# Towards lattice-gas description of low-temperature properties above the
Haldane and cluster-based Haldane ground states of a mixed spin-(1,1/2)
Heisenberg octahedral chain

Katarína Karl’ová<EMAIL_ADDRESS>Institute of Physics, Faculty of
Science, P. J. Šafárik University, Park Angelinum 9, 04001 Košice, Slovakia
Jozef Strečka  Institute of Physics, Faculty of Science, P. J. Šafárik
University, Park Angelinum 9, 04001 Košice, Slovakia  Johannes Richter
Institut für Physik, Otto-von-Guericke Universität in Magdeburg, 39016
Magdeburg, Germany  Max-Planck-Institut für Physik Komplexer Systeme,
Nöthnitzer Straße 38, D-01187 Dresden, Germany

###### Abstract

The rich ground-state phase diagram of the mixed spin-(1,1/2) Heisenberg
octahedral chain was previously elaborated from effective mixed-spin
Heisenberg chains, which were derived by employing a local conservation of a
total spin on square plaquettes of an octahedral chain. Here we present a
comprehensive analysis of the thermodynamic properties of this model. In the
highly-frustrated parameter region the lowest-energy eigenstates of the mixed-
spin Heisenberg octahedral chain belong to flat bands, which allow a precise
description of low-temperature magnetic properties within the localized-magnon
approach exploiting a classical lattice-gas model of hard-core monomers. The
present article provides a more comprehensive version of the localized-magnon
approach, which extends the range of its validity down to a less frustrated
parameter region involving the Haldane and cluster-based Haldane ground
states. A comparison between results of the developed localized-magnon theory
and accurate numerical methods such as full exact diagonalization and finite-
temperature Lanczos technique convincingly evidence that the low-temperature
magnetic properties above the Haldane and the cluster-based Haldane ground
states can be extracted from a classical lattice-gas model of hard-core
monomers and dimers, which is additionally supplemented by a hard-core
particle spanned over the whole lattice representing the gapped Haldane phase.

magnetization curves, thermodynamics, localized-magnon theory, octahedral
chain

###### pacs:

05.50.+q, 68.35.Rh, 75.10. Jm, 75.40.Cx, 75.50.Nr

##  I Introduction

Electron spin systems represent promising candidate for a design of quantum
computers, because a two-level character of the electron spin provides one of
the simplest platforms to encode a quantum bit [1]. However, the loss of
quantum information due to a quantum decoherence is regarded as the most
principal obstacle for the development of all quantum technologies exploiting
solid-state materials [2]. Molecular magnetic materials, which are composed
from discrete magnetic molecules, belong to the most perspective electron spin
systems for quantum computation and quantum information processing [3, 4]. A
targeted design of molecular magnets through a chemistry-based bottom-up
approach ensures their scalability, which allows not only the implementation
of a single qubit but also greater number of qubits integrated into a more
complex quantum circuit that can store and process quantum information [5].
The molecular magnetic materials generally possess a well defined pattern of
discrete energy levels, whereby the associated quantum states can be easily
tuned and coherently manipulated [6]. A coherence time of the molecular
magnets has been also significantly enhanced by suppressing a quantum
decoherence arising mostly from nuclear spins and dipolar forces [7]. The
molecular magnetic materials thus naturally satisfy most important
requirements imposed on basic building blocks of quantum computers [8, 9] when
proving their usefulness as prominent resources for the quantum computation
[10], the storage and processing of quantum information [11].

The concept of localized magnons [12] affords a powerful tool for a rigorous
assignment of quantum ground states of geometrically frustrated Heisenberg
spin systems at sufficiently high magnetic fields [13, 14, 15, 16]. Moreover,
the localized nature of eigenstates plays an important role in other flatband
systems [16, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 27, 28]. This concept can
be employed whenever destructive quantum interference traps magnon(s) within a
few lattice sites and hence, the frustrated quantum Heisenberg model can be
exactly mapped onto a classical lattice-gas model with a hard-core potential
[13, 14, 16]. Using this approach, the microscopic nature of the last
intermediate plateau emergent in a zero-temperature magnetization curve of the
quantum spin-1/2 Heisenberg kagome antiferromagnet has been, for instance,
elucidated along with the precise nature of a relevant second-order phase
transition emerging at low but nonzero temperatures [13, 29, 30]. This
exciting theoretical finding were experimentally verified by high-field
magnetization measurement on kagome-like compound Cd-kapellasite
CdCu3(OH)6(NO)${}_{3}\cdot$H2O [31]. The main advantage of the localized-
magnon approach lies in that it also provides, besides an exact ground state,
accurate description of the low-temperature thermodynamics due to a proper
counting of low-lying excited states [13, 14, 16].

In our recent papers, we have provided a proper description of the low-
temperature thermodynamics of the spin-1/2 Heisenberg octahedral chain [25,
27] in the full range of the magnetic fields within the highly frustrated
parameter region, because this frustrated quantum spin chain exhibits at
sufficiently low magnetic fields another exact ground state with the character
of the monomer-tetramer phase being composed from a localized two-magnon
state. Exactly the same ground state with the character of monomer-tetramer
phase appears in the ground-state phase diagram of the mixed spin-(1,1/2)
Heisenberg octahedral chain, for which we have also found a consistent
description of the low-temperature thermodynamics in the full range of the
magnetic fields in a highly frustrated parameter region [32]. It should be
mentioned that a less frustrated parameter region of the mixed spin-(1,1/2)
Heisenberg octahedral chain involves besides the monomer-tetramer phase three
additional fragmentized cluster-based Haldane phases, which are manifested in
the zero-temperature magnetization curve as fractional magnetization plateaus
at 1/6, 1/9 and 1/12 of the saturation magnetization. The cluster-based
Haldane phases appear due to a fragmentation of the octahedral chain, which is
caused by a plaquette-singlet state incorporating four spins from an
elementary square plaquette. Moreover, the plaquette-singlet state regularly
appears within the cluster-based Haldane phases at certain periods of the
octahedral chain and hence, smaller chain fragments separated from one another
by the plaquette-singlet state can be alternatively considered as bound
magnons effectively represented by hard-core dimers, trimers and tetramers,
respectively. This fact gives us hope for a proper description of the low-
temperature thermodynamics of the mixed spin-(1,1/2) Heisenberg octahedral
chain in the full range of the magnetic fields also in a less frustrated
parameter region from a mapping correspondence with the classical lattice-gas
model of hard-core monomers and at least dimers.

This paper is organized as follows. The model and basic steps of the
calculation procedure are reviewed in Section II. Section III deals with the
effective description of the model within the lattice-gas model of hard-core
particles. Then in Section IV we briefly illustrate our numerical tools,
namely the exact diagonalization (ED) and the finite-temperature Lanczos
method (FTLM). The most interesting results for the magnetization curves and
thermodynamics are presented in Section V together with numerical data
obtained from ED and FTLM. Finally, some conclusions and future outlooks are
mentioned in Section VI.

##  II Model and its ground states

Figure 1: A schematic illustration of the mixed spin-(1,1/2) Heisenberg
octahedral chain with spin-1 particles placed on monomeric sites and spin-1/2
particles on sites of square plaquettes. The coupling constant $J_{1}$
accounts for the antiferromagnetic interaction between nearest-neighbor
monomeric and square-plaquette spins, while the interaction $J_{2}$ stands for
the antiferromagnetic interaction between nearest-neighbor spins from square
plaquettes.

Let us consider the mixed spin-(1,1/2) Heisenberg octahedral chain, which is
schematically illustrated in Fig. 1 and given by the Hamiltonian

$\displaystyle\hat{\cal H}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\Bigl{[}J_{1}(\bm{\hat{S}}_{1,j}+\bm{\hat{S}}_{1,j+1})\\!\cdot\\!(\bm{\hat{S}}_{2,j}+\bm{\hat{S}}_{3,j}+\bm{\hat{S}}_{4,j}+\bm{\hat{S}}_{5,j})\Bigr{.}$
(1) $\displaystyle+$ $\displaystyle
J_{2}(\bm{\hat{S}}_{2,j}\\!\cdot\\!\bm{\hat{S}}_{3,j}+\bm{\hat{S}}_{3,j}\\!\cdot\\!\bm{\hat{S}}_{4,j}+\bm{\hat{S}}_{4,j}\\!\cdot\\!\bm{\hat{S}}_{5,j}+\bm{\hat{S}}_{5,j}\\!\cdot\\!\bm{\hat{S}}_{2,j})\Bigr{.}$
$\displaystyle-$
$\displaystyle\Bigl{.}h\sum_{i=1}^{5}\hat{S}_{i,j}^{z}\Bigr{]},$

where
$\bm{\hat{S}}_{i,j}\equiv(\hat{S}_{i,j}^{x},\hat{S}_{i,j}^{y},\hat{S}_{i,j}^{z})$
denotes spatial components of the spin-1 (spin-1/2) operator for the subscript
$i=1$ ($i=2,3,4,5$). The exchange interaction $J_{1}>0$ accounts for the
antiferromagnetic Heisenberg interaction between monomeric spins and spins
placed in vertices of square plaquette, while the spins belonging to the same
square plaquette are coupled through the antiferromagnetic exchange
interaction $J_{2}>0$. The last term in the Hamiltonian (1) accounts for the
Zeeman energy of magnetic moments in the external magnetic field $h\geq 0$ and
$N$ marks the total number of five-spin unit cells of the octahedral chain.
Periodic boundary conditions are assumed $\bm{S}_{1,N+1}\equiv\bm{S}_{1,1}$ in
order to eliminate boundary effects.

Before proceeding to a discussion of the calculation method to be used for
thermodynamic description, let us briefly recall all available ground states
of the mixed-spin Heisenberg octahedral chain, which were comprehensively
studied in our previous paper [32].

Figure 2: The ground-state phase diagram of the mixed spin-(1,1/2) Heisenberg
octahedral chain. The numbers in curly brackets denote the value of composite
spins of square plaquettes (all phases with more than one number correspond to
a phase with spontaneously broken symmetry). The tiny phase illustrated by red
dots corresponds to the 1/12-plateau in the zero-temperature magnetization
curve, which has each fourth square plaquette in a singlet state and other
three consecutive square plaquettes in a triplet state, i.e.
{$\dot{0}\dot{1}\dot{1}\dot{1}$}.

The ground-state phase diagram shown in Fig. 2 was obtained in our previous
work by the use of a few complementary analytical and numerical methods [32].
As one can see from Fig. 2, there are two ferrimagnetic phases corresponding
to 1/6- and 1/3-plateaus, two kinds of spin-liquid phases, four fragmentized
cluster-based Haldane phases corresponding to the 1/3-, 1/6-, 1/9- and
1/12-plateaus, a bound-magnon crystal phase corresponding to 2/3-plateau as
well as the Haldane phase. In our recent work [32] we have investigated
thermodynamics of the mixed-spin Heisenberg octahedral chain in the highly
frustrated parameter region $J_{2}/J_{1}>3$ by using of the mapping
correspondence with the classical lattice-gas model, which takes into
consideration bound two-magnon and one-magnon states of square plaquettes as
two different monomeric particles of the effective lattice-gas model of hard-
core monomers. More concretely, the highly frustrated parameter region
includes besides the trivial fully polarized ferromagnetic state also the
bound magnon-crystal phase schematically shown in Fig. 3(a) and given by the
eigenvector

$\displaystyle\\!\\!\\!\\!\\!|{\rm
BM}\rangle=\prod_{j=1}^{N}\\!|1\rangle_{1,j}\\!\otimes\\!\frac{1}{2}(|\\!\\!\downarrow_{2,j}\uparrow_{3,j}\uparrow_{4,j}\uparrow_{5,j}\rangle-|\\!\\!\uparrow_{2,j}\downarrow_{3,j}\uparrow_{4,j}\uparrow_{5,j}\rangle$
$\displaystyle+|\\!\\!\uparrow_{2,j}\uparrow_{3,j}\downarrow_{4,j}\uparrow_{5,j}\rangle-|\\!\\!\uparrow_{2,j}\uparrow_{3,j}\uparrow_{4,j}\downarrow_{5,j}\rangle\\!)\\!$
(2)

and the monomer-tetramer phase shown in Fig. 3(b) and given by the eigenvector

$\displaystyle|{\rm
MT}\rangle=\prod_{j=1}^{N}\\!|S\rangle_{1,j}\\!\otimes\\!\Bigl{[}\\!\frac{1}{\sqrt{3}}(|\\!\\!\uparrow_{2,j}\downarrow_{3,j}\uparrow_{4,j}\downarrow_{5,j}\rangle\\!\\!+\\!\\!|\\!\\!\downarrow_{2,j}\uparrow_{3,j}\downarrow_{4,j}\uparrow_{5,j}\rangle)$
$\displaystyle-\frac{1}{\sqrt{12}}(|\\!\\!\uparrow_{2,j}\uparrow_{3,j}\downarrow_{4,j}\downarrow_{5,j}\rangle+|\\!\\!\uparrow_{2,j}\downarrow_{3,j}\downarrow_{4,j}\uparrow_{5,j}\rangle$
$\displaystyle+|\\!\\!\downarrow_{2,j}\uparrow_{3,j}\uparrow_{4,j}\downarrow_{5,j}\rangle+|\\!\\!\downarrow_{2,j}\downarrow_{3,j}\uparrow_{4,j}\uparrow_{5,j}\rangle)\Bigr{]}\\!,$
(3)

where $|S\rangle_{1,j}$ denotes one out of three available states $|\pm
1\rangle_{i,j}$ and $|0\rangle_{i,j}$ of the monomeric spins. In the following
we will extend the calculation procedure in order to account for another two
ground states, the tetramer-hexamer phase and Haldane phase, which are
schematically shown in Fig. 3(c) and (d).

Figure 3: A schematic illustration of a few selected ground states of the
mixed-spin Heisenberg octahedral chain given by the Hamiltonian (1), which are
realized in the ground-state phase diagram in the moderately frustrated
regime: (a) the bound magnon-crystal phase; (b) the monomer-tetramer phase;
(c) the tetramer-hexamer phase; (d) the Haldane phase. Red arrows represent
polarized triplet state of a given cluster shaded by the different colors,
whereby spin clusters without arrows are in a singlet state.

##  III Effective lattice-gas model

In order to obtain magnetothermodynamics, we will use the mapping
correspondence between the Hamiltonian (1) of the mixed-spin Heisenberg
octahedral chain and the corresponding lattice-gas model. For the highly
frustrated parameter region $J_{2}/J_{1}>3$ we have presented thermodynamic
properties in our previous paper [32]. The Hamiltonian (1) can be in the
moderately frustrated regime $J_{2}/J_{1}\in(2.16;3)$ mapped to the effective
monomer-dimer lattice-gas model given by the Hamiltonian

$\displaystyle H_{\rm eff}$ $\displaystyle=$ $\displaystyle E_{\rm
FM}^{0}-3Nh-\mu_{H}^{(0)}n_{H}-\mu_{1}^{(2)}\sum_{j=1}^{N}m_{j}$ (4)
$\displaystyle-$
$\displaystyle\mu_{1}^{(1)}\sum_{j=1}^{N}n_{j}-\sum_{n=3}^{5}\mu_{2}^{(n)}\sum_{j=1}^{N}d_{j},$

which can be developed by the following construction. From the energy of the
fully polarized ferromagnetic state $E_{\rm FM}^{0}=N(4J_{1}+J_{2})$ in zero
field we have subtracted the energies of two kinds of monomeric particles
whose presence or absence is determined by occupation numbers $m_{j}$ and
$n_{j}$ and a dimeric particle determined through the occupation number
$d_{j}$. The chemical potential of the first monomeric particle
$\mu_{1}^{(1)}=2J_{1}+2J_{2}-h$ determines an energy cost, which is associated
with creation of bound one-magnon eigenstate on a square plaquette on a fully
polarized ferromagnetic background, while the chemical potential of the second
monomeric particle $\mu_{2}^{(2)}=4J_{1}+3J_{2}-2h$ determines an energy cost
connected with creation of the singlet-tetramer state on the ferromagnetic
background. The chemical potential of the dimeric particle
$\mu_{2}^{(n)}=7J_{1}+2J_{2}-nh\quad(n=3,4,5)$ represents an energy cost,
which relates to the creation of a single octahedron in one out of three
available lowest-energy triplet states. All possible spin values of the
triplet state $S_{T}^{z}=\pm 1,0$, were taken into consideration in order to
obtain correct degeneracy of the triplet-hexamer state in a zero-field limit.
The gapped Haldane phase has been introduced as a particle spread over the
whole chain as given by the occupation number $n_{H}$. The chemical potential
$\mu_{H}^{(0)}=2NJ_{2}+2NJ_{1}(2-\epsilon_{N})$ determines the energy
difference between the Haldane phase and the fully polarized ferromagnetic
state, where $\epsilon_{N}$ denotes ground-state energy of the Haldane phase
per spin for a spin-1 Heisenberg chain of $N$ sites with unit coupling
constant. The energy gap of the Haldane phase is well known
$\epsilon_{\infty}\approx-1.401484$ in the thermodynamic limit $N\to\infty$,
while finite-size results for the chain lengths of eight and twelve spins
corresponding to four and six unit cells of octahedral chain equal to
$\epsilon_{4}=-1.417119$ and $\epsilon_{6}=-1.405796$, respectively.

Except the trivial fully polarized ferromagnetic state the effective
Hamiltonian (4) takes into account a few lowest-energy states such as bound
one-magnon and two-magnon states of square plaquettes as well as a triplet
state of an elementary octahedron composed from a single square plaquette and
its two neighboring monomeric spins (see Fig. 4). In addition, the effective
Hamiltonian (4) also takes into account the Haldane phase spread over the
whole octahedral chain.

Figure 4: A schematic illustration of the mixed-spin Heisenberg octahedral
chain, in which spins from $i$-th elementary unit are either in bound one-
magnon or two-magnon states, triplet-hexamer state or are fully polarized.
Lower panel represents a schematic illustration of the corresponding lattice-
gas model with two kinds of monomeric particles and dimeric particles.

The partition function corresponding to the effective Hamiltonian (4) reads

$\displaystyle{\cal{Z}}$ $\displaystyle=$ $\displaystyle\exp(-\beta E_{\rm
FM}^{0}+3\beta
Nh)\sum_{\\{m_{j}\\}}\sum_{\\{n_{j}\\}}\sum_{\\{d_{j}\\}}\sum_{n_{H}^{0}=1,0}$
(5) $\displaystyle\times$
$\displaystyle\prod_{j=1}^{N}(1-d_{j}d_{j+1})(1-m_{j}d_{j})(1-n_{j}d_{j})(1-m_{j}n_{j})$
$\displaystyle\times$
$\displaystyle(1-n_{H}n_{j})(1-n_{H}m_{j})(1-n_{H}d_{j})$
$\displaystyle\\!\\!\\!\times$
$\displaystyle\sum_{n=3}^{5}\\!\exp\\!\\!\left(\beta\mu_{H}^{(0)}n_{h}\\!+\\!\beta\mu_{1}^{(2)}m_{j}\\!+\\!\beta\mu_{1}^{(1)}n_{j}\\!+\\!\beta\mu_{2}^{(n)}d_{j}\right)\\!\\!,$

where $\beta=1/(k_{\rm B}T)$, $k_{\rm B}$ is the Boltzmann’s constant, $T$ is
absolute temperature and the projection operators
$(1-d_{j}d_{j+1})(1-m_{j}d_{j})(1-n_{j}d_{j})(1-m_{j}n_{j})(1-n_{H}n_{j})(1-n_{H}m_{j})(1-n_{H}d_{j})$
forbids the multiple occupancy of the square plaquette with more than one
particle of the effective lattice-gas model. After tracing out degrees of
freedom of the monomeric particles $m_{j}$ and $n_{j}$, as well as, of the
Haldane phase $n_{H}$, the problem of finding the partition function reduces
to a problem of finding all eigenvalues of the transfer matrix

$\displaystyle{\cal{Z}}$ $\displaystyle=$ $\displaystyle\exp(-\beta
E_{H}\\!)\\!+\\!\exp(-\beta E_{\rm FM}^{0}\\!\\!+\\!3\beta
Nh)\\!\\!\sum_{\\{d_{j}\\}}\\!\prod_{j=1}^{N}\\!{\rm
T}(d_{j},\\!d_{j\\!+\\!1})$ (6) $\displaystyle=$ $\displaystyle\exp(-\beta
E_{H}\\!)\\!+\\!\exp(-\beta E_{\rm FM}^{0}\\!\\!+\\!3\beta Nh){\rm
Tr~{}T}^{N},$

where $E_{H}=-NJ_{2}+2NJ_{1}\epsilon_{N}$ is the energy of Haldane phase. The
expression ${\rm T}(d_{j},\\!d_{j\\!+\\!1})$ denotes the transfer matrix
depending only on occupation numbers of the dimeric particles from two
adjacent lattice sites satisfying the hard-core constraint

$\displaystyle{\rm T}(d_{j},\\!d_{j+1})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!(1-d_{j}d_{j+1})\sum_{n=3}^{5}\exp(\beta\mu_{2}^{(n)}d_{j})$
(7) $\displaystyle\times$
$\displaystyle\\!\\!\left[1+(1-d_{j})\exp\left(\beta\mu_{1}^{(2)}+\beta\mu_{1}^{(1)}\right)\right].$

The transfer matrix as defined by Eq. (7) has the following matrix
representation

$\displaystyle{\rm T}(d_{j},\\!d_{j+1})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\left(\\!\\!\\!\begin{array}[]{cc}1\\!+\\!\exp(\beta\mu_{1}^{(1)})\\!+\\!\exp(\beta\mu_{1}^{(2)})&1\\!+\\!\exp(\beta\mu_{1}^{(1)})\\!+\\!\exp(\beta\mu_{1}^{(2)})\\\
\exp(\beta\mu_{2}^{(4)})[1\\!\\!+\\!\\!2\cosh(\beta
h)]&0\end{array}\\!\\!\\!\right).$ (10) (11)

After the diagonalization of the transfer matrix (11), one gets two
eigenvalues

$\displaystyle\lambda_{\pm}\\!=\\!\frac{1}{2}\left(\Xi\\!\pm\\!\sqrt{\Xi^{2}\\!+\\!4\Xi\exp(\beta\mu_{2}^{(4)})[1\\!+\\!2\cosh(\beta
h)]}\right)\\!\\!,$ (12)

where $\Xi=1+\exp(\beta\mu_{1}^{(1)})+\exp(\beta\mu_{1}^{(2)})$. Then, the
partition function of the monomer-dimer lattice-gas model (4) is given by the
equation

$\displaystyle{\cal Z}\\!\\!=\\!\exp(-\beta E_{H})\\!+\\!\exp(-\beta E_{\rm
FM}^{0}\\!+\\!3\beta
Nh)\\!\\!\left(\lambda_{+}^{N}\\!\\!+\\!\lambda_{-}^{N}\right)\\!\\!.$ (13)

In the thermodynamic limit $N\to\infty$ the partition function is given only
by the higher eigenvalue of the transfer matrix

$\displaystyle{\cal Z}_{\infty}=\exp(-\beta E_{H})+\exp(-\beta E_{\rm
FM}^{0}+3\beta Nh)\lambda_{+}^{N}.$

From the partition function one can obtain the free energy, as well as the
magnetization, the susceptibility, the entropy and the specific heat.

Figure 5: A comparison of ED and FTLM data shown by symbols with the
analytical results obtained from the effective lattice-gas model shown by
solid lines for magnetic-field dependencies of (a)-(b) magnetization and
(c)-(d) susceptibility of the mixed-spin Heisenberg octahedral chain with
$N=4$ (left panel) and $N=6$ (right panel) unit cells, the fixed value of the
interaction ratio $J_{2}/J_{1}=2.7$ and a few selected values of temperature.
The magnetization is normalized with respect to its saturation value and
susceptibility is normalized per unit cell.

##  IV Exact-diagonalization and finite-temperature Lanczos methods

In order to verify the results obtained from the effective lattice-gas model,
we use the full exact numerical diagonalization (ED) and the finite
temperature Lanczos method (FTLM). Trivially the Hamiltonian (1) commutes with
the $z$-component of the total spin $\hat{S}_{T}^{z}$, i.e., the Hilbert space
splits into orthogonal subspaces related to the eigenvalues $S_{T}^{z}$ of
$\hat{S}_{T}^{z}$. In addition, we also exploit lattice symmetries to further
split the $S_{T}^{z}$ subspaces into smaller symmetry related subspaces. For
that we use Jörg Schulenburg’s spinpack code [33, 34] to perform the ED and
the FTLM calculations.

The ED is a well established quantum many-body technique which is widely
applied to frustrated quantum spin systems, see, e.g., Ref. [35]. The FTLM is
an unbiased accurate numerical approximation by which the partition function
${\cal{Z}}$ is determined using trace estimators [36, 37, 38, 39, 40, 41, 42,
43, 44, 45]. ${\cal{Z}}$ is then given by a Monte-Carlo like representation,
i.e., the sum over a complete set of basis states entering ${\cal{Z}}$ is
replaced by a much smaller sum over $R$ random vectors $|{\nu}\rangle$ for
each symmetry-related orthogonal subspace ${\mathcal{H}}(M,\gamma)$ of the
Hilbert space, where $\gamma$ labels the irreducible representations of the
employed symmetries.

In the present case we use full ED to calculate the partition function
${\cal{Z}}$ for chains of $N=4$ unit cells (i.e. 20 sites). For a longer chain
of $N=6$ unit cells (i.e. 30 sites) we combine both methods. We use the full
ED to determine the contribution to ${\cal{Z}}$ of the upper sectors of
$|S_{T}^{z}|=18,\ldots,12$ and the FTLM to calculate the contribution of the
lower sectors of $|S_{T}^{z}|\leq 11$ to ${\cal{Z}}$.

##  V Results and discussion

Figure 6: A comparison of ED and FTLM data shown by symbols with the
analytical results obtained from the effective lattice-gas model shown by
solid lines for magnetic-field dependencies of (a)-(b) entropy and (c)-(d)
specific heat of the mixed-spin Heisenberg octahedral chain with $N=4$ (left
panel) and $N=6$ (right panel) unit cells, the fixed value of the interaction
ratio $J_{2}/J_{1}=2.7$ and a few selected values of temperature. Both
quantities are normalized per unit cell.

Let us proceed to a discussion of the most interesting results. The
magnetization curves of the mixed spin-(1, 1/2) Heisenberg octahedral chain
for four and six elementary unit cells $N=4$ and $N=6$ (i.e. 20 spins and 30
spins in total, respectively) are plotted in Figs. 5(a)-(b) for the fixed
value of the interaction ratio $J_{2}/J_{1}=2.7$. In accordance with the
ground-state phase diagram, one can find in the magnetization process of the
mixed spin-(1, 1/2) Heisenberg octahedral chain three intermediate
magnetization plateaus at 1/6, 1/3 and 2/3 of its saturation magnetization. It
directly follows from a comparison of full ED and FTLM (both shown by symbols)
with the results obtained from the effective model (solid lines) that the
analytical results from the simplified lattice-gas model are in perfect
agreement with precise numerical results in the full range of magnetic fields
up to the temperature $k_{\rm B}T/J_{1}\lesssim 0.08$. The insets in Fig. 5(a)
and (b) are focused on magnetization curves calculated in the low-field region
for two low enough temperatures $k_{\rm B}T/J_{1}=0.02$ and 0.04, where a tiny
1/6-plateau due to the cluster-based Haldane phase with the character of a
tetramer-hexamer state is present. The presence of the 1/6-plateau is
satisfactorily described by the effective lattice-gas model including hard-
core monomeric and dimeric particles, the latter of which are crucial for its
correct description.

A perfect agreement of ED and FTLM data with the analytical results obtained
from the effective lattice-gas model is present also in the magnetic-field
dependence of the susceptibility depicted in Fig. 5(c) and 5(d) for $N=4$ and
$N=6$ elementary unit cells, the interaction ratio $J_{2}/J_{1}=2.7$ and three
selected values of temperature. The susceptibility as a function of magnetic
field exhibits peaks around each field-driven phase transition, which become
lower and rounder upon increasing of the temperature. Besides the perfect
agreement of numerical and analytical calculations, there is only a small
visible deviation of analytical calculations from the full ED data located in
the magnetic-field range $h/J_{1}\in(0.2,0.4)$ at temperature $k_{\rm
B}T/J_{1}=0.08$ for four unit cells [Fig. 5(c)], whereas a somewhat greater
discrepancy observable already for the same field range at lower temperature
$k_{\rm B}T/J_{1}=0.04$ for six unit cells [Fig. 5(d)] might be attributed to
the approximate nature of FTLM data.

Figure 7: A comparison of ED and FTLM data shown by symbols with the
analytical results obtained from the effective lattice-gas model shown by
solid lines for magnetic-field dependencies of (a)-(b) the magnetization and
(c)-(d) the susceptibility of the mixed-spin Heisenberg octahedral chain with
$N=4$ (left panel) and $N=6$ (right panel) unit cells, the fixed value of the
interaction ratio $J_{2}/J_{1}=2.3$ and a few selected values of temperature.
The magnetization is normalized with respect to its saturation value and
susceptibility is normalized per unit cell.

The magnetic-field dependence of the entropy and the specific heat normalized
per unit cell of the mixed-spin Heisenberg octahedral chain is shown in Fig. 6
for $N=4$ and $N=6$ elementary unit cells, the fixed value of the interaction
ratio $J_{2}/J_{1}=2.7$ and a few selected values of temperature. The entropy
of the mixed-spin Heisenberg octahedral chain displays a peak at each field-
driven phase transition, while the specific heat exhibits a double-peak
behavior in the vicinity of all field-driven phase transitions. It is clear
that the height of all peaks in the entropy and the specific heat is invariant
with respect to a small temperature change around the latter two critical
fields, but it changes significantly around a first critical field. It should
be stressed that the entropy and the specific heat are more sensitive for low-
lying excitations above the 1/6-plateau neglected in the effective monomer-
dimer lattice-gas model, which can be seen by more substantial discrepancies
between numerical ED data and the analytical results obtained from the
effective model. As a matter of fact, the more sizable differences are visible
in the low-field region even at lower temperature $k_{\rm B}T/J_{1}=0.04$ and
become higher for higher temperature $k_{\rm B}T/J_{1}=0.08$ for both system
sizes $N=4$ and $N=6$. Nevertheless, the effective monomer-dimer lattice-gas
model qualitatively describes the peak behavior of the entropy and the double-
peak behavior of the specific heat even in the low-field region up to moderate
temperatures $k_{\rm B}T/J_{1}\lesssim 0.08$. Moreover, the developed monomer-
dimer lattice-gas model shows in the high-field region a perfect agreement up
to relatively high temperatures $k_{\rm B}T/J_{1}\approx 0.1$. This fact was
comprehensively discussed in our previous work focusing on the description of
the highly frustrated parameter region, where the effective lattice-gas model
of two kinds of hard-core monomers satisfactorily captures the 1/3- and
2/3-plateau as well as the low-temperature thermodynamics above them [32].

Figure 8: A comparison of ED and FTLM data shown by symbols with the
analytical results obtained from the effective monomer-dimer lattice-gas model
shown by solid lines for field dependencies of (a)-(b) the entropy and (c)-(d)
the specific heat of the mixed-spin Heisenberg octahedral chain with $N=4$
(left panel) and $N=6$ (right panel) unit cells, the fixed value of the
interaction ratio $J_{2}/J_{1}=2.3$ and a few selected values of temperature.
Both quantities are normalized per unit cell.

In order to examine the less frustrated parameter region we have plotted in
Figs. 7 and 8 the same set of physical quantities of the mixed spin-(1,1/2)
Heisenberg octahedral chain for $N=4$ (left panels) and $N=6$ (right panels)
unit cells, the fixed value of the interaction ratio $J_{2}/J_{1}=2.3$ and a
few different values of temperature. In this parameter region, the mixed
spin-(1,1/2) Heisenberg octahedral chain additionally displays the Haldane
phase and the magnetization curve shown in Figs. 7(a) and (b) indeed exhibits
besides the 2/3-, 1/3- and 1/6-plateaus a zero magnetization plateau, which is
gradually blurred with increasing of the temperature. It can be seen from Fig.
7(a) that the effective monomer-dimer lattice-gas model coincides well with
the full ED data for four unit cells at low enough temperatures $k_{\rm
B}T/J_{1}\lesssim 0.04$, while there are more pronounced discrepancies between
numerical and analytical results in the low-field region at higher temperature
$k_{\rm B}T/J_{1}=0.08$. It should be noticed that the magnetization curves
shown in Fig. 7(b) for six unit cells show much greater discrepancy between
the analytical results obtained from the effective monomer-dimer lattice-gas
model and numerical data obtained by combining the full ED with FTLM (ED+FTLM)
due to presence of other cluster-based Haldane state manifested as the
1/9-plateau, which is neglected within the effective monomer-dimer lattice-gas
model and cannot appear in the mixed-spin Heisenberg octahedral chain with the
smaller number of unit cells $N=4$ due to insufficient system size required
for this higher-period ground state.

It is quite obvious from the magnetic-field dependencies of the susceptibility
shown in Fig.7(c) that analytical results obtained from monomer-dimer lattice-
gas effective model for $N=4$ unit cells are in excellent agreement with full
ED data above magnetic field $h/J_{1}>0.5$ up to relatively high temperatures
$k_{\rm B}T/J_{1}\lesssim 0.08$, while they start to deviate more
significantly at smaller magnetic fields $h/J_{1}\lesssim 0.5$. The effective
monomer-dimer lattice-gas model overestimates the height of the susceptibility
peak near the first field-induced transition between the Haldane and tetramer-
hexamer phases, while it underestimates the local minimum between the first
and second critical field $h/J_{1}\approx 0.5$ in comparison with precise
numerical results. This disagreement is caused by the construction of the
effective monomer-dimer lattice-gas model, which neglects low-lying
excitations above the Haldane state associated mainly with the emergence of
other fragmentized cluster-based Haldane phases with higher periods, which are
fully missing in the present effective monomer-dimer lattice-gas model that
would need to be supplemented by hard-core trimeric and tetrameric particles.
It actually turns out that the presence of other cluster-based Haldane phase
with period three being responsible for 1/9-plateau manifests itself in the
magnetic-field dependence of the susceptibility shown in Fig. 7(d) for the
mixed-spin octahedral chain with $N=6$ unit cells as additional peak with the
maximum emergent approximately around magnetic field $h/J_{1}\approx 0.356$.
This fact causes the quantitative disagreement of the height of the low-field
peaks as well as the qualitative discrepancies between results obtained from
the effective monomer-dimer lattice-gas model and numerical calculations based
on the combination of full ED and FTLM. The disagreement concerns thus with
the position of the first peak and the absence of a peak corresponding to
1/9-plateau in the analytical results obtained from the effective monomer-
dimer lattice-gas model.

The magnetic-field dependence of the entropy and the specific heat of the
mixed spin-(1,1/2) Heisenberg octahedral chain with $N=4$ and $N=6$ unit cells
are depicted in Fig. 8 for the interaction ratio $J_{2}/J_{1}=2.3$. It is
evident from Fig. 8(a)-(b) that the entropy exhibits sharp peaks in the
proximity of all critical fields at very low temperature $k_{\rm
B}T/J_{1}=0.02$, which become smeared out upon increasing of temperature. The
results obtained from the effective monomer-dimer lattice-gas model and the
full ED data for $N=4$ or by combining the full ED with FTLM for $N=6$, are in
reasonable agreement for magnetic fields higher than $h/J_{1}\gtrsim 0.75$ up
to moderate temperatures $k_{\rm B}T/J_{1}\approx 0.08$, while the observed
deviations in the low-field region are mainly caused by the neglecting low-
lying excited states above the Haldane phase in the effective monomer-dimer
lattice-gas model.

On the other hand, the specific heat as a function of the magnetic field of
the mixed spin-(1,1/2) Heisenberg octahedral chain for $N=4$ and $N=6$ unit
cells plotted in Fig. 8(c)-(d) exhibits a double-peak behavior near each
field-driven phase transition for the interaction ratio $J_{2}/J_{1}=2.3$. It
should be stressed that the analytical results obtained from the effective
monomer-dimer lattice-gas model and the numerical results obtained for the
specific heat from ED and FTLM are in a plausible accordance in the high-field
region $h/J_{1}\gtrsim 0.5$ up to relatively high temperatures $k_{\rm
B}T/J_{1}\approx 0.08$, while the disagreement at lower magnetic fields
$h/J_{1}<0.5$ relates to neglecting low-lying excited states above the Haldane
phase as well as neglecting the higher-period cluster-based Haldane ground
state corresponding to the 1/9-plateau emergent for larger system size $N=6$
[see Fig. 8(d)].

Table 1: Zero-field energy and degeneracy of the lowest-energy eigenstates with the given $z$-component of the total spin $S_{T}^{z}$ ranging between fully polarized state $S_{T}^{z}=18$ and the eigenstate corresponding to the 2/3-plateau of the mixed spin-(1,1/2) Heisenberg octahedral chain with $N=6$ unit cells (30 spins) by assuming two different values of the interaction ratio $J_{2}/J_{1}=2.3$ and $J_{2}/J_{1}=2.7$. The last column determines the total number of magnons. $S_{T}^{z}$ |  $J_{2}/J_{1}$=2.3 |  $J_{2}/J_{1}$=2.7 |  deg |  $n$
---|---|---|---|---
18 |  +37.8 |  +40.2 |  1 |  0
17 |  +31.2 |  +32.8 |  6 |  1
16 |  +24.6 |  +25.4 |  15 |  2
15 |  +18.0 |  +18.0 |  20 |  3
14 |  +11.4 |  +10.6 |  15 |  4
13 |  +4.8 |  +3.2 |  6 |  5
12 |  $-1.8$ |  $-4.2$ |  1 |  6

Figure 9: Magnetic-field dependencies of the entropy (a)-(b) and the specific
heat (c)-(d) of the mixed spin-(1,1/2) Heisenberg octahedral chain at low
temperature $k_{\rm B}T/J_{1}=0.02$ for three different system sizes with
$N=4$, $N=6$ and $N\to\infty$ unit cells. There are obvious finite-size
effects in the entropy and specific heat at low fields (left panel), but the
finite-size effects are totally absent in the high-field region (right panel).

Energies and degeneracies of the lowest-energy eigenstates of the mixed
spin-(1,1/2) Heisenberg octahedral chain with $N=6$ are presented in Tab. 1
for a few selected values of the total spin $S_{T}^{z}=12,13,...,18$. The
fully polarized ferromagnetic state with $S_{T}^{z}=18$ is nondegenerate,
while the degeneracy of the state with $n$-spin deviations from the fully
polarized ferromagnetic state equals to the combinatorial number
$6\choose{n}$. The respective degeneracy $6\choose{n}$ relates to the number
of all available combinations with $n$ localized magnons placed on 6 square
plaquettes of the octahedral chain. For instance, the lowest-energy eigenstate
from sector with $S_{T}^{z}=17$ corresponds to the state, in which all spins
are polarized except one square plaquette involving one localized magnon
($n=1$), etc. The lowest-energy eigenstate from the last sector with
$S_{T}^{z}=12$ corresponds to the bound magnon-crystal ground state (2), which
has one localized magnon on each square plaquette and is responsible for
magnon crystallization within 2/3-plateau.

To bring deeper insight into the finite-size effects of the mixed spin-(1,1/2)
Heisenberg octahedral chain, we have displayed in Fig. 9 the field dependence
of the entropy and the specific heat at relatively small temperature $k_{\rm
B}T/J_{1}=0.02$ with three different lattice sizes $N=4$, $N=6$ and
$N\to\infty$ and the interaction ratio $J_{2}/J_{1}=2.7$. A comparison between
the obtained analytical and numerical results, as well as behavior of these
quantities was comprehensively discussed above, let us therefore focus our
attention to finite-size effects only. It can be seen from Fig. 9 that sizable
finite-size effects are present in the entropy and specific heat only at low
fields (left panel in Fig. 9), while they are almost totally absent in the
high-field region (right panel in Fig. 9). More specifically, the entropy
differs for different system sizes from zero field nearly up to the magnetic-
field value $h/J_{1}\approx 0.25$. The higher the system size is, the lower is
the zero-field entropy. Similarly, finite-size effects of the specific heat
persists up to nearly the same value of the magnetic field $h/J_{1}\approx
0.25$ (see Fig. 9(c)).

Figure 10: Temperature dependencies of the entropy (a)-(b) and specific heat
(c)-(d) of the mixed spin-1 and 1/2 Heisenberg octahedral chain with $N=4$
(left panel) and $N=6$ (right panel) unit cells and the interaction ratio
$J_{2}/J_{1}=2.3$ just below ($h/J_{1}=0.25$) and above ($h/J_{1}=0.35$) the
critical field $h_{c}/J_{1}=0.3$. At the critical field monomer-tetramer phase
and tetramer-hexamer phase of mixed-spin 1/2 and spin 1 Heisenberg octahedral
chain coexist together.

Finally, let us discuss the temperature dependencies of the entropy and the
specific heat of the mixed spin-(1,1/2) Heisenberg octahedral chain with $N=4$
and $N=6$ unit cells at the interaction ratio $J_{2}/J_{1}=2.7$ just below and
just above the first critical field $h_{c}/J_{1}=0.3$, as displayed in Fig.
10. If the magnetic field is set to the value $h/J_{1}=0.25$, the ground state
of the mixed spin-(1,1/2) Heisenberg octahedral chain is two-fold degenerate
tetramer-hexamer phase. Indeed, the entropy asymptotically matches in the
limit of zero temperature the value $S/Nk_{\rm B}=\ln 2/4\doteq 0.172$ for
$N=4$ and $S/Nk_{\rm B}=\ln 2/6\doteq 0.116$ for $N=6$ unit cells (see Fig.
10(a) and (b), respectively). On the other hand, the ground state of the mixed
spin-(1,1/2) Heisenberg octahedral chain is for the other higher field value
$h/J_{1}=0.35$ the nondegenerate monomer-tetramer phase and thus, the zero-
temperature entropy tends trivially to zero for both system sizes ($N=4$ and
$N=6$). In any case, the entropy persist almost constant also at finite
temperatures up to the value $k_{\rm B}T/J_{1}\approx 0.01$ when it starts to
rise with the further increase of temperature. From the comparison of the
analytical results derived from the effective monomer-dimer lattice-gas model
with ED data for $N=4$ unit cells one may conclude that the effective monomer-
dimer lattice-gas model correctly predicts the behavior of the entropy
whenever the temperature is lower then $k_{\rm B}T/J_{1}\lesssim 0.05$, while
analytical and numerical data in Fig. 10(a) starts to deviate above this
temperature. The effective monomer-dimer lattice-gas model almost copies data
obtained from the full ED and FTLM method also for the entropy of the mixed
spin-(1,1/2) Heisenberg octahedral chain with $N=6$ unit cells up to $k_{\rm
B}T/J_{1}\lesssim 0.03$ (see Fig. 10(b)). Note that the tiny deviation of the
numerical FTLM data for the entropy from zero as $T\to 0$ can be attributed to
the approximate nature of the FTLM.

Temperature dependencies of the specific heat of the mixed spin-(1,1/2)
Heisenberg octahedral chain with $N=4$ and $N=6$ unit cells are plotted in
Fig. 10(c) and (d) for the interaction ratio $J_{2}/J_{1}=2.7$ and two
magnetic fields slightly above and slightly below the field-induced phase
transition between the tetramer-hexamer and monomer-tetramer phase. The
specific heat is zero up to $k_{\rm B}T/J_{1}\approx 0.005$ and then it
displays a striking temperature dependence with a pronounced low-temperature
maximum. From the comparison of results obtained from the effective monomer-
dimer lattice-gas model (solid lines) and numerical data acquired by ED and
FTLM (symbols) can be concluded that the effective monomer-dimer lattice-gas
model qualitatively as well as quantitatively describes the low-temperature
peak of the specific heat. The validity of the effective description seems to
be much better when considering the magnetic fields exceeding the first
critical field, because the analytical and numerical data coincide up to
higher temperatures.

##  VI Concluding remarks

In the present work we have investigated in detail the mixed spin-(1,1/2)
Heisenberg octahedral chain in presence of the external magnetic field using
the extended version of the localized-magnon approach, which establishes a
mapping relationship with the effective lattice-gas model of hard-core
monomers and dimers. By the use of monomeric and dimeric particles we have
afforded a classical description of the fully quantum mixed spin-(1,1/2)
Heisenberg octahedral chain. We have compared our analytical results obtained
from the effective monomer-dimer lattice-gas model with the numerical
calculations obtained from the full ED and FTLM of the mixed spin-(1,1/2)
Heisenberg octahedral chain for two system sizes with $N=4$ and $N=6$ unit
cells and we have shown that the effective monomer-dimer lattice-gas model
satisfactorily describes the 2/3-, 1/3-, as well as 1/6-plateau and the zero
magnetization plateau in the moderately frustrated parameter regime, which
involves the cluster-based Haldane phase (with character of a tetramer-hexamer
ground state) and the uniform Haldane phase. Moreover, the effective monomer-
dimer lattice-gas model qualitatively describes thermodynamic quantities of
the mixed spin-(1,1/2) Heisenberg octahedral chain such as susceptibility,
entropy and specific heat at very low temperatures and explains the degeneracy
of the lowest-energy eigenstates with given $S_{T}^{z}$ ranging from the fully
polarized state up to the eigenstate corresponding to the 2/3-plateau.

Besides this, we have proven that sizable finite-size effects appear in the
low-field region, while there are no finite-size effects in the high-field
region. The presented effective description is thus especially valuable,
because it provides reasonable results in the thermodynamic limit not
accessible to unbiased numerical methods. The theory based on the classical
description developed from the monomeric and dimeric hard-core particles lacks
low-lying excitations above the Haldane phase, which become relevant even at
very low temperatures. We consider this issue as a future challenging task
aimed at better quantitative description of thermodynamics of fully frustrated
quantum spin systems including the uniform and higher-period cluster-based
Haldane phases.

###### Acknowledgements.

This work was financially supported by the grant of The Ministry of Education,
Science, Research and Sport of the Slovak Republic under the contract No. VEGA
1/0531/19 and by the grants of the Slovak Research and Development Agency
under the contract No. APVV-20-0150. K.K. acknowledges kind hospitality during
summer 2021 at Max-Planck-Institut für Physik Komplexer Systeme in Dresden,
where the major part of this work was completed. J.R. thanks the Deutsche
Forschungsgemeinschaft for financial support (DFG RI 615/25-1).

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# Deep Learning Architectural Designs for
Super-Resolution of Noisy Images

###### Abstract

Recent advances in deep learning have led to significant improvements in
single image super-resolution (SR) research. However, due to the amplification
of noise during the upsampling steps, state-of-the-art methods often fail at
reconstructing high-resolution images from noisy versions of their low-
resolution counterparts. However, this is especially important for images from
unknown cameras with unseen types of image degradation. In this work, we
propose to jointly perform denoising and super-resolution. To this end, we
investigate two architectural designs: “in-network” combines both tasks at
feature level, while “pre-network” first performs denoising and then super-
resolution. Our experiments show that both variants have specific advantages:
The in-network design obtains the strongest results when the type of image
corruption is aligned in the training and testing dataset, for any choice of
denoiser. The pre-network design exhibits superior performance on unseen types
of image corruption, which is a pathological failure case of existing super-
resolution models. We hope that these findings help to enable super-resolution
also in less constrained scenarios where source camera or imaging conditions
are not well controlled. Source code and pretrained models are available at
https://github.com/angelvillar96/super-resolution-noisy-images.

Index Terms—  Super-resolution, Denoising, Deep learning, Image enhancement

##  1 Introduction

Single image super-resolution (SR) aims at recovering a high-resolution (HR)
image from its low-resolution (LR) counterpart, in which high-frequency
details have been lost due to degrading factors such as blur, hardware
limitations, or decimation.

Early SR approaches were based on upsampling and interpolation techniques [1,
2]. However, these methods are limited in their representational power, and
hence also limited in their ability to predict realistic high-resolution
images. More complex methods construct mapping functions between low- and
high-resolution images. Such a mapping function can be obtained from a variety
of different techniques, including sparse coding [3, 4], random forests [5, 6]
or embedding approaches [7, 8]. Recently, deep learning methods for super-
resolution lead to considerable performance improvements [9, 10]. ResNet-like
architectures [11] obtain state-of-the-art results for SR tasks while
maintaining low computational complexity [12, 13, 14]. Despite these
successes, it is still challenging to prevent the amplification of noise
during the upsampling steps, which often leads to loss of information and the
emergence of artifacts.

Fig. 1: Original WDSR architecture (top) in comparison to the pre-network
architectural design (center) and the in-network architectural design
(bottom). Pre-network cascades denoising and super-resolution. In-network
combines low-level features from the denoised input with high-level features
from the noisy input.

Several approaches have been considered to jointly perform super-resolution
and denoising. Image restoration can be formulated as an inverse problem. In
this approach, the data term for the respective objectives is specific to the
respective task. For the prior, a more generic function can be chosen that
applies to multiple tasks. For example, using deep learning models [15],
employing a denoiser as regularizer [16, 17] or a so-called plug-and-play
prior [18]. Furthermore, deep learning approaches have also been considered to
combine denoising with a SR model [19]. The authors in [20] propose cascading
a denoiser with a SR model so that the output of the denoiser is fed to the SR
network. The pre-network architectural design in our experiments is similar in
spirit, but instead of using a fixed convolutional neural network (CNN)
denoiser, our design allows for further flexibility regarding the choice of
the integrated denoiser.

In contrast to previous works, we compare two architectures that allow for
further flexibility regarding the choice of the integrated denoiser. This
flexibility can be used to incorporate domain knowledge into the network by
selecting a denoising technique optimized for the particular type of
degradation. For scenarios where domain knowledge is missing, we investigate
the generalization capability of different denoisers and the proposed
architectural designs. Especially for low-resolution images from cameras in-
the-wild, image degradation, such as unseen noise distributions, can lead to
artifacts in the reconstructed high-resolution images.

The first architecture, “pre-network”, includes a denoiser as a preprocessing
step to the super-resolution. The second architecture, “in-network”,
reconstructs the HR image by combining low-level features extracted from the
denoised input and high-level features extracted from the noisy input. To the
best of our knowledge, this type of network design has not yet been studied in
the context of denoising and super-resolution.

We evaluate our architectures with the Wide Activation Super-Resolution model
(WDSR) [13] on noisy versions of the images from the DIV2K [21] dataset. Both
architectures aim to suppress the noise corruptions while still recovering
most of the high-frequency information. We show that both architectures
achieve more realistic reconstructions and better PSNR values than WDSR only.

##  2 Methods

In this work, we employ the Wide Activation Super-Resolution (WDSR) model [13]
as a building block to investigate architectures for joint denoising and
super-resolution.

The original WDSR architecture is shown on top of Figure 1. It consists of two
paths. The main path is on top, consisting of a user-defined number $B$ of
residual blocks. Each block consists of two convolutional layers followed by
weight normalization [22] and ReLU activation. The lower path is a residual
connection. It provides low-level features from the input to the output, which
is critically important for SR tasks [14]. Both paths contain a pixel-shuffle
layer [23] , which performs the upsampling for image super-resolution. WDSR is
sensitive to input images that are corrupted by additive noise. However, we
show in this work that it can be paired with a denoiser in two different ways,
denoted as “pre-network” and “in-network”, which both considerably improve the
results.

Table 1: Average peak signal-to-noise ratio (PSNR) for varying levels of additive Gaussian noise $\mathbf{\sigma^{2}}$ on the low-resolution testing images. Baseline WDSR fine-tuned on the noisy data (“No denoiser”) performs best at low noise powers, while the in-net with an autoencoder denoiser achieves best the PSNR values for $\sigma^{2}\geq 0.20$. Baseline WDSR without fine-tuning (“No tuning”) performs worst (see text for details). |  |  |  |  |  | Median Filter |  | Wiener Filter |  | Autoencoder
---|---|---|---|---|---|---|---|---|---|---
$\mathbf{\sigma^{2}}$ |  | No tuning |  | No denoiser |  | In-net | Pre-net |  | In-net | Pre-net |  | In-net | Pre-net
0.00 |  | 32.16 |  | 32.29 |  | 32.62 | 29.23 |  | 32.37 | 28.95 |  | 32.15 | 24.58
0.05 |  | 23.54 |  | 29.99 |  | 29.76 | 28.30 |  | 29.32 | 27.26 |  | 29.74 | 24.02
0.10 |  | 18.78 |  | 28.28 |  | 28.09 | 27.02 |  | 27.73 | 26.31 |  | 28.04 | 23.94
0.15 |  | 15.79 |  | 27.00 |  | 26.67 | 26.03 |  | 26.34 | 25.69 |  | 26.69 | 23.24
0.20 |  | 13.81 |  | 25.28 |  | 25.54 | 25.32 |  | 25.45 | 25.08 |  | 25.88 | 23.05
0.25 |  | 12.40 |  | 25.12 |  | 24.90 | 24.83 |  | 24.83 | 24.33 |  | 25.64 | 22.85
0.30 |  | 11.32 |  | 24.45 |  | 24.40 | 24.26 |  | 24.34 | 23.67 |  | 24.64 | 22.86

Figure 1 shows both architectures. Pre-network (abbreviated pre-net) is shown
in the middle. It first passes the image through the denoiser prior to
branching into main path and skip connection. This is conceptually similar to
the denoiser and SR concatenation by Bei  _et al._ [20]. One potential
limitation of this approach is error propagation: if the denoiser removes
information that is relevant to super-resolution, it cannot be recovered
afterwards.

In-network (abbreviated in-net) is shown on the bottom of Fig. 1. Here, the
denoiser integrates into the residual connection. Hence, the SR model can
jointly combine low-level features from the denoised input and high-level
features from the noisy input.

Both designs are open for the choice of denoiser, which allows to choose a
task-specific denoiser, i.e., that performs best on an expected noise
distribution. In our experiments, we evaluate three popular denoisers of
varying complexity: median filter [24], Wiener filter [25], and denoising
autoencoders (DAE) [26].

OriginalNo tuningInputNo denoiserMedian pre-netMedian in-netWiener pre-
netWiener in-netDAE pre-netDAE in-net Fig. 2: Comparison of different
reconstructions of an image patch corrupted with Gaussian noise with
$\sigma^{2}=0.15$ using different denoisers and architectural designs. “No
tuning” corresponds to the reconstruction using the original WDSR trained on
clean images and “No denoiser” corresponds to the reconstruction using a WDSR
model fine-tuned on noisy patches.

##  3 Dataset Preparation and Training Procedure

The experiments use the DIV2K dataset. In our experiments, we consider low-
resolution images which have been downsampled by a factor of two. We use the
800 images from the training set for training or fine-tuning the models, and
the 100 validation images for evaluation. Closely following [13], we feed to
the model $96\times 96$ RGB image patches extracted from HR images, along with
their noisy bicubic downsampled counterparts.

We train our models using downsampled image patches corrupted with additive
Gaussian noise. The testing data is corrupted using additive Gaussian noise
with the same distribution as during training. Moreover, Poisson, speckle, and
salt-and-pepper noise are used to degrade the testing data for the robustness
analysis.

As a baseline, we consider two versions of WDSR. First, the publicly available
version of WDSR, pretrained on the DIV2K dataset, denoted as “No tuning”.
Second, WDSR fine-tuned on DIV2K images with added noise, denoted as “No
denoiser”. The model weights are initialized with the pretrained WDSR weights.
Each model is fine-tuned using the mean absolute error (MAE) loss function and
the ADAM update rule [27] with an initial learning rate of $10^{-4}$. To avoid
overfitting, the fine-tuning procedure is stopped after 100 epochs. Regarding
the median and Wiener filter, we use square kernel sizes with side length of
five pixels. For the denoising autoencoders, we construct a fully
convolutional DAE composed of three convolutional layers with 64, 128, and 256
kernels of size $5\times 5$ respectively. Each layer is followed by a ReLU
activation function and max-pooling. DAEs are trained for 80 epochs using the
MSE loss function and the ADAM update rule with an initial learning rate of
$10^{-4}$. For the integration into the WDSR model, the autoencoder parameters
are fixed during the fine-tuning of the network.

OriginalNo tuningInputNo denoiserMedian pre-netMedian in-netWiener pre-netWiener in-netDAE pre-netDAE in-net Fig. 3: Comparison of different reconstructions of an image patch with Poisson noise using different denoisers and architectures. “No tuning” corresponds to the reconstruction using the original WDSR trained on clean images and “No denoiser” corresponds to the reconstruction using a WDSR model fine-tuned on noisy patches. Table 2: Average PSNR values for noise distributions that differ from the training distribution. As a baseline, Gaussian noise is aligned with training distribution. Here, and for the mathematically similar speckle noise, WDSR without a denoiser (“No denoiser”) performs best. For the more challenging salt and pepper (S&P) noise and Poisson noise, the pre-network again generalizes best, by a large margin. |  |  |  |  | Median Filter |  | Wiener Filter |  | Autoencoder
---|---|---|---|---|---|---|---|---|---
Noise | No tuning |  | No denoiser |  | In-net | Pre-net |  | In-net | Pre-net |  | In-net | Pre-net
Gaussian | 18.78 |  | 28.28 |  | 28.09 | 27.02 |  | 27.73 | 26.31 |  | 28.04 | 23.94
Speckle | 24.28 |  | 27.86 |  | 27.63 | 26.78 |  | 26.88 | 24.48 |  | 27.54 | 23.20
Poisson | 14.34 |  | 19.47 |  | 21.30 | 26.69 |  | 20.49 | 20.50 |  | 20.59 | 18.16
S&P | 10.78 |  | 12.77 |  | 13.45 | 26.32 |  | 14.79 | 16.57 |  | 12.76 | 13.82

##  4 Evaluation

OriginalNo tuningInputNo denoiserMedian pre-netMedian in-netWiener pre-
netWiener in-netDAE pre-netDAE in-net Fig. 4: Comparison of reconstructions of
an image patch with salt-and-pepper noise using different denoisers and
architectures. “No tuning” corresponds to the reconstruction using the
original WDSR trained on clean images and “No denoiser” corresponds to the
reconstruction using WDSR fine-tuned on noisy patches. Only the pre-network
with the median filter succeeds at denoising and reconstructing the HR image.

###  4.1 Aligned Noise in Training and Testing

The noise in real images can often be approximated as additive Gaussian noise,
which we model in this experiment as a fixed power $\sigma^{2}$ for training
and testing. This evaluation assumes that noise distribution and strength are
known during training. This is a strong assumption for practical cases, but
this setup shows the fundamental capability of the models to capture non-ideal
images.

Table 1 shows the numerical performances for the peak signal-to-noise ratio
(PSNR). The experiment is performed for additive Gaussian noise with
$0.00~{}\leq~{}\sigma^{2}~{}\leq~{}0.30$. For low noise levels, the fine-tuned
WDSR (“No denoiser”) and in-network with Median filter perform best. In-
network with the denoising autoencoder outperforms the competing methods for
higher noise levels. A consistent decrease in PSNR can be observed with
increasing strength of the noise. This is expected, since image restoration
becomes increasingly difficult with increasing noise strength. Pre-net and in-
net are both affected by increasing noise, since the denoiser increasingly
removes useful information from the image.

In comparison, the original WDSR model (“No tuning”) is unable to accurately
recover the HR image from its noisy LR counterpart. However, fine-tuning WDSR
(“No denoiser”) considerably improves the PSNR, for example by about 10 dB for
$\sigma^{2}=0.1$. Thus, the fine-tuned WDSR without any integrated denoiser
learns to suppress noise and is on par with the best denoiser methods (i.e.,
in-net with autoencoder and median filter). Hence, WDSR’s 16 residual blocks
and 32 convolutional filters apparently possess sufficient representational
power to jointly learn denoising and SR. However, we will show in the next
section that these findings do not generalize to distortions that were not
seen during training.

Figure 2 shows a qualitative comparison of the reconstructions of an image
patch for a noise strength of $\sigma^{2}=0.1$. The in-network architecture
yields very similar results for all denoisers. It exhibits comparable results
to the fine-tuned WDSR and overall outperforms pre-network. Upon closer
examination, pre-net suffers from the denoiser’s removal of relevant
information at a very early processing state. This can be seen in the holes in
the hat. The holes appear blurry or are completely removed in all pre-net
configurations. In-net performs much better since it complements the denoised
image in the skip branch with original image information in the main branch.

In conclusion, when training and testing images are aligned in a fixed amount
of Gaussian noise, the fine-tuned WDSR without any denoiser and the in-net
exhibit the best PSNR results and generate the most accurate reconstructions.

###  4.2 Generalization to Unseen Noise Distributions

In this experiment, we investigate the generalization to noise distributions
that were not part of the training data. The models are trained on images with
additive Gaussian noise of strength $\sigma^{2}=0.1$. Testing is performed on
three different noise distributions. More specifically, we use speckle noise
with $\sigma^{2}=0.1$, Poisson noise with $\lambda=0.1$, and salt-and-pepper
(S&P) noise with a probability of $p=0.2$ to change a pixel.

Table 2 shows quantitative results for this experiment. As a reference, we
also include performances for testing on additive Gaussian noise with
$\sigma^{2}=0.1$, i.e., where training and testing are aligned. The fine-tuned
WDSR (“No denoiser”) performs best on images corrupted by Gaussian noise and
speckle noise. We attribute this to the fact that speckle noise is a
multiplicative distortion, but it follows the same probabilistic distribution
as additive Gaussian noise. In-network with the median filter and the
denoising autoencoder achieves similar results. However, pre-network with the
median filter outperforms the competing methods by a large margin on images
that are corrupted by salt-and-pepper noise and Poisson noise: an improvement
over the baselines of more than 13 dB is achieved.

For noise distributions that significantly differ from those seen during
training, it is highly beneficial to suppress as much of the input corruption
as possible prior to performing SR. For this reason, pre-network performs best
on Poisson noise and salt-and-pepper noise. The choice of the denoiser also
affects the results. To this end, Fig. 3 shows qualitative results on Poisson
noise. All configurations of in-net, as well as pre-net with Wiener filter,
and the fine-tuned WDSR (“No denoiser”) suppress noise in homogeneous areas,
such as the blue sky. However, the wheel exhibits more texture, and it still
contains noise. Pre-net with median filter removes the noise best, but the
reconstructed image appears slightly over-smoothed.

In Fig. 4, qualitative results on salt-and-pepper noise are shown. Here, the
median filter achieves the best reconstructions. The noise is completely
removed, but some details of the lemon peel are also removed. We attribute
both observations to to the fact that median filtering is particularly well-
suited for suppressing salt-and-pepper noise but does not preserve small
structures. The Wiener filter in the pre-net configuration generates unwanted
artifacts since it is optimal for Gaussian denoising. Finally, it is
interesting to note that the PSNR of the denoising autoencoder is oftentimes
outperformed by the other denoisers. This can be attributed to the fact that
the autoencoder is trained on additive Gaussian noise, and it apparently has
difficulties to generalize to other distributions without retraining.

In summary, models that are trained on additive Gaussian noise do not
generalize well beyond Gaussian and speckle noise. Pre-network combined with a
suitable denoiser generalizes considerably better to different noise
distributions than the competing methods.

##  5 Conclusion

In this paper, we compare two architectures to jointly perform image denoising
and single-image super-resolution. We combine the well-known WDSR model [13]
with three denoisers that can be chosen depending on the type of degradation.
Both networks have specific benefits. The “pre-network” architecture
sequentially removes the noise first, and then recovers the high-resolution
image. With a suitable denoiser, pre-network generalizes well to unseen noise
distributions. However, details in the image are removed and the
reconstructions appear slightly over-smoothed. The “in-network” architecture
reconstructs the high-resolution image by combining low-level features from
the denoiser with high-level features from the noisy input. This enables
better structure preservation and sharper reconstructed images, but is more
sensitive to unseen noise distributions and strength, independent of the
chosen denoiser. We hope that these findings are useful toward enabling super-
resolution in-the-wild, when camera and image conditions are not fully
controlled.

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