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Dynamic Response Functions

We define a dynamic correlation function as

(1)\chi_{AB}(\tau) = -(1/\hbar)\langle A(\tau)B(0)\rangle,

where A and B are operators.

Density-density response

Since pi-qmc uses a position basis, we often collect density fluctuations in real space. However, most textbook descriptions of density fluctioans are in k-space, and results for homogeneous systems are often best represented in k-space. Here we give a brief summary of common definitions for pedagogical purposes. For simplicity we write all formulas for spinless particles.

The dimensionsless Fourier transform of the density operator is (Eqs. 1.11 and 1.66 of Giuliani and Vignale)

(2)n_{\mathbf q}
&= \sum_j e^{-i\mathbf{k}\cdot\mathbf{r}_j} \\
&= \sum_{\mathbf{k}} a^\dagger_{\mathbf{k}-\mathbf{q}}a_{\mathbf{k}}.

Note that n_0=N, the total number of particles. To get back to real-space density use

(3)n(\mathbf{r}) = \frac{1}{V}
\sum_{\mathbf{q}} n_{\mathbf{q}} e^{i\mathbf{q}\cdot\mathbf{r}}.

For each of these, we define frequency-dependent density operators,

(4)n(\mathbf{r},i\omega_n)
= \int_0^{\beta\hbar} n(\mathbf{r},\tau) e^{i\omega_n\tau} d\tau,

and

(5)n_{\mathbf{q}}(i\omega_n) = \int_0^{\beta\hbar} n_{\mathbf{q}}
e^{i\omega_n\tau} d\tau,

where i\omega_n = 2\pi ink_BT/\hbar are the Matsubara frequencies. Within the pi-qmc code, these frequency-dependent densities are easily calculated with fast Fourier transforms, which are most efficient when the number of slices is a power of two.

Real-space response

The imaginary-frequency response of the density to an external perturbation is given by (Ch 3.3 of Guiliani and Vignale),

(6)\delta n(\mathbf{r},i\omega_n)
= \int d\mathbf{r} \chi_{nn}(\mathbf{r},\mathbf{r}'', i\omega_n)
V_{\text{ext}}(\mathbf{r}', i\omega_n).

In k-space this takes the convienent form,

(7)\delta n(\mathbf{q}, i\omega_n)
= \sum_{\mathbf{q}'} \chi_{nn}(\mathbf{q}, \mathbf{q}', i\omega_n)
V_{\text{ext}}(\mathbf{q}',i\omega_n).

where the external potential in k-space satisfies

(8)V_{\text{ext}}(\mathbf{r}')
= \frac{1}{V} \sum_{\mathbf{q}'} V_{\text{ext}}(\mathbf{q}')
e^{i\mathbf{q}'\cdot\mathbf{r}'},

and

(9)V_{\text{ext}}(\mathbf{q}') = \int d\mathbf{q}'
e^{-i\mathbf{q}'\cdot\mathbf{r}'} V_{\text{ext}}(\mathbf{r}').

These response functions are related to imaginary-frequency dynamic correlation functions,

(10)\chi_{nn}(\mathbf{r}, \mathbf{r}', i\omega_n)
= -\frac{1}{\beta\hbar^2}
\langle n(\mathbf{r} ,i\omega_n) n(\mathbf{r}',-i\omega_n)\rangle,

and

(11)\chi_{nn}(\mathbf{q}, \mathbf{q}', i\omega_n)
= -\frac{1}{\beta\hbar^2 V} \langle n_{\mathbf{q}}(i\omega_n)
n_{-\mathbf{q}'}(-i\omega_n)\rangle.

For a homogeneous system,

(12)\chi_{nn}(\mathbf{q}, \mathbf{q}',i\omega_n)
= -\frac{1}{\beta\hbar^2 V}
\langle n_{\mathbf{q}}(i\omega_n) n_{-\mathbf{q}}(-i\omega_n)\rangle
\delta_{\mathbf{q}\mathbf{q}'}.

Structure factor

The dynamic structure factor S(k,iωn) measures the density response of the system,

(13)S(\mathbf{k}, i\omega_n) = -\frac{V}{\hbar N}
\chi_{nn}(\mathbf{k}, \mathbf{k}, i\omega_n)

The static structure factor is defined for equal time, not for \omega_n \rightarrow 0,

(14)S(\mathbf{k})
= \frac{1}{N} \langle n_{\mathbf{k}}(\tau=0) n_{-\mathbf{k}}(\tau=0)\rangle.

In terms of \chi_{nn}(\mathbf{q}, \mathbf{q}', i\omega), the static structure factor is given by (prefactor is wrong)

(15)S(\mathbf{k}) = -\frac{V}{\hbar N} \sum_n \omega_n
\chi_{nn}(\mathbf{k}, \mathbf{k}, i\omega_n)
e^{-i\omega_n\tau}.

Polarizability