# Trade-off between Squashed Entanglement and Concurrence in Bipartite Quantum
States

Kapil K. Sharma∗ and Suprabhat Sinha†
∗DY Patil International University,
Sect-29, Nigdi Pradhikaran, Akurdi,
Pune, Maharashtra-411044, India
E-mail<EMAIL_ADDRESS>
†Department of Electronics,
West Bengal State University
Barasat, Kolkata-700126, India
Email<EMAIL_ADDRESS>
###### Abstract

In this article we investigate the unitary dynamics of squashed entanglement
and concurrence measures in Werner state and maximally entangled mixed states
(MEMS) under two different Hamiltonians. The aim of the present study is two
fold. The first part of the study deals with the dynamics under Heisenberg
Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian
which is the extension of first Hamiltonian. In both the parts we investigate
the dynamical trade off and balancing points for squashed entanglement and
concurrence. During the study, we also found the results of entanglement
sudden death (ESD) with Heisenberg Hamiltonian in Werner state under
concurrence measure. In second part, we investigate the special result for bi-
linear bi-quadratic Hamiltonian; it does not disturb squashed entanglement and
concurrence in both the states and exhibit the robust character for both of
the states.

##  I Introduction

Quantum information and computation is an emerging area which has potential
impact on future quantum technologies[1]. This area has done tremendous
progress in recent years and currently many commercial companies have interest
to explore the cloud services for Noisy Intermediate Scale Quantum Computers
(NISQ)[2] by 2025. These quantum computers are also rephrased as Near Term
Quantum Computers, which will be equipped with 50 to 100 qubits. Despite the
fact that quantum information and computation is very emerging but on the
other hand this area also have theoretical hindrances which are still
unexplored[3]. A physical system incorporating the superposition, prolonged
entanglement and minimum decoherence play an important role in quantum
information. The scientific community always have keen interest to discover
quantum materials[4] which can perform the quantum computation at room
temperature. To the date, quantum computer function at very low temperature
and dilute refrigerator is the primary requirement for these computers.

Exploring the entanglement dynamics in quantum information needs to consider
varieties of quantum states in the background; since the quantum states are
the building blocks for quantum technologies incorporating quantum
teleportation[5, 6, 7], quantum sensing[8], quantum cryptography[9, 10] etc.
Entanglement is very fragile phenomenon and very sensitive towards external
agents; also it is very difficult to maintain it for a long time in a physical
system. The experimental manifestation for prolonged entanglement in varieties
of quantum states are on the way. On the theoretical side, it is also
difficult to characterize the multipartite entanglement by following the
quantum resource theory[11]. The dynamical study of quantum correlations and
investigating their sustainability in varieties of quantum states is an
important regime for quantum information. The various theoretical entanglement
measures like Wootters concurrence[12] and quantum discord[13, 14, 15] are
tested in varieties of spin chains along with experimental manifestations. In
this direction it is also important to proceed theoretical studies with other
quantum correlations measures.

In the current article we focus on the dynamical aspects of quantum
correlations measures such as squashed entanglement[16] and concurrence[12] in
bipartite Werner state[17] and maximally entanglement mixed states (MEMS)[18].
The study is carried out under two different Hamiltonians called as Heisenberg
Hamiltonian[19] and bi-linear bi-quadratic Hamiltonian[20]. The bi-linear bi-
quadratic Hamiltonian is the non linear version of Heisenberg Hamiltonian and
both the Hamiltonians supports $SU(2)$ symmetry. The entanglement dynamics
under Heisenberg Hamiltonian has been studied in varieties of spin chains
configured in thermal and non-thermal conditions[22, 23]. But in literature
the study on bi-linear bi-quadratic Hamilton is very limited.

Following the literature, the present work pursue the study under two
different kinds of Hamiltonians and we have found an interesting property that
bi-linear bi-quadratic Hamiltonian does not effect the quantum correlations in
both the states (Werner and MEMS). In 2011, squashed entanglement[16, 24]
attracted much attention of quantum community, but the simulations of squashed
entanglement in larger Hilbert spaces are also missing to till date. So, in
this direction the present study is totally new in this direction to the best
of our knowledge.

The outlines of the paper is sketched in 8 sections. In section 2, we present
the brief introduction of Werner state and MEMS and characterization of
quantum correlations with squashed entanglement and concurrence. Section 3
highlights the Hamiltonians applied in the work and approach of unitary time
evolution. Section 4 is devoted to initial state preparation of quantum states
used in the work. In section 5, we obtain the mathematical functions of
squashed entanglement and concurrence with time evolution under Heisenberg
Hamiltonian. The dynamics of quantum correlations under Heisenberg Hamiltonian
is explored in section 6. Section 7 is devoted for the study under bi-linear
bi-quadratic Hamiltonian. The last section explore the conclusion of the work.

##  II Werner state, MEMS and quantum correlation measures

In this section, we present the description of the Werner state and MEMS with
their corresponding density matrices. Further we give the overview of two
quantum correlation measures named as squashed entanglement and concurrence,
which we have used in current work.

Werner state[17] is widely studied quantum state in quantum information theory
and it has its great importance in this field. This is a bipartite quantum
state in $d\times d$ dimensional Hilbert space which is invariant under all
unitary operators and satisfies the equation given below,

$\rho=(U\otimes U)\rho(U^{\dagger}\otimes U^{\dagger})$ (1)

Where $\rho$ is the density matrix of the system. Dealing with two qubits
Werner state in $2\times 2$ dimensional Hilbert space; the Werner state adopt
the the form as given below,

$\rho_{WS}=\gamma|\psi^{-}\rangle\langle\psi^{-}|+(1-\gamma)\frac{I}{4}$ (2)

Where $|\psi^{-}\rangle$ represents a singlet state and I is the $4\times 4$
dimensional identity matrix.

MEMS is an another bipartite quantum state, which has close connection with
the bipartite Werner state. The two qubits MEMS was investigated by Munro et
al.[18] which is more entangled than two qubit Werner state in terms of
concurrence measure and it is experimentally verified also[25]. The density
matrix of two qubits bipartite MEMS can be written as,

$\rho_{MEMS}=\left[\begin{array}[]{cccc}g(\gamma)&0&0&\frac{\gamma}{2}\\\
0&1-2g(\gamma)&0&0\\\ 0&0&0&0\\\
\frac{\gamma}{2}&0&0&g(\gamma)\end{array}\right]$ (3)

This state incorporate a function $g(\gamma)$ with the the following
conditions,

$g(\gamma)=\delta=\left\\{\begin{array}[]{cc}\frac{1}{3},&\quad
0\leq\gamma<\frac{2}{3}\\\ \frac{\gamma}{2},&\quad\frac{2}{3}\leq\gamma\leq
1\end{array}\right.$ (4)

For simplicity throughout this article we consider $g(\gamma)$ as $\delta$. In
continuation of the above discussion on quantum states, we would like to bring
the attention on quantum correlation measures that we use in our work.
Squashed entanglement is a well known quantum correlation measure in quantum
information theory. This multipartite entanglement measure satisfy convexity,
additivity and super additivity over tensor product in general[24]. Further
this measure can also be reduced to the entanglement entropy for pure states.
For a quantum state $\rho^{AB}$ defined over the bipartite Hilbert space;
squashed entanglement is defined as below,

$E_{sq}(\rho^{AB})=\textsf{inf}\left\\{\frac{1}{2}I(A;B|E):\rho^{ABE}\looparrowright\rho^{AB}\right\\}$
(5)

Here $\looparrowright$ represents the extension of $\rho^{AB}$. Further the
infimum will be taken over the set equipped with $\rho^{ABE}$. With the
definition of squashed entanglement the following condition is satisfied,

$\rho^{AB}=Tr_{E}(\rho^{ABE})$ (6)

The expression $I(A;B|E)$ in the definition of squashed entanglement reads,

$I(A;B|E)=S(AE)+S(BE)-S(ABE)-S(E)$ (7)

Where $S(.)$ represents the Von Neumann entropy in the above expression. Here
we mention that the squashed entanglement is invariant under the tripartite
exchange permutation symmetry[16].

Concurrence[12] is an another quantum correlation measure which is widely used
to determine the entanglement of bipartite quantum systems. Concurrence of a
bipartite quantum system of a density matrix $\rho$ can be defined as,

$C(\rho)=max\,\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}$ (8)

Where $\lambda_{i}$’s are the decreasing order square root of eigenvalues of
$\rho\tilde{\rho}$.

$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$
(9)

Where $\tilde{\rho}$ is the result of the spin-flip operation applied on
density matrix $\rho$. Here $\sigma_{y}$ is the Pauli Y matrix and $\rho^{*}$
is the complex conjugate of the density matrix $\rho$.

##  III Hamiltonian and Unitary time evolution

In this section, we present Heisenberg[19] and bi-linear bi-quadratic
Hamiltonian[20] used for the study. Further we also present the unitary time
evolution by using the Schrodinger time dependent equation.

In this paper we consider Hamiltonians with XXX model in one dimension without
magnetic field and following no boundary conditions. Under the above
assumption the Heisenberg Hamiltonian reads,

$H_{1}=-j\sum\limits_{i=1}^{N}\sigma_{i}^{z}\sigma_{i+1}^{z}$ (10)

Next we consider the non linear extension of the above Hamiltonian named as
bi-linear bi-quadratic Hamiltonian. This Hamiltonian is give below,

$H_{2}=j\sum\limits_{i=1}^{N}\left[\left(\sigma_{i}^{z}\sigma_{i+1}^{z}\right)-\left(\sigma_{i}^{z}\sigma_{i+1}^{z}\right)^{2}\right]$
(11)

Where $j$ is the coupling constant, $\sigma^{z}$ denotes the Pauli Z matrix
and $\left(\sigma_{i}^{z}\sigma_{i+1}^{z}\right)^{2}$ is the non linear term.
These Hamiltonians are used to investigate the unitary dynamics in bipartite
system.

As per the postulate of quantum mechanics, the unitary time evolution of the
physical system is governed by the time dependent Schrodinger equation given
as below,

$i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=E|\psi(t)\rangle$ (12)

Where $E$ is the real energies of the physical system. The solution of this
equation is obtained as,

$|\psi(t)\rangle=e^{\frac{-iHt}{\hbar}}|\psi(0)\rangle$ (13)

The density matrix version of the above equation can be framed as,

$\rho(t)=U(t).\rho(0).U(t)^{\dagger}$ (14)

Where $U(t)=e^{\frac{-iHt}{\hbar}}$ is the unitary matrix which include the
Hamiltonian $H$ in exponential. Here we assume $\hbar=1$ to maintain the
simplicity for the present study. This Eq.14 is used in current work to
develop the dynamics of quantum correlations in Werner state and MEMS.

##  IV Initial state preparation of Werner state and MEMS

In this section we prepare the initial states according to the requirement of
quantum correlations expressed in Eqs.$\ref{se},\ref{con}$.

As per the definition of squashed entanglement we add an additional qubit
$(E)$ to the bipartite system; this additional qubit produces the extended
density matrix $\rho^{ABE}$. We mention that larger the domain of extension
than better the accuracy in squashed entanglement measure. But larger domain
of the system makes the calculations of squashed entanglement very difficult,
infect it may be considered as NP hard problem[21]111In computer science the
problems whose domain is very large becomes very difficult to solve; such
problems are categorized as NP hard.. In our work we consider the simplest
case by considering the auxiliary qubit as external domain for squashed
entanglement. To begin with, we express the state vector of additional qubit
given as below,

$|E\rangle=\alpha|0\rangle+\beta|1\rangle$ (15)

with

$|\alpha|^{2}+|\beta|^{2}=1$ (16)

The density matrix of the additional qubit $|E\rangle$ is expressed as,

$\rho_{E}=\left[\begin{array}[]{cc}|\alpha|^{2}&\alpha\beta\\\
\alpha\beta&|\beta|^{2}\end{array}\right]$ (17)

By using the Eq.16, we can rewrite the above density matrix as below,

$\rho_{E}=\left[\begin{array}[]{cc}|\alpha|^{2}&\alpha\sqrt{1-|\alpha|^{2}}\\\
\alpha\sqrt{1-|\alpha|^{2}}&1-|\alpha|^{2}\end{array}\right]$ (18)

Here we recall that the additional qubit expressed in Eq.15 works as an
external domain for $\rho^{AB}$ which is the density metrics of bipartite
quantum state. Initially the state $\rho^{AB}$ is prepared in Werner State and
MEMS respectively, given in equations $\ref{w1},\ref{m1}$. After Heisenberg
interaction between qubits $B$ and $E$, the quantum correlations may change in
$\rho^{AB}$ with the advancement of time. The density matrices of the initial
tripartite system prepared in Werner state and MEMS can be expressed
respectively as below,

$\rho_{W}(0)=\rho_{WS}\otimes\rho_{E}$ (19)

and

$\rho_{M}(0)=\rho_{MEMS}\otimes\rho_{E}$ (20)

The above equations will be used for the study in the next sections to explore
the dynamics of quantum correlations expressed in Eqs.$\ref{se},\ref{con}$ for
Werner state and MEMS. The dynamical study is carried out under two different
Hamiltonians given in Eqs. $\ref{h1},\ref{h2}$ in two subsequent sections. In
the very next section we explore the mathematical expressions of the dynamics
of quantum correlations for quantum states and discuss our results under
Heisenberg Hamiltonian. In further section, we investigate the same study
under bi-linear bi-quadratic Hamiltonian.

##  V Quantum correlations under Heisenberg Hamilton: The mathematical
expressions

In this section, we investigate the quantum dynamics of quantum correlations
in tripartite system presented in Eqs.19,20. Further we use the time evolution
given in Eq.14 under Heisenberg Hamiltonian. Here we present the mathematical
expressions obtained for Squashed entanglement and concurrence in Werner
states and MEMS in two successive subsections.

###  V.0.1 Squashed entanglement and concurrence for Werner state

We obtained the time evolution of squashed entanglement for Werner state under
Heisenberg Hamiltonian as given below,

$\displaystyle SqE_{W}(H_{1})$ $\displaystyle=$ $\displaystyle
0.5+2a^{-1}b\left[(a-G)log(c)+(a+G)log(c^{{}^{\prime}})-(a-H)log(d)-(a+H)log(d^{{}^{\prime}})\right]$
(21)
$\displaystyle+3b\left[(f)log(0.25f)+(b^{{}^{\prime}}+\gamma)log(v)\right]$

Where,

$\displaystyle G=\sqrt{\alpha^{2}gg^{{}^{\prime}}+a^{2}(1-2\alpha^{2}g)},\quad
H=\sqrt{\alpha^{2}f^{{}^{\prime}}gg^{{}^{\prime}}+a^{2}(1-2\alpha^{2}f^{{}^{\prime}}g)},\quad
c=(0.5-0.5a^{-1}G),$ $\displaystyle c^{{}^{\prime}}=(0.5+0.5a^{-1}G),\quad
d=(0.25-0.25a^{-1}H),\quad d^{{}^{\prime}}=(0.25+0.25a^{-1}H)$

and,

$\displaystyle a=e^{2ijt},\quad b=0.180337,\quad
b^{{}^{\prime}}=0.333333,\quad f=(1-\gamma),\quad
f^{{}^{\prime}}=(1-\gamma^{2}),\quad
f^{{}^{\prime\prime}}=(1+\gamma^{*}),\quad
f^{{}^{\prime\prime\prime}}=(1+\gamma),$ $\displaystyle
f^{{}^{\prime\prime\prime\prime}}=(1-\gamma^{*}),\quad g=(1-\alpha^{2}),\quad
g^{{}^{\prime}}=(1+a^{4}),\quad h=(1-\delta),\quad
h^{{}^{\prime}}=(1-2\delta),\quad v=(0.25+0.75\gamma),$ $\displaystyle
m=(\gamma-2\delta),\quad m^{{}^{\prime}}=(\gamma+2\delta),\quad
n=(-0.5\gamma+\delta),\quad n^{{}^{\prime}}=(0.5\gamma+\delta).$ (22)

The symbols expressed in Eq.22 have been used throughout the paper. Further
for concurrence we need to obtain the spectrum of eigenvalues with unitary
time evolution under Heisenberg Hamiltonian. This spectrum for Werner state is
obtained below,

$\displaystyle\lambda_{W}(H_{1})=\left\\{\frac{1}{4}Xf,\frac{1}{4}Xf,\frac{1}{16}a^{-1}(p-p^{{}^{\prime}}),\frac{1}{16}a^{-1}(p+p^{{}^{\prime}})\right\\}$
(23)

Where,

$\displaystyle X=\frac{1}{4}(\alpha^{2}f+gf)^{*},\quad
Y=-\frac{1}{2}(a\alpha^{2}\gamma+a^{-1}g\gamma)^{*},\quad
p=-4\gamma(Z+Y\alpha^{2}-Z\alpha^{2})-4a^{2}\gamma(Y-Y\alpha^{2}+Z\alpha^{2})+af^{{}^{\prime\prime}}f^{{}^{\prime\prime\prime}},$
$\displaystyle Z=-\frac{1}{2}(a^{-1}\alpha^{2}\gamma+ag\gamma)^{*},\quad
p^{{}^{\prime}}=\sqrt{p^{2}-16\left[4\alpha^{2}\gamma^{2}gg^{{}^{\prime}}-a^{2}(1+2\gamma-(3-8\alpha^{2}g)\gamma^{2})\right]\left[YZ-\frac{1}{16}(f^{{}^{\prime\prime}})^{2}\right]}$

and rest of symbols are considered from Eq.22.

###  V.0.2 Squashed entanglement and concurrence in MEMS

In this section, we obtain the squashed entanglement for MEMS under unitary
time evolution with Heisenberg Hamiltonian. The mathematical expression is
obtained as below,

$\displaystyle SqE_{M}(H_{1})$ $\displaystyle=$ $\displaystyle
2a^{-1}b\left[(a-Q)log(k)+(a+Q)log(k^{{}^{\prime}})\right]+8b(0.5-\delta)log(h^{{}^{\prime}})-(4bh)log(h)-(4b\delta)log(\delta)+$
(24) $\displaystyle
a^{-1}\left[(4bP-2abh)log(l)-(4bP+2abh)log(l^{{}^{\prime}})-(4ab\delta)log(\delta)\right]-2b\left[(m)log(n)-(m^{{}^{\prime}})log(n^{{}^{\prime}})\right]$

Where,

$\displaystyle
P=\sqrt{a^{2}\left[0.25-(0.5+2\alpha^{2}g)\delta+(0.25+4\alpha^{2}g)\delta^{2}\right]+\alpha^{2}\delta
gg^{{}^{\prime}}h^{{}^{\prime}}},\quad Q=\sqrt{4\alpha^{2}h\delta
gg^{{}^{\prime}}+a^{2}(1-8\alpha^{2}gh\delta)},$ $\displaystyle
k=(0.5-0.5a^{-1}Q),\quad k^{{}^{\prime}}=(0.5+0.5a^{-1}Q),\quad
l=(0.5-a^{-1}P-0.5\delta),\quad l^{{}^{\prime}}=(0.5+a^{-1}P-0.5\delta)$

and other symbols taken from Eq.22. Further to obtain the concurrence we need
to calculate the eigenvalue spectrum; which is given below,

$\displaystyle\lambda_{M}(H_{1})=\left\\{0,0,0.25a^{-1}(q-2q^{{}^{\prime}}),0.25a^{-1}(q+2q^{{}^{\prime}})\right\\}$
(25)

Where,

$\displaystyle R=0.5\gamma(a\alpha^{2}\gamma+a^{-1}g\gamma)^{*},\quad
S=0.5\gamma(a^{-1}\alpha^{2}\gamma+ag\gamma)^{*},\quad
q=S+R\alpha^{2}-S\alpha^{2}+a^{2}(R-R\alpha^{2}+S\alpha^{2})+4a\delta\delta^{*}$
$\displaystyle
w=\alpha^{2}\gamma^{2}gg^{{}^{\prime}}+a^{2}[(1-2\alpha^{2}g)\gamma^{2}-4\delta^{2}],\quad
q^{{}^{\prime}}=\sqrt{-w(0.5a\alpha^{2}\gamma+0.5a^{-1}g\gamma)^{*}(0.5a^{-1}\alpha^{2}\gamma+0.5ag\gamma)^{*}+w(\delta^{*})^{2}+0.25q^{2}}$

and remaining symbols expressed in Eq.22.

##  VI Dynamics of Quantum correlations under Heisenberg Hamiltonian

In this section, we present the dynamical results of squashed entanglement and
concurrence for Werner state and MEMS under Heisenberg Hamiltonian. The
dynamics of Squashed entanglement for Werner State and MEMS is governed by the
Eqs.$\ref{sew1},\ref{sem1}$ respectively and for Concurrences it is governed
by the set of eigenvalues presented in Eqs.$\ref{l1},\ref{l2}$. We note that
the squashed entanglement and concurrence are the functions of the parameters
$\alpha,\gamma,j$ and $t$. Further we have shown the comparative study of
squashed entanglement and concurrence in graphical results with varying values
of the parameters $(\alpha,\gamma,j,t)$. We divide this study in different
cases to understand the dynamical behavior of quantum correlations with
respect to different parameters. During the discussion we consider the
coupling constant $j$ and time $t$ as a single variable $jt$. The study is
explored in five different cases, these cases are given in successive
subsections.

### Case 1: For the parameter values, $\alpha=0$ and $jt=0$

In this case, we present the study of quantum correlations in Werner state and
MEMS for initial condition i.e. at $\alpha=0,jt=0$. The values of the
parameters $(\alpha=0,jt=0)$ maps the tripartite system to the pure bipartite
initial state given as in Eqs.$\ref{w1},\ref{m1}$. The initial behavior of
quantum correlations with respect to $\gamma$ is shown in Fig.1. In this
figure $SE_{W}$ represents squashed entanglement of Werner state, $SE_{M}$
corresponds to squashed entanglement of MEMS, $C_{W}$ denotes the concurrence
of Werner state and $C_{M}$ represents the concurrence of MEMS which will be
followed by the next figures throughout this article.

Figure 1: Plot of quantum corrections (QC) vs. $\gamma$

To proceed the investigation, first we compare the squashed entanglement in
both the states; next we compare the concurrence in both the states and
finally squashed entanglement vs. concurrence comparison is done.

The squashed entanglement in both the states is plotted in the figure by solid
red and blue color. Looking at the graphs we have found that the squashed
entanglement in both of the states rise exponentially. The squashed
entanglement in Werner state achieve the maximum amplitude as $QC\simeq 1$;
while for MEMS it is $QC=0.4428$. It is investigated that, MEMS does not
exhibit the squashed entanglement after the value of the parameter
$\gamma=\frac{2}{3}$. The maximum squashed entanglement amplitude in MEMS is
always less than Werner state. At $\gamma=0$, there is no squashed
entanglement in Werner state; while for MEMS it has a certain value
$QC=0.126365$.

Now we compare the concurrence in both the states which are plotted in figure
by dashed purple and green color. Here it is noted that the concurrence grows
linearly. The figure shows that the Werner state does not have concurrence
between the limit $0\leq\gamma<\frac{1}{3}$; while on the other hand, MEMS has
the concurrence for all the range of the parameter $\gamma$. Both the states
achieve the maximum amplitude as $QC=1$.

Next, we discuss squashed entanglement and concurrence in both the states. For
Werner state, we have compared both the quantum correlations and found that
the squashed entanglement is rising exponentially for the range
$0\leq\gamma\leq 1$; while the concurrence is increasing in linear fashion
between the limit $\frac{1}{3}\leq\gamma\leq 1$. Whereas concurrence does not
exit for the limit $0\leq\gamma<\frac{1}{3}$. It is noted that at the initial
value of $\gamma$, the squashed entanglement dominates the concurrence. But
after the value of $\gamma=0.4630$ the trade off is reversed i.e. the
concurrence dominates the squashed entanglement and achieve the higher
amplitude for $0.4630<\gamma\leq 1$. At the value of $\gamma=0.4630$ both the
quantum correlations have the same amplitude and we call this point as
Squashed entanglement-Concurrence Equilibrium (SCE) point of Werner state.

For MEMS, we compare both the squashed entanglement and concurrence with
interesting trade-off. It is found that the concurrence exists for whole range
of $\gamma$; while the squashed entanglement exists in the the range
$0\leq\gamma\leq\frac{2}{3}$. Further for the range of $\frac{2}{3}<\gamma\leq
1$ squashed entanglement does not exist. It is also noticed that the SCE point
of MEMS exist at $\gamma=0.139$. We also observe that for MEMS with
$0\leq\gamma<0.139$; the squashed entanglement dominates the concurrence and
beyond $\gamma=0.139$, the concurrence dominates the squashed entanglement.

### Case 2: For a fixed value of $\alpha$ and different values of $jt$

In this case we fix the value of $\alpha$ and plot the graphical results with
varying values of $\gamma$ and $jt$. Few Important results are plotted in
Fig.2 with $\alpha=0.600001$.

The graphical plots show that as the value of $jt$ increases the maximum value
of all quantum correlations decreases and the rate of amplitude decrement of
concurrence is higher in comparison to the amplitude decrement of squashed
entanglement. It is noted that, the sustainability of squashed entanglement in
both the state w.r.t the axis of $\gamma$ does not change; it remains same as
described in case 1. On the other hand, the sustainability of the concurrence
along the axis of $\gamma$ in Werner state changes as the value of the
parameter $jt$ increases but for MEMS it does not change. Further we have
noticed that for $\gamma=0$, the squashed entanglement of MEMS have certain
value, but as time move forward this value decreases and the difference
between the squashed entanglement of both the states also decreases.

Figure 2: Plot of quantum corrections (QC) vs. $\gamma$ with different values
of $jt$.

### Case 3: For the fixed value of $jt$ and different values of $\alpha$

In this case, we describe the quantum correlations for constant value of $jt$
with respect to varying value of $\alpha$ and $\gamma$. We have shown few of
the plots for $jt=0.600001$ and different values of $\alpha$ in Fig.3.

Figure 3: Plot of quantum corrections (QC) vs. $\gamma$ with different values
of $\alpha$

Here we notice that as the value of $jt$ is fixed, amplitude of the
concurrence of quantum states become freeze at $QC=0.363468$. On the other
contrary, the increment of the value of $\alpha$ decreases the amplitude of
squashed entanglement. It is also noted that the amplitude of squashed
entanglement and their gap in both of the states become freeze at $\alpha\geq
0.600001$.

### Case 4: For the fixed value of $\alpha$ and different values of $\gamma$

In this section, we explore the study by fixing the value of the parameter
$\alpha=0.600001$ with the varying values of $\gamma$ and $jt$. First we
discuss the comparison of squashed entanglement in both the states, next the
comparison of concurrence in both of the states is explored. Finally the
squashed entanglement and concurrence is compared for each of the states. Here
we present the graphical results in Figure 4.

Figure 4: Plot of quantum corrections (QC) vs. $jt$ with different values of
$\gamma$.

The squashed entanglement is higher in MEMS than Werner state and amplified in
both the states as the value of the parameter $\gamma$ increases. As the value
of parameter $jt$ advances the amplitude of squashed entanglement decreases in
both the states.

Comparing the concurrence in both the states, we have found that the
concurrence is also higher in MEMS than werner states. On the other hand, we
have found the important result of Entanglement sudden death (ESD)[26, 27, 28,
29, 30, 31, 32, 33] for the concurrence in Werner state with the increasing
value of $\gamma$. But during the ESD, the concurrence exists in MEMS and this
state shows more robust character compared to Werner state. We have found as
the value of $\gamma$ increases the ESD zone squeezes.

Finally we compare the squashed entanglement and concurrence in both the
states. We have found at $jt=0$, the amplitude of saquashed entanglement is
always maximum. As the parameter $jt$ advances, both the quantum correlations
have decreasing tendency. It is also noted that at $jt=0.787579$, the
concurrence of MEMS becomes zero but at this point squashed entanglement
exists. For Werner state with $\gamma=0.200001$, the concurrence does not
exists, but advancing value of $\gamma$ produces the concurrence in the Werner
state with ESD effects. It is important to note that in the absence of
concurrence the squashed entanglement always exist in Werner state.

### Case 5: For the fixed value of $\gamma$ and different values of $\alpha$

In this case, we present the study of the dynamics of quantum correlations by
fixing the value of the parameter $\gamma$ as $\gamma=0.600001$ with the
varying values of $\alpha$ and $jt$. The graphical results are shown in Fig.5.

Figure 5: Plot of quantum corrections (QC) vs. $jt$ with different values of
$\alpha$

Here we have found that the maximum amplitude of all the quantum correlations
at $jt=0$ freeze for all values of $\alpha$. We also noticed that the ESD zone
of Werner state starts at $jt=0.623712$ and zone width freeze with the
advancement of time.

##  VII Quantum correlations under bi-linear bi-quadratic Hamiltonian: The
mathematical expressions

In this section, we explore the dynamics of quantum correlations in both the
states under the bi-linear bi-quadratic Hamiltonian presented in Eq.11. This
Hamiltonian incorporates a non-linear term as
$(\sigma_{i}^{Z}\sigma_{i+1}^{Z})^{2}$ along with the Heisenberg Hamilton.
This term is responsible to cancel out the parameter $jt$ while calculating
the time evolution of the system governed by Eq.14. Hence bi-linear bi-
quadratic Hamiltonian preserve the quantum correlations in the system. Here in
the following subsections, we present the mathematical functions calculated
for quantum correlations in both the states.

###  VII.0.1 Squashed entanglement and concurrence for Werner state

The expressions of squashed entanglement and eigenvalue spectrum for
concurrence under bi-linear bi-quadratic Hamiltonian with the Eq.14, is
obtained as,

$\displaystyle SqE_{W}(H_{2})=1+(3bf)log(0.25f)+b(1+3\gamma)log(v)$ (26)

and,

$\displaystyle\lambda_{W}(H_{2})=\left\\{\frac{1}{16}ff^{{}^{\prime\prime\prime\prime}},\frac{1}{16}ff^{{}^{\prime\prime\prime\prime}},\frac{1}{16}ff^{{}^{\prime\prime\prime\prime}},\frac{1}{16}(1+3\gamma)(1+3\gamma^{*})\right\\}$
(27)

The required symbols considered in Eq.22.

We have found, the above equations are independent from the parameters
$\alpha$ and $jt$. These equations maps to the Eqs.21 and 23 with the
condition $\alpha=0,jt=0$. Hence the dynamics of quantum correlations in
Werner state under bi-linear bi-quadratic Hamiltonian remain same as observed
in case 1 with the initial condition.

###  VII.0.2 Squashed entanglement and concurrence in MEMS

Here we obtain the expressions of squashed entanglement and eigenvalue
spectrum for concurrence under bi-linear bi-quadratic Hamiltonian with Eq.14.
These expressions are give below,

$\displaystyle
SqE_{M}(H_{2})=0.5\left[16b(0.5-\delta)log(h^{{}^{\prime}})-(16bh)log(h)-(16b\delta)log(\delta)-4b\left\\{(m)log(n)-(m^{{}^{\prime}})log(n^{{}^{\prime}})\right\\}\right]$
(28)

and,

$\displaystyle\lambda_{M}(H_{2})=\left\\{0,0,(0.25\gamma\gamma^{*}+\delta\delta^{*}-0.5T),0.5(T+0.5\gamma\gamma^{*}+2\delta\delta^{*})\right\\}$
(29)

Where,

$\displaystyle T=(\delta\gamma^{*}+\gamma\delta^{*})$

and other symbols expressed in Eq.22

Again it is found that the above equations are independent from the parameter
$\alpha$ and $jt$; it is happening because of the non-linear term present in
the bi-linear bi-quadratic Hamiltonian. The above equations maps to the Eqs.24
and 25 with the initial condition $\alpha=0,jt=0$. Hence the dynamics of
quantum correlations in MEMS remains same as explained in case 1 under the
section six with the initial condition.

##  VIII Conclusion

In this article we have studied the unitary dynamics of quantum correlations
for Werner state and MEMS under Heisenberg and bi-linear bi-quadratic
Hamiltonian. For initial condition with $\alpha=0,jt=0$, we have found a trade
off between squashed entanglement and concurrence in both of the states. With
the range $0\leq\gamma<\frac{1}{3}$, the concurrence vanish in Werner state,
while squashed entanglement exist. On the other hand, with the range
$\frac{2}{3}<\gamma\leq 1$ the squashed entanglement vanish in MEMS but
concurrence exist for the same range.

Dealing with Heisenberg Hamiltonian, we have explored this study with varying
the parameters $\alpha,\gamma$ and $jt$ in different cases and discuss the
impact of every parameters on quantum correlations. We have noticed that the
concurrence increases linearly in both of the states while squashed
entanglement increases in exponential fashion. In both the states, when either
one of the parameters $\alpha$ or $jt$ have been increased by keeping another
parameter fix; the amplitude of the quantum correlations are decreased.
Further the parameter $\gamma$ is responsible to amplify the quantum
correlations. We have also investigated the phenomenon of entanglement sudden
death (ESD) in Werner state with concurrence measure by varying the value of
$\gamma$. But as the value of parameter $\gamma$ is fixed the width of ESD
zone is also fixed but in the absence of concurrence the squashed entanglement
always exit. More interestingly, it has been observed that the Werner state is
more fragile than MEMS in terms of concurrence measure with the parameter
$\gamma$.

The most important result we have shown that, the bi-linear bi-quadratic
Hamiltonian does not contribute in the time evolution for both the quantum
states and hence it does not disturb the quantum correlations. The non-linear
term in this Hamiltonian plays an important role to preserve the quantum
correlations and both the states are quite robust under this Hamiltonian.

## References

  * [1] Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
  * [2] J. Preskil; Quantum Computing in the NISQ era and beyond; Quantum 2 79, 2018.
  * [3] Dyakonov M.I., More Problems with Quantum Computing, DOI https://doi.org/10.1007/978-3-030-42019-2_5 (2020).
  * [4] The rise of quantum materials. Nature Phys 12, 105 (2016).
  * [5] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters; Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels; Phys. Rev. Lett. 70, 1895 (1993).
  * [6] D. Boschi, S. Branca, F. De Martini, L. Hardy, S. Popescu; Experimental Realization of Teleporting an Unknown Pure Quantum State via Dual Classical and Einstein-Podolsky-Rosen Channels; Phys. Rev. Lett. 80, 1121 (1998).
  * [7] D. Bouwmeester, J. Pan, K. Mattle, M. Eibl, H. Weinfurter, A. Zeilinger; Experimental quantum teleportation; Nature 390, 575–579 (1997).
  * [8] C. L. Degen, F. Reinhard, P. Cappellaro; Quantum sensing; Rev. Mod. Phys. 89, 035002 (2017).
  * [9] A. K. Ekert; Quantum cryptography based on Bell’s theorem; Phys. Rev. Lett. 67, 661 (1991).
  * [10] C. H. Bennett, F. Bessette, G. Brassard, L. Salvail, J. Smolin; Experimental quantum cryptography; J. Cryptology 5, 3–28 (1992).
  * [11] E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001, (2019).
  * [12] W. K. Wootters; Entanglement of Formation of an Arbitrary State of Two Qubits; Phys. Rev. Lett. 80, 2245 (1998).
  * [13] H. Ollivier,W. H. Zurek; Quantum Discord: A Measure of the Quantumness of Correlations; Phys. Rev. Lett. 88, 017901 (2001).
  * [14] B. Dakić, V. Vedral, C. Brukner; Necessary and Sufficient Condition for Nonzero Quantum Discord; Phys. Rev. Lett. 105, 190502 (2010).
  * [15] P. Giorda1, M. G. A. Paris; Gaussian quantum discord; arXiv:1003.3207v2 (2010).
  * [16] F. Brandao, M. Christandl and J. Yard, Faithful Squashed Entanglement, Communications in Mathematical Physics. 306 3, 805 (2011).
  * [17] R. F. Werner; Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model; Phys. Rev. A 40, 4277 (1989).
  * [18] W. J. Munro, D. F. V. James, A. G. White, P. G. Kwiat; Maximizing the entanglement of two mixed qubits; Phys. Rev. A 64, 030302(R) (2001).
  * [19] X. Wang, H. Li, Z. Sun, Y. Li; Entanglement in spin-1 Heisenberg chains; J. Phys. A: Math. Gen. 38, 8703 (2005).
  * [20] https://core.ac.uk/download/pdf/31157826.pdf
  * [21] Knuth, Donald, Postscript about NP-hard problems, ACM SIGACT News. 6s 15 (1974).
  * [22] O. Salberger and V. Korepin, Entangled spin chain, Reviews in Mathematical Physics 29, 10, 1750031 (2017).
  * [23] S. S Gong and G. Su, Thermal entanglement in one-dimensional Heisenberg quantum spin chains under magnetic fields, Phys. Rev. A 80, 012323 (2009).
  * [24] M. Christandl, A. Winter, Squashed entanglement: An additive entanglement measure, J. Math. Phys. 45, 829 (2004).
  * [25] N. A. Peters, J. B. Altepeter, D. Branning, E. R. Jeffrey, T. Wei, and P. G. Kwiat; Maximally Entangled Mixed States: Creation and Concentration; Phys. Rev. Lett. 92, 133601 (2004).
  * [26] Sharma, K.K., Awasthi, S.K., Pandey, S.N.: Entanglement sudden death and birth in qubit-qutrit systems under Dzyaloshinskii-Moriya interaction. Quantum Inf. Process. 12, 3437 (2013)
  * [27] Sharma, K.K., Pandey, S.N.: Entanglement Dynamics in two parameter qubit-qutrit states under Dzyaloshinskii-Moriya interaction. Quantum Inf. Process. 13, 2017 (2014)
  * [28] Sharma, K.K., Pandey, S.N.: Influence of Dzyaloshinshkii-Moriya interaction on quantum correlations in two qubit Werner states and MEMS. Quantum. Info. Process. 14, 1361 (2015)
  * [29] Sharma, K.K., Pandey, S.N.: Dzyaloshinshkii-Moriya interaction as an agent to free the bound entangled states. Quantum. Info. Process. 15, 1539 (2016)
  * [30] Sharma, K.K., Pandey, S.N.: Dynamics of entanglement in two parameter qubit-qutrit states with x-component of DM interaction. Commun. Theor. Phys. 65, 278 (2016)
  * [31] Sharma, K.K.,Herring-Flicker coupling and thermal quantum correlations in bipartite system, Quantum. Info. Process.,17, 321 (2018).
  * [32] Sharma, K.K., Pandey SN, Robustness of Greenberger- Horne-Zeilinger and W states against Dzyaloshinskii-Moriya interaction, Quantum. Info. Process., 15, 12, 4995 (2016).
  * [33] Sharma, K.K., Gerdt Vladimir P., Entanglement sudden death and birth effects in two qubits maximally entangled mixed states under quantum channels, Int. J. Theor. Phys, 59, 403 (2020.)


#  Metal-insulator transition in CaCu3Fe4O12

Atsushi Hariki  Institute of Solid State Physics, TU Wien, 1040 Vienna,
Austria  Mathias Winder  Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria  Jan Kuneš  Institute of Solid State Physics, TU Wien, 1040
Vienna, Austria  Institute of Physics, Czech Academy of Sciences, Na Slovance
2, 182 21 Praha 8, Czechia

###### Abstract

We study structurally-triggered metal-insulator transition in CaCu3Fe4O12 by
means of local density approximation (LDA) +$U$ and LDA+dynamical mean-field
theory (DMFT). The ferrimagnetic insulating phase is essentially the same
within both approaches. While LDA+$U$ describes the metal-insulator transition
as a Peierls-like instability driven by Fermi surface nesting in the
magnetically ordered phase, LDA+DMFT allows also the site-selective Mott
transition without magnetic ordering as well as smooth crossover between the
two pictures. We point out similarities and differences to rare-earth
nickelates.

##  I Introduction

Synergy of the charge, spin, orbital or lattice degrees of freedom is known to
give rise to a plethora of different thermodynamic phases and transitions
between them in transition-metal oxides (TMOs) Imada  _et al._ (1998);
Khomskii (2014) with perovskite structure. Competition between crystal field,
spin-orbit coupling and Hund’s exchange leads to spin-state crossovers Tanabe
and Sugano (1954); Werner and Millis (2007); Gavriliuk  _et al._ (2008) and
possibly to spin-state ordered phases Kuneš and Křápek (2011); Afonso  _et
al._ (2019) or excitonic magnetism Khaliullin (2013); Kuneš and Augustinský
(2014); Yamaguchi  _et al._ (2017). Different bonding between the TM $e_{g}$
and $t_{2g}$ orbitals with oxygen $p$ orbitals leads to coexistence of local
moments with itinerant electrons in some materials, e.g. doped manganites
Zener (1951); Tokura and Tomioka (1999); van den Brink  _et al._ (1999), while
others are Mott insulators described by pseudospin models of Kugel-Khomskii
type Kugel and Khomskiĭ (1982); Khomskii (2014). The perovskite structure
built out of corner-sharing oxygen octahedra allows various structural
deformations. Rotations/tilting of the oxygen octahedra Glazer (1972);
Woodward (1997); Howard and Stokes (1998) affect the electronic bandwidths.
Cooperative Jahn-Teller distortions Jahn  _et al._ (1937); Goodenough (1998);
Khomskii (2014); Bersuker (2014) strongly couple to orbital orders Kanamori
(1960); Mazin  _et al._ (2007); Tokura and Nagaosa (2000); Pavarini and Koch
(2010), while breathing distortions couple to charge order or site-selective
Mott state Park  _et al._ (2012); Greenberg  _et al._ (2018).

In high-valence TMOs, which are metallic or close to metal-insulator
transition (MIT), the charge degree of freedom adds to the complexity.
Concomitant MIT and structural transition followed by unusual magnetic
ordering attract much attention to rare-earth nickelates RNiO3 Mazin  _et al._
(2007); Lee  _et al._ (2011); Park  _et al._ (2012); Lau and Millis (2013);
Johnston  _et al._ (2014); Jaramillo  _et al._ (2014); Subedi  _et al._
(2015); Bisogni  _et al._ (2016); Mercy  _et al._ (2017); Varignon  _et al._
(2017). Starting from the ionic picture, Mazin et al. Mazin  _et al._ (2007)
proposed the charge disproportionation (CD) $2d^{7}\rightarrow d^{6}+d^{8}$ to
be the origin of a breathing distortion of NiO6 octahedra and MIT. Park et al.
Park  _et al._ (2012) used LDA+DMFT and introduced the concept of the site-
selective Mott transition. Recently, the CD was shown to emerge as an
effective low-energy description of the site-selective Mott transition from
LDA+DMFT calculations Seth  _et al._ (2017). Lee et al. Lee  _et al._ (2011)
took the band picture to argue that a Peierls-like distortion originating in
Fermi surface nesting explains the physics of nickelates. This view is
supported by LDA+U calculations of Mercy et al. Mercy  _et al._ (2017) in the
magenetically ordered state who pointed out the importance of octahedral
rotations for tuning the Peierls-like instability. In this picture, the metal-
insulator transition is connected with the magnetic ordering, which is at odds
with the paramagnet-to-paramagnet MIT transition in a number of nickelates.
The question of Peierls vs site-selective Mott mechanism Ruppen  _et al._
(2015) remains a subject of active debate. Recently, the site-selective Mott
transition was reported also in Fe2O3 Greenberg  _et al._ (2018) under
pressure.

In this Article, we study the iron oxide CaCu3Fe4O12 (CCFO), which belongs to
A-site ordered TM perovskites AA${}_{3}^{\prime}$B4O12 Shimakawa (2008); Cheng
_et al._ (2013); Long  _et al._ (2009); Meyers  _et al._ (2013); Shimakawa
(2015); Yamada  _et al._ (2014). Iron in CCFO is proposed to have anomalously
high Fe4+ valence Yamada  _et al._ (2008). In addition to common parameters in
TMOs, such as intra-atomic Coulomb interaction, band filling or structural
distortions, the interaction between TM ions on A′ and B sites adds another
control parameter in this group of materials.

The high-temperature crystal structure of CCFO belongs to $Im\overline{3}$
space group with a sizable rotation of FeO6 octahedra, which can be viewed as
a frozen $M_{2}^{+}$ mode (viewed as a rotation of the octahedra along the
[111] axis) with a minor $M_{1}^{+}$ contribution (elongation of the octahedra
along the [111] axis), see Figs. 1c,d and Supplementary Material (SM) sm . The
$M_{2}^{+}+M_{1}^{+}$ distortion (with the experimental amplitudes) is
referred to as the $Q_{\rm R}$ distortion hereafter. The $Q_{\rm R}$
distortion, which leads to a square-planar coordination of Cu with the four O
neighbors (Fig. 1a), is common in the AA${}_{3}^{\prime}$B4O12 series. At 210
K, a structural phase transition to $Pn\overline{3}$ structure takes place
simultaneously with MIT and magnetic transition. Although the Fe4+ ions may be
expected to adopt the Jahn-Teller-active high-spin $t_{2g}^{3}e^{1}_{g}$
configuration, a Jahn-Teller distortion has not been reported. Instead, a
breathing distortion $Q_{\rm B}$ of the FeO6 octahedra was observed in the
insulating phase (Fig. 1b). This gives rise to two distinct Fe sites with
short and long Fe–O bonds (Fig. 1a).

The concomitant electronic and structural deformation in CCFO is reminiscent
of MIT in RNiO3 (formally Ni3+: $t^{6}_{2g}e^{1}_{g}$) accompanied by the
breathing distortion of NiO6 octahedra. Parallel to the debate on RNiO3
Johnston  _et al._ (2014); Park  _et al._ (2012); Lau and Millis (2013); Mazin
_et al._ (2007); Mercy  _et al._ (2017); Jaramillo  _et al._ (2014); Bisogni
_et al._ (2016); Subedi  _et al._ (2015); Varignon  _et al._ (2017), it has
been suggested that a CD of the type: 2$(d^{5}\underline{L})\rightarrow
d^{5}\underline{L}^{2}+d^{5}$ ($\underline{L}$: oxygen hole) takes place in
the insulating phase of CCFO Yamada  _et al._ (2008); Hao  _et al._ (2009);
Ueda  _et al._ (2013); Shimakawa (2015). It is worth pointing out that in
nickelates the $t_{2g}$ electrons form a spin singlet ($t^{6}_{2g}$), while in
CCFO those tend to form a stable high-spin $t^{3}_{2g}$ state with $S=3/2$
moment, and thus rather different magnetic behavior is to be expected. Indeed,
along with at the MIT, a ferrimagnetic (FM) order with a large magnetization
of 9.7$\mu_{\rm B}$/f.u. is observed Yamada  _et al._ (2008); Mizumaki  _et
al._ (2011). The large magnetization indicates a ferromagnetic alignment of Fe
moments antiferromagnetically coupled to the Cu2+ ($S=1/2$) sublattice in
CCFO.

We use the local-density approximation (LDA) + $U$ and LDA + dynamical mean-
filed theory (DMFT) approaches to investigate the electronic response to the
$Q_{\rm R}$ and $Q_{\rm B}$ distortion. We characterize the local state of Fe
ions in the uniform and disproportionated phases and determine the preferred
magnetic order. We discuss the similarities and differences of CCFO to RNiO3
in the context of the Peierls and site-selective Mott scenarios.

Figure 1: (a) Low-temperature structure of CaCu3Fe4O12 (space group
$Pn\overline{3}$). (b) breathing distortion of oxygen atoms. (c) $M^{+}_{2}$
mode. (d) $M^{+}_{1}$ mode. The crystal structure is visualized using VESTA3
Momma and Izumi (2011). The mode analysis is performed using AMPLIMODE Perez-
Mato  _et al._ (2010). Cu (Ca) atoms sitting on 3/4 (1/4) of the A sites in
the ABO3 perovskite lattice show no displacements from the high-symmetry
positions in all temperatures. Thus, Co and Ca atoms are not shown in panels
(b)–(d).

##  II Computational Method

The LDA+$U$ calculations are performed using the augmented plane wave and
local orbital (APW+lo) method implemented in the Wien2k package Blaha  _et
al._ . Following previous GGA+$U$ studies of CCFO Hao  _et al._ (2009); Ueda
_et al._ (2013), the effective $U$ values ($U_{\rm eff}=U-J$) are chosen as
$U_{\rm eff}=$ 4.0 eV and 7.0 eV for Fe and Cu ions, respectively. We have
checked the robustness of the present result with respect to the choice of the
exchange-correlation potential and $U_{\rm eff}$ parameters, see SM sm . The
reciprocal space cut-off $K_{\rm max}$ for the basis functions was chosen such
that $R_{\rm MT}K_{\rm max}=7.0$. The Brillouin zone was sampled with an
$8\times 8\times 8$ mesh. To perform LDA+DMFT calculations Kotliar  _et al._
(2006); Kuneš  _et al._ (2009); Georges  _et al._ (1996), a tight-binding
model spanning Cu 3$d$, Fe 3$d$ and O 2$p$ bands, constructed using the
wien2wannier Kuneš  _et al._ (2010) and wannier90 Mostofi  _et al._ (2014)
codes, is augmented with the local Coulomb interaction on the Fe and Cu sites.
We use the interaction parameters (in eV) $(U,J)=(7.50,0.98)$ and
$(6.80,0.80)$ for the Cu and Fe 3$d$ shells, respectively Anisimov  _et al._
(1991); Kuneš  _et al._ (2009). To correct for the double counting of the
Coulomb interaction, we shift the site energies of the Fe and Cu orbitals by
the respective shell-averaged high-frequency limit (Hartree part) of the self-
energy Kuneš  _et al._ (2009). We show that our conclusions hold for a wide
range of the double-counting corrections (treated as adjustable parameters),
see SM sm . In the DMFT self-consistent calculation, the continuous-time
quantum Monte Carlo solver with density-density approximation to the on-site
interaction was used to solve the auxiliary Anderson impurity model Werner
_et al._ (2006); Boehnke  _et al._ (2011); Hafermann  _et al._ (2012); Hariki
_et al._ (2015).

Figure 2: Evolution of the LDA+$U$ total energy of the ferrimagnetic state
with the breathing distortion $Q_{\rm B}$ for fixed $Q_{\rm R}$ rotation mode.
Note that 100% corresponds to the experimental amplitude.

##  III Results

###  A LDA+$U$ and Peierls mechanism

Since the LDA+$U$ approach cannot describe fluctuating local moments, we
restrict ourselves to magnetically ordered phases (at $T=0$). Overall, the FM
solution has lower energy than antiferromagnetic or ferromagnetic solutions,
except for $Q_{\rm R}=0$ % where ferromagnetic alignment of all Fe and Cu
moments is favored. Fig. 2 shows the LDA+$U$ total energies for various
amplitudes of the breathing $Q_{\rm B}$ and rotation $Q_{\rm R}$ distortions.
For small $Q_{\rm R}$ the breathing distortion is unstable, while for larger
$Q_{\rm R}$ a new local minimum appears at a finite $Q_{\rm B}$ amplitude. We
find the global energy minimum close to the experimental $Q_{\rm R}$ and
$Q_{\rm B}$ amplitudes (i.e. $Q_{\rm R}=Q_{\rm B}=$ 100 %) irrespective of the
interaction parameters within a realistic range sm .

Figure 3: (a-f) Low-energy bands for the majority spin channel calculated by
LDA+$U$ for selected $Q_{\rm R}$ and $Q_{\rm B}$ amplitudes. The energy origin
is taken at the Fermi energy (red line). (g) LDA+DMFT spectra in the
ferrimagnetic solution (majority spin). The minority-spin bands can be found
in SM sm .  Figure 4: The $k$-integrated LDA+DMFT spectra in (a) ferrimagnetic
insulating, (b) paramagnetic insulating and (c) paramagnetic metallic
solution. The two insulating solutions are obtained with the experimental low-
temperature distorted structure (with $Q_{\rm B}$), while the metallic one is
obtained with the experimental high-temperature structure ($Im\overline{3}$).

In Figs. 3a-f we show the band structure in the vicinity of $E_{F}$ at
selected $Q_{\rm R}$ and $Q_{\rm B}$. The depicted bands have dominantly
majority-spin Fe 3$d$ $e_{g}$ character. The breathing distortion $Q_{\rm
B}$=100 % opens a gap for all $Q_{\rm R}$. However, only for $Q_{\rm
R}\gtrsim$ 75 % the Fermi level falls into the gap, allowing the system to
gain electronic energy and thus stabilize the breathing distortion. The
$Q_{\rm R}$ distortion reduces the $e_{\rm g}$ band width, which in turn
pushes the band crossing ($Q_{\rm B}$=0) at $R$-point to $E_{F}$, see Fig. 3.
While at $Q_{\rm B}$=100 % the band crossing locates at the optimal position
on the Fermi level, Fig. 3d, the insulating band structure is obtained already
for $Q_{\rm B}$=75 %. This picture is the Peierls-like instability for opening
the gap, described in Refs. Lee  _et al._ , 2011; Mercy  _et al._ , 2017, in
rare-earth nickelates.

The problem with this picture is that saturated spin polarization of the Fe
$e_{g}$ bands is necessary to open a gap at $E_{F}$ Mercy  _et al._ (2017),
while experimentally the normal phase is paramagnetic. This criticism in CCFO
is weaker than in case of the paramagnetic-metal to paramagnetic-insulator
transition in RNiO3 since short range magnetic order is expected above $T_{c}$
in CCFO, where MIT and magnetic ordering take place simultaneously.

Figure 5: LDA+DMFT valence spectra projected onto Fe $e_{g\sigma}$-states on
(a,c) long-bond (LB) and (b,d) short-bond (SB) site. The panels (a,b) and
(c,d) are for the majority-spin and minority-spin channel in the ferrimagnetic
solution, respectively. Total as well as Cu $d$-projected spectra can be found
in SM. The imaginary part of the self-energies in the real-frequency domain
for (e) LB and (f) SB site. The real part of the self-energies is shown in the
inset. Figure 6: LDA+DMFT valence spectra projected onto Fe
$e_{g\sigma}$-states on (a) long-bond (LB) and (b) short-bond (SB) site in the
paramagnetic insulating solution obtained with the distorted structure (i.e.
$Q_{\rm B}=Q_{\rm R}=100$ %) (c) Fe $e_{g\sigma}$-states in the paramagnetic
metallic solution obtained with the experimental structure above the MI
transition ($Im\overline{3}$). The imaginary part of the self-energies in the
real-frequency domain is shown in panels (d-f). The inset in panel (f) shows
the self-energy in the Matsubara-frequency domain.

###  B LDA+DMFT and site-selective Mott transition

The LDA+DMFT approach can capture the magnetically ordered phase as well as
the paramagnetic state with fluctuating local moments. Our implementation of
LDA+DMFT does not allow a reliable calculation of the free energy. Therefore
we do not search for the most stable state as with LDA+U, but only investigate
the electronic structure above and below the experimental structural
transition. In the following we analyze the ferrimagnetic and paramagnetic
solutions. We find the ferrimagnetic solution to be stable for both distorted
and undistorted structure up to 1200 K, which grossly overestimates the
experimental $T_{c}$. We attribute it largely to the density-density
approximation, in which the large Fe moments are treated as Ising rather than
Heisenberg moments. The PM solutions are obtained by constraint.

Figs. 4ab show the $k$-integrated spectra of the ferrimagnetic and
paramagnetic LDA+DMFT solutions for the experimental crystal structure. In
agreement with experiment, we find a metallic solution in the undistorted
structure (the FM solution not shown here is also metallic). The states around
$E_{F}$ have dominatly Fe $e_{g}$ character. Unlike in RNiO3, we find a rather
strong damping of the $e_{g}$ states due to interaction with fluctuating
$t_{2g}$ moments (absent in RNiO3). This explains smaller conductivity of
metallic CCFO.

The LDA+DMFT solution in the distorted structure is insulating irrespective of
the presence of the magnetic order. The FM state at the simulation temperature
is fully saturated. The absence of spin fluctuations leads to a band structure
that closely resembles the LDA+U one, see Figs. 3fg. This is not surprising
since the self-energy on both the SB and LB site, shown in Figs. 5ef, is
dominated by the frequency-independent Hartree term with vanishing imaginary
part around $E_{F}$. The gap then has a Peierls character between the LB-
dominated valence and SB-dominated conduction band, Figs. 5ab, and the
minority bands removed from the vicinity of $E_{F}$, Figs. 5cd.

The PM spectra in the distorted structure offer a rather different picture,
Fig. 6abde. The bands are strongly damped and renormalized due to fluctuating
$t_{2g}$ moments. However, the main difference is the nature of the gap that
is opened due to a pole in the LB self-energy, while the SB self-energy has a
Fermi liquid character. The MIT in the PM phase thus has the site-selective
Mott character Park  _et al._ (2012); Ruppen  _et al._ (2015).

Figure 7: Local spin/charge susceptibility in (a)/(c) long-bond and (b)/(d)
short-bond Fe site calculated by LDA+DMFT. The orbital-diagonal contributions
in the local spin susceptibility of (e) long-bond and (f) short-bond Fe site.
Note that the rotation of the FeO6 octahedra splits $t_{2g}$ states into
$a_{1g}$ and doubly-degenerated $e_{g\pi}$ states. Square symbols in (e) and
(d) mark the value at $\tau$=0.

##  IV Discussion

We start our discussion by analysing the local physics on Fe sites in the
site-selective Mott state. The LDA+DMFT calculation yields similar
$d$–occupations (per site) of 5.18 and 5.08 on the LB and SB Fe sites,
respectively. Although the charge fluctuation is slightly larger on the SB
site, the local charge susceptibilities (charge-charge correlation function)
on the two Fe sites, Figs. 7cd, resemble each other with almost incompressible
Fe $3d$ states. The different nature of LB and SB sites is revealed by the
local spin susceptibilities in Figs. 7ab, which yield the effective screened
moments $m_{\rm scr}=\sqrt{T\chi_{\rm loc}}$ of 4.37 and 3.10 $\mu_{\rm B}$
for the LB and SB sites, respectively. The long-lived moment on the SB site is
in qualitative contrast with that in RNiO3 with vanishing $m_{\rm scr}$ on the
SB Ni site Park  _et al._ (2012); Subedi  _et al._ (2015). The orbital-
diagonal contributions in the local spin susceptibility in Figs. 7ef reveal
that, while all $d$ electrons contribute to the LB local moment, only the
$t_{2g}$ ones contribute on the SB site. This leads to the following site-
disproportionation picture of CCFO: 2$(d^{5}\underline{L})\rightarrow
t^{3}_{2g}[e^{2}_{g}\underline{L}_{\sigma}^{2}]\ (S=3/2)+t^{3}_{2g}e^{2}_{g}\
(S=5/2)$, where $\underline{L}_{\sigma}$ denotes a ligand hole that couples
with the $e_{g}$ states.

This picture is corroborated by the weights of atomic configurations (diagonal
element of the site-reduced density matrix) in Fig. 8. The high-spin ($S=5/2$)
state with $e^{2}_{g}t^{3}_{2g}$ configuration is dominant on the LB site. The
SB Fe site shows a broader distribution of the atomic weights. This is to be
attributed to the stronger covalent bonding with O sites. While this affects
mostly the $e_{g}$ orbitals, the abundance of the $e^{1}_{g}t^{4}_{2g}$
configuration shows that reducing the role of $t_{2g}$ electrons to building
of a rigid local moment is an approximation.

Figure 8: Weights of dominant atomic states in $N$=4 (green), 5 (red) and 6
(blue) occupations on (a) long-bond and (b) short-bond Fe site. The atomic
weights integrated in $N$=4, 5 and 6 sectors are shown in inset.

Next, we discuss the origin of the magnetic ordering. We find that both in the
non-disproportionated metallic state and the disproportionated insulating
state the Fe moment favor parallel (ferro- or ferrimagnetic) orientation. To
understand this, we refer to the general picture of Zener double-exchange
model Zener (1951) in which antiferromagneticaly arranged local ($t_{2g}$)
moments generate, via intra-atomic Hund’s coupling, a staggered potential for
the electrons in partially filled $e_{g}$ bands. This increases the kinetic
energy of the $e_{g}$ electrons and thus favors FM alignment of the local
moments. This mechanism, which is traditionally used to describe doped
manganites, survives also in metallic systems with moderate fluctuations of
$t_{2g}$ charge Sotnikov  _et al._ (2018) such as SrCoO3 Kuneš  _et al._
(2012). The metallic phase of CCFO is in this respect similar to the high-
valence SrCoO3.

In the disproportionated state, the (staggered) antiferromagnetic potential
leads to closing of the gap and thus insulator-compatible FM order is favored.
The antiferromagnetic coupling of the Cu ($S=1/2$) and Fe moments observed in
our calculations is not the glue of the ferromagnetic coupling of the Fe
moments. A DMFT calculation with a paramagnetic constraint on the Cu site
yields a ferrimagnetic insulating solution with almost unchanged Fe bands, see
SM sm .

The key question concerning materials exhibiting a bond disproportionation
such as RNiO3 or the present CCFO is the site-selective Mott vs Peierls
mechanism of the MIT. In the context of RNiO3, this question was addressed in
a number of recent studies Park  _et al._ (2012); Ruppen  _et al._ (2015);
Mercy  _et al._ (2017). Due to the limitation of our present approach we are
not able to provide a definite answer - magnetic ordering and site
disproportionation take place simultaneously in CCFO. Nevertheless, the
strikingly different band structures of FM and PM states in Fig. 5 and Fig. 6
open a question whether these are, in principle, compatible with each other,
i.e., can one go from the PM band structure to the FM one without closing the
gap? The gap in the PM state relies on the pole in the self-energy of the LB
site, while in the FM state the Hartree (energy independent) shift of the
spin-minority band above the chemical potential is crucial.

To understand how the PM Mott state can evolve in the FM one without closing
the gap, we invoke a toy model of Ref. Ruppen  _et al._ , 2015 that captures
essentials of the site-disproportionated Mott state and the behavior of the
self-energy pole when going from PM to spin-polarized Mott insulator. The
authors of Ref. Sangiovanni  _et al._ , 2006 showed that the magnetic ordering
(in one-band Hubbard model) introduces, in addition to the spin-dependent
Hartree contribution to the self-energy, a spin dependent shift of the self-
energy along the frequencies. This way the LB and SB site self-energies evolve
from the PM state, Figs. 6de, to the FM state, Figs. 5ef. Introducing this
behavior by hand to the toy model, one can go smoothly from the PM to the FM
state without moving the gap away from the $E_{F}$ as demonstrated by a movie
in SM sm .

##  V Conclusion

By means of LDA+$U$ and LDA+DMFT methods, we investigated insulating and
magnetic properties in CaCu3Fe4O12, consisting of high Fe4+ valence. By a
systematic LDA+$U$ calculation combined with a distortion-mode analysis, we
revealed an underlying Peierls instability to open the electronic gap in the
ferrimagnetic ordered phase. The LDA+DMFT simulation indicates that the site-
selective Mott mechanism also opens the insulating gap in the experimental
distorted structure. In the site-disproportionated state of CaCu3Fe4O12, the
two $e_{g}$ electrons on the short-bond Fe site couples with oxygen $p$ holes,
forming a singlet, while $t_{2g}$ moments are robust in both long- and short-
bond Fe sites, leading to the site-disproportionation picture:
2$(d^{5}\underline{L})\rightarrow
t^{3}_{2g}[e^{2}_{g}\underline{L}_{\sigma}^{2}]\ (S=3/2)+t^{3}_{2g}e^{2}_{g}\
(S=5/2)$. In contrast to rare-earth nickelates RNiO3, the paramagnetic site-
selective Mott phase is not observed experimentally. The active Fe $t_{2g}$
moment ($S=3/2$), absent in RNiO3, stabilizes the ferrimagnetic order via the
double-exchange mechanism. We discussed the spectral change from the site-
selective Mott to ferrimagnetic Peierls-like insulator.

Figure 9: Magnetic moment $M_{\rm Fe}$ on Fe 3$d$ states as a function of
temperature obtained by LDA+DMFT. The red (blue) symbol shows the $M_{\rm Fe}$
value calculated for $Im\overline{3}$ ($T_{\rm latt}>T_{\rm MI}$, no $Q_{\rm
B}$) and $Pn\overline{3}$ structure ($T_{\rm latt}<T_{\rm MI}$, finite $Q_{\rm
B}$).

###### Acknowledgements.

We thank A. Georges, I. Yamada, M. Mizumaki, Y. Uchimura, J. Fernández Afonso
and A. Sotnikov for fruitful discussions. A.H, M.W, and J.K are supported by
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant Agreement No. 646807-EXMAG). The
computational calculations were performed at the Vienna Scientific Cluster
(VSC).

## References

  * Imada  _et al._ (1998) M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
  * Khomskii (2014) D. I. Khomskii,  _Transition Metal Compounds_ (Cambridge University Press, Cambridge, England, 2014).
  * Tanabe and Sugano (1954) Y. Tanabe and S. Sugano, J. Phys. Soc. Jpn. 9, 753 (1954).
  * Werner and Millis (2007) P. Werner and A. J. Millis, Phys. Rev. Lett. 99, 126405 (2007).
  * Gavriliuk  _et al._ (2008) A. G. Gavriliuk, V. V. Struzhkin, I. S. Lyubutin, S. G. Ovchinnikov, M. Y. Hu, and P. Chow, Phys. Rev. B 77, 155112 (2008).
  * Kuneš and Křápek (2011) J. Kuneš and V. Křápek, Phys. Rev. Lett. 106, 256401 (2011).
  * Afonso  _et al._ (2019) J. F. Afonso, A. Sotnikov, A. Hariki, and J. Kuneš, Phys. Rev. B 99, 205118 (2019).
  * Khaliullin (2013) G. Khaliullin, Phys. Rev. Lett. 111, 197201 (2013).
  * Kuneš and Augustinský (2014) J. Kuneš and P. Augustinský, Phys. Rev. B 90, 235112 (2014).
  * Yamaguchi  _et al._ (2017) T. Yamaguchi, K. Sugimoto, and Y. Ohta, J. Phys. Soc. Jpn. 86, 043701 (2017).
  * Zener (1951) C. Zener, Phys. Rev. 82, 403 (1951).
  * Tokura and Tomioka (1999) Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200, 1 (1999).
  * van den Brink  _et al._ (1999) J. van den Brink, G. Khaliullin, and D. Khomskii, Phys. Rev. Lett. 83, 5118 (1999).
  * Kugel and Khomskiĭ (1982) K. I. Kugel and D. I. Khomskiĭ, Phys.-Uspekhi 25, 231 (1982).
  * Glazer (1972) A. M. Glazer, Acta Crystallogr. B 28, 3384 (1972).
  * Woodward (1997) P. M. Woodward, Acta Crystallogr. B 53, 32 (1997).
  * Howard and Stokes (1998) C. J. Howard and H. T. Stokes, Acta Crystallogr. B 54, 782 (1998).
  * Jahn  _et al._ (1937) H. A. Jahn, E. Teller, and F. G. Donnan, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences 161, 220 (1937).
  * Goodenough (1998) J. B. Goodenough, Annu. Rev. Mater. Sci. 28, 1 (1998).
  * Bersuker (2014) I. B. Bersuker,  _The Jahn-Teller Effect_ (Cambridge University Press, Cambridge, England, 2014).
  * Kanamori (1960) J. Kanamori, J. Appl. Phys. 31, S14 (1960).
  * Mazin  _et al._ (2007) I. I. Mazin, D. I. Khomskii, R. Lengsdorf, J. A. Alonso, W. G. Marshall, R. M. Ibberson, A. Podlesnyak, M. J. Martínez-Lope, and M. M. Abd-Elmeguid, Phys. Rev. Lett. 98, 176406 (2007).
  * Tokura and Nagaosa (2000) Y. Tokura and N. Nagaosa, Science 288, 462 (2000).
  * Pavarini and Koch (2010) E. Pavarini and E. Koch, Phys. Rev. Lett. 104, 086402 (2010).
  * Park  _et al._ (2012) H. Park, A. J. Millis, and C. A. Marianetti, Phys. Rev. Lett. 109, 156402 (2012).
  * Greenberg  _et al._ (2018) E. Greenberg, I. Leonov, S. Layek, Z. Konopkova, M. P. Pasternak, L. Dubrovinsky, R. Jeanloz, I. A. Abrikosov, and G. K. Rozenberg, Phys. Rev. X 8, 031059 (2018).
  * Lee  _et al._ (2011) S. Lee, R. Chen, and L. Balents, Phys. Rev. Lett. 106, 016405 (2011).
  * Lau and Millis (2013) B. Lau and A. J. Millis, Phys. Rev. Lett. 110, 126404 (2013).
  * Johnston  _et al._ (2014) S. Johnston, A. Mukherjee, I. Elfimov, M. Berciu, and G. A. Sawatzky, Phys. Rev. Lett. 112, 106404 (2014).
  * Jaramillo  _et al._ (2014) R. Jaramillo, S. D. Ha, D. M. Silevitch, and S. Ramanathan, Nat. Phys. 10, 304 EP (2014).
  * Subedi  _et al._ (2015) A. Subedi, O. E. Peil, and A. Georges, Phys. Rev. B 91, 075128 (2015).
  * Bisogni  _et al._ (2016) V. Bisogni, S. Catalano, R. J. Green, M. Gibert, R. Scherwitzl, Y. Huang, V. N. Strocov, P. Zubko, S. Balandeh, J.-M. Triscone, G. Sawatzky, and T. Schmitt, Nat. Commun. 7, 13017 (2016).
  * Mercy  _et al._ (2017) A. Mercy, J. Bieder, J. Íñiguez, and P. Ghosez, Nat. Commun. 8, 1677 (2017).
  * Varignon  _et al._ (2017) J. Varignon, M. N. Grisolia, J. Íñiguez, A. Barthélémy, and M. Bibes, npj Quantum Materials 2, 21 (2017).
  * Seth  _et al._ (2017) P. Seth, O. E. Peil, L. Pourovskii, M. Betzinger, C. Friedrich, O. Parcollet, S. Biermann, F. Aryasetiawan, and A. Georges, Phys. Rev. B 96, 205139 (2017).
  * Ruppen  _et al._ (2015) J. Ruppen, J. Teyssier, O. E. Peil, S. Catalano, M. Gibert, J. Mravlje, J.-M. Triscone, A. Georges, and D. van der Marel, Phys. Rev. B 92, 155145 (2015).
  * Shimakawa (2008) Y. Shimakawa, Inorg. Chem. 47, 8562 (2008).
  * Cheng  _et al._ (2013) J.-G. Cheng, J.-S. Zhou, Y.-F. Yang, H. D. Zhou, K. Matsubayashi, Y. Uwatoko, A. MacDonald, and J. B. Goodenough, Phys. Rev. Lett. 111, 176403 (2013).
  * Long  _et al._ (2009) Y. W. Long, N. Hayashi, T. Saito, M. Azuma, S. Muranaka, and Y. Shimakawa, Nature 458, 60 EP (2009).
  * Meyers  _et al._ (2013) D. Meyers, S. Mukherjee, J. G. Cheng, S. Middey, J. S. Zhou, J. B. Goodenough, B. A. Gray, J. W. Freeland, T. Saha-Dasgupta, and J. Chakhalian, Sci. Rep. 3, 1834 (2013).
  * Shimakawa (2015) Y. Shimakawa, J. Phys. D: Appl. Phys. 48, 504006 (2015).
  * Yamada  _et al._ (2014) I. Yamada, M. Ochi, M. Mizumaki, A. Hariki, T. Uozumi, R. Takahashi, and T. Irifune, Inorg. Chem. 53, 7089 (2014).
  * Yamada  _et al._ (2008) I. Yamada, K. Takata, N. Hayashi, S. Shinohara, M. Azuma, S. Mori, S. Muranaka, Y. Shimakawa, and M. Takano, Angew. Chem. Int. Ed. 47, 7032 (2008).
  * (44) See Supplementary Material for details of computations and structural-mode analysis, which includes Ref. Perdew  _et al._ (1996); Mizumaki  _et al._ (2011).
  * Hao  _et al._ (2009) X. Hao, Y. Xu, F. Gao, D. Zhou, and J. Meng, Phys. Rev. B 79, 113101 (2009).
  * Ueda  _et al._ (2013) T. Ueda, M. Kodera, K. Yamauchi, and T. Oguchi, J. Phys. Soc. Jpn. 82, 094718 (2013).
  * Mizumaki  _et al._ (2011) M. Mizumaki, W. T. Chen, T. Saito, I. Yamada, J. P. Attfield, and Y. Shimakawa, Phys. Rev. B 84, 094418 (2011).
  * Momma and Izumi (2011) K. Momma and F. Izumi, J. Appl. Crystall. 44, 1272 (2011).
  * Perez-Mato  _et al._ (2010) J. M. Perez-Mato, D. Orobengoa, and M. I. Aroyo, Acta Crystallogr. A 66, 558 (2010).
  * (50) P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz,  _WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties_ (Karlheinz Schwarz, Techn. Universitat Wien, Austria, 2001, ISBN 3-9501031-1-2).
  * Kotliar  _et al._ (2006) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006).
  * Kuneš  _et al._ (2009) J. Kuneš, I. Leonov, M. Kollar, K. Byczuk, V. I. Anisimov, and D. Vollhardt, Eur. Phys. J. Spec. Top. 180, 5 (2009).
  * Georges  _et al._ (1996) A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * Kuneš  _et al._ (2010) J. Kuneš, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, and K. Held, Comput. Phys. Commun. 181, 1888 (2010).
  * Mostofi  _et al._ (2014) A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Comput. Phys. Commun. 185, 2309 (2014).
  * Anisimov  _et al._ (1991) V. I. Anisimov, J. Zaanen, and O. K. Andersen, Phys. Rev. B 44, 943 (1991).
  * Kuneš  _et al._ (2009) J. Kuneš, D. M. Korotin, M. A. Korotin, V. I. Anisimov, and P. Werner, Phys. Rev. Lett. 102, 146402 (2009).
  * Werner  _et al._ (2006) P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97, 076405 (2006).
  * Boehnke  _et al._ (2011) L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann, and O. Parcollet, Phys. Rev. B 84, 075145 (2011).
  * Hafermann  _et al._ (2012) H. Hafermann, K. R. Patton, and P. Werner, Phys. Rev. B 85, 205106 (2012).
  * Hariki  _et al._ (2015) A. Hariki, A. Yamanaka, and T. Uozumi, J. Phys. Soc. Jpn. 84, 073706 (2015).
  * Sotnikov  _et al._ (2018) A. Sotnikov, A. Cichy, and J. Kuneš, Phys. Rev. B 97, 235157 (2018).
  * Kuneš  _et al._ (2012) J. Kuneš, V. Křápek, N. Parragh, G. Sangiovanni, A. Toschi, and A. V. Kozhevnikov, Phys. Rev. Lett. 109, 117206 (2012).
  * Sangiovanni  _et al._ (2006) G. Sangiovanni, A. Toschi, E. Koch, K. Held, M. Capone, C. Castellani, O. Gunnarsson, S.-K. Mo, J. W. Allen, H.-D. Kim, A. Sekiyama, A. Yamasaki, S. Suga, and P. Metcalf, Phys. Rev. B 73, 205121 (2006).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).