Seven $\acute{E}tudes$ on dynamical Keldysh Model

D.V. Efremov1 and M.N. Kiselev2*

1 IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany

2 The Abdus Salam International Centre for Theoretical Physics, Strada
Costiera 11, I-34151, Trieste, Italy

*<EMAIL_ADDRESS>
## Abstract

We present a comprehensive pedagogical discussion of a family of models
describing the propagation of a single particle in a multicomponent non-
Markovian Gaussian random field. We report some exact results for single-
particle Green’s functions, self-energy, vertex part and T-matrix. These
results are based on a closed form solution of the Dyson equation combined
with the Ward identity. Analytical properties of the solution are discussed.
Further we describe the combinatorics of the Feynman diagrams for the Green’s
function and the skeleton diagrams for the self-energy and vertex, using
recurrence relations between the Taylor expansion coefficients of the self-
energy. Asymptotically exact equations for the number of skeleton diagrams in
the limit of large $N$ are derived. Finally, we consider possible realizations
of a multicomponent Gaussian random potential in quantum transport via complex
quantum dot experiments.

###### Contents

  1. 1 $Prelude.$ Keldysh model
  2. 2 $Intermezzo.$ Realization of dynamical multi-component Keldysh models with complex quantum dots
  3. 3 $\acute{E}tude$ No 1. Keldysh model with one-component noise
  4. 4 $\acute{E}tude$ No 2. Keldysh model with two-component noise
  5. 5 $\acute{E}tude$ No 3. Keldysh model with three-component noise
  6. 6 $\acute{E}tude$ No 4. Keldysh model with $D\gg 1$-component noise
  7. 7 $\acute{E}tude$ No 5. Perturbative expansion for the bold Green’s function
  8. 8 $\acute{E}tude$ No 6. Enumeration of ”skeleton” Feynman diagrams in a single-component Keldysh model: square recursion
  9. 9 $\acute{E}tude$ No 7. Enumeration of ”skeleton” Feynman diagrams in a multi-component Keldysh model: cubic recursion
  10. 10 $Coda.$ Concluding remarks
  11. 11 $Acknowledgements$
  12. 12 $Mathematical\;\;Instrumentation$

##  1 $Prelude.$ Keldysh model

The original Keldysh Model (KM) was proposed by Keldysh in 1965 [1].

Figure 1: Diagrammatic expansion for the Green’s function. The double line
corresponds to bold Green’s function. The single lines denote bare Green’s
functions. The wavy lines stand for the correlator of the Gaussian classical
random field.

He showed that a single-particle problem of electron’s propagation in a
single-component random Gaussian field $V(r)$ with forward scattering which is
given by the correlator

$D({\bf r}-{\bf r^{\prime}})=\langle V({\bf r})V({\bf
r^{\prime}})\rangle=W^{2}\to D({\bf q})=(2\pi)^{3}W^{2}\delta({\bf q}),$ (1)

can be solved exactly by the Feynman diagrammatic technique.

The idea behind Keldysh’s solution is as follows. According to the
diagrammatics rules, the diagram of order $N$ contains $N$ lines of the
Gaussian random field (denoted by wavy lines), $N+2$ Green’s functions
(denoted by solid lines) and $2N$ vertices [2]. The total number of diagrams
is given by the number $A_{N}$ corresponding to the total number of
possibilities to connect $2N$ vertices with $N$\- lines:

$A_{N}=(2N-1)!!=\frac{(2N-1)!}{2^{N-1}(N-1)!}.$ (2)

An example of the diagrammatic expansion corresponding to the single electron
Green’s function $G$ is shown in Fig.1.

For the chosen Gaussian random field Eq.(1) all Feynman graphs in the order
$N$ give the same contribution to the Green’s function (GF). As a result, the
GF is represented by the following form:

$G(E)=G_{0}(E)\left\\{1+\sum_{N=1}^{\infty}(2N-1)!!G_{0}^{2N}(E)W^{2N}\right\\}.$
(3)

Here we use the shorthand notation $E=\epsilon-{\bf p}^{2}/(2m)$ and
$G_{0}(E)=1/E$ is the bare Green function. Using the representation for the
Euler Gamma-function:

$(2N-1)!!=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dtt^{2N-2}e^{-t^{2}/2}$
(4)

and summing up the geometric progression results in a very simple expression
for the full GF:

$\boxed{G^{R}(E)=\frac{1}{\sqrt{2\pi
W^{2}}}\int_{-\infty}^{\infty}dV\frac{e^{-V^{2}/2W^{2}}}{E-V+i\delta}.}$ (5)

This equation has a very simple physical meaning. Propagation of an electron
in a Gaussian random field corresponds to an ensemble Gaussian averaging of
the single particle Green’s function over Gaussian realizations of the random
potential. This is so-called zero-dimensional realization of the Gaussian
disorder.

The next very important contribution to the physics of the Keldysh model was
made by Efros in [3]. He built a theory of semiconductors with very shallow
charge impurities located at random positions. The potential created by the
charge impurities is the screened Coulomb potential characterized by the
screening radius $r_{0}$. If the electron’s density $n_{0}$ satisfies the
condition $n_{0}r_{0}^{3}\gg 1$, then the electrons experience the potential
only at the point at which they are located. The correlations of the impurity
potential can be represented as a superposition of the short-scale and large-
range parts. The large-scale fluctuations are exactly accounted in the
Gaussian field approximation. The short-range correlations are accounted for
by the perturbation theory. It was shown that the single particle Green’s
function satisfies the Dyson equation which for this model becomes an ordinary
differential equation:

$\displaystyle W^{2}\frac{dG(E)}{dE}+E\cdot G(E)=1.$ (6)

The Dyson equation is obtained using the Ward identity [3], which has a very
simple form due to $\delta({\bf q})$-shape of the Gaussian random potential
correlator (1). One of the direct consequences of the Keldysh-Efros theory is
the emergence of a tail in the single-particle density of states (DOS) at
$\epsilon<0$. The formation of such tails in DOS is one of the general
properties of disordered systems. For a detailed description of the original
Keldysh model and its properties we refer to a remarkable book by M.V.
Sadovskii [4].

Recently, it has been suggested that the Keldysh model can be realized not
only in systems with long-range spatial variations of disordered potentials,
but also in a system characterized by slow temporal fluctuations of the
confining potential [5]. Such behaviour can be realized, for example, in
functional nano-structures [6]. Generalized dynamical Keldysh models pave the
way towards understanding of systems characterized by multi-component Gaussian
non-Markovian random fields. A few-component dynamical Keldysh model have been
proposed in [5, 6, 7, 8] to describe effects of a slow electric gate noise in
double quantum dots. In this paper, we develop the ideas of [5] in connection
with implementation of the multi-component Keldysh model in the complex
quantum dots.

These Notes are organized as follows: In $Intermezzo$ we discuss the
realizations of dynamical (time-dependent) multi-component Keldysh models in
quantum circuits containing multiple quantum dots. We demonstrate that
fluctuations of two potentials: the confinement potential and the inter-nodal
barrier potential, which arise due to fluctuations of the corresponding
electric gates, can create all the necessary ingredients for a multicomponent
Keldysh model. The first $\acute{E}tude$ is a very simple exercise detailing
the ”orthodox” single-component Keldysh model; the second $\acute{E}tude$ uses
the same ”theme” to extend the analysis to a two-component non-Markovian
random Gaussian field; the third $\acute{E}tude$ is devoted to the three-
component model and contains two variations involving two different
realizations of a three component slow noise model; the fourth short
$\acute{E}tude$ summarizes the technique studied in all previous
$\acute{E}tudes$. The fifth $\acute{E}tude$ begins a new melody of Feynman
diagram combinatorics; the sixth and seventh $\acute{E}tudes$, being very
advanced, are devoted to enumeration of skeleton diagrams for the one-
component model ($\acute{E}tude\;6$) including a new derivation of the
Sadovskii - Kuchinskii - Suslov [9, 10] equation and the multi-component
models ($\acute{E}tude\;7$) described by the generalized recurrence equation.
The necessary mathematical instrumentation are presented in the Appendices.

##  2 $Intermezzo.$ Realization of dynamical multi-component
Keldysh models with complex quantum dots

The goal of this Section is to discuss possible realizations of time-dependent
Keldysh models in functional nano-structure devices (see Fig. 2 for an
illustration). The key elements of the electric circuit consist of quantum
dots (see details below), gates controlling the shape of confining potential,
gates controlling the shape of the inter-dot barriers and a global back gate
operating uniformly on the dot’s levels. The nano-devices are connected by the
leads (contacts) to the rest of the quantum circuit.

Figure 2: Experimental realization of the Keldysh model in quantum dots: a)
single QD; b) serial double QD; c) linear (serial) triple QD; d) triangular
triple QD.

### Single QD

We start with a single quantum dot (QD) Fig. 2a which is an island of two- or
three- dimensional electron gas confined by an electrostatic potential (see
Fig. 3). We assume that QD operates in the Coulomb Blockade regime [6].

Figure 3: Left: single quantum dot with the noisy back gate. Right: quantum
well with fluctuating energy level.

There is a global noisy gate attached to QD. This gate slowly changes the
confining potential.

The single-particle Hamiltonian describing the dot (we do not discuss the
contacts attached to it) is given by the following expression:

$H=\left[\epsilon_{0}+\lambda\right]n.$ (7)

Here $n=c^{\dagger}c$ with $c$ being annihilation operator of the single
electron state in QD. For simplicity we consider spinless (spin polarized)
electrons. We refer to [6] for the Coulomb blockade effects. The classical
Gaussian random potential $\lambda$ is defined by its mean value and the two-
point correlation function:

$\langle\lambda(t)\rangle=0,\;\;\;\;\langle\lambda(t)\lambda(t^{\prime})\rangle=D_{1}(t-t^{\prime}).$
(8)

We use a color noise to describe the two-point correlator of the classical
Gaussian noise.

$D_{1}(t-t^{\prime})=W^{2}e^{-\gamma|t-t^{\prime}|},$ (9)

where $\gamma=1/t_{\rm noise}$ defines the characteristic time for the noise
correlation function and $W$ is the amplitude of the noise. There are two
important limits:

$\displaystyle\gamma$ $\displaystyle\to$
$\displaystyle\infty:\;\;\;\;D_{1}(t-t^{\prime})\to\delta(t-t^{\prime}),$ (10)
$\displaystyle\gamma$ $\displaystyle\to$ $\displaystyle
0:\;\;\;\;D_{1}(\omega)\to\delta(\omega).$ (11)

Here $D_{1}(\omega)$ is the Fourier transform of $D_{1}(t-t^{\prime})$. The
equation corresponds to a fast noise (it’s ”fastest” realization is given by
the white noise) and describes Markovian processes. The second equation
corresponds to the a slow noise (it’s ”slowest” realization is known as the
Keldysh model) representing the Gaussian disorder with a long (infinite)
memory. In $\acute{E}tudes$ we focus on a discussion of the physics described
by the Keldysh model.

The single particle Green’s function in given realization of the quasi-static
disorder is:

$\boxed{G^{R}(\epsilon)=\frac{1}{\epsilon-\epsilon_{0}-\lambda+i\delta}.}$
(12)

Averaging over the realizations of the Gaussian disorder can be done
equivalently by means of the distribution function $P(\lambda)=1/\sqrt{2\pi
W^{2}}\exp(-\lambda^{2}/(2W^{2})$.

### Double QD

Figure 4: Slow noise produced by the back gate in double QD. Top: ”diagonal”
noise describing fluctuations of the level positions; Bottom: $U(1)$
fluctuations of the barrier. Three $2\times 2$ Pauli matrices constitute the
basis for the fundamental representation of the $SU(2)$ group.

The single-particle Hamiltonian describing a two-level system (double well
potential, double quantum dot Fig. 2b) consists of two terms: the diagonal
$H_{Z}$ and the off-diagonal (tunneling) part $H_{\rm tun}$. The diagonal part
is defined as follows:

$H_{Z}=\left[(\epsilon_{0}+\lambda)\right]n_{\alpha}+\Delta_{3}S^{z},$ (13)

where $\alpha=l,r$ stand for the left and right quantum well respectively,
$S^{z}=n_{l}-n_{r}$. Thus, the diagonal part consists of a uniform term acting
on the levels of each dot similar to that discussed in the previous subsection
and an alternating term acting similarly to the longitudinal random synthetic
magnetic field (see Fig.4 upper panel) 111There is no real magnetic field in
the model since all fields are associated with noisy electric gates.. Since
the effects of the uniform noise $\lambda$ have been discussed above, in this
subsection we focus on the effects of the synthetic magnetic field. The
correlator of the longitudinal random Gaussian potential is given by:

$\langle\Delta_{3}(t)\rangle=0,\;\;\;\;\langle\Delta_{3}(t)\Delta_{3}(t^{\prime})\rangle=D_{1}(t-t^{\prime}).$
(14)

The off-diagonal (tunneling) Hamiltonian accounting for the tunneling through
the barrier separating the two quantum wells (see Fig.4 lower panel) is given
by:

$H_{\rm tun}=\Delta S^{-}+\Delta^{\ast}S^{+},$ (15)

where $\Delta=\Delta_{1}+i\Delta_{2}$. We introduced the pseudo-spin operators
$S^{+}=c^{\dagger}_{r}c_{l}$, $S^{-}=c^{\dagger}_{l}c_{r}$ such that the
tunneling from one quantum well to another represents the pseudo-spin flip
process. The two-component Gaussian random field describing slow non-Markovian
fluctuations of the barrier is defined as:

$\langle\Delta(t)\rangle=0,\;\;\;\;\langle\Delta(t)\Delta^{\ast}(t^{\prime})\rangle=D_{2}(t-t^{\prime})$
(16)

with

$D_{2}(t-t^{\prime})=2W^{2}e^{-\gamma|t-t^{\prime}|}.$ (17)

Thus, the Hamiltonian describing the double well potential is expressed in
terms of the basis of the Pauli matrices - the fundamental representation of
the group $SU(2)$ 222The sum of the Hamiltonians (13) and (15) can be written
as
$H=\left[(\epsilon_{0}+\lambda)\tau^{0}+\Delta_{3}\tau^{z}+\Delta\tau^{-}+\Delta^{\ast}\tau^{+}\right]_{\alpha\beta}c^{\dagger}_{\alpha}c_{\beta},$
(18) where $\tau^{i}$ are the Pauli matrices ($i=x,y,z$) and $\tau^{0}$ is the
unit $2\times 2$ matrix.  The single particle Green’s function in a given
realization of the off-diagonal disorder (only) is:

$\boxed{G_{l}^{R}(\epsilon)=G_{r}^{R}(\epsilon)=\frac{\epsilon-\epsilon_{0}}{(\epsilon-\epsilon_{0}+i\delta)^{2}-|\Delta|^{2}}.}$
(19)

The lines $|\Delta|^{2}=\Delta_{1}^{2}+\Delta_{2}^{2}=const$ define circles
$S_{1}$ in the $d=2$ parametric space.

The single particle Green’s functions in the presence of both the diagonal and
the off-diagonal disorders are given by:

$\boxed{G_{r/l}^{R}(\epsilon)=\frac{\epsilon-\epsilon_{0}\pm\Delta_{3}}{(\epsilon-\epsilon_{0}+i\delta)^{2}-\Delta_{3}^{2}-|\Delta|^{2}}.}$
(20)

The surfaces
$|\Delta|^{2}+\Delta_{3}^{2}=\Delta_{1}^{2}+\Delta_{2}^{2}+\Delta_{3}^{2}=const$
define spheres $S_{2}$ in the $d=3$ parametric space. When both uniform and
staggered diagonal noises are present in addition to the off-diagonal noise,
the GF in the given realization of quasi-static (infinite memory) Gaussian
random potential casts the form:

$\boxed{G_{r/l}^{R}(\epsilon)=\frac{\epsilon-\epsilon_{0}-\lambda\pm\Delta_{3}}{(\epsilon-\epsilon_{0}-\lambda+i\delta)^{2}-\Delta_{3}^{2}-|\Delta|^{2}}.}$
(21)

### Triple QD

Figure 5: Slow noise produced by the back gate in the serial triple QD. First
and second rows: fluctuations of the level positions in each of three QD.
Third row: symmetric fluctuations of the barriers. Fourth row: anti-symmetric
fluctuations of the barriers. Bottom row: fluctuations of the co-tunneling.
The same processes will naturally appear in the triangular QD Fig. 2d and
describe direct tunneling between the first and the third dot. Eight $3\times
3$ Gell-Mann matrices constitute the basis for the fundamental representation
of the $SU(3)$ group.

Triple quantum dots (TQD) consist of three electron gas islands and either two
(serial TQD, Fig. 2c) or three (triangular TQD, Fig. 2d) barriers separating
the QDs. For the purposes of our $\acute{E}tudes$ we consider a serial TQD
only to avoid a discussion of magnetic flux effects. The triangular TQD will
be considered elsewhere. A cartoon illustrating the triple well potentials is
shown in Fig. 5.

A natural basis for representing single-electron processes in the serial
triple quantum dots is the basis of eight Gell-Mann matrices - the fundamental
representation of the $SU(3)$ group (see Appendix A). The classification of
all possible contributions to the Hamiltonian is shown on Fig. 5. Note, that
we use a ”rotated” representation of the Gell-Mann matrices [11, 12, 13]
different from the conventional $SU(3)$ basis. This basis embeds three
$SU(2)$, $S=1$ generators as the first three Gell-Mann matrices (see Appendix
A). The last five Gell-Mann matrices represent quadrupole moments expressed as
bilinear combinations of $S^{x}$, $S^{y}$ and $S^{z}$ [11, 12, 13, 14].

First, there are two types of the non-uniform diagonal processes described by
two diagonal Gell-Mann matrices $\mu_{3}$ and $\mu_{8}$ (see Appendix A)
333The uniform random potential identically acting on all levels can be
straightforwardly added, as explained in the previous subsection. forming the
Cartan basis:

$H_{\rm
diag}=\left[\epsilon_{0}+\eta_{1}(t)\mu_{3}^{\alpha}+\eta_{2}(t)\mu_{8}^{\alpha}\right]n_{\alpha}.$
(22)

We introduce two scalar random Gaussian fields $\eta_{1}$ and $\eta_{2}$ to
account for the diagonal noise (see first two rows in Fig. 5).

The off-diagonal random processes can be classified as follows:

i) Processes which can be considered as independent tunneling processes in the
double well potential. These terms describe slow symmetric (in-phase)
fluctuations of the two barriers (see the third row in Fig. 5):

$H_{1}=\Delta_{12}S^{-}+\Delta_{12}^{\ast}S^{+}.$ (23)

Here $\Delta_{12}=\Delta_{1}+i\Delta_{2}$, $\hat{S}^{\pm}=(\hat{S}^{x}\pm
i\hat{S}^{y})/\sqrt{2}$ and $S^{+}=c^{\dagger}_{r}c_{c}+c^{\dagger}_{c}c_{l}$,
$(S^{+})^{\dagger}=S^{-}$ (see Appendix A), and we label the states in the dot
$c_{\alpha}$ with $\alpha=l,c,r$ for the left, central and right quantum well
correspondingly.

ii) Processes which can be considered as anti-symmetric (out-of-phase)
fluctuations of the barriers (see the fourth row in Fig. 5).

$H_{2}=\Delta_{34}T^{-}+\Delta_{34}^{\ast}T^{+}.$ (24)

Here $\hat{T}^{\pm}=(\hat{T}^{x}\pm i\hat{T}^{y})/\sqrt{2}$,
$T^{+}=c^{\dagger}_{c}c_{l}-c^{\dagger}_{r}c_{c}$, $(T^{+})^{\dagger}=T^{-}$
(see Appendix A). We defined the two-component random Gaussian field as
$\Delta_{34}=\Delta_{3}+i\Delta_{4}$.

iii) Processes directly connecting the left and right quantum wells (see fifth
and the last row in Fig. 5).

$H_{3}=\Delta_{56}P^{-}+\Delta^{\ast}_{56}P^{+}.$ (25)

Here $\hat{P}^{\pm}=(\hat{P}^{x}\pm i\hat{P}^{y})/2$, and
$P^{+}=c^{\dagger}_{r}c_{l}$, $(P^{+})^{\dagger}=P^{-}$ (see Appendix A). We
define the two-component random Gaussian field as
$\Delta_{56}=\Delta_{5}+i\Delta_{6}$. For a linear TQD such processes require
co-tunneling and therefore are proportional to square of the tunneling matrix
element (and therefore, smaller compared to the off-diagonal processes
described by $H_{1}$ and $H_{2}$). For triangular TQD however, each dot is
connected to its two tunnel partners and therefore there is no additional
smallness in $H_{3}$ tunneling compared to $H_{1}$ and $H_{2}$.

Summarizing, the five gates acting on the TQD (three for the quantum wells and
two for the barriers) create eight random Gaussian fields (two single
component random field for diagonal processes and three two-component random
fields for the off-diagonal processes). The uniform noise gate acting
simultaneously on all levels (not shown on the Fig. 5) can be added
straightforwardly by an extra random field component.

The single electron Green’s function in a particular realization of off-
diagonal disorder given by the Hamiltonians $H_{1}$ and $H_{2}$ is:

$\boxed{G^{R}_{r}(\epsilon)=G^{R}_{l}(\epsilon)=\frac{(\epsilon-\epsilon_{0})^{2}-\frac{1}{2}\left[\Delta_{12}|^{2}+|\Delta_{34}|^{2}+2{\rm
Re}\left\\{\Delta_{12}^{\ast}\Delta_{34}\right\\}\right]}{(\epsilon-\epsilon_{0}+i\delta)\left\\{(\epsilon-\epsilon_{0}+i\delta)^{2}-\left[|\Delta_{12}|^{2}+|\Delta_{34}|^{2}\right]\right\\}}.}$
(26)
$\boxed{G^{R}_{c}(\epsilon)=\frac{\epsilon-\epsilon_{0}}{(\epsilon-\epsilon_{0}+i\delta)^{2}-|\Delta_{12}|^{2}-|\Delta_{34}|^{2}}.}$
(27)

The surfaces
$|\Delta_{12}|^{2}+|\Delta_{34}|^{2}=\Delta_{1}^{2}+\Delta_{2}^{2}+\Delta_{3}^{2}+\Delta_{4}^{2}=const$
define the spheres $S_{3}$ in $d=4$ parametric space.

The most general form of the single-particle Hamiltonian is:

$\boxed{H=\left[\epsilon_{0}\cdot\hat{\mathbb{I}}+\vec{h}\cdot\hat{\vec{M}}\right]_{\alpha\beta}c^{\dagger}_{\alpha}c_{\beta}.}$
(28)

where $\vec{h}$ is the eight component Gaussian random potential (synthetic
magnetic field) and $\hat{\vec{M}}$ is the eight-component vector consisting
of eight Gell-Mann matrices.

These arguments can be repeated for the serial complex Quantum dots consisting
of $M$ gated quantum wells separated by $M-1$ fluctuating barriers. Using the
basis of $SU(M)$ group one straightforwardly obtains the Hamiltonian
describing a particle in $(M^{2}-1)$ \- component random synthetic magnetic
field.

In $\acute{E}tudes$ we analyse the disorder-averaged Green’s function
$\bar{G}(\epsilon)$ for Keldysh models with a multi-component Gaussian random
potential. Since all models discussed in this manuscript are $0+1$ dimensional
with zero stands for spatial dimensions, DOS $\nu(\epsilon)$ is related to the
single particle Green’s functions as follows 444We omit the degeneracy
associated with the spin assuming for example existence of a strong polarizing
magnetic field which lifts out such degeneracy.:

$\nu(\epsilon)=-\frac{1}{\pi}{\rm Im}\bar{G}^{R}(\epsilon).$ (29)

The single particle T-matrix is expressed in terms of the self-energy
$\Sigma(\epsilon)=G^{-1}_{0}-\bar{G}^{-1}(\epsilon)$:

$T(\epsilon)=G^{-1}_{0}(\epsilon)\Sigma(\epsilon)\bar{G}(\epsilon)=\epsilon^{2}\bar{G}(\epsilon)-\epsilon.$
(30)

Knowing the exact form of $\bar{G}^{R}$ gives access to all physical single
particle characteristics of QDs.

##  3 $\acute{E}tude$ No 1. Keldysh model with one-component noise

We start our first $\acute{E}tude$ by considering the original Keldysh model
in the time domain [5]. The model describes, for example, single electron
states in a single well potential (Single Quantum Dot) under the action of a
fluctuating back gate (see the the $Intermezzo$). The ”scalar” single
component classical Gaussian random field (noise) is associated with the slow
fluctuation of the confining potential. For the purposes of our notes we
assume that the characteristic time associated with the change of the
confining potential is greater than any other characteristic time of the
system (e.g. Zener time, dwell time etc). Since we are dealing with single
particle physics, statistics do not play any role. We will consider Fermi and
Bose many-body models in a separate publication. One of the goals of our
$\acute{E}tudes$ is to demonstrate the properties of the model and the
response functions. To achieve this we will allow ourselves to show sketches
of derivation and providing some minimal details of our calculations.

The single particle Green’s function (GF) is defined as follows 555To simplify
our notations we omit bar in the notations of the Green’s function. Everywhere
below we denote by $G$ the disorder averaged GF.:

$G_{1}^{-1}(\epsilon)=\epsilon-\Sigma_{1}(\epsilon)=G_{0}^{-1}(\epsilon)-\Sigma_{1}(\epsilon).$
(31)

Here the index (subscript) ”1” refers to the one-component model. The inverse
bare GF is $G_{0}^{-1}(\epsilon)=\epsilon$ and $\Sigma_{1}(\epsilon)$ is the
self-energy part. As already mentioned, the classical Gaussian noise
correlator is

$D_{1}(\omega)=2\pi W^{2}\delta(\omega).$ (32)

is quasi-static as the characteristic noise time $t_{\rm noise}\to\infty$.
Parameter $W^{2}$ is the variance of the Gaussian distribution. The self-
energy (SE) part is defined by the following equation

$\Sigma_{1}(\epsilon)=\epsilon-
G_{1}^{-1}(\epsilon)=W^{2}G_{1}(\epsilon)\Gamma_{1}(\epsilon).$ (33)

The first and second order Feynman diagrams for the self-energy are shown on
Figure 6 a-c. SE is connected to the ”triangular” vertex $\Gamma_{1}$ (Fig. 6
d-e).

The Ward Identity (WI)

$\Gamma_{1}(\epsilon)=\frac{dG_{1}^{-1}(\epsilon)}{d\epsilon}=1-\frac{d\Sigma_{1}(\epsilon)}{d\epsilon}$
(34)

establishes an explicit relation between $\Gamma_{1}$ and $\Sigma_{1}$ as the
”transmitted” through the wavy line frequency is zero.

Combining the Dyson equation (31 \- 33) with the Ward Identity (34)

$\displaystyle W^{2}\frac{dG_{1}(\epsilon)}{d\epsilon}+\epsilon
G_{1}(\epsilon)$ $\displaystyle=$ $\displaystyle 1,$ (35) $\displaystyle
W^{2}\frac{d\Sigma_{1}(\epsilon)}{d\epsilon}+\epsilon\Sigma_{1}(\epsilon)-\Sigma^{2}_{1}(\epsilon)$
$\displaystyle=$ $\displaystyle W^{2}$ (36)

we get a closed set of linear first order ordinary differential equations for
GF (35) and a non-linear first order Riccati-type differential equation (36)
for SE. 666Note that both equations originate from the Dyson equation. The
equation for $\Sigma_{1}$ does not contain any new information. Therefore, it
is sufficient to solve just one equation (e.g. for $G_{1}$) and use a non-
linear connection between $G_{1}$ and $\Sigma_{1}$ to obtain the self-energy
part. However in this $\acute{E}tude$ we show how to solve each equation to
demonstrate the existence of a non-trivial differential identity between GF
and SE.

Figure 6: Self-energy part diagrams a) first order perturbation theory; b) and
c) second order of perturbation theory; d) first order vertex correction; e)
bold self-energy part. Single straight line denoted bare Green’s function,
double solid line denotes bold (dressed) Green’s function. Wavy line stands
for the Gaussian classical random field correlator. Bold triangle denotes the
full vertex $\Gamma$.

Boundary condition

$G_{1}(\epsilon\to\infty)=\frac{1}{\epsilon}$ (37)

determines the large frequency behaviour of GF.

It is convenient to introduce the following dimensionless variables and
functions:

$\displaystyle
G_{1}(\epsilon)=\frac{1}{W}\Psi_{1}\left(z=\frac{\epsilon}{W}\right),\;\;\;\;\;\Sigma_{1}(\epsilon)=W\sigma_{1}\left(z=\frac{\epsilon}{W}\right).$
(38)

As a result, the dimensionless equations for GF and SE

$\boxed{\frac{d\Psi_{1}(z)}{dz}+z\Psi_{1}(z)=1,}$ (39)
$\boxed{\frac{d\sigma_{1}(z)}{dz}+z\sigma_{1}(z)-\sigma_{1}^{2}(z)=1}$ (40)

are accompanied by the boundary conditions:

$\displaystyle\Psi_{1}(z\to\infty)=\frac{1}{z},\;\;\;\;\;\;\;\sigma_{1}(z\to\infty)=\frac{1}{z}.$
(41)

The non-linear Riccati equation (40) can be transferred to a linear second
order ordinary differential equation by substitution

$\sigma_{1}(z)=-\frac{1}{\Phi_{1}(z)}\frac{d\Phi_{1}(z)}{dz}=-\frac{d}{dz}\log\left[\Phi_{1}(z)\right].$
(42)

Corresponding boundary condition for the new function $\Phi_{1}(z)$ is

$\Phi_{1}(z\to\infty)=\frac{1}{z}.$ (43)

The equation for $\Phi_{1}(z)$ reads as follows:

$\frac{d^{2}\Phi_{1}(z)}{dz^{2}}+z\frac{d\Phi_{1}(z)}{dz}+\Phi_{1}(z)=0,$ (44)

or, equivalently,

$\frac{d}{dz}\left[\frac{d\Phi_{1}(z)}{dz}+z\Phi_{1}(z)\right]=0.$ (45)

Using the boundary condition Eq.(43) we re-write the equations in the
following form:

$\frac{d\Phi_{1}(z)}{dz}+z\Phi_{1}(z)=1$ (46)

and therefore $\Phi_{1}(z)\equiv\Psi_{1}(z)$. Finally, the Ward Identity casts
the form

$\displaystyle\sigma_{1}(z)=-\frac{d}{dz}\log\left[\Psi_{1}(z)\right],\;\;\;\;\;\Gamma_{1}(z)=1-\frac{d\sigma_{1}(z)}{dz}=1+\frac{d^{2}}{dz^{2}}\log\left[\Psi_{1}(z)\right],$
(47)

or, equivalently,

$\displaystyle\sigma_{1}(z)=z-\frac{1}{\Psi_{1}(z)},\;\;\;\;\;\;\Gamma_{1}(z)=\frac{\sigma_{1}(z)}{\Psi_{1}(z)}=\frac{z\Psi_{1}(z)-1}{\Psi_{1}^{2}(z)}.$
(48)

Obviously, these equations can be obtained directly from the definition of
$\sigma_{1}(z)$, the Ward identity and Eq.(39).

### Analytic properties

This subsection is devoted to a discussion of the analytic properties of the
single particle Green’s function.

Figure 7: Left: ${\rm Re}[\Psi^{R}_{1}(z)]$ (red curve), ${\rm
Im}[\Psi^{R}_{1}(z)]$ (blue curve). Right: ${\rm Re}[\sigma^{R}_{1}(z)]$ (red
curve), ${\rm Im}[\sigma^{R}_{1}(z)]$ (blue curve).

The retarded dimensionless Green’s function $\Psi_{1}^{R}(z)$ can be obtained
with the help of the Weierstrass transformation:

$\displaystyle\Psi^{R}_{1}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}du\psi^{R}_{1}(u-z)e^{-u^{2}/2}.$
(49)

Substituting it to the Dyson equation we get:

$\displaystyle\Psi_{1}^{R}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}du\frac{e^{-u^{2}/2}}{z-u+i\delta}.$
(50)

The imaginary part ${\rm Im}\Psi_{1}^{R}(z)$ can be easily computed:

$\boxed{{\rm
Im}\Psi_{1}^{R}(z)=-\sqrt{\frac{\pi}{2}}e^{-z^{2}/2}\xrightarrow[]{z\to
0}-\sqrt{\frac{\pi}{2}}\left[1-\frac{z^{2}}{2}+O(z^{4})\right].}$ (51)

The real part ${\rm Re}\Psi_{1}^{R}(z)$ can be determined by the Kramers -
Kronig relations

${\rm Re}\Psi_{1}^{R}(z)=\frac{1}{\pi}{\rm
P.V.}\int_{-\infty}^{\infty}du\frac{{\rm Im}\Psi_{1}^{R}(u)}{u-z}.$ (52)

Here P.V. denotes Cauchy principle value of the integral. The above
transformation is connected to the Hilbert transform (HT) of the Gaussian
function $H[e^{-u^{2}}](z)=H_{0}(z)$ [15] as shown in Appendix B:

$H_{0}(z)=\frac{1}{\pi}{\rm
P.V.}\int_{-\infty}^{\infty}\frac{e^{-u^{2}}}{z-u}du.$ (53)

Evaluating Eq. (53) at $z=0$ we obtain the obvious property ${\rm
Re}\Psi_{1}^{R}(z=0)=0$ which can be used as a new initial condition for the
linear ordinary differential equation for the function $\psi_{1}(z)={\rm
Re}\Psi_{1}^{R}(z)$ 777As, the Green’s function $\Psi_{1}$ satisfies a linear
differential equation, both real and imaginary parts of $\Psi_{1}$ also
satisfy linear differential equation. The equation for $\phi_{1}(z)={\rm
Im}\Psi_{1}^{R}(z)$ reads as
$\frac{d\phi_{1}}{dz}+z\phi_{1}=0,\;\;\;\;\phi_{1}(0)=-\sqrt{\frac{\pi}{2}}.$
(54) :

$\frac{d\psi_{1}}{dz}+z\psi_{1}=1,\;\;\;\;\psi_{1}(0)=0.$ (55)

The solution of this equation is obtained by the method of indefinite
coefficients and finally reads as:

$\psi_{1}(z)=C\cdot e^{-z^{2}/2}+e^{-z^{2}/2}\int_{0}^{z}e^{u^{2}/2}du.$ (56)

Implementing the initial condition $\psi_{1}(0)=0$ we obtain $C=0$. Finally,
${\rm Re}\Psi_{1}^{R}(z)$ can be expressed in terms of the Dawson function
$D_{+}(z)=F(z)$ [16]:

${\rm
Re}\Psi_{1}^{R}(z)=\sqrt{2}F\left(\frac{z}{\sqrt{2}}\right),\;\;\;\;F(z)=\frac{\sqrt{\pi}}{2}e^{-z^{2}}{\rm
erfi}(z),$ (57)

where ${\rm erfi}(z)$ is imaginary error function [17, 18].

One can represent the Dawson function as the $\sin$\- Fourier-Laplace
transform of the Gaussian function:

$F(z)=\frac{1}{2}\int_{0}^{\infty}dse^{-s^{2}/4}\sin zs.$ (58)

The Dawson function can be expanded as follows for small values of the
argument $z\to 0$:

$F(z\to 0)=\sum_{n=0}^{\infty}\frac{(-1)^{n}2^{n}}{(2n+1)!!}z^{2n+1}$ (59)

and in the limit $z\to\infty$ the expansion reads:

$F(z\to\infty)=\sum_{n=0}^{\infty}\frac{(2n-1)!!}{2^{n+1}z^{2n+1}}.$ (60)

Keeping first few terms of the Dawson function for small/large values of its
argument, we get:

$\displaystyle{\rm
Re}\Psi_{1}^{R}(z)=\sqrt{2}F\left(\frac{z}{\sqrt{2}}\right)$
$\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle\left[z-\frac{1}{3}z^{3}+O\left(z^{5}\right)\right]\;\;\;\;$
(61) $\displaystyle\xrightarrow[]{z\to\infty}$
$\displaystyle\left[\frac{1}{z}+\frac{1}{z^{3}}+O\left(\frac{1}{z^{5}}\right)\right].\;\;\;\;$
(62)

Obviously, since GF is purely imaginary in the zero frequency limit
$\Psi_{1}^{R}(0)=-i\sqrt{\frac{\pi}{2}}$, the modulus square is
$|\Psi_{1}^{R}(0)|^{2}=\frac{\pi}{2}$. We plot the frequency dependence of the
${\rm Re}\Psi_{1}$ and ${\rm Im}\Psi_{1}$ in the left panel of Fig.7 888The
real-time GF (dimensionless time $s=t\cdot W$) is defined through the Fourier
transformation:
$\Psi_{1}^{R}(s)=\int_{-\infty}^{\infty}\frac{dz}{2\pi}\Psi_{1}^{R}(z)e^{-izs}=-ie^{-s^{2}/2}\Theta(s).$
(63) Here $\Theta(s)$ is a step (Heaviside) function. The real time GF
satisfies the differential equation
$\frac{d\Psi_{1}^{R}(s)}{ds}+s\cdot\Psi_{1}^{R}(s)=-i\delta(s).$ (64) One can
notice a duality of Dyson equations in the time and frequency domain. This
duality is associated with the Gaussian form of the random field..

Substituting the Taylor expansion for $\Psi_{1}^{R}(z)$ into the equation for
the self-energy

$\sigma_{1}^{R}(z)=z-\frac{{\rm Re}\Psi_{1}^{R}(z)-i{\rm
Im}\Psi_{1}^{R}(z)}{|\Psi_{1}^{R}(z)|^{2}}$ (65)

we obtain the asymptotic behaviour of the self-energy at $z\to 0$

$\displaystyle{\rm Re}\sigma^{R}_{1}(z)$ $\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle\left(1-\frac{2}{\pi}\right)z+\frac{4(\pi-3)z^{3}}{3\pi^{2}}+O\left(z^{4}\right),$
(66) $\displaystyle{\rm Im}\sigma^{R}_{1}(z)$
$\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle-\sqrt{\frac{2}{\pi}}-\frac{(\pi-4)z^{2}}{\sqrt{2}\pi^{3/2}}+O\left(z^{4}\right).$
(67)

In the $z\to\infty$ limit

$\displaystyle{\rm
Re}\sigma^{R}_{1}(z)\xrightarrow[]{z\to\infty}\frac{1}{z}+\frac{2}{z^{3}}+O\left(\frac{1}{z^{5}}\right).$
(68)

The frequency dependencies of ${\rm Re}\sigma_{1}$ and ${\rm Im}\sigma_{1}$
are shown in the right panel of Fig.7. The DOS for the single component
Keldysh model is represented by a single Gaussian-shaped peak centred at
$z=0$.

##  4 $\acute{E}tude$ No 2. Keldysh model with two-component noise

The two-component time-dependent Keldysh model describes, for example, the
single electron states in a double well potential under action of slow
fluctuations of the barrier separating the wells (see the $Intermezzo$) [5].
The complex tunnel matrix element has two independent components representing
two Gaussian random fields (noise) in the x- and y- pseudospin directions.

The single particle Green’s function of the two-component time dependent
Keldysh model is defined as follows

$G_{2}^{-1}(\epsilon)=\epsilon-\Sigma_{2}(\epsilon)=G_{0}^{-1}(\epsilon)-\Sigma_{2}(\epsilon),$
(69)

where $G_{0}^{-1}(\epsilon)=\epsilon$ is the inverse barer GF. The two-
component Gaussian random field correlator

$D_{2}(\omega)=2\times 2\pi W^{2}\delta(\omega)$ (70)

describes the two-component slow noise. For simplicity, we assume that two
independent amplitudes of the noise coincide. Feynman diagrams for the Green’s
function before Gaussian averaging are shown on Fig. 8. At each vertex of the
diagram the ”pseudo-spin” is flipped and therefore black and white vertices
form the staggered pattern. Dashed lines denote the Gaussian random potential.

The self-energy part is proportional to the product of the bold GF $G_{2}$ and
the triangular vertex $\Gamma_{2}$

$\Sigma_{2}(\epsilon)=\epsilon-
G_{2}^{-1}(\epsilon)=W^{2}G_{2}(\epsilon)\Gamma_{2}(\epsilon).$ (71)

Several first irreducible diagrams for the self-energy part are shown in Fig.
9. Each ”black” vertex is connected only to a ”white” vertex and vice-versa.
As a result, there exists no second order irreducible diagram 999The reducible
second order diagram and other reducible diagrams in all orders result in
replacement of the bare GF in diagram Fig. 9 to bold GF.

Before we derive the differential equation for $G_{2}$ we show that GF can be
obtained by direct re-summation of the Feynman diagrams of the ”formal”
perturbation theory

$G^{R}_{2}(\epsilon)=G_{0}(\epsilon)+\sum_{n=1}^{\infty}A^{(2)}_{n}(W)^{2n}G_{0}^{2n+1}(\epsilon),$
(72)

where the coefficients $A^{(2)}_{n}=(2n)!!$ reflect the fact that all diagrams
at a given order of the perturbation theory have the same value. To proceed
with the re-summation we use the integral representation for the Euler’s
Gamma-function. As the next step, we compute explicitly the sum over $n$ to
transform the integral to the form

$G^{R}_{2}(\epsilon)=\int_{0}^{\infty}2tdt\frac{G_{0}(\epsilon)}{1-2t^{2}W^{2}G_{0}^{2}(\epsilon)}e^{-t^{2}}.$
(73)

As a result, the Green’s function is expressed in terms of the double
integrals:

$\displaystyle
G^{R}_{2}(\epsilon)=\frac{1}{2}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{dxdye^{-(x^{2}+y^{2})/2W^{2}}}{2\pi
W^{2}}\left[\frac{1}{\epsilon-\sqrt{x^{2}+y^{2}}+i\delta}+\frac{1}{\epsilon+\sqrt{x^{2}+y^{2}}+i\delta}\right].$
(74)

Transforming this integral from teh ”Cartesian” coordinates to the ”polar”
coordinate system one gets:

$G^{R}_{2}(\epsilon)=\int_{0}^{\infty}\frac{udu}{2W^{2}}\left(\frac{1}{\epsilon-u+i\delta}+\frac{1}{\epsilon+u+i\delta}\right)e^{-u^{2}/2W^{2}}.$
(75)

Due to the $SO(2)$ symmetry of the noise the integral is independent on the
polar angle. Here we would like to note that in fact the Eq. (75) describes an
averaging of GF with the known Rayleigh distribution. Finally, by
differentiating Eq. (75) with respect to the frequency $\epsilon$ we obtain a
linear differential equation for the single particle Green’s function.
101010To simplify our notations we omit below the index $R$ (retarded) in the
superscript of the GF. Basically, it means that we will be dealing with the
real part of GF. We restore the index $R$ in the next subsection when
considering analytic properties of GF and SE.

$W^{2}\frac{dG_{2}(\epsilon)}{d\epsilon}+\epsilon
G_{2}(\epsilon)\left(1-\frac{W^{2}}{\epsilon^{2}}\right)=1.$ (76)

The same equation can be derived with the help of the Ward identity

$\Gamma_{2}(\epsilon)=\frac{1}{\epsilon}\frac{d}{d\epsilon}\left[\epsilon
G_{2}^{-1}(\epsilon)\right].$ (77)

The corresponding Dyson equation for the self-energy functions reads as
follows:

Figure 8: Feynman diagram for the GF before taking averaging over two-
component Gaussian random field.

$\displaystyle
W^{2}\frac{d\Sigma_{2}(\epsilon)}{d\epsilon}+W\left[\frac{\epsilon}{W}+\frac{W}{\epsilon}\right]\Sigma_{2}(\epsilon)-\Sigma^{2}_{2}(\epsilon)=2W^{2}.\;\;\;\;\;$
(78)

The boundary condition reads:

$G_{2}(\epsilon\to\infty)=\frac{1}{\epsilon},\;\;\;\;\;\;\;\;\Sigma_{2}(\epsilon\to\infty)=\frac{2}{\epsilon}.$
(79)

Figure 9: Irreducible self-energy part diagram in the first a) and third b)
orders of the perturbation theory. Second order diagram is topological
impossible. c) first non-trivial diagram for the vertex.

Similarly to the first $\acute{E}tude$ we introduce dimensionless variables
and functions:

$\displaystyle
G_{2}(\epsilon)=\frac{1}{W}\Psi_{2}\left(z=\frac{\epsilon}{W}\right),\;\;\;\;\;\;\Sigma_{2}(\epsilon)=W\sigma_{2}\left(z=\frac{\epsilon}{W}\right).$
(80)

As a result, the dimensionless equations cast the form:

$\boxed{\frac{d\Psi_{2}(z)}{dz}+\left[z-\frac{1}{z}\right]\Psi_{2}(z)=1,}$
(81)
$\boxed{\frac{d\sigma_{2}(z)}{dz}+\left[z+\frac{1}{z}\right]\sigma_{2}(z)-\sigma_{2}^{2}(z)=2.}$
(82)

being accompanied by the corresponding boundary conditions:

$\displaystyle\Psi_{2}(z\to\infty)=\frac{1}{z},\;\;\;\;\;\sigma_{2}(z\to\infty)=\frac{2}{z}.$
(83)

It can be seen that in contrast the the one-component Keldysh model, an
additional $1/z$ ”centrifugal” term appears in the Dyson equation. As we will
see below, this term is responsible for level repulsion.

The boundary condition for $\sigma_{2}$ follows from the Ward Identity and the
boundary condition for $\Psi_{2}(z)$:

$\left.\Gamma_{2}(z)\right|_{z\to\infty}=\frac{1}{z}\frac{d}{dz}\left[z\Psi^{-1}_{2}(z)\right]=\frac{\Psi^{-1}_{2}(z)}{z}+\frac{d\Psi^{-1}_{2}(z)}{dz}=2.$
(84)

Therefore we have

$\sigma_{2}(z)=\Psi_{2}(z)\Gamma_{2}(z)\xrightarrow[]{z\to\infty}\frac{2}{z}$
(85)

The solution of Eq. 81 with the boundary condition Eq. (83) is given by

$\Psi_{2}(z)=\frac{1}{2}\int_{0}^{\infty}ue^{-u^{2}/2}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right].$
(86)

The non-linear equation (82) for the SE is a Riccati-type equation.
Substituting

$\sigma_{2}(z)=-\frac{1}{\Phi_{2}(z)}\frac{d\Phi_{2}(z)}{dz}=-\frac{d}{dz}\log\left[\Phi_{2}(z)\right]$
(87)

and taking into account the boundary condition Eq. (83) we get the
corresponding boundary condition for the auxiliary function $\Phi_{2}$

$\Phi_{2}(z\to\infty)=\frac{1}{z^{2}}.$ (88)

The second-order linear ordinary differential equation (ODE) for $\Phi_{2}(z)$
reads as follows:

$\displaystyle\frac{d^{2}\Phi_{2}(z)}{dz^{2}}+\left[z+\frac{1}{z}\right]\frac{d\Phi_{2}(z)}{dz}+2\Phi_{2}(z)=0,$
(89)

which can be re-written in more compact form:

$\frac{d}{dz}R(z)+\frac{1}{z}R(z)=0.$ (90)

Here we introduced a new function

$R(z)=\frac{d\Phi_{2}(z)}{dz}+z\Phi_{2}(z).$ (91)

The corresponding boundary condition for $R(z)$ reads as:

$R(z\to\infty)=\frac{1}{z}.$ (92)

The general solution of the equation (90) is $R(z)=A/z$ and the boundary
condition (92) defines the constant $A=1$.

Now we use the definition of $R(z)$ to obtain a first order linear ODE for
$\Phi_{2}(z)$:

$\frac{d\Phi_{2}(z)}{dz}+z\Phi_{2}(z)=\frac{1}{z}.$ (93)

Substituting $\Phi_{2}(z)=\frac{\Theta_{2}(z)}{z}$ we obtain the ODE for
$\Theta_{2}$:

$\frac{d\Theta_{2}(z)}{dz}+\left[z-\frac{1}{z}\right]\Theta_{2}(z)=1$ (94)

with the boundary condition

$\Theta_{2}(z\to\infty)=\frac{1}{z}.$ (95)

By comparing Eq. (94) with Eq. (81) we conclude that
$\Theta_{2}(z)\equiv\Psi_{2}(z)$. In addition we obtain new differential
identities:

$\displaystyle\sigma_{2}(z)=-\frac{d}{dz}\log\left[\frac{\Psi_{2}(z)}{z}\right],\;\;\;\;\Gamma_{2}(z)=1-\frac{d\sigma_{2}(z)}{dz}=1+\frac{d^{2}}{dz^{2}}\log\left[\frac{\Psi_{2}(z)}{z}\right].$
(96)

The connection between the functions $\sigma_{2}(z)$, $\Gamma_{2}(z)$ and
$\Psi_{2}(z)$ is given by the equations identical to Eq. (48):

$\displaystyle\sigma_{2}(z)=\frac{z\Psi_{2}(z)-1}{\Psi_{2}(z)},\;\;\;\;\;\Gamma_{2}(z)=\frac{z\Psi_{2}(z)-1}{\Psi_{2}^{2}(z)}.$
(97)

### Analytic properties

Figure 10: Left: ${\rm Re}[\Psi^{R}_{2}(z)]$ (red curve), ${\rm
Im}[\Psi^{R}_{2}(z)]$ (blue curve). Right: ${\rm Re}[\sigma^{R}_{2}(z)]$ (red
curve), ${\rm Im}[\sigma^{R}_{2}(z)]$ (blue curve).

The retarded dimensionless Green’s function $\Psi_{2}^{R}(z)$ is given by

$\Psi^{R}_{2}(z)=\frac{1}{2}\int_{0}^{\infty}ue^{-u^{2}/2}du\left[\frac{1}{z-u+i\delta}+\frac{1}{z+u+i\delta}\right].$
(98)

The imaginary part ${\rm Im}\Psi_{2}^{R}(z)$ can be easily computed as:

$\boxed{{\rm Im}\Psi_{2}^{R}(z)=-\frac{\pi}{2}z\;{\rm
sgn}\;z\;e^{-z^{2}/2}\xrightarrow[]{z\to 0}-\frac{\pi}{2}z\;{\rm
sgn}\;z\;\left[1-\frac{z^{2}}{2}+O(z^{4})\right].}$ (99)

The identity ${\rm Re}\Psi_{2}^{R}(z=0)=0$ follows directly from the analytic
properties of $\Psi_{2}$. The function $\psi_{2}(z)={\rm Re}\Psi_{2}^{R}(z)$
satisfies the ordinary linear differential equation:

$\frac{d\psi_{2}}{dz}+\left[z-\frac{1}{z}\right]\psi_{2}=1,\;\;\;\;\psi_{2}(0)=0.$
(100)

The homogeneous solution is given by:

$\psi_{2}^{(h)}=C\cdot z\;e^{-z^{2}/2}.$ (101)

The particular solution is obtained by the method of indefinite coefficients
which leads to another differential equation for the function $C(z)$:

$\frac{dC(z)}{dz}=\frac{e^{z^{2}/2}}{z}.$ (102)

Finally, the solution of the boundary problem is given by:

$\psi_{2}(z)=\frac{1}{2}z\;e^{-z^{2}/2}{\rm Ei}\left[z^{2}/2\right].$ (103)

Here ${\rm Ei}(u)$ is the exponential integral function [17, 18]. The real
part ${\rm Re}\Psi_{2}^{R}(z)$ is expressed through the Hilbert transform
$H_{1}[ue^{-u^{2}}]$ (see details in Appendix B):

$\displaystyle{\rm Re}[\Psi^{R}_{2}(z)]=\frac{1}{2}{\rm
P.V.}\int_{0}^{\infty}ue^{-u^{2}/2}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right]=\frac{1}{2}z\;e^{-z^{2}/2}{\rm
Ei}\left[z^{2}/2\right]\;\;\;\;$ (104) $\displaystyle\xrightarrow[]{z\to
0}z\log|z|+\frac{1}{2}\left[\gamma-\log 2\right]z+O(z^{3}),$ (105)

where $\gamma$ is the Euler constant, $|\Psi_{2}(z\to 0)|^{2}\propto
z^{2}\log^{2}[z]$. Note, the ${\rm Im}\Psi_{2}^{R}(z\to 0)\propto|z|$ is not
differentiable at $z=0$. Furthermore, ${\rm Re}\Psi_{2}^{R}(z\to 0)\propto
z\log|z|$ is not analytic in the limit $z\to 0$ and have a branch cut. So GF
cannot be decompose into a Taylor expansion in the limit $z\to 0$. The
frequency dependencies of the real and imaginary parts of GF $\Psi_{2}$ are
shown in the left panel of Fig. 10 111111Note that the real-time GF obtained
by $\sin$ \- Fourier transform can be written in terms of the Dawson function:
$\psi_{2}(s)=\int_{-\infty}^{\infty}dz\psi_{2}(z)\sin
zs=\pi\left(1-s\sqrt{2}F\left(\frac{s}{2}\right)\right).$ (106) .

The self-energy is diverging and non-analytic in the limit $z\to 0$:

$\displaystyle{\rm Re}\sigma^{R}_{2}(z)$ $\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle-\frac{1}{z\log|z|},$ (107) $\displaystyle{\rm
Im}\sigma^{R}_{2}(z)$ $\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle\frac{\pi}{2|z|\log^{2}|z|}.$ (108)

The asymptotic behaviour of GF in the limit $z\to\infty$ is:

$\displaystyle{\rm
Re}[\Psi^{R}_{2}(z)]\xrightarrow[]{z\to\infty}\frac{1}{z}+\frac{2}{z^{3}}+O\left(\frac{1}{z^{5}}\right)$
(109)

and $\sigma_{2}(z\to\infty)\to 2/z+O(1/z^{3})$. The frequency dependencies of
the real and imaginary parts of the self-energy $\sigma_{2}$ are shown on the
right panel of Fig. 10.

The DOS for the two -component Keldysh model is represented by two Gaussian
peaks separated by a pseudo-gap with linear energy dependence in the vicinity
$z=0$.

##  5 $\acute{E}tude$ No 3. Keldysh model with three-component noise

We start this $\acute{E}tude$ by considering a Double Quantum Dot realization
of the Keldysh model. We assume that there exists a random Gaussian field
proportional to $\tau_{x}$, $\tau_{y}$ Pauli matrices, which is associated
with the barrier fluctuations, and an additional alternating diagonal random
Gaussian field acting on the levels in each dot. The last field is
proportional to $\tau_{z}$. A realization of the alternating diagonal noise is
done by applying two separate back gates acting to each QD. All three
generators of SU($2$) group are contained in the Hamiltonian. Without losing
generality, we consider the case of equal noise amplitudes similarly to
previous $\acute{E}tude$. Breaking the rotational symmetry is discussed in
$Variation\;No.\;1$. Another realization of the three component random field
can be implemented when barrier fluctuations are accompanied by a uniform
random Gaussian field acting on the levels of each QD through a single back
gate applied to both QD in the double well potential. This noise is
proportional to the unit matrix $\tau_{0}$. We discuss it in
$Variation\;No.\;2$.

The general equations relating the dimensionless SE $\sigma_{3}(z)$ and the
dimensionless triangular vertex $\Gamma_{3}(z)$ with the single particle GF
$\Psi_{3}(z)$ are:

$\displaystyle\sigma_{3}(z)=\frac{z\Psi_{3}(z)-1}{\Psi_{3}(z)},\;\;\;\;\;\;\Gamma_{3}(z)=\frac{z\Psi_{3}(z)-1}{\Psi_{3}^{2}(z)}.$
(110)

These equations are obtained directly from the definition of $\sigma_{3}(z)$
and the Ward Identity

$\boxed{\sigma_{3}(z)=\Psi_{3}(z)\Gamma_{3}(z),\;\;\;\;\;\Gamma_{3}(z)=\frac{1}{z^{2}}\frac{d}{dz}\left[z^{2}\Psi^{-1}_{3}(z)\right].}$
(111)

The differential equation for the $\Psi_{3}(z)$ and $\sigma_{3}(z)$ and the
boundary conditions are given by

$\boxed{\frac{d\Psi_{3}(z)}{dz}+\left[z-\frac{2}{z}\right]\Psi_{3}(z)=1,\;\;\;\;\Psi_{3}(z\to\infty)=\frac{1}{z},}$
(112)
$\boxed{\frac{d\sigma_{2}(z)}{dz}+\left[z+\frac{2}{z}\right]\sigma_{2}(z)-\sigma_{2}^{2}(z)=3.}$
(113)

The solution of the linear ODE for $\Psi_{3}(z)$ casts the form:

$\Psi_{3}(z)=\frac{1}{2\Omega_{3}}\int_{0}^{\infty}u^{2}e^{-u^{2}/2}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right].$
(114)

Here $\Omega_{3}=\sqrt{\pi/2}$ is the normalizing coefficient. Equivalently,
the solution can be written in terms of the Hilbert transform
$H[u^{2}e^{-u^{2}/2}](z)$ (see Appendix B) as:

$\Psi_{3}(z)=\frac{1}{2\Omega_{3}}\int_{-\infty}^{\infty}du\frac{u^{2}e^{-u^{2}/2}}{z-u}.$
(115)

Less trivial identities

$\boxed{\sigma_{3}(z)=-\frac{d}{dz}\log\left[\frac{\Psi_{3}(z)}{z^{2}}\right],\;\;\;\;\;\Gamma_{3}(z)=1-\frac{d\sigma_{3}(z)}{dz}=1+\frac{d^{2}}{dz^{2}}\log\left[\frac{\Psi_{3}(z)}{z^{2}}\right].}$
(116)

are obtained by solving a non-linear Riccati-type equation for
$\sigma_{3}(z)$.

### Analytic properties

Figure 11: Left: ${\rm Re}[\Psi^{R}_{3}(z)]$ (red curve), ${\rm
Im}[\Psi^{R}_{3}(z)]$ (blue curve). Right: ${\rm Re}[\sigma^{R}_{3}(z)]$ (red
curve), ${\rm Im}[\sigma^{R}_{3}(z)]$ (blue curve).

The retarded dimensionless Green’s function $\Psi_{3}^{R}(z)$ is given by the
equation

$\displaystyle\Psi_{3}^{R}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}du\frac{u^{2}e^{-u^{2}/2}}{z-u+i\delta}.$
(117)

The imaginary part ${\rm Im}\Psi_{3}^{R}(z)$ is easily computed

$\boxed{{\rm
Im}\Psi_{3}^{R}(z)=-\sqrt{\frac{\pi}{2}}z^{2}e^{-z^{2}/2}\xrightarrow[]{z\to
0}-\sqrt{\frac{\pi}{2}}\left[z^{2}-\frac{z^{4}}{2}+O(z^{4})\right].}$ (118)

The real part of the analytic in the upper half-plane of $z$ function ${\rm
Re}\Psi_{3}^{R}(z)$ can be determined by the Kramers - Kronig relations:

${\rm Re}\Psi_{3}^{R}(z)=\frac{1}{\pi}{\rm
P.V.}\int_{-\infty}^{\infty}du\frac{{\rm Im}\Psi_{3}^{R}(u)}{u-z}.$ (119)

It can be also expressed by the Hilbert transform
$H[u^{2}e^{-u^{2}}](z)=H_{2}(z)$ [15] (see Appendix B) which, in turn, is re-
written in terms of the Dawson function as follows:

$H_{2}(z)=-\frac{1}{\sqrt{\pi}}z+\frac{2}{\sqrt{\pi}}z^{2}F(z).$ (120)

Substituting $H_{2}(z)$ to the definition of ${\rm Re}\Psi_{3}^{R}(z)$ we get:

$\displaystyle{\rm Re}\Psi_{3}^{R}(z)$ $\displaystyle=$
$\displaystyle-z+\sqrt{2}z^{2}F\left(\frac{z}{\sqrt{2}}\right),$ (121)
$\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle-z+z^{3}+O\left(z^{5}\right),$ (122)
$\displaystyle\xrightarrow[]{z\to\infty}$
$\displaystyle\frac{1}{z}+\frac{3}{z^{3}}+O\left(\frac{1}{z^{5}}\right).$
(123)

Using the properties of the Dawson function for small values of its argument,
we obtain $|\Psi_{3}^{R}(z\to 0)|^{2}\propto z^{2}$. The frequency dependence
of the real and imaginary parts of the GF $\Psi_{3}$ is shown in the left
panel of Fig. 11.

Substituting the Taylor expansion for $\Psi_{3}^{R}(z)$ into the equation for
the self-energy

$\sigma_{3}^{R}(z)=z-\frac{{\rm Re}\Psi_{3}^{R}(z)-i{\rm
Im}\Psi_{3}^{R}(z)}{|\Psi_{3}^{R}(z)|^{2}}$ (124)

we obtain the real and imaginary parts of the self-energy:

$\displaystyle{\rm Re}\sigma^{R}_{3}(z)$ $\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle\frac{1}{z}+\left(2-\frac{\pi}{2}\right)z+\left(\frac{2}{3}-\pi+\frac{\pi^{2}}{4}\right)z^{3}+O\left(z^{4}\right),$
(125) $\displaystyle{\rm Im}\sigma^{R}_{3}(z)$
$\displaystyle\xrightarrow[]{z\to 0}$
$\displaystyle-\sqrt{\frac{\pi}{2}}+\frac{1}{2}(\pi-3)\sqrt{\frac{\pi}{2}}z^{2}+O\left(z^{4}\right).$
(126)

The frequency dependencies of the real and imaginary parts of the SE
$\sigma_{3}$ are shown on the right panel of Fig. 10.

The DOS for the three-component Keldysh model is represented by two Gaussian
peaks separated by a pseudo-gap with the energy-square dependence at the
vicinity $z=0$.

###  $Variation$ No. 1: Anisotropic $two+one$ Keldysh model:
Breaking rotational symmetry.

In this $Variation$ we assume that the amplitudes of the random complex
Gaussian field associated with the fluctuations of the inter-dot barrier
(which we denote below by $W_{\perp}$) and the amplitude of the random scalar
Gaussian field associated with the fluctuations of the level position in each
well of the double well potential, denoted by $W_{\parallel}$ are different.
In this case it is not possible any longer to introduce the dimensionless
functions of the dimensionless argument with a unique re-scaling parameter.
For convenience of the reader we will not rescale the variables in this
Subsection. The simplest way to calculate the Green function of the
anisotropic three parametric Keldysh model is to use the path integral
representation by introducing a generating functional (see details in [6]). In
the extreme non-Markovian case corresponding to the ”infinite memory” limit
[5], the Green function of the anisotropic three-component Keldysh model
(which we denote $G_{2+1}(\varepsilon)$) is expressed as follows:

$\displaystyle
G_{2+1}(\varepsilon)=\frac{1}{W_{\parallel}W_{\perp}^{2}(2\pi)^{3/2}}\int_{-\infty}^{\infty}dze^{-z^{2}/2W_{\parallel}^{2}}\int
dw^{*}dwe^{-|w|^{2}/2W_{\perp}^{2}}\frac{\varepsilon}{(\varepsilon+i\delta)^{2}-z^{2}-|w|^{2}}.$
(127)

Re-writing the integral in the spherical coordinate system and performing the
integration over the angles we obtain after some simplification:

$\displaystyle
G_{2+1}(\varepsilon)=\frac{1}{2W_{\perp}}\int_{0}^{\infty}d\rho\rho\exp\left(-\frac{\rho^{2}}{2W_{\perp}^{2}}\right)\frac{{\rm
erf}\left(\rho\sqrt{\frac{W_{\perp}^{2}-W_{\parallel}^{2}}{2W_{\perp}^{2}W_{\parallel}^{2}}}\right)}{\sqrt{W_{\perp}^{2}-W_{\parallel}^{2}}}\left(\frac{1}{\varepsilon-\rho+i\delta}+\frac{1}{\varepsilon+\rho+i\delta}\right).$
(128)

Here ${\rm erf}(z)$ is the error function. [17, 18] Let us analyse a
particular case of anisotropy, namely $W_{\perp}>W_{\parallel}$. 121212The
complimentary anisotropic limit $W_{\perp}<W_{\parallel}$ can be considered in
a similar manner.  The constant energy surface for the three-component
anisotropic Keldysh model in both cases is an ellipsoid which transforms into
a sphere if $W_{\perp}=W_{\parallel}$ and isotropy is restored. Two extreme
cases of the two-component Keldysh model $W_{\parallel}\to 0$ of the easy-
plane anisotropy and one-component Keldysh model $W_{\perp}\to 0$ of the easy
axis anisotropy can be obtained straightforwardly. To proceed with remaining
integration over $\rho$ it is convenient to use the series expansion of the
error function

${\rm
erf}(z)=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}.$
(129)

As a result, Eq. (128) is transformed into

$\displaystyle
G_{2+1}(\varepsilon)=2\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}C_{nm}\alpha^{2m}\frac{W_{\perp}^{2n+1}}{W_{\parallel}^{2m+1}}G_{0}^{2n+1}(\varepsilon).$
(130)

Here $\alpha=\sqrt{(W_{\perp}^{2}-W_{\parallel}^{2})/2}$, and
$G_{0}(\epsilon)=(\epsilon+i\delta)^{-1}$. The coefficients

$C_{nm}=\frac{[2(m+n)+1]!!}{m!(2m+1)}$ (131)

are the combinatorial coefficients in the $n$-th order of the ”three-color”
diagrammatic expansion for the full anisotropic three-component Keldysh model.
In the isotropic limit $\alpha=0$ and only the term $m=0$ survives. The
coefficient $C_{n0}=A_{n}^{(3)}=(2n+1)!!.$

###  $Variation$ No. 2: Combined $one+two$ Keldysh model

Let us assume that in addition to the slow barrier fluctuations in the double
QD there exists a uniform noise to both QDs, which is produced by back gate.
For simplicity we suppose that the noise amplitudes for the one-component
uniform diagonal Gaussian random potential and the two-component off-diagonal
Gaussian random potential are the same. The dimensionless GF $\Psi_{1+2}(z)$
is defined as:

$\Psi_{1+2}(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}dx\;e^{-(x-z)^{2}/2}\Psi_{2}(x),$
(132)

which is, in fact, the Weierstrass transform of the two-component Keldysh
model Green’s function.

Green’s function $\Psi_{1+2}(z)$ satisfies the second order ODE which can be
obtained by differentiating $\Psi_{1+2}(z)$ given by Eq. (132) with respect to
it’s argument, integrating by part and using the differential equation for
$\Psi_{2}(z)$ Eq. (81):

$\boxed{\frac{2}{z^{2}}\frac{d^{2}}{dz^{2}}\Psi_{1+2}(z)+\frac{3}{z}\frac{d}{dz}\Psi_{1+2}(z)+\Psi_{1+2}(z)=\frac{1}{z}.}$
(133)

The corresponding boundary conditions follow from fixing two coefficients in
the $z\to\infty$ asymptotic:

$\Psi_{1+2}(z\to\infty)=\frac{1}{z}+\frac{1+2}{z^{3}}.$ (134)

In fact, these boundary conditions (134) correspond to two conditions:

$\Psi_{1+2}(z\to\infty)=\frac{1}{z},\;\left.\left[z\cdot\Psi_{1+2}(z)-1\right]\right|_{z\to\infty}=\frac{3}{z^{2}}.$
(135)

The analytic properties of $\Psi_{1+2}(z)$ are fully defined by the
corresponding properties of $\Psi_{2}(z)$ (see previous $\acute{E}tude$).

##  6 $\acute{E}tude$ No 4. Keldysh model with $D\gg 1$-component noise

Let us consider the Keldysh model with a $d$-color Gaussian random potential
having both diagonal (longitudinal) and off-diagonal (transverse) components.
The GF in a given realization of disorder (before averaging) has a pole which
defines a $D=d-1$ -dimensional sphere $S_{D}$ (see $Intermezzo$).

For consistency with the notations of the previous $\acute{E}tudes$: $d=1$
corresponds to the one-component KM, $d=2$ describes the two-component KM on a
circle $S_{1}$ and $d=3$ corresponds to the three-component SU($2$) KM on
$S_{2}$. The solution for $\sigma_{d}(z)$ and $\Gamma_{d}(z)$ is given by:

$\sigma_{d}(z)=\frac{z\Psi_{d}(z)-1}{\Psi_{d}(z)},\;\;\;\;\;\Gamma_{d}(z)=\frac{z\Psi_{d}(z)-1}{\Psi_{d}^{2}(z)}.$
(136)

These equations can be obtained directly from the definition of
$\sigma_{d}(z)$ and the Ward Identity

$\boxed{\sigma_{d}(z)=\Psi_{d}(z)\Gamma_{d}(z),\;\;\;\;\;\Gamma_{d}(z)=\frac{1}{z^{d-1}}\frac{d}{dz}\left[z^{d-1}\Psi^{-1}_{d}(z)\right]={\rm
div}[\vec{\Upsilon}_{d}],\;\;\;\vec{\Upsilon}_{d}=\Psi_{d}^{-1}\vec{e}_{r}.}$
(137)

Here we assumed that the GF has only ”radial” component in d-dimensional
spherical coordinate system.

The Dyson equations for the GF and SE are accompanied by the corresponding
boundary conditions:

$\boxed{\frac{d\Psi_{d}(z)}{dz}+\left[z-\frac{d-1}{z}\right]\Psi_{d}(z)=1,\;\;\;\;\Psi_{d}(z\to\infty)=\frac{1}{z},}$
(138)
$\boxed{\frac{d\sigma_{d}(z)}{dz}+\left[z+\frac{d-1}{z}\right]\sigma_{d}(z)-\sigma_{d}^{2}(z)=d,\;\;\;\;\sigma_{d}(z\to\infty)=\frac{d}{z}.}$
(139)

The straightforward solution of the Dyson equation can be written by the
Hilbert transform $H[u^{d-1}e^{-u^{2}}](z)$ (see Appendix B):

$\Psi_{d}(z)=\frac{1}{2\Omega_{d}}\int_{0}^{\infty}u^{d-1}e^{-u^{2}/2}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right].$
(140)

Here $\Omega_{d}=2^{\frac{d}{2}-1}\Gamma(\frac{d}{2})$ is the normalizing
coefficient and $\Gamma(u)$ is the Euler’s Gamma-function. The $(d-1)/z$
”centrifugal” term is responsible for the level repulsion.

The differential identities connecting GF and SE are given by:

$\displaystyle\sigma_{d}(z)=-\frac{d}{dz}\log\left[\frac{\Psi_{d}(z)}{z^{d-1}}\right],\;\;\;\;\;\Gamma_{d}(z)=1-\frac{d\sigma_{3}(z)}{dz}=1+\frac{d^{2}}{dz^{2}}\log\left[\frac{\Psi_{d}(z)}{z^{d-1}}\right].$
(141)

These identities follow from the solution of Riccati-type equation for
$\sigma_{d}(z)$.

### Large $D$-limit

Differential equations describing the single-particle Green’s function of the
Keldysh model with a few-component Gaussian random potential contain
coefficients $O(1)$. Keldysh models with a $D+1\gg 1$-component random
potential contain a natural small parameter $1/D$ which can be used to find an
approximate but reliable solution based on the $1/D$ expansion. To obtain the
differential equation in the form convenient for the expansion we first
proceed with re-scaling of the functions and the arguments: new GF
$\phi=\Psi_{D}/\sqrt{D}$, new self-energy $\rho=\sigma_{D}/\sqrt{D}$ and new
variable $w=z/\sqrt{D}$.

The differential equations for new GF and SE read:

$\boxed{\frac{1}{D}\frac{d\phi(w)}{dw}+\left[w-\frac{1}{w}\right]\phi(w)=\frac{1}{D},\;\;\;\;\phi(w\to\infty)=\frac{1}{Dw},}$
(142)
$\boxed{\frac{1}{D}\frac{d\rho(z)}{dw}+\left[w+\frac{1}{w}\right]\rho(w)-\rho^{2}(w)=1+\frac{1}{D},\;\;\;\;\rho(w\to\infty)=\frac{1+1/D}{w}.}$
(143)

The Ward Identity acquires the form:

$\boxed{\Gamma(w)=\frac{1}{w^{D}}\frac{d}{dw}\left[w^{D}\phi^{-1}(w)\right]={\rm
div}\left[\vec{\upsilon}(w)\right],\;\;\;\;\vec{\upsilon}(w)=\phi^{-1}(w)\vec{e}_{r}.}$
(144)

It is straightforward now to find an approximate solution of these equations.
In particular, one can see that the boundary condition gives a reliable
approximation for both the GF and SE at $z>D$:

$\Psi^{R}_{D}(z)=\frac{1}{z-D/z+i\delta},\;\;\;\;\;\;\sigma_{D}(z)=\frac{D}{z}.$
(145)

The imaginary part of the $\Psi^{R}_{D}(z)$ in this approximation consists of
two $\delta$-functions separated by $\Delta z=2\sqrt{D}$.

To summarize, adding components of the random Gaussian field allows to
engineer the gap in the DOS: few-component Keldysh models are characterized by
a ”soft” (pseudo) gap which becomes ”harder” when the number of components
increases.

##  7 $\acute{E}tude$ No 5. Perturbative expansion for the bold Green’s
function

The three last advanced $\acute{E}tudes$ are devoted to combinatorics of the
Feynman diagrams for the bold Green’s function and derivation of recurrence
relations allowing to find a number of skeleton Feynman diagrams in a given
order of the perturbation expansion. The ”skeleton” means that we are looking
for irreducible diagrams containing the bold GF (see e.g. Fig 12). As it has
been mentioned before, the advantage of slow non-Markovian classical random
potential models is that all diagrams in a given order of the perturbation
theory have the same value. Thus, the coefficients of the expansion give the
total number of the diagrams. 131313The situation is completely different for
fast (Markovian) random potentials. In this case, the selection rule says that
the rainbow-type diagrams give much large contribution in comparison to the
diagrams with self-crossing. $\acute{E}tude\;6$ is based on the works of
Sadovskii - Kuchinskii [4, 9] and Suslov [10] (SKS) who considered the
enumeration of skeleton diagrams for the ”orthodox” single-component Keldysh
model. In this $\acute{E}tude$ we present a simple pedagogical derivation of
the SKS recurrence equation using a method different from the original SKS
works. This method proves to be very convinient for deriving a general
equation giving access to enumeration of skeleton diagrams for classical
random Gaussian field models with arbitrary number of the random field
components. These equations will be derived and discussed in
$\acute{E}tude\;7$. We will show that the SKS equations is just a particular
limit of the general theory.

In this preparatory $\acute{E}tude$ we create a basis for answering questions
about the combinatorics of Feynman diagrams. The first question is: ”How many
Feynman diagrams exist in n-th order of the perturbative expansion?”. Or, put
more simply ”How many Feynman diagrams contain n-wavy lines for $d$-color
random Gaussian field models?” This question is trivial as we already know the
answers for one-, two- and three- component models. To answer this question we
need to extend the general solution for the $d$-color GF $\Psi_{d}$ in terms
of the bare GF in the limit $z\to\infty$:

$\displaystyle\Psi_{d}(z)=\frac{1}{\Omega_{d}}\int_{0}^{\infty}\sum_{n=0}^{\infty}\frac{u^{2n+d-1}}{z^{2n+1}}e^{-u^{2}/2}du=\frac{1}{\Omega_{d}}\sum_{n=0}^{\infty}\frac{1}{z^{2n+1}}\int_{0}^{\infty}u^{2n+d-1}e^{-u^{2}/2}du=\sum_{n=0}^{\infty}\frac{A^{(d)}_{n}}{z^{2n+1}}.$

Using equation for $\Omega_{d}=2^{\frac{d}{2}-1}\Gamma(\frac{d}{2})$ from the
previous $\acute{E}tude$ we get

$\boxed{A^{(d)}_{n}=\frac{2^{n}\Gamma\left(n+\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)}.}$
(146)

This coefficient defines the number of Feynman diagrams in the $n$-th order of
GF expansion.

To check that the general answer is correct, we write down explicitly the
values of $A^{(d)}_{n}$ for $d=1,2,3$:

$A^{(1)}_{n}=(2n-1)!!,\;\;\;\;A^{(2)}_{n}=(2n)!!,\;\;\;\;A^{(3)}_{n}=(2n+1)!!$
(147)

Another important check is the number of diagrams in the lowest (first) order
of the perturbative expansion:

$A^{(d)}_{1}=\frac{2\Gamma\left(1+\frac{d}{2}\right)}{\Gamma\left(\frac{d}{2}\right)}=d.$
(148)

Finally, for the limit $d\gg 1$:

$A^{(n\ll d/2)}_{n}\to 2^{n}.$ (149)

The numbers given by Eq. (7) being plugged into the equations for the self-
energies and the triangular vertex give the answer about the corresponding
combinatorics in terms of bare GF.

##  8 $\acute{E}tude$ No 6. Enumeration of ”skeleton” Feynman diagrams in a
single-component Keldysh model: square recursion

Summation and re-summation methods of Feynman diagrams are widely used in
analytical (quantum field theories) and numerical (bold diagrammatic Quantum
Monte-Carlo methods, see, e.g. [27, 28, 29]) solutions in various problems of
modern many-body physics. Counting the number of Feynman diagrams is another
challenging problem, which is important, for example, for benchmarking
numerical techniques based on the diagrammatic expansion. Hedin in his
pioneering work [19] formulated a set of equations (Hedin’s equations)
connecting the single particle propagator $G$ with the effective potential
$W$, self-energy $\Sigma$, polarization $\Pi$ and vertex $\Gamma$. Another
additional equation follows from the functional-derivative connection between
$\Gamma$, $\Sigma$ and $G$. In zero dimension of the space-time the Hedin
equations become algebraic. By searching the solution of the Hedin equations
in powers of the bare interacting potential and taking into account the number
of closed loops, one gets the number of Feynman diagrams from the expansion
coefficients [20, 21, 22]. Further development of these methods made it
possible to formulate an iterative algorithm for counting Feynman diagrams via
many-body relations [23] .

Figure 12: Diagrammatic expansion for the self-energy. Double line corresponds
to bold Green’s function. Wavy line stands for the correlator of Gaussian
classical random field.

In this $\acute{E}tude$ we develop analytical methods for counting skeleton
Feynman diagrams for self-energy, triangular vertex and single-particle
T-matrix for models with $d$-component classical (no loops) Gaussian random
field. We are dealing with $0+1$ models and therefore will end up with some
differential (not algebraic, contrast to the Hedin’s theory) equations. The
methods discussed in $\acute{E}tudes\;6\;and\;7$ can also be generalized to
enumerate diagrams for the two-particle Green’s function and the full four-
point correlator. In order to find the combinatorics for the self-energies and
the triangular vertices of the single-component Keldysh model in terms of the
bold GF we start with the general equations Eq.(141) formally defining
$\sigma_{1}$ and $\Gamma_{1}$ as Taylor series in terms of the bold GF
$\Psi_{1}$:

$\displaystyle\sigma_{1}[\Psi_{1}]$ $\displaystyle=$
$\displaystyle\frac{z[\Psi_{1}]\Psi_{1}-1}{\Psi_{1}}=\sum_{n=0}^{\infty}a_{n}^{(1)}\Psi_{1}^{2n+1},$
(150) $\displaystyle\Gamma_{1}[\Psi_{1}]$ $\displaystyle=$
$\displaystyle\frac{\sigma_{1}[\Psi_{1}]}{\Psi_{1}}=\frac{z[\Psi_{1}]\Psi_{1}-1}{\Psi^{2}_{1}}=\sum_{n=0}^{\infty}a_{n}^{(1)}\Psi_{1}^{2n}.$
(151)

Thus, the goal is to find $z[\Psi_{1}]$. Since $\Psi_{1}(z)$ is expressed in
terms of the Dawson function, we aim to construct the inverse Dawson function
[16]. The coefficients $a_{n}^{(1)}$ in the asymptotic series fully determine
the number of irreducible Feynman diagrams for the self-energy and the vertex
correspondingly.

Using equation (150) we write:

$z[\Psi_{1}]=\frac{1}{\Psi_{1}}+\sigma_{1}[\Psi_{1}]=\frac{1}{\Psi_{1}}+\sum_{n=0}^{\infty}a_{n}^{(1)}\Psi_{1}^{2n+1}.$
(152)

We outline a simple and elegant way to construct the inverse Dawson function
below. The function $z[\Psi_{1}]={\rm invD}_{+}[\Psi_{1}]$ satisfies the
equation

$\frac{dz[\Psi_{1}]}{d\Psi_{1}}\left(1-\Psi_{1}z[\Psi_{1}]\right)=1,$ (153)

which is obtained by inversion of Eq. (39). The boundary condition for this
equation reads $\left.\Psi_{1}\cdot z[\Psi_{1}]\right|_{z\to\infty}=1$
141414The self energy $\sigma_{1}$ satisfies Abel’s second kind non-linear
differential equation [24]
$\sigma_{1}\cdot\frac{d\sigma_{1}}{d\psi}=\frac{\sigma_{1}}{\psi^{2}}-\frac{1}{\psi},$
(154) where $\psi$$=$$\Psi_{1}$ is the bold GF and the initial condition is
defined by behaviour of $\sigma_{1}$ (67) and $\Psi_{1}$ (62) at $z\to 0$ [9].
Strictly speaking as both $\sigma_{1}$ and $\Psi_{1}$ are complex functions of
a parameter $z$ (see $\acute{E}tude\;1$), the Eq. (154) represents a system of
two coupled non-linear partial differential equations. We consider Eq. (154)
assuming $\psi$$\to$$x$$=$${\rm Re}$$\psi$ and $s$$={\rm Re}$$\sigma_{1}$
while ${\rm Im}$$\psi$$\to 0$ and ${\rm Im}$$\sigma_{1}$$\to$$0$. It
corresponds to parametric regime $z$$\gg$$1$ where both ${\rm
Im}$$\Psi_{1}$$/$$({\rm Re}$$\Psi_{1})$$\to 0$ and ${\rm Im\sigma_{1}}/({\rm
Re}\sigma_{1})\to 0$. The Eq. (154) is transferred to a normal form:
$s\cdot\frac{ds}{d\xi}-s=\frac{1}{\xi}$ (155) by the change of variables
$\xi=-1/x$ [24, 25]. The solution $s(\xi)$ of this non-linear equation is
defined in implicit form by a transcendental equation containing Dawson
function $F$ (57) in the l.h.s. (see $\acute{E}tude\;1$):
$\sqrt{2}F\left(\frac{s-\xi}{\sqrt{2}}\right)+\frac{1}{\xi}=0$ (156)
Introducing a new function $u(\xi)=-1/s$ allows to convert the non-linear ODE
Eq. (155) to Abel’s first kind non-linear differential equation [26]:
$\frac{du}{d\xi}=-\frac{u^{3}}{\xi}+u^{2}.$ (157) To see the connection with
Sadovskii - Kuchinskii (SK) theory [9] we redefine $\sigma_{1}(\psi)=\psi\cdot
Q(\psi)$. As a result, substituting $\sigma_{1}(\psi)$ into Eq. (154) we
obtain the non-linear differential equation for the function $Q$ which was
derived by a different method by Sadovskii and Kuchinskii and analysed in [9]:
$Q=1+\psi^{2}\cdot Q^{2}+\psi^{3}\cdot
Q\cdot\frac{dQ}{d\psi}\;\;\;\;\;\;\;\;\;\xrightarrow[]{y=\psi^{2}}\;\;\;\;\;\;\;\;\;Q=1+y\cdot\frac{d}{dy}\left(y\cdot
Q^{2}\right).$ (158) The solution of this equation formally written in terms
of the Taylor series with zero convergence radius
$Q(y)=\sum_{n=0}^{\infty}a^{(1)}_{n}y^{n}$ (159) defines the recurrence
relations (163). . Substituting expansion for $z[\Psi_{1}]$ into (153) we get:

$\left[1-\sum_{n=0}^{\infty}(2n+1)a_{n}^{(1)}\Psi_{1}^{2n+2}\right]\left[\sum_{n=0}^{\infty}a_{n}^{(1)}\Psi_{1}^{2n}\right]=1,$
(160)

which after opening the brackets is re-written as follows:

$\displaystyle
a_{0}^{(1)}+\sum_{n=1}^{\infty}a_{n}^{(1)}\Psi_{1}^{2n}-\sum_{n=1}^{\infty}\Psi_{1}^{2n}\sum_{m=0}^{n-1}\left[2(n-m)-1\right]a_{m}^{(1)}a_{n-m-1}^{(1)}=1.\;\;\;\;\;\;\;\;$
(161)

Equating the coefficients in front of the $n$-th power of the bold GF
$\Psi_{1}^{n}$ we obtain the recurrence relations for the coefficients
$a_{n}^{(1)}$:

$a_{n}^{(1)}=\sum_{m=0}^{n-1}\left[2(n-m)-1\right]a_{m}^{(1)}a_{n-m-1}^{(1)},\;\;\;\;\;a_{0}^{(1)}=1.$
(162)

The equation (162) can be written in more compact form

$\boxed{a_{n}^{(1)}=n\sum_{m=0}^{n-1}a_{m}^{(1)}a_{n-m-1}^{(1)},\;\;\;\;\;a_{0}^{(1)}=1.}$
(163)

The number of the irreducible diagrams in the first ten orders of the
perturbation theory is presented in the Table 1.

The recurrence relations for the diagram combinatorics of the single-component
Keldysh model have been obtained by Sadovskii and Kuchinskii [9] and Suslov
[10] using a bit different technique. The large-n asymptotic of the
coefficients $a_{n}^{(1)}$ has been analysed by Sadovskii and Kuchinskii in
[9]. In the limit $n\gg 1$ SK transferred the recurrence equation (163) into
an ordinary differential equation. Its solution gives the asymptotic:

$\boxed{a_{n}^{(1)}\xrightarrow[]{n\gg
1}\frac{1}{e}\left[1-\frac{5}{4n}+O\left(\frac{1}{n^{2}}\right)\right](2n+1)!!}$
(164)

We plot in Fig. 13 (left panel) the normalized number of irreducible Feynman
diagrams for the first $n=50$ orders of the perturbation theory (dots). The
red solid line shows asymptotic behaviour given by Eq.164 accounting for the
$1/n$ correction.

The SK method, while powerful enough to handle $1/n$ corrections, does not,
however, allow the coefficient ($1/e$) before $(2n+1)!!!$ to be determined
analytically. This coefficient was calculated numerically by SK and derived
analytically by Suslov [10]. In the next and the last $\acute{E}tude$ we will
derive the recurrence relations for the self-energy and the triangular vertex
combinatorics considering the multi-component Keldysh model. We will derive
general recurrence equations which give the number of the skeleton diagrams in
any order of expansion. To illustrate how to get the asymptotic behaviour of
the coefficients of $d$-component models at the limit $n\gg 1$ we apply
Sadovskii and Kuchinskii method.

##  9 $\acute{E}tude$ No 7. Enumeration of ”skeleton” Feynman diagrams in a
multi-component Keldysh model: cubic recursion

This $\acute{E}tude$ begins with derivation of the recurrence equation for the
two-component Keldysh model. Following the key steps of the derivation we
present detailed discussion for computing the large - $n$ asymptotic for the
number of skeleton diagrams. After ”learning” how the method works for a two-
component random Gaussian field we proceed with the general derivation for the
multi-component case. At the end of this $\acute{E}tude$ we briefly discuss
how the combinatorics of skeleton diagrams for self-energy can be used for the
combinatorics of the diagrams for the $T$-matrix.

### Enumeration of ”skeleton” diagrams for the two-component Keldysh model

Figure 13: Asymptotic of the skeleton Feynman diagrams number for the self-energy for $d=1$ KM model, red solid line is $1-5/(4n)$ (left) and $d=2$ KM model, red solid line is $1-1/n$ (right). $N$ | $a_{N-1}^{(1)}$ | $a_{N-1}^{(2)}$ | $b_{N}^{(2)}=a^{(2)}_{N-1}/2^{N}$ | $a_{N-1}^{(3)}$
---|---|---|---|---
1 | 1 | 2 | 1 | 3
2 | 1 | 0 | 0 | -3
3 | 4 | 8 | 1 | 24
4 | 27 | 32 | 2 | -45
5 | 248 | 416 | 13 | 1044
6 | 2830 | 4736 | 74 | 3078
7 | 38232 | 69632 | 544 | 119232
8 | 593859 | 1141248 | 4458 | 1517427
9 | 10401712 | 21105152 | 41221 | 33013980
10 | 202601898 | 431525888 | 421412 | 670457250
$N\gg 1$ | $\frac{1}{e}(2N-1)!!$ | $\frac{1}{e^{2}}(2N)!!$ | $\frac{1}{e^{2}}N!$ | $\frac{1}{e^{3}}(2N+1)!!$
Table 1: Expansion coefficients for the self-energy dependence on the bold GF
for $d=1,2,3$ KM.

The differential equation for the inverted function $z[\Psi_{2}]$ is obtained
by inversion the differential equation for $\Psi_{2}(z)$:

$\displaystyle\frac{dz[\Psi_{2}]}{d\Psi_{2}}\left(z[\Psi_{2}]-\Psi_{2}\left[(z[\Psi_{2}])^{2}-1\right]\right)$
$\displaystyle=$ $\displaystyle z[\Psi_{2}].$

We substitute the Taylor expansion for $\sigma_{2}[\Psi_{2}]$

$z[\Psi_{2}]=\frac{1}{\Psi_{2}}+\sigma_{2}[\Psi_{2}]=\frac{1}{\Psi_{2}}+\sum_{n=0}^{\infty}a_{n}^{(2)}\Psi_{2}^{2n+1}$
(165)

to get the recurrence relations for the coefficients $a_{n}^{(2)}$:

$\displaystyle\left[1-\sum_{n=0}^{\infty}(2n+1)a_{n}^{(2)}\Psi_{2}^{2n+2}\right]\left[\sum_{n=0}^{\infty}a_{n}^{(2)}\Psi_{2}^{2n}-1+\Psi_{2}^{2}\left(\sum_{n=0}^{\infty}a_{n}^{(2)}\Psi_{2}^{2n}\right)^{2}\right]=1+\sum_{n=0}^{\infty}a_{n}^{(2)}\Psi_{2}^{2n+2}.$

This equation finally leads to the following recurrence relations:

$\displaystyle a_{0}^{(2)}=2,\;\;\;\;\;\;a_{1}^{(2)}=0,$ (166) $\displaystyle
a_{n}^{(2)}=-2(n-1)a_{n-1}^{(2)}+2\sum_{m=0}^{n-1}\left[n-m-1\right]a_{m}^{(2)}a_{n-m-1}^{(2)}$
$\displaystyle+\sum_{m=0}^{n-2}\sum_{p=0}^{n-m-2}\left[2(n-m-p-2)+1\right]a_{m}^{(2)}a_{p}^{(2)}a_{n-m-p-2}^{(2)}.\;\;\;\;\;\;$

We notice an interesting property of the case $d=2$, namely, that the
coefficient $a_{1}^{(2)}=0$. It reflects the fact that the second order
irreducible Feynman diagram vanishes due to topology of the two-component
model (see $\acute{E}tude\;2$). The equation (9) can be further simplified:

$\boxed{a_{n}^{(2)}=-2(n-1)a_{n-1}^{(2)}+(n-1)\sum_{m=0}^{n-1}a_{m}^{(2)}a_{n-m-1}^{(2)}+\sum_{m=0}^{n-2}\sum_{p=0}^{n-m-2}\left[n-m-1\right]a_{m}^{(2)}a_{p}^{(2)}a_{n-m-p-2}^{(2)}.}$

This recurrence equation generalizes the combinatorics of the Keldysh model to
the case of the two-component random Gaussian field. The main difference from
SKS equation is that in addition to the bilinear recurrence, a cubic term
appears in the recurrence equation.

To find the asymptotic behaviour of $a_{n}^{(2)}$ at $n\gg 1$ we will follow
the method developed by Sadovskii and Kuchinskii in [9]. We look for the
asymptotic solution of (9) in the form

$a_{n}^{(2)}\approx\left(\alpha n+\beta\right)a_{n-1}^{(2)}.$ (167)

Substituting Eq. (167) into Eq. (9) we find that at the limit $n\gg 1$

$\alpha=\beta=2.$ (168)

We can therefore re-parametrize Eq. (9) by introducing the new coefficients as
follows:

$a_{n}^{(2)}=2^{n+1}(n+1)!c_{n}^{(2)}.$ (169)

This re-parametrization gives the asymptotic behaviour for the number of the
skeleton diagrams for the self-energy in the large-$n$ limit. In order to find
the $1/n$ correction we need to deal with the equations for the coefficients
$c_{n}^{(2)}$. By plugging in Eq. (169) into Eq. (9) we get a new recurrence
relations:

$\displaystyle
c_{n}^{(2)}=-\frac{n-1}{n+1}c_{n-1}^{(2)}+\frac{n-1}{2}\sum_{m=0}^{n-1}\frac{(m+1)!(n-m-1)!}{(n+1)!}c_{m}^{(2)}c_{n-m-1}^{(2)}$
$\displaystyle+\sum_{m=0}^{n-2}\sum_{p=0}^{n-m-2}\left[n-m-1\right]\frac{(m+1)!(p+1)!(n-p-m-2)!}{(n+1)!}c_{m}^{(2)}c_{p}^{(2)}c_{n-m-p-2}^{(2)}.$

Taking the limit of large $n$ and retaining only the leading $1/n^{2}$ terms
in (9) under assumption $c=c_{n}\approx c_{n-1}\approx c_{n-2}$ we finally
obtain a simple differential equation:

$\frac{dc}{dn}=\frac{1}{n^{2}}c.$ (170)

The solution of this equation is given by

$c={\rm const}\cdot\exp\left(-\frac{1}{n}\right)\xrightarrow[]{n\gg 1}{\rm
const}\left(1-\frac{1}{n}+O\left(\frac{1}{n^{2}}\right)\right).$ (171)

As it has been discussed before, Sadovskii-Kuchinskii method does not provide
information about the coefficient in front of the exponential function. To
obtain correctly this coefficient we follow the logic of Suslov’s work [10].
Let’s begin with $d=1$ case. For all $d\neq 1$, since the noise components are
independent (no cross-correlations), we can repeat the same arguments for each
component independently and hence the constant found for the $d=1$ case will
be replaced by the constant to the power $d$. First, we observe that the
coefficient $B^{(1)}_{n}=a_{n-1}^{(1)}=(2n)!!c_{n-1}^{(1)}$ has the same
$n$-dependence as the coefficient $A^{(1)}_{n}$ which defines the total number
of the Feynman diagrams for GF 151515It is easy to see that the number of
skeleton diagrams in the $n$-th order of the perturbation theory is defined
strictly speaking by the coefficients $B^{(1)}_{n}$ while the total number of
diagrams is defined by $A^{(1)}_{n}$. Second, we see that the diagrams
contributing to $A^{(1)}_{n}$ consist of all irreducible diagrams contributing
to $B^{(1)}_{n}$ plus all reducible diagrams $B^{(1)}_{n-1}=B^{(2)}_{n}/1!$
with one wavy line insert. Third, there are diagrams of order $n-2$ with
$B^{(1)}_{n-2}=B^{(2)}_{n}/2!$ and so on. As a result, 161616We omit $O(1/n!)$
corrections to the constant in the large $n$ limit. we get the relation:

$A^{(1)}_{n}=B^{(1)}_{n}\left(1+\frac{1}{1!}+\frac{1}{2!}+...\right)\xrightarrow[]{n\gg
1}eB^{(1)}_{n}.$ (172)

Similarly, for $d=2$ we obtain

$A^{(2)}_{n}=B^{(2)}_{n}\left(1+\frac{1}{1!}+\frac{1}{2!}+...\right)^{2}\xrightarrow[]{n\gg
1}e^{2}B^{(2)}_{n}.$ (173)

Finally, using ${\rm const}=1/e^{2}$ we get:

$\boxed{a_{n}^{(2)}\xrightarrow[]{n\gg
1}\frac{1}{e^{2}}\left[1-\frac{1}{n}+O\left(\frac{1}{n^{2}}\right)\right]2^{n+1}(n+1)!}$
(174)

Note that for $d=2$ each random Gaussian field line gives the factor $2$
because the correlator
$\langle\tau^{+}\tau^{-}\rangle=\langle\tau^{x}\tau^{x}\rangle+\langle\tau^{y}\tau^{y}\rangle$
and the correlations in the $x$\- and $y$\- pseudospin directions are
statistically independent. The second column of the Table 1 contains
coefficient $a_{N-1}^{(2)}$ computed for the first ten orders of the
perturbation theory. The number of irreducible diagrams in the $n$-th order is
given by the coefficient $b^{(2)}_{N}=a^{(2)}_{N-1}/2^{N}$. Note that the
coefficient $b^{(2)}_{2}=0$ since the second order irreducible Feynman diagram
is absent (see $\acute{E}tude\;2$).

We plot in Fig. 13 (right panel) the normalized number of irreducible Feynman
diagrams for the first $n=50$ orders of the perturbation theory (dots). The
red solid line shows the asymptotic behaviour given by Eq. (174) accounting
for the $1/n$ correction.

Before proceeding to the discussion of the general $d$-component
combinatorics, let us make an interesting remark. In the last column of the
Table 1 we present the coefficients $a_{N-1}^{(3)}$ for the $d=3$ component
Keldysh model. For the two previously considered cases, namely, $d=1$ and
$d=2$ we observe that all coefficients $a_{N-1}^{(d<3)}$ are positive. For
$d=3$, however, we see that the coefficients for $N=2$ and $N=4$ become
negative. This is the generic situation for Keldysh models with $d>2$. Since
the coefficients $a_{N-1}^{(d)}$ determine not only the number of the skeleton
diagrams, but also the total sign, we briefly illustrate the source of the
negative sign in Fig. 14.

### General $d>1$ \- component isotropic Keldysh model

Let us now consider a general multi-component Keldysh model with the number of
Gaussian random field components $d>1$. The differential equation for the
inverse function $z[\Psi_{d}]$ 171717The self energy $\sigma_{d}(x={\rm
Re}\Psi_{d})$ satisfies the non-linear differential equation
$\left(\frac{1}{x}-x\cdot\frac{d\sigma_{d}}{dx}\right)\cdot\left(\sigma_{d}-\frac{x\cdot(d-1)}{1+x\cdot\sigma_{d}}\right)=1,$
(175) The initial condition for $\sigma_{d}(\Psi_{d})$ can be deduced from
parametric equations $\sigma_{d}(z\to 0)$ and $\Psi_{d}(z\to 0)$ (see
$\acute{E}tude\;2$, $\acute{E}tude\;3$ and $\acute{E}tude\;6$). After a
standard substitution $x=-1/\xi$ we get:
$\left(\frac{d\sigma_{d}}{d\xi}-1\right)\cdot\left(\sigma_{d}-\frac{d-1}{\sigma_{d}-\xi}\right)=\frac{1}{\xi}.$
(176) The solution of this non-linear differential equation $\sigma_{d}(\xi)$
satisfies a transcendental equation  ${\rm
Re}\Psi_{d}\left(\sigma_{d}-\xi\right)+\frac{1}{\xi}=0,$ (177) where ${\rm
Re}\Psi_{2}$ is given by Eq. (103), ${\rm Re}\Psi_{3}$ is defined by Eq. (123)
(see Appendix B for explicit form of ${\rm Re}\Psi_{d}$ for arbitrary $d$).
Introducing generalized SK function $Q_{d}$ defined as
$\sigma_{d}=\Psi_{d}\cdot Q_{d}$ and depending on a new variable
$y=\Psi_{d}^{2}$ allows to derive the non-linear differential equation:
$Q_{d}=d+y\left(y\cdot Q_{d}^{2}\right)^{\prime}-(d-1)\frac{2y}{1+y\cdot
Q_{d}}\cdot\left(y\cdot Q_{d}\right)^{\prime}$ (178) which is the
generalization of the SK equation for the case $d>1$. We used notation
$\displaystyle(\bullet)^{\prime}=\frac{d\bullet}{dy}$. Similarly to $d=1$ case
we formally write the solution of this equation in terms of the Taylor series
with zero convergence radius:  $Q_{d}(y)=\sum_{n=0}^{\infty}a^{(d)}_{n}y^{n}.$
(179) As a result, the recurrence relations (9) are obtained.  is given by

$\displaystyle\frac{dz[\Psi_{d}]}{d\Psi_{d}}\left(z[\Psi_{d}]-\Psi_{d}\left[(z[\Psi_{d}])^{2}-(d-1)\right]\right)$
$\displaystyle=$ $\displaystyle z[\Psi_{d}].$

Substituting the Taylor expansion for the self-energy in terms of the bold GF
$\sigma_{d}[\Psi_{d}]$

$z[\Psi_{d}]=\frac{1}{\Psi_{d}}+\sigma_{d}[\Psi_{d}]=\frac{1}{\Psi_{d}}+\sum_{n=0}^{\infty}a_{n}^{(d)}\Psi_{d}^{2n+1}$
(180)

we obtain recurrence relations between the coefficients $a_{n}^{(d)}$.

$\displaystyle\left[1-\sum_{n=0}^{\infty}(2n+1)a_{n}^{(d)}\Psi_{d}^{2n+2}\right]\left[\sum_{n=0}^{\infty}a_{n}^{(d)}\Psi_{d}^{2n}-(d-1)+\Psi_{d}^{2}\left(\sum_{n=0}^{\infty}a_{n}^{(d)}\Psi_{d}^{2n}\right)^{2}\right]=1+\sum_{n=0}^{\infty}a_{n}^{(d)}\Psi_{d}^{2n+2}.$

Finally, following the steps outlined in the previous $\acute{E}tude$ we get:

$\displaystyle a_{0}^{(d)}=d,\;\;\;\;\;\;a_{1}^{(d)}=d(2-d),$ (181)
$\displaystyle
a_{n}^{(d)}=-\left\\{(d-1)(2n-1)-1\right\\}a_{n-1}^{(d)}+2\sum_{m=0}^{n-1}\left[n-m-1\right]a_{m}^{(d)}a_{n-m-1}^{(d)}$
$\displaystyle+\sum_{m=0}^{n-2}\sum_{p=0}^{n-m-2}\left[2(n-m-p-2)+1\right]a_{m}^{(d)}a_{p}^{(d)}a_{n-m-p-2}^{(d)}.\;\;\;\;\;\;$

The last equation can be re-written as follows:

$\boxed{a_{n}^{(d)}=-\left\\{2n(d-1)-d\right\\}a_{n-1}^{(d)}+(n-1)\sum_{m=0}^{n-1}a_{m}^{(d)}a_{n-m-1}^{(d)}+\sum_{m=0}^{n-2}\sum_{p=0}^{n-m-2}\left[n-m-1\right]a_{m}^{(d)}a_{p}^{(d)}a_{n-m-p-2}^{(d)}.}$

Figure 14:  We illustrate why for $d>2$ the coefficient
$a_{1}^{(d)}=d(2-d)<0$. Let us take an example $d=3$. The Feynman diagrams
contain wavy line denoting $z-z$ noise correlator and dashed line denoting
$+-=xx+yy$ noise correlator. It is straightforward to show that three 3rd
order irreducible diagrams come with factors $-2$ another $-2$ and $+1$ which
gives finally overall factor $-3$. Sign $-$ is associated with the
multiplication rule ${\rm det}[\tau^{z}]=\tau^{z}_{11}\tau^{z}_{22}=-1$.

This equation is the central point of this $\acute{E}tude$ and the central
result of the paper.

The large-$n$ asymptotes and the $1/n$ expansion for the coefficients
$a_{n}^{(d)}$ can be straightforwardly performed using the Sadovskii-
Kuchinskii-Suslov [4, 9, 10] method which leads to the following final result:

$\boxed{a_{n}^{(d)}\xrightarrow[]{n\gg
1}\frac{1}{e^{d}}\left[1-\frac{f(d)}{n}+O\left(\frac{1}{n^{2}}\right)\right]\frac{2^{n+1}\Gamma(n+1+d/2)}{\Gamma(d/2)},}$
(182)

where the function $f(d)$ is computed in the previous subsection for two
particular cases: $f(d=1)=5/4$ and $f(d=2)=1$. We leave the problem of
determining $f(d)$ for arbitrary $d$ as an exercise for readers.

### Enumeration of ”skeleton” diagrams for the T-matrix

The T-matrix is related to the self-energy by the following relation:
$T_{d}\cdot G_{0}=\Sigma\cdot G$. Equivalently, for the dimensionless T-matrix
function of the dimensionless variable $z$, $\textgoth{T}_{d}\cdot
z^{-1}=\sigma_{d}\cdot\Psi_{d}$:

$\textgoth{T}_{d}(z)=z\Psi_{d}(z)\sigma_{d}(z)=z^{2}\Psi_{d}(z)-z.$ (183)

Combinatorics of the diagrams for the T-matrix is defined through the
equation:

$\boxed{\textgoth{T}_{d}[\Psi_{d}]=\sum_{n=0}^{\infty}a^{(d)}_{n}\Psi_{d}^{2n+1}+\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}a^{(d)}_{n}a^{(d)}_{m}\Psi_{d}^{2(n+m)+3}.}$
(184)

Therefore, there is no need to construct independent equations for the
T-matrix combinatorics since it can be determined from the number given by the
coefficients $a^{(d)}_{n}$:

$\displaystyle{\cal
T}_{d}[\Psi_{d}]=\sum_{n=0}^{\infty}\left[a^{(2)}_{n}+\sum_{m=0}^{\infty}a^{(d)}_{m}\Psi_{d}^{2(m+1)}\right]\Psi_{d}^{2n+1}=\sum_{n=0}^{\infty}r^{(d)}_{n}\Psi_{d}^{2n+1}$
(185)

with

$\boxed{r^{(d)}_{n}=a^{(d)}_{n}+\sum_{m=0}^{n-1}a^{(d)}_{m}a^{(d)}_{n-m-1}.}$
(186)

Thus, the combinatorics of the coefficients $a^{(d)}_{n}$ for the self-energy
fully determines the combinatorics of the vertex part and the single particle
T-matrix.

##  10 $Coda.$ Concluding remarks

To summarize, let us discuss what we can learn from playing
$Seven\;\acute{E}tudes$ on dynamical Keldysh model.

  * •

The single particle Green’s function of the Keldysh models with a
$d$-component non-Markovian classical random Gaussian field satisfies the
first order ordinary linear differential equation. The Dyson equations can be
obtained in the closed form due to the Ward Identities and solved exactly for
an arbitrary number of the random field components.

  * •

The exact solution for the single-particle Green’s function corresponds to an
ensemble Gaussian averaging of the single particle Green’s function over the
Gaussian realizations of the random potential.

  * •

The imaginary part of the single particle GF (proportional to the Density of
States) has the Gaussian shape.

  * •

If the number of components is $d>1$, there exists a ”centrifugal” term in the
Dyson equation responsible for level repulsion. The level repulsion becomes
more pronounced when the number component of the random fields grows.

  * •

All Keldysh models with even numbers of components contain intrinsic non-
analyticity in the single particle correlators in the zero frequency limit due
to the Rayleigh-type distribution. The single particle correlators of the
Keldysh models with odd numbers of random field components are analytic at all
frequencies due to the Gaussian distribution.

  * •

The total number of skeleton Feynman diagrams in a given order of expansion
with respect to a multi-component random potential satisfy some recurrence
equations containing linear, bilinear and cubic terms. The single component
Keldysh model has some additional degeneracy as both linear and cubic terms
are absent.

  * •

The combinatorics of the skeleton Feynman diagrams for the self-energy
completely determines the corresponding combinatorics of the full vertices and
the single-particle T-matrices.

Multi-component Keldysh models describe the behaviour of complex (double,
triple- etc) quantum dots subject to slowly fluctuating electric potentials
which determine the shape of the quantum wells and inter-dot barriers. The
application of the Keldysh model to computing dynamical correlation functions
of many-body Fermi and Bose systems, as well as some other realizations of the
Keldysh model in theories with multi-component synthetic gauge fields [30, 34,
35, 31, 32, 33], polaron problem [36, 37, 38] and Sachdev-Ye-Kitaev [39] model
as well as applications to Weyl semimetals [40, 41, 42] opens interesting
directions for future research.

##  11 $Acknowledgements$

The Authors are thankful to late Kostya Kikoin for many years of fruitful
collaboration on numerous topics of modern condensed matter physics including
but not limited by dynamical symmetries and the Keldysh model. We are grateful
to Boris Altshuler for very fruitful discussions on disordered systems and
Nikolay Prokof’ev and Boris Svistunov for teaching the basics of the bold
diagrammatic Quantum Monte Carlo method. DE thanks DFG (project number
405940956) for the partial financial support. The work of MK is conducted
within the framework of the Trieste Institute for Theoretical Quantum
Technologies (TQT).

##  12 $Mathematical\;\;Instrumentation$

### Gell-Mann matrices

The Gell-Mann matrices constitute the fundamental representation of the group
$SU(3)$. The eight generators satisfy the commutation relations:

$[\Lambda_{\alpha},\Lambda_{\beta}]=if^{\lambda}_{\alpha\beta\gamma}\Lambda_{\gamma},$
(A1)
$\\{\Lambda_{\alpha},\Lambda_{\beta}\\}=\frac{1}{3}\delta_{\alpha\beta}\mathbb{I}+d^{\lambda}_{\alpha\beta\gamma}\Lambda_{\gamma}.$
(A2)

We refer to the conventional form of the Gell-Mann matrices as the
$\lambda$-basis, $\Lambda_{\alpha}=\lambda_{\alpha}/2$.:

$\displaystyle\lambda_{1}=\left({\begin{array}[]{*{20}c}0&1&0\\\ 1&0&0\\\
0&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace}\lambda_{2}=\left({\begin{array}[]{*{20}c}0&-i&0\\\
i&0&0\\\ 0&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\lambda_{3}=\left({\begin{array}[]{*{20}c}1&0&0\\\
0&-1&0\\\ 0&0&0\\\ \end{array}}\right),$ (A12)
$\displaystyle\lambda_{4}=\left({\begin{array}[]{*{20}c}0&0&1\\\ 0&0&0\\\
1&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\lambda_{5}=\left({\begin{array}[]{*{20}c}0&0&-i\\\
0&0&0\\\ i&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\lambda_{6}=\left({\begin{array}[]{*{20}c}0&0&0\\\
0&0&1\\\ 0&1&0\\\ \end{array}}\right),$ (A22)
$\displaystyle\lambda_{7}=\left({\begin{array}[]{*{20}c}0&0&0\\\ 0&0&-i\\\
0&i&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\lambda_{8}=\frac{1}{\sqrt{3}}\left({\begin{array}[]{*{20}c}1&0&0\\\
0&1&0\\\ 0&0&-2\\\ \end{array}}\right).$ (A29)

The $\lambda$\- matrices are normalized by the identity

$\displaystyle Tr(\lambda_{\alpha}\lambda_{\beta})=2\delta_{\alpha\beta}.$
(A30)

The structure constants are

$\displaystyle f^{\lambda}_{\alpha\beta\gamma}=\frac{1}{4i}{\rm
Tr}([\lambda_{\alpha},\lambda_{\beta}]\cdot\lambda_{\gamma}),$ (A31)
$\displaystyle d^{\lambda}_{\alpha\beta\gamma}=\frac{1}{4}{\rm
Tr}(\\{\lambda_{\alpha},\lambda_{\beta}\\}\cdot\lambda_{\gamma}).$ (A32)

It is convenient to use the rotated basis of the Gell-Mann matrices [12]. The
rotation is performed to embed the representation $S=1$ of the group $SU(2)$
as first three generators of the group $SU(3)$. We refer to the rotated
representation of Gell-Mann matrices as the $\mu$\- basis,
$M_{\alpha}=\mu_{\alpha}/2$:

$[M_{\alpha},M_{\beta}]=if^{\mu}_{\alpha\beta\gamma}M_{\gamma},$ (A33)
$\\{M_{\alpha},M_{\beta}\\}=\frac{1}{3}\delta_{\alpha\beta}\mathbb{I}+d^{\mu}_{\alpha\beta\gamma}M_{\gamma},$
(A34)

where

$\displaystyle\mu_{1}=\frac{1}{\sqrt{2}}\left({\begin{array}[]{*{20}c}0&1&0\\\
1&0&1\\\ 0&1&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace}\mu_{2}=\frac{1}{\sqrt{2}}\left({\begin{array}[]{*{20}c}0&-i&0\\\
i&0&-i\\\ 0&i&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\mu_{3}=\left({\begin{array}[]{*{20}c}1&0&0\\\
0&0&0\\\ 0&0&-1\\\ \end{array}}\right),$ (A44)
$\displaystyle\mu_{4}=\left({\begin{array}[]{*{20}c}0&0&1\\\ 0&0&0\\\ 1&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\mu_{5}=\left({\begin{array}[]{*{20}c}0&0&-i\\\
0&0&0\\\ i&0&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\mu_{6}=-\frac{1}{\sqrt{2}}\left({\begin{array}[]{*{20}c}0&1&0\\\
1&0&-1\\\ 0&-1&0\\\ \end{array}}\right),$ (A54)
$\displaystyle\mu_{7}=-\frac{1}{\sqrt{2}}\left({\begin{array}[]{*{20}c}0&-i&0\\\
i&0&i\\\ 0&-i&0\\\
\end{array}}\right)\mathrm{,\thinspace\thinspace\thinspace}\mu_{8}=-\frac{1}{\sqrt{3}}\left({\begin{array}[]{*{20}c}1&0&0\\\
0&-2&0\\\ 0&0&1\\\ \end{array}}\right).$ (A61)

The $\mu$\- matrices are normalized by identity

$\displaystyle Tr(\mu_{\alpha}\mu_{\beta})=2\delta_{\alpha\beta}.$ (A62)

The structure constants are

$\displaystyle f^{\mu}_{\alpha\beta\gamma}=\frac{1}{4i}{\rm
Tr}([\mu_{\alpha},\mu_{\beta}]\cdot\mu_{\gamma}),$ (A63) $\displaystyle
d^{\mu}_{\alpha\beta\gamma}=\frac{1}{4}{\rm
Tr}(\\{\mu_{\alpha},\mu_{\beta}\\}\cdot\mu_{\gamma}).$ (A64)

The connections between two basis are

$\displaystyle\mu_{1}=(\lambda_{1}+\lambda_{6})/\sqrt{2},\;\;\;\mu_{2}=(\lambda_{2}+\lambda_{7})/\sqrt{2},\;\;\;\mu_{3}=(\sqrt{3}\lambda_{8}+\lambda_{3})/2,\;\;\;\mu_{4}=\lambda_{4},$
$\displaystyle\mu_{5}=\lambda_{5},\;\;\;\mu_{6}=-(\lambda_{1}-\lambda_{6})/\sqrt{2},\;\;\;\mu_{7}=-(\lambda_{2}-\lambda_{7})/\sqrt{2},\;\;\;\;\mu_{8}=(\lambda_{8}-\sqrt{3}\lambda_{3})/2.$
(A65)

### Hilbert transform, Dawson function and exponential integral

In this Instrumentation we summarize the equations for the Hilbert transform
representation of the $d$\- component Keldysh model Green’s functions

  * •

Odd $d=2n+1$ number of the random Gaussian field components
The Hilbert transform $H_{n}[u^{2n}e^{-u^{2}}](z)$ is defined on a line
$-\infty\infty$

$H_{n}(z)=\frac{1}{\pi}{\rm
P.V.}\int_{-\infty}^{\infty}\frac{u^{2n}e^{-u^{2}}}{z-u}du.$ (B1)

Introducing

$H^{a}(z)=\frac{1}{\pi}{\rm
P.V.}\int_{-\infty}^{\infty}\frac{e^{-ax^{2}}}{z-x}dx=\frac{2}{\sqrt{\pi}}F(z\sqrt{a})$
(B2)

we relate $H_{n}[u^{2n}e^{-u^{2}}](z)$ to the Dawson function

$H_{n}(z)=(-1)^{n}\left.\frac{\partial^{n}H^{a}(z)}{\partial
a^{n}}\right|_{a=1}.$ (B3)

Alternatively $H_{n}$ can be calculated using the recurrence relations

$H_{n+1}(z)=z^{2}H_{n}(z)-z\frac{(2n-1)!!}{2^{n}\sqrt{\pi}}.$ (B4)

  * •

Even $d=2n$ number of the random Gaussian field components

$\psi_{n}(z)=\frac{1}{2}{\rm
P.V.}\int_{0}^{\infty}u^{2n+1}e^{-u^{2}}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right].$
(B5)

We define

$\displaystyle\psi^{a}(z)=\frac{1}{2}{\rm
P.V.}\int_{0}^{\infty}ue^{-au^{2}}du\left[\frac{1}{z-u}+\frac{1}{z+u}\right]=\frac{1}{2}z\;e^{-az^{2}}{\rm
Ei}\left[az^{2}\right].$ (B6)

As a result we relate $\psi_{n}$ and Exponential Integral function derivatives

$\psi_{n}(z)=(-1)^{n}\left.\frac{\partial^{n}\psi^{a}(z)}{\partial
a^{n}}\right|_{a=1}.$ (B7)

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