# Doped Mott-Hubbard materials with a low quasiparticle transparency

V. A. Gavrichkov  L. V. Kirensky Institute of Physics, Siberian Branch of
Russian Academy of Sciences, 660036 Krasnoyarsk, Russia

###### Abstract

Based on the Wilson’s criterion metal-insulator, extended to materials with
strong electronic correlations, we have identified a specific class of the
materials, which is not associated with their usual classification into Mott-
Hubbard and charge transfer dielectrics. The local symmetry of these materials
leads to a disappearance of quasiparticle states (so-called first removal or
fist extra states) in the Hubbard’s gap. It’s especially unusual for the doped
materials, where quasiparticles - charge carriers can disappear or appear
under external factors, but the Mott transition has not yet been achieved. In
the work we introduced the so-called a ”quasiparticle transparency”, and also
provided specific experiments to identify the materials with the low
quasiparticle transparency. Some examples of these materials with a spin
crossover under high pressure and showing Jahn-Teller nature have been
observed.

###### pacs:

71.30.+h

##  I Introduction

The satisfiability of known metal-insulator(MI) criterion
$W\mathord{\left/{\vphantom{W{U\sim\left({R+1}\right)}}}\right.\kern-1.2pt}{U\sim\left({R+1}\right)}$
(where $R$ \- orbital degeneracy) of the Mott-Hubbard transition depends on
the characteristic material scale, because a width $W$ of the quasiparticle
band decreases with decreasing scale of materials, and the Coulomb interaction
$U$ increases due to the weakening of screening effects. The MI transition
itself can also be controlled by external effects, which change physical
properties of the material (thermal expansion, pressure, optical pumping,
etc.). Of particular interest is the transition of Mott-Hubbard insulators to
the metallic state induced by a doping effect, since the doped materials have
unique properties, e.g. high temperature superconductivity in the 2D
perovskite cuprates Kohsaka  _et al._ (2008); Yang  _et al._ (2008) and
colossal magnetoresistance in 3D manganites Saitoh  _et al._ (2000); Mannella
_et al._ (2005). Moreover, in both materials the pseudogap effect is observed.
However, it’s hard to imagine that the initial MI criterion is valid in all
various cases.

To see the origins of the problem, it is enough to look at the ”band
structure” within a framework of the formalism of Hubbard operators Hubbard
(1963) in the zero-hopping approximation. The formalism is necessary to detect
the effects of many-electron local states in the number of quasiparticles in
the band. In the zero-hopping approximation, according to the initial MI
criterion, the material must be an insulator ($W=0$), but the MI criterion
based on Wilson’s ideas Wilson (1931) for the doped Mott-Hubbard materials
shows non-trivial scale invariant results, independent on band width $W$. The
purpose of our work is to construct and apply Wilson’s MI criterion concerning
a system of itinerant electrons in the analytical form for the doped Mott-
Hubbard materials. The approach includes a key statement that if an electron
system consists of completely occupied and empty bands, it is an insulator,
otherwise, it is a metal, where however the spectral density of quasiparticles
depends on the doped carrier concentration $x$ due to many-electron effects.

##  II  Wilson’s MI criterion for the doped materials

To extend the Wilson’s MI criterion, we will further follow the work
Gavrichkov (2015) where it was demonstrated that doped 2D perovskite cuprates
have metallic conductivity, and there are no forrbidden quasiparticle states.
Although there are some features here (e.g. an impurity potential effect and
associated states), we will consider the criterion taking into account the
many-electron effects only. Our extended approach uses the fact that the
optical intracell $dd$-transitions with their ($l$-orbital,
$S$-spin)-selection rules in the transparency window and optical charge
transfer transitions in the oxides can be observed at the same $3d$-states.
Krinchik and Chetkin (1969); Eremenko and Belyaeva (1969)

In the first approximation we can assume that the quasiparticles are unit cell
excitations which can be represented graphically as the single-particle
transitions between different sectors
$N_{h}=...(N_{h0}-1),N_{h0},(N_{h0}+1),...$ of the configuration space of unit
cell ($N_{h0}$-hole number per cell in undoped material, see Fig.1).
Ovchinnikov  _et al._ (2012) Each of these transitions forms the $r$-th
quasiparticle band, where the root vector $r=\\{ii^{\prime}\\}$ in the
configuration space numerates the initial $i$ and final $i^{\prime}$ many-
electron states in the transition. The quasiparticle transitions with
increasing or decreasing the electron is formed the conduction or valence
bands respectively. Here, it is convenient to start with Lehmann’s
representation for Green’s function $G_{fg\sigma}^{\lambda\lambda}$ of the
intracell Hamiltonian $H_{0}$ with respect to the family of single-particle
operators $c_{f\lambda\sigma}^{(+)}$ and their matrix elements in the basis of
${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$ \- eigenstates of $H_{0}$ ($S$
and $M$\- spin and spin projection of the multielectron cell eigenstate):

$\sum\limits_{\lambda\sigma}{G_{fg\sigma}^{\lambda\lambda}}=\sum\limits_{\lambda\sigma}{\left\langle{\left\langle{{c_{f\lambda\sigma}}}\mathrel{\left|{\vphantom{{c_{f\lambda\sigma}}{c_{g\lambda\sigma}^{+}}}}\right.\kern-1.2pt}{{c_{g\lambda\sigma}^{+}}}\right\rangle}\right\rangle}=\delta_{fg}\sum\limits_{rr^{\prime}}{\sum\limits_{\lambda\sigma}{\gamma_{f\lambda\sigma}\left(r\right)\gamma_{f\lambda\sigma}\left({r^{\prime}}\right)D_{fg}^{rr^{\prime}}\left(E\right)}=}=\delta_{fg}\sum\limits_{rr^{\prime}}{\delta_{rr^{\prime}}\sum\limits_{\sigma}{\frac{{F_{r}\left(x\right)\chi_{rr}^{\sigma}}}{{E-\Omega_{r}}}}},$
(1)

where matrix elements:

$\displaystyle{\gamma_{\lambda\sigma}}\left({{r}}\right)$ $\displaystyle=$
$\displaystyle\left\langle({{N_{h}+1,{{M^{\prime}}_{S^{\prime}}}})_{\tau}}\right|c_{f\lambda\sigma}\left|({{N_{h},{M_{S}}})_{\mu}}\right\rangle\times$
(2) $\displaystyle\times$
$\displaystyle\delta\left({S^{\prime},S\pm|\sigma|}\right)\delta\left({M^{\prime},M+\sigma}\right),$

and

$\chi_{rr^{\prime}}^{\sigma}=\sum\limits_{\lambda}{\gamma_{\lambda\sigma}^{*}\left(r\right)\gamma_{\lambda\sigma}\left({r^{\prime}}\right)}$
(3)

Figure 1:  $E_{i}({N_{h}},{M_{S}})$ \- energy level scheme of the
configuration space based on cell states with hole numbers per cell
$N_{h}=N_{h0}-1,N_{h0},N_{h0}+1,..$, where $i=\mu,\tau,\eta$ and $N_{h0}$ is a
hole number per cell in the undoped material. The $e$ \- and $h$ \- circles
point to the occupied ground cell states of electron and hole doped material
respectively. A solid line with arrows corresponds to the $fes$ and $frs$
quasiparticles.

for the $p$ \- and $n$ \- quasiparticle states in the valence and conduction
bands respectively, where the total space of root vectors
$\\{r\\}=...+\\{r_{12}\\}+\\{r_{23}\\}+...,(\\{r_{12}\\}=\\{\mu\tau\\},\\{r_{23}\\}=\\{\tau\eta\\}$
and so on, see Fig.1). An occupation factor $F_{r}\left(x\right)$ is the
probability to detect a cell in any of the $i,i^{\prime}$ states participating
in the $r$-th transition, and
$\Omega^{v}_{r}=E_{i}({N_{h}},{M_{S}})-E_{i^{\prime}}({N_{h}+1},{{M^{\prime}}_{S^{\prime}}})$
and
$\Omega^{c}_{r}=E_{i}({N_{h}-1},{M_{S}})-E_{i^{\prime}}({N_{h}},{{M^{\prime}}_{S^{\prime}}})$
are the quasiparticle energies in the $r$-th valence and conduction bands
respectively. In the PM phase of doped material factor of ocuppation has a
form:

$F_{r_{12}}\left(x\right)=\frac{{1-\alpha x}}{{2S+1}},$ (4)

where $\alpha=1-({2S+1})/({2S^{\prime}+1})$ is proportional to the ratio of
the spin multiplets of $i,i^{\prime}$ states participating in the $r_{12}$ \-
(from the subspace $\\{r_{12}\\}$) transition between the ground states
${\left|{({N_{h0}},{M_{S}})_{i=0}}\right\rangle}$ and
${\left|{({N_{h0}+1},{M^{\prime}_{S^{\prime}}})_{i^{\prime}=0}}\right\rangle}$
indicated by the arrow in Fig.1

The Green’s function in Eq.(1) is yet free from the shortcomings of the
hydrogen-like (s-)representation and low-energy approximations, because we do
not restrict ourselves to choose the intracell Hamiltonian $H_{0}$, and are
ready to work with all of the ${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$
states. Taking into account the specifics of the cuprates, we will consider
the p - doped materials, where a valence states number is equal to the sum
over all quasiparicle states:

$N_{v}(x)={\sum\limits_{\lambda\sigma}{\sum\limits_{r}{{\gamma_{\lambda\sigma}}^{2}\left(r\right)}{{\int\limits{dE\left({-\tfrac{1}{\pi}}\right){\operatorname{Im}{D_{0}^{r}}\left({E}\right)}}_{E+i0}}}}}=N_{v}^{12}(x)+N_{v}^{23}(x),$
(5)

where $N_{v}^{12}(x)$ and $N_{v}^{23}(x)$ are the contributions from the
quasiparticles with the root vectors $r$ from the $\\{r_{12}\\}$ and
$\\{r_{23}\\}$ subspaces because the other states of
${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$ in p - doped material are not
occupied, and there is a zero probability $F_{r}\left(x\right)=0$ to detect a
cell in these states at a low temperature. Wilson’s condition at the
insulating state, which we are interested, it is

$N_{e}-x=N_{v}(x),$ (6)

where $(N_{e}-x)$ is a total electron number per cell of hole doped material.
That is, if the number of electrons in cell equals the number of valence
quasiparticle states, the doped material is an insulator.

To obtain the Fermi level position in the degenerate doped material at zero
temperature we could carry out the integration on the right side of the
equation

$x={\sum\limits_{\lambda\sigma}{\sum\limits_{r}{{\gamma_{\lambda\sigma}}^{2}\left(r\right)}{{\int\limits_{E_{F}}{dE\left({-\tfrac{1}{\pi}}\right){\operatorname{Im}{D_{0}^{r}}\left({E}\right)}}_{E+i0}}}}},$
(7)

over the top valence band of the first removal electron states ($frs$) with
the lowest binding energy (see Fig.1), and this is sufficient at the actual
concentrations $x\sim 0.1$ as a rule. However, this is not sufficient, when
the hole concentration $x$ exceeds the number of quasiparticle states in the
top valence band $x\gg{N_{frs}}$, because the number of $frs$ quasiparticle
states $N_{frs}$ may be very small. Therefore, the solution Eq.(7) has the
features at $N_{frs}\rightarrow 0$. To understand this, we will obtain the a
total number of valence quasiparticle states $N_{v}$ as a function of both the
doping concentration $x$, and $N_{frs}$:

$N_{v}(x,N_{frs})=N_{v}^{12}+N_{v}^{23},$ (8)

where the contributions $N_{v}^{12}$ and $N_{v}^{23}$ from the subspaces
$\\{r_{12}\\}$ and $\\{r_{23}\\}$ are calculated in the Appendix A, and the
root vectors $r$ characterizes a specific quasiparticle band: if
$r=\left\\{{\nu_{0},l_{0}}\right\\}$ or $r=\left\\{{\tau_{0},l_{0}}\right\\}$
in Eqs.(2) and (3), then we are dealing with $fes$ or $frs$ quasiparticles in
Fig.1, respectively. By following this approach, we obtained a MI criterion:

$N_{v}(x)=N_{e}-x(1-N_{frs})$ (9)

which is characterized by a condition: $N_{frs}=0$ ( - insulator) or
$N_{frs}\neq 0$ ( - metal), and $N_{frs}$ is calculated in the Appendix A.
Indeed, at conditions $N_{\lambda}=1$ and $N_{e}=(1-x)$ we always obtain a
simple metal with $N_{frs}=2$ and $N_{v}(x)=(1+x)$ valence states, as it takes
place in the Hubbard model, where the highly spin (triplet) states are simply
not available. The criterion is based only on the properties of completeness
of a set of states ${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$ in the
configuration space of the cell, and the number of states $N_{frs}$ depends on
their spin and orbital nature.

##  III  Quasiparticles in doped material with low quasiparticle transparency

From Eq.(9) it follows that the doped material can show both the metallic
$N_{v}(x)>(N_{e}-x)$, and dielectric properties $N_{v}(x)=(N_{e}-x)$ at the
$N_{frs}>0$ or $N_{frs}=0$ respectively. The physical meaning of the MI
criterion lies in the matrix value $\chi_{rr^{\prime}}^{\sigma}$ (see Eq.(3))
that in the Hubbard operators representation corresponds to the quasiparticle
as a sequence of intracell transitions between multielectron cell states. If
the single-particle transitions are forbidden by any symmetry, the charge
carriers is missing. The doped particles (electron or holes) are in local
multielectron states ${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$, but there
is no peak in the single-particle density of states, and the matrix value
$\chi_{rr^{\prime}}^{\sigma}$ defined in the root vectors space $\\{r\\}$ can
be called ”quasiparticle transparency” of the doped material.

###  III.1  Pressure induced effects

The $frs$ states can be prohibited at the $\delta(S^{\prime},S\pm|\sigma|)=0$
in Eq.(2) and Eq.(3) (s-forbidden $frs$ quasiparticles). However, can
forbidden $frs$($fes$) states really exist in any doped Mott-Hubbard
materials? A review of Tanabe-Sugano diagrams Tanabe and Sugano (1956) shows
that the low transparency effects in the materials with $3d$ elements in an
octahedral environment are unlikely. Indeed, the ground states in different
sectors of the configuration space are connected by non-zero matrix elements
(2) of single-particle operators. However, the nature of the ground state of
the transition element ion depends on the applied pressure, while some
materials with $3d^{k}$ ions at $3<k<8$: the transition metal oxides,
transition metal complexes, metal-organic molecules and molecular assemblies
exhibit spin crossover, with increasing pressure from ambient pressure
Lyubutin and Gavriliuk (2009); Gavrichkov  _et al._ (2020); Halder  _et al._
(2002); Ohkoshi  _et al._ (2011); Wentzcovitch  _et al._ (2009); Hsu  _et al._
(2010); Liu  _et al._ (2014); Sinmyo  _et al._ (2017); Marbeuf  _et al._
(2013); Nishino  _et al._ (2007); Ovchinnikov and Zabluda (2004); Saha-
Dasgupta and Oppeneer (2014). The spin crosover ocurrs due to competition
between the energy crystal field $10Dq$ and the intra-atomic Hund exchange
$J_{H}$, for example, in the interesting material FeBO3 Knyazev  _et al._
(2022), where the Fe3+ ion is in a high-spin configuration
$3d^{5}\left({t_{2g}^{3}e_{g}^{2}}\right)$ at the ambient pressure. The energy
of the high and low spin states in the $N_{0}(3d^{5})$ sector can be presented
in the form Ovchinnikov (2008):

$\displaystyle E_{hs}$ $\displaystyle=$ $\displaystyle E_{c}(d^{5})-10J_{H}$
$\displaystyle E_{ls}$ $\displaystyle=$ $\displaystyle
E_{c}(d^{5})-20Dq-4J_{H}$ (10)

Eq.(10) shows that spin crossover
$\left.S\right|_{P<P_{S}}={5\mathord{\left/{\vphantom{5{2\leftrightarrow\left.S\right|_{P>P_{S}}={1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}}\right.\kern-1.2pt}{2\leftrightarrow\left.S\right|_{P>P_{S}}={1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}}$
in the ground state is possible at a certain pressure $P_{S}$ ($=48-54$ GPa
for the iron borate Knyazev  _et al._ (2022)) corresponding to the crystal
field $10Dq=3J_{H}$. Here and below, $E_{c}$ is a part of the term energy
independent of the Hund exchange $J_{H}$ and the crystal field $10Dq$
Ovchinnikov (2013). Similarly in sector $N_{+}$ for the $3d^{6}$ configuration

$\displaystyle E_{hs}$ $\displaystyle=$ $\displaystyle
E_{c}(d^{6})-4Dq-10J_{H}$ $\displaystyle E_{ls}$ $\displaystyle=$
$\displaystyle E_{c}(d^{6})-24Dq-6J_{H}$ (11)

for spins $S_{\nu_{0}}=2$ and $S_{\nu_{0}}=0$, respectively. This shows that
crossover in the ground state of the term is possible, at the condition
$10Dq=2J_{H}$, in accordance with the pressure $P_{min}$. Similarly, in sector
$N_{-}$ for the $3d^{4}$ configuration

$\displaystyle E_{hs}$ $\displaystyle=$ $\displaystyle
E_{c}(d^{4})-6Dq-6J_{H}$ $\displaystyle E_{ls}$ $\displaystyle=$
$\displaystyle E_{c}(d^{4})-16Dq-2J_{H}$ (12)

Figure 2: Ground state crossover $S_{\nu_{0}}=2\leftrightarrow 0$ in the
$N_{h0}-1$ sector FeBO3 under the pressure $P_{min}<P<P_{S}$. A semiellipse
with a cross shows forbidden $fes$ quasiparticles.

Eq.(12) shows that crossover in the ground state of the term is possible at
the same crystal field $10Dq=3J_{H}$ as for the $3d^{5}$ configuration. The
energy of ground states in Eqs.(10) and (12) in the pressure range
$P_{min}<P<P_{S}$ promote the forbidden $fes$ quasiparticles (see Fig.2), and
the $n$ \- doped iron borate, can turn out to be an insulator (semiconductor),
where

$\chi_{r_{0}r_{0}}^{\sigma}={\sum\limits_{\lambda}{\gamma_{\lambda\sigma}^{*}\left({{}^{6}A_{1},{}^{5}E}\right)\gamma_{\lambda\sigma}\left({{}^{5}E,{}^{6}A_{1}}\right)}}=0$
(13)

at
$\delta\left({S^{\prime}=0,S=5/2\pm|\sigma|}\right)\delta\left({M^{\prime},M+\sigma}\right)=0$.
The disappearance of $n$ quasiparticles precedes the spin crossover at the
pressure $P_{S}$. Note, however, the iron borate has so far not been able to
be converted into the metallic state with increasing pressure.

###  III.2  Quasiparticles induced by pseudo JT effect

Figure 3:  $E_{i}({N_{h}},{M_{S}})$ \- energy level scheme of the
configuration space based on the cell states (15) in CuO2 layer. The solid
line with arrows corresponds to the $frs$ quasiparticles. (a) The tilting and
orientation angles $\varphi$ and $\theta$ of the CuO6 octahedron are saved by
the $frs$ quasiparticles. (b) The wavy line shows the states active in the
pseudo JT effect. The angles do not have specific magnitudes and are changed
by $frs$ quasiparticles

A second possibility to observe the effects of low quasiparticle transparency
is the the material with a dynamic JT effect, where the doped hole changes the
initial orbital configuration of
$\left|({N_{h0},{M_{S}}})_{\mu=0}\right\rangle$ ground cell states ($l$ \-
forbidden $frs$ quasiparticles). Indeed single-particle operators
$c_{f\lambda\sigma}^{(+)}$ in the matrix elements (2) and quasiparticle
transparence (3) are non-diagonal operators, and quenching effects can be
expected.

As follows from the works Grissonnanche  _et al._ (2020, 2019) the phonons
have zero termal Hall response outside the pseudogap phase in the hole doped
2D perovskite cuprates. However, inside the pseudogap phase, the phonons
become chiral to generate the Hall response. It’s show a specific symmetrical
nature of the electron-lattice coupling. Next we apply the MI criterion, to
understand the symmetric nature of phonons in the pseudogap effect of the hole
spectrum in doped 2D perovskite cuprates. At first, we will check the MI
criterion for doped cuprates in the frameworks of usual Russell-Saunders
scheme. Let’s calculate the magnitude of $N^{s(t)}_{frs}$ in the cuprates,
where $r=\\{^{2}b_{1},A_{1g}\\}$ \- root vector are relevant in Eq.(26) at
$\mu=0$ and $\tau=0$, Feiner  _et al._ (1996); Gavrichkov  _et al._ (2001)
i.e. it corresponds to only the $A_{1g}$ singlet $frs$ state. It’s the well
known Zhang-Rice singlet Zhang and Rice (1988). Using exact diagonalization
procedure with LDA parameters from the work Korshunov  _et al._ (2005), we
obtain the relation:

$\displaystyle N_{frs}^{s}$ $\displaystyle=$ $\displaystyle
1+[\beta^{2}_{0}(h_{b})-\beta^{2}_{0}(h_{d_{x}})]\times$ (14)
$\displaystyle\times$
$\displaystyle[B^{2}_{0}(h^{2}_{b})-B^{2}_{0}(h^{2}_{d_{x}})]\approx 0.97$

for the singlet $frs$ states, where the doublet and singlet ground states (5)
and (6) are

$\displaystyle|^{2}b_{1}\rangle_{0}$ $\displaystyle=$
$\displaystyle\beta_{0}(h_{b})|h_{b},\sigma_{\frac{1}{2}}\rangle+\beta_{0}(h_{d_{x}})|h_{d_{x}},\sigma_{\frac{1}{2}}\rangle$
$\displaystyle|A_{1g}\rangle_{0}$ $\displaystyle=$ $\displaystyle
B_{0}(h^{2}_{b})|h^{2}_{b},0_{0}\rangle+B_{0}(h^{2}_{d_{x}})|h^{2}_{d_{x}},0_{0}\rangle+$
(15) $\displaystyle+$ $\displaystyle
B_{0}(h_{d_{x}},h_{b})|h_{d_{x}},h_{b},0_{0}\rangle,$

$h_{b}$ and $h_{d_{x}}$ \- holes in the $b$-symmetrized cell states of oxygen
and $d_{x^{2}-y^{2}}$ cooper states of CuO2 layer respectively. Thus, there is
no any $l$ \- forbidden quasiparticles in the Russell-Saunders scheme, and the
number of valence states is practically a constant: $N_{v}(x)\approx
N_{e}-0.03x$. Here, the $frs$ quasiparticles are associated with single-hole
transitions in Fig.3a, where the lattice of CuO2 layer is unchanged, and
adiabatic approximation is correct.

In the dynamic JT effect, the CuO6 octahedra can be in both the U stripe, and
the D stripe states with different tilting and orientation angles
$\varphi_{U(D)}$ and $\theta_{U(D)}$ respectively Gavrichkov  _et al._ (2019).
The hole concentration $x$ in them is also different, and the dynamic JT
effect in the CuO2 layer as a whole would be possible only if its total charge
and regular stripe structure were saved. Let’s assume that the tilts
$\varphi_{U(D)}$ and orientations $\theta_{U(D)}$ of the CuO6 octahedra with
respect to the spacer rock salt layers are the active JT distortions, and a
rotation of all tilted CuO6 octahedra around the $c$ axis (i.e. changing their
orientation $\theta$) in a stripe U/D/U/D… structure fits within these
limitations Gavrichkov and Polukeev (2022). However, a scale of the novel JT
cell in the stripe set U/D/U/D… exceeds the initial cell (i.e. the single CuO6
octahedron). As a result, in the dynamic JT effect, the hole number $x$ are
not saved in the single CuO6 octahedron under rotation around the $c$ axis.

In Eqs.(2) and (3), where matrix elements $\gamma_{\lambda\sigma}(r)$ were
calculated still in local Russell-Saunders scheme, the overlapping phonon
parts of the initial cell functions (18) and (19) arises. In meaning, it’s
Ham’s reduction factor Ham (1965) in the quasiparticle transparence, because
the JT cell has a fourfold degeneracy with different $\theta_{U(D)}$
orientations of tilted CuO6 octahedra:

$\chi_{r_{0}r_{0}}^{\sigma}(\delta\theta,\delta\varphi)={\gamma_{x^{2}\sigma}^{*}\left({{}^{1}A_{1g},{}^{2}b_{1g}}\right)\gamma_{x^{2}\sigma}\left({{}^{2}b_{1g}{},^{1}A_{1g}}\right)}\cdot{\sum\limits_{\delta\varphi\delta\theta}\alpha(\delta\varphi,\delta\theta)},$
(16)

where $r_{0}=\\{^{1}A_{1g},^{2}b_{1g}\\}$, $\theta=\theta_{U},\theta_{D}$ and
$\varphi=\varphi_{U},\varphi_{D}$ with the indices U and D related to the
stripe affiliation for the single CuO6 octahedron in the harmonic oscillator
states $\left|{\psi_{\tau(\mu)}(\theta)}\right\rangle$ and
$\left|{\psi_{\tau(\mu)}(\varphi)}\right\rangle$ of the displaced $2D$
oscillator Gavrichkov and Polukeev (2022):

$\alpha(\delta\varphi,\delta\theta)=\left\langle{\psi_{{}^{1}A_{1g}}(\theta_{D})}\right|\left.{\psi_{{}^{2}b_{1g}}(\theta_{U})}\right\rangle^{2}\cdot\left\langle{\psi_{{}^{1}A_{1g}}(\varphi_{D})}\right|\left.{\psi_{{}^{2}b_{1g}}(\varphi_{U})}\right\rangle^{2}\approx\exp\left\\{{-\nu{{\left({\delta\theta^{2}+\delta\varphi^{2}}\right)}\mathord{\left/{\vphantom{{\left({\delta\theta^{2}+\delta\alpha^{2}}\right)}2}}\right.\kern-1.2pt}2}}\right\\}$
(17)

where $\delta\varphi=\varphi_{D}-\varphi_{U}\approx 9^{\circ}-13^{\circ}$,
$\delta\theta=\theta_{D}-\theta_{U}=\pm 45^{\circ}$ and
$\nu=K/\hbar\omega_{D}$ with a Debye frequency $\omega_{D}$ and force
coefficient $K$. Formally, the hole-doped 2D cuprate becomes an insulator only
at $\omega_{D}\rightarrow 0$. We will obtain the reduction effects in Eq.(17),
if the charge inhomogeneity of the dynamical U/D/U/D… stripe structure (see
Fig.(3b) occurs. To illustrate the last conclusion we considered a simplified
model in the Appendix B.

##  IV  Conclusions

In accordance with Wilson’s MI criterion for the materials with strong
electronic correlations, a class of materials with the forbidden $frs(fes)$
quasiparticles in the Hubbard gap was identified. In order to highlight the
nature of the specific materials, we introduced the term ”quasiparticle
transparency” for the matrix $\chi_{rr^{\prime}}^{\sigma}$ (see Eq.(3)), which
in the materials can have a zero magnitude, and the doping does not lead to
the charge carriers generation. Actually, with the term we note the analogy of
the $frs(fes)$ quasiparticles with a propagation of light in a material with
optical intracell absorption at frequencies of the $dd$ transitions.

A first tentative search for materials with the low quasiparticle transparency
allowed us to identify the following materials as candidates:

(i) the $3d$ oxides with a spin crossover (e.g. FeBO3 and other transition
element oxides: LaMnO3($3d^{4}$), Fe2O3($3d^{5}$), MnO($3d^{5}$),
LaCoO3($3d^{6}$), CoO($3d^{7}$), LaCoO4($3d^{7}$), LaNiO3($3d^{7}$) …, where
the low quasiparticle transparency effects can be observed in the vicinity of
the crossover at high pressure (e.g. $P_{S}=48-54$ GPa for the iron borate).
The doped charge carriers are generated and disappear with increasing
pressure.

(ii) the materials with dynamic JT effect (2D perovskite cuprates), where in
the hole-doped materials the overlapping of phonon functions in the different
sectors $N_{h0}$ and $N_{h0}+1$ of cell configuration space leads to partial
quenching of charge carriers. An essence of the quenching is that even in the
dynamic JT effect, the doped holes avoid the U stripes, and the threshold
nature of the pseudo Jahn Teller effect in the doping concentration
$x_{D}>x_{c}$ in the D stripes continues to support the charge inhomogeneous
U/D/U/D… stripe structure. Otherwise, the hole concentration drops below
critical $x<x_{c}$ and the pseudo JT effect disappears Gavrichkov (2024).
However, we are forced to pay for the dynamic stripe structure by introducing
the bifurcation nature of the adiabatic potential in Fig.3b.

The real question is how can the doped materials with low quasiparticle
transparency be identified? In undoped materials, the signatures of forbidden
$frs(fes)$ quasiparticles in the single-particle density of states are
missing. However, they can be enhanced by a resonant optical excitation, since
a forrbiden on the nonzero magnitudes (2) does not apply to the optical matrix
elements. Therefore, it is possible to detect the low quasiparticle
transparency by studying a difference between the optical gap and
photoconductivity measurements for a mobility gap(see e.g. the work Yu  _et
al._ (1992)). The first corresponds to a charge transfer gap (so called a CT
gap) in the cuprates, where light-induced $frs$ ”quasiparticles” are
localized, and the second corresponds to a gap in the spectrum of charge
carriers. As follows from our work, the difference in the undoped cuprates is
directly related to the pseudogap effect in the hole-doped cuprates. The
photoconductivity has been mainly a topic of superconductor research during
the 1990s, the explanation for the effect is still under some debate Destraz
(2015).

###### Acknowledgements.

We acknowledge the support of the Russian Science Foundation through grant RSF
No.22-22-00298.

##  Appendix A  The number of frs(fes) states in doped Mott-Hubbard materials

In a case of one hole per cell, ${\left|{({N_{h}},{M_{S}})_{i}}\right\rangle}$
cell states are a superposition of different hole configurations of the same
orbital ($l$ -)symmetry:

$\left|({N_{h0},{M_{S}}})_{\mu}\right\rangle=\sum\limits_{\lambda}{{\beta_{\mu}}\left({{h_{\lambda}}}\right)\left|{{h_{\lambda}},{M_{S}}}\right\rangle}|\psi_{\mu}(\varphi)\psi_{\mu}(\theta)\rangle$
(18)

Thus, there are $C_{{2N_{\lambda}}}^{1}={2N_{\lambda}}$ the one-hole spin
doublet states, where $C^{k}_{n}$ \- number of combinations. Altogether there
are $C^{2}_{2N_{\lambda}}=N_{S}+3N_{T}$ of the spin singlets
$N_{S}=C_{N_{\lambda}}^{2}+N_{\lambda}$ and triplets
$N_{T}=C_{N_{\lambda}}^{2}$:

$\displaystyle\left|({N_{h0}+1,{M^{\prime}_{S^{\prime}}}})_{\tau}\right\rangle=$
$\displaystyle\sum\limits_{\nu\nu^{\prime}}{{B_{\tau}}\left({{h_{\nu}},{h_{\nu^{\prime}}}}\right)\left|{{h_{\nu}},{h_{\nu^{\prime}}},{M^{\prime}_{S^{\prime}}}}\right\rangle}\times$
(19) $\displaystyle\times|\psi_{\tau}(\varphi)\psi_{\tau}(\theta)\rangle$

in the two-hole sector (see Fig.1) of the ${N_{\lambda}}$ \- orbital approach,
where the harmonic oscillator wave function
$\left|\psi_{\mu(\tau)}(\varphi)\psi_{(\mu(\tau)}(\theta)\right\rangle=\left|\psi_{\mu(\tau)}(\varphi)\right\rangle\left|\psi_{\mu(\tau)}(\theta)\right\rangle$
associated with the (non-)displaced $2D$ oscillator Gavrichkov and Polukeev
(2022). Using the intracell Hamiltonian $H_{0}$ in the cell functions
representation the configuration weights $\beta_{\mu}(h_{\lambda})$ and
$B_{\tau}(h_{\lambda},h_{\lambda^{\prime}})$ can be obtained by the exact
diagonalization procedure for the matrix
$(\hat{H_{0}})_{\lambda\lambda^{\prime}}$ and
$(\hat{H_{0}})^{\nu\nu^{\prime}}_{\lambda\lambda^{\prime}}$ in the
$E_{i}({N_{h},{M_{S}}})$-eigenvalue problem at the different sectors $N_{h}$.
Ovchinnikov  _et al._ (2012)

A sum (5) over all the $r$-th excited states with $\mu\neq 0$ in the sector
$N_{h}=N_{h0}$ is omitted, and only the excited states with any $\tau$($\eta$)
index in the nearest $N_{h}=(N_{h0}+1)$ and $(N_{h0}+2)$ sectors are summed
up. The expressions for high- and low-spin two-hole partner states (with
$S^{\prime}=S\pm|\sigma|$) can be combined into a single expression:

$\left|{h_{\lambda},{h_{\lambda^{\prime}},M^{\prime}_{S^{\prime}}}}\right\rangle=\\{\Gamma_{\uparrow}\left({S^{\prime}_{M^{\prime}},S}\right)c_{\lambda^{\prime}\downarrow}\left|{{h_{\lambda}},{M^{\prime}-\textstyle{1\over
2}}}\right\rangle+{\mathop{\rm sgn}}(\Delta
S)\Gamma_{\downarrow}\left({S^{\prime}_{M^{\prime}},S}\right)c_{\lambda^{\prime}\uparrow}\left|{{h_{\lambda}},{M^{\prime}+\textstyle{1\over
2}}}\right\rangle\\}$ (20)

where $\Delta S=S^{\prime}-S=\pm|\sigma|$, and the coefficients

$\Gamma_{\sigma}^{2}\left({S^{\prime}_{M^{\prime}},S}\right)=\frac{{S+\eta\left(\sigma\right){\mathop{\rm
sgn}}\left({\Delta S}\right)M^{\prime}+{\textstyle{1\over 2}}}}{{2S+1}}$ (21)

have the completenessn property for contributions from the identical spin
states of doped hole to the different high- and low-spin two-hole partners:

$\sum\limits_{\Delta
S=-\left|\sigma\right|}^{+\left|\sigma\right|}{\Gamma_{\sigma}^{2}\left({S^{\prime}_{M^{\prime}},S}\right)}=\sum\limits_{\sigma}{\Gamma_{\sigma}^{2}\left({S^{\prime}_{M^{\prime}},S}\right)=1},$
(22)

and also

$\sum\limits_{M=-S}^{S}{{\Gamma_{\sigma}^{2}\left({{{S^{\prime}}_{M^{\prime}}},S}\right)}}=S+\frac{1}{2}$
(23)

Taking into account relations (18), (19) and (23) we can determine the matrix
element in Eq.(5) by the sum:

$\displaystyle\left\langle({{N_{h0}}+1,{{M^{\prime}}_{S^{\prime}}}})_{\tau}\right|c_{\nu\sigma}\left|({{N_{h0}},{M_{S}}})_{\mu}\right\rangle=$
$\displaystyle\sum\limits_{\lambda,\lambda^{\prime},\lambda^{\prime\prime}}{\left\langle{{h_{\lambda^{\prime}}},{h_{\lambda^{\prime\prime}}},M^{\prime}_{S^{\prime}}\left|{c_{\nu\sigma}}\right|{h_{\lambda}},M_{S}}\right\rangle{\beta_{\mu}}\left({{h_{\lambda}}}\right){B_{\tau}}\left({{h_{\lambda^{\prime}}},{h_{\lambda^{\prime\prime}}}}\right)\Gamma_{\sigma}\left({S^{\prime}_{M^{\prime}},S}\right)}\times$
(24)
$\displaystyle\times\left\langle{\psi_{\tau}(\varphi)\psi_{\tau}(\theta)}\right|\left.{\psi_{\mu}(\varphi)\psi_{\mu}(\theta)}\right\rangle.$

After substituting Green’s function (1), the expressions (2) and (23) to the
Eq.(5) we obtain:

$N_{v}(x)=N_{v}^{12}+N_{v}^{23}={N^{12}_{s,v}}+3{N^{12}_{t,v}}+N_{v}^{23},$
(25)

where instead of the sum over root vectors ${{r}}$, we used the summation over
a physically meaningful indices $\tau$, $M$ and $\Delta S$ (i.e. the sum over
all low ($s$) - and high-spin ($t$) two-hole states). Here is

$\displaystyle N_{s(t),v}^{12}$ $\displaystyle=$
$\displaystyle\sum\limits_{\nu\sigma}{\sum\limits_{\tau}{{F^{s(t)}_{r=\left\\{{0,\tau}\right\\}}}\left(x\right)\sum\limits_{MM^{\prime}}{{{\left\\{{\sum\limits_{\lambda}{\Gamma_{\sigma}\left({S^{\prime}_{M^{\prime}},S}\right){\beta_{\mu=0}}\left({{h_{\lambda}}}\right)}{B_{\tau}}\left({{h_{\lambda}},{h_{\nu}}}\right)\delta\left({S^{\prime},S\pm|\sigma|}\right)\delta\left({M^{\prime},M+\sigma}\right)}\right\\}}^{2}}}}}\times$
(26)
$\displaystyle\times{\sum\limits_{\varphi\theta}\left\langle{\psi_{\tau}(\varphi)\psi_{\tau}(\theta)}\right|\left.{\psi_{\mu=0}(\varphi)\psi_{\mu=0}(\theta)}\right\rangle^{2}},$

where the $(+)$ and $(-)$ on the right side are used with indices $t$ and $s$
respectively, occupation factor in the PM phase:

${F^{s(t)}_{\left\\{{0,\tau}\right\\}}}(x)=\left\\{{\begin{array}[]{*{20}{c}}{\tfrac{1}{2}\left({1-\alpha_{s(t)}x}\right),\tau=0}\\\
{\tfrac{1}{2}\left({1-x}\right),\tau\neq 0}\\\ \end{array}}\right.,$ (27)

with $\alpha_{s(t)}=1-{2}/{(2S^{\prime}+1)}$ and $S^{\prime}=0,1$; $S=1/2$.
Let’s start with the contribution from the spin singlet $frs$ states
$N^{s}_{frs}$:

$N_{v}(x)=(2N_{\lambda}-1)-x(1-N^{s}_{frs})=N_{e}-x(1-N^{s}_{frs}),$ (28)

where the low and high spin contributions are

$N^{12}_{s,v}=(1/2)[(N_{\lambda}+1)(1-x)+2xN^{s}_{frs}]$ (29)

and

$N^{12}_{t,v}=(1/2)(N_{\lambda}-1)(1-x)$ (30)

respectively. In the static case,

$\displaystyle\sum\limits_{\varphi\theta}\left\langle{\psi_{\tau}(\varphi)\psi_{\tau}(\theta)}\right|\left.{\psi_{\mu}(\varphi)\psi_{\mu}(\theta)}\right\rangle^{2}$
$\displaystyle=$ (31)
$\displaystyle=\left\langle{\psi_{\tau}(\varphi_{U,D})\psi_{\tau}(\theta_{U,D})}\right|$
$\displaystyle\left.{\psi_{\mu}(\varphi_{U,D})\psi_{\mu}(\theta_{U,D})}\right\rangle^{2}=1,$

at the any $|\tau\rangle$ and $|\mu\rangle$ electron states because the
tilting $\varphi=\varphi_{U,D}$ and orientation $\theta=\theta_{U,D}$ angles
are fixed and associated with the minimum of the single adiabatic potential in
the U and D stripes. The relation $N_{v}^{23}=x(2N_{\lambda}-2)$ for the
contributions from the quasiparticle with root vectors from $\\{r_{23}\\}$
subspace is derived similarly to the previous expression for the contribution
(26). The number of possible singlet $frs$ states is in the range $0\leq
N^{s}_{frs}\leq 2$, where

$\displaystyle N_{frs}^{s}$ $\displaystyle=$ $\displaystyle
1-\sum\limits_{\lambda}{\beta_{0}^{2}\left({{h_{\lambda}}}\right)}\times$ (32)
$\displaystyle\sum\limits_{\lambda^{\prime},\lambda^{\prime\prime}}{\left[{1-{\delta_{\lambda\lambda^{\prime}}}-{\delta_{\lambda\lambda"}}}\right]B_{\tau=0}^{2}\left({{h_{\lambda^{\prime}}}{h_{\lambda^{\prime\prime}}}}\right)},$

and $\tau=0$ corresponds to the $frs$-quasiparticles. In deriving Eq.(28) we
also used the Eq.(23) and identity
$\sum\limits_{\lambda}{\beta_{\mu}^{2}\left({{h_{\lambda}}}\right)}\sum\limits_{\lambda^{\prime},\lambda^{\prime\prime}}{B_{\tau}^{2}\left({{h_{\lambda^{\prime}}}{h_{\lambda^{\prime\prime}}}}\right)}=1$
at any $\mu$ and $\tau$. Because the sum

$\sum\limits_{\tau}{\left[{{\beta_{0}}\left({{h_{\lambda}}}\right){B_{\tau}}\left({{h_{\lambda}},{h_{\nu}}}\right)}\right]\left[{{\beta_{0}}\left({{h_{\lambda^{\prime}}}}\right){B_{\tau}}\left({{h_{\lambda^{\prime}}},{h_{\nu}}}\right)}\right]}=0$
(33)

at the any $\nu$ and $\lambda\neq\lambda^{\prime}$, the contribution from
cross-term from (26) to the total number of the valence states is missing. In
a case of triplet nature of the $frs$ states, we obtain a similar expression
to (28) with the contribution

$N_{frs}^{t}=1-\sum\limits_{\lambda}{\beta_{0}^{2}\left({{h_{\lambda}}}\right)}\sum\limits_{\lambda^{\prime}\neq\lambda^{\prime\prime}\neq\lambda}{B_{\tau=0}^{2}\left({{h_{\lambda^{\prime}}}{h_{\lambda^{\prime\prime}}}}\right)},$
(34)

where $0\leq{N_{frs}^{t}}\leq 1$.

##  Appendix B  The number of frs states in doped Mott-Hubbard materials with
pseudo JT effect

To show a role of the charge inhomogeneity, let’s obtain the MI criterion in a
simpler three-orbital model ($\lambda=a,b,c$) of a semiconductor with two
spinless electrons and hole doping $N_{e}=2-x$. Here are
$\left|\mu\right\rangle=a_{\mu}\left|a\right\rangle+b_{\mu}\left|b\right\rangle+c_{\mu}\left|c\right\rangle$
and
$\left|\tau\right\rangle=\left({ab}\right)_{\tau}\left|a\right\rangle+\left({bc}\right)_{\tau}\left|{bc}\right\rangle+\left({ac}\right)_{\tau}\left|{ac}\right\rangle$,
where $\left|\mu\right\rangle$ and $\left|\tau\right\rangle$ are the states
with $\mu\left(\tau\right)=0-2$ in the sectors $N_{h0}$ and $N_{h0}+1$
respectively. The coefficients satisfy the completeness relations
$a_{\mu}^{2}+b_{\mu}^{2}+c_{\mu}^{2}=1$ and .
$\left({ab}\right)_{\tau}^{2}+\left({bc}\right)_{\tau}^{2}+\left({ac}\right)_{\tau}^{2}=1$.
In the homogeneous undoped case $\left({x=0}\right)$, the number of valence
states

$N_{v}=N_{v}^{12}=\sum\limits_{\tau}{\left\\{{1-a^{2}\left({bc}\right)_{\tau}^{2}-b^{2}\left({ac}\right)_{\tau}^{2}-c^{2}\left({ab}\right)_{\tau}^{2}}\right\\}}=2$
(35)

since
$\sum\limits_{\tau}{\left({bc}\right)_{\tau}^{2}}=\sum\limits_{\tau}{\left({ab}\right)_{\tau}^{2}}=\sum\limits_{\tau}{\left({ac}\right)_{\tau}^{2}}=1$
the material, according to the criterion, is a semiconductor, at $N_{e}=2$.
Let’s now, $x\neq 0$ , then

$N_{v}\left(x\right)=N_{v}^{12}+N_{v}^{23}=N_{frs}+\left({1-x}\right)\sum\limits_{\tau=1}^{2}\left\\{1-a^{2}_{0}\left({bc}\right)_{\tau}^{2}-b^{2}_{0}\left({ac}\right)_{\tau}^{2}-c^{2}_{0}\left({ab}\right)_{\tau}^{2}\right\\}+N_{v}^{23}=2-\left\\{{1-N_{frs}}\right\\}x,$
(36)

where
$N_{frs}=1-a^{2}_{0}\left({bc}\right)_{0}^{2}-b^{2}_{0}\left({ac}\right)_{0}^{2}-c^{2}_{0}\left({ac}\right)_{0}^{2}$,
and $N_{v}^{23}=x$.

Now let us introduce into this model the JT instability in the two-particle
sector, as is assumed for cuprates Gavrichkov and Polukeev (2022), where the
JT distortions are the angles $\varphi$ and $\theta$ of the tilting and
orientation of the CuO6 octahedra. In the dynamic state, all octahedra do not
have specific tilts $\varphi$ and orientations $\theta$, and all states
$\left|{\tau\left(\mu\right)}\right\rangle\left|{\psi_{\mu\left(\tau\right)}\left(\varphi\right)\psi_{\mu\left(\tau\right)}\left(\theta\right)}\right\rangle$
cannot be occupied with the factors $\left({1-x}\right)$ and $x$ in both
sectors $N_{h0}$ and $N_{h0}+1$ of the configuration space. Indeed, in this
case the charge inhomogeneity disappears along with the pseudo JT effect at
the concentration below the threshold $x<x_{c}$. We can choose the bifurcation
potential in Fig.3b, at which $x$ doped carriers still avoid U stripes, so
that the number of valence states at $x\neq 0$ is

$N_{v}\left(x\right)\approx
2-\left\\{{1-N_{frs}}\right\\}x\cdot\alpha\left({\delta\varphi,\delta\theta}\right),$
(37)

where

$\alpha\left({\delta\varphi,\delta\theta}\right)={\sum\limits_{\varphi_{D}\theta_{D}}{\sum\limits_{\varphi_{U}\theta_{U}}{\left\langle{\psi_{\mu=0}\left({\varphi_{D}}\right)\psi_{\mu=0}\left({\theta_{D}}\right)\left|{\psi_{\tau=0}\left({\varphi_{U}}\right)\psi_{\tau=0}\left({\theta_{U}}\right)}\right.}\right\rangle^{2}}}}\leq
1,$ (38)

and the number of valence states $N_{v}\left(x\right)$ decreases. Depending on
the magnitude of $\alpha\left({\delta\varphi,\delta\theta}\right)$, the
criterion can detect the ground state of doped JT material close to
insulating.

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