Three-band Emery Model Research Articles

Dressing due to correlations strongly reduces the effect of electron-phonon coupling

We investigate the difference between the coupling of a bare carrier to phonons versus the coupling of a correlations-dressed quasiparticle to phonons, and show that latter may be weak even if the former is strong. Specifically, we analyze the effect of the hole-phonon coupling on the dispersion of the quasiparticle that forms when a single hole is doped into a cuprate layer. To model this, we start from the three-band Emery model supplemented by the Peierls modulation of the $p$-$d$ and $p$-$p$ hoppings due to the motion of O ions. We then project onto the strongly correlated $U_{dd}\rightarrow \infty$ limit where charge fluctuations are frozen on the Cu site. The resulting effective Hamiltonian describes the motion of a doped hole on the O sublattice, and its interactions with Cu spins and O phonons. We show that even though the hole-phonon coupling is moderate to strong, it leads to only a very minor increase of the quasiparticle's effective mass as compared to its mass in the absence of coupling to phonons, consistent with a weak coupling to phonons of the correlations-dressed quasiparticle. We explain the reasons for this suppression, revealing why it is expected to happen in any systems with strong correlations.

Superconductivity in the three-band model of cuprates: nodal direction characteristics and influence of intersite interactions

The three-band Emery model is applied to study the selected principal features of the d-wave superconducting phase in the copper-based compounds. The electron–electron correlations are taken into account by the use of the diagrammatic expansion of the Guztwiller wave function (DE-GWF method). The nodal Fermi velocity, Fermi momentum, and effective mass are all determined in the paired state and show relatively good agreement with the available experimental data, as well as with the corresponding single-band calculations. Additionally, the influence of the next-nearest neighbor oxygen–oxygen hopping and intersite Coulomb repulsion terms on the superconducting phase is analyzed.

Investigation of the effective interactions for the Emery model by the constrained random-phase approximation and constrained functional renormalization group

The effective interaction of downfolded low-energy models for electrons in solids can be obtained by integrating out the high energy bands away from the target band near the Fermi level. Here we apply the constrained random-phase approximation (cRPA) and constrained functional renormalization group (cfRG), which can go beyond cRPA by including all one-loop diagrams, to calculate and compare the effective interactions of the three-band Emery model, which is often used to investigate cuprate high-temperature superconductors. At half-filling we find that the effective interaction increases as the charge transfer energy (${\mathrm{\ensuremath{\Delta}}}_{dp}$) increases and similar behavior is obtained as a function of the interatomic $2p\text{\ensuremath{-}}3d$ interaction (${U}_{dp}$). However, the effective interaction is more sensitive to ${\mathrm{\ensuremath{\Delta}}}_{dp}$ than ${U}_{dp}$. For most of the parameter sets, the effective static interaction is overscreened in cRPA compared to cfRG. The low-energy models at half-filling are solved within dynamical mean-field theory (DMFT). The results show that despite the different static interactions, the systems with cRPA and cfRG interaction exhibit a Mott transition at similar values of ${\mathrm{\ensuremath{\Delta}}}_{dp}$. We also investigate the effective interaction as a function of doping. The cfRG effective interaction decreases as the electron number increases and displays a trend opposite to that of cRPA. Antiscreening is observed for the hole-doped case. For all the cases studied, the near cancellation of the direct particle-hole channel is observed. This indicates that at least for the downfolding of the on-site interaction terms, methods beyond cRPA may be required.

Superconductivity and intra-unit-cell electronic nematic phase in the three-band model of cuprates

The intra-unit-cell nematic phase is studied within the three-band Emery model of the cuprates by using the diagrammatic expansion of the Gutzwiller wave function (DE-GWF). According to our analysis a spontaneous rotational (C4) symmetry breaking of the electronic wave function, leading to the nematic behavior, can appear due to electron correlations induced mainly by the onsite Coulomb repulsion, even in the absence of the corresponding intersite oxygen–oxygen repulsion term. The latter has been considered as the triggering factor of the nematic state formation in a number of previous studies. Also, we show that at the transition to the nematic phase, electron concentration transfer from d- to p-orbitals takes place, apart from the usually discussed px∕py polarization. The nematicity appears in a similar doping range as the paired phase, showing that both phases may have a common origin, even though they compete. As we show a coexistence region of both superconductivity and nematicity appears in a relatively wide doping range. The results are discussed in view of the experimental findings corresponding to the relation between nematicity and pseudogap behavior.Graphical abstract

Multiband nonequilibriumGW+EDMFTformalism for correlated insulators

We study the dynamics of charge-transfer insulators after a photo-excitation using the three-band Emery model which is relevant for the description of cuprate superconductors. We provide a detailed derivation of the nonequilibrium extension of the multi-band GW+EDMFT formalism and the corresponding downfolding procedure. The Peierls construction of the electron-light coupling is generalized to the multi-band case resulting in a gauge invariant combination of the Peierls intra-band acceleration and dipolar intra-band transitions. We apply the formalism to the study of momentum-dependent (inverse) photo-emission spectra and optical conductivities. The time-resolved spectral function shows a strong renormalization of the charge-transfer gap and a substantial broadening of some of the bands. While the upper Hubbard band exhibits a momentum-dependent broadening, an almost rigid band shift is observed for the ligand bands. The inverse photo-emission spectrum reveals that the inclusion of the non-local and inter-band charge fluctuations lead to a very fast relaxation of holes into the lower Hubbard band. Consistent with the changes in the spectral function, the optical conductivity shows a renormalization of the charge-transfer gap, which is proportional to the photo-doping. The details of the photo-induced changes strongly depend on the dipolar matrix elements, which calls for an ab-initio determination of these parameters.

Dynamics of photodoped charge transfer insulators

We study the dynamics of charge-transfer insulators after photo-excitation using the three-band Emery model and a nonequilibrium extension of Hartree-Fock+EDMFT and GW+EDMFT. While the equilibrium properties are accurately reproduced by the Hartree-Fock treatment of the full $p$ bands, dynamical correlations are essential for a proper description of the photo-doped state. The insertion of doublons and holons leads to a renormalization of the charge transfer gap %and $p$ bands and to a substantial broadening of the bands. We calculate the time-resolved photoemission spectrum and optical conductivity and find qualitative agreement with experiments. Our formalism enables the realistic description of nonequilibrium phenomena in a large class of charge-transfer insulators, and provides a tool to explore the optical manipulation of interaction and correlation effects.

Generalized Kondo lattice model and its spin-polaron realization by the projection method for cuprates

The spin–fermion model, which is an effective low-energy realization of the three-band Emery model after passing to the Wannier representation for the px and py orbitals of the subsystem of oxygen ions, reduces to the generalized Kondo lattice model. A specific feature of this model is the existence of spin-correlated hoppings of the current carriers between distant cells. Numerical calculations of the spectrum of spin-electron excitations highlight the important role of the long-range spin-correlated hoppings.

Coexistence ofΘII-loop-current order with checkerboardd-wave CDW/PDW order in a hot-spot model for cuprate superconductors

We investigate the strong influence of the $\Theta_{II}$-loop-current order on both unidirectional and bidirectional d-wave charge-density-wave/pair-density-wave (CDW/PDW) composite orders along axial momenta $(\pm Q_0,0)$ and $(0,\pm Q_0)$ that emerge in an effective hot spot model departing from the three-band Emery model relevant to the phenomenology of the cuprate superconductors. This study is motivated by the compelling evidence that the $\Theta_{II}$-loop-current order described by this model may explain groundbreaking experiments such as spin-polarized neutron scattering performed in these materials. Here, we demonstrate, within a saddle-point approximation, that the $\Theta_{II}$-loop-current order clearly coexists with bidirectional (i.e. checkerboard) d-wave CDW and PDW orders along axial momenta, but is visibly detrimental to the unidirectional (i.e. stripe) case. This result has potentially far-reaching implications for the physics of the cuprates and agrees well with very recent x-ray experiments on YBCO that indicate that at higher dopings the CDW order has indeed a tendency to be bidirectional.

Non-Zhang-Rice Singlet Character of the First Ionization State of T-CuO.

We argue that tetragonal CuO (T-CuO) has the potential to finally settle long-standing modeling issues for cuprate physics. We compare the one-hole quasiparticle (qp) dispersion of T-CuO to that of cuprates, in the framework of the strongly correlated (U_{dd}→∞) limit of the three-band Emery model. Unlike in CuO_{2}, magnetic frustration in T-CuO breaks the C_{4} rotational symmetry and leads to strong deviations from the Zhang-Rice singlet picture in parts of the reciprocal space. Our results are consistent with angle-resolved photoemission spectroscopy data but in sharp contradiction to those of a one-band model previously suggested for them. These differences identify T-CuO as an ideal material to test a variety of scenarios proposed for explaining cuprate phenomenology.

Differences between the insulating limit quasiparticles of one-band and three-band cuprate models

We study the charge dynamics of the quasiparticle that forms when a single hole is doped in a two-dimensional antiferromagnet as described by the one-band t---J model, using a variational approximation that includes spin fluctuations in the vicinity of the hole. We explain why the spin fluctuations and the longer range hopping have complementary contributions to the quasiparticle dynamics, and thus why both are essential to obtain a dispersion in agreement with that measured experimentally. This is very different from the three-band Emery model in the strongly-correlated limit, where the same variational approximation shows that spin fluctuations have a minor effect on the quasiparticle dynamics. This difference proves that these one-band and three-band models describe qualitatively different quasiparticles in the insulating limit, and therefore that they cannot both be suitable to describe the physics of very underdoped cuprates.

Strong competition betweenΘII-loop-current order andd-wave charge order along the diagonal direction in a two-dimensional hot spot model

We study the fate of the so-called $\Theta_{II}$-loop-current order that breaks both time-reversal and parity symmetries in a two-dimensional hot spot model with antiferromagnetically mediated interactions, using Fermi surfaces relevant to the phenomenology of the cuprate superconductors. We start from a three-band Emery model describing the hopping of holes in the CuO$_{2}$ plane that includes two hopping parameters $t_{pp}$ and $t_{pd}$, local on-site Coulomb interactions $U_{d}$ and $U_{p}$ and nearest-neighbor $V_{pd}$ couplings between the fermions in the copper [Cu$(3d_{x^{2}-y^{2}})$] and oxygen [O$(2p_{x})$ and O$(2p_{y})$] orbitals. By focusing on the lowest-energy band, we proceed to decouple the local interaction $U_{d}$ of the Cu orbital in the spin channel using a Hubbard-Stratonovich transformation to arrive at the interacting part of the so-called spin-fermion model. We also decouple the nearest-neighbor interaction $V_{pd}$ to introduce the order parameter of the $\Theta_{II}$-loop-current order. In this way, we are able to construct a consistent mean-field theory that describes the strong competition between the composite order parameter made of a quadrupole-density-wave and $d$-wave pairing fluctuations proposed in Efetov \emph{et al.} [Nat. Phys. \textbf{9}, 442 (2013)] with the $\Theta_{II}$-loop-current order parameter that is argued to be relevant for explaining important aspects of the physics of the pseudogap phase displayed in the underdoped cuprates.

Spatially modulated electronic nematicity in the three-band model of cuprate superconductors

Charge order in cuprate superconductors is a possible source of anomalous electronic properties in the underdoped regime. Intra-unit cell charge ordering tendencies point to electronic nematic order involving oxygen orbitals. In this context we investigate charge instabilities in the Emery model and calculate the charge susceptibility within diagrammatic perturbation theory. In this approach, the onset of charge order is signalled by a divergence of the susceptibility. Our calculations reveal three different kinds of order: a commensurate ($q=0$) nematic order, and two incommensurate nematic phases with modulation wavevectors that are either axial or oriented along the Brillouin zone diagonal. We examine the nematic phase diagram as a function of the filling, the interaction parameters, and the band structure. We also present results for the excitation spectrum near the nematic instability, and show that a soft nematic mode emerges from the particle-hole continuum at the transition. The Fermi surface reconstructions that accompany the modulated nematic phases are discussed with respect to their relevance for magneto-oscillation and photoemission measurements. The modulated nematic phases that emerge from the three-band Emery model are compared to those found previously in one-band models.

Mean-field analysis of intra-unit-cell order in the Emery model of the CuO2plane

Motivated by recent experiments on high-$T_c$ cuprate superconductors pointing towards intra-unit-cell order in the pseudogap phase, we investigate three distinct intra-unit-cell-ordering possibilities: nematic, nematic-spin-nematic, and current-loop order. The first two are Fermi-surface instabilities involving a spontaneous charge and magnetization imbalance between the two oxygen sites in the unit cell, respectively, while the third describes circulating currents within the unit cell. We analyze the three-band Emery model of a single CuO$_2$ layer including various on-site and nearest-neighbor interactions within a self-consistent mean-field approach. We show how these on-site and further-neighbor repulsions suppress or enhance particular IUC orders. In particular, we show that the attractive interactions necessary for nematic and nematic-spin-nematic orders in one-band models have their natural microscopic origin in the O-O on-site and nearest-neighbor repulsions in the three-band model. Finally, we find that while the nematic and nematic-spin-nematic orders cannot coexist in this framework, the loop-current order can coexist with nematic order.

The Fermi surface and the role of electronic correlations inSm2−xCexCuO4

Using a LDA+GTB (local density approximation+generalized tight-binding) hybrid scheme we investigate the band structure of the electron-doped high-Tc materialSm2−xCexCuO4. Parameters of the minimal tight-binding model for this system (theso-called three-band Emery model) were obtained within the NMTO (Nth-order muffin-tin orbital) method. The doping evolution of the dispersion and the Fermisurface in the presence of electronic correlations was investigated in two regimes ofmagnetic order: short-range (spin-liquid) and long-range (antiferromagnetic metal). Eachregime is characterized by the specific topologies of the Fermi surfaces and we discuss theirrelation to recent experimental data.

Energy structure of high-temperature superconductors with the intersite Coulomb interaction

The effect of the intersite Coulomb interaction of fermions on the ground state and energy structure of high-temperature superconductors is considered within the three-band Emery model using an extended basis of the irreducible operators orthogonal on Mori. The proposed theory allows prediction of bands of the fluctuation states whose spectral intensity grows with an increase in the root-mean-square fluctuations of the occupation numbers. The increase in these fluctuations in high-temperature superconductors upon doping plays a key role in redistribution of the integrated density of the Fermi states.

Electronic pseudogap of optimally doped high-temperatureNd2−xCexCuO4superconductors

We study the effect of antiferromagnetic correlations in the three-band Emery model, in comparison with the experimental angle-resolved photoemission spectra (ARPES) in optimally doped ${\mathrm{Nd}}_{2\ensuremath{-}x}{\mathrm{Ce}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$ (NCCO). The same calculation, formerly used to describe ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}\mathrm{Ca}{\mathrm{Cu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ (BSCCO), is applied here, but in contrast to BSCCO, where quantum paramagnon fluctuations are important, the characteristic energy of the dispersive paramagnons in NCCO is of the order of ${T}_{c}$. The wide dispersing features of the single-electron spectrum in NCCO are analogous to the BSCCO hump. The Fermi surface is pseudogapped in both the nodal and antinodal directions, although the detailed features differ, being dominated by loss of intensity in the nodal direction, and loss of coherence in the antinodal one. Direct oxygen-oxygen hopping is important in NCCO as well as in BSCCO, in order to obtain overall agreement with the measured ARPES spectra.

Narrow antiadiabatic peak in optimally doped and underdoped high-Tc superconductors

Abstract We study the effect of antiferromagnetic (AF) correlations in the three-band Emery model, with respect to the experimental situation in weakly underdoped and optimally doped BSCCO. In the vicinity of the vH singularity of the conduction band there appears a central peak in the middle of a pseudogap, due to an antiadiabatic quasiparticle, which is insensitive to the energy scale of the mechanism responsible for the pseudogap. We find a quantum low-temperature regime, in which the pseudogap profile is created by zero-point motion of the magnons. Detailed analysis of the spectral functions along the (π,0)–(π,π) line shows significant agreement with experiment. We conclude that optimally doped BSCCO has a well-developed pseudogap of the order of 1000 K. This is only masked by the antiadiabatic quasiparticle, which provides the experimentally observed small binding energy scale.

Central peak in the pseudogap of high T c superconductors

We study the effect of antiferromagnetic (AF) correlations in the three-band Emery model, with respect to the experimental situation in weakly underdoped and optimally doped BSCCO. In the vicinity of the vH singularity of the conduction band there appears a central peak in the middle of a pseudogap, which is in an antiadiabatic regime, insensitive to the time scale of the mechanism responsible for the pseudogap. We find a quantum low-temperature regime corresponding to experiment, in which the pseudogap is created by zero-point motion of the magnons, as opposed to the usual semiclassical derivation, where it is due to a divergence of the magnon occupation number. Detailed analysis of the spectral functions along the (pi,0)-(pi,pi) line show significant agreement with experiment, both qualitative and, in the principal scales, quantitative. The observed slight approaching-then-receding of both the wide and narrow peaks with respect to the Fermi energy is also reproduced. We conclude that optimally doped BSCCO has a well-developed pseudogap of the order of 1000 K. This is only masked by the narrow antiadiabatic peak, which provides a small energy scale, unrelated to the AF scale, and primarily controlled by the position of the chemical potential.

Fermion kinetics in the Falicov-Kimball limit of the three-band Emery model

The three-band Emery model is reduced to a single-particle quantum model of Falicov-Kimball type, by allowing only up-spins to hop, and forbidding double occupation by projection. It is used to study the effects of geometric obstruction on mobile fermions in thermodynamic equilibrium. For low hopping overlap, there appears a plateau in the entropy, due to charge correlations, and related to real-space disorder. For large overlap, the equilibrium thermopower susceptibility remains anomalous, with a sign opposite to the one predicted from the single-particle density of states. The heat capacity and non-Fermi liquid response are discussed in the context of similar results in the literature. All results are obtained by evaluation of an effective single-particle free-energy operator in closed form. The method to obtain this operator is described in detail.

Density of states and dispersion of quasiparticles in the Emery model: Nonlocal path-integral Monte Carlo algorithm

The spectral density of states for a 108-site Cu36O72 cluster in the two-dimensional three-band Emery model is reconstructed with the aid of a path integral Monte Carlo algorithm. Dispersion relations are obtained for quasiparticles in the upper Hubbard band and in the correlated-states band; this corresponds to electron and hole doping of high-T c superconductors. The form of the isoenergy surfaces is close to the experimentally observed form and confirms the existence of singularities in the density of states near the Fermi level.