1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
|
---
title: "Random phase approximation"
sort_title: "Random phase approximation"
date: 2021-12-01
categories:
- Physics
- Quantum mechanics
layout: "concept"
---
Recall that the [self-energy](/know/concept/self-energy/) $$\Sigma$$
is defined as a sum of [Feynman diagrams](/know/concept/feynman-diagram/),
which each have an order $$n$$ equal to the number of interaction lines.
We consider the self-energy in the context of [jellium](/know/concept/jellium/),
so the interaction lines $$W$$ represent Coulomb repulsion,
and we use [imaginary time](/know/concept/imaginary-time/).
Let us non-dimensionalize the Feynman diagrams in the self-energy,
by measuring momenta in units of $$\hbar k_F$$,
and energies in $$\epsilon_F = \hbar^2 k_F^2 / (2 m)$$.
Each internal variable then gives a factor $$k_F^5$$,
where $$k_F^3$$ comes from the 3D momentum integral,
and $$k_F^2$$ from the energy $$1 / \beta$$:
$$\begin{aligned}
\frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}}
\quad\sim\quad
k_F^5
\end{aligned}$$
Meanwhile, every line gives a factor $$1 / k_F^2$$.
The [Matsubara Green's function](/know/concept/matsubara-greens-function/) $$G^0$$
for a system with continuous translational symmetry
is found from [equation-of-motion theory](/know/concept/equation-of-motion-theory/):
$$\begin{aligned}
W(\vb{k}) = \frac{e^2}{\varepsilon_0 |\vb{k}|^2}
\:\:\sim\:\:
\frac{1}{k_F^2}
\qquad \qquad
G_s^0(\vb{k}, i \omega_n^F)
= \frac{1}{i \hbar \omega_n^F - \varepsilon_\vb{k}}
\:\:\sim\:\:
\frac{1}{k_F^2}
\end{aligned}$$
An $$n$$th-order diagram in $$\Sigma$$ contains $$n$$ interaction lines,
$$2n\!-\!1$$ fermion lines, and $$n$$ integrals,
so in total it evolves as $$1 / k_F^{n-2}$$.
In jellium, we know that the electron density is proportional to $$k_F^3$$,
so for high densities we can rest assured that higher-order terms in $$\Sigma$$
converge to zero faster than lower-order terms.
However, at a given order $$n$$, not all diagrams are equally important.
In a given diagram, due to momentum conservation,
some interaction lines carry the same momentum variable.
Because $$W(\vb{k}) \propto 1 / |\vb{k}|^2$$,
small $$\vb{k}$$ make a large contribution,
and the more interaction lines depend on the same $$\vb{k}$$,
the larger the contribution becomes.
In other words, each diagram is dominated by contributions
from the momentum carried by the largest number of interactions.
At order $$n$$, there is one diagram
where all $$n$$ interactions carry the same momentum,
and this one dominates all others at this order.
The **random phase approximation** consists of removing most diagrams
from the defintion of the full self-energy $$\Sigma$$,
leaving only the single most divergent one at each order $$n$$,
i.e. the ones where all $$n$$ interaction lines
carry the same momentum and energy:
{% include image.html file="self-energy.png" width="92%" alt="RPA self-energy definition" %}
Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$,
denoted by a double wavy line:
{% include image.html file="interaction.png" width="95%" alt="RPA screened interaction definition" %}
Rearranging the above sequence of diagrams quickly leads to the following
[Dyson equation](/know/concept/dyson-equation/):
{% include image.html file="dyson.png" width="55%" alt="Dyson equation for screened interaction" %}
In Fourier space, this equation's linear shape
means it is algebraic, so we can write it out:
$$\begin{aligned}
\boxed{
W^\mathrm{RPA}
= W + W \Pi_0 W^\mathrm{RPA}
}
\end{aligned}$$
Where we have defined the **pair-bubble** $$\Pi_0$$ as follows,
with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, and spin $$s$$.
Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$
and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$:
{% include image.html file="pairbubble.png" width="45%" alt="Internal variables of pair-bubble diagram" %}
We isolate the Dyson equation for $$W^\mathrm{RPA}$$,
which reveals its physical interpretation as a *screened* interaction:
the "raw" interaction $$W \!=\! e^2 / (\varepsilon_0 |\vb{k}|^2)$$
is weakened by a term containing $$\Pi_0$$:
$$\begin{aligned}
W^\mathrm{RPA}(\vb{k}, i \omega_n^B)
= \frac{W(\vb{k})}{1 - W(\vb{k}) \: \Pi_0(\vb{k}, i \omega_n^B)}
= \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, i \omega_n^B)}
\end{aligned}$$
Let us evaluate the pair-bubble $$\Pi_0$$ more concretely.
The Feynman diagram translates to:
$$\begin{aligned}
-\hbar \Pi_0(\vb{k}, i \omega_n^B)
&= - \sum_{s} \frac{1}{(2 \pi)^3} \int \frac{1}{\hbar \beta} \sum_{m = -\infty}^\infty
\hbar G_s(\vb{k} \!+\! \vb{q}, i \omega_n^B \!+\! i \omega_m^F) \: \hbar G_s(\vb{q}, i \omega_m^F) \dd{\vb{q}}
\\
&= - \frac{2 \hbar}{(2 \pi)^3} \int \frac{1}{\beta} \sum_{m = -\infty}^\infty
\frac{1}{i \hbar \omega_n^B + i \hbar \omega_m^F - \varepsilon_{\vb{k}+\vb{q}}} \: \frac{1}{i \hbar \omega_m^F - \varepsilon_{\vb{q}}} \dd{\vb{q}}
\end{aligned}$$
Here we recognize a [Matsubara sum](/know/concept/matsubara-sum/),
and rewrite accordingly.
Note that the residues of $$n_F$$ are $$1 / (\hbar \beta)$$
when it is a function of frequency,
and $$1 / \beta$$ when it is a function of energy, so:
$$\begin{aligned}
\Pi_0(\vb{k}, i \omega_n^B)
&= \frac{2}{(2 \pi)^3} \int
\frac{n_F(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B)}{(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B) - \varepsilon_{\vb{q}}}
+ \frac{n_F(\varepsilon_{\vb{q}})}{i \hbar \omega_n^B + (\varepsilon_{\vb{q}}) - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
\\
&= \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})}
{i \hbar \omega_n^B + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
\end{aligned}$$
Where we have used that $$n_F(\varepsilon \!+\! i \hbar \omega_n^B) = n_F(\varepsilon)$$.
Analogously to extracting the retarded Green's function $$G^R(\omega)$$
from the Matsubara Green's function $$G^0(i \omega_n^F)$$,
we replace $$i \omega_n^F \to \omega \!+\! i \eta$$,
where $$\eta \to 0^+$$ is a positive infinitesimal,
yielding the retarded pair-bubble $$\Pi_0^R$$:
$$\begin{aligned}
\boxed{
\Pi_0^R(\vb{k}, \omega)
= \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})}
{\hbar (\omega + i \eta) + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
}
\end{aligned}$$
This is as far as we can go before making simplifying assumptions.
Therefore, we leave it at:
$$\begin{aligned}
\boxed{
W^\mathrm{RPA}(\vb{k}, \omega)
= \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, \omega)}
}
\end{aligned}$$
## References
1. H. Bruus, K. Flensberg,
*Many-body quantum theory in condensed matter physics*,
2016, Oxford.
|
