# Three-body correlations in nonlinear response of correlated quantum liquid

Tokuro Hata111Correspondence and requests for materials should be addressed to
T.H. (email<EMAIL_ADDRESS>or K.K. (email:
<EMAIL_ADDRESS>) Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan.  Yoshimichi Teratani  Department of Physics,
Osaka City University, Osaka 558-8585, Japan.  Tomonori Arakawa  Graduate
School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan.  Center
for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531,
Japan.  Sanghyun Lee  Graduate School of Science, Osaka University, Toyonaka,
Osaka 560-0043, Japan.  Meydi Ferrier  Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan.  Université Paris-Saclay, CNRS,
Laboratoire de Physique des Solides, 91405, Orsay, France.  Richard Deblock
Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France.  Rui Sakano  The Institute for Solid State Physics, The
University of Tokyo, Chiba 277-8581, Japan.  Akira Oguri  Department of
Physics, Osaka City University, Osaka 558-8585, Japan.  Nambu Yoichiro
Institute of Theoretical and Experimental Physics, Osaka City University,
Osaka 558-8585, Japan.  Kensuke Kobayashi  Graduate School of Science, Osaka
University, Toyonaka, Osaka 560-0043, Japan.  Institute for Physics of
Intelligence and Department of Physics, The University of Tokyo, Bunkyo-ku,
Tokyo 113-0033, Japan.  Trans-scale Quantum Science Institute, The University
of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan.

###### Abstract

### Abstract

Behavior of quantum liquids is a fascinating topic in physics. Even in a
strongly correlated case, the linear response of a given system to an external
field is described by the fluctuation-dissipation relations based on the two-
body correlations in the equilibrium. However, to explore nonlinear non-
equilibrium behaviors of the system beyond this well-established regime, the
role of higher order correlations starting from the three-body correlations
must be revealed. In this work, we experimentally investigate a controllable
quantum liquid realized in a Kondo-correlated quantum dot and prove the
relevance of the three-body correlations in the nonlinear conductance at
finite magnetic field, which validates the recent Fermi liquid theory extended
to the non-equilibrium regime.

### Introduction

Understanding the properties of correlated quantum liquids is a fundamental
issue of condensed matter physics. Even in a strongly correlated case,
fascinatingly, we can tell that the equilibrium fluctuations of the system
govern its linear response to an external field, relying on the fluctuation-
dissipation relations based on the two-body correlations. Going beyond this
well-established regime, the three-body correlations are known to be of
importance for van der Waals force Axilrod1943 , the three-body force in
nuclei Otsuka2010 , the Efimov state Efimov ; Kraemer2006 , the ring exchange
interaction in solid 3He Thouless_1965 ; Roger1983 , and frustrated spin
systems Lacroix2011 .

A quantum dot (QD) shows the Kondo effect, when a localized spin in the QD is
coupled with conduction electrons in the reservoirs to form a spin singlet
state GoldhaberGordonNature1998 ; Cronenwett540 ; SCHMID1998182 ;
vanderWielScience2000 . The QD in the Kondo regime is an ideal realization of
a strongly correlated quantum liquid with high controllability and
accessibility to the spin degree of freedom. The essential physics can be
captured by the Anderson impurity model, which describes the single impurity
state with spin $\sigma~{}(=\uparrow,\downarrow$ or $\pm 1)$ and energy
$\varepsilon_{\sigma}$ coupled to the two metallic reservoirs. The Fermi
liquid theory for the Anderson model, which has been developed
phenomenologically NozieresJLTP1974  and microscopically YosidaPTPS1970 ;
YamadaPTP1975_1 ; YamadaPTP1975_2 ; ShibaPTP1975 ; YoshimoriPTP1976 , can
explain the low-energy physics of this highly correlated quantum system as the
properties of an ensemble of quasi-particles interacting with each other via
residual interaction.

Figure 1: Three-body correlations and symmetry breaking in a quantum dot in
the Kondo regime. a Schematic view of the correlations between the three
electrons accounting for the Fermi liquid correction. b Schematic view of the
three-body correlation $\chi_{\uparrow\uparrow\downarrow}$. Electrons
fluctuate between the quantum dot (QD) and the leads in the equilibrium and
all those processes are summed up in $\chi_{\uparrow\uparrow\downarrow}$. c
(center) QD with the time-reversal symmetry (TRS) and particle-hole symmetry
(PHS). The energy level is $\varepsilon_{\sigma}=-U/2$. The two-body
correlations $\chi_{\sigma_{1}\sigma_{2}}$ are finite, while the three-body
correlations are quenched. (left) QD with broken TRS, where the two spins are
separated by $g\mu_{\rm B}B$. (right) QD with broken PHS, where the energy
level is $\varepsilon\neq-U/2$. Experimentally, the gate voltage is applied to
the QD in order to tune the energy level. In these broken symmetry cases, the
three-body correlations $\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$ are finite,
which we detect in this paper.

At zero magnetic field, namely when the time-reversal symmetry (TRS) is
present, the linear response of the Kondo state can be explained in terms of
only a few parameters; the phase shift $\delta_{\sigma}$ of the electron
passing through the impurity state, and the two-body correlations
($\chi_{\sigma_{1}\sigma_{2}}$) between spin $\sigma_{1}$ and spin
$\sigma_{2}$. Here, the two-body correlation is identical to the conventional
linear susceptibility, which is defined as the differential of the free energy
$\Omega$ of the total system regarding the energies of two electrons,
$\chi_{\sigma_{1}\sigma_{2}}\equiv-\partial^{2}\Omega/\partial\varepsilon_{\sigma_{1}}\partial\varepsilon_{\sigma_{2}}$
(see Supplementary Note 1). These parameters determine the two crucial
physical quantities of the Kondo problem: the Kondo temperature $k_{\rm
B}T_{\rm K}\equiv 1/(4\chi_{\uparrow\uparrow})$ and the Wilson ratio $R\equiv
1-\chi_{\uparrow\downarrow}/\chi_{\uparrow\uparrow}$ NozieresJLTP1974 ;
YosidaPTPS1970 .

To go beyond this well-established regime, nonlinear susceptibility, that is,
three-body correlation $\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$ comes into the
game (see Fig. 1a). Similarly as above, it is defined as
$\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}\equiv-\partial^{3}\Omega/\partial\varepsilon_{\sigma_{1}}\partial\varepsilon_{\sigma_{2}}\partial\varepsilon_{\sigma_{3}}$.
To give an intuitive picture, consider a situation where three electrons with
spin $\sigma_{1}$, $\sigma_{2}$, and $\sigma_{3}$ pass through the QD level in
sequence as schematically shown in Fig. 1b. The electron with $\sigma_{1}$
first occupies and leaves the level, and then the second electron with
$\sigma_{2}$ passes through it. Finally, the third electron with $\sigma_{3}$
occupies it. This process describes that the spin in the QD level fluctuates
from $\sigma_{1}$ to $\sigma_{2}$ to $\sigma_{3}$ (from $\uparrow$ to
$\uparrow$ to $\downarrow$, in this specific case shown in Fig. 1b) and that
the spin of the first electron affects that of the third electron. The
correlation among $\sigma_{1}$ and $\sigma_{3}$ via $\sigma_{2}$, or three-
spin exchange process, gives the most naive picture of the three-body
correlations, which constitute the leading term of the nonlinear response of
the system as characterized by $\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$
($\chi_{\uparrow\uparrow\downarrow}$ in this case).

In this work, using a carbon nanotube QD in the $SU(2)$ Kondo regime, we
experimentally prove that the three-body correlations
$\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$ indeed contribute to the nonlinear
conductance. Thanks to the quality of our sample, where the Kondo effect in
the unitary limit is achieved, we quantitatively measure the three-body
correlations, in perfect agreement with recent results of the Fermi liquid
theory MoraPRB2015 ; FilipponePRB2018 ; OguriPRB2018 ; OguriPRL2018 ;
TerataniPRB2020 ; TerataniPRL2020 , and verify their role in the non-
equilibrium regime. In particular, we demonstrate their importance when TRS is
broken, and solve a long-standing puzzle of the Kondo system under the
magnetic field FilipponePRB2018 . The demonstrated method to relate three-body
correlations and non-equilibrium transport opens up a way for further
investigation of the dynamics of quantum many-body systems.

### Results

Nonlinear conductance in Kondo regime. We first outline the basic idea of the
present study. We examine the differential conductance ($dI/dV$) of the Kondo-
correlated QD with the following expansion in terms of the bias voltage ($V$)
PustilnikJPCM2004 ; OguriJPSJ2005 ; RinconPRB2009 ; RinconPRB2010err ;
RouraBasPRB2010 ; SelaPRB2009 ; MoraPRB2009 :

$\displaystyle\dfrac{dI}{dV}=G_{0}-{\alpha_{\rm V}}\left(\dfrac{eV}{k_{\rm
B}T_{\rm K}}\right)^{2}+\cdots,$ (1)

at $T\ll T_{\rm K}$. Here $G_{0}$, $e$, and $k_{\rm B}$ are the zero-bias
conductance, elementary charge, and the Boltzmann constant, respectively. The
coefficient $\alpha_{\rm V}$ of the order $V^{2}$ consists of two- and three-
body terms, $W_{2}$ and $W_{3}$, respectively. As discussed later, $W_{2}$ is
defined using $\delta_{\sigma}$ and $\chi_{\sigma_{1}\sigma_{2}}$ (and hence
$R$) [see Equation (2)], which was previously addressed and established
experimentally GrobisPRL2008 ; ScottPRB2009 ; YamauchiPRL2011 ;
KretininPRB2011 ; FerrierNatPhys2016 . In this work, we newly focus on
$W_{3}$, which is the function of $\delta_{\sigma}$,
$\chi_{\uparrow\uparrow}$, $\chi_{\uparrow\uparrow\uparrow}$, and
$\chi_{\uparrow\uparrow\downarrow}$ for specific cases [see Equation (7),
Methods, and Supplementary Note 1] MoraPRB2015 ; FilipponePRB2018 ;
OguriPRB2018 ; OguriPRL2018 ; TerataniPRB2020 ; TerataniPRL2020 . In the
analysis, we rely on the fact that
$G_{0}=e^{2}/h\times(\sin^{2}\delta_{\uparrow}+\sin^{2}\delta_{\downarrow})$
in the Kondo effect in the unitary limit. This treatment is valid in this
experimental work as we have achieved the unitary limit in our QD (see Fig.
2a) FerrierNatPhys2016 ; FerrierPRL2017 .

Importantly, when the on-site Coulomb interaction ($U$) in the QD is zero, a
complete analytical form for Equation (1) is obtained by replacing $k_{\rm
B}T_{\rm K}$ with $2\gamma_{0}$ (see Methods and Supplementary Note 1). Here,
$2\gamma_{0}$ virtually corresponds to the half width of a resonance peak.
Note that the three-body correlations for the same spins
$(\chi_{\uparrow\uparrow\uparrow}$ and
$\chi_{\downarrow\downarrow\downarrow})$ are finite even in this $U=0$ case
due to the Pauli exclusion principle. On the other hand, only for the $U\neq
0$ case, the Kondo effect makes other components such as
$\chi_{\uparrow\uparrow\downarrow}$ finite. Thus, the experimental observation
of the deviation from the $U=0$ case, which we refer as the free particle (FP)
model later, directly tells us the relevance of those correlations in the
nonlinear Kondo regime. In addition, recently, the expression of the
coefficient $\alpha_{\rm V}$ in the interacting case was theoretically
obtained by expanding the Fermi liquid theory to the non-equilibrium regime
MoraPRB2015 ; FilipponePRB2018 ; OguriPRB2018 ; OguriPRL2018 ; TerataniPRB2020
; TerataniPRL2020 . Our present work, relying on a precise experiment, aims to
perform an accurate test of this theory.

To this end, we experimentally examine $\alpha_{\rm V}$ when TRS or the
particle-hole symmetry (PHS) is broken. In the presence of the magnetic filed
($B$), a single electron energy with spin $\sigma$ in the QD is expressed as
$\varepsilon_{\sigma}=\varepsilon-g\sigma\mu_{\rm B}B/2$, where $\mu_{\rm B}$
and $g\approx 2$ are the Bohr magneton and the $g-$factor in the nanotube,
respectively. When both TRS and PHS hold, the energy level is located at
$\varepsilon_{\sigma}=-U/2$ (center of Fig. 1c), the situation that many
experimental works have addressed GrobisPRL2008 ; ScottPRB2009 ;
YamauchiPRL2011 ; KretininPRB2011 ; FerrierNatPhys2016 . In this symmetrical
point, the two-body correlations are finite, while the three-body correlations
are quenched due to the symmetry. We systematically tune
$\varepsilon_{\sigma}$ to investigate the behavior of $\alpha_{\rm V}$ by
varying either $B$ or the gate voltage to vary $\varepsilon$. As is known in
the Kondo physics, the magnetic field splits the degenerate level by a
quantity $g\mu_{\rm B}B$, which breaks TRS (see the left panel of Fig. 1c).
The gate voltage, on the other hand, moves $\varepsilon$ from $-U/2$ and
breaks PHS (see the right panel of Fig. 1c) MoraPRB2015 ; FilipponePRB2018 ;
OguriPRB2018 ; OguriPRL2018 .

Sample and measurement setup. We perform conductance measurement for a carbon
nanotube QD connected to two Pd/Al electrodes in the dilution fridge, using a
standard low-frequency lock-in technique. We focus on the conventional $SU(2)$
Kondo state previously reported in Refs. FerrierNatPhys2016 ; FerrierPRL2017 ;
HataArxiv2018  . The details are presented in those references and in
Supplementary Note 4. The shape of the Coulomb diamond and the shot noise
measurement experimentally determines whether the symmetry of the Kondo state
is $SU(2)$ or $SU(4)$ FerrierNatPhys2016 ; FerrierPRL2017 ; HataArxiv2018  .
The base temperature of the dilution fridge is $16\ {\rm mK}$. A small
magnetic field $0.1\ {\rm T}$, which is sufficiently small compared to the
Kondo temperature ($T_{\rm K}=1.6\ {\rm K}$ at the PHS point), is always
applied to suppress the superconductivity of the electrodes. In this paper,
for simplicity, we refer to this lowest field $0.1\ {\rm T}$ as zero field
($0\ {\rm T}$), except when we mark experimental points at $B=0.1\ {\rm T}$ in
the figures.

Figure 2: Energy-level and magnetic-field dependence of the Kondo state. a
$\varepsilon/U$ dependence of $G_{0}$ for three temperatures ($16$, $380$, and
$780\ \rm{mK}$). The experimental $T_{\rm K}^{\ast}$ shown on the right axis
is fitted with the formula shown in Ref. FerrierNatPhys2016 . b Differential
conductance at the PHS point as a function of $eV/k_{\rm B}T_{\rm K}$ for
different magnetic fields from $0$ to $2.5\ \rm{T}$ with $0.1\ \rm{T}$ steps.
Here, the lowest field $0.1\ {\rm T}$ is defined as $0\ {\rm T}$ (see text).
(inset) $2\Delta$, which is the width of Zeeman splitting, as a function of
$\mu_{\rm B}B/k_{\rm B}T_{\rm K}$.

Basic characteristics of the Kondo state. Figure 2a shows the zero-bias
conductance $G_{0}$ as a function of $\varepsilon/U$ at zero field for three
different temperatures. In this paper, we define $T_{\rm K}$ as the Kondo
temperature at the TRS and PHS point ($B=0\ {\rm T}$ and
$\varepsilon/U=-0.5$). We also define $T_{\rm K}^{\ast}(\varepsilon,B)$ as the
Kondo temperature for general cases. By definition, $T_{\rm
K}^{\ast}(\varepsilon/U=-0.5,B=0\ {\rm T})\equiv T_{\rm K}=1.6\ {\rm K}$. The
black points in Fig. 2a are the Kondo temperatures $T_{\rm
K}^{\ast}(\varepsilon,B=0\ {\rm T})$, which we evaluate by analyzing the
temperature dependence of $G_{0}$ FerrierNatPhys2016 . We obtain $U=6.4\pm
0.8\ \rm{meV}$ and $\Gamma=1.9\pm 0.2\ \rm{meV}$ by fitting the gate voltage
dependence of $T_{\rm K}^{\ast}(\varepsilon,B=0\ {\rm T})$ (dashed curve),
where $\Gamma$ is the width of energy levels due to coupling between the
electron in the QD and the lead electrons FerrierNatPhys2016 . Together with
$U/\Gamma=3.4\pm 0.6$, $R=1.95\pm 0.05$, which we independently determined
from the shot noise experiment FerrierNatPhys2016 , indicates that the Kondo
state is very close to the limit of the strong correlation OguriJPSJ2005 . As
the two-body correlation $W_{2}$ is defined OguriPRB2018 ; OguriPRL2018 ;
TerataniPRB2020 ; TerataniPRL2020  by

$\displaystyle
W_{2}=-\left[1+5\left(R-1\right)^{2}\right]\sum_{\sigma}\dfrac{\cos{2\delta_{\sigma}}}{2}.$
(2)

$W_{2}=5.5$ at $R=1.95$ and the symmetric point ($\delta_{\sigma}=\pi/2$).

Figure 2b shows the differential conductance obtained at $16\ \rm{mK}$ and the
PHS point ($\varepsilon/U=-0.5$) as a function of $eV/k_{\rm B}T_{\rm K}$ for
different $B$ from $0$ to $2.5\ \rm{T}$. When $B$ increases, the Kondo
resonance gets split, and its amplitude is reduced. Inset in Fig. 2b shows the
Zeeman splitting $2\Delta$ as a function of $\mu_{\rm B}B/k_{\rm B}T_{\rm K}$.
The linear fitting shows that the $g$-factor is $2.0\pm 0.05$ just as expected
for carbon nanotube and the splitting seems to start at $\mu_{\rm B}B/k_{\rm
B}T_{\rm K}=0.23\pm 0.02$ ($B=0.6\ \rm{T}$). Although this splitting has
served as a hallmark of the Kondo effect GoldhaberGordonNature1998 ;
Cronenwett540 ; SCHMID1998182 ; vanderWielScience2000 , the microscopic
mechanism has been theoretically revealed only recently FilipponePRB2018 ;
OguriPRB2018 ; OguriPRL2018 ; TerataniPRB2020 ; TerataniPRL2020 .

Figure 3: Two- and three-body correlations as a function of magnetic field. a
Two-body correlations $W_{2}$ and magnetization $m_{d}$ as a function of
$\mu_{\rm B}B/k_{\rm B}T_{\rm K}$. The scaling factor $k_{\rm B}T_{\rm K}$ is
$138\ {\rm\mu eV}$. $\mu_{\rm B}B/k_{\rm B}T_{\rm K}=1.0$ corresponds to
$B=2.4\ {\rm T}$. The marks indicate the results deduced from the experiment.
The dotted and dashed curves for $W_{2}$ are given by the free particle (FP)
model and the NRG calculations ($U/\Gamma=3.5$), respectively. b Curvature
$\alpha_{\rm V}$ as a function of $\mu_{\rm B}B/k_{\rm B}T_{\rm K}$. The
dotted and dashed curves are derived by the free particle (FP) model and the
NRG calculations ($U/\Gamma=3.5$), respectively. (Inset) $dI/dV$ as a function
of ${\rm sign}(V)\times(eV/k_{\rm B}T_{\rm K})^{2}$ at different magnetic
fields. The curves are offset for clarity. We obtained $\alpha_{\rm V}$ with
linear fitting. Error bars correspond to the uncertainty of the linear fit
performed on slightly different ranges. c Three-body correlations $W_{3}$ as a
function of $\mu_{\rm B}B/k_{\rm B}T_{\rm K}$. In the analysis, we have
compared the experimental results with the FP model (dotted curves) and the
NRG results for the Anderson model with $U/\Gamma=3.5$ (dashed curves)
OguriPRB2018 ; OguriPRL2018 ; TerataniPRB2020 ; TerataniPRL2020 . The error
bars are determined using error propagation.

Two- and three-body correlations in TRS broken case. We analyze the magnetic
field dependence of the Kondo peak at the PHS point with Equation (1) (the
analysis procedure is detailed in the Supplementary Note 3). At a finite
magnetic field, the phase shift relates to the induced magnetization of the
electron inside the QD ($m_{d}$) such that

$m_{d}=\frac{\delta_{\uparrow}-\delta_{\downarrow}}{\pi}.$ (3)

Then, $G_{0}$ and $W_{2}$ are expressed as:

$G_{0}=\dfrac{2e^{2}}{h}\cos^{2}{\left(\frac{\pi m_{d}}{2}\right)}$ (4)

and

$W_{2}=\left[1+5\left(R-1\right)^{2}\right]\cos{\left(\pi m_{d}\right)},$ (5)

respectively OguriPRB2018 . Note that to derive them we use the Friedel sum
rule, $1=\left(\delta_{\uparrow}+\delta_{\downarrow}\right)/\pi,$ which holds
even under the magnetic field. The gradual increase of $m_{d}$ deduced from
the experimental $G_{0}$ is shown on the right axis in Fig. 3a. The numerical
renormalization group (NRG) calculations tell that the Wilson ratio is almost
insensitive to the magnetic field up to $\mu_{\rm B}B/k_{\rm B}T_{\rm K}\sim
1$ (see Methods and Supplementary Note 2). Thus, as shown in Fig. 3a, the
magnetic-field dependence of $W_{2}$ is directly deduced using the
experimentally obtained $m_{d}$ and zero-field Wilson ratio $R=1.95$
FerrierNatPhys2016 . To the best of our knowledge, such a quantitative
measurement of the two-body correlation of the Kondo state under magnetic
fields has never been reported.

Let us now analyze the non-equilibrium case of Fig. 2b in detail based on
Equation (1). The inset of Fig. 3b shows $dI/dV$ as a function of ${\rm
sign}(V)\times(eV/k_{\rm B}T_{\rm K})^{2}$ at $B=0,\ 0.5,\ 1,\ 1.5$, and $2.5\
\rm{T}$, respectively. The “V-shaped” curves indicate that the conductance
shows a parabolic behavior around zero bias, and importantly, the sign of
their curvature changes around $1\ {\rm T}$ when $B$ increases. This sign
reversal can also define the splitting of the Kondo peak, on which we focus
here, instead of the above $2\Delta$ Hewson2005 . We obtain the curvature,
that is, the coefficient $\alpha_{\rm V}$ from the fitting with Equation (1)
(see Methods and Supplementary Note 2). Figure 3b represents $\alpha_{\rm V}$
as a function of the normalized magnetic field $\mu_{\rm B}B/k_{\rm B}T_{\rm
K}$. ${\alpha_{\rm V}}$ crosses zero at $\mu_{\rm B}B/k_{\rm B}T_{\rm K}=0.38$
($B=0.9\ \rm{T}$).

In order to elucidate the physical meaning of the magnetic field dependence of
${\alpha_{\rm V}}$, we consider the FP model with a single resonant level (see
Methods and Supplementary Note 1). In this case, analytic solutions of
${\alpha_{\rm V}}$ are given only by using a single parameter, coupling
constant ($\gamma_{0}$). $2\gamma_{0}$ corresponds to the half width of
$dI/dV$ at zero field. When $\gamma_{0}$ is given, the differential
conductance in the magnetic field can be straightforwardly calculated for the
FP case. $\alpha_{\rm V}(B)$ thus obtained is shown in the dotted curve in
Fig. 3b. Clearly, this dotted curve fails to explain the experimental
observation, indicating the relevance of the Kondo correlation.

Now, using NRG calculations, we obtain $\alpha_{\rm V}(B)$ for the Kondo-
correlated QD with $U/\Gamma=3.5$ OguriPRB2018 . As shown in Fig. 3b, the
theoretical curve (red dashed curve) satisfactorily reproduces the experiment.

Then what can we learn from this agreement between the experiment and the
theory? As recently discussed FilipponePRB2018 ; OguriPRB2018 ; OguriPRL2018 ;
TerataniPRB2020 ; TerataniPRL2020 , $\alpha_{\rm V}$ is related to the three-
body correlations $W_{3}$ as well as $W_{2}$:

$\displaystyle\alpha_{\rm V}(B)$
$\displaystyle=\dfrac{2e^{2}}{h}\dfrac{\pi^{2}}{64}({W_{2}}+{W_{3}})\times\left(\dfrac{T_{\rm
K}}{T_{\rm K}^{\ast}}\right)^{2}.$ (6)

Here, $W_{3}$ is defined as follows (see Methods and Supplementary Note 1),

$W_{3}=\frac{1}{2\chi_{\uparrow\uparrow}^{2}}\sum_{\sigma}\dfrac{\sin{2\delta_{\sigma}}}{2\pi}\left(\chi_{\sigma\sigma\sigma}+3\chi_{\sigma-\sigma-\sigma}\right).$
(7)

As shown in Fig. 3a, we have already obtained the magnetic field dependence of
$W_{2}$. In addition, $T_{\rm K}^{\ast}(\varepsilon=0,B)$ as a function of $B$
is obtained from the Lorentzian fitting of $dI/dV$ (see Supplementary Note 2).
Thus, we can finally derive the magnetic-field dependence of $W_{3}$ using
Equation (6) as presented in Fig. 3c. $W_{3}$ is zero at zero field and
becomes finite as $B$ is increased, i.e., as the TRS is broken.

We can also analytically derive $W_{3}$ in the FP model as shown in the dotted
curve in Fig. 3c [see Methods and Supplementary Note 1]. Importantly, as is
clear in this figure, the absolute value of the experimental $W_{3}$ is much
larger than the theoretical value of the FP model. The behavior of $W_{3}$ is
clearly due to the enhancement of the three-body correlations
$\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$ by the Kondo effect. Indeed, NRG
calculations with $U/\Gamma=3.5$ shown in a red dashed curve nicely reproduce
the experimental $W_{3}$. This result assures that the residual interaction
manifests itself in the non-equilibrium transport in the symmetry-breaking
regime. The experimental determination of the three-body correlations is our
central achievement of this work.

Figure 4: Two- and three-body correlations as a function of the energy level.
a Zero-bias conductance $G_{0}$ and phase shift $\delta_{\sigma}$ as a
function of $\varepsilon/U$. The marks are experimental results obtained at
$B=0$. The dashed curve for $G_{0}$ is given by the NRG calculations with
$U/\Gamma=3.5$. b Curvature $\alpha_{\rm V}$ as a function of $\varepsilon/U$.
The dashed curve is the NRG results with $U/\Gamma=3.5$. (Inset) $dI/dV$ as a
function of ${\rm sign}(V)\times(eV/k_{\rm B}T_{\rm K})^{2}$ at
$\varepsilon/U=-0.51$ and $-0.65$. Error bars correspond to the uncertainty of
the linear fit performed on slightly different ranges. c Three-body
correlations $W_{3}$ as a function of $\varepsilon/U$. We compare the
experimental results with the NRG results with $U/\Gamma=3.5$ (dashed curve)
OguriPRB2018 ; OguriPRL2018 ; TerataniPRB2020 ; TerataniPRL2020 . The error
bars are determined using error propagation.

Two- and three-body correlations in PHS broken case. So far, we have focused
on the PHS point ($\varepsilon/U=-0.5$, see Fig. 1c). Next, we address the PHS
breaking regime by tuning the gate voltage ($\varepsilon$) at $B=0$. The
square mark in Fig. 4a shows $\varepsilon/U$ dependence of $G_{0}$, which
decreases as $\varepsilon$ deviates from the PHS point. The dashed curve given
by the NRG calculations again agrees with the experimental results well. We
also show the phase shift,
$\delta_{\sigma}(=\delta_{\uparrow}=\delta_{\downarrow})$, evaluated from
Equation (1) (circles in Fig. 4a). As expected, the phase is locked around
$\pi/2$ and develops a plateau TakadaPRL2014 ; TakadaPRB2016 . The NRG
calculations indicate that the Wilson ratio does not decrease in this region
(see Supplementary Note 2). Relying on this fact, we evaluate the
$\varepsilon/U$-dependence of the two-body correlation $W_{2}$ with Equation
(2) and the experimental value of the Wilson ratio $R=1.95$ at $B=0$ (For the
obtained $W_{2}$, see Supplementary Figure 5).

The inset of Fig. 4b shows $dI/dV$ as a function of ${\rm
sign}(V)\times(eV/k_{\rm B}T_{\rm K})^{2}$ at $\varepsilon/U=-0.51$ and
$-0.65$. The slope at the former (around the PHS point) is larger than that at
the latter (PHS broken). Figure 4b is the deduced curvature $\alpha_{\rm V}$
as a function of $\varepsilon/U$. Here, we investigate the region within
$|\varepsilon/U+0.5|<0.15$, where we experimentally obtain the Kondo
temperature, $T_{\rm K}^{\ast}(\varepsilon,B=0\ {\rm T})$ (see Fig. 2a). In
Fig. 4c, we plot $W_{3}$ evaluated with Equation (6), showing that $W_{3}$
remains almost zero in the entire region of $|\varepsilon/U+0.5|<0.15$, which
is in sharp contrast with the above result in the magnetic field. This
indicates that three-body correlations are more sensitive to magnetic field
modulation than the gate voltage modulation (energy level modulation). This
reflects that the spin degree of freedom of the Kondo state is fluctuating
while the charge part is frozen. The NRG calculations with $U/\Gamma=3.5$
shown in a dashed curve agree with the experimental $\alpha_{\rm V}$ (Fig. 4b)
and $W_{3}$ (Fig. 4c). This agreement again guarantees that the three-body
correlations can be estimated experimentally.

### Discussion

In conclusion, we report the experimental determination of many-body
correlations in the Kondo QD. We demonstrate that the curvatures of the
differential conductance in the TRS- and PHS- broken regimes are well
reproduced by the NRG calculations with strong interactions, enabling us to
successfully evaluate the two-body and three-body correlations. The
demonstrated quantitative analysis confirms the validity of the recent
development of the Fermi liquid theory in the non-equilibrium regime, which
should inspire future work, for example, to cover the temperature region
around and above $T_{\rm K}$. In particular, their sensitivity to TRS and PHS
should make the three-body correlations an efficient tool to explore
correlated quantum liquids such as topological spin liquids.

## Methods

Fabrication process We show the outline of the fabrication process DelagPhD .

  1. 1.

Sputtering Fe catalyst with $1\ \rm{nm}$ thickness on intrinsic Si substrate.

  2. 2.

Placing the substrate in a quartz tube and heat it in a oven with low pressure
($\sim 1.1\times 10^{-4}\ \rm{mbar}$) and stabilized temperature between $800$
and $1000\ \rm{{}^{\circ}C}$.

  3. 3.

Injecting $\rm{C_{2}H_{2}}$ into the quartz tube for 9 seconds and pumping it
out. After this process, we can get carbon nanotubes.

  4. 4.

Connecting the carbon nanotubes with metallic leads by the electron beam
lithography. We used Pd/Al bilayer for electrodes in the experiment
[Supplementary Figure 6 (a)].

Analysis in the $U=0$ case The complete analytical form for Equation (1) is
obtained in the $U=0$ case (see Supplementary Note 1), which is used in the FP
model in this work. The experimental Kondo peak at zero field is treated as if
it were a resonance peak with $U=0$. We obtain $\gamma_{0}=65\ {\rm\mu eV}$ by
fitting the peak with the following formula:

$\displaystyle\dfrac{dI}{dV}=\dfrac{2e^{2}}{h}\dfrac{\gamma_{0}^{2}}{(eV/2)^{2}+\gamma_{0}^{2}}.$
(8)

As $k_{\rm B}T_{\rm K}=138\ {\rm\mu eV}$, we see $k_{\rm B}T_{\rm
K}/\gamma_{0}\sim 2$, which is explained in Supplementary Note 1.

We present the magnetic field dependence of $W_{2}$, $\alpha_{\rm V}$, and
$W_{3}$ in Figs. 3a, 3b, and 3c, respectively, by using:

$\displaystyle W_{2}$ $\displaystyle=\dfrac{\gamma_{0}^{2}-(\mu_{\rm
B}B)^{2}}{(\mu_{\rm B}B)^{2}+\gamma_{0}^{2}},$ (9) $\displaystyle\alpha_{\rm
V}$ $\displaystyle=\dfrac{2e^{2}}{h}\dfrac{1-3(\mu_{\rm
B}B/\gamma_{0})^{2}}{\left\\{1+(\mu_{\rm B}B/\gamma_{0})^{2}\right\\}^{3}},$
(10) $\displaystyle W_{3}$ $\displaystyle=\dfrac{-2(\mu_{\rm
B}B)^{2}}{(\mu_{\rm B}B)^{2}+\gamma_{0}^{2}}.$ (11)

We treat $\mu_{\rm B}B/\gamma_{0}=(k_{\rm B}T_{\rm
K}/\gamma_{0})\times(\mu_{\rm B}B/k_{\rm B}T_{\rm K})$. In Figs. 3a, 3b, and
3c, $\mu_{\rm B}B/k_{\rm B}T_{\rm K}$ is taken as the horizontal axis.

Properties of $\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}$ Three-body correlations
have permutation symmetry for the spin indexes:

$\displaystyle\begin{split}&\chi_{\sigma_{1}\sigma_{2}\sigma_{3}}=\chi_{\sigma_{2}\sigma_{3}\sigma_{1}}=\chi_{\sigma_{3}\sigma_{1}\sigma_{2}}\\\
&=\chi_{\sigma_{1}\sigma_{3}\sigma_{2}}=\chi_{\sigma_{2}\sigma_{1}\sigma_{3}}=\chi_{\sigma_{3}\sigma_{2}\sigma_{1}}.\end{split}$
(12)

At zero magnetic field (the TRS point),
$\chi_{\uparrow\uparrow\uparrow}=\chi_{\downarrow\downarrow\downarrow}$ and
$\chi_{\uparrow\downarrow\downarrow}=\chi_{\uparrow\uparrow\downarrow}$ hold.
At the PHS point,
$\chi_{\uparrow\uparrow\uparrow}=-\chi_{\downarrow\downarrow\downarrow}$ and
$\chi_{\uparrow\downarrow\downarrow}=-\chi_{\uparrow\uparrow\downarrow}$ hold.
These properties are used in the discussion of $W_{3}$ defined by Equation
(7).

Analysis of differential conductance In the analysis shown in Fig. 2b, we do
not take the data obtained at very low bias region (typically
$|eV/k_{B}T_{K}|^{2}<0.01$), where a slight deviation from the parabolicity is
observed. This effect is most probably due to some other lower energy physics
such as two-stage Kondo effect (see Supplementary Note 2 and Ref.
vanderWielPRL2002 ).

Numerical calculations NRG results shown in Figs. 3 and 4 have been calculated
using the Anderson impurity model in the form
$\mathcal{H}=\mathcal{H}_{d}+\mathcal{H}_{c}+\mathcal{H}_{T}$,

$\displaystyle\mathcal{H}_{d}\,$
$\displaystyle=\sum_{\sigma=\uparrow,\downarrow}\,\varepsilon_{\sigma}\,d_{\sigma}^{\dagger}d_{\sigma}\,+\,U\,d_{\uparrow}^{\dagger}d_{\uparrow}\,d_{\downarrow}^{\dagger}d_{\downarrow}\,,$
$\displaystyle\mathcal{H}_{c}\,$
$\displaystyle=\sum_{\sigma=\uparrow,\downarrow}\sum_{\lambda=L,R}\int_{-D}^{D}\\!d\xi\,\xi\,c^{\dagger}_{\xi\lambda\sigma}c_{\xi\lambda\sigma}\,,$
$\displaystyle\mathcal{H}_{T}\,$
$\displaystyle=\sum_{\sigma=\uparrow,\downarrow}\sum_{\lambda=L,R}v_{\lambda}\int_{-D}^{D}d\xi\,\sqrt{\rho_{c}}\left(c^{\dagger}_{\xi\lambda\sigma}d_{\sigma}+\mathrm{H.c.}\right).$

Here, $\varepsilon_{\sigma}=\varepsilon-\sigma\mu_{B}B$ is the energy of a
single electron with spin $\sigma$ and $U$ is the Coulomb interaction between
electrons in the QD. The $g$-factor is taken as 2. $\mathcal{H}_{c}$ describes
the conduction bands of the leads on the left ($L$) and right ($R$) with the
constant density of states $\rho_{c}\equiv 1/(2D)$. When $U=0$, the resonance
width of the local level due to the tunnel couplings in $\mathcal{H}_{T}$ is
given by $\Gamma/2=\pi\rho_{c}\left(v_{L}^{2}+v_{R}^{2}\right)$. We assume
that the couplings to be symmetric $v_{L}=v_{R}$.

The NRG calculations have been carried out choosing the discretization
parameter to be $\Lambda=2.0$ and keeping $N_{\mathrm{trunc}}=3600$ low-lying
energy states at each step of the iterative procedure Bulla2008 . Our NRG code
uses the global $\mathrm{U}\left(1\right)\otimes\mathrm{SU}\left(2\right)$
symmetries and the method for deducing the correlation functions described in
Ref. TerataniPRL2020 .

Data availability The data that support the findings of this study are
available from the corresponding author upon reasonable request.

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Acknowledgment We thank Raphaëlle Delagrange for helpful discussions and for
sample fabrication. K.K. acknowledges the stimulating discussion in the
meeting of the Cooperative Research Project of the RIEC, Tohoku University.
This work was partially supported by JSPS KAKENHI Grant (No. JP19K14630, No.
JP19H05826, No. JP19H00656, No. JP18K03495, JP18H01815, and No. JP18J10205),
JST CREST Grant (No. JPMJCR1876), and the French program ANR JETS
(ANR-16-CE30-0029-01).

This is a preprint of an article published in Nature Communications. The final
authenticated version is available online at:
https://doi.org/10.1038/s41467-021-23467-4.

Author contributions M.F. and R.D. fabricated the sample. T.H., M.F., T.A.,
and S.L. performed the experiments. T.H. analysed the data. Y.T., R.S., and
A.O. performed the theoretical simulations. T.H. and K.K. wrote the paper with
the input of all authors. K.K. and A.O. supervised the research.

Competing interests The authors declare no competing interests.


BO      -0.64$\;\pm$0.44     -0.2$\;\pm$0.78     0.17$\;\pm$0.81     0.7$\;\pm$0.43     0.46$\;\pm$0.8     0.59$\;\pm$0.57     -0.26$\;\pm$0.62     -0.28$\;\pm$0.72      -0.4$\;\pm$0.67      -0.32$\;\pm$0.75

[0.05cm]    $w_{11}$            $w_{12}$           $w_{13}$            $w_{14}$           $w_{15}$            $w_{16}$            $w_{17}$            $w_{18}$            $w_{19}$            $w_{20}$
LIPO     0.38$\;\pm$0.67     0.89$\;\pm$0.2     -0.15$\;\pm$0.72     0.47$\;\pm$0.55     -0.11$\;\pm$0.80     -0.22$\;\pm$0.65     -0.99$\;\pm$0.04     -0.53$\;\pm$0.39     0.76$\;\pm$0.32     1.0$\;\pm$0.0

BO       0.44$\;\pm$0.75     0.61$\;\pm$0.69     0.11$\;\pm$0.81     0.31$\;\pm$0.62     -0.26$\;\pm$0.75     -0.5$\;\pm$0.73     -0.4$\;\pm$0.92     0.23$\;\pm$0.73      0.73$\;\pm$0.43      0.4$\;\pm$0.92

Mean value and half standard deviation of the von Karman vortex street feedback control law coefficients

natexlab#1#1#1#1 #1#1 #1#1#1#1#1#1
#1#1#1#1 #1#1 #1#1

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# Diffusion-based image inpainting with internal learning

Nicolas Cherel  LTCI, Télécom Paris
IP Paris Andrés Almansa  MAP5, CNRS & Université Paris Cité Yann Gousseau
LTCI, Télécom Paris
IP Paris Alasdair Newson  ISIR, Sorbonne Université

###### Abstract

Diffusion models are now the undisputed state-of-the-art for image generation
and image restoration. However, they require large amounts of computational
power for training and inference. In this paper, we propose lightweight
diffusion models for image inpainting that can be trained on a single image,
or a few images. We show that our approach competes with large state-of-the-
art models in specific cases. We also show that training a model on a single
image is particularly relevant for image acquisition modality that differ from
the RGB images of standard learning databases. We show results in three
different contexts: texture images, line drawing images, and materials BRDF,
for which we achieve state-of-the-art results in terms of realism, with a
computational load that is greatly reduced compared to concurrent methods.

###### Index Terms:

inpainting, internal, single image

##  I Introduction

Inpainting, or image completion, is the task of filling in an unknown zone in
an image in a manner which is coherent with the surrounding context of this
image. Some of the first methods for this task include exemplar-based methods
[1, 2, 3]. These techniques rely on the powerful self-similarity property of
natural images to fill in the occlusion. They are particularly good at
synthesizing convincing textures that fit in the image.

Deep learning, on the other hand, takes the approach of training a model on a
large external dataset, leading to impressive results. Assuming similarity
between the training data and testing data, the network can thus efficiently
inpaint missing content [4]. The advantage of this approach, compared to
patch-based methods, is that it can reconstruct semantic content that is not
found in the test data. This is particularly useful, for example, in the case
of facial images where the relevant content (eyes, mouth) is hidden by the
occlusion.

Even more recently, new incredible results have been produced by diffusion
models [5], surpassing previous generative approaches like Generative
Adversarial Networks (GANs).

However, these require extremely large models, making the approach
computationally expensive. The central reason why these models are so large is
that many different sub-tasks must be carried out during diffusion-based
inpainting. At the beginning of the diffusion, the global geometry of the
image must be reconstructed, followed by finer details at the end of the
diffusion.

Training large models on large datasets incontestably yields great results,
however, in this work we show that these “external” methods, trained on
external data, can be matched by internal methods, which use only the test
data or very limited training data. In this context, we propose to use a
lightweight neural network based on the diffusion framework for the diverse
inpainting of images. Our neural network is trained only on the test image,
using the available visible data. We thus overfit on this single example to
have a specialized model. We also show that in specific contexts such as
material inpainting, there is no existing solution available off-the-shelf,
and an internal method is well-suited given the specifics of such modalities,
and the limited availability of large datasets. On material BRDF datasets, our
model produces very good results and is perfectly adapted to the specificities
of these images, maintaining the correlation between the different maps.

##  II Related works

Classical methods for image inpainting have long relied on the self-similarity
of natural images, building solutions on patches from the inpainted image. Two
classical approaches include the greedy exemplar-based method of Criminisi
_et al_. [1] and the global optimization method by Wexler  _et al_. [2].
Several improvements have been proposed to make this last approach faster
(Barnes  _et al_. [6]) or to include texture features (Newson  _et al_. [3]).

With the development of deep learning, it has become common to train a neural
network on a large database of images and to perform the inpainting with a
simple forward pass. The seminal work of Pathak  _et al_. [4] has been refined
with global and local discriminators by Iizuka  _et al_. [7]. Yu  _et al_. [8]
adds an attention layer which mimicks the classical patch-based approaches.
The method LAMA [9] uses Fourier convolutions to avoid expensive attention
layers.

Recently, diffusion models [5], acclaimed for their generative performances,
have been used for image inpainting. Lugmayr  _et al_. [10] derive an
inpainting algorithm from a purely unconditional generative model. Saharia
_et al_. [11] train a diffusion model specifically for image inpainting. These
approaches enable state-of-the-art inpainting results but require massive
training procedures.

A series of work have explored the training of deep networks on very small
datasets and even single images, we refer to these methods as internal
learning. Deep Image Prior [12] has been proposed to use a neural network as
the regularization component for various image restoration methods. Following
SinGAN [13] for single image generation, Alkobi  _et al_. [14] proposed a
network trained on a single image for image inpainting. It can generate
diverse image completions. A preliminary version of our work was presented in
[15].

##  III Method

###  III-A Framework

We adapt the general framework of diffusion models [5] to image inpainting.
Starting from the unknown distribution of natural images $q(x_{0}|y)$
conditioned on the observations $y$, we define a Markov chain
$q(x_{0}|y),...,q(x_{T}|y)$ with the forward transition kernel for
$t\in[1,T]$:

$q(x_{t}|x_{t-1},y)=\mathcal{N}\left(x_{t};\sqrt{1-\beta_{t}}x_{t-1},\beta_{t}\mathbf{I}\right),$
(1)

with the variance schedule linearly increasing from
$\beta_{1}=1\mathrm{e}{-4}$ to $\beta_{T}=0.02$. We introduce the variables
$\alpha_{t}=1-\beta_{t}$ and $\bar{\alpha}_{t}=\prod_{s=1}^{t}\alpha_{s}$
which will come in handy because
$q(x_{t}|x_{0})=\mathcal{N}\left(x_{t};\sqrt{\bar{\alpha}_{t}}x_{0},(1-\bar{\alpha}_{t})\mathbf{I}\right)$.
In this case, the reverse process defines a conditional Gaussian distribution
$p_{\theta}(x_{t}|x_{t+1},y)$ of learnable mean and variance:

$p_{\theta}\left(x_{t}|x_{t+1},y\right)=\mathcal{N}\left(x_{t};\mu_{\theta}(x_{t+1},y,t),\sigma^{2}_{t}\mathbf{I}\right)$

As is the standard approach, we use a neural network (detailed below) to
predict the conditional mean $\mu_{\theta}(x_{t+1},y,t)$. We use a fixed
variance schedule for the reverse process:
$\sigma^{2}_{t}=\frac{1-\bar{\alpha}_{t-1}}{1-\bar{\alpha}_{t}}\beta_{t}$. The
parameters $\theta$ are optimized to maximize a lower bound of the log-
likelihood of the data in [5].

In the case of inpainting, as mentioned above, we are interested in the data
distribution conditioned on our observations _i.e_. the known region of the
image. The observations are introduced in the network as incomplete images via
concatenation in the input layer (similarly to [11]). We also add the mask
information as input. We train our network by minimizing the reweighted
$\ell_{2}$ loss of the $x_{0}$-parametrization:

$\mathbb{E}_{x_{0},\epsilon,t,M}\lVert
M\odot\left(x_{0}-f_{\theta}(x_{t},y,t,M)\right)\rVert_{2}^{2},$ (2)

where $M\sim\mathcal{P}_{\text{mask}}$ is a binary mask with $0$ for the known
region and $1$ elsewhere, $y$ is the masked image _i.e_. $y=x_{0}\odot(1-M)$,
and $x_{t}$ is obtained by the forward diffusion of $x_{0}$ by the following:

$x_{t}=\sqrt{\bar{\alpha}_{t}}x_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon$

where $\epsilon\sim\mathcal{N}(0,\mathbf{I})$.

The parameter $t$ controls the noise level. Our network $f_{\theta}$ takes as
input the noisy image $x_{t}$, the observation $y$, the mask $M$ and the noise
level at step $t$.

###  III-B Single image inpainting

In the case of single image inpainting, we are given a large image
$x_{\text{test}}$ and its corresponding mask $M_{\text{test}}$. When possible,
we take sub-regions of the large image that do not intersect with the testing
mask, and train on these smaller images: we generate synthetic masks, pick a
random timestep, and sample noise to be added. After training, the model is
then applied to the test data.

When the image is too small to take sub-regions, we still use the same
training tactic but mask the input data with both the training mask and the
testing mask. The loss for the pixels in the testing mask is zero.

###  III-C Architecture

In practice, our conditional network takes as input the masked image $y$, the
noisy image $x_{t}$, the inpainting binary mask $M$, and the temporal
information $t$. We use a much smaller network architecture than in the
classical approach [5]. We also use a UNet-type network, but without attention
layers. The network entries are concatenated before the first convolution and
the number of channels is limited to 32. As a result, our network has 160k
parameters, compared to 450M for the state-of-the-art RePaint method [10]. The
architecture is represented in Figure 1.

Figure 1: UNet architecture used for our experiments. The different inputs are
first concatenated before being processed by fully convolutional layers.

.

Contrary to [5], we get better results by directly predicting the initial
image $x_{0}$ rather than the noise $\epsilon$. We find that our simple
formulation, similar to that used in some denoising networks, gives us
equivalent results and simplifies the network. Training is performed by
randomly drawing regions of size 256x256 from our reference image, which are
then noised and masked. The $M$ mask is generated synthetically as in [8]. The
algorithm is designed to mimick an eraser used for removing a stain with back
and forth motions in a local region. Starting from a random starting point, a
sequence of lengths and angles are drawn randomly and defines the
brushstrokes. This is repeated multiple times.

It is necessary to train the network to perform denoising for all time steps,
so they are drawn uniformly in $[1,T]$.

The UNet architecture allows for an adaptive receptive field: depending on the
task, the multi-scale nature of the network can be exploited or not. This is a
clear advantage over patch-based methods, for which the patch size and the
number of scales in the Gaussian pyramid are two very important parameters
that can be difficult to set.

##  IV Experiments

###  IV-A Methods

For all experiments, we train our diffusion model for 15k iterations for each
image. The initial learning rate is 1e-4 and is lowered to 1e-5 after 10k
iterations. Training time is 15 minutes on a NVidia V100 GPU. We compare our
method to the internal patch-based method of Newson  _et al_. [3]. We also
include comparisons with a very large, state-of-the-art external diffusion
model trained on natural images for image inpainting: RePaint [10]. Our method
can also produce diverse solutions but we do not evaluate this property here.

###  IV-B Textures

In order to show the capacity of the method to leverage internal regularity,
we test it on 100 images from the texture dataset of [16]. Many of these
images have self-similar structures and are well suited for internal learning
and patch-based methods.

We show some visual results in Figure 2. We see that the diffusion model
produces very crisp results, on par with the results from RePaint. RePaint may
suffer from hallucination and bias from the external datasets despite
performing very well on many challenging textures. This can be seen with the
appearance of red squares in one of the examples. We measure the inpainting
error with PSNR, and LPIPS which are not optimal given the ill-defined nature
of inpainting but reasonable for textures (a lot less for non-stationary
images with many degrees of freedom) in Table I. We also include a comparison
with a lighter external method, DeepFill [8]. This approach produces
relatively poor results in comparisons to the others. RePaint produces very
good results, at the cost of a huge number of parameters (3 orders of
magnitude more than our approach). The internal methods, patch-based and ours,
yield the best performances on this database.

TABLE I: Reconstruction error using different metrics on our texture inpainting dataset. | Method | PSNR (dB) ↑ | SSIM ↑ | LPIPS ↓ | #Param.
---|---|---|---|---|---
External | DeepFill | 21.55 | 0.870 | 0.105 | 3M
RePaint | 26.77 | 0.899 | 0.053 | 450M
Internal | Patch | 28.12 | 0.902 | 0.046 | -
Ours | 28.02 | 0.908 | 0.057 | 160k
Patch |  |  |  |  |
---|---|---|---|---|---
DeepFill |  |  |  |  |
RePaint |  |  |  |  |
Ours |  |  |  |  |
Figure 2: Inpainting results for 4 methods (see text). Results from DeepFill
are the least satisfying. Results from RePaint are good but sometime lacks
sharpness, produce results with wrong scales or hallucinate content (4th
column). Patch and our method yield the best results in these cases.

###  IV-C Line drawing

We also include an evaluation on line drawing inpainting, using the dataset
from Sasaki  _et al_. [17]. Our method is trained on the corresponding
training set. We generate a testing set using subregions of the test images
(100 of them). For the patch-based method of Newson  _et al_., we only use the
reference image as there is no easy way to use the rest of the training set.
We show some results in Figure 3. The results are visually very good for all
three methods, with sharp lines and no blurring. We measure the results
quantitatively using binary cross-entropy which is well suited to binary
modalities and the LPIPS metric. For this task, we show that our method is
competitive with RePaint (Table II), but with only a fraction of the inference
time (3 seconds for our method compared to 10 minutes for RePaint). The
diversity, speed, and quality of results make our method interesting for
interactive uses.

TABLE II: Reconstruction error using different metrics on line drawing Method | Cross-entropy ↓ | LPIPS ↓ | Training time | Inference time
---|---|---|---|---
Patch | 0.269 | 0.110 | - | 2 min
RePaint | 0.113 | 0.042 | 24 days | 10 min
Ours | 0.080 | 0.047 | 15 min | 3 sec
|  |
---|---|---
|  |
Patch | RePaint | Ours
Figure 3: For simple completions (top), our method performs as well as RePaint
and better than the method of Newson  _et al_. For complex completions with no
obvious ground-truth, all results are good.

###  IV-D SVBRDF

Spatially Varying Bidirectional Reflectance Distribution Functions (SVBRDF)
are a common representation of materials used to render textures on 3D
objects. These spatial functions encode the appearance and physical properties
of each material to model their interaction with light for rendering purposes.
These maps are highly correlated, and their inpainting must be done jointly.
We use 50 examples from the Deschaintre  _et al_. [18] dataset, which includes
the diffuse color map, the normal map, the roughness map and the specular map.
Other types of maps (metallic, displacement, etc.) may be included depending
on the material, which complicates the development of a single inpainting
model. SVBRDFs are used to decorate objects and structures in 3D worlds, and
have similar properties to image textures. An internal method thus appears
well-suited to inpaint SVBRDFs.

We adapt the patch-based method of Newson  _et al_. to use more than 3 input
channels and to reconstruct all channels together, simply by modifying the
distance used to compare patches. For our method, we simply add more input and
output channels. We do not reweight the channels. Finally, we compare with an
external method (RePaint), which is used to inpaint each map independently. In
details, we perform 4 separate inpaintings for the diffuse color, normal map,
roughness map, and specular map. Observe that the retraining of the
corresponding diffusion model would imply massive training times and very
large databases.

The results are shown in Figure 4. It can be seen that RePaint produces
incoherent completions despite good individual inpainting. On the other hand,
the patch-based method and our internal method preserve much better the
correlations between maps. This is important for rendering the materials as
seen in the last column. The lack of correlation produces results which lacks
simulated depth, which is the original goal of SVBRDF. Quantitatively, we
simply measure the mean square error between the ground truth and the
reconstruction. A better evaluation would be to evaluate the renderings in
different lighting conditions, but that is beyond the scope of this work. We
see in Table III that the MSE is lower for our method.

TABLE III: Quantitative evaluation of material inpainting for all the maps using mean-squared error. Method | Diffuse ↓ | Normals ↓ | Roughness ↓ | Specular ↓
---|---|---|---|---
RePaint | 0.0024 | 0.0058 | 0.0069 | 0.00021
Patch | 0.0031 | 0.0071 | 0.0078 | 0.00020
Ours | 0.0017 | 0.0050 | 0.0045 | 0.00012
Patch |  |  |  |  |
---|---|---|---|---|---
RePaint |  |  |  |  |
Ours |  |  |  |  |
| Diffuse | Normal | Roughness | Zoom-In | Render
Figure 4: Inpainting results for SVBRDFs. The properties of the materials are
correlated across maps, so inpainting should maintain this correlation. Our
method preserves this property while non-specific methods like RePaint does
not (see zoom-in with borders circled in red, where the short lines are not in
coherent positions). RePaint’s render looks flat. The specular map is omitted
as this material has a uniform specularity which is correctly inpainted by all
approaches.

##  V Conclusion

In this paper, we have shown that a lightweight diffusion model trained on a
single image or a few images is very efficient for inpainting self-similar
images. In such cases, results are on par or better than results produced by
massive diffusion models implying training or inference times that are several
order of magnitude larger. As a result, our method can be used in a variety of
contexts to achieve high-quality and diverse inpainting results. this would be
particularly interesting for interactive use. We have shown on material data
that image-like data can be handled much better by specific diffusion models
than by large standard networks. This is of great interest in cases where data
availability is low and the training of large networks is infeasible.

Our approach could benefit from the most common techniques for speeding up
diffusion models, as well as any quantization or pruning technique, to make it
lighter and faster.

## Acknowledgements

The authors acknowledge support from the French Research Agency through the
PostProdLEAP project (ANR-19-CE23-0027-01) and Mistic project
(ANR-19-CE40-005).

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