High temperature superconductivity from realistic Coulomb and Fröhlich interactions

# High temperature superconductivity from realistic Coulomb and Fröhlich interactions

## Abstract

In the last years ample experimental evidence has shown that charge carriers in high-temperature superconductors are strongly correlated but also coupled with lattice vibrations (phonons), signaling that the true origin of high-T superconductivity can only be found in a proper combination of Coulomb and electron-phonon interactions. On this basis, we propose and study a model for high-T superconductivity, which accounts for realistic Coulomb repulsion, strong electron-phonon (Fröhlich) interaction and residual on-site (Hubbard ) correlations without any ad-hoc assumptions on their relative strength and interaction range. In the framework of this model, which exhibits a phase transition to a superconducting state with a critical temperature T well in excess of K, we emphasize the role of as the driving parameter for a BEC/BCS crossover. Our model lays a microscopic foundation for the polaron-bipolaron theory of superconductivity. We argue that the high-T phenomenon originates in competing Coulomb and Fröhlich interactions beyond the conventional BCS description.

###### pacs:
74.20.-z, 74.72.-h, 71.38.-k

## I Introduction

Unconventional symmetries of the order parameter allowed some researchers to maintain that a purely repulsive interaction between electrons (Hubbard ) accounts for superconductivity without phonons in a number of high-temperature superconductors pla (). However, recent analytical alekab2011 () and numerical (Monte-Carlo) MC (); MCHardy () studies shed doubts on the possibility of high temperature superconductivity from repulsive interactions only.

Also a growing number of experimental and theoretical results suggest that strong electron correlations and significant electron-phonon interaction (EPI) are the unavoidable features for a microscopic theory of high T superconductivity aledev, ; alespecial, . In particular the doping dependent oxygen-isotope effects on the critical temperature and on the in-plane supercarrier mass (ref. ZhaoYBCO95 (); ZhaoLSCO95 (); ZhaoNature97 (); ZhaoJPCM98 (); Zhao01 (); Keller1 ()), provide direct evidence for a significant EPI and bipolaronic carriers asazhaoiso () in high-temperature cuprate superconductors. Angle-resolved photoemission spectra (ARPES) lanzara (); shen () provide further evidence for the strong EPI apparently with c-axis-polarised optical phonons meev (). Some theoretical models show that detailed understanding of ARPES requires EPI alechris (); mish2 (); hoh (); gun (); shen () and the lattice disorder alekim (); ale () to be taken into account along with strong correlations. These results as well as neutron scattering ega (); rez (), tunneling Zhao07 (); Boz08 (); ZhaoPRL09 (), pump-probe Gadermaier2010 (), earlier mih () and more recent mish () optical specroscopies unambiguously show that lattice vibrations play a significant but unconventional role in high-temperature superconductivity.

Since first proposed AlexandrovRanninger (), much attention has been paid to the strong EPI as a mechanism of superconductivity providing effective on-site and inter-site attractions between small polarons (electrons dressed by a cloud of phonons) AlexandrovMott (). In the framework of negative Hubbard- and extended negative Hubbard-U models, the strong electron-phonon coupling results in a bound state of two polarons that condense with a Bose-Einstein critical temperature strictly related to the mobility of the pairs Loktev (). However, the failure of these models in predicting a high critical temperature, due to localization of pairs in the strong coupling regime with some particular (Holstein) EPI, led to a better understanding of more realistic EPIs as the competing interactions with respect to Coulomb repulsion.

Analytical and numerical calculations alebra2 (); bau () clarified that EPI with high-frequency optical phonons in ionic solids remains poorly screened signaling the presence of long-range (Fröhlich) EPI at any density of polarons with a remarkable reduction of the polaron effective mass. Consistently, studies on the so called “Fröhlich-Coulomb” model (FCM) FCM (); SCinFCM (), in which strong long-range EPI and long-range Coulomb repulsion are treated on equal footing, predict light polarons and bipolarons (bound state of two polarons) in cuprates with a remarkably high superconducting critical temperature in the range in which all the interactions are strong compared with the kinetic energy of carriers. The interpretation of the optical spectra of high-T materials as the polaron absorption mih (); dev (); aledev () strengthens the view FCM () that the Fröhlich EPI is important in those compounds.

In most analytical and numerical models of high-temperature superconductivity, proposed so far, one or both Coulomb and electron-phonon interactions have been introduced as input parameters not directly related to the material. Different from those studies an analytical multi-polaron model of high-temperature superconductivity in highly polarisable ionic lattices has been recently proposed Alexandrov2011 () and numerically studied (for two-particle states) sica () with generic (bare) Coulomb and Fröhlich interactions avoiding any ad-hoc assumptions on their range and relative magnitude. It has been shown that the generic Hamiltonian comprising any-range Coulomb repulsion and the Fröhlich EPI can be reduced to a short-range model at very large lattice dielectric constant, , for the moderate and strong EPI. In this limit the bare static Coulomb repulsion and EPI negate each other giving rise to a novel physics described by the polaronic model with a short-range polaronic spin-exchange of phononic origin Alexandrov2011 ().

The cancelation of the bare Coulomb repulsion by the Fröhlich EPI is accurate up to corrections. At finite a residual on-site repulsion of polarons, , could be substantial if the size of the Wannier (atomic) orbitals is small enough. Here we study the effect of this on-site repulsion on the ground state of the extended -- model accounting for all essential correlations in high-temperature superconductors. It is worth emphasizing that the effect of the on-site does not follows as a mere generalization of the - model. The residual Hubbard in fact leads not only to the suppression of on-site pairs but also to the reduction of the exchange interaction and to the Bose-Einstein condensation (BEC) to BCS (BEC/BCS) crossover.

## Ii Bare Hamiltonian

Keeping major terms in both interactions, diagonal with respect to sites, yield our generic Hamiltonian in the site representation,

 H =− ∑i,j(Tijδσσ′+μδij)c†icj+α2ϵ∞¯∑i≠j^ni^nj|m−n|+ (2) ∑q,ν,iℏωq^ni[ui(q)dq+H.c.]+Hph.

Here is the bare hopping integral, if , or the site energy, if , is the chemical potential, and include both site and spin quantum numbers, are electron and phonon operators respectively, is a site occupation operator, ( F/m is the vacuum permittivity), and with the phonon frequency .

The EPI matrix element is

 ui(q)=(2N)−1/2γ(q)exp(iq⋅m) (3)

with the dimensionless EPI coupling, ( is the number of unit cells ). Deriving the generic Hamiltonian in the site representation Alexandrov2011 () we approximate the Wannier orbitals as the delta-functions, which is justified as long as the characteristic wave-length of doped carriers significantly exceeds the orbital size . A singular on-site () Coulomb repulsion of two carriers with the opposite spins (the Hubbard ) is infinite in this approximation. In fact, it should be cut at as indicated by the bar above the sum, . Also for mathematical transparency we consider a single electron band dropping the electron band index.

Quantitative calculations of the EPI matrix elements in semiconductors and metals have to be performed numerically from pseodopotentials. Fortunately one can parametrize EPI rather than compute it in many physically important cases mahan (). EPI in ionic lattices such as the cuprates is dominated by coupling with polar optical phonons. This dipole interaction is much stronger than the deformation potential coupling to acoustic phonons and other multipole EPIs. While the EPI matrix elements are ill-defined in metals, they are well defined in doped insulators, which have their parent dielectric compounds with well-defined phonon frequencies and the electron band dispersion.

To parameterize EPI one can calculate the lowest order two-particle vertex function comprising the direct Coulomb repulsion and a phonon exchange mahan (),

 Γ(q,Ωn)=4παϵ∞V0q2+|γ(q)|2(ℏωq)2D(q,Ωn). (4)

Here , are the momentum and energy transfer in a scattering process of two carriers with the initial momenta and the Matsubara frequencies and , respectively, and is the propagator of a phonon of frequency , and is the unit cell volume. In the static limit, , Eq.(4) yields the Fourier component of the particle-particle interaction as

 Γ(q,0)=4παϵ∞V0q2−|γ(q)|2ℏωq. (5)

On the other hand, two static carriers localised on sites and in the ionic lattice repel each other with the Coulomb potential

 vij=αϵ0|m−n|, (6)

where the static dielectric constant, accounts for the screening by both core electrons and ions. Comparing Eq.(5) and Eq.(6) we find

 |γ(q)|2ℏωq=ακ¯∑meiq⋅mm≈4πακV0q2, (7)

at relatively small . Here is the lattice constant and with the high-frequency dielectric constant . The static dielectric constant and the high-frequency dielectric constant are readily measured by putting the parent insulator in a capacitor and as the square of the refractive index of the insulator, respectively. Hence, different from many models of high-temperature superconductors proposed so far, our generic Hamiltonian with the bare Coulomb and Fröhlich interactions is defined through the measurable material parameters.

## Iii t−Jp and t−Jp−~U models

Using the Lang-Firsov (LF) canonical transformation LF () one can integrate out most of both interactions in the transformed Hamiltonian Alexandrov2011 (),

 ~H=−∑i,j(^σijδσσ′+~μδij)c†icj+Hph+12∑i≠jvijninj, (8)

since the residual repulsion, , is substantially diminished by the large dielectric constant of the polar lattice [see Eq.(6)]. Here

 ^σij=Tij^X†i^Xj (9)

is the renormalised hopping integral involving multi-phonon transitions with and is the chemical potential shifted by the polaron binding energy.

Then using the Schrieffer-Wolf (SW) canonical transformation sw () and neglecting the transformed Hamiltonian Eq.(8) is reduced to the Hamiltonian as Alexandrov2011 ()

 HtJp = −∑i,jtijδσσ′c†icj+ (11) 2∑m≠nJp(m−n)(Sm⋅Sn+14nmnn).

Here the sum over counts each pair once only, is the spin operator ( are the Pauli matrices), , and are site occupation operators.

All quantities in the polaronic - Hamiltonian (11) are defined through the material parameters, in particular the polaron hopping integral, with the polaron band-narrowing exponent

 g2(m)=2πe2κℏω0NV0∑q1−cos(q⋅m)q2, (12)

and

 Jp(m)=T2(m)/2g2(m)ℏω0, (13)

It has been proposed that the - Hamiltonian, Eq.(11), has a high-T superconducting ground state protected from clustering Alexandrov2011 (). The polaronic exchange is attractive for polarons in the singlet channel and repulsive for polarons in the triplet channel. The origin of this exchange attraction is illustrated in Fig.1. If two polarons with opposite spins occupy nearest-neighbor sites, they can exchange sites without any potential barrier between them, which lowers their energy by proportional to the unrenormalised hopping integral squared.

Importantly the LF trasfomation Eq.(8) is exact, and the SW transformation is accurate for the intermediate and strong EPI coupling, , where is the BCS coupling constant and is the lattice coordination number as discussed in details in Ref. Alexandrov2011, . The residual repulsion of polarons, in the transformed Hamiltonian, Eq.(8), is small compared with the exchange inter-site polaron attraction and the short-range bipolaron-bipolaron repulsion of about the same magnitude, as long as . With the typical parameters of the cuprates is about 1 eV and eV, so that the residual inter-site repulsion is small if , which is well satisfied in all relevant compounds alebra ().

Nevertheless the on-site term in , Eq.(6), could be substantial, if the size of the Wannier orbitals is small enough . This renormalised is strongly diminished by the lattice polarization with respect to the bare on-site repulsion. We have emphasised in Refs.Alexandrov2011, ; sica, that our model describes carriers doped into the charge-transfer Mott-Hubbard (or any polar) insulator, rather than the insulator itself, different from the conventional Hubbard U or t-J models. The bare Hubbard- on the oxygen orbitals (where doped holes reside) in a rigid cuprate lattice is of the same order of magnitude as the on-site attraction induced by the Fröhlich EPI ( to eV alebra ()), so that the residual Hubbard could be as large as a few hundred meV. We now take it into account in the energy of a virtual double occupied state with two opposite spins on the same site,

 (14)

Then performing the SW transformation the exchange attraction is found as

 Jp(u,m−n)=t2ℏω0∞∑k=1(2g2(m−n))kk!(k+u), (15)

where . The reduction with respect to is moderate as long as the relative is less than , but becomes substantial for , Fig.2, which puts the characteristic bipolaron binding energy in the range of a hundred meV comparable with the double pseudogap in the cuprates aledev (). Importantly remains large or comparable with the polaron hopping integral since the spin exchange of the model , Eq.(13), does not contain the small polaron narrowing exponent .

Hence our extended -- model including major correlations effects reads as follows

 H = −∑i,jtijδσσ′c†icj+~U∑mnm↑nm↓+ (17) +2∑m≠nJp(u,m−n)(Sm⋅Sn+14nmnn).

## Iv Low density limit and high Tc

As in Refs.Alexandrov2011, ; sica, we adopt here the strong-coupling approach to the multi-polaron problem described by the Hamiltonian, Eq.(17), solving first a two-particle problem and then projecting the Hamiltonian on the repulsive Bose gas of small inter-site bipolarons. Such projection allows for a reliable estimate of the superconducting critical temperature for low carrier density as long as bipolarons remain small.

If we neglect the polaronic hopping taking , then the ground and the highest energy states are bipolaronic spin-singlet and spin-triplet, respectively, made up of two polarons on neighboring sites. The zero-energy states [in the nearest-neighbor (NN) approximation] are pairs of polarons separated by more than one lattice spacing. The on-site bipolaron has energy .

For our exact diagonalization (ED) results on finite clusters show that the probability to find NN bipolarons falls as we increase the hopping or the strength of the on-site repulsion as shown in Fig.3 for a square lattice. Consistently, the bipolaron size increases but remains on the order of the lattice spacing in a wide domain of the parameters (see Fig.4). Importantly, although the small bipolaronic configuration persists for any values of the hopping at , for and large values of up to a critical value , the presence of a finite on-site interaction leads to the crossover from a small to a large bipolaronic configuration. Finally, for further increasing the system undergoes a phase transition to an unbound state at . The crossover from a small to a large bipolaronic configuration is also confirmed by the calculation of the bipolaron to polaron effective mass ratio with in the large bipolaron regime, as shown in Fig.5.

In the small bipolaron regime, the kinetic energy operator in Eq.(17) connects singlet configurations in the first and higher orders with respect to the polaronic hopping integrals. Taking into account only the singlet bipolaron band and discarding all other configurations one can map the Hamiltonian on the hard-core charged Bose gas as described in Ref. Alexandrov2011, . This gas is superfluid in 2D and higher dimensions. In particular, its 2D critical temperature in the dilute limit is given by Fisher1988 ()

 Tc=2πℏ2nbkBm∗∗lnln(1/nba2), (18)

where is the boson density per unit area.

The occurrence of superconductivity in this regime is not controlled by a pairing strength, but by the phase coherence among small bipolarons AlexandrovRanninger (). At low enough density the Bose-Einstein condensation (BEC) temperature in 3D or its Berezinsky-Kosterlitz-Thouless (BKT) analog in 2D, Eq.(18) should not significantly depend on the bipolaron size as long as it remains small. On the other hand increasing in our model finally results in a bipolaron overlap, where the bipolaron condensation should appear in the form of the polaronic Cooper pairs in momentum space Alexandrov1983 () with a lower critical temperature, rather than in real space (BEC-BCS crossover eagles (); legget (); Alexandrov1983 (); nozieres (); mic (); levin ()). Hence, we can safely estimate the BEC critical temperature by weighting Eq.18 with the probability to find NN polarons as sica (). As shown in Fig.6, despite a low carrier density, for a physical choice of the parameters (eV, and eV Alexandrov2011 ()) the critical temperature is found to be well in excess of K for and rapidly decreases with increasing .

It is worth noting that, unlike in other theories, the strength of the on-site interaction term reduces . However, we recall that our residual on site interaction is defined as the difference between bare Hubbard and on-site Fröhlich EPI therefore at , when is maximized, we have a strong bare on-site interaction with eV.

## V Conclusions

In conclusion, we have introduced and studied the polaronic -- model, defined through the bare material parameters. The model, being an essential generalization of the - model Alexandrov2011 (), includes all electron-electron and electron-phonon correlations providing a microscopic explanation of the high-T phenomenon without any ad-hoc approximations. We show that the inclusion of the residual on-site interaction (neglected in the - model Alexandrov2011 (); sica ()), drives the system to a BEC/BCS crossover that reconciles the polaron-bipolaron theory of superconductivity with the observation of a large Fermi surface in overdoped cuprate superconductors. We offer an explanation, on microscopic grounds, of the high-T phenomenon as a consequence of competing Coulomb and Fröhlich interactions in highly polarizable ionic lattices beyond the conventional BCS description.

## Acknowledgements

We gratefully acknowledge enlightening discussions with Antonio Bianconi, Slaven Barisic, Ivan Bozovic, Victor Kabanov, Ferdinando Mancini, Dragan Mihailovic, Nikolay Plakida, and support from the UNICAMP visiting professorship program and ROBOCON (Campinas, Brasil).

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