# InvGAN: Invertible GANs

Partha Ghosh $\,{}^{\dagger}$ &Dominik Zietlow∗† &Michael J. Black‡ Larry S.
Davis ‡ &Xiaochen Hu ‡
$\dagger$ MPI for Intelligent Systems, Tübingen ${\ddagger}$ Amazon.com, Inc.
{pghosh<EMAIL_ADDRESS>{mjblack, lrrydav<EMAIL_ADDRESS>
This work was completed while the author was an intern with Amazon

###### Abstract

Generation of photo-realistic images, semantic editing and representation
learning are a few of many potential applications of high resolution
generative models. Recent progress in GANs have established them as an
excellent choice for such tasks. However, since they do not provide an
inference model, image editing or downstream tasks such as classification can
not be done on real images using the GAN latent space. Despite numerous
efforts to train an inference model or design an iterative method to invert a
pre-trained generator, previous methods are dataset (e.g. human face images)
and architecture (e.g. StyleGAN) specific. These methods are nontrivial to
extend to novel datasets or architectures. We propose a general framework that
is agnostic to architecture and datasets. Our key insight is that, by training
the inference and the generative model together, we allow them to adapt to
each other and to converge to a better quality model. Our InvGAN, short for
Invertible GAN, successfully embeds real images to the latent space of a high
quality generative model. This allows us to perform image inpainting, merging,
interpolation and online data augmentation. We demonstrate this with extensive
qualitative and quantitative experiments.

##  1 Introduction

The ability to generate photo-realistic images of objects such as human faces
or fully clothed bodies has wide applications in computer graphics and
computer vision. Traditional computer graphics, based on physical simulation,
often fails to produce photo-realistic images of objects with complicated
geometry and material properties. In contrast, modern data-driven methods,
such as deep learning based generative models show great promise for realistic
image synthesis [22, 23]. Among the four major categories of generative
models, generative adversarial networks (GANs), variational auto-encoders
(VAEs), normalizing flow networks and autoregressive models, GANs deliver
images with the best visual quality. Although recent efforts in VAEs [11, 33]
have tremendously improved their generation quality, they still use larger
latent space dimensions and deliver lower quality images. Autoregressive
models are very slow to sample from and do not provide a latent representation
for the trained data. Finally, flow-based methods do not perform
dimensionality reduction and hence produce large models and latent
representations. On the other hand, GANs do not provide a mechanism to embed
real images into the latent space. This limits them as a tool for image
editing and manipulation. Specifically, while several methods exist, there is
no method that trains a GAN so that it can be efficiently and effectively
inverted. To that end, we propose InvGAN, an invertible GAN with an inference
module that can embed real images into the latent space. InvGAN has wide range
of applications.

Representation Learning. GANs learn a latent representation of the training
data. This representation has been shown to be well structured [8, 22, 23],
allowing GANs to be employed for a variety of downstream tasks (e.g.
classification, regression and other supervised tasks) [27, 32]. We extend the
GAN framework to include an inference model that embeds real images into the
latent space. InvGAN addresses this problem and can be used to support
representation learning [9, 26], data augmentation [8, 14] and algorithmic
fairness [5, 36, 37]. Previous methods of inversion rely on computationally
expensive optimization of inversion processes, limiting their scope to offline
applications, e.g. data augmentation has to happen before training starts.
Efficient, photo-realistic, semantically consistent, and model based inversion
is the key to online and adaptive use-cases.

Conditional Image Editing. Recent work shows that even unsupervised GAN
training isolates several desirable generative characteristics [28, 41].
Prominent examples are correspondences between latent space directions and
e.g. hair style, skin tone and other visual characteristics. Recent works
provide empirical evidence suggesting that one can find paths in the latent
space (albeit non-linear) that allow for editing individual semantic aspects.
GANs therefor have the potential to become a high-quality graphics editing
tool [16, 39]. However without a reliable mechanism for projecting real images
into the latent space of the generative model, editing of real data is
impossible. InvGAN take a step towards addressing this problem.

##  2 Related work

The GAN inversion task has been addressed in two primary ways (1) using an
inversion model (often a deep neural network), (2) embedding real images into
the latent space of a trained generator using an iterative optimization based
method, typically initialized with a deep model.

Optimization based: iGAN [49] optimizes for a latent code while minimizing the
distance between a generated image and a source image. To ensure uniqueness of
the preimage of a GAN-generated data point, Zachary et al. [25] employ
stochastic clipping. As the complexity of the GAN generators increases, an
inversion process based on gradient descent and pixel space MSE is
insufficient. Addressing this, Rameen et al. specifically target StyleGAN
generators and optimize for perceptual loss [2, 1]. However, they invert into
the $W+$ space, the so called extended $w$ space of StyleGAN. This results in
high dimensional latent codes and consequently prolongs inversion time. This
can also produce out-of-distribution latent representations which makes them
unsuitable for downstream tasks. Contrary to these drawbacks InvGAN offers
fast inference embedding in the non-extended latent space.

Model based: BiGAN [12] and ALI [15] invert the generator of a GAN during the
training process by learning the joint distribution of the latent vector and
the data in a completely adversarial setting. However, the quality is limited,
partially because of the choice of DCGAN [31] and partially because of the
significant dimensionality and distribution diversity between the latent
variable and the data domain [13]. More recent models target the StyleGAN
architecture [34, 42, 48] and achieve impressive results. Most leverage
StyleGAN peculiarities, i.e. they invert in the $W+$ space – so adaptation to
other GAN backbones is non-trivial. Adversarial latent auto-encoders [30], are
closest to our current work. Our model and adversarial autoencoders can be
made equivalent with a few alterations to the architecture and to the
optimization objective. We discuss this more in detail in Section 3.2. Our
method in contrast to previous works discussed in this section, neither uses
any data set specific loss nor does it depend upon any specific network
architecture.

Hybrid optimization and regression based: Guan et al. [19] train a regressor
that is used to initialize an optimization-based method to refine the
regressor’s guess. However, to achieve good results, this method uses an
identity loss to guide the refinement procedure making it specific to human
face datasets. Zhu et al. [47] modify the general hybrid approach with an
additional criterion such that the recovered latent code must belong to the
semantically meaningful region learned by the generator. It is thereby assumed
that the real image can be reconstructed more faithfully in the immediate
neighbourhood of this initial guess. Yuval et al. [4] replace gradient-based
optimization with an iterative encoder that encodes the current generation and
target image to the estimated latent code difference. They empirically show
that this iterative process converges and the recovered image improves over
iterations. However, this method requires multiple forward passes in order to
achieve a suitable latent code. In contrast to the work above, the inference
module obtained by our method infers the latent code in one shot. Hence it is
much faster and does not run the risk of finding a non-meaningful latent code.

The inversion mechanisms presented so far do not directly influence the
generative process. In most of the cases, they are conducted on a pre-trained
frozen generator. Although in the case of ALI [15] and BiGAN [12], the
inference model loosely interacts with the generative model at training time.
However, the interaction is only indirect; i.e. through the discriminator. In
our work we tightly couple the inference module with the generative module,
resulting in better reconstruction quality.

Joint training of generator and inference model: We postulate that jointly
training an inference module will help regularize GAN generators towards
invertability. This is inspired by the difficulty of inverting a pre-trained
high-performance GAN. For instance Bau et al. [6] invert PGAN [21], but for
best results a two-stage mechanism is needed. Similarly Image2StyleGAN [3]
projects real images into the extend $w^{+}$ space of StyleGAN, whereas,
arguably, all the generated images can be generated from the more compact $z$
or $w$ space. This is further evident from Wulff et al. [43] who find an
intermediate latent space in StyleGAN that is more Gaussian than the assumed
prior. However, they too use an optimization-based method and, hence, it is
computationally expensive and at the same time specific to both the StyleGAN
backend and the specific data set. Finally we refer the readers to ‘GAN
Inversion: A Survey’ [44] for a comprehensive review of relate work.

##  3 Method

Goal: Our goal is to learn an inversion module alongside the generator during
GAN training. Specifically we find a generator
$G:\mathbb{W}\rightarrow\mathbb{X}$ and an inference model
$D:\mathbb{X}\rightarrow\mathbb{W}$ such that $x\approx
G(D(x\sim\mathbb{X}))$. Where $\mathbb{X}$ denotes the data domain and
$\mathbb{W}$ denotes the latent space. By doing so we (1) unlock semantic
editability of real images, (2) allow semi-supervised learning, (3) encourage
latent space smoothness, within the GAN framework.

###  3.1 Architecture

We demonstrate InvGAN using DC-GAN, BigGAN and StyleGAN as the underlying
architectures. Figure 1 represents the schematic of our model. We follow the
traditional alternative generator-discriminator training mechanism. The
generative part consists of three steps $z\sim\mathcal{N}(0,I);w=M(z);x=G(w)$,
where $M$ is a mapping network, $G$ is the generator, $D$ is the
discriminator, and $\mathcal{N}(0,I)$ is the standard normal distribution. In
the generator, we use the standard $8$-layer mapping network with StyleGAN and
add a $2$-layer mapping network to BigGAN and DC-GAN. The discriminator,
besides outputing real/fake score, also outputs inferred $w$ parameter. From
here on we use $\tilde{w},c=D(x)$ to denote the inferred latent code
($\tilde{w}$) using the discriminator $D$ and $c$ to denote the real-fake
classification decision for the sample $x\in\mathbb{X}$. Wherever obvious we
simply use $D(x)$ to refer to $c$, the discrimination decision only.

###  3.2 Objective

GAN Objective: The min-max game between the discriminator network and the
generator network of vanilla GAN training is described as

$\min_{G,M}\max_{D}\mathcal{L}_{\text{GAN}}=\min_{G,M}\max_{D}\left[\mathbb{E}_{x\in\mathbb{X}}[\log
D(x)]+\mathbb{E}_{z\in\mathbb{Z}}[\log(1-D(G(M(z))))]\right].$ (1)

A naive attempt at an approximately invertible GAN would perform
$\min_{G}\max_{D}L_{\text{GAN}}+\min_{D}\left\lVert
w-\tilde{w}\right\rVert_{p}$, where $\left\lVert\bullet\right\rVert_{p}$
denotes an $L_{p}$ norm. This loss function can be interpreted as an optimal
transport cost. We discuss this in more detail at the end of this section.
However, this arrangement, coined the "naive model", does not yield
satisfactory results, cf. Section 4.4. This can be attributed to two factors:
(1) $w$ corresponding to real images are never seen by the generator; (2) no
training signal is provided to the discriminator for inferring the latent code
corresponding to real images ($w_{R}$); (3) the distribution of $w_{R}$ might
differ from prior distribution of $w$. We address each of these concerns with
a specific design choice. Our naive model corresponds to the adversarial
autoencoders [30] if the real-fake decision is derived from a common latent
representation. However, this forces the encoding of real and generated images
to be linearly separable and contributes to degraded inference performance.

Minimizing latent space extrapolation: Since, in the naive version, neither
the generator nor the discriminator gets trained with $w_{R}$, it relies
completely upon its extrapolation characteristics. In order to reduce the
distribution mismatch for the generator we draw half the mini batch of latent
codes from the prior and the other half consists of $w_{R}$; i.e.
$w_{\text{total}}=w\mathbin{+\mkern-10.0mu+}w_{R},\ w\sim P(W)$ where
$\mathbin{+\mkern-10.0mu+}$ denotes a batch concatenation operation. By $w\sim
P(W)$ we denote the two stage process given by the following $w=M(z\sim
P(Z))$.

Pixel space reconstruction loss: Since latent codes for real images are not
given, the discriminator cannot be trained directly. However, we recover a
self-supervised training signal by allowing the gradients from the generator
to flow into the discriminator. Intuitively, the discriminator tries to infer
latent codes from real images that help the generator reproduce that image. As
shown in Section 4.4, this helps improving real image inversion tremendously.
We enforce further consistency by imposing an image domain reconstruction loss
between input and reconstructed real images. However, designing a meaningful
distance function for images is a non-trivial task. Ideally we would like a
feature extractor function $f$ that extracts low- and high-level features from
the image such that two images can be compared meaningfully. Given such a
function a reconstruction loss can be constructed as

$\mathcal{L}_{\text{fm}}=\left\lVert\mathbb{E}_{x\in\mathbb{X}}(f(x)-f(G(w\sim
P(W|x)))\right\rVert_{p}$ (2)

A common practice in the literature is to use a pre-trained VGG [20, 46]
network as a feature extractor $f$. However it is well known that deep neural
networks are susceptible to adversarial perturbations. Given this weakness,
optimizing for perceptual loss is error prone. Hence a combination of a pixel-
domain $L_{2}$ and feature-space loss is typically used. This often results in
degraded quality. Consequently, we take the discriminator itself as the
feature extractor function $f$. Due to the min-max setting of GAN training, we
are guaranteed to avoid the perils of adversarial and fooling samples. The
feature loss is shown in the second half of Figure 1. Although this resembles
the feature matching described by Salimans et al. [35], it has a crucial
difference. As seen in Equation 2 the latent code fed into the generator is
drawn from the conditional distribution $P(W|x):=\delta_{D(x)}(w)$ rather than
the prior $P(W)$, where $\delta(x)$ represents the Dirac delta function
located at $x$. This forces the distribution of the features to match more
precisely as compared to the simple first-moment matching proposed by Salimans
et al. in [35].

Addressing mismatch between prior and posterior: Finally, we address the
possibility of mismatch between inferred and prior latent distributions (point
(3) described above), by imposing a maximum mean discrepancy (MMD) loss
between the sets of samples of the said two distributions. We use an RBF
kernel to compute this loss. This loss improves the random sampling quality by
providing a direct learning signal to the mapping network.

Figure 1: We train InvGAN following a regular GAN. We use a second output head
in the discriminator besides the real fake decision head, to infer the latent-
code $z$ of a given image. Here $\varobslash$ denotes no gradient propagation
during back propagation step. It also denotes ‘no training’ when it is placed
on a model. We use red color to show data flow corresponding to real images.

Finally we summarize the objective of our complete model as in Equation 3.
Here and in the rest of the paper we use a plus operator $+$, between two
optimization process to indicate that both of them are performed
simultaneously.

$\min_{G,M}\left[\max_{D}\mathcal{L}_{\text{GAN}}+\min_{D}\left[\mathbb{E}_{w\mathbin{+\mkern-10.0mu+}w_{R}}\left[\left\lVert
M(z)-\tilde{w}\right\rVert_{2}^{2}+\mathcal{L}_{\mathrm{fm}}+\left\lVert\tilde{w}-\tilde{\tilde{w}}\right\rVert_{2}^{2}+\mathrm{MMD}\\{w,w_{R}\\}\right]\right]\right]$
(3)

An optimal transport based interpretation: Neglecting the last three terms
described in Equation 3, our method can be interpreted as a Wasserstein
autoencoder-GAN (WAE-GAN) [40]. Considering a WAE with its data domain set to
our latent space and its latent space assigned to our image domain, if the
encoder and the discriminator share weights the analogy is complete. Our model
can, hence, be thought of as learning the latent variable model $P(W)$ by
randomly sampling a data point $x\sim\mathbb{X}$ from the training set and
mapping it to a latent code $w$ via a deterministic transformation. In terms
of density it can be written as in Equation 4.

$P(W):=\int_{x\in\mathbb{X}}P(w|x)P(x)\mathrm{d}x.$ (4)

As proven by Olivier, al. [7], under this model the optimal transport problem
$W_{c}(P(W),P_{D}(W)):=\inf_{\Gamma\in P(w_{1}\sim P(W),w_{2}\sim
P_{D}(W))}\left[\mathbb{E}_{w_{1},w_{2}\sim\Gamma}\left[c(w_{1},w_{2})\right]\right]$
can be solved by finding a generative model $G(X|W)$ such that its $X$
marginal, $P_{G}(X)=\mathbb{E}_{w\sim P(W)}G(X|w)$ matches the image
distribution $P(X)$. We ensure this by considering the Jensen–Shannon
divergence $D_{\text{JS}}(P_{G}(X),P(X))$ using a GAN framework. This leads to
the cost function given in Equation 5, when we choose the ground cost function
$c(w_{1},w_{2})$ to be squared $L_{2}$ norm.

$\min_{G,M}\max_{D}\mathcal{L}_{\mathrm{GAN}}+\min_{G,M}\min_{D}\left\lVert
w-\tilde{w}\right\rVert_{2}^{2}$ (5)

Finally we find that by running the encoding/decoding cycle one more time we
can impose several constraints that improve the quality of the encoder and the
decoder network in practice. This leads to our full optimization criterion as
described in Equation 3.

###  3.3 Dealing with resolutions higher than training resolution

Although StyleGAN [23] and BigGAN [8] have shown that it is possible to
generate relatively high resolution images, in the range of $1024\times 1024$
and $512\times 512$, their training is resource intense and the models are
difficult to tune for new data sets. Equipped with invertability, we explore a
tiling strategy to improve output resolution. First, we train an invertible
GAN at a lower resolution ($m\times m$) and simply tile them $n\times n$ times
with $n^{2}$ latent codes to obtain a higher resolution ($mn\times mn$) final
output image. The new latent space containing $n^{2}$ latent codes obtained
using the inference mechanism of the invertible GAN can now be used for
various purposes as described in Section 4.3 and reconstructions are
visualized in Figure 5. This process correlates in spirit somewhat to COCO-GAN
[24]. The main difference however is that our model at no point learns to
assemble neighbouring patches. Indeed the seams are visible if one squints at
the generated images, e.g in figure 5. However, a detailed study of tiling for
generation of higher resolution images than input domain is beyond the scope
of our paper. We simply explore some naive settings and their applications in
section 4.3.

##  4 Experiments

We test InvGAN on several diverse datasets (MNIST, ImageNet, CelebA, FFHQ) and
multiple backbone architectures (DC-GAN, BigGAN, StyleGAN). Our method is
evaluated both qualitatively (via style mixing, image inpainting etc.) and
quantitatively (via the FID score and the suitability for data augmentation
for discriminative tasks such as classification). Table 1 shows random sample
FIDs, middle point linear interpolation FIDs and test set reconstruction mean
absolute errors (MAEs) of our generative model. We intend to provide a
definition, baseline and understanding of inversion of a high quality
generator. Specifically we highlight model-based inversion, joint training of
generative and inference model and its usability in downstream tasks. We
demonstrate that our method generalizes across architectures, datasets and
types of downstream task.

Models RandFID
RandRecFID TsRecFID IntTsFID MAE$\pm 1$ Run Time FFHQ [45] 49.65/14.59
56.71/23.93 -/13.73 68.45/38.01 0.129 0.045 FFHQ Enc.[47] 46.82/14.38 -/-
88.48 / - -/- 0.460 FFHQ MSE opt.[47] 46.82/14.38 -/- 58.04 / - -/- 0.106 FFHQ
In-D. Inv.[47] 46.82/14.38 52.02/- 42.64 / - 71.83/- 0.115 99.76 DCGAN, MNIST
$17.44/6.10$ $16.76/4.25$ $17.77/4.70$ $26.04/11.44$ $0.070$ $3.3\cdot
10^{-5}$ StyleGAN, CelebA $26.63/4.81$ $24.35/3.51$ $24.37/4.14$ $32.37/15.60$
$0.150$ $1.0\cdot 10^{-3}$ StyleGAN, FFHQ $49.14/12.12$ $44.42/8.85$
$41.14/7.15$ $49.52/14.36$ $0.255$ $2.0\cdot 10^{-3}$

Table 1: Here we report random sample FID (RandFID), FID of reconstructed
random samples (RandRecFID), FID of reconstructed test set samples (TsRecFID),
FID of the linear middle interpolation of test set images (IntTsFID) and
reconstruction per pixel per color channel mean absolute error when images are
normalized between $\pm 1$, also from test set. All FID scores are here
evaluated against train set using 500 and 50000 samples. They are separated by
‘/’. For the traditional MSE optimization based and In-Domain GAN inversion,
the MSE errors are converted to MAE by taking square root and averaging over
the color channels and accounting for the re-normalization of pixel values
between $\pm 1$. Runtime is given in seconds per image. We ran them on a V100
32GB GPU and measured wall clock time.

Training data and tasks: We start with a StyleGAN-based architecture on FFHQ
and CelebA for image editing. Then we train a BigGAN-based architecture on
ImageNet, and show super resolution and video key-framing by tiling in the
latent domain to work with images and videos that have higher resolution than
training data. We also show ablation studies with a DC-GAN-based architecture
on MNIST. This variety of architecture, dataset, and task provide insights
into the method and its generality. In the following sections we evaluate
qualitatively by visualizing semantic editing of real images and
quantitatively on various downstream tasks including classification fairness,
image super resolution, image mixing, etc.

###  4.1 Semantically consistent inversion using InvGAN

GANs can be used to augment training data and substantially improve downstream
tasks learning. Improving fairness of classifiers on human face images is a
prominent example [37, 36, 5, 32]. There is an important shortcoming in using
existing GAN approaches for such tasks: the labeling of augmented data relies
on methods that are trained independently on the original data set, using
human annotators or compute-expensive optimization-based inversion. This is
due to the fact that most generative models used are unconditional and so
generate unlabeled synthetic data. A typical example is data-set de-biasing by
Ramaswamy et al. [32]. For each training image, an altered example that
differs in some attribute (e.g. age, hair color, …) has to be generated. This
has previously been done in one of two ways, e.g. by finding the latent
representation of the ground truth image via optimization or by labeling
random samples using pre-trained classifiers on the biased data set.
Optimization-based methods are slow and not a viable option for on-
demand/adaptive data augmentation. Methods using pre-trained classifiers
inherit their flaws like correlation induced dependencies.

Here we focus on the subproblem of reliably encoding face images to the latent
space in a semantically consistent manner using InvGAN. For this we train
ResNet50 attribute classifiers on the CelebA dataset. We validate that the
encoding and decoding of InvGAN results in a semantically consistent
reconstruction by training the classifier only on reconstructions of the full
training set. As a baseline, we use the same classifier trained on the
original CelebA. We produce two reconstructed training sets by using the
tiling based inversion (trained on ImageNet) and by training InvGAN on CelebA
(without tiling). For each attribute a separate classifier has been trained
for $20$ epochs. The resulting mean average precisions are reported in Table
2. We see that training on the reconstructions allows for very good domain
transfer to real images, indicating that the reconstruction process maintained
the semantics of the images.

Evaluated on | Original | Tile Reconstruction | Full Reconstruction
---|---|---|---
Original | $0.81\pm 0.15$ | $0.77\pm 0.16$ | $0.79\pm 0.15$
Tile recon. | $0.79\pm 0.16$ | $0.80\pm 0.15$ | $0.78\pm 0.16$
Full recon. | $0.81\pm 0.15$ | $0.78\pm 0.16$ | $0.81\pm 0.14$
Recon. Vis. |  |  |
Table 2: Mean average precision for a ResNet50 attribute classifier on CelebA,
averaged over 20 attributes. We report the performance for training on the
original dataset, the reconstructed dataset using the tiling based method pre-
trained on ImageNet and the reconstruction on InvGAN trained on the CelebA
training set directly.

###  4.2 Suitability for image editing

GAN inversion methods have been proposed for machine supported photo editing
tasks [47, 10, 29]. Although there is hardly any quantitative evaluation for
the suitability of a specific inversion algorithm or model, a variety of
representative operations have been reported [1, 2, 47]. Amongst those are in-
painting cut out regions of an image, image-merging and improving on
amateurish manual photo editing. Figures 2 and 8 in appendix visualize those
operations performed on FFHQ and CelebaA images respectively. We demonstrate
in-painting by zeroing out a randomly positioned square patch and then simply
reconstructing the image. This can be interpreted as a image-repair
operation/correcting imperfections in unseen data. The image-merging is
performed by reconstructing an image which is composed out of two images by
simply placing them together. By reconstructing an image that has undergone
manual photo editing, higher degrees of photo-realism are achieved.
Quantitative metric for such tasks are hard to define and hence is scarcely
found in prior art, since they depend upon visual quality of the results. We
report reconstruction and interpolation FIDs in Table 1, in an effort to
establish a baseline for future research. However, wee do acknowledge that a
boost in pixel fidelity in our reconstruction will greatly boost the
performance of InvGAN on photo editing tasks. The experiments clearly show the
general suitability of the learned representations to project out of
distribution images to the learned posterior manifold via reconstruction.

Style mixing In-painting Editing Merging I Merging II

Figure 2: Benchmark image editing tasks on FFHQ ($128\,\mathrm{px}$). Style
mixing: We transfer the first $0,1,2,\dots,11$ style vectors from one image to
another. For the other image editing tasks, pairs of images are input image
(left) and reconstruction (right).

###  4.3 Tiling to boost resolution

Limitations in video RAM and instability of high resolution GANs are prominent
obstacles in generative model training. One way to bypass such difficulties is
to generate the image in parts. Here we train our invertible generative model,
a BigGAN architecture on $32\times 32$ random patches from ImageNet. Once the
inversion mechanism and the generator are trained to satisfactory quality, we
reconstruct both FFHQ and ImageNet images. We use $256\times 256$ resolution
and tiling $64$ patches in an $8\times 8$ grid for FFHQ images, and $128\times
128$ resolution and tiling $16$ patches in a $4\times 4$ grid for ImageNet
images. The reconstruction results are shown in Figure 5. Given the successful
reconstruction process, we explore the tiled latent space for tasks such as
image deblurring and time interpolation of video frames.

Image de-blurring: Here we take a low resolution image, scale it to the
intended resolution using bicubic interpolation, invert it patch by patch,
gaussian blur it, invert it again and linearly extrapolate it in the
deblurring direction. The deblurring direction is simply obtained by
subtracting the latent code of the given low resolution but bicubic up sampled
image from the latent code of the blurred version of it at the same
resolution. The exact amount of extrapolation desired is left up to the user.
As shown in Figure 3 we show the effect of $3$ different levels of
extrapolation. Although our method is not trained for the task of super
resolution, by virtue of a meaningful latent space we can enhance image
quality.

Figure 3: Super resolution using extrapolation in the tiled latent space. From
left we visualize the original image, the low resolution version of it, the
reconstruction of the low resolution version, and progressive extrapolation to
achieve debluring.

Temporal interpolation of video frames: Here we boost the frame rate of a
video post capture. We infer the tiled latent space of consecutive frames in a
video and linearly interpolate each tile to generate one or more intermediate
frames. Results are shown in the accompanying videos in the supplementary
material and in Figure 6 in the appendix. This can be used to create slow
motion video, post capture. We find the latent code of each frame in a video
sequence and then derive intermediate latent codes by weighted averaging
neighbouring latent codes using a Gaussian window. Even though this
effectively interpolates between latent codes for background patch with
foreground patch for fast moving small objects leading to blur, it results in
smooth slow motion video, as can be seen in the supplementary material . We
use the UCF101 data set [38] for this task.

(a)

(b)

Figure 4: Tiled reconstruction of random (a) FFHQ Images and (b) ImageNet
images. Left column shows the real images, the second shows the patch by patch
reconstructions and the third shows the absolute pixel-wise differences. Note
that interestingly though the patches are reconstructed independent of each
other, the errors lie mostly on the edges of the objects in the images,
arguably the most information dense region of the images.

###  4.4 Ablation studies

Recall that the naive model defined in Section 3.2 uses the following
optimization $\min_{G,M}\left[\max_{D}L_{\text{GAN}}+\min_{D}\mathbb{E}_{z\sim
P(Z)}\left\lVert M(z)-\tilde{w}\right\rVert_{p}\right]$ to train (also given
in Equation 5), (results in Figure 5(a)). Here we progressively show how our
three main components contributes on the naive model. As is apparent from the
method section, the first major improvement comes from exposing the generator
to the latent code inferred from real images. This is primarily due to the
difference in the prior and the induced posterior distribution. This is
especially true during early training, which imparts a lasting impact. The
corresponding optimization is
$\min_{G,M}\left[\max_{D}L_{\text{GAN}}+\min_{D}\mathbb{E}_{w=M(z\sim
P(Z))\mathbin{+\mkern-10.0mu+}w_{R}}\left\lVert
w-\tilde{w}\right\rVert_{p}\right]$. Simply reducing the distribution mismatch
between prior and posterior by injecting inferred latent codes improves
inversion quality. This is visualized in Figure 5(b). We shall call this model
as the augmented naive model. However, this modification unlocks the
possibility to enforce back propagation of generator loss gradients to the
discriminator and real-image, generated-image pairing as detailed in Section
3.2. This leads to our full model and the results are visualized in Figure
5(c)

(a)

(b)

(c)

Figure 5: Inversion of held out test samples. Columns are in groups of three:
first column holds real images, second their reconstruction and third the
absolute pixel-wise difference. (a) Inversion using naive model, i.e. only z
reconstruction loss is used, (b) inversion using model that uses latent codes
from real samples, i.e. the augmented naive model. (c) our full model. Notice
how the imperfections in the reconstructions highlighted with red boxes
gradually vanishes as the model improves.

##  5 Discussion and future work

While InvGAN can reliably invert the generator of a GAN, it still can benefit
from an improved reconstruction fidelity for tasks such as, image compression,
image segmentation, etc. We observe that the reconstruction of rare features,
such as microphones, hats or background features, tend to have lower quality,
as seen in appendix in Figure 7 bottom row 3rd and 4th column. This combined
with the fact that reconstruction loss during training tends to saturate even
when the weights are sufficiently high indicates that even well-engineered
architectures such as StyleGAN and BigGAN lack representative power to provide
sufficient data coverage.

Strong inductive biases in the generative model have the potential to improve
the quality of the inference module. For instance GIF [16] and hologan [28]
among others introduce strong inductive bias from the underlying 3D geometry
and lighting properties of a 2D image. Hence an inverse module of these
generative mechanism has the potential to outperform their counterparts, which
are trained fully supervised on the labelled training data alone at estimating
3D face parameters from 2D images.

As was shown by the success of RAEs [18], there is often a mismatch between
the induced posterior and the prior of generative models which can be removed
by an ex-post density estimator. InvGAN is also aminable to ex-post density
estimation. When applied to the tiled latent codes, it estimates a joint
density of the tiles for unseen data. This would recover a generative model
without going through the unstable GAN training.

We have shown that our method scales to large datasets such as ImageNet,
CelebA, and FFHQ. A future work that is able to improve upon reconstruction
fidelity, would be able explore adversarial robustness by extending [17] to
larger datasets.

##  6 Conclusion

We presented InvGAN, an inference framework for the latent generative
parameters used by a GAN generator. InvGAN enjoys several advantages compared
to state-of-the-art inversion mechanisms. The inversion mechanism is
integrated into the training phase of the generator, potentially contributing
to the disentanglement of the latent space. Beyond the computational advantage
of model-based inversion, our mechanism can reconstruct images that are larger
than the training images by tiling with no additional merging needed. We
further demonstrated that the inferred latent code for a given image is
semantically meaningful i.e. it falls inside the structured part of the latent
space learned by the generator.

## Acknowledgments and Disclosure of Funding

We thank Alex Vorobiov, Javier Romero, Betty Mohler Tesch and Soubhik Sanyal
for their insightful comments and intriguing discussions. This research was
fully funded and supported by Amazon. While PG and DZ are affiliated with Max
Planck Institute for Intelligent Systems, this project was completed during
PG’s and DZ’s internship at Amazon. MJB is affiliated to both Amazon and MPI.
This work however was carried out solely with Amazon’s support.

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## Supplementary Material (InvGAN: Invertible GANs)

### Video Key Framing

Figure 6: Tiled reconstruction of a video sequence. (a) original sequence, (b)
reconstructed sequence, (c) up-sampled in time sequence.

### CelebA Reconstructions

|  |  |  |  |  |  |
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |
|  |  |  |  |  |  |

Figure 7: InvGAN reconstructions of CelebA at $128\times 128$ resolution.
Alternating (from left to right) original and reconstructed images.

### Additional FID Evaluations

We additionally evaluate the FID scores on reconstructions using the approach
of ReStyle [4]. The resulting scores are presented in Table 3. Unfortunately
we can not compute test set MAE and FIDs since ReStyle has been trained on the
whole FFHQ set.

models | RandFID
| RandRecFID
---|---|---
1itr FFHQ ReStyle[4] | 40.33/4.71 | 57.63/29.56
2itr FFHQ ReStyle[4] | 40.33/4.71 | 53.09/22.88
3itr FFHQ ReStyle[4] | 40.33/4.71 | 51.68/20.80
4itr FFHQ ReStyle[4] | 40.33/4.71 | 51.49/19.93
Table 3: Random sample FID (RandFID), FID of reconstructed random samples
(RandRecFID). FID scores are here evaluated using 500 and 50000 samples. They
are separated by ‘/’.

### Additional Baseline for Semantic Reconstruction

We conduct the same experiment as presented in Section 4.1 with
reconstructions using ReStyle [4]. The resulting mean average precision
evaluated on the reconstructed evaluation set is $0.80\pm 0.15$. Evaluated on
original evaluation set images, the performance drops to $0.78\pm 0.17$, which
indicates a weaker transfer as compared to both the tiled reconstruction and
the full reconstruction using InvGAN.

### CelebA Image Editing

We conducted the same image editing operations shown in Figure 2 on CelebA.
The results are shown in Figure 8.

Style mixing In-painting Editing Merging I Merging II

Figure 8: Benchmark image editing tasks on CelebA ($128\,\mathrm{px}$). Style
mixing: We transfer the first $0,1,2,\dots,11$ style vectors from one image to
another. For the other image editing tasks, pairs of images are input image
(left) and reconstruction (right).


# Method of successive separation and summing of multiplicative diagrams of
mass operator for the multi-level quasiparticle interacting with polarization
phonons

M.V. Tkach111E-mail<EMAIL_ADDRESS>Ju.O. Seti, O.M. Voitsekhivska, V.V.
Gutiv

(Received June 17, 2019, in final form August 20, 2019)

###### Abstract

The theory of renormalized energy spectrum of a multi-level quasiparticle
interacting with polarization phonons at $T=0$ K is developed within the
Feynman-Pines diagram technique in a new approach. It permits a successive
separation of multiplicative diagrams from non-multiplicative ones for all
orders of mass operator, and their partial summing as well. The obtained mass
operator is presented as a sum of branched chain fractions, which effectively
take into account the multi-phonon processes. For the two-level quasiparticle
it is shown that just the interlevel (non-diagonal) interaction with phonons
fundamentally changes the properties of the spectrum. In the vicinity of all
threshold energies, the quasi-equidistant phonon satellite bands (i.e., groups
of energy levels) are formed. They correspond to the complexes of bound states
of a quasi-particle with many phonons.

Key words: diagram technique, quasiparticle, mass operator, phonon, spectrum

PACS: 71.38.-k, 72.10.Di, 63.20.kk

###### Abstract

Íà îñíîâ äàãðàìíî¿ òåõíêè Ôåéíìàíà-Ïàéíñà ó íîâîìó ïäõîä ðîçâèíåíà òåîðÿ
ïåðåíîðìîâàíîãî ñïåêòðà ñèñòåì áàãàòîðâíåâèõ êâàçàñòèíîê âçàìîäþèõ ç
ïîëÿðèçàöéíèìè ôîíîíàìè ïðè $T=0$ K, ÿêèé äîçâîëÿ ïîñëäîâíî âèäëÿòè
ìóëüòèïëêàòèâí ç íåìóëüòèïëêàòèâíèõ äàãðàì óñõ ïîðÿäêâ ìàñîâîãî îïåðàòîðà
çäéñíþâàòè ¿õ ïàðöàëüíå ïäñóìîâóâàííÿ. Îòðèìàíèé ó öüîìó ïäõîä ìàñîâèé
îïåðàòîð ñóìîþ ðîçãàëóæåíèõ ëàíöþãîâèõ äðîáâ, ÿê åôåêòèâíî âðàõîâóþòü
áàãàòîôîíîíí ïðîöåñè. Íà ïðèêëàä ñèñòåìè äâîðâíåâèõ êâàçàñòèíîê ïîêàçàíî, ùî
ñàìå ìæðâíåâà (íåäàãîíàëüíà) âçàìîäÿ ç ôîíîíàìè êàðäèíàëüíî çìíþ âëàñòèâîñò
ïåðåíîðìîâàíîãî ñïåêòðà. Ó îêîëàõ óñõ ïîðîãîâèõ åíåðãé óòâîðþþòüñÿ
êâàçåêâäèñòàíòí ôîíîíí ñàòåëòí çîíè — ãðóïè åíåðãåòèíèõ ðâíâ, ÿê âäïîâäàþòü
êîìïëåêñàì çâÿçàíèõ ñòàíâ êâàçàñòèíêè ç áàãàòüìà ôîíîíàìè.

Ключов слова: äàãðàìíà òåõíêà, êâàçàñòèíêà, ìàñîâèé îïåðàòîð, ôîíîí, ñïåêòð

##  1 Introduction

Recently, new physical phenomena (in particular, high temperature
superconductivity) in 3D and low-dimensional structures associated with the
effects of spatial quantization have greatly urged an intensive development of
new theoretical approaches to the study of interaction of quasiparticles
(electron, hole, exciton) with different types of phonons. All these various
of interactions play an important role in the formation and in temperature
evolution of the renormalized spectra and, hence, in the observed physical
phenomena.

Generally, during the recent two decades, new powerful and phighly promissing
methods have been scrupulously developed in order to calculate the spectra of
quasiparticles renormalized due to interaction with phonons in polaron models
(Fröhlich, Holstein, Su-Schrieffer-Heeger) which have different spatial
dimension. The progress in the solution of polaron problems has been reached
due to the use of modern analytical and analytically-numerical methods: exact
diagonalization (ED) [1], modern variational method (VM) [2, 3], dynamical
mean-field theory (DMFT) [4], density-matrix renormalization group (DMRG) [5,
6], quantum Monte Carlo (QMC) [7, 8, 9], diagrammatic Monte Carlo (DMC) [10,
11, 12, 13, 14], the bold diagrammatic Monte Carlo (BDQMC) [15, 16, 17, 18],
the momentum average approximation (MA) [19, 20, 21, 22, 23].

The results presented in the papers where these non-perturbed methods have
been used, proved to be well correlating for different models of the
structures where the interaction between quasiparticles and phonons is
important in a wide range of their coupling (from weak to strong). In the
majority of papers, the one-band models for quasiparticles were usually
studied. However, recently, the systems of quasiparticles with a more
complicated spectrum were observed [22, 23]. The arising interest to the
multi-level (multi-band) quasiparticles interacting with phonons is caused by
the rapid development of physics of nanoheterostructures, where the energy
spectrum of even uncoupled quasiparticles is multi-level (multi-band) due to
the effect of spatial quantization.

The theory of renormalized energy spectrum of multi-band quasiparticles
interacting with different phonon modes (confined, interface, propagating and
others) [24, 25, 26, 27] in nanoheterostructures is not only of theoretical
importance but it has an essential practical significance. Although the
multilayered nanostructures have been long used as components of unique modern
devices [quantum cascade lasers (QCL) and detectors (QCD)] operating in the
actual infra-red range of electromagnetic waves, the consistent theory of
their physical phenomena is far from being complete.

It is well known [28, 29, 30, 31] that phonons play an important positive role
for the QCDs functioning in the middle and in the far IR-range because their
extractors operate within the so-called “complete phonon ladder”. The quantum
wells and barriers of the extractors have got the composition and design of
sizes which provide the needed number of equidistant electron energy levels
(with the distance of one phonon energy) that create this “ladder”. Herein,
the tunneling electron performs consistent radiationless one-phonon quantum
transitions from the upper level of the cascade active region exactly to the
lower (ground) level of the active region of the next cascade through this
ladder, which ensures an appropriate QCD operation.

The described mechanism of extractor operating within the “complete phonon
ladder”, which relaxes the electron energy between the active regions of the
neighbouring cascades, is not the only one. Recently, it turned out that the
cascades of newly produced and appropriately operating QCD of near IR-range
[32, 33, 34, 35] work with the extractors having a “torn phonon ladder”,
lacking almost a half of the lower energy levels! A detailed review and
analysis of experimental papers show that in some QCD operating in the other
ranges [36, 37] the extractors were imperfect with respect to the equivalent
distance between the levels that create a “phonon ladder”.

There is still no theory of electron-phonon interaction that would reliably
describe the physical phenomena in multilayered nanostructures. However, the
satellite states with equidistant energy spectrum arising due to the
interaction between one-level quasiparticles and phonons were studied in
theoretical papers [38, 39, 40]. It gives reason to expect that multi-level
uncoupled electrons interacting with phonons in quasi-two-dimensional
multilayered nanostructures are capable of creating quasi-equidistant
satellite mini-bands, which would compensate both for the energy levels
lacking in the “phonon ladder” and for their imperfect equidistant
characteristics.

For the simplest model of an extractor with the “torn phonon ladder”, created
by two energy levels of a quasiparticle and by their phonon satellites, in the
papers [41, 42] there was presented a theory of an energy spectrum of two-
level localized quasiparticle interacting with polarization phonons at $T=0$
K. Using the modified method of Feynman-Pines diagram technique, it was shown
that in the first approximation of renormalized mass operator (MO) (where all
multiplicative diagrams are taken into account completely), the satellite
mini-bands are really observed in the spectrum.

In this paper, we develop a generalized theory for the spectrum of a multi-
level quasiparticle renormalized due to the interaction with phonons, taking
into account not only the main multiplicative diagrams in the mass operator
but also the diagrams that are successively separated from its non-
multiplicative block. For the same model as in [41, 42], it is shown that the
renormalized spectrum of the system, apart from the known properties, also
prossesses some new important ones.

##  2 Hamiltonian of the system. Mass operator of quasiparticle Green’s
function at $T=0$ K

To obtain a renormalized spectrum of the system consisting of a multi-level
localized quasiparticle interacting with polarization phonons, we use the
Fröhlich Hamiltonian, like in [41, 42, 39]

$\hat{H}=\sum_{\mu=1}^{\tau}E_{\mu}a_{\mu}^{+}a_{\mu}+\sum_{\vec{q}}\Omega(\vec{q})\left(b_{\vec{q}}^{+}b_{\vec{q}}+\frac{1}{2}\right)+\sum_{\vec{q}}\sum_{\mu_{1},\mu_{2}=1}^{\tau}\varphi_{\mu_{1}\mu_{2}}a_{\mu_{1}}^{+}a_{\mu_{2}}\big{(}b_{\vec{q}}+b_{-\vec{q}}^{+}\big{)}.$
(1)

Here, $E_{\mu=1,\ldots,\tau}$ are the energy levels of an uncoupled
quasiparticle. As the energy of optical phonon typically weakly depends on its
quasimomentum ($\vec{q}$), we further put $\Omega(\vec{q})=\Omega$, neglecting
the dispersion. Coupling constants ($\varphi_{\mu_{1}\mu_{2}}$), which
describe quasiparticle-phonons interaction, are assumed to be the known
parameters and characterize either intra-level interaction at
$\mu_{1}=\mu_{2}$ or the inter-level interaction at $\mu_{1}\neq\mu_{2}$.

We should note that Hamiltonian like (1) can describe a wide range of
3D-models (for example, impurity centers) and spatially confined 1D- or
2D-models of low dimensional structures containing multi-level localized
quasiparticles interacting with confined or interface phonons.

The energy spectrum of the system renormalized due to the interaction at
cryogenic temperature (formally $T=0$ K) is obtained within the method of
Feynman-Pines diagram technique [42, 43, 44] for the Fourier image of
quasiparticle casual Green’s function $G_{\mu\mu^{\prime}}(\omega)$. Moreover,
we use the approach proposed in papers [41, 42], modified for the case of a
quasiparticle [with an arbitrary number ($\tau$) of levels] interacting with
phonons. Taking into account the Hamiltonian (1), the Green’s functions
$G_{\mu\mu^{\prime}}(\omega)$ satisfy the system of $\tau^{2}$ Dyson equations

$G_{\mu\mu^{\prime}}(\omega)=(\omega-
E_{\mu}+i\eta)^{-1}\bigg{[}\delta_{\mu\mu^{\prime}}+\sum_{\mu_{1}=1}^{\tau}M_{\mu\mu_{1}}(\omega)G_{\mu_{1}\mu^{\prime}}(\omega)\bigg{]},\,\,\,\,\,\mu,\,\mu^{\prime}=1,\,2,\,\ldots,\tau;\,\,\,\,(\eta\to
0),$ (2)

where $M_{\mu\mu_{1}}$ is the complete matrix MO and
$E_{\mu=2,\ldots,\tau}=E_{1}+\Delta E_{\mu=2,\ldots,\tau}$. This system is
conveniently solved in dimensionless functions, variables and constants:

$\begin{array}[]{c}{g_{\mu}(\xi)=g_{\mu\mu}(\xi)=\Omega G_{\mu\mu}\,,\qquad
m_{\mu}(\xi)=M_{\mu}\Omega^{-1},\qquad
m_{\mu\mu}(\xi)=M_{\mu\mu}\Omega^{-1},}\\\ \\\ {\xi=(\omega-
E_{1})\Omega^{-1},\,\,\,\,\,\xi_{\mu_{l}}=(\omega-
E_{\mu_{l}})\Omega^{-1},\,\,\,\,\,\alpha_{\mu_{1}\mu_{2}}=\varphi_{\mu_{1}\mu_{2}}\Omega^{-1},\,\,\,\,\,\delta_{\mu=2,\,\ldots,\tau}=\Delta
E_{\mu=2,\,\ldots,\tau}\Omega^{-1}.}\end{array}$ (3)

The system of $\tau^{2}$ dimensionless equations obtained from (2) is
sophisticated (for big $\tau$ numbers), though it is solved exactly. For the
dimensionless Green’s function of $\mu$-th level
[$g_{\mu}(\xi)=g_{\mu\mu}(\xi)$], the solution is defined by Dyson equation
(at $\hbar=1$)

$g_{\mu}(\xi)=\\{\xi_{\mu}-m_{\mu}(\xi)\\}^{-1}$ (4)

within the complete MO $m_{\mu}(\xi)$, which can be written in the form

$m_{\mu}(\xi)=m_{\mu\mu}(\xi)+m_{\mu\mu}^{g}(\xi)\,,$ (5)

where $m_{\mu\mu}$ is a diagonal component, which describes all intra-level
interactions with phonons but without taking into account non-diagonal
elements of $g_{\mu\mu^{\prime}}$ matrix. The components $m_{\mu\mu}^{g}$,
produced by non-diagonal elements of $g_{\mu\mu^{\prime}}$, are defined
exactly. However, they have a sophisticated structure for big $\tau$ numbers,
presented, for example, in appendix A at $\tau=3,4,5$. For the two-level
system ($\tau=2$), the diagonal elements are given by simple formulae

$m_{11}^{g(2)}(\xi)=\frac{m_{12}(\xi)m_{21}(\xi)}{\xi_{2}-m_{22}(\xi)}\,,\qquad\qquad
m_{22}^{g(2)}(\xi)=\frac{m_{21}(\xi)m_{12}(\xi)}{\xi_{1}-m_{11}(\xi)}\,.$ (6)

The matrix of complete MO $m_{\mu\mu^{\prime}}$ is defined by the rules of
Feynman-Pines diagram technique [44], which are generalized for the case of
multi-level systems. Since the energies of quasiparticle and phonons are
dispersionless, the equivalent diagrams with and without crossing the phonon
lines, correspond to the identical analytical expressions, such as

$\begin{array}[]{c}\includegraphics[width=158.99377pt,keepaspectratio={true}]{diagr7}\raisebox{12.0pt}{$\hskip
2.84526pt.$}\end{array}$ (7)

Thus, contrary to the classic Fröhlich one-band Hamiltonian [42, 44], where
the energy of an uncoupled quasiparticle is the function of quasimomentum, the
dimensionless MO $m_{\mu\mu^{\prime}}$ is obtained in such diagrammatic
representation that contains only all non-equivalent diagrams without crossing
the phonon lines. The number of equivalent diagrams of this type is fixed by
the integers prior to the respective diagrams.

(8)

All MO diagrams in an arbitrary $p$-th order over the power of the pairs of
coupling constants, which are the same as the number of phonon lines in a
respective diagram, are definitely calculated by a computer program. It would
be clear further on that in order to establish the exact rules of partial
summing of infinite ranges of diagrams, it is quite sufficient to take into
account all the first 128 non-equivalent diagrams to the eighth order
inclusive.

The analytical expression for an arbitrary diagram is obtained as the sum over
all inner indices ($\mu_{1},\mu_{2},\ldots,\mu_{N}$), except the outer ones
($\mu,\mu^{\prime}$), of the products of all vertices and solid lines

$\begin{array}[]{c}\hfil\includegraphics[width=108.405pt,keepaspectratio={true}]{diagr9a}\qquad\qquad\qquad\includegraphics[width=108.405pt,keepaspectratio={true}]{diagr9b}\end{array}$
(9)

where $p$ is a number of dashed (phonon) lines situated above the solid
(quasiparticle) line with index $\mu_{l}$. For example, the first two
components of $m_{\mu\mu^{\prime}}$ are of the following diagrammatic and
analytical forms:

$\begin{array}[]{c}\includegraphics[width=166.2212pt,keepaspectratio={true}]{diagr10}\end{array}$
(10)
$\begin{array}[]{c}\includegraphics[width=289.07999pt,keepaspectratio={true}]{diagr11}\end{array}$
(11)

The renormalized spectrum of the system is calculated in a wide energy range,
containing not only the ground level but the levels of the excited states,
which arise in the vicinity of the thresholds of the radiation of phonons.
Thus, it is necessary to represent the complete MO (8) in the form which
effectively takes into account both the inter-level interaction and multi-
phonon processes. However, the classic form (8) does not satisfy the second
condition, in particular, due to the so-called ”problem of sign”. In order to
solve it, one should perform a successive partial summing of infinite ranges
of diagrams in the complete MO.

##  3 Method of successive partial summing of MO diagrams

To perform the partial summing of diagrams in the complete MO
$m_{\mu\mu^{\prime}}$, it is convenient to group them into the classes which
are written in brackets in expression (8). It is clear that the $p$-th class
of renormalized diagrams (except the first one) together with the factor $p!$
contains an infinite number of only those diagrams (together with numerical
factors), whose arbitrary blocks, in their turn, contain not more than $p$
dashed lines over any of the solid ones. This class of diagrams is further
referred to as a renormalized $p$-phonon MO and is denoted by
$m_{\mu\mu^{\prime}}^{[p]}$. We should note that, as it is clear from (8),
only one-phonon diagram (10) in its first class of partial $p$-phonon diagrams
is single, hence, $m_{\mu\mu^{\prime}}^{[1]}=m_{\mu\mu^{\prime}}^{(1)}$. All
other classes of renormalized MO $\big{(}m_{\mu\mu^{\prime}}^{[p\geq
2]}\big{)}$ are given by the respective infinite ranges of diagrams. For
example, the renormalized two-phonon MO is

$\begin{array}[]{c}\includegraphics[width=252.94499pt,keepaspectratio={true}]{diagr12}\raisebox{12.0pt}{$\hskip
2.84526pt.$}\end{array}$ (12)

Then, the complete MOs $m_{\mu\mu^{\prime}}$ can be written as a sum of
renormalized MO of all orders

$m_{\mu\mu^{\prime}}(\xi)=\sum_{p=1}^{\infty}m_{\mu\mu^{\prime}}^{[p]}(\xi)\,.$
(13)

In each MO diagram, the analytical contributions of its elements are summed up
over all $\mu_{l}$ indices, except the first one
($m_{\mu\mu^{\prime}}^{[1]}$). Hence, none of them is multiplicative, in
general form. However, the analytical expressions show that each diagram can
be expressed as a sum of two components: multiplicative (m) and non-
multiplicative (nm). To this end, in any diagram, in which summing is
performed over all inner indices [for example, $\mu_{1},~{}\mu_{2},~{}\mu_{3}$
in the first diagram of (12)], one should separate the multiplicative
component having certain equal indices (for example,
$\mu_{1},~{}\mu_{2},~{}\mu_{3}=\mu_{1}$), where the summing is performed only
over the indices $\mu_{1},~{}\mu_{2}$ and the non-multiplicative component
having the indices ($\mu_{1},~{}\mu_{2},~{}\mu_{3}\neq\mu_{1}$), where the
summing is performed over all these indices. In this approach, the non-
renormalized MO of the second order over the number of phonon lines is
expressed in the following diagrammatic and analytical forms:

$\begin{array}[]{c}\includegraphics[height=72.26999pt,keepaspectratio={true}]{diagr14}\\\
\displaystyle=\sum_{\mu_{1}=1}^{\tau}\frac{\alpha_{\mu\mu_{1}}m_{\mu_{1}\mu_{1}}^{(1)}(\xi-1)\,\alpha_{\mu_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)^{2}}+\sum_{\mu_{1}=1}^{\tau}\,\frac{\alpha_{\mu\mu_{1}}}{\xi_{\mu_{1}}-1}\,\sum_{\mu_{3}\neq\mu_{1}}^{\tau}\frac{m_{\mu_{1}\mu_{3}}^{(1)}(\xi-1)\,\alpha_{\mu_{3}\mu^{\prime}}}{\xi_{\mu_{3}}-1}\,.\end{array}$
(14)

The non-renormalized MO of the third order is as follows:

$\begin{array}[]{c}\displaystyle=\sum_{\mu_{1},\mu_{2}=1}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}m_{\mu_{2}\mu_{2}}^{(1)}(\xi-2)\,\alpha_{\mu_{2}\mu_{1}}\alpha_{\mu_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)^{2}(\xi_{\mu_{2}}-2)^{2}}+\left[\sum_{\scriptsize{\begin{array}[]{l}{\mu_{1},\mu_{2},}\\\
{\mu_{5}\neq\mu_{1}}\end{array}}}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}m_{\mu_{2}\mu_{2}}^{(1)}(\xi-2)\,\alpha_{\mu_{2}\mu_{5}}\alpha_{\mu_{5}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)^{2}(\xi_{\mu_{5}}-1)}+\right.\\\
\displaystyle\left.+\sum_{\scriptsize{\begin{array}[]{l}{\mu_{1},\mu_{2},\mu_{5},}\\\
{\mu_{4}\neq\mu_{2}}\end{array}}}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}m_{\mu_{2}\mu_{4}}^{(1)}(\xi-2)\,\alpha_{\mu_{4}\mu_{5}}\alpha_{\mu_{5}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)(\xi_{\mu_{4}}-2)(\xi_{\mu_{5}}-1)}\right].\end{array}$
(15)

Having separated all the diagrams of all orders at m- and nm- classes in the
complete MO (8) and taking into account that one-phonon MO
$m_{\mu\mu^{\prime}}^{(1)}$ belongs to the m-class of diagrams, the complete
MO (13) can be written as a sum of two components

$m_{\mu\mu^{\prime}}(\xi)=m_{\mu\mu^{\prime}}^{1,m}(\xi)+m_{\mu\mu^{\prime}}^{2,nm}(\xi)=\sum_{p=1}^{\infty}m_{\mu\mu^{\prime}}^{p,[m]}(\xi)+\sum_{p=2}^{\infty}m_{\mu\mu^{\prime}}^{p,[nm]}(\xi)\,.$
(16)

The first component ($m_{\mu\mu^{\prime}}^{1,m}$) contains one-phonon diagram
renormalized by all m-diagrams of all orders and the second one
($m_{\mu\mu^{\prime}}^{2,nm}$) contains the rest nm-diagrams of all orders
beginning from the second, in which one should further successively separate
other classes of m- and nm-diagrams of higher orders over the powers of
coupling constants (phonon lines). The multiplicative structure of all terms
in MO $m_{\mu\mu^{\prime}}^{1,m}$ makes it possible to perform an exact
partial summing. This was described in detail in papers [41, 42], where the
latter was denoted as $m_{\mu\mu^{\prime}}^{[m]}$.

As a result, being renormalized by all the multiphonon process, one-phonon MO
in the first approximation $m_{\mu\mu^{\prime}}^{1,m}$ is obtained in the form
of infinite branched chain fraction

$\begin{array}[]{c}\includegraphics[height=72.26999pt,keepaspectratio={true}]{diagr17}\end{array}$
(17)

For further calculations it is convenient to represent the latter expression
as identical functional equation

$m_{\mu\mu^{\prime}}^{1,m}(\xi)=\sum_{\mu_{1}=1}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu^{\prime}}}{\xi_{\mu_{1}}-1-m_{\mu_{1}\mu_{1}}^{2,m}(\xi)}\,.$
(18)

Thus, for an arbitrary $p=1,2,\ldots,\infty$, the relationship is valid:

$\displaystyle
m_{\mu_{p-1}\mu^{\prime}_{p-1}}^{p,m}(\xi)=\sum_{\mu_{p}=1}^{\tau}\frac{p\alpha_{\mu_{p-1}\mu_{p}}\alpha_{\mu_{p}\mu^{\prime}_{p-1}}}{\xi_{\mu_{p}}-p-m_{\mu_{p}\mu_{p}}^{p+1,m}(\xi)}\,.$
(19)

The next stage in calculating the complete MO ($m_{\mu\mu^{\prime}}$) is to
separate all m-diagrams from the whole infinite number of nm-diagrams
$m_{\mu\mu^{\prime}}^{2,nm}$, starting from the two-phonon , which were not
logged in $m_{\mu\mu^{\prime}}^{1,m}$ and are renormalized due to all multi-
phonon processes by two-phonon MO $m_{\mu\mu^{\prime}}^{2,[m]}$ (being the
correction of the second approximation for the complete MO). Then, all other
diagrams of the complete MO, which were not logged in
$m_{\mu\mu^{\prime}}^{1,m}$ and $m_{\mu\mu^{\prime}}^{2,[m]}$, starting from
the three-phonon ones, produce the non-multiplicative MO
$m_{\mu\mu^{\prime}}^{3,[nm]}$. As for $m_{\mu\mu^{\prime}}^{2,[m]}$, it is
partially summed up exactly in the same way as $m_{\mu\mu^{\prime}}^{1,m}$.
Thus, having separated from $m_{\mu\mu^{\prime}}^{2,nm}$ the infinite ranges
of only m-diagrams of the second approximation

(20)

and having performed their exact partial summing according to the presented
rules of diagram technique (9), the following analytical expression is
obtained

$m_{\mu\mu^{\prime}}^{2,[m]}(\xi)=\sum_{\scriptsize{\begin{array}[]{l}{\mu_{1},\mu_{2},}\\\
{\mu^{\prime}_{1}\neq\mu_{1}}\end{array}}}^{\tau}\frac{2\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu^{\prime}_{1}}\alpha_{\mu^{\prime}_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu^{\prime}_{1}}-1)[\xi_{\mu_{2}}-2-m_{\mu_{2}\mu_{2}}^{3,m}(\xi)]}\sum_{s=0}^{\infty}\bigg{\\{}\sum_{\mu^{\prime}_{2}}^{\tau}\frac{2\alpha_{\mu_{1}\mu^{\prime}_{2}}\alpha_{\mu^{\prime}_{2}\mu_{1}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)\Big{[}1-\frac{m_{\mu_{2}\mu_{2}}^{3,m}(\xi)}{(\xi_{\mu_{2}}-2)}\Big{]}}\bigg{\\}}^{S},$
(21)

Partially summing the ranges of m-diagrams or, which is the same, converting
the last row of geometric progression in (21), final diagrammatic and
analytical expressions for a completely renormalized (due to multi-phonon
processes) two-phonon MO $m_{\mu\mu^{\prime}}^{2,[m]}$ (in the second
approximation) are obtained

$\begin{array}[]{c}\includegraphics[height=32.52127pt,keepaspectratio={true}]{diagr22}\\\
\displaystyle=\sum_{\mu_{1},\mu_{2},\mu^{\prime}_{1}\neq\mu_{1}}^{\tau}\frac{2\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu^{\prime}_{1}}\alpha_{\mu^{\prime}_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)(\xi_{\mu^{\prime}_{1}}-1)\left[1-\frac{2\,m_{\mu_{1}\mu_{1}}^{(1)}(\xi-1)}{\xi_{\mu_{1}}-1}-\frac{m_{\mu_{2}\mu_{2}}^{3,m}(\xi)}{\xi_{\mu_{2}}-2}\right]}\,.\end{array}$
(22)

In the same way, final diagrammatic and analytical expressions for the
completely renormalized (due to multi-phonon processes) three-phonon MO
$m_{\mu\mu^{\prime}}^{3,[m]}$ (in the third approximation) are obtained

$\begin{array}[]{c}\includegraphics[height=75.88371pt,keepaspectratio={true}]{diagr23}\\\
\displaystyle{=2^{2}\sum_{\scriptsize{\begin{array}[]{l}{\mu_{1},\mu_{2},\mu^{\prime}_{2},}\\\
{\mu^{\prime\prime}_{1},\mu^{\prime}_{1}\neq\mu_{1}}\end{array}}}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu^{\prime}_{1}}\alpha_{\mu^{\prime}_{1}\mu^{\prime}_{2}}\alpha_{\mu^{\prime}_{2}\mu^{\prime\prime}_{1}}\alpha_{\mu^{\prime\prime}_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)(\xi_{\mu^{\prime}_{1}}-1)(\xi_{\mu^{\prime}_{2}}-2)(\xi_{\mu^{\prime\prime}_{1}}-1)}\left[1-\frac{2\,m_{\mu_{1}\mu_{1}}^{(1)}(\xi-1)}{\xi_{\mu_{1}}-1}-\frac{m_{\mu^{\prime}_{2}\mu^{\prime}_{2}}^{3,m}(\xi)}{\xi_{\mu^{\prime}_{2}}-2}\right]^{-2}+}\\\
\displaystyle{+2\cdot
3\sum_{\scriptsize{\begin{array}[]{l}{\mu_{1},\mu_{2},\mu_{3},}\\\
{\mu^{\prime}_{1},\mu^{\prime}_{2}\neq\mu_{2}}\end{array}}}^{\tau}\frac{\alpha_{\mu\mu_{1}}\alpha_{\mu_{1}\mu_{2}}\alpha_{\mu_{2}\mu_{3}}\alpha_{\mu_{3}\mu^{\prime}_{2}}\alpha_{\mu_{2}\mu^{\prime}_{1}}\alpha_{\mu^{\prime}_{1}\mu^{\prime}}}{(\xi_{\mu_{1}}-1)(\xi_{\mu_{2}}-2)(\xi_{\mu_{3}}-3)(\xi_{\mu^{\prime}_{2}}-2)(\xi_{\mu^{\prime}_{1}}-1)}\left[1-\frac{2\,m_{\mu^{\prime}_{1}\mu_{1}}^{(1)}(\xi-1)}{\xi_{\mu^{\prime}_{1}}-1}\right]^{-2}\left[1-m_{\mu_{3}\mu_{3}}^{4,m}(\xi)\right]^{-1}}.\end{array}$
(23)

A similar cyclic procedure of successive separation of m-diagrams from nm-
diagrams with their subsequent summing up in the proposed way is totally
applicable in all higher orders for MO. It is clear that with each next cycle,
the number ($s$) of terms $m_{\mu\mu^{\prime}}^{s,[m]}$ renormalized by
m-diagrams and, consequently, their contribution into $m_{\mu\mu^{\prime}}$
will be bigger while the number of terms $m_{\mu\mu^{\prime}}^{s,nm}$ produced
by nm-diagrams and their contribution will become smaller.

In the limiting case of infinite number of cycles ($s\rightarrow\infty$),
$m_{\mu\mu^{\prime}}^{s,nm}\rightarrow 0$. Thus, in $m_{\mu\mu^{\prime}}$
there remains only the sum of all terms $m_{\mu\mu^{\prime}}^{s,m}$ produced
by all m-diagrams. Now it is possible to formally present the complete MO in
the form

$m_{\mu\mu^{\prime}}(\xi)={\lim\limits_{s_{\max}\to\infty}}\sum_{s=1}^{s_{\max}}\left[m_{\mu\mu^{\prime}}^{s,[m]}(\xi)+m_{\mu\mu^{\prime}}^{s,nm}(\xi)\right]=\sum_{s=1}^{\infty}m_{\mu\mu^{\prime}}^{s,[m]}(\xi)\,,$
(24)

where, in order to unambiguously understand this formula, we use the denotion
$m_{\mu\mu^{\prime}}^{1,[m]}\equiv m_{\mu\mu^{\prime}}^{1,m}$ at $s=1$.

The algorithm of successive separation of m-diagrams from the nm-diagrams with
their partial summing is completely clear but if the number of terms ($s$)
increases, the analytical expressions for $m_{\mu\mu^{\prime}}^{s,[nm]}$
become more sophisticated. Therefore, when MO (24) is practically used for the
calculation of renormalized spectrum of the system, one should take into
account this number of terms ($s$) in $m_{\mu\mu^{\prime}}^{s,[nm]}$, which
provides sufficient convergence (saturation) of the spectrum.

In our previous paper [41], we studied only the first
($m_{\mu\mu^{\prime}}^{1,m}$) approximation for a complete MO and revealed the
main property of a renormalized spectrum of two-level localized quasiparticle
interacting with phonons, in a wide range of energies. In this paper, using
the obtained MO in third approximation

$m_{\mu\mu^{\prime}}^{\text{III}}(\xi)=m_{\mu\mu^{\prime}}^{1,[m]}(\xi)+m_{\mu\mu^{\prime}}^{2,[m]}(\xi)+m_{\mu\mu^{\prime}}^{3,[m]}(\xi)$
(25)

we shall clarify some new properties of this spectrum, which were not revealed
in the first paper.

##  4 Properties of renormalized energy spectrum of the system at $T=0$ K
taking into account the higher successive approximations of mass
operator

In our paper [41], we studied the renormalized spectrum of a two-level
quasiparticle interacting with dispersionless phonons within zero ($m^{0}$)
and first ($m^{\text{I}}$) approximation for the MO. In this paper, using the
analytic results presented in the previous section, we shall take into account
the terms of MO in the second ($m^{\text{II}}$) and the third
($m^{\text{III}}$) approximations, which essentially affect some properties of
the energy spectrum. As far as there are no imaginary terms of MOs
[$m_{1}(\xi),m_{2}(\xi)$], in both Green’s functions [$g_{1}(\xi),g_{2}(\xi)$]
all their poles are the same and, according to (4) and (6), define the
dispersion equation

$[\xi-m_{11}(\xi)][\xi-\delta-
m_{22}(\xi)]=m_{12}^{2}(\xi),\qquad[\delta\equiv\delta_{2}=(E_{2}-E_{1})\Omega^{-1}].$

Its solutions determine all renormalized energy levels both for the main and
satellite states of the system.

Figure 1: (Colour online) Renormalized spectrum $e_{\mu,l,k}$ as function of
$\delta$ at $\alpha_{11}=0.3$, $\alpha_{22}=0.2$, $\alpha_{12}=0.05$.

The spectrum is numerically calculated and analyzed for the system which is
characterized by rather small parameters of intra-level
($\alpha_{11}=0.3,\alpha_{22}=0.2$) and inter-level ($\alpha_{12}=0.05$)
coupling constants of quasiparticle and phonons. Figure 1 shows the spectrum
obtained in four successive approximations for MO, as a function of energy
distance ($\delta$) between both levels of uncoupled quasiparticle. We use
convenient three-index symbols for the energy levels ($e_{\mu,l,k}$) and two-
index symbols for the widths of satellite groups ($\Delta e_{\mu,l}$), which
correspond to their physical origin. Thus, in $e_{\mu,l,k}$: $\mu=1,2$ –
numerates the state of uncoupled quasiparticle, $l$ is the number of a
satellite group of the levels which fixes the threshold energy of radiation
accompanied by the creation of $l$ phonons, $k$ is the number of the level in
the $l$-th group. The width of $\mu l$-th satellite group ($\Delta
e_{\mu,l}=e_{\mu,l}^{>}-e_{\mu,l}^{<}$) is fixed by the difference between the
upper $e_{\mu,l}^{>}=e_{\mu,l,\textrm{max}k}$ and lower
$e_{\mu,l}^{<}=e_{\mu,l,\textrm{min}k}$ levels of this group. Moreover, we
should note that the renormalized energies of the first and second main levels
are written as $e_{1,0}$ and $e_{2,0}$, while the complicated groups of levels
in resonant regions are allocated by energy lines marked by two digital
indices ($l_{1},l_{2}$), where $l_{1}=1,2,3\ldots$; $l_{2}=0,1,2\ldots$
numerate the satellite groups of the first and second main levels,
respectively.

The energy levels $e_{\mu,l,k}$ calculated in different approximations for MO
are presented in figure 1. Herein, in the panels, the following curves
correspond to: $e_{1,l,k}$ — dots (a) $m^{0}$; solid red (a) $m^{\text{I}}$,
(b) $m^{\text{II}}$, (c) $m^{\text{III}}$; $e_{2,l,k}$ — triangles (a)
$m^{0}$; solid green (a) $m^{\text{I}}$, (b) $m^{\text{II}}$, (c)
$m^{\text{III}}$. Solid blue curves in resonant regions present the energy
levels of satellite complexes which are produced by the superposition of
multi-phonon states bound to the both states of uncoupled quasiparticle. Now,
let us analyse how the main properties of renormalized spectrum, calculated in
different successive approximations for the MO, change.

In figure 1 (a), one can see the same dependence of the spectrum on $\delta$
as obtained in approximations $m^{0}$ and $m^{\text{I}}$ and analyzed in
detail in our paper [41]. The results in these two approximations are quite
different. In zero approximation ($m^{0}$), neglecting the inter-level
interaction ($\alpha_{12}=0$), the spectrum contains two infinite series of
equidistant levels (the distance is of one-phonon energy). If $\delta$
increases, these levels intersect (except the ground one $e_{10}$) in all
resonant regions of energies ($l_{1},l_{2}$). On the contrary, in the first
approximation ($m^{\text{I}}$), the anti-crossing of levels is observed in the
first resonant region (1,0) while in the other resonant regions
($l_{1}=2,3,\ldots;l_{2}=0,1,\ldots$) there are multi-anti-crossings. It is
due to this that in all non-resonant regions one can see two first satellite
levels ($e_{1,1,1}$ and $e_{2,1,1}$) and, besides, the bands of satellite
levels with the widths $\Delta e_{\mu,l=2,3,\ldots}$.

Comparing the panels (a) and (b) in figure 1, one can see two qualitatively
new features of the spectra calculated in the first ($m^{\text{I}}$) and
second ($m^{\text{II}}$) approximations for the MO. In the second
approximation: i) in all resonant regions ($l_{1},l_{2}$) there are complexes
of satellite levels which do not intersect at increasing $\delta$, but some
neighbouring pairs degenerate and disappear or appear. Due to the scale of
figure 1, this phenomenon is weakly visible, therefore, in figure 2 it will be
shown more in detail. ii) in non-resonant regions, the number of levels in all
satellite groups increases and their widths become a little bigger ($\Delta
e_{\mu,l=2,3,\ldots}$). Two first satellite groups with the respective widths
($\Delta e_{1,1}$, $\Delta e_{2,1}$) are created instead of two first
satellite levels ($e_{1,1,1}$ and $e_{2,1,1}$).

Figure 2: (Colour online) Renormalized spectrum $e_{\mu,l,k}$ as function of
$\delta$ at $\alpha_{11}=0.1$, $\alpha_{22}=0.08$, $\alpha_{12}=0.05$
calculated in the third approximation for the mass operator
$m^{\text{III}}(\xi)$ (a) and bigger scales of three resonant regions (1,0) –
b, (2,0) – c, (3,0) – $d_{1},d_{2},d_{3}$.

From the panels (b) and (c) in figure 1 it is clear that comparing the second
and third approximations for the MO, the properties of the spectra are
qualitatively the same. Though the number of levels in satellite complexes and
groups becomes bigger, their widths increase weakly.

In figure 2 (a), the spectrum calculated in the third approximation for the MO
is shown for the system with smaller constants of intra-level coupling
($\alpha_{11}=0.1$, $\alpha_{22}=0.008$) but at the same magnitude of the
inter-level coupling constant ($\alpha_{12}=0.005$) as for the system
described in figure 1. Figure 2 (a) shows the same behaviour of the spectrum
as figure 1 (c) but here, due to the smaller $\alpha_{11}$ and $\alpha_{22}$,
the sizes of the respective satellite complexes in resonant regions are
smaller the same as widths of satellite bands in non-resonance regions.

For example, in the panels (b), (c), (d), figure 2 (with a bigger scale), the
satellite levels in complexes (1,0), (2,0), (3,0), produced due to the
superposition of the second main state with phonon satellites of the first
main state are shown. In these panels, instead of the corresponding blue lines
in figure 2 (a), the lines are painted in three colors in the bottom-up
sequence: red, green, blue and so on, for better visualization. In the panel
(c), figure 2 in two circles, one can see green and blue lines which visually
seem touching each other, although in reality they are just very close. The
rest properties of the energy levels in satellite complexes are typical and
well seen in the panels (a), (b), (c), (d).

Figure 3: (Colour online) Widths of satellite bands ($\Delta e_{\mu,l}$) as
functions of the inter-level coupling constant ($\alpha_{12}$) at different
intra-level coupling constants ($\alpha_{11},\alpha_{22}$), presented in the
panels.

The widths of satellite bands are important parameters of the spectrum in the
non-resonant energy region, depending on intra- and inter-level coupling
constants. The results of their calculations in the third approximation are
presented in figure 3, with $\alpha_{11}$, $\alpha_{22}$, $\alpha_{12}$ values
at the respective panels. The figure proves that the widths of all satellite
bands quasi-linearly depend on the magnitude of inter-level coupling in the
range $0<\alpha_{12}\leqslant 0.1$. The panels (a), (b), (c) show that the
widths of three satellite bands of the first level ($\Delta e_{1,l}$) are
mainly determined by the magnitude of intra-level coupling ($\alpha_{11}$) of
quasiparticle in the same state with phonons, while increasing $\alpha_{22}$
or $\alpha_{12}$ weakly increases the widths $\Delta e_{1,1}$ and $\Delta
e_{1,2}$. The width $\Delta e_{1,3}$ almost does not depend on $\alpha_{22}$
and $\alpha_{12}$. The widths of three satellite bands of the second level
($\Delta e_{2,l}$), panels (d), (e), (f), are mainly determined by
$\alpha_{22}$ magnitude while $\alpha_{11}$ and $\alpha_{12}$ affect weakly.

##  5 Main results and conclusions

To calculate the energy spectrum of multi-level quasiparticle interacting with
polarization phonons described by Fröhlich-type Hamiltonian at $T=0$ K, the
Feynman-Pines diagram technique is generalized for an effective
renormalization of mass operator due to multi-phonon processes.

Within the approach of successive separation of infinite classes of
multiplicative diagrams from the blocks of non-multiplicative diagrams of MO
and their partial summing, the MO is finally expressed as the sum of
continuous branch fractions with typical links. Such a representation
effectively takes into account multi-phonon processes and avoids the so-called
“problem of sign”.

The renormalized energy spectrum is calculated using four successive
approximations for the MO within the example of a two-level quasiparticle
interacting with phonons. The analysis of this spectrum in different
approximations shows that its properties essentially depend on the inter-level
interaction.

In zero approximation for the MO (neglecting inter-level interaction), the
spectrum contains two infinite series of equidistant satellite levels, which
intersect in all resonant regions when the energy distance between two levels
of uncoupled quasiparticle varies. In the first and further approximations for
MO, instead of two series of single satellite levels, the complexes of
satellite levels are observed in resonant regions while the bands of satellite
levels are observed in non-resonant regions. In higher approximations, the
number of levels in satellite complexes and bands becomes bigger though their
widths vary weakly. At increasing $\delta$, the levels do not intersect but
some neighbouring levels degenerate (disappear or appear), which causes an
unstable spectrum. It is shown that the widths of satellite bands are mainly
proportional to the magnitudes of the intra-coupling constants
($\alpha_{\mu\mu}$) of those main levels ($\mu$) of an uncoupled
quasiparticle, by which they are produced due to the interaction with phonons.
The sizes of satellite complexes are characterized by both widths of those two
bands that produce them by intersecting.

Observing the system under research as the simplest (rough) model of electron-
phonon interaction in basic elements of modern nano-devices (QCD and QCL), one
can expect that the revealed bands and complexes of satellite levels (which
were not taken into account in the previous theoretical papers) would
essentially affect their operation. In particular, the satellite complexes and
bands of the levels can replace the missing “steps” of the “torn phonon
ladder” or imperfect “complete phonon ladder” in extractors of QCD operating
in IR-range.

Of course, for a further development of a consistent theory of electron-phonon
interaction in multi-layered nanostructures, due to the effect of spatial
quantization, it is necessary to take into account both the multi-band
quasiparticles and all modes of the phonon spectrum that are present in real
structures. This will generally make the theory much more complicated, but
will make it possible to make a research into the new interesting and
important properties and physical phenomena in nanoheterostructures.

##  Appendix A MO matrix elements $m^{g(\tau)}_{\mu\mu}$ at ($\tau=2,3,4,5$)

$m_{11}^{g(2)}=m_{12}m_{21}\bar{\xi}_{2}^{\ -1},$ (26)

$m_{11}^{g(3)}=\big{(}m_{12}^{2}\bar{\xi}_{3}+m_{13}^{2}\bar{\xi}_{2}+2m_{12}m_{23}m_{31}\big{)}\left(\bar{\xi}_{2}\bar{\xi}_{3}-m_{23}m_{32}\right)^{-1},$
(27)

$\begin{array}[]{c}m_{11}^{g(4)}=\Big{\\{}m_{12}^{2}\left(\bar{\xi}_{3}\bar{\xi}_{4}-m_{34}^{2}\right)+m_{13}^{2}\left(\bar{\xi}_{2}\bar{\xi}_{4}-m_{24}^{2}\right)+m_{14}^{2}\left(\bar{\xi}_{2}\bar{\xi}_{3}-m_{23}^{2}\right)\\\
\\\
+2\left[m_{12}m_{13}\left(m_{23}\bar{\xi}_{4}+m_{24}m_{34}\right)+m_{12}m_{14}\left(m_{24}\bar{\xi}_{3}+m_{23}m_{43}\right)+m_{13}m_{14}\left(m_{34}\bar{\xi}_{2}+m_{32}m_{42}\right)\right]\Big{\\}}\\\
\\\
\times\left(\bar{\xi}_{2}\bar{\xi}_{3}\bar{\xi}_{4}-m_{23}^{2}\bar{\xi}_{4}-m_{34}^{2}\bar{\xi}_{2}-m_{42}^{2}\bar{\xi}_{3}-2m_{23}m_{34}m_{42}\right)^{-1},\end{array}$
(28)

$\begin{array}[]{c}m_{11}^{g(5)}=\Big{\\{}m_{12}^{2}\big{(}\bar{\xi}_{3}\bar{\xi}_{4}\bar{\xi}_{5}-m_{34}^{2}\bar{\xi}_{5}-m_{45}^{2}\bar{\xi}_{3}-m_{53}^{2}\bar{\xi}_{4}-2m_{34}m_{45}m_{53}\big{)}\\\
\\\
+m_{13}^{2}\big{(}\bar{\xi}_{2}\bar{\xi}_{4}\bar{\xi}_{5}-m_{24}^{2}\bar{\xi}_{5}-m_{45}^{2}\bar{\xi}_{2}-m_{52}^{2}\bar{\xi}_{4}-2m_{24}m_{45}m_{52}\big{)}\\\
\\\
+m_{14}^{2}\big{(}\bar{\xi}_{2}\bar{\xi}_{3}\bar{\xi}_{5}-m_{23}^{2}\bar{\xi}_{5}-m_{35}^{2}\bar{\xi}_{2}-m_{52}^{2}\bar{\xi}_{3}-2m_{23}m_{35}m_{52}\big{)}\\\
\\\
+m_{15}^{2}\left(\bar{\xi}_{2}\bar{\xi}_{3}\bar{\xi}_{4}-m_{23}^{2}\bar{\xi}_{4}-m_{34}^{2}\bar{\xi}_{2}-m_{42}^{2}\bar{\xi}_{3}-2m_{23}m_{34}m_{42}\right)\\\
\\\
+2m_{12}m_{13}\left[m_{23}\big{(}\bar{\xi}_{4}\bar{\xi}_{5}-m_{45}^{2}\big{)}+m_{24}m_{34}\bar{\xi}_{5}+m_{25}m_{35}\bar{\xi}_{4}+m_{45}\left(m_{25}m_{34}+m_{24}m_{35}\right)\right]\\\
\\\
+2m_{12}m_{14}\left[m_{24}\big{(}\bar{\xi}_{3}\bar{\xi}_{5}-m_{35}^{2}\big{)}+m_{23}m_{43}\bar{\xi}_{5}+m_{25}m_{45}\bar{\xi}_{3}+m_{35}\left(m_{25}m_{43}+m_{23}m_{45}\right)\right]\\\
\\\
+2m_{12}m_{15}\left[m_{25}\big{(}\bar{\xi}_{3}\bar{\xi}_{4}-m_{34}^{2}\big{)}+m_{23}m_{53}\bar{\xi}_{4}+m_{24}m_{54}\bar{\xi}_{3}+m_{34}\left(m_{24}m_{53}+m_{23}m_{54}\right)\right]\\\
\\\
+2m_{13}m_{14}\left[m_{34}\big{(}\bar{\xi}_{2}\bar{\xi}_{5}-m_{25}^{2}\big{)}+m_{32}m_{42}\bar{\xi}_{5}+m_{35}m_{45}\bar{\xi}_{2}+m_{25}\left(m_{35}m_{42}+m_{32}m_{45}\right)\right]\\\
\\\
+2m_{13}m_{15}\left[m_{35}\big{(}\bar{\xi}_{2}\bar{\xi}_{4}-m_{24}^{2}\big{)}+m_{32}m_{52}\bar{\xi}_{4}+m_{34}m_{54}\bar{\xi}_{2}+m_{24}\left(m_{34}m_{52}+m_{32}m_{54}\right)\right]\\\
\\\
+2m_{14}m_{15}\left[m_{45}\big{(}\bar{\xi}_{2}\bar{\xi}_{3}-m_{23}^{2}\big{)}+m_{42}m_{52}\bar{\xi}_{3}+m_{43}m_{53}\bar{\xi}_{2}+m_{23}\left(m_{43}m_{52}+m_{42}m_{53}\right)\right]\Big{\\}}\\\
\end{array}$
$\begin{array}[]{c}\times\Big{\\{}\bar{\xi}_{2}\bar{\xi}_{3}\bar{\xi}_{4}\bar{\xi}_{5}-m_{23}^{2}\bar{\xi}_{4}\bar{\xi}_{5}-m_{24}^{2}\bar{\xi}_{3}\bar{\xi}_{5}-m_{25}^{2}\bar{\xi}_{3}\bar{\xi}_{4}-m_{34}^{2}\bar{\xi}_{2}\bar{\xi}_{5}-m_{35}^{2}\bar{\xi}_{2}\bar{\xi}_{4}-m_{45}^{2}\bar{\xi}_{2}\bar{\xi}_{3}\\\
\\\
-2\left[m_{23}m_{34}\left(m_{24}\bar{\xi}_{5}+m_{25}m_{45}\right)+m_{24}m_{45}\left(m_{25}\bar{\xi}_{3}+m_{23}m_{53}\right)+m_{25}m_{53}\left(m_{32}\bar{\xi}_{4}+m_{24}m_{34}\right)+m_{34}m_{45}m_{53}\bar{\xi}_{2}\right]\\\
\\\
+m_{23}^{2}m_{45}^{2}+m_{24}^{2}m_{53}^{2}+m_{25}^{2}m_{34}^{2}\Big{\\}}^{-1},\end{array}$
(29)

where $\bar{\xi}_{\mu}=\xi_{\mu}-m_{\mu\mu}(\xi)$.

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