We discuss a formalism that allows for the calculation of a higher-harmonic-current response to a strong applied electric field for disordered superconducting systems described on the basis of tight-binding models with on- and/or intersite interactions. The theory is based on an expansion of the density matrix in powers of the field amplitudes, where we solve the equation of motion for the individual components. This allows the evaluation of higher-order response functions on significantly larger lattices than one can achieve with a previously used approach, which is based on a direct temporal integration of the equation of motion for the complete density matrix. In the case of small lattices, where both methods can be applied by including also the contribution of collective modes, we demonstrate the agreement of the corresponding results.
Within the time-dependent Gutzwiller approximation applied to the negative-U Hubbard model, we investigate the dynamics of a superconductor after an interaction quench for different values of the final interaction from weak to strong coupling. The equilibrium BCS-BEC crossover becomes a sharp transition in the out-of-equilibrium dynamics between weak- and strong-coupling dynamical phases. Two different frequencies (ΩJ<ΩU) dominate the order parameter dynamics. In the weak coupling phase, ΩJ follows twice the asymptotic average value of the gap (ΩJ≈2Δ∞) and ΩU is much larger but hardy visible in the anomalous density. At long times the BCS dynamics is recovered but surprisingly differences remain during the transient phase. In the strong coupling phase, the dynamics decouples from the asymptotic value of the order parameter except at exactly half filling and close to the dynamical transition where ΩU≈2Δ∞. The out-of-equilibrium transient spectral density and optical conductivity are presented and discussed in relation to pump probe experiments. Both ΩJ and ΩU give rise to a complex structure of self-driven slow Rabi oscillations which are visible in the nonequilibrium optical conductivity where also sidebands appear due to the modulation of the double occupancy by superconducting amplitude oscillations. Analogous results apply to CDW and SDW systems. Our results show that in systems with long coherence times, pump-probe experiments allow us to characterize the regime (BCS vs preformed pairs) through very specific out-of-equilibrium fingerprints.
Although charge density waves (CDWs) are omnipresent in cuprate high-temperature superconductors, they occur at significantly different wave vectors, confounding efforts to understand their formation mechanism. Here, we use resonant inelastic x-ray scattering to investigate the doping- and temperature-dependent CDW evolution in La₂₋ₓBaₓCuO₄ (x=0.115–0.155). We discover that the CDW develops in two stages with decreasing temperature. A precursor CDW with a quasicommensurate wave vector emerges first at high temperature. This doping-independent precursor CDW correlation originates from the CDW phase mode coupled with a phonon and “seeds” the low-temperature CDW with a strongly doping-dependent wave vector. Our observation reveals the precursor CDW and its phase mode as the building blocks of the highly intertwined electronic ground state in the cuprates.
We calculate the dynamics of tax evasion within a multi-agent econophysics model which is adopted from the theory of magnetism and previously has been shown to capture the main characteristics from agent-based based models which build on the standard Allingham and Sandmo approach. In particular, we implement a feedback of public goods provision on the decision-making of selfish agents which aim to pursue their self interest. Our results imply that such a feedback enhances the moral attitude of selfish agents thus reducing the percentage of tax evasion. Two parameters govern the behavior of selfish agents,
(i) the rate of adaption to changes in public goods provision and (ii) the threshold of perception of public goods provision. Furtheron we analyze the tax evasion
dynamics for different agent compositions and under the feedback of public goods provision. We conclude that policymakers may enhance tax compliance behavior via the threshold of perception by means of targeted public relations.
We investigate an inhomogeneous Ising model in the context of tax evasion dynamics where different types of agents are parameterized via local temperatures and magnetic fields. In particular, we analyze the impact of lapse of time effects (i.e. backauditing) and endogenously determined penalty rates on tax compliance. Both features contribute to a microfoundation of agent-based econophysics models of tax evasion.
Electronic Inhomogeneities and Pairing from Unscreened Interactions in High-Tc Superconductors
(2007)
Electronic inhomogeneities, electron-lattice and pairinginteractions in high-Tc superconductors
(1997)
The spin galvanic effect (SGE) describes the conversion of a nonequilibrium spin polarization into a transverse charge current. Recent experiments have demonstrated a large conversion efficiency for the two-dimensional electron gas formed at the interface between two insulating oxides, LaAlO3 and SrTiO3. Here, we analyze the SGE for oxide interfaces within a three-band model for the Ti t2g orbitals which displays an interesting variety of effective spin-orbit couplings in the individual bands that contribute differently to the spin-charge conversion. Our analytical approach is supplemented by a numerical treatment where we also investigate the influence of disorder and temperature, which turns out to be crucial to providing an appropriate description of the experimental data.
There is steadily increasing evidence that the two-dimensional electron gas (2DEG) formed at the interface of some insulating oxides like LaAlO3/SrTiO3 and LaTiO3/SrTiO3 is strongly inhomogeneous. The inhomogeneous distribution of electron density is accompanied by an inhomogeneous distribution of the (self-consistent) electric field confining the electrons at the interface. In turn this inhomogeneous transverse electric field induces an inhomogeneous Rashba spin-orbit coupling (RSOC). After an introductory summary on two mechanisms possibly giving rise to an electronic phase separation accounting for the above inhomogeneity, we introduce a phenomenological model to describe the density-dependent RSOC and its consequences. Besides being itself a possible source of inhomogeneity or charge-density waves, the density-dependent RSOC gives rise to interesting physical effects like the occurrence of inhomogeneous spin-current distributions and inhomogeneous quantum-Hall states with chiral “edge” states taking place in the bulk of the 2DEG. The inhomogeneous RSOC can also be exploited for spintronic devices since it can be used to produce a disorder-robust spin Hall effect.
On the Evaluation of the Spin Galvanic Effect in Lattice Models with Rashba Spin-Orbit Coupling
(2018)
The spin galvanic effect (SGE) describes the conversion of a non-equilibrium spin polarization into a charge current and has recently attracted renewed interest due to the large conversion efficiency observed in oxide interfaces. An important factor in the SGE theory is disorder which ensures the stationarity of the conversion. Through this paper, we propose a procedure for the evaluation of the SGE on disordered lattices which can also be readily implemented for multiband systems. We demonstrate the performance of the method for a single-band Rashba model and compare our results with those obtained within the self-consistent Born approximation for a continuum model.
In the last few years, charge density waves (CDWs) have been ubiquitously observed in high-temperature superconducting cuprates and are now the most investigated among the competing orders in the still hot debate on these systems. A wealth of new experimental data raises several fundamental issues that challenge the various theoretical proposals. We here relate our mean-field instability line T0CDW of a strongly correlated Fermi liquid to the pseudogap T∗(p) line, marking in this way the onset of CDW-fluctuations. These fluctuations reduce strongly the mean-field critical line. Controlling this reduction via an infrared frequency cutoff related to the characteristic time of the probes, we account for the complex experimental temperature versus doping phase diagram. We provide a coherent scenario explaining why different CDW onset curves are observed by different experimental probes and seem to extrapolate at zero temperature into seemingly different quantum critical points (QCPs) in the intermediate and overdoped region. The nearly singular anisotropic scattering mediated by these fluctuations also accounts for the rapid changes of the Hall number seen in experiments and provides the first necessary step for a possible Fermi surface reconstruction fully establishing at lower doping. Finally, we show that phase fluctuations of the CDWs, which are enhanced in the presence of strong correlations near the Mott insulating phase, naturally account for the disappearance of the CDWs at low doping with yet another QCP as seen by the experiments.
We evaluate the non-equilibrium spin polarization induced by an applied electric field for a tight-binding model of electron states at oxides interfaces in LAO/STO heterostructures. By a combination of analytic and numerical approaches we investigate how the spin texture of the electron eigenstates due to the interplay of spin-orbit coupling and inversion asymmetry determines the sign of the induced spin polarization as a function of the chemical potential or band filling, both in the absence and presence of local disorder. With the latter, we find that the induced spin polarization evolves from a non monotonous behavior at zero temperature to a monotonous one at higher temperature. Our results may provide a sound framework for the interpretation of recent experiments.
LaAlO 3/SrTiO 3 and LaTiO 3/SrTiO 3 (LXO / STO) interfaces are known to host a strongly inhomogeneous (nearly) two-dimensional electron gas (2DEG). In this work, we present three unconventional electronic mechanisms of electronic phase separation (EPS) in a 2DEG as a possible source of inhomogeneity in oxide interfaces. Common to all three mechanisms is the dependence of some (interaction) potential on the 2DEGs density. We first consider a mechanism resulting from a sizable density-dependent Rashba spin-orbit coupling. Next, we point out that an EPS may also occur in the case of a density-dependent superconducting pairing interaction. Finally, we show that the confinement of the 2DEG to the interface by a density-dependent, self-consistent electrostatic potential can by itself cause an EPS.
Recent experiments with strong THz fields in both conventional and unconventional superconductors have clearly evidenced a marked third-harmonic generation below the superconducting temperature Tc. Its interpretation challenged substantial theoretical work aimed at establishing the relative efficiency of quasiparticle excitations and collective modes in triggering such a resonant response. Here we compute the nonlinear current by implementing a time-dependent Bogoljubov–de Gennes approach, with the twofold aim to account nonperturbatively for the effect of local disorder, and to include the contribution of all collective modes, i.e., superconducting amplitude (Higgs) and phase fluctuations, and charge fluctuations. We show that, in agreement with previous work, already at small disorder the quasiparticle response is dominated by paramagnetic effects. We further demonstrate that paramagnetic processes mediate also the response of all collective modes, with a substantial contribution of charge/phase fluctuations. These processes, which have been overlooked so far, turn out to dominate the third-order current at strong disorder. In addition, we show that disorder strongly influences the polarization dependence of the nonlinear response, with a marked difference between the clean and the disordered case. Our results are particularly relevant for recent experiments in cuprates, whose band structure is in a first approximation reproduced by our lattice model.
We investigate the physics of an adiabatic transition from a BCS superconductor to a Fermi liquid for an exponentially slow decreasing pairing interaction. We show that, depending on the order of the thermodynamic limit and large times, a situation can arise in which the Fermi liquid keeps a memory of the parent BCS state. Furthermore, a time inversion of the interaction, supplemented by a manipulation analogous to a spin-/photon-echo experiment, allows us to recover the parent BCS state. Moreover, we study the evolution of the order parameter phase ϕ in transforming the BCS superconductor to a conventional metal. Since the global phase is the conjugate variable of the density, we explicitly show how to use the dynamics of ϕ together with gauge invariance to build up the noninteracting chemical potential away from particle-hole symmetry. We further analyze the role of ϕ in restoring the gauge-invariant current response when the noninteracting Fermi liquid is approached starting from a BCS superconductor in the presence of an external vector field.
Anomalous metallic properties are often observed in the proximity of quantum critical points, with violation of the Fermi Liquid paradigm. We propose a scenario where, near the quantum critical point, dynamical fluctuations of the order parameter with finite correlation length mediate a nearly isotropic scattering among the quasiparticles over the entire Fermi surface. This scattering produces a strange metallic behavior, which is extended to the lowest temperatures by an increase of the damping of the fluctuations. We phenomenologically identify one single parameter ruling this increasing damping when the temperature decreases, accounting for both the linear-in-temperature resistivity and the seemingly divergent specific heat observed, e.g., in high-temperature superconducting cuprates and some heavy-fermion metals.
Besides the mechanism responsible for high critical temperature superconductivity, the grand unresolved issue of the cuprates is the occurrence of a strange metallic state above the so-called pseudogap temperature T*. Even though such state has been successfully described within a phenomenological scheme, the so-called Marginal Fermi-Liquid theory, a microscopic explanation is still missing. However, recent resonant X-ray scattering experiments identified a new class of charge density fluctuations characterized by low characteristic energies and short correlation lengths, which are related to the well-known charge density waves. These fluctuations are present over a wide region of the temperature-vs-doping phase diagram and extend well above T*. Here we investigate the consequences of charge density fluctuations on the electron and transport properties and find that they can explain the strange metal phenomenology. Therefore, charge density fluctuations are likely the long-sought microscopic mechanism underlying the peculiarities of the metallic state of cuprates.
The low-energy optical conductivity of conventional superconductors is usually well described by Mattis-Bardeen (MB) theory, which predicts the onset of absorption above an energy corresponding to twice the superconducing (SC) gap parameter Δ. Recent experiments on strongly disordered superconductors have challenged the application of the MB formulas due to the occurrence of additional spectral weight at low energies below 2Δ. Here we identify three crucial items that have to be included in the analysis of optical-conductivity data for these systems: (a) the correct identification of the optical threshold in the Mattis-Bardeen theory and its relation with the gap value extracted from the measured density of states, (b) the gauge-invariant evaluation of the current-current response function needed to account for the optical absorption by SC collective modes, and (c) the inclusion into the MB formula of the energy dependence of the density of states present already above Tc. By computing the optical conductivity in the disordered attractive Hubbard model, we analyze the relevance of all these items, and we provide a compelling scheme for the analysis and interpretation of the optical data in real materials.
Defects play an important role for surface reconstructions and therefore also influence the substrate growth. In this work we present a first principle calculation for the Ge(001) surface without and with tin impurities incorporated into the top surface layer. By mapping the system onto an Ising-type model, with interaction constants taken from density functional theory, the stability of the surface reconstructions under the influence of different concentrations of tin impurities is explored. This approach allows us to simulate the possible phase transitions for the different surface reconstructions including the local structure around the tin impurity atoms. In addition, we compare our theoretical results with experimental STM images on clean and Sn-doped Ge(100) surfaces.
Recent resonant X-ray scattering experiments on cuprates allowed to identify a new kind of collective excitations, known as charge density fluctuations, which have finite characteristic wave vector, short correlation length and small characteristic energy. It was then shown that these fluctuations provide a microscopic scattering mechanism that accounts for the anomalous transport properties of cuprates in the so-called strange-metal phase and are a source of anomalies in the specific heat. In this work, we retrace the main steps that led us to attributing a central role to charge density fluctuations in the strange-metal phase of cuprates, discuss the state of the art on the issue and provide an in-depth analysis of the contribution of charge density fluctuations to the specific heat.
We investigate theoretically and experimentally the statistical properties of the inhomogeneous order-parameter distribution (OPD) at the verge of the superconductor-insulator transition (SIT). We find within two prototype fermionic and bosonic models for disordered superconductors that one can identify a universal rescaling of the OPD. By performing scanning-tunneling microscopy experiments in three samples of NbN with increasing disorder we show that such a rescaling also describes the experimental data with excellent accuracy. These results can provide a breakthrough in our understanding of the SIT.
The microscopic mechanism of charge instabilities and the formation of inhomogeneous states in systems with strong electron correlations is investigated. We demonstrate that within a strong coupling expansion the single-band Hubbard model shows an instability towards phase separation and extend the approach also for an analysis of phase separation in the Hubbard-Kanamori hamiltonian as a prototypical multiband model. We study the pairing fluctuations on top of an inhomogeneous stripe state where superconducting correlations in the extended s-wave and d-wave channels correspond to (anti)bound states in the two-particle spectra. Whereas extended s-wave fluctuations are relevant on the scale of the local interaction parameter U, we find that d-wave fluctuations are pronounced in the energy range of the active subband which crosses the Fermi level. As a result, low energy spin and charge fluctuations can transfer the d-wave correlations from the bound states to the low energy quasiparticle bands. Our investigations therefore help to understand the coexistence of stripe correlations and d-wave superconductivity in cuprates.
We study a model for low-doped cuprates where holes aggregate into oriented stripe segments which have a magnetic vortex and antivortex at the extremes. We argue that due to the interaction between segments a ferronematic state with macroscopic polarization is stabilized. This state can be characterized as a charge nematic which, due to the net polarization, breaks inversion symmetry and also exhibits an incommensurate spin modulation. Our calculation can reproduce the doping-dependent spin structure factor of lanthanum cuprates in excellent agreement with experiment and allows to rationalize experiments in which the incommensurability has an order-parameter-like temperature dependence.
The time-dependent Gutzwiller approximation is rederived on the basis of a variational wave function which is a direct product of a Slater determinant and a coherent state. The latter can be related to the slave-boson formulation by Kotliar and Ruckenstein, and our approach offers a convenient way to generalize the theory towards the implementation of symmetry-broken states and the study of non-equilibrium phenomena. We discuss the concept with regard to applications in the linear response limit and for quenched antiferromagnetic order for which we evaluate the optical conductivity in non-equilibrium situations.
High-temperature superconductors exhibit a characteristic hourglass-shaped spectrum of magnetic fluctuations which most likely contribute to the pairing glue in the cuprates. Recent neutron scattering experiments in strongly underdoped compounds have revealed a significant low energy anisotropy of these fluctuations which we explain by a model in which topological defects of the antiferromagnet clump to producing domain wall segments with ferronematic order. This state does not invoke global charge order but breaks C4 rotational and inversion symmetry. The incommensurability of the low doping charge-disordered state is in good agreement with experiment and interpolates smoothly with the incommensurability of the stripe phase at higher doping. Within linear spin-wave theory the dynamic structure factor is in very good agreement with inelastic neutron scattering data and can account for the observed energy dependent anisotropy.
Within a Lagrangian formalism, we derive the time-dependent Gutzwiller approximation for general multi-band Hubbard models. Our approach explicitly incorporates the coupling between time-dependent variational parameters and a time-dependent density matrix from which we obtain dynamical correlation functions in the linear-response regime. Our results are illustrated for the one-band model where we show that the interacting system can be mapped to an effective problem of fermionic quasiparticles coupled to 'doublon' (double occupancy) bosonic fluctuations. The latter have an energy on the scale of the on-site Hubbard repulsion U in the dilute limit but become soft at the Brinkman–Rice transition, which is shown to be related to an emerging conservation law of doublon charge and the associated gauge invariance. Coupling with the boson mode produces a structure in the charge response and we find that a similar structure appears in dynamical mean-field theory.
We present a short account of the present experimental situation of stripes in cuprates followed by a review of our present understanding of their ground state and excited state properties. Collective modes, the dynamical structure factor, and the optical conductivity of stripes are computed using the time-dependent Gutzwiller approximation applied to realistic one band and three band Hubbard models, and are found to be in excellent agreement with experiment. On the other hand, experiments like angle-resolved photoemission and scanning tunneling microscopy show the coexistence of stripes at high energies with Fermi liquid quasiparticles at low energies. We show that a phenomenological model going beyond mean-field can reconcile this dynamic dichotomy.
Electronic inhomogeneities are nowadays a well established feature of high-temperature superconductors, most prominently exemplified by the observation of charge-ordered states in a large number of cuprate materials. In this contribution we review the particular contribution of the Stuttgart-Tartu-Cottbus group to this field, which has been largely stimulated by intense discussion with K. Alex Müller. The original scenario, which was based on the formation of a percolative electronic network, is substantiated by a microscopic analysis of the clustering of magnetic polarons within the one-band Hubbard model.
We present a detailed theoretical investigation of the linear and nonlinear optical response in a model system for a disordered d-wave superconductor. By evaluating the quasiparticle contribution (BCS response) we show that for both quantities the gap symmetry considerably changes the paradigm of the optical response as compared to the conventional s-wave case. For what concerns the linear response our findings agree with previous work showing that in strongly disordered d-wave superconductors a large fraction of uncondensed spectral weight survives below Tc, making the optical absorption around the gap-frequency scale almost unchanged with respect to the normal state. Our numerical results are in excellent quantitative agreement with experiments in overdoped cuprates. In the nonlinear regime we focus on the third-harmonic generation (THG), finding that, as already established for the s-wave case, in general a large THG is triggered by disorder-activated paramagnetic processes. However, in the d-wave case the BCS response is monotonously increasing in frequency, losing any signature of THG enhancement when the THz pump frequency ω matches the gap maximum Δ, a hallmark of previous experiments in conventional s-wave superconductors. Our findings, along with the mild polarization dependence of the response, provide an explanation for recent THG measurements in cuprates, setting the framework for the theoretical understanding of nonlinear effects in unconventional cuprates.
Particle-particle response function as a probe for electronic correlations in the p-d Hubbard model
(2010)
Domain wall structures in the two-dimensional Hubbard modelwith long-range Coulomb interaction
(2000)
In underdoped cuprates, both stripes and spiral states may account for the incommensurate spin response observed by elastic neutron scattering experiments. Here, we investigate the respective stability of both textures within the framework of the three-band Hubbard model which we treat within the unrestricted Gutzwiller approximation. Our calculations indicate that for parameter sets appropriate for lanthanum cuprates and small doping nor purely longitudinal stripes nor uniform spirals are stable but stripes with significant spin canting. Indeed at small doping uniform spirals are unstable toward nanoscale phase separation.
It is shown that the magnetic structure of high-T c superconductors is strongly influenced by the next-nearest neighbor hopping parameter t' which distinguishes different families of cuprates. Our investigations indicate that uniform spirals get favored by a large t'=t ratio but are unstable at small doping towards stripes with spin canting. For large /t'/t/ spirals can be stabilized under certain conditions in the overdoped regime which may explain the elastic incommensurate magnetic response recently observed in iron-co-doped Bi2201 materials.
Empirical evidence in heavy fermion, pnictide and other systems suggests that unconventional superconductivity appears associated to some form of real-space electronic order. For the cuprates, despite several proposals, the emergence of order in the phase diagram between the commensurate antiferromagnetic state and the superconducting state is not well understood. Here we show that in this regime doped holes assemble in ‘electronic polymers’. Within a Monte Carlo study, we find that in clean systems by lowering the temperature the polymer melt condenses first in a smectic state and then in a Wigner crystal both with the addition of inversion symmetry breaking. Disorder blurs the positional order leaving a robust inversion symmetry breaking and a nematic order, accompanied by vector chiral spin order and with the persistence of a thermodynamic transition. Such electronic phases, whose properties are reminiscent of soft-matter physics, produce charge and spin responses in good accord with experiments.
According to the Goldstone theorem the breaking of a continuous U(1) symmetry comes along with the existence of low-energy collective modes. In the context of superconductivity these excitations are related to the phase of the superconducting (SC) order parameter and for clean systems are optically inactive; that is, single-mode excitations do not directly couple to light. Here we show that for strongly disordered superconductors phase modes acquire a dipole moment and appear as a subgap spectral feature in the optical conductivity. This finding is obtained with both a gauge-invariant random-phase approximation scheme based on a fermionic Bogoliubov–de Gennes state and a prototypical bosonic model for disordered superconductors. In the strongly disordered regime, where the system displays an effective granularity of the SC properties, the optically active dipoles are linked to the isolated SC islands, offering a new perspective for realizing microwave optical devices.
We investigate the disorder dependence of the static density, amplitude, and current correlations within the attractive Hubbard model supplemented with onsite disorder. It is found that strong disorder favors a decoupling of density and amplitude correlations due to the formation of superconducting (SC) islands. This emergent granularity also induces an enhancement of the density correlations on the SC islands whereas amplitude fluctuations are most pronounced in the “insulating” regions. While density and amplitude correlations are short ranged at strong disorder, we show that current correlations have a long-range tail due to the formation of percolative current paths in agreement with the constant behavior expected from the analysis of one-dimensional models.
Pseudogap and (An)isotropic Scattering in the Fluctuating Charge-Density Wave Phase of Cuprates
(2017)
We present a general scenario for high-temperature superconducting cuprates, based on the presence of dynamical charge density waves (CDWs) and to the occurrence of a CDW quantum critical point, which occurs, e.g., at doping p ≈ 0.16 in YBa2Cu3O6 + δ (YBCO). In this framework, the pseudogap temperature T∗ is interpreted in terms of a reduction of the density of states due to incipient CDW and, at lower temperature to the possible formation of incoherent superconducting pairs. The dynamically fluctuating character of CDW accounts for the different temperatures at which the CDW onset revealed by X-ray scattering (Tons(p)), and the static three-dimensional CDW ordering appear. We also investigate the anisotropic character of the CDW-mediated scattering. We find that this is strongly anisotropic only close to the CDW quantum critical point (QCP) at low temperature and very low energy. It rapidly becomes nearly isotropic and marginal-Fermi-liquid-like away from the CDW QCP and at finite (even rather small) energies. This may reconcile the interpretation of Hall measurements in terms of anisotropic CDW scattering with recent photoemission experiments Bok, J.M., et al. Sci. Adv. 2, e1501329 (2016).
Phase separation and proximity effects in itinerant ferromagnet/superconductor heterostructures
(2018)
Heterostructures made of itinerant ferromagnets and superconductors are studied. In contrast to most previous models, ferromagnetism is not enforced by an effective Zeeman field but induced in a correlated single-band model (CSBM) that displays itinerant ferromagnetism as a mean-field ground state. In this model superconductivity and magnetism are both calculated self-consistently. We calculate the magnitude of the magnetization, the superconducting correlations, and variations of the charge density self-consistently for a superconducting-magnetic bilayer by solving the Bogoliubov–de Gennes equations on a two-dimensional lattice. We determine all three quantities as a function of the Coulomb repulsion U and the ferromagnetic exchange interaction J. The CSBM displays a variety of features not present in the Zeeman exchange model—for example, the occurrence of electronic phase separation and the competition of magnetic and superconducting orders far away from the interface.
Nanoscale electronic inhomogeneities are nowadays a well established feature of high-temperature superconductors, most prominently exemplified by the observation of charge-ordered states in a large number of cuprate materials. These inhomogeneities have profound consequences for the electron–phonon coupling since they allow for long-range pairing interactions due to the reduced screening supplemented by the enhancement of the electron–phonon vertex due to strong correlations. In this chapter we review the particular contribution of the Stuttgart-Tartu-Cottbus group to this field and discuss how the current status of experimental results provides affirmation to their ideas.