# Coupling strongly correlated electron systems to a tunable electronic
reservoir

S. Bag1,+, L. Fratino1, A. Camjayi2, M. Civelli1, M. Rozenberg1 1Université
Paris-Saclay, CNRS Laboratoire de Physique des Solides, 91405, Orsay, France.
2 Universidad de Buenos Aires, Ciclo Básico Común and CONICET - Universidad de
Buenos Aires, Instituto de Física de Buenos Aires (IFIBA), Buenos Aires,
Argentina.  +Department of Physics, Arizona State University, Tempe, AZ 85287,
USA

(August 28, 2024)

###### Abstract

We study the effect of coupling an electronic reservoir to a Hubbard model and
to a Dimer Hubbard Model. This is motivated by recent experiments on the
effect of illumination on the insulator-metal transition in a vanadium oxides
and photo-conductive cadmium sulfide heterostructure. We model the system as
an electronic reservoir hybridized to the correlated system. We assume that
the light intensity controls the hybridization coupling strength. We find that
the light intensity acts similarly as the temperature in the weak interaction
regime. This is consistent with the role played by electronic reservoirs in
out-of-equilibrium systems. In contrast, qualitative differences appear at
strong coupling. We show that modeling the V2O3 compound with a Hubbard model,
our results describe qualitatively well the observed illumination-driven
suppression of the insulator-metal transition. In contrast, in the DHM results
fail to capture the mild suppression observed in the case of VO2. This
indicates that the lattice may play an important role in this case.

##  I Introduction

Mott insulator materials are interesting for neuromorphic devices, such as
artificial spiking neuronsDel Valle et al. (2018); Stoliar et al. (2017);
Pickett et al. (2013). The several orders-of-magnitude changes of the
resistivity across the metal-to-insulator-transition (MIT) of Mott insulators
can be exploited in artificial neuromorphic devices to recreate the all-or-
nothing excitation of an action potential Stoliar et al. (2017); Pickett et
al. (2013); Luo et al. (2022); Qiu et al. (2023); Cheng et al. (2021).
Furthermore, there are several ways to control the Mott MIT, namely by
changing doping, pressure, electric field, etc. Kumar et al. (2013); Chen et
al. (2017); Yang et al. (2016); Schlom et al. (2014); Aetukuri et al. (2013);
Matsuda et al. (2020); Wu et al. (2011); Jeong et al. (2013); Wan et al.
(2018); Dong et al. (2018); Navarro et al. (2021); Ke et al. (2018); Singer et
al. (2018). A particularly interesting possibility, in the context of
neuromorphic applications, is to control the transition by illumination with
light in hybrid thin-film structures where one changes the coupling between
the Mott material and its environment Navarro et al. (2021, 2023); Adda et al.
(2022)

In recent years, the study of phenomena that emerge from the control of the
coupling between a system to a reservoir is attracting significant attention.
In experiments, changing the environment coupling can lead, for instance, to
line-width broadeningAman et al. (2016), decoherence, and a finite lifetime of
statesZeiher et al. (2016). Moreover, it can allow tailoring desired states,
like entangledKraus et al. (2008), antiferromagnetically (AFM)
orderedKaczmarczyk et al. (2016), Bose condensed Diehl et al. (2008), etc.
While this has been so far considered mostly in the context of cold atoms, it
is also potentially relevant for electronic properties of solid state systems,
including strongly correlated ones. For instance, an important case is that of
systems with metal-insulator transitions that are driven out of equilibrium.
In those cases, it has been recognized that coupling to an electronic
reservoir is an essential physical ingredient to achieve a steady state in a
static electric field-driven Hubbard Model (HM) Amaricci et al. (2012); Li et
al. (2015); Del Valle et al. (2021, 2019); Diener et al. (2018); Rocco et al.
(2022). Hence, it is an important question to consider the systematic effect
that the coupling to an electronic reservoir may have on a given strongly
correlated system.

In addition to the previous motivation, the effect of a (semi)metallic
reservoir coupled to a strongly correlated system is also relevant for hetero-
structures. Indeed, the ability to grow multi-layers including high quality
strongly correlated oxides, prompted the interest in understanding the fate of
the Mott metal-insulator transition in metal and semi-metal / Mott-insulator
hybrid systems Thiel et al. (2006).

For example, in the study of hetero-structures of a metal/AFM-Mott model, a
suppression of AFM structure factor is observed in the AFM-Mott layer due to
its proximity to the metalEuverte et al. (2012); Jiang et al. (2012). However,
it was also reported that the effect of an additional conduction band in the
periodic Anderson model can lead to stabilization of the AF orderHu et al.
(2017); Sen and Vidhyadhiraja (2016); Peters et al. (2013) in the system.
Furthermore, in the hetero-structure of a paramagnetic (PM) metal and a PM-
Mott insulator, the metal state may penetrate into the Mott insulator side
Helmes et al. (2008); Borghi et al. (2010); Zenia et al. (2009).

A particularly interesting type of Mott insulator materials are the vanadium
oxides VO2 and V2O3, which have temperature-driven MITs Imada et al. (1998).

In a recent experiment, the effects on the MIT was studied in the
heterostructure of vanadium oxide and the photoconductive semiconductor
Cadmium Sulfide (CdS), for various levels of illumination Navarro et al.
(2021). Interestingly, both VO2 and V2O3 showed a suppression of MIT with the
power of light illuminated on CdS. However, the modulation of the MITs in VO2
and V2O3 were dramatically different. The illumination quickly suppressed the
$T_{\rm MIT}$ of V2O3, driving it down toward zero temperature. While it only
had a minor suppression effect on $T_{\rm MIT}$ of VO2, shifting it down by
just a few degrees.

The present study is motivated by those experiments. We aim to understand the
suppression of $T_{\rm MIT}$ in Vanadium Oxides/CdS heterostructures in the
presence of light. To that goal, we need to model both the effect of the CdS
and also the MITs in V2O3 and VO2. In the case of V2O3, the MIT occurs between
an AFM insulator to a PM metallic state. In this initial study, for the sake
of simplicity, we shall follow the recent study Trastoy et al. (2020) to model
the transition. In that work, it was shown that the single-band HM treated
within DMFT was able to capture non-trivial features of the T driven MIT with
AFM symmetry breaking at half-filling, such as the anomalous enhancement of
the negative magneto-resistance. On the other hand, the MIT in the case of VO2
is qualitatively different, since it is non-magnetic. Moreover, the Mott
insulator phase of VO2 is monoclinic, where pairs of V atoms are dimerized,
presumably forming ”dynamical singlets” Biermann et al. (2005). Fortunately,
in this case we can also count on a simple model that captures basic non-
trivial features of the MIT, which we can adopt in this initial study. This is
the Dimer Hubbard Model (DHM), which is an extension of the Hubbard model
where each unit cell contains a dimer. It was recently shown that the DMFT
solution of the model exhibits a T-driven first order non-magnetic MIT at
half-filling, where the ground-state is a dimerized Mott insulator with
dynamical singlets Nájera et al. (2018); Stinson et al. (2018a); Ocampo
(2017); Nájera et al. (2017). Hence, in this initial study we shall adopt
these two simple models as zero-order approximation to describe the MITs of
V2O3 and VO2 in the presence of a tunable hybridization.

One of our main results is that the reservoir may qualitatively act as a
temperature and can may suppress the TMIT in agreement with the experimental
observations on V2O3. However, the comparison with observations in VO2 there
seems to remain some significant quantitative differences. We shall argue
later on that this may point to a relevant role played by the lattice degree
of freedom that are not included in our model Grandi et al. (2020).

This paper is organized as follows: In Section II we describe the model and
the numerical approaches. Then, in Section III we shall describe the results.
Firstly, we discuss the systematic effect of $T$ and $\Gamma$ on the AFM phase
and the MIT of the single band HM. Secondly, we shall describe the effects of
those parameters for the interaction-driven PM insulator-metal transition in
both the HM and the DHM. While the former is not directly relevant to the
physics of the vanadates, we find it useful to include that study as a
reference case, since it is the most widely studied MIT within DMFT. In the
last Section V, we discuss the comparison of our findings with experimental
results, including the possible origin of some standing discrepancies.

##  II Models and Method

Our task is to model the hybridization effect of CdS, which is a good
photoconductive semiconductorDeng and Li (2014); Cheng et al. (2018). In dark
it is highly insulating, i.e. more than the Mott insulator states in the
vanadium oxides. However, when illuminated by light it creates a substantial
amount of electron-hole pairs and, in consequence, shows a drop in the
resistivity. Nevertheless, CdS never has truly metallic character but remains
a semiconductor, i.e. does not have metallic conduction. Therefore, we propose
to model the CdS as a set of electronic reservoirs that that are locally
coupled with the V sites. In other words, we consider that the excited
electron-hole pairs are described as incoherent metallic states that locally
hybridize with the vanadium oxide’s. Since the vanadium oxides are thin films
of the order of 10 nm, we shall also assume that the electric conduction is
dominated by the physics of the interface, which we model as a layer of HM or
DHM hybridized with electronic reservoirs at each lattice site. Since
increasing the light intensity on the CdS one can tune the number of
(photo)carriers, we model this effect through the strength of the
hybridization parameter $\Gamma=\gamma^{2}\rho(E_{F})$. $\gamma$ is a
geometrical parameter independent of the light intensity as it describes the
hybridization hopping amplitude between the sites that represent the CdS and
the vanadium oxide layers, respectively. On the other hand, $\rho(E_{F})$
represents the density of states of the CdS, which increases with light
intensity. However, the detailed dependence of $\rho(E_{F})$ and the intensity
of illumination, that is experimentally controlled by the power on the LED
light, is not trivial. Therefore, here for the sake of definiteness and
simplicity, we assume that the control parameter $\Gamma$ (through the effect
of $\rho$) is proportional to the power of the LED.

###  II.1 Hubbard Model (HM)

The Hamiltonian of the single-band HM coupled with an incoherent electronic
reservoir reads as

$\displaystyle\hat{\mathcal{H}}=$ $\displaystyle-t\sum_{\langle
i,j\rangle,\sigma}(\hat{c}^{\dagger}_{i\sigma}\hat{c}_{j\sigma}+h.c)$
$\displaystyle+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}-\mu\sum_{i,\sigma}\hat{c}^{\dagger}_{i\sigma}\hat{c}_{i,\sigma}+\sum_{i,\sigma}\hat{\mathcal{H}}^{i\sigma}_{c},$
(1)

where $\hat{c}^{\dagger}_{i\sigma}$ and $\hat{c}_{i\sigma}$ are respectively
the fermionic creation and annihilation operators at the $i$-th site of the
lattice and $\hat{n}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{c}_{i\sigma}$
is the number operator. Here $t$ is the hopping amplitude of an electron
between nearest-neighbor sites, $U$ is the on-site repulsion energy if two
electrons (of opposite spins) occupy the site, and $\mu$ is the chemical
potential of the system, controlling filling. We fixed $\mu$ at $U/2$ for the
half-filled case. Last term of the Hamiltonian,
$\sum_{\sigma}\hat{\mathcal{H}}^{i\sigma}_{c}$, represents the coupling of an
electronic reservoir with $i$-th site of the lattice. Details about this
coupling term are given in the later part of this section. The top-left panel
of Fig. 1 displays a schematic representation of HM with the electronic
reservoir connected to each site of the lattice.

Figure 1: schematic experimental setup for CdS/VnO2n-1 heterostructures with
light on CdS where $n=2$ corresponds to CdS/V2O3 and $n=\infty$ corresponds to
CdS/VO2. Schematic representation of HM (top-right) and DHM (bottom-right)
with electronic reservoir connected to each site of the lattice. $t$ is the
hopping amplitude of an electron between nearest-neighbor sites. $\gamma$ is
the hopping amplitude of an electron between the site and the electronic
reservoir. $t_{\perp}$ is the hopping amplitude of an electron between dimer
sites.

We solve this model using DMFT with a semicircular density of states:
$D(\epsilon)=\frac{2}{\pi D^{2}}\sqrt{(D^{2}-\epsilon^{2})}$ with half band
width $D=1$. DMFT provides the exact solution of the modelGeorges et al.
(1996); Trastoy et al. (2020) on the Bethe lattice. We solve the impurity
problem of DMFT using Hybridization Expansion-Continuous-Time Quantum Monte
Carlo (HYB-CTQMC) Gull et al. (2011); Werner and Millis (2006); Haule (2007).
We solve the model both with and without AFM symmetry. We present the result
of the HYB-CTQMC in the main text.

###  II.2 Dimer Hubbard Model (DHM)

An interesting extension of single-band HM is the DHM Moeller et al. (1999).
This schematic model was shown to capture some key qualitative features of the
temperature-driven MIT transition of VO2 Nájera et al. (2017, 2018); Stinson
et al. (2018b). The DHM Hamiltonian, coupled with an incoherent electronic
reservoir reads,

$\displaystyle\hat{\mathcal{H}}$ $\displaystyle=-t\sum_{\langle
i,j\rangle,\sigma,\alpha}\left(\hat{c}^{\dagger}_{i\alpha\sigma}\,\hat{c}_{j\alpha\sigma}+\mathrm{h.c}\right)+$
$\displaystyle+t_{\perp}\sum_{i,\sigma}\left(\hat{c}^{\dagger}_{i1\sigma}\,\hat{c}_{i2\sigma}+\mathrm{h.c}\right)+$
$\displaystyle+U\sum_{i,\alpha}\hat{n}_{i\alpha\uparrow}\,\hat{n}_{i\alpha\downarrow}-\mu\sum_{i,\alpha,\sigma}\hat{n}_{i\alpha\sigma}+\sum_{i,\alpha,\sigma}\hat{\mathcal{H}}^{i\alpha\sigma}_{c},$
(2)

where the index $i$ and $j$ denote the lattice cells, $\alpha=1,2$ denotes the
dimer sites within a given cell and $\sigma$ labels the spin. The hopping $t$
and $t_{\perp}$ correspond to the amplitudes between nearest-neighbor lattice
cells and between dimer sites, respectively. The last term of the Hamiltonian,
$\sum_{\sigma}\hat{\mathcal{H}}^{i\sigma\alpha}_{c}$, represent the electronic
reservoir coupled to $i$-th lattice cell’s dimer site $\alpha$. Note that we
use an independent electronic reservoir for each site of the dimer as can be
seen in the schematic representation of the model in the bottom-right panel of
Fig. 1. Details about the electronic reservoir are given later in this
section. The local Green function of the lattice becomes diagonal in the Bond
($B$) and Anti-Bond ($AB$) basis, where the creation operator in the $B$ and
$AB$ basis is related to the site basis by the following equation:

$\displaystyle\hat{c}^{\dagger}_{iAB/B\sigma}=\frac{\hat{c}^{\dagger}_{i1\sigma}\pm\hat{c}^{\dagger}_{i2\sigma}}{\sqrt{2}}$
(3)

Therefore, we find it practical to solve the model on this basis.

###  II.3 Electronic Reservoirs

The Hamiltonian of the electronic reservoir coupled to each lattice site of
the models (HM or DHM) reads,

$\sum_{\sigma}\hat{\mathcal{H}}^{i\sigma}_{c}=\sum_{k,\sigma}\epsilon_{k}\hat{a}^{\dagger}_{ik\sigma}\hat{a}_{ik\sigma}+\gamma\sum_{l,\sigma}\left(\hat{a}^{\dagger}_{il\sigma}\hat{c}_{i\sigma}+h.c\right)$
(4)

where $\hat{c}_{i\sigma}$ denotes the fermionic operator at site $i$ for the
HM (or similarly, adding a label $\alpha$, at the $i$-th cell’s $\alpha$ site
for DHM) and $\hat{a}_{il\sigma}$ is the fermionic operator for reservoir’s
electron. $\epsilon_{l}$ are the reservoir’s electron energy levels. For
simplicity, we assume that the reservoir’s electrons hybridize with the
lattice fermions with constant amplitude $\gamma$. We can integrate out the
non-interacting reservoir’s electrons, and their effect appears as an
additional effective hybridization for the $c$ fermions at each impurity site,
which remains fixed in the DMFT calculation. This effective hybridization due
to the reservoir’s electrons reads as

$\Delta_{\Gamma}(\omega_{n})=\sum_{l}\frac{\gamma^{2}}{i\omega_{n}-\epsilon_{l}}=\int
d\epsilon\rho_{l}(\epsilon)\frac{\gamma^{2}}{i\omega_{n}-\epsilon}$ (5)

For simplicity, we adopt a semi-circular Density of States for the reservoir
electrons, $\rho_{l}(\epsilon)=\frac{2}{\pi D^{\prime 2}}\sqrt{(D^{\prime
2}-\epsilon^{2})}$, so the above integration takes the closed form

$\Delta_{\Gamma}(\omega_{n})=-\frac{2D^{\prime}i\Gamma}{\omega_{n}+sig(\omega_{n})\sqrt{\omega_{n}^{2}+D^{\prime
2}}}$ (6)

where $\Gamma=\gamma^{2}/D^{\prime}$ controls the strength of the effect of
hybridization with a reservoir. For convenience, we choose half the bandwidth
of the electronic reservoir $D^{\prime}=1.0$.

###  II.4 Details on the DMFT method

We describe here how the hybridization of the associated impurity problem is
modified by the presence of the reservoirs.

In the case of the HM we shall consider both solutions with and without AFM
symmetry. The effect of the reservoir within DMFT is most clearly observed
through the self-consistency condition of the associated quantum impurity
problem Georges et al. (1996). The total hybridization function reads,

$\Delta_{\sigma}(\omega)=t^{2}G_{loc,\bar{\sigma}}(\omega)+\Delta_{\Gamma}(\omega)$
(7)

where we observe that the reservoir is simply added to the quantum single site
environment. For further details on the DMFT method and self-consistent
equations for the AFM-HM see Refs.Georges et al., 1996; Trastoy et al., 2020.

In the case of the DHM, we consider the $AB/B$ basis, hence the total
hybridization entering the DMFT self-consistency equation becomes

$\Delta_{AB/B\sigma}(\omega)=t^{2}G_{loc,AB/B\sigma}(\omega)+\Delta_{\Gamma}(\omega)$
(8)

For further details on the DMFT self-consistent equations for the DHM see
Refs.Nájera et al., 2017, 2018; Fratino et al., 2022.

As model parameters, we adopt in the case of the HM $t=0.5$ (since we already
set $D=1$), and we fix onsite local repulsion $U=1.7$, which was found
adequate to describe the AFM-PM MIT in V2O3Trastoy et al. (2020). For the DHM
we also adopt $t=0.5$, and then fix $t_{\perp}=0.3$ and the onsite local
repulsion $U=2.2$, as where already shown to be adequate to qualitatively
describe the PM MIT in VO2Nájera et al. (2017, 2018). In addition, we shall
also consider for both models and for the sake of obtaining a full qualitative
picture of the behavior, a large value of $U$=4. This sets the systems well
into their Mott insulator states, so the MIT actually would become an
“insulator-insulator” transition. However, as we shall see, the presence of
the reservoirs will change this naif expectation.

The DMFT calculation with the HYB-CTQMC impurity solver produces data in
Matsubara Frequency. However, to obtain the behavior of the density of state
(DOS) and the DC resistivity, one requires real-frequency data. Therefore, we
use the maximum entropy method to get the analytical continued Green’s
functionHaule (2007); Jarrell and Gubernatis (1996). From the Green’s function
in real frequency, one can then calculate the local DOS
($\mathcal{A}(\omega)$), which reads

$\displaystyle\mathcal{A}(\omega)=-\frac{1}{\pi}Im\,G(\omega^{+})$ (9)

For the HM we shall show the total DOS, i.e. the average of the two spin
projections,
$\mathcal{A}_{HM}(\omega)=\frac{1}{2}\sum_{\sigma}\mathcal{A}_{\sigma}(\omega)$.
Whereas, for the DHM, the total DOS results from the average of spin and B/AB-
symmetry projections,
$\mathcal{A}_{DHM}(\omega)=\frac{1}{4}\sum_{\sigma}\left(\mathcal{A}_{B,\sigma}(\omega)+\mathcal{A}_{AB,\sigma}(\omega)\right)$.
Expressions for the conductivity of the HM and the DHM are given in the
supplementary material.

##  III Results and Discussion

###  III.1 Hubbard Model with antiferromagnetic symmetry

Figure 2: (a) and (b) the variation of the staggered magnetization as a
function of $T$ and $\Gamma$, respectively, for HM at $U=1.7$. (c) and (d) the
same at $U=4.0$. We observe that both thermal fluctuations and $\Gamma$ can
suppress the AFM order, seemingly playing a qualitatively similar role
($T$=0.01 in panels (b) and (d)).

We begin with the MIT in the single-band HM model case, which occurs in the
presence of antiferromagnetism. In the absence of an electronic bath, the low-
temperature state is an AFM insulator, and there is a $U$-dependent $T_{N}$
where magnetization goes to zero, and the gap closes, and one has the
$T$-driven MIT. The presence of an AFM state is signaled by the staggered
magnetization ($m_{s}=|n_{\uparrow}-n_{\downarrow}|$) order parameter.

Fig.2 shows how $m_{s}$ continuously goes to zero as a function of $T$ for
$U=1.7$ and for $U=4.0$, when $\Gamma$=0. Interestingly, similarly as the
effect of $T$, we observe that $\Gamma$ can also suppress the AFM ordering in
HM at both low and high $U$, when $T$ is fixed at a small value. One may be
tempted to conclude that $\Gamma$ behaves qualitatively as $T$. However, the
systematic results shown in Fig. 3 shows otherwise. They display the variation
of Neel’s temperature ($T_{N}$) and, similarly the $\Gamma_{\rm N}$ that
signals the critical $\Gamma$ value that drives the magnetization to zero, as
a function of $U$. The data reveals that the dependence is qualitatively
different. In the absence of reservoirs, $T_{N}$ has the well-known non-
monotonous dependence on $U$Georges et al. (1996); Bag et al. (2015); Fratino
et al. (2017). For small $U$, $T_{N}$ increases with increasing $U$, as the
AFM gap increases with both $m_{s}$ and $U$. While at large $U$, $T_{N}$
changes its behavior and decreases with increasing $U$, since the magnetic
exchange coupling decreases as $\sim t^{2}/U$. For the case of $\Gamma$ at low
$T$ we observe that for low values of interaction $U$, it seems to produce an
effect similar to $T$, as was expected from the results of Fig.2. However, in
stark contrast, we observe that at larger values of $U$, the critical
$\Gamma_{\rm N}$ increases monotonously with the interaction strength. This
means that the reservoir is no longer as effective in suppressing the
magnetism.

The physics driving the suppression of $m_{s}$ with increasing in $\Gamma$ is
qualitatively different from the thermal disorder induced by $T$. It can be
attributed to the Kondo screening of magnetic moment at each site of HM due to
the electronic reservoir. This can be most clearly seen if, for a moment, we
set the hopping $t$ of the HM to zero, and we obtain a collection of single
Anderson impurity models (AIM), at every site. The bath of the AIM is
controlled by the strength of $\Gamma$. Thus, the local moment of the lattice
sites are induced by $U$, but screened by $\Gamma$ through the Kondo effect.
Therefore, with increasing $U$, one needs a larger $\Gamma_{N}$ to screen
$m_{S}$ and suppress the magnetic order.

To gain a deeper understanding of the observed contrast between similar
behaviors of $\Gamma_{N}$ and $T_{N}$ at small $U$ but different at large $U$,
we study the evolution of the local DOS. We have selected specific points
across the $T-$ and $\Gamma-$driven transitions indicated by the color spots
in Fig.2. The panel (a) and (b) of Fig.4 displays the variation of the local
DOS with $T$ and $\Gamma$, respectively, for $U=1.7$. We recall that the DOS
looks symmetric because we are showing the total DOS, which is the sum of the
up and down spin components, since we are mostly interested in its qualitative
evolution. At small $T$, the HM has AFM order and, consequently, the DOS has a
gap at the Fermi energy. As one increases $T$, $m_{s}$ decreases, and the gap
gradually decreases and fills up. For large enough $T$, $T\geq 0.09$, $m_{s}$
vanishes and the DOS develops a prominent quasi-particle peak at Fermi energy.
This is the familiar Kondo-peak that develops in the associated impurity
problem Georges et al. (1996). A qualitatively similar behavior is observed in
the variation of DOS as one increases $\Gamma$ while keeping $T$ fixed.
However, some quantitative differences are apparent. For instance, the Hubbard
side-bands are not so well resolved as the system becomes metallic. This can
be understood from the fact that the metallic quasi-particle peak is the
manifestation of the Kondo peak in the associated impurity problem. In
contrast to the previous $\Gamma$=0 case, now, the finite value of $\Gamma$
implies that there is a non-renormalizable component of the impurity bath
which does not change under the self-consistency condition 7. This “wide”
hybridization increases as $\Gamma$ increases, driving the width of the Kondo
peak larger and larger. Hence, the quasi-particle peak is wider, taking
significant spectral weight from the Hubbard band components.

In contrast, at large $U=4.0$, the evolution of the local DOS with increasing
$T$ and $\Gamma$ differs qualitatively from the low $U$ case. The DOS shown in
the panel (a) of Fig. 5 hardly changes as $T$ is increased, even though the
system undergoes an AFM to PM transition Rozenberg et al. (1995); Camjayi et
al. (2006). In this case, therefore, instead of an insulator-to-metal
transition we have a insulator-to-insulator one. At this large $U$, the charge
degrees of freedom are frozen, which is signaled by the large Mott gap that
can be observed. Hence, only the spin degrees of freedom respond to increasing
$T$, which leads to the suppression of the (antiferro-)magnetic polarization,
but does not affect the charge gap in the DOS, as typical in the Heisenberg
regime.

In contrast, the evolution as a function of $\Gamma$ for low $T$=0.01 is
qualitatively different. We observe that in this case, the local DOS results
do reveal an insulator-metal transition. Nevertheless, this transition is also
qualitatively different from that of the lower $U$ case. The main feature that
emerges in the DOS upon the increase of $\Gamma$ is the emergence of a two in-
gap peak structure around zero frequency, within the Mott-Hubbard band gap. We
can understand these two features in terms of the associated impurity problem.
Since the state is AF, the spin symmetry is broken into a Néel type of order
into two sublattices. Thus, at a given lattice site, the one type of spin has
a larger occupation than the opposite, and vice-versa on the neighboring site.
Thus, one sublattice will have a, say, positive magnetic moment with a higher
occupation of the spin-up electrons, while the other sublattice will have the
opposite. Thus, each associated impurities, one corresponding at either
lattice, will realize a different state. Of course, there is up-down or
particle-hole symmetry relating those two states, due to the Néel order.

The crucial point is, nevertheless, that both impurities are in an environment
that is not fully gapped as before, but has low energy states due to the
finite $\Gamma$. This can be directly seen from the hybridization term
associated to $\Gamma$ in Eq.8. The consequence for this is that there will be
a small Kondo peak emerging near the Fermi energy in the solution of the
associated impurity. Since we have two inequivalent impurity sites, one for
each sublattice, they lead to the two small quasi-particle peaks that appear
in the local DOS in Fig.5. As we discussed before, the contribution to the
effective bath associated to $\Gamma$ does not renormalize and remains the
same for both spin projections and for both sublattices. As $\Gamma$ increases
we see that the strength of the Kondo peaks increases. Eventually, the
screening becomes so strong that the magnetic order collapses as $m_{s}\to 0$
at $\Gamma_{N}$. At that point, there is a sole quasiparticle peak in the
local DOS since the spin symmetry is restored, and the system has gone through
an insulator to strongly correlate metal transition.

Figure 3: (a) $T$-$U$ phase diagram for $\Gamma=0.0$ and (b) $\Gamma$-$U$
phase diagram for $T=0.01$ of the half-filled HM with AFM symmetry. M, I, and
B label Metal, Insulator, and Bad-metal, respectively. Dashed-lines
approximately denote the phase boundaries. Figure 4: Comparison of the
evolution of the total (i.e. spin averaged) local DOS as a function of $T$ at
fixed $\Gamma=0.0$ (a) and of $\Gamma$ at fixed $T=0.01$ (b), at interaction
$U$=1.7, for the AFM-HM across the AFM to PM. The line colors correspond to
the spots along the $m_{s}$ curve of Fig.2. Figure 5: Comparison of the
evolution of the total (i.e. spin averaged) local DOS as a function of $T$ at
fixed $\Gamma=0.0$ (a) and of $\Gamma$ at fixed $T=0.01$ (b), at interaction
$U$=4.0, for the AFM-HM across the AFM to PM. The line colors correspond to
the spots along the $m_{s}$ curve of Fig.2. Figure 6: (a) $T$-$U$ phase
diagram at $\Gamma=0.0$ (b) $\Gamma$-$U$ phase the diagram at $T=0.01$ of the
half-filled HM with PM symmetry. (c) $T$-$U$ phase diagram at $\Gamma=0.0$ (d)
$\Gamma$-$U$ Phase diagram at $T=0.005$ of the DHM. In the $T$-$U$
($\Gamma$-$U$) phase diagram, while increasing $U$ at a given $T$ ($\Gamma$),
the system goes from metal to insulator at $T_{MIT}$(U) ( $\Gamma_{MIT}$(U))
and while decreasing $U$ system goes from insulator to metal at $T_{IMT}$(U)
($\Gamma_{IMT}$(U)). The triangular regions denote the coexistence of
solutions which is consistent with the first order nature of the
experimentally observed transitions.

###  III.2 Hubbard and Dimer Hubbard models in the paramagnetic state

In previous sections, we have compared the effects of $T$ and $\Gamma$ on the
AFM-PM transition of the HM, which are insulator-to-metal at low $U$ and
insulator-to-insulator at large $U$. We observed that the effects of bath
coupling can be considered qualitatively similar at low interaction $U$, but
become substantially different at large $U$. We point out that both, the $T-$
and $\Gamma-$driven transitions, start out from an insulator state at all $U$,
which is gapped. While this first study was motivated by the experimental
results on the transition in the V2O3 heterostructure, it may be interesting
to complete the picture to consider the case of the effect of $\Gamma$ in the
well studied paramagnetic MIT in the Hubbard model as well Georges et al.
(1996). In fact, as in this transition the AF order is absent, the system
starts in a correlated metal without a gap. So in this section we shall
consider the interesting question of whether $\Gamma$ may also play an
analogous role as $T$ does, but this time within paramagnetic states. We shall
consider two paradigmatic strongly correlated MIT, that of the HM and the DHM
one.

####  III.2.1 Phase Diagrams

The MIT in the Hubbard model within the paramagnetic phase is by now a well
known a paradigm of a strongly correlated phenomena Georges et al. (1996). The
phase diagram in the $T-U$ plane shows a metallic region at low $U$ and low
$T$, a Mott insulator at high $U$ and low $T$, and a bad metal at higher $T$
in the intermediate $U$ regime. One of the most interesting features is the
existence of a coexistence region in between the correlated metal and the Mott
insulator, which gives a first order character to the transition. These well
known results are shown in the phase diagram at the panel (a) of Fig. 6.
Following a similar strategy as before, we compared the $T-$driven behavior
with that of $\Gamma$. The results are shown in the panel (b) of the same
figure. We strikingly observe that the behavior of the two parameters is
qualitatively similar and the main feature of a triangular coexistence region
is preserved. However, there is a qualitative difference, as it can be
observed that the tilting of the triangular region has changed.

To gain further insight we turned to the other basic model that we are
considering in the present work, the DHM, which may be relevant, as a first
approximation, to the experiments in VO2 hetero-structures. Indeed, the DHM
also presents a coexistence region in the $T-U$ phase diagram within the
paramagnetic phase Nájera et al. (2017, 2018); Stinson et al. (2018b). The
phase diagram obtained with CT-QMC simulations is shown in the panel(c) of
Fig. 6. We observe in this case that the triangular region is tilted to the
right, which is a common feature of cluster DMFT models, and the DHM can be
considered as the simplest of those ones Nájera et al. (2017). We explored by
numerical simulations the fate of the triangular region at the lowest $T$ and
increasing $\Gamma$. The obtained phase diagram is shown in the panel (d) of
the figure. Once again, we observe the feature that the coupling to the
electronic reservoir seems to play a similar role as the $T$, as now even the
triangular coexistence region shows a similar tilt.

Figure 7: Side-by-side comparison of the evolution of DOS as a function of $T$
with fixed $\Gamma=0.0$(a, c and e) and $\Gamma$ with fixed $T$(b, d and f) in
weak interaction region for PM-HM and PM-DHM. (a and b) and (c and d) are for
PM-HM and PM-DHM at $U$=1.7, respectively. (e) and (f) is for PM-DHM at $U$ =
2.2. foo

####  III.2.2 Density of States

To understand further the effect of $T$ and $\Gamma$ at small $U$ and at large
$U$, we study the systematic changes of DOS for the two models at small and
large $U$. We begin with the results at low $U$ which are obtained by CT-QMC
plus maximal entropy analytic continuation. In Fig.7 we show the DOS at the
small $U$=1.7 (i.e metallic side of the transition) for the HM and DHM.
Interestingly, we observe that increasing both, $T$ and $\Gamma$ have a
qualitatively similar effect in the DOS. They can both control the intensity
of the quasiparticle peak, however, increasing T provokes a more significant
transfer of weight toward the Hubbard bands with respect to $\Gamma$.

Figure 8: (Side-by-side comparison of the evolution of DOS as a function of
$T$ (a and c) and $\Gamma$ (b and d) at strong interaction $U=4.0$ for HM and
DHM. (a) and (b) DOS at various $T$ for fixed $\Gamma=0.0$ and at various
$\Gamma$ for fixed $T$, respectively, of HM without AFM symmetry. (c) and (d)
shows DOS of DHM at $U$ = 4.0 at various $T$ with fixed $\Gamma=0.0$ and at
various $\Gamma$ for fixed $T$=0.05 respectively.

We also explore a stronger correlated state for the DHM, namely at $U$=2.2
within the coexistence region, which is considered relevant for the VO2
compound Nájera et al. (2017). In this case, we start from the insulator state
within the coexistence, which as a function of $T$ exhibits a first-order
insulator-to-metal transition in qualitative agreement with the mentioned
compound. The results for the evolution of the DOS are shown in the panel (e)
of Fig. 7, where we observe the metallization of the correlated insulator. In
the panel (f), we report the behavior as a function of $\Gamma$ for the same
starting state. Quite strikingly, we observe a similar behavior. The gap is
filled and a prominent quasi-particle peak emerges at the Fermi energy,
reminiscent of Kondo physics. Nevertheless, we should also mention that a
relatively small difference seems to be present in the evolution of the
respective spectra, since at intermediate values of $\Gamma$ we observe two
small quasi-particle peaks at the inner edges of the Hubbard bands, which are
not present (or very subtle) in the $T$-driven case. Another difference is
that in the correlated metal state driven by $\Gamma$ the Hubbard bands have
relative smaller spectral intensity as compared to the $T$-driven transition,
which is similar to the effect noted at smaller $U$.

For completeness, we also considered the large $U$ region, where both systems
are deep in the Mott insulator phase. We adopt the relatively large value
$U$=4.0. The results are shown in Fig.8, where we track changes in the DOS for
the two models (HM and DHM) as a function of $T$ and $\Gamma$. With increasing
temperature at $\Gamma$=0 on the left hand side we observe that the DOS hardly
changes for both, HM and DHM. Which is natural since the excitation gap at
$U$=4 is much larger than $T$. However, in striking contrast, the respective
DOSs at low $T$ and increasing $\Gamma$, shown on the right hand of the
figure, develop a prominent quasi-particle peak at the Fermi energy.

The interpretation of these behavior also is rooted in the Kondo effect.
Indeed, the large value of $U$ create a strong local magnetic model, which at
$\Gamma$=0 is essentially an uncoupled spin, which is a peculiarity of the
Mott paramagnet Georges et al. (1996). When the coupling with the electronic
bath is switched on, the bath electrons immediately screen the local spins
forming Kondo resonances. We may note that a emergence of a Kondo peak in a
large Mott gap was reported in systems with a narrow correlated band coupled
with a wide conduction band de’ Medici et al. (2005) and in the hetero-
structure of a metal coupled to a Mott insulatorHelmes et al. (2008); Borghi
et al. (2010).

Comparing Figs. 7 and 8, and consistent with the Kondo origin of the
phenomenon, we observe that the quasi-particle-peak weight increases with
$\Gamma$ at constant $U$, but decreases with increasing $U$ at constant
$\Gamma$. At large $U$, both the HM and the DHM can be driven across an MIT by
the intensity of the coupling to the reservoir, $\Gamma$, but not by $T$. We
may also point out that at large $U$ we also observe in the DOS of the DHM the
two small edge quasiparticle peaks that develop as the gap closes.

Figure 9:  Comparing temperature-dependent experimental resistivity with
theoretically calculated resistivity for various light intensities (P) on
CdS/V2O3 and CdS/VO2. (top-left) and (bottom-left) experimental resistivity
($\rho$) for CdS/V2O3 and CdS/VO2, respectivelyNavarro et al. (2021). (top-
right) and (bottom-right) variation of $\rho$ of HM-AFM at $U$ = 1.7 and DHM
at $U$=2.2, respectively, as a function of $T$ for various values of $\Gamma$.
We scale the $T$ with the $T_{MIT}$ of CdS/V2O3 with no light on CdS, which is
148 K.

From the systematic study of the model done so far, with and without magnetic
order and at low and large values of $U$, we observe that there is an emergent
pattern. Namely, we can condense our results noting that to zero order, the
effect of $T$ and $\Gamma$ are qualitatively similar when the interaction $U$
is relatively low. One may be tempted to say that this is in the metallic
state and the reason is that the local coupling to the incoherent electron
bath plays an analogous role as thermal scattering. This is qualitatively
true, however, the argument has even more general validity, since in the AFM
insulator at low $U$, both $T$ and $\Gamma$ also have a qualitatively similar
effect. Interestingly, in a recent paper Foulquier, Paul et al. (2023) the
qualitative difference of the behavior of the AFM Hubbard model at low and
large $U$ was investigated. The results showed that in the former case the
system is described by a Slater AFM insulator where the bands are splitted by
a gap but retain their coherent (i.e. quasiparticle) character. In contrast,
at large $U$ the system is a Heisenberg AFM insulator where the (Hubbard)
bands are fully incoherent. Hence, in the low $U$ AFM the coupling with the
local electron reservoirs disturb the coherent character of the quasi-particle
propagation, pretty much as $T$ does.

For completeness, we have constructed a full 3D $T-U-\Gamma$ phase diagram of
the HM and DHM where the global behavior in parameter space may be appreciated
and is presented in the Supplementary Material.

##  IV Comparison with experiments

We now compare our results with the experimentally observed effect of light
illumination on the MIT of vanadate thin films with a deposited layer of
photoconductor CDS. The panel (a) of Fig. 9 shows the variation of resistivity
($\rho$) of CdS/V2O3 as a function of $T$ for various values of illumination
intensity $P$ (that we qualitatively identify with $\Gamma$ see Section II).
The system shows a sharp resistance change across the insulating to metal
transition. Significantly, we also note that the transition temperature
decreases with increasing illumination intensity. The system’s $T$MIT is
driven down to zero temperature for high enough light intensity. We may
favourably compare those experimental results with our calculations of the
variation of resistivity $\rho(T)$ in the AFM-HM coupled with the electronic
reservoir at $U$=1.7 for various values of $\Gamma$, that we show in the panel
(b) of Fig. 9. For better comparison, we scaled the $T$ with Néel temperature
($T_{N}$) at $\Gamma$=0, which is 0.09. The resistivity of our model
calculation also shows a sharp decrease of resistivity across MIT, which is
associated with the AFM to PM transition. Importantly, one can also note that
transition temperature decreases with increasing $\Gamma$. Thus, AFM Hubbard
model coupled with an electronic reservoir can qualitatively capture the
suppression of $T_{MIT}$ observed in CdS/V2O3. While the simple one band
Hubbard model may seem a simplistic model of V2O3, and in many aspects it is,
it is worthy to note that the same model for the same value of the parameter
$U$ was adopted to successfully account for non-trivial large negative
magneto-resistance effects observed in thin films of the same compound Trastoy
et al. (2020)

We now turn to the experiments done on VO2. The panel (c) of Fig.9 shows the
experimental resistivity of CdS/VO2 for various values of illumination
intensity. The system shows a sharp insulating to metal transition as a
function of temperature with sharp resistance change at $T_{MIT}$. Similarly
as before, we scale the $T$ with the $T_{MIT}$=308K of CdS/VO2 with no light
on CdS. Rather surprisingly, there is also a reduction in $T_{MIT}$ upon
illumination as seen in CdS/V2O3, but the magnitude of the effect for the same
light intensity is much less sensitive. We turn to the DHM and set the
parameter as were adopted in Ref. Nájera et al. (2017); Stinson et al.
(2018b), namely $t_{\perp}$= 0.3 and $U$=2.2 (near the coexistence region) at
half-filling ($\mu$ = $U$/2) where that model successfully accounted for some
experimentally observed features in VO2. The panel (d) of Fig. 9 shows the
calculated resistivity $\rho(T)$ of the DHM coupled with the electronic
reservoir for various values of $\Gamma$. For better comparison, we scaled $T$
with the critical temperature $T_{c}$=0.03 at $\Gamma=0.0$. One can note that
transition temperature decreases with increasing $\Gamma$ as observed for
CdS/VO2 and in the previous case. However, the effect seems too strong as
compared to the experimental findings. We have tried varying different
parameters in different regions of the phase diagram, but did not find any
better agreement.

Since the comparably simple Hubbard model has accounted for V2O3 and in view
of past success of the DHM model to account for aspects of VO2, we should try
to speculate on the origin of the discrepancy. One natural missing ingredient
in the model may be that the structural transition may play a significant
role. VO2 has a larger structural transition that is concurrent with the MIT
than V2O3, so its relative effect may be more significant Budai et al. (2014).
In fact, one may argue that since a structural transition can be understood as
due to the condensation of phonon fluctuations, therefore a purely electronic
reservoir may not be effective in driving the system sufficiently. Therefore
the effect is still present but in a much smaller magnitude as the MIT is
always arrested by the inability of the CdS to promote phononic excitations.

##  V Conclusion

We study the effect of the photoconductor CdS on two strongly correlated
vanadate thin films. The theoretical modeling that we introduced consists of a
collection of incoherent electronic reservoirs coupled at every lattice site
of a Hubbard and a Dimer Hubbard model. We note that besides the motivation to
account for the experimental observations, these models are also interesting
from a fundamental physics point. Indeed, the coupling with electronic
reservoirs is a required feature in order to describe electrons on lattices,
driven out-of-equilibrium by external electric fields. The currents that
develop produce heat that need to be evacuated via reservoirs to allow the
system to thermally equilibrate Díaz et al. (2023).

The coupling strength of those models with the electronic reservoirs is
characterized by a parameter $\Gamma$, which we argued may play the similar
role as the illumination power. These models are solved using DMFT with HYB-
CTQMC as an impurity solvers.

From the systematic study we observed that $\Gamma$ and $T$ played a
qualitatively similar role, so long the underlying electronic structure could
be described with quasi-particles, which is at low to moderate $U$, regardless
whether the system is a metal or an insulator. We trace that to the fact that
the local coupling with the reservoir produces a source of electronic
scattering, similarly as the $T$ does.

This observation provided a natural explanation to the dramatic suppression of
the transition temperature in V2O3, where the insulator state is due to the
opening of an antiferromagnetic gap, i.e. of electronic origin. Hence the
electronic reservoir was efficient to suppress it. In contrast, the agreement
was not so successful for the VO2 experiments, which show a significant
smaller effect of reduction in the transition temperature. We argue that this
points to a prominent role of the structural transition, which is couple to
lattice vibration degrees of freedom Grandi et al. (2020), hence the
electronic reservoir is less efficient.

It will be interesting in future work to consider the extension of our hybrid
model to other strongly correlated models, such as the double exchange model
and the Periodic Anderson model, which are relevant for other material
compounds and may open the way to new forms of control of strongly correlated
phenomena.

##  VI Acknowledgements

We acknowledge support from the French ANR “MoMA” project ANR-19-CE30-0020. LF
is supported by the Quantum Materials for Energy Efficient Neuromorphic
Computing, an Energy Frontier Research Center funded by the US Department of
Energy, Office of Science, Basic Energy Sciences under Award DE-SC0019273. MC
is supported by French ANR “Neptun” ANR-19-CE30-0019-01. We thank H. Navarro
of UCSD for providing the figure with experimental data and for useful
discussions.

##  Appendix A DC conductivity

The Drude conductivity of HM on Bethe lattice was calculated using Eqn.

$\displaystyle\Re\sigma_{HM}$ $\displaystyle(\omega=0)=4\int
d\epsilon\rho(\epsilon)\int d\omega\left(-\frac{\partial
f(\omega)}{\partial\omega}\right)_{\omega=0}\times$ (10)
$\displaystyle\left(\frac{D^{2}-\epsilon^{2}}{3}\right)\left(\mathcal{A}_{\uparrow}(\epsilon,\omega)\mathcal{A}_{\downarrow}(\epsilon,\omega)+\mathcal{A}^{2}_{off}(\epsilon,\omega)\right)$

where
$\mathcal{A}_{\sigma}(\epsilon,\omega)=\frac{-1}{\pi}Im\frac{z_{\bar{\sigma}}(\omega)}{z_{\bar{\sigma}}(\omega)z_{\sigma}(\omega)-\epsilon^{2}}$
and
$\mathcal{A}_{off}(\epsilon,\omega)=\frac{-1}{\pi}Im\frac{\epsilon}{z_{\bar{\sigma}}(\omega)z_{\sigma}(\omega)-\epsilon^{2}}$
with
$z_{\sigma}(\omega)=\omega+\mu-\Sigma_{\sigma}(\omega)-\Delta_{\Gamma}(\omega)$.
$\frac{D^{2}-\epsilon^{2}}{3}$ is the velocity of electron of energy
$\epsilon$ in Bethe lattice Chung and Freericks (1998).

The DC conductivity of DHM on Bethe lattice was calculated using Eqn.Ocampo
(2017)

$\displaystyle\Re\sigma_{DHM}$ $\displaystyle(\omega=0)=2\int
d\epsilon\rho(\epsilon)\int d\omega\left(-\frac{\partial
f(\omega)}{\partial\omega}\right)_{\omega=0}\times$ (11)
$\displaystyle\left(\frac{D^{2}-\epsilon^{2}}{3}\right)\left(\mathcal{A}^{2}_{AB}(\epsilon,\omega)+\mathcal{A}^{2}_{B}(\epsilon,\omega)\right)$

Where $\mathcal{A}^{2}_{AB}(\epsilon,\omega)$ and
$\mathcal{A}^{2}_{B}(\epsilon,\omega)$ are DOS of Anti-bond and Bond
respectively.

##  Appendix B Metal-Insulator Coexistence regions

We display here the metal-insulator coexistence region in the full
$T-U-\Gamma$ phase diagram. Notice the difference in the tilting between the
Hubbard (HM) and Dimer Hubbard (DHM) models by cutting $T-U$ planes.

Figure 10: $T$-$\Gamma$-$U$ phase diagram of HM at half filling. The
coexistence region (colored region) in the $T-U$ plane of the phase diagram
shifted toward the right as one increases $\Gamma$ and Tc decreases with
increasing $\Gamma$ Figure 11:  $T$-$\Gamma$-$U$ phase diagram of DHM at half
filling. The coexistence region (colored region) in the $T-U$ plane of the
phase diagram shifted toward the right as one increases $\Gamma$ and Tc
decreases with increasing $\Gamma$

,

## References

  * Del Valle et al. [2018] Javier Del Valle, Juan Gabriel Ramírez, Marcelo J Rozenberg, and Ivan K Schuller.  Challenges in materials and devices for resistive-switching-based neuromorphic computing.  _Journal of Applied Physics_ , 124(21), 2018.
  * Stoliar et al. [2017] Pablo Stoliar, Julien Tranchant, Benoit Corraze, Etienne Janod, Marie-Paule Besland, Federico Tesler, Marcelo Rozenberg, and Laurent Cario.  A leaky-integrate-and-fire neuron analog realized with a mott insulator.  _Advanced Functional Materials_ , 27(11):1604740, 2017.
  * Pickett et al. [2013] Matthew D Pickett, Gilberto Medeiros-Ribeiro, and R Stanley Williams.  A scalable neuristor built with mott memristors.  _Nature materials_ , 12(2):114–117, 2013.
  * Luo et al. [2022] Renjie Luo, Xuanhan Zhao, Liyang Chen, Tanner J Legvold, Henry Navarro, Ivan K Schuller, and Douglas Natelson.  Spin seebeck effect at low temperatures in the nominally paramagnetic insulating state of vanadium dioxide.  _Applied Physics Letters_ , 121(10):102404, 2022\.
  * Qiu et al. [2023] Erbin Qiu, Pavel Salev, Lorenzo Fratino, Rodolfo Rocco, Henry Navarro, Coline Adda, Junjie Li, Min-Han Lee, Yoav Kalcheim, Marcelo Rozenberg, et al.  Stochasticity in the synchronization of strongly coupled spiking oscillators.  _Applied Physics Letters_ , 122(9), 2023.
  * Cheng et al. [2021] Shaobo Cheng, Min-Han Lee, Richard Tran, Yin Shi, Xing Li, Henry Navarro, Coline Adda, Qingping Meng, Long-Qing Chen, RC Dynes, et al.  Inherent stochasticity during insulator–metal transition in vo2.  _Proceedings of the National Academy of Sciences_ , 118(37):e2105895118, 2021.
  * Kumar et al. [2013] Suhas Kumar, Matthew D Pickett, John Paul Strachan, Gary Gibson, Yoshio Nishi, and R Stanley Williams.  Local temperature redistribution and structural transition during joule-heating-driven conductance switching in vo2.  _Advanced Materials_ , 25(42):6128–6132, 2013\.
  * Chen et al. [2017] Shi Chen, Zhaowu Wang, Lele Fan, Yuliang Chen, Hui Ren, Heng Ji, Douglas Natelson, Yingying Huang, Jun Jiang, and Chongwen Zou.  Sequential insulator-metal-insulator phase transitions of vo2 triggered by hydrogen doping.  _Physical Review B_ , 96(12):125130, 2017.
  * Yang et al. [2016] Mengmeng Yang, Yuanjun Yang, Bin Hong, Liangxin Wang, Kai Hu, Yongqi Dong, Han Xu, Haoliang Huang, Jiangtao Zhao, Haiping Chen, et al.  Suppression of structural phase transition in vo2 by epitaxial strain in vicinity of metal-insulator transition.  _Scientific reports_ , 6(1):1–10, 2016.
  * Schlom et al. [2014] Darrell G Schlom, Long-Qing Chen, Craig J Fennie, Venkatraman Gopalan, David A Muller, Xiaoqing Pan, Ramamoorthy Ramesh, and Reinhard Uecker.  Elastic strain engineering of ferroic oxides.  _Mrs Bulletin_ , 39(2):118–130, 2014.
  * Aetukuri et al. [2013] Nagaphani B Aetukuri, Alexander X Gray, Marc Drouard, Matteo Cossale, Li Gao, Alexander H Reid, Roopali Kukreja, Hendrik Ohldag, Catherine A Jenkins, Elke Arenholz, et al.  Control of the metal–insulator transition in vanadium dioxide by modifying orbital occupancy.  _Nature Physics_ , 9(10):661–666, 2013.
  * Matsuda et al. [2020] Yasuhiro H Matsuda, Daisuke Nakamura, Akihiko Ikeda, Shojiro Takeyama, Yuki Suga, Hayato Nakahara, and Yuji Muraoka.  Magnetic-field-induced insulator–metal transition in w-doped vo2 at 500 t.  _Nature communications_ , 11(1):1–7, 2020.
  * Wu et al. [2011] B Wu, A Zimmers, H Aubin, R Ghosh, Y Liu, and R Lopez.  Electric-field-driven phase transition in vanadium dioxide.  _Physical Review B_ , 84(24):241410, 2011.
  * Jeong et al. [2013] Jaewoo Jeong, Nagaphani Aetukuri, Tanja Graf, Thomas D Schladt, Mahesh G Samant, and Stuart SP Parkin.  Suppression of metal-insulator transition in vo2 by electric field–induced oxygen vacancy formation.  _Science_ , 339(6126):1402–1405, 2013.
  * Wan et al. [2018] Chenghao Wan, Erik H Horak, Jonathan King, Jad Salman, Zhen Zhang, You Zhou, Patrick Roney, Bradley Gundlach, Shriram Ramanathan, Randall H Goldsmith, et al.  Limiting optical diodes enabled by the phase transition of vanadium dioxide.  _ACS Photonics_ , 5(7):2688–2692, 2018.
  * Dong et al. [2018] Kaichen Dong, Sukjoon Hong, Yang Deng, He Ma, Jiachen Li, Xi Wang, Junyeob Yeo, Letian Wang, Shuai Lou, Kyle B Tom, et al.  A lithography-free and field-programmable photonic metacanvas.  _Advanced Materials_ , 30(5):1703878, 2018.
  * Navarro et al. [2021] Henry Navarro, Javier del Valle, Yoav Kalcheim, Nicolas M Vargas, Coline Adda, M-H Lee, Pavel Lapa, Alberto Rivera-Calzada, IA Zaluzhnyy, Erbin Qiu, et al.  A hybrid optoelectronic mott insulator.  _Applied Physics Letters_ , 118(14):141901, 2021\.
  * Ke et al. [2018] Yujie Ke, Shancheng Wang, Guowei Liu, Ming Li, Timothy J White, and Yi Long.  Vanadium dioxide: The multistimuli responsive material and its applications.  _Small_ , 14(39):1802025, 2018.
  * Singer et al. [2018] Andrej Singer, Juan Gabriel Ramirez, Ilya Valmianski, Devin Cela, Nelson Hua, Roopali Kukreja, James Wingert, Olesya Kovalchuk, James M Glownia, Marcin Sikorski, et al.  Nonequilibrium phase precursors during a photoexcited insulator-to-metal transition in v2o3.  _Physical review letters_ , 120(20):207601, 2018\.
  * Navarro et al. [2023] Henry Navarro, Ali C Basaran, Fernando Ajejas, Lorenzo Fratino, Soumen Bag, Tianxing D Wang, Erbin Qiu, Victor Rouco, Isabel Tenreiro, Felipe Torres, et al.  Light-induced decoupling of electronic and magnetic properties in manganites.  _Physical Review Applied_ , 19(4):044077, 2023\.
  * Adda et al. [2022] C Adda, H Navarro, J Kaur, M-H Lee, C Chen, M Rozenberg, SP Ong, and Ivan K Schuller.  An optoelectronic heterostructure for neuromorphic computing: Cds/v3o5.  _Applied Physics Letters_ , 121(4):041901, 2022\.
  * Aman et al. [2016] JA Aman, Brian Joseph DeSalvo, FB Dunning, TC Killian, S Yoshida, and J Burgdörfer.  Trap losses induced by near-resonant rydberg dressing of cold atomic gases.  _Physical Review A_ , 93(4):043425, 2016.
  * Zeiher et al. [2016] Johannes Zeiher, Rick Van Bijnen, Peter Schauß, Sebastian Hild, Jae-yoon Choi, Thomas Pohl, Immanuel Bloch, and Christian Gross.  Many-body interferometry of a rydberg-dressed spin lattice.  _Nature Physics_ , 12(12):1095–1099, 2016.
  * Kraus et al. [2008] B. Kraus, H. P. Büchler, S. Diehl, A. Kantian, A. Micheli, and P. Zoller.  Preparation of entangled states by quantum markov processes.  _Phys. Rev. A_ , 78:042307, Oct 2008.
  * Kaczmarczyk et al. [2016] J Kaczmarczyk, H Weimer, and M Lemeshko.  Dissipative preparation of antiferromagnetic order in the fermi-hubbard model.  _New Journal of Physics_ , 18(9):093042, sep 2016\.
  * Diehl et al. [2008] Sebastian Diehl, A Micheli, Adrian Kantian, B Kraus, HP Büchler, and Peter Zoller.  Quantum states and phases in driven open quantum systems with cold atoms.  _Nature Physics_ , 4(11):878–883, 2008.
  * Amaricci et al. [2012] A. Amaricci, C. Weber, M. Capone, and G. Kotliar.  Approach to a stationary state in a driven hubbard model coupled to a thermostat.  _Phys. Rev. B_ , 86:085110, Aug 2012.
  * Li et al. [2015] Jiajun Li, Camille Aron, Gabriel Kotliar, and Jong E. Han.  Electric-field-driven resistive switching in the dissipative hubbard model.  _Phys. Rev. Lett._ , 114:226403, Jun 2015.
  * Del Valle et al. [2021] Javier Del Valle, Nicolas M Vargas, Rodolfo Rocco, Pavel Salev, Yoav Kalcheim, Pavel N Lapa, Coline Adda, Min-Han Lee, Paul Y Wang, Lorenzo Fratino, et al.  Spatiotemporal characterization of the field-induced insulator-to-metal transition.  _Science_ , 373(6557):907–911, 2021.
  * Del Valle et al. [2019] Javier Del Valle, Pavel Salev, Federico Tesler, Nicolás M Vargas, Yoav Kalcheim, Paul Wang, Juan Trastoy, Min-Han Lee, George Kassabian, Juan Gabriel Ramírez, et al.  Subthreshold firing in mott nanodevices.  _Nature_ , 569(7756):388–392, 2019.
  * Diener et al. [2018] Pascale Diener, Etienne Janod, B Corraze, M Querré, C Adda, Maryline Guilloux-Viry, Stéphane Cordier, Alberto Camjayi, M Rozenberg, MP Besland, et al.  How a dc electric field drives mott insulators out of equilibrium.  _Physical Review Letters_ , 121(1):016601, 2018\.
  * Rocco et al. [2022] Rodolfo Rocco, Javier del Valle, Henry Navarro, Pavel Salev, Ivan K. Schuller, and Marcelo Rozenberg.  Exponential escape rate of filamentary incubation in mott spiking neurons.  _Phys. Rev. Applied_ , 17:024028, Feb 2022.
  * Thiel et al. [2006] Stefan Thiel, German Hammerl, Andreas Schmehl, Christof W Schneider, and Jochen Mannhart.  Tunable quasi-two-dimensional electron gases in oxide heterostructures.  _Science_ , 313(5795):1942–1945, 2006.
  * Euverte et al. [2012] A. Euverte, F. Hébert, S. Chiesa, R. T. Scalettar, and G. G. Batrouni.  Kondo screening and magnetism at interfaces.  _Phys. Rev. Lett._ , 108:246401, Jun 2012.
  * Jiang et al. [2012] M. Jiang, G. G. Batrouni, and R. T. Scalettar.  Density of states and magnetic correlations at a metal-mott insulator interface.  _Phys. Rev. B_ , 86:195117, Nov 2012.
  * Hu et al. [2017] Wenjian Hu, Richard T. Scalettar, Edwin W. Huang, and Brian Moritz.  Effects of an additional conduction band on the singlet-antiferromagnet competition in the periodic anderson model.  _Phys. Rev. B_ , 95:235122, Jun 2017.
  * Sen and Vidhyadhiraja [2016] Sudeshna Sen and N. S. Vidhyadhiraja.  Quantum critical mott transitions in a bilayer kondo insulator-metal model system.  _Phys. Rev. B_ , 93:155136, Apr 2016.
  * Peters et al. [2013] Robert Peters, Yasuhiro Tada, and Norio Kawakami.  Kondo effect in $f$-electron superlattices.  _Phys. Rev. B_ , 88:155134, Oct 2013.
  * Helmes et al. [2008] R. W. Helmes, T. A. Costi, and A. Rosch.  Kondo proximity effect: How does a metal penetrate into a mott insulator?  _Phys. Rev. Lett._ , 101:066802, Aug 2008.
  * Borghi et al. [2010] Giovanni Borghi, Michele Fabrizio, and Erio Tosatti.  Strongly correlated metal interfaces in the gutzwiller approximation.  _Phys. Rev. B_ , 81:115134, Mar 2010.
  * Zenia et al. [2009] H Zenia, JK Freericks, HR Krishnamurthy, and Th Pruschke.  Appearance of “fragile” fermi liquids in finite-width mott insulators sandwiched between metallic leads.  _Physical review letters_ , 103(11):116402, 2009\.
  * Imada et al. [1998] Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura.  Metal-insulator transitions.  _Reviews of modern physics_ , 70(4):1039, 1998\.
  * Trastoy et al. [2020] J. Trastoy, A. Camjayi, J. del Valle, Y. Kalcheim, J.-P. Crocombette, D. A. Gilbert, J. A. Borchers, J. E. Villegas, D. Ravelosona, M. J. Rozenberg, and Ivan K. Schuller.  Magnetic field frustration of the metal-insulator transition in ${V}_{2}{O}_{3}$.  _Phys. Rev. B_ , 101:245109, Jun 2020.
  * Biermann et al. [2005] Silke Biermann, A Poteryaev, AI Lichtenstein, and A Georges.  Dynamical singlets and correlation-assisted peierls transition in VO2.  _Physical review letters_ , 94(2):026404, 2005\.
  * Nájera et al. [2018] O Nájera, M Civelli, V Dobrosavljević, and MJ Rozenberg.  Multiple crossovers and coherent states in a mott-peierls insulator.  _Physical Review B_ , 97(4):045108, 2018.
  * Stinson et al. [2018a] HT Stinson, A Sternbach, O Najera, R Jing, AS Mcleod, TV Slusar, A Mueller, L Anderegg, HT Kim, M Rozenberg, et al.  Imaging the nanoscale phase separation in vanadium dioxide thin films at terahertz frequencies.  _Nature communications_ , 9(1):1–9, 2018a.
  * Ocampo [2017] Oscar Nájera Ocampo.  _Study of the dimer Hubbard Model within Dynamical Mean Field Theory and its application to $VO_{2}$_.  PhD thesis, 2017.
  * Nájera et al. [2017] O Nájera, Marcello Civelli, Vladimir Dobrosavljević, and Marcelo J Rozenberg.  Resolving the VO2 controversy: Mott mechanism dominates the insulator-to-metal transition.  _Physical Review B_ , 95(3):035113, 2017.
  * Grandi et al. [2020] F. Grandi, A. Amaricci, and M. Fabrizio.  Unraveling the mott-peierls intrigue in vanadium dioxide.  _Phys. Rev. Res._ , 2:013298, Mar 2020.
  * Deng and Li [2014] Kaimo Deng and Liang Li.  Cds nanoscale photodetectors.  _Advanced Materials_ , 26(17):2619–2635, 2014\.
  * Cheng et al. [2018] Lei Cheng, Quanjun Xiang, Yulong Liao, and Huaiwu Zhang.  Cds-based photocatalysts.  _Energy & Environmental Science_, 11(6):1362–1391, 2018.
  * Georges et al. [1996] Antoine Georges, Gabriel Kotliar, Werner Krauth, and Marcelo J. Rozenberg.  Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions.  _Rev. Mod. Phys._ , 68:13–125, Jan 1996.
  * Gull et al. [2011] Emanuel Gull, Andrew J. Millis, Alexander I. Lichtenstein, Alexey N. Rubtsov, Matthias Troyer, and Philipp Werner.  Continuous-time monte carlo methods for quantum impurity models.  _Rev. Mod. Phys._ , 83:349–404, May 2011.
  * Werner and Millis [2006] Philipp Werner and Andrew J. Millis.  Hybridization expansion impurity solver: General formulation and application to kondo lattice and two-orbital models.  _Phys. Rev. B_ , 74(15):155107, Oct 2006.
  * Haule [2007] Kristjan Haule.  Quantum Monte Carlo impurity solver for cluster dynamical mean-field theory and electronic structure calculations with adjustable cluster base.  _Phys. Rev. B_ , 75:155113, Apr 2007.
  * Moeller et al. [1999] G. Moeller, V. Dobrosavljević, and A. E. Ruckenstein.  Rkky interactions and the mott transition.  _Phys. Rev. B_ , 59:6846–6854, Mar 1999.
  * Stinson et al. [2018b] HT Stinson, A Sternbach, O Najera, R Jing, AS Mcleod, TV Slusar, A Mueller, L Anderegg, HT Kim, M Rozenberg, et al.  Imaging the nanoscale phase separation in vanadium dioxide thin films at terahertz frequencies.  _Nature communications_ , 9(1):1–9, 2018b.
  * Fratino et al. [2022] L. Fratino, S. Bag, A. Camjayi, M. Civelli, and M. Rozenberg.  Doping-driven resistive collapse of the mott insulator in a minimal model for ${\mathrm{vo}}_{2}$.  _Phys. Rev. B_ , 105:125140, Mar 2022.
  * Jarrell and Gubernatis [1996] Mark Jarrell and James E Gubernatis.  Bayesian inference and the analytic continuation of imaginary-time quantum monte carlo data.  _Physics Reports_ , 269(3):133–195, 1996.
  * Bag et al. [2015] Soumen Bag, Arti Garg, and H. R. Krishnamurthy.  Phase diagram of the half-filled ionic hubbard model.  _Phys. Rev. B_ , 91:235108, Jun 2015.
  * Fratino et al. [2017] L. Fratino, P. Sémon, M. Charlebois, G. Sordi, and A.-M. S. Tremblay.  Signatures of the mott transition in the antiferromagnetic state of the two-dimensional hubbard model.  _Phys. Rev. B_ , 95:235109, Jun 2017.
  * Rozenberg et al. [1995] M. J. Rozenberg, G. Kotliar, H. Kajueter, G. A. Thomas, D. H. Rapkine, J. M. Honig, and P. Metcalf.  Optical conductivity in mott-hubbard systems.  _Phys. Rev. Lett._ , 75:105–108, Jul 1995.  doi: 10.1103/PhysRevLett.75.105.
  * Camjayi et al. [2006] A. Camjayi, R. Chitra, and M. J. Rozenberg.  Electronic state of a doped mott-hubbard insulator at finite temperatures studied using the dynamical mean-field theory.  _Phys. Rev. B_ , 73:041103, Jan 2006.  doi: 10.1103/PhysRevB.73.041103.
  * [64] The MaxEnt formalism tends to produce spurious noise around the $\omega$=0 point. this is usually not problematic for insulating solutions, but not desirable for metallic spectral functions. J. Skilling, Fundamentals of MaxEnt in data analysis, in Maximum Entropy in Action, edited by B. Buck and V. A. Macaulay (Clarendon Press, Oxford, UK, 1991), p. 19; https://triqs.github.io/maxent/latest/guide/preblur.html.
  * de’ Medici et al. [2005] L. de’ Medici, A. Georges, G. Kotliar, and S. Biermann.  Mott transition and kondo screening in $f$-electron metals.  _Phys. Rev. Lett._ , 95:066402, Aug 2005.
  * Foulquier, Paul et al. [2023] Foulquier, Paul, Civelli, Marcello, Rozenberg, Marcelo, Camjayi, Alberto, Bobadilla, Joel, Colson, Dorothée, Forget, Anne, Thuéry, Pierre, Bertran, François, Fèvre, Patrick Le, and Brouet, Véronique.  Evolution of the spectral lineshape at the magnetic transition in Sr2IrO4 and Sr3Ir2O7.  _Eur. Phys. J. B_ , 96(4):42, 2023.
  * Budai et al. [2014] John D Budai, Jiawang Hong, Michael E Manley, Eliot D Specht, Chen W Li, Jonathan Z Tischler, Douglas L Abernathy, Ayman H Said, Bogdan M Leu, Lynn A Boatner, et al.  Metallization of vanadium dioxide driven by large phonon entropy.  _Nature_ , 515(7528):535–539, 2014.
  * Díaz et al. [2023] Manuel I Díaz, Jong E Han, and Camille Aron.  Electrically driven insulator-to-metal transition in a correlated insulator: Electronic mechanism and thermal description.  _Physical Review B_ , 107(19):195148, 2023.
  * Chung and Freericks [1998] Woonki Chung and J. K. Freericks.  Charge-transfer metal-insulator transitions in the spin-$\frac{1}{2}$ falicov-kimball model.  _Phys. Rev. B_ , 57:11955–11961, May 1998.  doi: 10.1103/PhysRevB.57.11955.