Physical observable

Many physical observables are measured in our CT-HYB quantum impurity solvers. Here we provide a list of them.

Single-particle Green's function $G(\tau)$

The most important observable is the single-particle Green's function $G(\tau)$, which is measured using the elements of the matrix $\mathcal{M}$,

\[\begin{align} G(\tau) = \left\langle \frac{1}{\beta} \sum_{ij}\delta^{-}(\tau, \tau_i - \tau_j) \mathcal{M}_{ji}\right\rangle, \end{align}\]

with

\[\begin{align} \delta^{-}(\tau, \tau') = \begin{cases} \delta(\tau - \tau'), & \tau' > 0, \\ -\delta(\tau - \tau' + \beta), & \tau' < 0. \end{cases} \end{align}\]

Single-particle Green's function $G(i\omega_n)$

Note that in the iQIST software package, the Matsubara Green's function $G(i\omega_n)$ is also measured directly, instead of being calculated from $G(\tau)$ using Fourier transformation.

\[G(i\omega_n) = -\frac{1}{\beta} \sum_{ij} e^{i\omega_n\tau_i}\mathcal{M}_{ij}e^{-i\omega_n\tau_j}\]

Two-particle correlation function $\chi_{\alpha\beta}(\omega,\omega',\nu)$

The two-particle correlation functions are often used to construct lattice susceptibilities within DMFT and diagrammatic extensions of DMFT. However, the measurements of two-particle correlation functions are a nontrivial task^[1] as it is very time-consuming to obtain good quality data, and most of the previous publications in this field are restricted to measurements of two-particle correlation functions in one-band models. Thanks to the development of efficient CT-HYB algorithms, the calculation of two-particle correlation functions for multi-orbital impurity models now become affordable^[2][3][4]. In the iQIST software package, we implemented the measurement for the two-particle correlation function $\chi_{\alpha\beta}(\tau_a,\tau_b,\tau_c,\tau_d)$, which is defined as follows:

\[\begin{equation} \chi_{\alpha\beta}(\tau_a,\tau_b,\tau_c,\tau_d) = \langle c_{\alpha}(\tau_a)c^{\dagger}_{\alpha}(\tau_b)c_{\beta}(\tau_c)c^{\dagger}_{\beta}(\tau_d)\rangle. \end{equation}\]

Due to the memory restrictions, the actual measurement is performed in the frequency space, for which we use the following definition of the Fourier transform:

\[\begin{align} \chi_{\alpha\beta}(\omega,\omega',\nu) &= \frac{1}{\beta} \int^{\beta}_{0}d\tau_a\int^{\beta}_{0}d\tau_b\int^{\beta}_{0}d\tau_c\int^{\beta}_{0}d\tau_d \\ &\times \chi_{\alpha\beta}(\tau_a,\tau_b,\tau_c,\tau_d) e^{i(\omega+\nu)\tau_a}e^{-i\omega\tau_b}e^{-i\omega'\tau_c}e^{-i(\omega'+\nu)\tau_d}. \end{align}\]

where $\omega$ and $\omega'$ [$\equiv (2n+1)\pi\beta$] are fermionic frequencies, and $\nu$ is bosonic ($\equiv 2n\pi/\beta$).

Local irreducible vertex functions $\Gamma_{\alpha\beta}(\omega,\omega',\nu)$

From the two-particle Green's function $\chi_{\alpha\beta}(\omega,\omega',\nu)$, the local irreducible vertex function $\Gamma_{\alpha\beta}(\omega,\omega',\nu)$ can be calculated easily, via the Bethe-Salpeter equation^[3][4][5]:

\[\begin{equation} \Gamma_{\alpha\beta}(\omega,\omega',\nu) = \frac{\chi_{\alpha\beta}(\omega,\omega',\nu) - \beta[G_\alpha(\omega+\nu)G_\beta(\omega')\delta_{\nu,0} - G_\alpha(\omega+\nu) G_\beta(\omega') \delta_{\alpha\beta}\delta_{\omega\omega'}]} {G_\alpha(\omega+\nu)G_\alpha(\omega)G_\beta(\omega')G_\beta(\omega'+\nu)}. \end{equation}\]

The $G(i\omega_n)$ and $\Gamma_{\alpha\beta}(\omega,\omega',\nu)$ are essential inputs for the ladder dual fermion code ROSEMARY, see section Ladder dual fermions for more details.

Pair susceptibility

Impurity self-energy function $\Sigma(i\omega_n)$

The self-energy $\Sigma(i\omega_n)$ is calculated using Dyson's equation directly

\[\begin{equation} \Sigma(i\omega_n) = G^{-1}_{0}(i\omega_n) - G^{-1}(i\omega_n), \end{equation}\]

or measured using the so-called improved estimator^[3][4]. Noted that now the latter approach only works when the segment representation is used.

Histogram of the perturbation expansion order

We record the histogram of the perturbation expansion order $k$, which can be used to evaluate the kinetic energy.

Kurtosis and skewness of perturbation expansion order

Skewness

\[\gamma_1 = \frac{E[(k - \langle k \rangle)^3]}{(E[(k - \langle k \rangle)^2])^{3/2}}\]

Kurtosis

\[\frac{E[(k - \langle k \rangle)^4]}{(E[(k - \langle k \rangle)^2])^{2}}\]

Actually, in the iQIST software package, only the $\langle k \rangle$, $\langle k^2 \rangle$, $\langle k^3 \rangle$, and $\langle k^4 \rangle$ are measured. And then they are used to evaluate the skewness and kurtosis.

Kinetic energy

The expression for the system kinetic energy reads

\[E_{\text{kin}} = -\frac{1}{\beta} \langle k \rangle,\]

where $k$ is the perturbation expansion order.

Potential energy

Occupation number and double occupation number

The orbital occupation number $\langle n_\alpha\rangle$ and double occupation number $\langle n_\alpha n_\beta \rangle$ are measured. From them we can calculate for example the charge fluctuation $\sqrt{\langle N^2 \rangle - \langle N \rangle^2}$, where $N$ is the total occupation number:

\[\begin{equation} N = \sum_{\alpha} n_{\alpha}. \end{equation}\]

Magnetic moment

Actually, we only measure $\langle S_{z} \rangle$.

Spin-spin correlation function

For a system with spin rotational symmetry, the expression for the spin-spin correlation function reads

\[\begin{equation} \chi_{ss}(\tau) = \langle S_{z}(\tau) S_{z}(0) \rangle, \end{equation}\]

where $S_{z} = n_{\uparrow} - n_{\downarrow}$. From it we can calculate the effective magnetic moment:

\[\begin{equation} \mu_{\text{eff}} = \int^{\beta}_{0}d\tau \chi_{ss}(\tau). \end{equation}\]

Orbital-orbital correlation function

The expression for the orbital-orbital correlation function reads

\[\begin{equation} \chi^{nn}_{\alpha\beta}(\tau) = \langle n_{\alpha}(\tau) n_{\beta}(0) \rangle. \end{equation}\]

Atomic state probability

The expression for the atomic state probability is

\[\begin{equation} p_{\Gamma} = \langle |\Gamma \rangle \langle \Gamma| \rangle, \end{equation}\]

where $\Gamma$ is the atomic state.

Fidelity susceptibility

\[\chi_{\text{FS}} = \langle k_{L} k_{R} \rangle - \langle k_{L} \rangle \langle k_{R} \rangle\]

Kinetic energy fluctuation

\[\chi_{\text{k}} = \langle k^2 \rangle - \langle k \rangle^2 - \langle k \rangle\]

Reference