Universal Matsubara time decay of quantum autocorrelations for Boltzmann particles

The general properties of time dependent autocorrelations in many-body quantum systems are here analyzed at thermodynamic equilibrium in the Boltzmann canonical ensemble at temperature $T$, by means of the exponential expansion theory (EET). It is shown that the Kubo-Martin-Schwinger (KMS) symmetry applied to the exponential expansion of the correlation leads to the existence of two different sets of decay modes (channels) here indicated as “Matsubara modes” and “system modes,” respectively. The Matsubara modes are a series of pure decay channels with time constants representing a direct action of the thermostat upon the correlation, with a characteristic principal decay time ${\tau }_1=\hslash /\left(2\pi k_{\mathrm{B}}T\right)$, where $\hslash$ and $k_{\mathrm{B}}$ are the Planck and Boltzmann constants, and $T$ is the temperature. Moreover, the KMS condition implies that the amplitudes pertaining to the even and odd contribution of the system modes to the quantum correlation are not independent. These two properties are quantum mechanical in nature and “universal,” in the sense that they are present for any autocorrelation of a quantum system at equilibrium at a temperature $T$. The Matsubara modes' contribution to the time behavior of a quantum correlation is limited to times of the order of ${\tau }_1$, which however can be comparable with some of the characteristic decay times of the system modes. In addition, since the parameters representing the overall time behavior of the quantum correlation can be given in terms of the parameters of its Kubo transform, the EET representation turns out to be useful in calculations exploiting the outputs of some widespread quantum simulation methods. A discussion of the properties of these relations is described in detail with numerical examples. The case of the velocity autocorrelation function of para hydrogen at low temperature is also reported as a final example for a real system.

Figure 1

First five terms, as a function of $\mathrm{Re}z/{\omega }_1$, for the overall amplitudes $M_n=M_n^{(+)}+M_n^{(-)}$ of the Matsubara series in Eq. (48). The color code is specified in the legend.

Figure 2

Dependence on $\mathrm{Re}z/{\omega }_1$ of the individual terms contributing to the amplitude [see Eq. (49)] of an $M$ mode and of an $S$ mode in ${\tilde{c}}_{\mathrm{e}}\left(t\right)$, for real $z$ ($z<0$) . For the $M$ mode only the leading term ($n=1$) is displayed (red curves). The two terms of the $S$ mode are shown with black curves. Solid curves are used for $S^{(+)}$ and $M^{(+)}$, while dashed curves are used for $S^{(-)}$ and $M^{(-)}$.

Figure 3

As in Fig. 2, but for an $M$ pair and an $S$ pair with complex $z$ ($\mathrm{Re}z<0$) in ${\tilde{c}}_{\mathrm{e}}\left(t\right)$.

Figure 4

${\tilde{c}}_{\mathrm{e}}\left(t\right)$ as a function of time (cyan thick curve) for a Kubo correlation characterized by one real mode and one complex pair, for different values of $z/{\omega }_1$ (see text): 0.2 (left frame), 0.5 (central frame), and 0.7 (right frame). The global Matsubara (dashed red curve) and system (black thin curve) contributions are also shown separately.

Figure 5

Real part of the velocity autocorrelation function of liquid para ${\mathrm{H}}_2$ at $T=$ 30 K obtained by RPMD simulations [9] (blue circles) compared with the result of Eqs. (44, 45, 46) (cyan curve) using the parameters of the Kubo $z\left(t\right)$, fitted as described in Ref. [9]. As before, the global Matsubara (dashed red curve) and system (black curve) contributions are also shown separately. The inset is a zoom of the plot at short times, and extends up to the value of the Matsubara time constant ${\tau }_1$ at the chosen temperature.

Figure 6

Fourier transform of $c\left(t\right)/z\left(0\right)=[z\left(t\right)+{\tilde{c}}_{\mathrm{e}}\left(t\right)+i{c}_{\mathrm{o}}\left(t\right)]/z\left(0\right)$ in the same example case analyzed in the right frame of Fig. 4 for ${\tilde{c}}_{\mathrm{e}}\left(t\right)/z\left(0\right)$. The total asymmetric spectrum (blue curve) is shown together with its components: the Matsubara part (red dashed curve), and the system symmetric (black thin curve) and antisymmetric (green dot-dashed curve) parts. The magenta circles correspond to the Kubo-asymmetrized spectrum of $z\left(\omega \right)/z(t=0)$, i.e., $\beta \hslash \omega /(1-e^{-\beta \hslash \omega })z\left(\omega \right)/z\left(0\right)$, perfectly coinciding with the total asymmetric spectrum in blue. The magenta spectrum is simply Eq. (15) for real frequency.