Effective Electron-electron Interaction Research Articles

Topological origin of flat bands as pseudo-Landau levels in uniaxial strained graphene nanoribbons and induced magnetic ordering due to electron-electron interactions

Flat bands play a central role in the presence of correlated phases in moir\'e and other modulated two-dimensional systems. In this paper, flat bands are shown to exist in uniaxially periodic strained graphene. Such strain should be produced for example by a substrate. The model is thus mapped into a one-dimensional effective Hamiltonian and this allows to find the conditions for having flat bands, i.e., a long-wavelength modulation only on each one of the bipartite graphene sublattices, while having a tagged strain field between neighboring carbon atoms. The origin of such flat bands is thus tracked down to the existence of topological localized wavefunctions at domain walls separating different regions, each with a nonuniform Su-Schriffer-Hegger model (SSH) type of coupling. Thereafter, the system is mapped into a continuum model allowing to explain the numerical results in terms of the Jackiw-Rebbi model and of pseudo-Landau levels. Finally, the interplay between the obtained flat bands and electron-electron interaction is explored through the Hubbard model. The numerical results within the mean-field approximation indicate that the flat bands induce N\'eel antiferromagnetic and ferromagnetic domains even for a very weak Hubbard interaction, while the repulsive Hubbard interaction results in an effective electron-electron attraction. Thus this paper provides a simple, analytically solvable but realistic model to understand the physical origin of flat bands, pseudo-Landau levels and their effects on the effective electron-electron interaction.

Topological Superconductivity Mediated by Skyrmionic Magnons.

Topological superconductors are associated with the appearance of Majorana bound states, with promising applications in topologically protected quantum computing. In this Letter, we study a system where a skyrmion crystal is interfaced with a normal metal. Through interfacial exchange coupling, spin fluctuations in the skyrmion crystal mediate an effective electron-electron interaction in the normal metal. We study superconductivity within a weak-coupling approach and solve gap equations both close to the critical temperature and at zero temperature. Special features in the effective electron-electron interaction due to the noncolinearity of the magnetic ground state yield topological superconductivity at the interface.

Quantitative spin-dependent electron-electron interaction to calculate the superconducting parameters μ and λ

The spin-dependent Kukkonen-Overhauser (KO) effective electron-electron interaction in electron gas with a deformable background is used to calculate the BCS superconducting parameters $\ensuremath{\mu}$, ${\ensuremath{\mu}}^{*}$, and $\ensuremath{\lambda}$. The density and spin local field factors are utilized to incorporate the quantitative effects of exchange and correlation. The repulsive parameter $\ensuremath{\mu}$ is compared to results using the historical Thomas-Fermi or nearly equivalent random phase approximation (RPA) interactions. The resulting $\ensuremath{\mu}$ using the KO interaction is $45%$ larger at ${r}_{s}=1.65$; rising to $153%$ larger at ${r}_{s}=5.62$. Retardation reduces the effective repulsion, but ${\ensuremath{\mu}}^{*}$ is still $20\ensuremath{-}25%$ larger. The predicted superconducting transition temperature would be reduced by $5\ensuremath{-}20%$ using the McMillan formula. The attractive superconducting parameter $\ensuremath{\lambda}$, which depends on the electron-test charge interaction and the phonon spectrum, is also calculated using simple Debye-based models for phonon dispersion. This leads to a larger value of $\ensuremath{\lambda}$ than the same calculation using Thomas-Fermi or RPA interactions. Modern calculations of the phonon dispersion relations are often done using density functional theory. With the proper exchange and correlation kernel, the self-consistency of the method should yield the correct phonon dispersion relation and electron-phonon matrix elements. If the Thomas-Fermi or RPA interactions are used at any stage, the results for $\ensuremath{\lambda}$ are probably quantitatively inaccurate.

Asymptotic Pomeranchuk instability of Fermi liquids in half-filled Landau levels

We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form nu =2 , n+1/2). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the corresponding susceptibility never diverges, though the system is asymptotically unstable in the n rightarrow infty limit.

Theory of superconductivity mediated by Rashba coupling in incipient ferroelectrics

Experimental evidence suggests that superconductivity in ${\mathrm{SrTiO}}_{3}$ is mediated by a soft transverse ferroelectric mode which, according to conventional theories, has negligible coupling with electrons. A phenomenological Rashba-type coupling has been proposed on symmetry arguments but a microscopic derivation is lacking. Here, we fill this gap and obtain a linear coupling directly from a minimal microscopic model of the electronic structure. We find that the effective electron-electron pairing interaction has a strong momentum dependence. This yields an unusual situation in which the leading $s$-wave channel is followed by a subleading $p$-wave state which shows a stronger pairing instability than the $d$-wave state. The bare Rashba coupling constant is estimated for the lowest band of doped ${\mathrm{SrTiO}}_{3}$ with the aid of first-principles computations. Extrapolating the estimation to the multiband regime we show that it can explain bulk superconductivity. For densities where the Meissner effect has not been observed an extra softening due to inhomogeneities is needed to explain the zero resistance state.

Superconductivity from repulsive interactions in rhombohedral trilayer graphene: A Kohn-Luttinger-like mechanism

We study the emergence of superconductivity in rhombohedral trilayer graphene due purely to the long-range Coulomb repulsion. This repulsive-interaction-driven phase in rhombohedral trilayer graphene is significantly different from those found in twisted bilayer and trilayer graphenes. In the latter case, the nontrivial momentum-space geometry of the Bloch wavefunctions leads to an effective attractive electron-electron interaction; this allows for less modulated order parameters and for spin-singlet pairing. In rhombohedral trilayer graphene, we instead find spin-triplet superconductivity with critical temperatures up to 0.15 K. The critical temperatures strongly depend on electron filling and peak where the density of states diverge. The order parameter shows a significant modulation within each valley pocket of the Fermi surface.

Quantitative electron-electron interaction using local field factors from quantum Monte Carlo calculations

The effective electron-electron interaction in electron gas depends on both the density and spin local field factors. Variational diagrammatic quantum Monte Carlo calculations of the spin local field factor are reported. These are used together with the charge local field factor from previous diffusion quantum Monte Carlo calculations to quantitatively present the full effective spin-dependent electron-electron interaction in three-dimensional electron gas. Very simple quadratic formulas are presented for the local field factors that quantitatively produce all of the response functions of the electron gas at metallic densities. Exchange and correlation become increasingly important at low densities. At the compressibility divergence at ${r}_{s}=5.25$, both the direct (screened Coulomb) term and the charge-dependent exchange term in the electron-electron interaction at $q=0$ are separately divergent. However, due to large cancellations, their difference is finite, well behaved, and much smaller than either term separately. As a result, the spin contribution to the electron-electron interaction becomes an important factor. The static electron-electron interaction is repulsive as a function of density but is less repulsive for electrons with parallel spins. The effect of allowing a deformable, rather than rigid, positive background is shown to be as quantitatively important as exchange and correlation. As a simple concrete example, the electron-electron interaction is calculated using the measured bulk modulus of the alkali metals with a linear phonon dispersion. The net electron-electron interaction in lithium is attractive for wave vectors $0\ensuremath{-}2{k}_{F}$, which suggests superconductivity, and is mostly repulsive for the other alkali metals.

Vibrational effects on the formation of quantum W states

We theoretically investigate the formation of W states in a tripartite system composed of three charge qubits coupled to vibrational modes. The electromechanical coupling is responsible for second-order virtual processes that result in an effective electron-electron interaction between neighbor qubits, which leads to the formation of W states. Based on the Lang-Firsov transformation and perturbation theory, we analytically solve the quantum dynamics, providing a mathematical expression for the maximally entangled W state. Dephasing is also taken into account, paying particular attention to the robustness of bipartite entanglement against local dephasing processes.

Moiré versus Mott: Incommensuration and Interaction in One-Dimensional Bichromatic Lattices.

Inspired by the rich physics of twisted 2D bilayer moiré systems, we study Coulomb interacting systems subjected to two overlapping finite 1D lattice potentials of unequal periods through exact numerical diagonalization. Unmatching underlying lattice periods lead to a 1D bichromatic "moiré" superlattice with a large unit cell and consequently a strongly flattened band, exponentially enhancing the effective dimensionless electron-electron interaction strength and manifesting clear signatures of enhanced Mott gaps at discrete fillings. An important nonperturbative finding is a remarkable fine-tuning effect of the precise lattice commensuration, where slight variations in the relative lattice periods may lead to a suppression of the correlated insulating phase, in qualitative agreement with the observed fragility of the correlated insulating phase in twisted bilayer graphene. Our predictions, which should be directly verifiable in bichromatic optical lattices, establish that the competition between interaction and incommensuration is a key element of the physics of moiré superlattices.

Interplay between superconductivity and non-Fermi liquid at a quantum critical point in a metal. IV. The γ model and its phase diagram at 1<γ<2

In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical (QC) systems with an effective dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model). In previous papers we studied the cases $0<\gamma <1$ and $\gamma \approx 1$. We argued that the pairing by a gapless boson is fundamentally different from BCS/Eliashberg pairing by a massive boson as for the former there exists an infinite number of topologically distinct solutions for the gap function $\Delta_n (\omega_m)$ at $T=0$ ($n=0,1,2...$), each with its own condensation energy $E_{c,n}$. Here we extend the analysis to larger $1< \gamma <2$. We argue that the discrete set of solutions survives, and the spectrum of $E_{c,n}$ gets progressively denser as $\gamma$ approaches $2$ and eventually becomes continuous at $\gamma \to 2$. This increases the strength of "longitudinal" gap fluctuations, which tend to reduce the actual superconducting $T_c$ and give rise to a pseudogap region of preformed pairs. We also detect two features on the real axis for $\gamma >1$ which become critical at $\gamma\to 2$. First, the density of states evolves towards a set of discrete $\delta-$functions. Second, an array of dynamical vortices emerges in the upper frequency half-plane. These two features come about because on a real axis, the real part of the interaction, $V' (\Omega) \propto \cos(\pi \gamma/2)/|\Omega|^\gamma$, becomes repulsive for $\gamma >1$, and the imaginary $V^{''} (\Omega) \propto \sin(\pi \gamma/2)/|\Omega|^\gamma$, gets progressively smaller at $\gamma \to 2$. The features on the real axis are consistent with the development of a continuum spectrum of $E_{c,n}$ obtained using $\Delta_n (\omega_m)$ on the Matsubara axis. We consider the case $\gamma =2$ separately in the next paper.

Schwinger boson study of superconductivity mediated by antiferromagnetic spin fluctuations

We study superconductivity in a normal metal, arising from effective electron-electron interactions mediated by spin-fluctuations in a neighboring antiferromagnetic insulator. Introducing a frustrating next-nearest neighbor interaction in a N\'eel antiferromagnet with an uncompensated interface, the superconducting critical temperature is found to be enhanced as the frustration is increased. Further, for sufficiently large next-nearest neighbor interaction, the antiferromagnet is driven into a stripe phase, which can also give rise to attractive electron-electron interactions. For the stripe phase, as previously reported for the N\'eel phase, the superconducting critical temperature is found to be amplified for an uncompensated interface where the normal metal conduction electrons are coupled to only one of the two sublattices of the magnet. The superconducting critical temperature arising from fluctuations in the stripe phase antiferromagnet can be further enhanced by approaching the transition back to the N\'eel phase.

Interplay between superconductivity and non-Fermi liquid behavior at a quantum critical point in a metal. III. The γ model and its phase diagram across γ=1

In this paper we continue our analysis of the interplay between the pairing and the non-Fermi liquid behavior in a metal for a set of quantum-critical models with an effective dynamical electron-electron interaction $V(\Omega_m) \propto 1/|\Omega_m|^\gamma$ (the $\gamma$-model). We analyze both the original model and its extension, in which we introduce an extra parameter $N$ to account for non-equal interactions in the particle-hole and particle-particle channel. In two previous papers(arXiv:2004.13220 and arXiv:2006.02968), we considered the case $0 < \gamma <1$ and argued that (i) at $T=0$, there exists an infinite discrete set of topologically different gap functions, $\Delta_n (\omega_m)$, all with the same spatial symmetry, and (ii) each $\Delta_n$ evolves with temperature and terminates at a particular $T_{p,n}$. In this paper, we analyze how the system behavior changes between $\gamma <1$ and $\gamma >1$, both at $T=0$ and a finite $T$. The limit $\gamma \to 1$ is singular due to infra-red divergence of $\int d \omega_m V(\Omega_m)$, and the system behavior is highly sensitive to how this limit is taken. We show that for $N =1$, the divergencies in the gap equation cancel out, and $\Delta_n (\omega_m)$ gradually evolve through $\gamma=1$ both at $T=0$ and a finite $T$. For $N \neq 1$, divergent terms do not cancel, and a qualitatively new behavior emerges for $\gamma >1$. Namely, the form of $\Delta_n (\omega_m)$ changes qualitatively, and the spectrum of condensation energies, $E_{c,n}$ becomes continuous at $T=0$. We introduce different extension of the model, which is free from singularities for $\gamma >1$.

Dimensionality of metallic atomic wires on surfaces

We investigate the low-energy collective charge excitations (plasmons, holons) in metallic atomic wires deposited on semiconducting substrates. These systems are described by two-dimensional correlated models representing strongly anisotropic lattices or weakly coupled chains. Well-established theoretical approaches and results are used to study their properties: random phase approximation for anisotropic Fermi liquids and bosonization for coupled Tomonaga-Luttinger liquids as well as Bethe Ansatz and density-matrix renormalization group methods for ladder models. We show that the Fermi and Tomonaga-Luttinger liquid theories predict the same qualitative behavior for the dispersion of excitations at long wave lengths. Moreover, their scaling depends on the choice of the effective electron-electron interaction but does not characterize the dimensionality of the metallic state. Our results also suggest that such anisotropic correlated systems can exhibit two-dimensional dispersions due to the coupling between wires but remain quasi-one-dimensional strongly anisotropic conductors or retain typical features of Tomonaga-Luttinger liquids such as the power-law behaviour of the density of states at the Fermi energy. Thus it is possible that atomic wire materials such as Au/Ge(100) exhibit a mixture of features associated with one and two dimensional metals.

Quantum entanglement driven by electron-vibrational mode coupling

In this work, we provided a proof-of-principle of efficient production of maximally entangled states using charged quantum dots coupled to vibrational modes. The physical system consists of two pairs of quantum dots, each pair with a single electron able to tunnel between the dots, thus encoding a qubit. The electrons, initially not coupled, interact with two bosonic vibrational modes. It is demonstrated that the electron-vibrational mode coupling drives to an effective electron-electron interaction, which is the main mechanism behind the formation of maximally quantum entangled electronic states. The effect of this coupling follows a non-monotonic behavior, which is explained through an effective hamiltonian which takes into account high order transition processes.

Electron pairing by remote-phonon scattering in oxide-supported graphene

Using first-principles calculations we have previously shown that placing graphene on a (111)-oriented perovskite $\mathrm{SrTi}{\mathrm{O}}_{3}$ (STO) surface provides a possible doping mechanism [D. Shin and A. A. Demkov, Phys. Rev. B 97, 075423 (2018)]. Further theoretical analysis presented here suggests that coupling of electrons in graphene to interfacial hybrid plasmon/optical modes via remote-phonon scattering may result in an effective attractive electron-electron interaction that, in turn, could lead to electron pairing and superconductivity. Specifically, we consider top-gated graphene supported by STO. Using the full dynamic polarizability within the random phase approximation for the entire system (including the hybrid modes arising from the coupling of the graphene plasmons to the optical phonons of the STO substrate and gate insulator), we estimate the superconducting transition temperature in the strong-coupling limit.

Superconductivity near a Ferroelectric Quantum Critical Point in Ultralow-Density Dirac Materials

The experimental observation of superconductivity in doped semimetals and semiconductors, where the Fermi energy is comparable to or smaller than the characteristic phonon frequencies, is not captured by the conventional theory. In this paper, we propose a mechanism for superconductivity in ultralow-density three-dimensional Dirac materials based on the proximity to a ferroelectric quantum critical point. We derive a low-energy theory that takes into account both the strong Coulomb interaction and the direct coupling between the electrons and the soft phonon modes. We show that the Coulomb repulsion is strongly screened by the lattice polarization near the critical point even in the case of vanishing carrier density. Using a renormalization group analysis, we demonstrate that the effective electron-electron interaction is dominantly mediated by the transverse phonon mode. We find that the system generically flows towards strong electron-phonon coupling. Hence, we propose a new mechanism to simultaneously produce an attractive interaction and suppress strong Coulomb repulsion, which does not require retardation. For comparison, we perform same analysis for covalent crystals, where lattice polarization is negligible. We obtain qualitatively similar results, though the screening of the Coulomb repulsion is much weaker. We then apply our results to study superconductivity in the low-density limit. We find strong enhancement of the transition temperature upon approaching the quantum critical point. Finally, we also discuss scenarios to realize a topological $p$-wave superconducting state in covalent crystals close to the critical point.

Position representation of effective electron-electron interactions in solids

An essential ingredient in many model Hamiltonians, such as the Hubbard model, is the effective electron-electron interaction $U$, which enters as matrix elements in some localized basis. These matrix elements provide the necessary information in the model, but the localized basis is incomplete for describing $U$. We present a systematic scheme for computing the manifestly basis-independent dynamical interaction in position representation, $U({\bf r},{\bf r}';\omega)$, and its Fourier transform to time domain, $U({\bf r},{\bf r}';\tau)$. These functions can serve as an unbiased tool for the construction of model Hamiltonians. For illustration we apply the scheme within the constrained random-phase approximation to the cuprate parent compounds La$_2$CuO$_4$ and HgBa$_2$CuO$_4$ within the commonly used 1- and 3-band models, and to non-superconducting SrVO$_{3}$ within the $t_{2g}$ model. Our method is used to investigate the shape and strength of screening channels in the compounds. We show that the O 2$p_{x,y}-$Cu 3$d_{x^2-y^2}$ screening gives rise to regions with strong attractive static interaction in the minimal (1-band) model in both cuprates. On the other hand, in the minimal ($t_{2g}$) model of SrVO$_3$ only regions with a minute attractive interaction are found. The temporal interaction exhibits generic damped oscillations in all compounds, and its time-integral is shown to be the potential caused by inserting a frozen point charge at $\tau=0$. When studying the latter within the three-band model for the cuprates, short time intervals are found to produce a negative potential.

Comparative study of nonequilibrium insulator-to-metal transitions in electron-phonon systems

We study equilibrium and nonequilibrium properties of electron-phonon systems described by the Hubbard-Holstein model using the dynamical mean-field theory. In equilibrium, we benchmark the results for impurity solvers based on the one-crossing approximation and slave-rotor approximation against non-perturbative numerical renormalization group reference data. We also examine how well the low energy properties of the electron-boson coupled systems can be reproduced by an effective static electron-electron interaction. The one-crossing and slave-rotor approximations are then used to simulate insulator-to-metal transitions induced by a sudden switch-on of the electron-phonon interaction. The slave-rotor results suggest the existence of a critical electron-phonon coupling above which the system is transiently trapped in a non-thermal metallic state with coherent quasiparticles. The same quench protocol in the one-crossing approximation results in a bad metallic state.

All-optical nonequilibrium pathway to stabilising magnetic Weyl semimetals in pyrochlore iridates

Nonequilibrium many-body dynamics is becoming a central topic in condensed matter physics. Floquet topological states were suggested to emerge in photodressed bands under periodic laser driving. Here we propose a viable nonequilibrium route without requiring coherent Floquet states to reach the elusive magnetic Weyl semimetallic phase in pyrochlore iridates by ultrafast modification of the effective electron-electron interaction with short laser pulses. Combining ab initio calculations for a time-dependent self-consistent light-reduced Hubbard U and nonequilibrium magnetism simulations for quantum quenches, we find dynamically modified magnetic order giving rise to transiently emerging Weyl cones that can be probed by time- and angle-resolved photoemission spectroscopy. Our work offers a unique and realistic pathway for nonequilibrium materials engineering beyond Floquet physics to create and sustain Weyl semimetals. This may lead to ultrafast, tens-of-femtoseconds switching protocols for light-engineered Berry curvature in combination with ultrafast magnetism.

Effect of the film thickness on the effective electron-electron interaction in a metal film

Effect of the film thickness on the effective electron-electron interaction in a metal film