Strongly correlated metals often display anomalous transport, including $T$-linear resistivity above the Mott-Ioffe-Regel limit. We introduce a tractable microscopic model for such bad metals, by supplementing the well-known Hubbard model --- with hopping $t$ and on-site repulsion $U$ --- with a `screened Coulomb' interaction between charge densities that decays exponentially with spatial separation. This interaction entirely lifts the extensive degeneracy in the spectrum of the $t=0$ Hubbard model, allowing us to fully characterize the small $t$ electric, thermal and thermoelectric transport in our strongly correlated model. Throughout the phase diagram we observe $T$-linear resistivity above the Mott-Ioffe-Regel limit, together with strong violation of the Weidemann-Franz law and a large thermopower that can undergo sign change. At intermediate temperatures $t \ll k_B T \lesssim U$, the approximate $T$-linear resistivity arises from a cancellation between the nontrivial temperature dependence of both diffusivities and thermodynamic susceptibilities, as observed in recent transport experiments on cold atomic gases.
We study the finite-temperature properties of the half-filled two-band Hubbard model in the presence of Hund's rule coupling and orbital anisotropy. We use the mean-field treatment of the $Z_2$ slave-spin theory with a finite-temperature extension of the zero-temperature gauge variable previously developed by Hassan and de' Medici http://dx.doi.org/10.1103/PhysRevB.81.035106}{Phys. Rev. B {\bf 81}, 035106 (2010)}]. We consider the instability of the Fermi liquid phases and how it is enhanced by the Hund's rule. We identify paramagnetic solutions that have zero quasi-particle weight with bad me…
Bad-Metal (BM) behavior featuring linear temperature dependence of the resistivity extending to well above the Mott-Ioffe-Regel (MIR) limit is often viewed as one of the key unresolved signatures of strong correlation. Here we associate the BM behavior with the Mott quantum criticality by examining a fully frustrated Hubbard model where all long-range magnetic orders are suppressed, and the Mott problem can be rigorously solved through Dynamical Mean-Field Theory. We show that for the doped Mott insulator regime, the coexistence dome and the associated first-order Mott metal-insulator transiti…
We analyze the paramagnetic state of a two-band Hubbard model with finite Hund's coupling close to integer filling at $n=2$ in two spacial dimensions. Previously, a Mott metal-insulator transition was established at $n=2$ with a coexistence region of a metallic and a bad metal state in the vicinity of that integer filling. The coexistence region ends at a critical point beyond which a charge instability persists. Here we investigate the transition into negative electronic compressibility states for an extended filling range close to $n=2$ within a slave boson setup. We analyze the separate con…
Bad metals display transport behavior that differs from what is commonly seen in ordinary metals. One of the most significant differences is a resistivity that is linear in temperature and rises to well above the Ioffe-Regel limit (where the mean-free path is equal to the lattice spacing). Using an exact Kubo formula, we show that a linear resistivity naturally occurs for many systems when they are in an incoherent intermediate-temperature state. We verify the analytic arguments with numerical calculations for a simplified version of the Hubbard model which is solved with dynamical mean-field …
"Bad metals" have a large linear resistivity at high-T that is universally seen in oxides close to the Mott-Hubbard insulating phase. They also have an universal thermopower alpha(T): (i) at very low doping (lightly doped) alpha(T) has a pronounced low-T peak that shifts to higher-T with doping; (ii) at moderate doping (underdoped) alpha(T) has a small low-T peak that shifts to lower-T with doping and has a high-T sign change; and (iii) at the highest doping (overdoped) alpha(T) is negative and depends monotonically on T. Here we show that the simplified Hubbard model provides an easy to under…
We investigate with angle resolved photoelectron spectroscopy the change of the Fermi Surface (FS) and the main bands from the paramagnetic (PM) state to the antiferromagnetic (AFM) occurring below 72 K in Fe_{1.06}Te. The evolution is completely different from that observed in iron-pnictides as nesting is absent. The AFM state is a rather good metal, in agreement with our magnetic band structure calculation. On the other hand, the PM state is very anomalous with a large pseudogap on the electron pocket that closes in the AFM state. We discuss this behavior in connection with spin fluctuations…
Bad metallicity in the semi-quantum regime of the Hubbard model
Stanford University（斯坦福大学）；The Graduate Center, CUNY（纽约市立大学研究生中心）；College of Staten Island, CUNY（纽约市立大学斯塔顿岛学院）；SLAC National Accelerator Laboratory（斯坦福直线加速器中心）
The interplay between long-range and local Coulomb repulsion in strongly interacting electron systems is explored through a two-dimensional Hubbard-Wigner model. An unconventional metallic state is found in which collective low-energy excitations characteristic of the Wigner crystal induce a flow of electrical current despite the absence of one-electron spectral weight at the Fermi surface. Photoemission experiments on certain quarter-filled layered molecular crystals should observe a gap in the excitation spectrum whereas optical spectroscopy should find a finite Drude weight indicating metal…
Strange or bad metallic transport, defined by its incompatibility with conventional quasiparticle pictures, is a theme common to strongly correlated materials and ubiquitous in many high temperature superconductors. The Hubbard model represents a minimal starting point for modeling strongly correlated systems. Here we demonstrate strange metallic transport in the doped two-dimensional Hubbard model using determinantal quantum Monte Carlo calculations. Over a wide range of doping, we observe resistivities exceeding the Mott-Ioffe-Regel limit with linear temperature dependence. The temperatures …
In the present work ferromagnetic ordering in the Hubbard model generalized by taking into account the inter-atomic exchange interaction and correlated hopping in partially filled narrow band is considered. In the case of weak electron-electron interaction the ground state energy and condition of ferromagnetic state realization are found using the Green function technique. The obtained results indicate the important role of correlated hopping.
We study a range of thermodynamic properties (charge susceptibility, specific heat, entropy and spin susceptibility) of the Hubbard model on the anisotropic triangular lattice at half filling by means of the numerical finite-temperature Lanczos method. We observe clear signatures of a metal-Mott insulator transition in the charge susceptibility, and show that the metallic phase is characterized by a small charge susceptibility, large entropy, low coherence temperature, large renormalized quasiparticle mass, and large spin susceptibility. We show that the local magnetic moment in the metallic p…
We re-examine the Peierls insulator to Mott insulator transition scenario in the one-dimensional Holstein-Hubbard model where, at half-filling, electron-phonon and electron-electron interactions compete for establishing charge- and spin-density-wave states, respectively. By means of large-scale density-matrix renormalization group calculations we determine the spin, single-particle and two-particle excitation gaps and prove--in the course of a careful finite-size scaling analysis--recent claims for an intervening metallic phase in the weak-coupling regime. We show that for large phonon frequen…
In the paper a possibility of metallic ferromagnetic state realization in a generalized Hubbard model with more complete accounting of electron-electron interactions, in particular, the correlated hopping and exchange interaction integrals is investigated. Recently obtained by means of mean-field approximation single electron energy spectrum is used for the description of finite temperature properties of the system. In the paper the expression for the critical temperature of ferromagnet-paramagnet transition is found, the behaviour of temperature dependencies of magnetization and paramagnetic …
The interaction between itinerant and Mott localized electronic states in strongly correlated materials is studied within dynamical mean field theory in combination with the numerical renormalization group method. A novel nonmagnetic zero temperature quantum phase transition is found in the bad-metallic orbital-selective Mott phase of the two-band Hubbard model, for values of the Hund's exchange which are relevant to typical transition metal oxides.
We investigate the phases of the ionic Hubbard model in a two-dimensional square lattice using determinant quantum Monte Carlo (DQMC). At half-filling, when the interaction strength or the staggered potential dominate we find Mott and band insulators, respectively. When these two energies are of the same order we find a metallic region. Charge and magnetic structure factors demonstrate the presence of antiferromagnetism only in the Mott region, although the externally imposed density modulation is present everywhere in the phase diagram. Away from half-filling, other insulating phases are foun…
Electron correlations in the two-orbital Hubbard model at half-filling are investigated by combining dynamical mean field theory with the exact diagonalization method. We systematically study how the interplay of the intra- and inter-band Coulomb interactions, together with the Hund coupling, affects the metal-insulator transition. It is found that if the intra- and inter-band Coulomb interactions are nearly equal, the Fermi-liquid state is stabilized due to orbital fluctuations up to fairly large interactions, while the system is immediately driven to the Mott insulating phase away from this …
We propose one possible mechanism for an anomalous metallic phase appearing frequently in two spatial dimensions, that is, local pairing fluctuations. Introducing a pair-rotor representation to decompose bare electrons into collective pairing excitations and renormalized electrons, we derive an SU(2) gauge theory of the Hubbard model as an extended version of its U(1) gauge theory\cite{Florens,LeeLee} to allow only local density fluctuations. Since our effective SU(2) gauge theory admits two kinds of collective bosons corresponding to pair excitations and density fluctuations respectively, an …
The strong coupling diagram technique is used for investigating states near the metal-insulator transition in the half-filled two-dimensional repulsive Hubbard model. The nonlocal third-order term is included in the irreducible part along with local terms of lower orders. Derived equations for the electron Green's function are solved by iteration for moderate Hubbard repulsions and temperatures. Starting iteration from Green's functions of the Hubbard-I approximation with various distances of poles from the real frequency axis continua of different metallic and insulating solutions are obtaine…
We study the transport properties for a family of geometrically frustrated models on the triangular lattice with an interaction scale far exceeding the single-particle bandwidth. Starting from the interaction-only limit, which can be solved exactly, we analyze the transport and thermodynamic behavior as a function of filling and temperature at the leading non-trivial order in the single-particle hopping. Over a broad range of intermediate temperatures, we find evidence of a dc resistivity scaling linearly with temperature and with typical values far exceeding the quantum of resistance, $h/e^2$…
Employing Nernst-Einstein decomposition $σ=e^2χ_c D$ of the conductivity $σ$ onto charge susceptibility (compressibility) $χ_c$ and diffusion constant $D$, we argue that the bad-metallic behavior of $σ$ in the regime of high temperatures and lightly doped insulator is dominated by the strong temperature and doping dependence of $χ_c$. In particular, we show how at small dopings $χ_c$ strongly decreases towards undoped-insulating values with increasing temperature and discuss simple picture leading to the linear-in-temperature resistivity with the prefactor increasing inversely with decreasing …
Exact ground states of interacting electrons on the diamond Hubbard chain in a magnetic field are constructed which exhibit a wide range of properties such as flat-band ferromagnetism and correlation induced metallic, half-metallic or insulating behavior. The properties of these ground states can be tuned by changing the magnetic flux, local potentials, or electron density.
We construct a multi-band Hubbard model on the lattice obtained by "decorating" a closely packed $d$-dimensional lattice $\mathcal{M}$ (such as the triangular lattice) where $d\ge2$. We take the limits in which the Coulomb interaction and the band gap become infinitely large. Then there remains only a single band with finite energy, on which electrons are supported. Let the electron number be $N_\mathrm{e}=|\mathcal{M}|-N_\mathrm{h}$, where $|\mathcal{M}|$ corresponds to the electron number which makes the lowest (finite energy) band half-filled, and $N_\mathrm{h}$ is the number of "holes". It…
We present the first rigorous example of the Hubbard model in any dimensions which exhibits metallic ferromagnetism. The model is a genuine Hubbard model with short-range hopping and on-site Coulomb repulsion, and has multi single-electron bands. In the limit where the band gap and the Coulomb repulsion become infinite, we prove that the ground states are completely ferromagnetic and at the same time conducting.
Motivated by the recent experimental evidence of commensurate surface CDW in Pb/Ge(111) and Sn/Ge(111) $\sqrt{3}$-adlayer structures, as well as by the insulating states found on K/Si(111):B and SiC(0001), we have investigated the role of electron-electron interactions, and also of electron-phonon coupling, on the narrow surface state band originating from the dangling bond orbitals of the adsorbate. We model the problem by an extended two-dimensional Hubbard model at half-filling on a triangular lattice. We include an on-site Hubbard repulsion U and a nearest-neighbor V, plus a long-ranged Co…
Stability of a metallic state in the two-orbital Hubbard model at half-filling is investigated. We clarify how spin and orbital fluctuations are enhanced to stabilize the formation of quasi-particles by combining dynamical mean field theory with the quantum Monte Carlo simulations. These analyses shed some light on the reason why the metallic phase is particularly stable when the intra- and inter-band Coulomb interactions are nearly equal.
We study the dynamics of doublon in the half-filled Hubbard model on the triangular lattice by using the cellular dynamical mean field theory. Investigating the nearest-neighbor dynamical correlations, we demonstrate that a nearest-neighbor doublon-holon pair shows a strong attraction, in particular in the insulating phase. We also calculate the on-site dynamical correlation of doublon and find that the life time of doublon is longer in the metallic phase than in the insulating phase. In the long-time region, the metallic phase has persistent fluctuations in various nearest-neighbor configurat…
The interaction-driven insulator-to-metal transition has been reported in the ionic Hubbard model (IHM) for moderate interaction $U$, while its metallic phase only occupies a narrow region in the phase diagram. To explore the enlargement of the metallic regime, we extend the ionic Hubbard model to two coupled layers and study the interplay of interlayer hybridization $V$ and two types of intralayer staggered potentials $Δ$: one with the same (in-phase) and the other with a $π$-phase shift (anti-phase) potential between layers. Our determinant Quantum Monte Carlo (DQMC) simulations at lowest ac…
We study the electronic state of the doped Mott-Hubbard insulator within Dynamical Mean Field Theory. The evolution of the finite temperature spectral functions as a function of doping show large redistributions of spectral weight in both antiferromagnetic and paramagnetic phases. In particular, a metallic antiferromagnetic state is obtained with a low frequency Slater-splitted quasiparticle peak coexisting with Hubbard bands. In the high temperature paramagnetic metallic phase, upon reducing doping, the system has a crossover through a ``bad metal'' state characterized by an anomalous shift o…
The Mott transition in a multi-orbital Hubbard model involving subbands of different widths is studied within the dynamical mean field theory. Using the iterated perturbation theory for the quantum impurity problem it is shown that at low temperatures inter-orbital Coulomb interactions give rise to a single first-order transition rather than a sequence of orbital selective transitions. Impurity calculations based on the Quantum Monte Carlo method confirm this qualitative behavior. Nevertheless, at finite temperatures, the degree of metallic or insulating behavior of the subbands differs greatl…
We study the "phase diagram" of a Hubbard plaquette with Rashba spin-orbit coupling. We show that the peculiar way in which Rashba coupling breaks the spin-rotational symmetry of the Hubbard model allows a mixing of singlet and triplet components in the ground-state that slows down and it changes the nature of the Mott transition and of the Mott insulating phases.
The one-electron spectral-weight function of the half-filled ionic Hubbard model is calculated by means of Quantum Monte Carlo. A metallic regimen occurs between two values of the coupling constant (Hubbard U) U1 < U2 . The system is a band insulator below U1, and an increasingly antiferromagnetic (AF) insulator above U2 evolving into a Mott insulator as U tends to infinity . The intermediate regimen which is both AF and dimerized, is caused by a two-peak structure at k = 0,5 pi (the Fermi surface). As U increases both peaks approach each other, overlap, and separate again, the system becoming…
The metal-insulator transition of the quarter-filled Hubbard model on triangular lattice is studied at the mean field level. We find a quasi-one dimensional metallic state with a collinear magnetic order competes closely with an insulating state with a non-coplanar magnetic order for both signs of the hopping integral $t$. In the strong correlation regime($U/|t|\gg1$), it is found that the metal-insulator transition of the system occurs in a two-step manner. The quasi-one dimensional metallic state with collinear magnetic order is found to be stable in an intermediate temperature region betwee…
We model a quantum dot-array (with one electron per dot) comprising of two (or more than two) coupled dots by an extended Hubbard Hamiltonian to investigate the role played by the inter-dot tunneling amplitude td, together with intra-dot (U) and inter-dot(U1) coulomb repulsions, in the singlet / triplet bound state formation and evolution of the system from the Mott-insulator-like state to a correlated semi-metallic state via charge-bond-order route. In the presence of magnetic field, td is complex due to the appearance of Peierls phase factor. We introduce a short-ranged inter-dot capacitive …
The frequency-dependent conductivity is studied for the one-dimensional Hubbard model, using a selection rule, the Bethe ansatz, and symmetries associated with conservation laws. For densities where the system is metallic the absorption spectrum has two contributions, a Drude peak at $ω= 0$ separated by a pseudogap from a broad absorption band whose lower edge is characterized by a non-classical critical exponent. Our findings shed new light on the ``far infrared puzzle'' and other optical properties of metallic organic chain compounds.
A detailed study of electronic phase transitions in the ionic Hubbard model at half filling is presented. Within the dynamical mean field approximation a series of transitions from the band insulator via a metallic state to a Mott-Hubbard insulating phase is found at intermediate values of the one-body potential $Δ$ with increasing the Coulomb interaction $U$. We obtain a critical region in which the metallic phase disappears and a {\it novel} coexistence phase between the band and the Mott insulating state sets in. Our results are consistent with those obtained at low dimensions, thus they pr…
We investigate paramagnetic metal-insulator transitions in the infinite-dimensional ionic Hubbard model at finite temperatures. By means of the dynamical mean-field theory with an impurity solver of the continuous-time quantum Monte Carlo method, we show that an increase in the interaction strength brings about a crossover from a band insulating phase to a metallic one, followed by a first-order transition to a Mott insulating phase. The first-order transition turns into a crossover above a certain critical temperature, which becomes higher as the staggered lattice potential is increased. Furt…
Scattering by slowly fluctuating degrees of freedom can cause a transient localization of the current-carrying electrons in metals, driving the system away from normal metallic behavior. We illustrate and characterize this general phenomenon by studying how signatures of localization emerge in the optical conductivity of electrons interacting with slow bosonic fluctuations. The buildup of quantum localization corrections manifests itself in the emergence of a displaced Drude peak (DDP), whose existence strongly alters the low frequency optical response and suppresses the d.c. conductivity. We …
We employ the dynamical mean field approximation to study the effects of ionic potential ($Δ$) on the square lattice Hubbard model. At half-filling when the staggered potential ($Δ$) dominates the on-site Hubbard interaction ($U$), the system is in the band insulator phase. We find that competition between $U$ and $Δ$ can suppress the gap to zero and leading to an intermediate metallic region. At the large-$U$ limit, we identify a Mott insulator phase where the gap opens again and increases upon increasing the Hubbard interaction U. For $U\leq 9t$ and $Δ\approx 0$ the phase of the system is me…
The effect of the Hubbard interaction among conduction electrons on the double exchange model is investigated in a ferromagnetic metallic phase. Applying iterative perturbation theory to the Hubbard interaction within dynamical mean field theory, we calculate the one-particle spectral function and the optical conductivity, in which coherent-potential approximation is further used to treat the ferromagnetic Hund coupling between conduction electrons and localized spins. Identifying the decrease of the magnetization for the localized spin with the increase of the temperature, we discuss the temp…
A systematic study of the effect of magnetic field (h) on Hubbard model has been carried out at half filling within dynamical mean field theory. In agreement with previous studies, we find a zero temperature itinerant metamagnetic transition, reflected in the discontinuous changes in magnetization as well as in the hysteresis, from a paramagnetic (PM) metallic state to a polarized quasi-ferromagnetic (QFM) state, at intermediate and large interaction strength ($U$). The jump in magnetization vanishes smoothly with decreasing interaction strength, and at a critical $U$, the transition becomes c…
We investigate the characteristics of the metallic phase near the Mott transition in the Kagomé lattice Hubbard model using the cellular dynamical mean field theory. By calculating the specific heat and spin correlation functions, we demonstrate that the quasiparticles show anomalous properties in the metallic phase close to the Mott transition. We find clear evidence for the multi-band heavy quasiparticles in the specific heat, which gives rise to unusual temperature dependence of the spin correlation functions.
Motivated by the current interest in the understanding of the Mott insulators away from half filling, observed in many perovskite oxides, we study the Mott metal-insulator transition (MIT) in the doped Hubbard-Holstein model using the Hatree-Fock mean field theory. The Hubbard-Holstein model is the simplest model containing both the Coulomb and the electron-lattice interactions, which are important ingredients in the physics of the perovskite oxides. In contrast to the half-filled Hubbard model, which always results in a single phase (either metallic or insulating), our results show that away …
The existence of a novel metal-insulator transition in the ferromagnetic state of models for undoped manganites is here discussed using numerical techniques applied to the $e_{\rm g}$-orbital degenerate Hubbard model tightly coupled with Jahn-Teller distortions. The ground-state phase diagram is presented in the plane defined by the electron-phonon coupling $λ$ and Coulomb interaction $u$. In contrast to the standard one-band Hubbard model for cuprates, the metallic phase is found to exist for finite values of both $λ$ and $u$ in the present $e_{\rm g}$-orbital Hubbard model even at half-filli…
The two-orbital Hubbard model with the Hund coupling is investigated in a metallic phase close to the Mott insulator. We calculate the one-particle spectral function and the optical conductivity within dynamical mean field theory, for which the effective impurity problem is solved by using the non-crossing approximation. For a metallic system close to quarter filling, a heavy quasi-particle band is formed by the Hubbard interaction, the effective mass of which is not so sensitive to the orbital splitting and the Hund coupling. In contrast, a heavy quasi-particle band near half filling disappea…
Based on an unrestricted Gutzwiller approximation (GA) we investigate the stripe orientation and periodicity in an extended one-band Hubbard model. A negative ratio between next-nearest and nearest neighbor hopping t'/t, as appropriate for cuprates, favors partially filled (metallic) stripes for both vertical and diagonal configurations. At around optimal doping diagonal stripes, site centered (SC) and bond centered (BC) vertical stripes become degenerate suggesting strong lateral and orientational fluctuations. We find that within the GA the resulting phase diagram is in agreement with experi…
The Quantum Monte Carlo simulations of the ionic Hubbard model on a two-dimensional square lattice at half filling were performed. The method based on the direct-space, proposed by Suzuki and al., Hirsch and al., was used. Cycles of increasing and decreasing values of the Coulomb interaction $ U $ were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at low temperature, the two insulator phases are separated by a metallic phase for weak to intermediate values of the staggered potential $ Δ$. For large Coulomb repulsion the system is in a Mott insulator with an antiferromag…
The electronic structure of small Hubbard molecules coupled between two non-interacting semi-infinite leads is studied in the low bias-voltage limit. To calculate the finite-temperature Green's function of the system, each lead is simulated by a small cluster, so that the problem is reduced to that of a finite-size system comprising the molecule and clusters on both sides. The Hamiltonian parameters of the lead clusters are chosen such that their embedding potentials coincide with those of the semi-infinite leads on Matsubara frequencies. Exact diagonalization is used to evaluate the effect of…
We investigate the dynamical superlattice correlation in the two-dimensional three-band Hubbard model on the basis of the unrestricted fluctuation exchange approximation. We calculate the one-particle spectral function, the spin correlation function and the charge correlation function at finite temperature. We find that some experimental results can be reproduced consistenly by taking inhomogenous distribution of Cu 3d electrons into account. The correlation functions suggest that several kinds of instabilities with spatial inhomogenieties exist in some regions, where these instabilities signi…
We studied the changes in the electronic structure of V2O3 using a cluster model. The calculations included fluctuations from the coherent band in the metallic phase, and non-local Mott-Hubbard fluctuations in the insulating phase. The incoherent structure is mostly related to the usual ligand screening channel (3d2L). The coherent peak in the metallic phase corresponds to coherent band fluctuations (3d2C). The non-local screened state in the insulating phase (3d2D) appears at higher energies, opening the band gap. The photon energy dependence of the spectra is mostly due to the relative V 3d …
We discuss examples of two dimensional metallic states with charge fractionalization, and we will demonstrate that the mechanism of charge fractionalization leads to exotic metallic behaviors at low and intermediate temperature. The simplest example of such state is constructed by fermionic partons at finite density coupled to a $Z_N$ gauge field, whose properties can be studied through rudimentary methods. This simple state has the following exotic features: (1) at low temperature this state is a "bad metal" whose resistivity can exceed the Mott-Ioffe-Regel limit; (2) while increasing tempera…
The effect of disorder on a class of transition metal oxides described by a single orbital Hubbard model at half filling is investigated. The phases are characterized by the nature of the electronic and spin excitations. The frequency and temperature-dependent conductivity and spin susceptibility as functions of disorder are calculated. The interplay of disorder and electron-electron interaction produces unusual behavior in this system. For example, the dc conductivity, which is vanishingly small at low disorder in the Mott phase and at high disorder in the localized phase, gets surprisingly e…
A new two-pole approximation, which allows to describe the transition from an insulating state to a metallic one at increase of bandwidth, and also the observable in some compounds transition from a metalic state to an insulating one with increasing temperature, is presented. The case of a paramagnetic narrow-band material at half-filling is considered. Single-particle Green function and energy spectrum of electron system are calculated. The expressions for energy gap width and the concentration of polar states (holes or doublons) are obtained. The conditions for metallic and insulating states…
