)\right|/2{\rho }_{\alpha \beta }$ [Eq. (59)]. The interlayer spacing: $0.3\le d\le 0.5$. Simulation parameters: $D=1$, $\mu =20$, $V\left(L^2\right)=81$, and $T=1$. In panels (b) and (e) the dotted (dashed) lines, on the top of the data for $S_{\alpha \beta }\left(q\right)$ (symbols with error bars), represent the Fourier transform (56) of the interpolated radial intra(inter)layer correlation functions $g_{\alpha \beta }\left(r\right)$ [Eqs. (22), (54), and (62)].

Figure 2

The $d$ dependence of the inlayer superfluid fraction ${\gamma }_s(d,T)$ at temperatures $T=1\left(0.5\right)$. Coupling $D=1$. The values ${\gamma }_{s\alpha }\left(T\right)$, measured in each layer $\alpha$, coincide within the statistical errors. The simulated system contains $〈N〉\sim 160$ particles, with $〈N_{\alpha }〉\sim 80$ particles per layer. The finite-size corrections can change absolute values, but not the observed trend in ${\gamma }_s\left(d\right)$. The insets show instantaneous density snapshots from PIMC at $d=0.3,0.4$, and 0.6 (each quantum particle is presented as a cloud of $\sim 100$ beads). Black (brown) color is used to distinguish the top (bottom) layer.

Figure 3

The $d$ dependence of total ${\epsilon }_N$, kinetic $k_N$, and potential energy $v_N$ (per particle) from the many-body simulations at $T=1$. For comparison, the solution of a single-dimer problem is presented by ${\epsilon }^T,v^T$, and $k^T$ (the dimer energies are divided by $N=2$). The upper index $T$ denotes the temperature argument in the matrix squaring technique [56].

Figure 4

(Left) Dimer energy ${\epsilon }_d=\left|\epsilon \right|/{\epsilon }_0$ as a function of the coupling strength $U_0=D/d$ and in the weak ($w$) and strong ($s$) coupling approximation; see Eqs. (63) and (64). (Right) The mean dimer size, ${\sigma }_d=\sqrt{\langle r^2\rangle /a^2}$, and the ratio of kinetic to potential energy, $k_d/\left|v_d\right|$, versus $U_0$.

Figure 5

The inter- and intralayer PDFs. Coupling $D=1$. The many-body results $g_{12}\left(r\right),g_{11}\left(r\right)$, and $\mathrm{\Delta }g\left(r\right)=g_{12}\left(r\right)-g_{11}\left(r\right)$ are compared with the single-dimer solution $g_d\left(r\right)$. The range of layer spacing $0.25\le d\le 0.5$ corresponds to $2\le U_0\le 4$ in Fig. 4.

Figure 6

The $d$ dependence of the inlayer density ${\rho }_{\alpha }$ and the compressibility ${\kappa }_{\alpha }\left[{10}^2\right]$ for $D=1$.

Figure 7

The inter- and intralayer PDFs at several spacing for $D=5.5$. The many-body results $g_{12}\left(r\right),g_{11}\left(r\right)$, and $\mathrm{\Delta }g\left(r\right)=g_{12}\left(r\right)-g_{11}\left(r\right)$ are compared with the single-dimer solution $g_d\left(r\right)$. The range $0.6\le d\le 1$ corresponds to $5.5\le U_0\le 9.17$ in Fig. 4.

Figure 8

The $d$-dependence of the superfluid fraction ${\gamma }_s(d,T)$ at temperature $T=1$ and $D=5.5$. The system size is $〈N〉\sim 160$ with $〈N_{\alpha }〉\sim 80$ particles per layer. The insets show instantaneous density snapshots at $d=0.7,0.8$, and 1.0. Black (brown) color distinguishes the top (bottom) layer.

Figure 9

(a),(d) Intra- and interlayer PDFs $g_{\alpha \beta }\left(r\right)$. (b),(e) Static structure factor $S_{\alpha \beta }\left(q\right)$. (c),(f) Density response function ${\chi }_{\alpha \beta }\left(q\right)\equiv \left|Re{\chi }_{\alpha \beta }(q,\omega =0)\right|/2{\rho }_{\alpha \beta }$. The interlayer spacing: $0.6\le d\le 0.8$. Simulation parameters: $D=5.5$, $\mu =70$, $V\left(L^2\right)=81$, and $T=1$.

Figure 10

The $d$ dependence of total, kinetic, and potential energies, ${\epsilon }_N,k_N,v_N$ (per particle), from many-body simulations at $T=1$ for $D=5.5$. For comparison, the solution of a single-dimer problem is presented by ${\epsilon }^T,v^T$, and $k^T$.

Figure 11

The $d$ dependence of the inlayer density ${\rho }_{\alpha }$ and the compressibility ${\kappa }_{\alpha }\left[{10}^2\right]$ for $D=5.5$.

Figure 12

Inter- and intralayer pair distribution functions for $D=0.1$. The many-body results—$g_{12}\left(r\right),g_{11}\left(r\right)$, and $\mathrm{\Delta }g\left(r\right)=g_{12}\left(r\right)-g_{11}\left(r\right)$—are compared with a single-dimer solution $g_d\left(r\right)$. Two additional curves take into account finite-density effects via an effective external potential: (i) a hard wall of a radius $r_c$ (the curve “$r_c$”); (ii) a soft potential $V_b\left(r\right)=\gamma r^3$ which mimics the intralayer dipole repulsion. The considered layer spacing, $0.1\le d\le 0.4$, corresponds to $0.25\le U_0\le 1$ in Fig. 4.

Figure 13

The $d$ dependence of the superfluid fraction ${\gamma }_s(d,T)$ for $T=1$ and $D=0.1$. The insets show the instantaneous density snapshots at $d=0.1,0.15$, and 0.2. Black (brown) color distinguishes the top (bottom) layer.

Figure 14

(a),(d) Intra- and interlayer PDFs $g_{\alpha \beta }\left(r\right)$. (b),(e) Static structure factor $S_{\alpha \beta }\left(q\right)$. (c),(f) Density response function ${\chi }_{\alpha \beta }\left(q\right)\equiv \left|Re{\chi }_{\alpha \beta }(q,\omega =0)\right|/2{\rho }_{\alpha \beta }$ [Eq. (59)]. The interlayer spacing: $0.1\le d\le 0.2$. Simulation parameters: $D=0.1$, $\mu =4.8$, $V\left(L^2\right)=81$, and $T=1$.

Figure 15

The $d$ dependence of total, kinetic, and potential energies, ${\epsilon }_N,k_N,v_N$ (per particle), from many-body simulations at $T=1$. For comparison, the solution of a single-dimer problem with the boundary conditions $\{r_c,\gamma r^3\}$ (see the text) is presented along with a free-dimer case $\{\epsilon ,v,k\}$.

Figure 16

The $d$ dependence of inlayer density ${\rho }_{\alpha }$ and compressibility ${\kappa }_{\alpha }\left[10\right]$ for $D=0.1$.

Figure 17

(Left) The low-frequency symmetric and antisymmetric mode ${\omega }_{\pm }^L\left(q\right)$ from the two-mode ansatz. (Right) Spectral weights $S_{\pm }^L\left(q\right)$. Coupling $D=1$. Both solutions demonstrate an acoustic behavior in the long-wavelength limit for $d\ge 0.35$. An optical branch in ${\omega }_{-}^L\left(q\right)$ is observed for $d\le 0.30$. For $qa<2$ the gap value is better described by the Feynman ansatz ${\omega }_{-}^f\left(q\right)$ (denoted by solid dots for $d=0.25,0.27,0.30$). The legend indicates the layer spacing. Green (black) color is used to distinguish a normal (superfluid) phase.

Figure 18

Rescaled dynamic structure factor for the symmetric (left panel), $S_{+}(q,\omega )/S_{+}\left(q\right)$, and antisymmetric (right panel), $S_{-}(q,\omega )/S_{-}\left(q\right)$, modes. Coupling $D=1$. The legend indicates the layer spacing, $0.25\le d\le 0.35$, and the superfluid fraction ${\gamma }_s$. For comparison, several upper bounds for the dispersion relation are shown: ${\omega }_{\pm }\left(q\right)\le {\omega }_{\pm }^{\chi }\left(q\right)\le {\omega }_{\pm }^f\left(q\right)\le {\omega }_{\pm }^{{\mu }_3}\left(q\right)$; see Eq. (31). The ansatz ${\omega }_{\pm }^L$ (indicated by blue symbols) provides best agreement with the low-frequency resonances in $S_{\pm }(q,\omega )$. The solid gray line denotes the free-particle dispersion ${\epsilon }_q=q^2/2m$.

Figure 19

The $d$ dependence of $S_{-}(q,\omega )$ for $D=1$ at two wave numbers $q=q_{0}n\left(q_0=2\pi /L\right)$: $n=1$ (left) and $n=2$ (right). The legend indicates the layer spacing, $0.25\le d\le 0.35$.

Figure 20

Comparison of the antisymmetric mode spectra for Bose bilayers at $T=1\left({\gamma }_{\text{s}}=0.42\right)$, $T=2\left({\gamma }_{\text{s}}=0\right)$ and the bilayer with Boltzmann statistics at $T=1\left({\gamma }_{\text{s}}=0\right)$. Layer spacing $d=0.32$ and $D=1$. The right panels show the results on the log scale. The legend indicates the wave numbers $qa$. In the partially superfluid phase (${\gamma }_{\text{s}}=0.42$) the spectra is dominated by the acoustic branch. The optical branch is suppressed and strongly overlaps with the acoustic one. In contrast, in the simulations with ${\gamma }_{\text{s}}=0$ the spectral weight is carried by the optical branch. At low frequencies some additional resonance (but with a much smaller spectral weight) is observed for $T_{\text{Boltz}}=1$. The same resonance (but significantly enhanced) is observed also for $T_{\text{Bose}}=2$. This additional dispersionless branch is formed in the wide range of wave numbers ($0<qa<9$) and can be interpreted as intrinsic excitations of the dimer states.

Figure 21

Comparison of the static properties for $d=0.32$ and $D=1$. Compared are Bose bilayers at $T=1,2$ and the bilayer of boltzmannons at $T=1$.

Figure 22

Decomposition of the renormalized dynamic structure factor, ${\tilde{S}}_{-}(q,\omega )=S_{-}(q,\omega )/S_{-}\left(q\right)$ at $qa=2\pi /L$, into the acoustic and the optical branch: ${\tilde{S}}_{-}(q,\omega )=S_A(q,\omega )+S_O(q,\omega )$. The high-frequency optical branch is fitted with the MM ansatz: $S_O=S_{\text{M}}$ [Eq. (77)]. The spectral weight of the acoustic mode $S_A$ is compared in Table 1 with the superfluid fraction ${\gamma }_s$. The legend indicates ${\gamma }_s$ at different layer spacing $d$.

Figure 23

(a),(c) The $d$-dependence of the renormalized static structure factor, $S_{\pm }(q_1,d)=S_{\pm }(q_1,d)/S_{\pm }(q_1,d=0.7)$, the static response function ${\tilde{\chi }}_{\pm }(q_1,d)=Re{\chi }_{\pm }(q_1,d)/Re{\chi }_{\pm }(q_1,d=0.7)$ for the (anti)symmetric mode, and of the square of the superfluid fraction ${\gamma }_s^2\left(d\right)$. The normalization factors: $S_{\pm }(q_1,d=0.7)=0.07663\left(0.06741\right)$ and $\frac{1}{2}Re{\chi }_{\pm }(q_1,d=0.7)=0.02112\left(0.01704\right)$. (b) The integrated spectral weight of the acoustic branch, $S_A(q_n,d)$, versus ${\gamma }_s$. The considered wave numbers: $q_n=2\pi n/L\left(n=1,2\right)$. The system size: $L=9$. (d) Same as in panel (c) plotted versus ${\gamma }_s^2\left(d\right)$ to demonstrate a linear dependence.

Figure 24

The low-frequency modes, ${\omega }_{\pm }^L\left(q\right)$, and their spectral weights, $S_{\pm }^L(q,\omega )$, for $D=5.5$. Both solutions demonstrate an acoustic behavior in the long-wavelength limit for $d>0.75$. An optical branch is present in the out-of-phase mode, ${\omega }_{-}^L\left(q\right)$, for $d\le 0.75$. Its behavior for $qa<2$ and the gap value is better described by the Feynman ansatz ${\omega }_{-}^f\left(q\right)$ (denoted by solid dots for $d=0.6,0.65,0.7$). The legend indicates the layer spacing. Different colors are used to distinguish a superfluid (${\gamma }_s\ge 0.9$ and $d\ge 0.8$) and a normal phase (${\gamma }_s=0$ and $d\le 0.65$).

Figure 25

The same as in Fig. 18 for $D=5.5$. The legend indicates the layer spacing, $0.65\le d\le 0.75$, and the superfluid fraction ${\gamma }_s$. The symbols show three upper bounds $\{{\omega }_{\pm }^{\chi },{\omega }_{\pm }^f,{\omega }_{\pm }^{{\mu }_3}\}$ and the low-energy mode ${\omega }_{\pm }^L$ is indicated by the blue symbols. The solid gray line denotes the free-particle dispersion ${\epsilon }_q=q^2/2m$.

Figure 26

The $d$ dependence of $S_{-}(q,\omega )$ for the asymmetric mode spectra at two wave numbers $q=q_{0}n:n=1$ (left) and $n=2$ (right). The legend indicates the layer spacings: $0.60\le d\le 0.725$. Dipole coupling $D=5.5$.

Figure 27

(a),(c) The $d$ dependence of the renormalized static structure factor, $S_{\pm }(q_1,d)=S_{\pm }(q_1,d)/S_{\pm }(q_1,d=0.7)$, the static response function ${\tilde{\chi }}_{\pm }(q_1,d)=Re{\chi }_{\pm }(q_1,d)/Re{\chi }_{\pm }(q_1,d=0.7)$ for the in-phase and out-of-phase modes, and the square of the superfluid fraction ${\gamma }_s^2\left(d\right)$. (b) The integrated spectral weight of $S_A(q_1,d)$ vs ${\gamma }_s$. (d) Same as in panel (c) plotted vs ${\gamma }_s^2\left(d\right)$ to demonstrate a linear dependence. The wave number $q_1$ corresponds to the smallest wave number in the simulation $2\pi /L\left(L=9\right)$. The normalization values: $S_{\pm }(q_1,d=1.0)=0.03775\left(0.03334\right)$ and $\frac{1}{2}Re{\chi }_{\pm }(q_1,d=1.0)=0.5874\times {10}^{-2}(0.4629\times {10}^{-2})$.

Figure 28

Comparison of the antisymmetric mode spectra for a bosonic bilayer at $T_{\text{Bose}}=1[{\gamma }_{\text{s}}=0.625\left(5\right)]$ and $T_{\text{Bose}}=2\left({\gamma }_{\text{s}}=0\right)$ and a bilayer with Boltzmann statistics at $T_{\text{Boltz}}=1\left({\gamma }_{\text{s}}=0\right)$. Layer spacing $d=0.725$ and $D=5.5$. The right panels show the results on the log scale. The legend indicates the wave numbers $qa$. In the superfluid system ($T_{\text{Bose}}=1$) the spectra are dominated by the acoustic branch. For the Boltzmann case (${\gamma }_{\text{s}}=0$) the spectral weight is carried by the optical branch. For $T_{\text{Bose}}=2$ a new dispersionless branch is formed around $\hslash \omega /E_0\sim 6$, most probably due to intrinsic excitations of the dimer states.

Figure 29

Comparison of the static properties for $d=0.725$ and $D=5.5$. Compared are the bilayers at $T=1$ with Bose and Boltzmann statistics.

Figure 30

The $d$ dependence of the phonon and roton parameters of the symmetric mode $S_{+}(q,\omega )$ for $D=1.0$. (a) The position of the maximum in $S_{+}$ with its half-width (shown as error bars). The solid line is the prediction from the two-mode ansatz ${\omega }_{+}^L$. (b),(c) The energy resonances and their half-width (symbols with error bars) in the roton region, $q_n=2\pi n/L\left(n=9,10\right)$. The solid line with bold symbols is a fit to ${\omega }_{+}^L\left(q\right)$ by the roton-ansatz ${\epsilon }_{+}^r$ (84). (d) The $d$ dependence of the roton wave number $q_{+}^r$ from (84).

Figure 31

Similar analyses as in Fig. 30 for $D=5.5$.

Figure 32

Dynamic structure factors $S_{\pm }(q,\omega )$ and $S_{11}(q,\omega )=\frac{1}{2}[S_{+}(q,\omega )+S_{-}(q,\omega )]$ for $D=1$. Each spectral density is identified by a color as specified in the legend. Left and right columns show the results for the phonon ($qa=0.70$) and roton ($qa=6.28$) wave numbers, respectively. Vertical arrows indicate positions of the peaks predicted by the sum-rules analyses: ${\omega }_{+}^L\left(q\right)$ (brown), ${\omega }_{-}^L\left(q\right)$ (green), and ${\omega }_{-}^f\left(q\right)$ (gray, shown only for $qa=0.70$).

Figure 33

Same as in Fig. 32 for $D=5.5$.

Figure 34

The low-frequency modes, ${\omega }_{\pm }^L\left(q\right)$, and their spectral weights, $S_{\pm }^L(q,\omega )$, for $D=0.1$. Both solutions demonstrate an acoustic behavior in the long wavelength limit for $d\ge 0.1$. Formation of an optical branch at these layer spacings is not observed due to a high superfluid fraction, ${\gamma }_s>0.9$.

Figure 35

Rescaled dynamic structure factor for the symmetric (left), $S_{+}(q,\omega )/S_{+}\left(q\right)$, and antisymmetric (right), $S_{-}(q,\omega )/S_{-}\left(q\right)$, modes. Coupling: $D=0.1$. The legend indicates the layer spacing, $0.1\le d\le 0.2$, and the superfluid fraction ${\gamma }_s$. For comparison, several upper bounds for the dispersion relation are shown: ${\omega }_{\pm }\left(q\right)\le {\omega }_{\pm }^{\chi }\left(q\right)\le {\omega }_{\pm }^f\left(q\right)\le {\omega }_{\pm }^{{\mu }_3}\left(q\right)$; see Eq. (31). The ansatz ${\omega }_{\pm }^L$ (indicated by blue symbols) provides best agreement with the low-frequency resonances in $S_{\pm }(q,\omega )$. The solid gray line denotes the free-particle dispersion ${\epsilon }_q=q^2/2m$.

Figure 36

The $d$ dependence of $S_{-}(q,\omega )$ for $D=0.1$. Used wave numbers $q=q_{0}n:n=1$ (left) and $n=2$ (right). The legend indicates the layer spacing: $0.1\le d\le 0.2$.

Figure 37

Comparison of the antisymmetric mode spectra for a bosonic bilayer at $T_{\text{Bose}}=1\left({\gamma }_{\text{s}}=0.93\right)$ and $T_{\text{Bose}}=3.3\left({\gamma }_{\text{s}}=0\right)$ and a bilayer with Boltzmann statistics at $T_{\text{Boltz}}=1\left({\gamma }_{\text{s}}=0\right)$. Layer spacing: $d=0.1$ and $D=0.1$. The right panels show the results on the log scale. The legend indicates the wave numbers $qa$. For the Bose statistics only the acoustic branch is observed. Both the acoustic and the optical branch are present in the Boltzmann case.

Figure 38

Same as in Fig. 37 for the symmetric mode.

Figure 39

Comparison of the static properties for $d=0.1$ and $D=0.1$. Compared are a bosonic bilayer at the three temperatures, $T_{\text{Bose}}=\{1,2,3.3\}$, and a bilayer with Boltzmann statistics at $T_{\text{Boltz}}=1$.

Figure 40

The $d$ dependence of the low-frequency resonances ${\omega }_{\pm }\left(q\right)$ [maximum in $S_{\pm }(q,\omega )$] and their half-width for $D=0.1$. The wave numbers $q_n=2\pi n/L$ in the phonon ($n=1,2$) and “roton” ($n=9$) domain are considered. Predictions from the two-mode ansatz ${\omega }_{\pm }^L$ are indicated by solid lines.

Figure 41

(a),(c) The $d$ dependence of the renormalized static structure factor, $S_{\pm }(q_1,d)=S_{\pm }(q_1,d)/S_{\pm }(q_1,d=0.5)$, the static response function ${\tilde{\chi }}_{\pm }(q_1,d)=Re{\chi }_{\pm }(q_1,d)/Re{\chi }_{\pm }(q_1,d=0.5)$ for the symmetric and antisymmetric mode, and the square of the superfluid fraction ${\gamma }_s^2\left(d\right)$. (b) The dispersion ${\omega }_{\pm }^L\left(q\right)$ from the two-mode ansatz at two layer spacings. (d) The $d$ dependence of the acoustic sound speed obtained from the linear fit, ${\omega }_{\pm }^L(q,d)=c_{\pm }\left(d\right)q$, for $qa\le 2$. The increase of $d$ has a larger effect on the antisymmetric mode (see also Fig. 34). The normalization values: $S_{\pm }(q_1,d=0.5)=0.2239\left(0.2045\right)$ and $Re{\chi }_{\pm }(q_1,d=0.5)/2=0.09209\left(0.08296\right)$.

Figure 42

Dynamic structure factors $S_{\pm }(q,\omega )$ and $S_{11}\left(q\right)$ for $D=0.1$. Each spectral density is identified by the color as specified by the legend. Left (right) column shows the results for the wave number $qa=0.70\left(qa=6.28\right)$. Vertical arrows indicate positions of the peaks predicted from the sum rules: ${\omega }_{+}^L\left(q\right)$, ${\omega }_{-}^L\left(q\right)$, and ${\omega }_{-}^f\left(q\right)$.

Figure 43

(a)–(c) Comparison of the effective potentials in the RPA-type ansatz for the symmetric (lower set of curves) and antisymmetric (upper set of curves) mode for $D=0.1$ and the layer spacing $d=0.24,0.15,0.10$. (d)–(f) The reference value of the static density response function, ${\chi }_{\pm }\left(q\right)=\left|Re{\chi }_{\pm }(q,0)\right|/2\rho$, versus the RPA-type approximations, ${\chi }_{\pm }^{\text{ex,}d}\left(q\right)$. Note that in all cases ${\chi }_{+}\left(q\right)>{\chi }_{-}\left(q\right)$.

Figure 44

Same as in Fig. 43 for $D=1.0$ and the layer spacing $d=0.35,0.30,0.27$.

Figure 45

Same as in Fig. 43 for $D=5.5$ and the layer spacing $d=0.75,0.70,0.65$.

Figure 46

Comparison of the dynamic structure factor $S_{\alpha \alpha }(q_n,\omega )\left(\alpha =1,2\right)$ in the mass-symmetric bilayer for a set of wave numbers $q_n=2\pi n/L\left(L=9\right)$. The left (right) panel is without (with) minimization of ${\delta }^r〈{\omega }^k〉\left(q\right)\left(k=1,3\right)$. The reconstruction on the left panel demonstrates much larger deviations between $S_{11}(q,\omega )$ and $S_{22}(q,\omega )$. Here the statistical errors in the density correlation function allow for a broader class of different trial solutions which satisfy (A5).

Figure 47

Absolute relative deviations of the spectral power moments (A11, A12, A13, A14) from the reference values (27). The upper bound is given the relative errors of (27) shown by open gray triangles. The largest deviations are observed in the third power moment (shown by open circles), up to ${\delta }^r〈{\omega }^3〉\sim 0.38\left(38\%\right)$.

Figure 48

The same as in Fig. 47 with additional minimization of ${\delta }^r〈{\omega }^k〉\left(q\right)\left(k=1,3\right)$.