A possible origin of the high-temprature superconductivity in cuprates has been suggested. It is supposed that electron-phonon interaction determines the strong correlation narrowing of the electron band. It provides the conditions for the formation of a singlet electron pair coupled by exchange interaction. For the pure t-J model it has been proved that these electron pairs are destroyed by a strong effective kinematic field. The detailed analysis of an influence of the Holstein polaron excitations upon normal and superconducting properties of the strongly correlated electrons was made. A calculated critical temperature of the superconductivity and gap function are in good agreement with experimental data for cuprates.
The interplay between electron-electron correlations and disorder has been a central theme of condensed matter physics over the last several decades, with particular interest in the possibility that interactions might cause delocalization of an Anderson insulator into a metallic state, and the disrupting effects of randomness on magnetic order and the Mott phase. Here we extend this physics to explore electron-phonon interactions and show, via exact quantum Monte Carlo simulations, that the suppression of the charge density wave correlations in the half-filled Holstein model by disorder can st…
We present a theoretical study of superconductivity of polarons in the Hubbard-Holstein model. A residual kinematic interaction proportional to the square of the polaron hopping energy between polarons and phonons provides a pairing field for the polarons. We find that superconducting instability in the d-wave channel is possible with small transition temperatures which is maximum in the large to small polaron cross-over region. An s-wave instability is found to be not possible when the effective on-site interaction between polarons is repulsive.
We examine the superconducting solution in the Holstein model, where the conduction electrons couple to the dispersionless Boson fields, using the Migdal-Eliashberg theory and Dynamical Mean Field Theory. Although different in numerical values, both methods imply the existence of an optimal Boson energy for superconductivity at a given electron-Boson coupling. This non-monotonous behavior can be understood as an interplay between the polaron and superconducting physics, as the electron-Boson coupling is the origin of the superconductor, but at the same time traps the conduction electrons makin…
The electron-phonon (e-ph) interaction remains of great interest in condensed matter physics and plays a vital role in realizing superconductors, charge-density-waves (CDW), and polarons. We study the two-dimensional Holstein model for e-ph coupling using determinant quantum Monte Carlo across a wide range of its phase diagram as a function of temperature, electron density, dimensionless e-ph coupling strength, and the adiabatic ratio of the phonon frequency to the Fermi energy. We describe the behavior of the CDW correlations, the competition between superconducting and CDW orders and polaron…
In one dimension the coupling of electrons to phonons leads to a transition from a metallic to a Peierls distorted insulated state if the coupling exceeds a critical value. On the other hand, in two dimensions the electron-phonon interaction may also lead to the formation of Cooper pairs. This competition of superconductivity and charge order (in conjunction with a lattice distortion) is studied in this letter by means of the projector-based renormalization method (PRM). Increasing the electron-phonon interaction, we find a crossover behavior between a purely superconducting state and a charge…
The Holstein Hamiltonian describes fermions hopping on a lattice and interacting locally with dispersionless phonon degrees of freedom. In the low density limit, dressed quasiparticles, polarons and bipolarons, propagate with an effective mass. At higher densities, pairs can condense into a low temperature superconducting phase and, at or near commensurate filling on a bipartite lattice, to charge density wave (CDW) order. CDW formation breaks a discrete symmetry and hence occurs via a second order (Ising) transition, and therefore at a finite $T_{\rm cdw}$ in two dimensions. Quantum Monte Car…
In iron pnictides, we find that the moderate electron-phonon interaction due to the Fe-ion oscillation can induce the critical d-orbital fluctuations, without being prohibited by the Coulomb interaction. These fluctuations give rise to the strong pairing interaction for the s-wave superconducting (SC) state without sign reversal ($s_{++}$-wave state), which is consistent with experimentally observed robustness of superconductivity against impurities. When the magnetic fluctuations due to Coulomb interaction are also strong, the SC state shows a smooth crossover from the s-wave state with sign …
Quantum Monte Carlo Simulation of Bipolaron Superconductivity in Extended Hubbard--Holstein models on Face-Centered-Cubic and Body-Centered-Cubic Lattices
Tuning superconductivity and charge density wave order by next-nearest-neighbor hopping integral in honeycomb Holstein model
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理学院）；Key Laboratory of Multiscale Spin Physics (Beijing Normal University), Ministry of Education（教育部多尺度自旋物理重点实验室（北京师范大学））；School of Physical Science and Technology, Beijing University of Posts and Telecommunications（北京邮电大学理学院）
By using variational wave functions and quantum Monte Carlo techniques, we investigate the interplay between electron-electron and electron-phonon interactions in the two-dimensional Hubbard-Holstein model. Here, the ground-state phase diagram is triggered by several energy scales, i.e., the electron hopping $t$, the on-site electron-electron interaction $U$, the phonon energy $ω_0$, and the electron-phonon coupling $g$. At half filling, the ground state is an antiferromagnetic insulator for $U \gtrsim 2g^2/ω_0$, while it is a charge-density-wave (or bi-polaronic) insulator for $U \lesssim 2g^…
The search for realistic one-dimensional (1D) models that exhibit dominant superconducting (SC) fluctuations effects has a long history. In these 1D systems, the effects of commensurate band fillings--strongest at half-filling--and electronic repulsions typically lead to a finite charge gap and the favoring of insulating density wave ordering over superconductivity. Accordingly, recent proposals suggesting a gapless metallic state in the Holstein-Hubbard (HH) model, possibly superconducting, have generated considerable interest and controversy, with the most recent work demonstrating that the …
We investigate a two-orbital model for iron-based superconductors to elucidate the effect of interplay between electron correlation and Jahn-Teller electron-phonon coupling by using the dynamical mean-field theory combined with the exact diagonalization method. When the intra- and inter-orbital Coulomb interactions, $U$ and $U'$, increase with $U=U'$, both the local spin and orbital susceptibilities, $χ_{s}$ and $χ_{o}$, increase with $χ_{s}=χ_{o}$ in the absence of the Hund's rule coupling $J$ and the electron-phonon coupling $g$. In the presence of $J$ and $g$, there are distinct two regimes…
Looking for superconductors with higher transition temperature requires a guiding principle. In conventional superconductors, electrons pair up into Cooper pairs via the retarded attraction mediated by electron-phonon coupling. Higher-frequency phonon (or smaller atomic mass) leads to higher superconducting transition temperature, known as the isotope effect. Furthermore, superconductivity is the only instability channel of the metallic normal state. In correlated systems, the above simple scenario could be easily violated. The strong local interaction is poorly screened, and this conspires wi…
The two-dimensional many-body Holstein-Hubbard model in the T=0 normal state is examined within the framework of the self-consistent coupling of charge fluctuation correlations to the vibrational ones. The parameters of our model are the adiabaticity, the electron concentration, as well as the electron-phonon and the Coulomb interaction strengths. A fluctuation-based effective cumulant approach is introduced to examine the T=0 normal-state fluctuations and an analytic approximation to the true dynamical entangled ground state is suggested. Our results for the effective charge-transfer amplitud…
We propose a numerical method which embeds the variational non-Gaussian wavefunction approach within exact diagonalization, allowing for efficient treatment of correlated systems with both electron-electron and electron-phonon interactions. Using a generalized polaron transformation, we construct a variational wavefunction that absorbs entanglement between electrons and phonons into a variational non-Gaussian transformation; exact diagonalization is then used to treat the electronic part of the wavefunction exactly, thus taking into account high-order correlation effects beyond the Gaussian le…
We study the competition between an instantaneous local Coulomb repulsion and a boson mediated retarded attraction, as described by the Hubbard-Holstein model. Restricting to the case of half filling, the ground-state phase diagram and the transitions from antiferromagnetically ordered states to charge ordered states are analyzed. The calculations are based on the model in large dimensions, so that dynamical mean field theory can be applied, and the associated impurity problem is solved using the numerical renormalization group method. The transition is found to occur when electron-electron co…
I present results from an extended Migdal-Eliashberg theory of electron-phonon interactions and superconductivity. The history of the electron-phonon problem is introduced, and then study of the intermediate parameter regime is justified from the energy scales in the cuprate superconductors. The Holstein model is detailed, and limiting cases are examined to demonstrate the need for an extended theory of superconductivity. Results of the extended approximation are shown, including spectral functions and phase diagrams. These are discussed with reference to Hohenberg's theorem, the Bardeen-Coope…
I examine electron-phonon mediated superconductivity in the intermediate coupling and phonon frequency regime of the quasi-2D Holstein model. I use an extended Migdal-Eliashberg theory which includes vertex corrections and spatial fluctuations. I find a d-wave superconducting state that is unique close to half-filling. The order parameter undergoes a transition to s-wave superconductivity on increasing filling. I explain how the inclusion of both vertex corrections and spatial fluctuations is essential for the prediction of a d-wave order parameter. I then discuss the effects of a large Coulom…
The theory of electron-phonon superconductivity depends on retardation drastically reducing effects of the strong Coulomb repulsion. The standard theory only treats the lowest order diagram, which is an uncontrolled approximation. We study retardation in the Hubbard-Holstein model in a controlled way using perturbation theory and dynamical mean-field theory. We calculate analytically second order results for the pseudopotential $μ^*$ and demonstrate the validity up to intermediate couplings by comparison with non-perturbative results. Retardation effects are still operative, but less efficient…
Electron-lattice interactions play a prominent role in quantum materials, making a deeper understanding of direct routes to phonon-mediated high-transition-temperature ($T_{\mathrm{c}}$) superconductivity desirable. However, it has been known for decades that weak electron-phonon coupling gives rise to low values of $T_{\mathrm{c}}$, while strong electron-phonon coupling leads to lattice instability or formation of bipolarons, generally assumed to be detrimental to superconductivity. Thus, the route to high-$T_{\mathrm{c}}$ materials from phonon-mediated mechanisms has heretofore appeared to b…
We study the five-orbital Hubbard-Holstein model for iron pnictides with small electron-phonon interaction due to Fe-ion Einstein oscillators. Using the fluctuation-exchange (FLEX) approximation, orbital fluctuations evolve inversely proportional to the temperature, and therefore the resistivity shows linear or convex T-dependence for wide range of temperatures. We also analyze the Eliashberg gap equation, and show that s-wave superconducting state without sign reversal (s_{++}-wave state) emerges when the orbital fluctuations dominate the spin fluctuations. When both fluctuations are comparab…
The normal state of the Holstein model is studied at half-filling in infinite dimensions and in the adiabatic regime. The dynamical mean-field equations are solved using perturbation expansions around the extremal paths of the effective action for the atoms. We find that the Migdal-Eliashberg expansion breaks down in the metallic state if the electron-phonon coupling $λ$ exceeds a value of about 1.3 in spite of the fact that the formal expansion parameter $λω_0/E_F$ ($ω_0$ is the phonon frequency, $E_F$ the Fermi energy) is much smaller than 1. The breakdown is due to the appearance of more th…
Determinant quantum Monte Carlo (DQMC) simulations are used to study non-linear electron-phonon interactions in a two-dimensional Holstein-like model on a square lattice. We examine the impact of non-linear electron-lattice interactions on superconductivity and on Peierls charge-density-wave (CDW) correlations at finite temperatures and carrier concentrations. We find that the CDW correlations are dramatically suppressed with the inclusion of even a small non-linear interaction. Conversely, the effect of the non-linearity on superconductivity is found to be less dramatic at high temperatures; …
In the theory of electron-phonon superconductivity both the magnitude of the electron-phonon coupling $λ$ as well as the Coulomb pseudopotential $μ^*$ are important to determine the transition temperature $T_c$ and other properties. We calculate corrections to the conventional result for the Coulomb pseudopotential. Our calculation are based on the Hubbard-Holstein model, where electron-electron and electron-phonon interactions are local. We develop a perturbation expansion, which accounts for the important renormalization effects for the electrons, the phonons, and the electron-phonon vertex.…
I apply the newly developed dynamical cluster approximation (DCA) to the calculation of the electron and phonon dispersions in the two dimensional Holstein model. In contrast to previous work, the DCA enables the effects of spatial fluctuations (non-local corrections) to be examined. Approximations neglecting and incorporating lowest-order vertex corrections are investigated. I calculate the phonon density of states, the renormalised phonon dispersion, the electron dispersion and electron spectral functions. I demonstrate how vertex corrections stabilise the solution, stopping a catastrophic s…
High-temperature superconductivity induced by the Su-Schrieffer-Heeger electron-phonon coupling
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Institute for Advanced Study, Tsinghua University（清华大学高等研究院）
A possibility of the odd-frequency pairing in the strong-coupling electron-phonon systems is discussed. Using the Holstein-Hubbard model, we demonstrate that the anomalously soft Einstein mode with the frequency $ω_{\rm E}\llω_{c}$ ($ω_{c}$ is the order of the renormalized bandwidth) mediates the s-wave odd-frequency triplet pairing against the ordinary even-frequency singlet pairing. It is necessary for the emergence of the odd-frequency pairing that the pairing interaction is strongly retarded as well as the strong coupling, since the pairing interaction for the odd-frequency pairing is effe…
Recently developed numerical methods have enabled the explicit construction of the superconducting state of the Hubbard model of strongly correlated electrons in parameter regimes where the model also exhibits a pseudogap and a Mott insulating phase. $d_{x^2-y^2}$ symmetry superconductivity is found to occur in proximity to the Mott insulator, but separated from it by a pseudogapped nonsuperconducting phase. The superconducting transition temperature and order parameter amplitude are found to be maximal at the onset of the normal-state pseudogap. The emergence of superconductivity from the nor…
Motivated by the remarkable discovery of superconductivity in elemental Bismuth, we study its normal state in detail using a combination of tight-binding (TB) and-structural supplemented by dynamical mean-field theory (DMFT). We show that a two-fluid model composed of preformed and dynamically fluctuating excitons coupled to a tiny number of carriers provides a unified rationalization of a range of ill-understood normal state spectral and transport data. Based on these, we propose that resonant scattering involving a very low density of renormalized carriers and the excitonic liquid drives log…
Recent experiments revealed that the plain $s$-wave state without any sign-reversal emerges in various metals near the magnetic criticality. To understand this counter-intuitive phenomenon, we study the gap equation for the multiorbital Hubbard-Holstein model, by analyzing the vertex correction (VC) due to the higher-order electron-correlation effects. We find that the phonon-mediated orbital fluctuations are magnified by the VC for the susceptibility ($χ$-VC). In addition, the charge-channel attractive interaction is enlarged by the VC for the coupling-constant ($U$-VC), which is significant …
Extending the BCS theory towards the strong electron-phonon interaction (EPI), a charged Bose liquid of small bipolarons has been predicted by us with a further prediction that the highest superconducting critical temperature is found in the crossover region of the EPI strength from the BCS-like to bipolaronic superconductivity. Later on we have shown that the unscreened (infinite-range) Froehlich EPI combined with the strong Coulomb repulsion create \emph{superlight} small bipolarons, which are several orders of magnitude lighter than small bipolarons in the Holstein-Hubbard model (HHM) with …
After three decades of enormous scientific inquiry, the emergence of superconductivity in the cuprates remains an unsolved puzzle. One major challenge has been to arrive at a satisfactory understanding of the unusual metallic normal state from which the superconducting state emerges upon cooling. A second challenge has been to achieve a unified understanding of hole- and electron-doped compounds. Here we report detailed magnetoresistivity measurements for the archetypal electron-doped cuprate Nd$_{2-x}$Ce$_x$CuO$_{4+δ}$ that, in combination with prior data, provide crucial links between the no…
We report Hall measurement of the normal state in K- and Co-doped BaFe$_2$As$_2$, as well NaFe$_{1-x}$Co$_x$As. We found that a power-law temperature dependence of Hall angle, cot$θ_{\rm H}$$\propto$ $T^β$, prevails in normal state with temperature range well above the structural, spin-density-wave and superconducting transitions for the all samples with various doping levels. The power $β$ is nearly 4 for the parent compounds and the heavily underdoped samples, while around 3 for the superconducting samples. The $β$ suddenly changes from 4 to 3 at a doping level that is close to the emergence…
We have measured the magnetic field and temperature dependence of specific heat on $Bi_2Sr_{2-x}La_xCuO_{6+δ}$ single crystals in wide doping and temperature regions. The superconductivity related specific heat coefficient $γ_{sc}$ and entropy $S_{sc}$ are determined. It is found that $γ_{sc}$ has a hump-like anomaly at $T_c$ and behaves as a long tail which persists far into the normal state for the underdoped samples, but for the heavily overdoped samples the anomaly ends sharply just near $T_c$. Interestingly, we found that the entropy associated with superconductivity is roughly conserved …
UTe2 is a promising candidate for spin-triplet superconductors, in which a paramagnetic normal state becomes superconducting due to spin fluctuations. The subsequent discovery of various unusual superconducting properties has promoted the use of UTe2 as an exciting playground to study unconventional superconductivity, but fathoming the normal state fermiology and its influence on the superconductivity still requires further investigation. Here, we theoretically show that electron correlation induces a dramatic change in the normal state fermiology with an emergent correlated Fermi surface (FS)…
We derive a phase diagram for the pseudogap onset temperature $T^*$ (associated with the breakdown of the Fermi liquid state, due to strong pairing correlations) and the superconducting instability, $T_c$, as a function of variable pairing strength. Our diagrammatic approach to the BCS - Bose-Einstein cross-over problem self consistently treats the coupling between the single particle and pair propagators, and leads to a continuous evolution of these propagators into the standard $T<T_c$ counterparts. A rich structure is found in $T_c$ which reflects the way in which the superconducting instab…
We analyze a minimal model of SrTiO3, such as has been historically used to analyze the superconducting state. Treating the electron-phonon and electron-electron interactions on an equal footing, we calculate the normal state properties, focusing on the spectral function. We find a density-driven crossover caused mostly by the adiabaticity of the bosonic modes, but the spectral features of our theory are dominated by coupled plasmon-phonon oscillations rather than the bare phonon features seen in experiment.
In this paper, we study superconductivity of nonrelativistic fermions at finite-density coupled to a transverse $U(1)$ gauge field, with the effective interaction including the Landau-damping. This model, first studied by Holstein, Norton, and Pincus [Phys. Rev B, {\bf 8}, 2649 (1973)] has been known as an example of a non-Fermi liquid, {ı.e.} a metallic state in which the decay rate of a quasiparticle is large compared to the characteristic quasiparticle energy; other examples of the non-Fermi liquid includes the 2d electron gas in a magnetic field at $ν=1/2$ and the normal state of optimally…
We carried out a comparative study of the in-plane resistivity and optical spectrum of doped BaFe2As2 and investigated the doping evolution of the charge dynamics. For BaFe2As2, charge dynamics is incoherent at high temperatures. Electron (Co) and isovalent (P) doping into BaFe2As2 increase coherence of the system and transform the incoherent charge dynamics into highly coherent one. On the other hand, charge dynamics remains incoherent for hole (K) doping. It is found in common with any type of doping that superconductivity with high transition temperature emerges when the normal-state charge…
The search for topological superconductors is one of the most pressing and challenging questions in condensed matter and material research. Despite some early suggestions that doping a topological insulator might be a successful recipe to find topological superconductors, until today there is no general understanding of the relationship of the topology of the superconductor and the topology of its underlying normal state system. One of the major obstacles is the strong effect of the Fermi surface and its subsequent pairing tendencies within the Hubbard model framework, usually preventing a det…
We report both 77Se and 87Rb NMR studies on Tl0.47Rb0.34Fe1.63Se2 single-crystalline superconductors (T_c about 32 K). Singlet superconductivity is decisively determined by a sharp drop of the Knight shift K(T) below T_c, after subtracting the superconducting diamagnetic effect. However, the Hebel-Slichter coherence peak below T_c is not observed in the spin-lattice relaxation rate 1/T_1, even with a low in-plane NMR field of 2.6 Tesla. Just above T_c, no evidence of low-energy spin fluctuation is found in the spin-lattice relaxation rate on both the 77Se and the 87Rb sites. Upon warming, howe…
In unconventional superconductors, it is generally believed that understanding the physical properties of the normal state is a pre-requisite for understanding the superconductivity mechanism. In conventional superconductors like niobium or lead, the normal state is a Fermi liquid with a well-defined Fermi surface and well-defined quasipartcles along the Fermi surface. Superconductivity is realized in this case by the Fermi surface instability in the superconducting state and the formation and condensation of the electron pairs (Cooper pairing). The high temperature cuprate superconductors, on…
Photoemission and optical experiments indicate that the transition to superconductivity in cuprates is an 'undressing' transition . In photoemission this is seen as a coherent quasiparticle peak emerging from an incoherent background, in optics as violation of the Ferrell-Glover-Tinkham sum rule indicating effective mass reduction of superconducting carriers. We propose that this is a manifestation of the fundamental electron-hole asymmetry of condensed matter described by the theory of hole superconductivity. The theory asserts that electrons in nearly empty bands and holes in nearly full ban…
We report a systematic investigation, together with a theoretical analysis, of the resistivity and Hall effect in single crystals of Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$, over a wide doping range. We find a surprisingly great disparity between the relaxation rates of the holes and the electrons, in excess of an order of magnitude in the low-doping, low-temperature regime. The ratio of the electron to hole mobilities diminishes with temperature and doping (away from the magnetically ordered state) and becomes more conventional. We also find a straightforward explanation of the large asymmetry (…
We study the Kondo lattice model with additional attractive interaction between the conduction electrons within the dynamical mean-field theory using the numerical renormalization group to solve the effective quantum impurity problem. In addition to normal-state and magnetic phases we also allow for the occurrence of a superconducting phase. In the normal phase we observe a very sensitive dependence of the low-energy scale on the conduction-electron interaction. We discuss the dependence of the superconducting transition on the interplay between attractive interaction and Kondo exchange.
I describe the issues of the competing orders in normal state of a parent and Ni-doped iron pnictide superconductor, SrF e 2 As 2 , using LMTO band structure calculation plus multi orbital dynamical mean-field theory. Strong, electronic correlations along with minimal antiferromagnetic instability create a partially gapped Fermi surface, in qualitative agreement with earlier experiments. A good semiquantitative similarity in both normal and ordered state with the earlier experimental transport data is observed along with suppression of magnetic ordering and explained within a correlated, mul- …
An understanding of the normal state in the high-temperature superconducting cuprates is crucial to the ultimate understanding of the long-standing problem of the origin of the superconductivity itself. This so-called strange metal state is thought to be associated with a quantum critical point (QCP) hidden beneath the superconductivity(1,2). In electron-doped cuprates in contrast to hole-doped cuprates it is possible to access the normal state at very low temperatures and low magnetic fields to study this putative QCP and to probe the T~0 K state of these materials(3,4). We report measurement…
