Huai-Shuang Zhu(祝懷霜) and Qiang Han(韓強)

Department of Physics,Renmin University of China,Beijing 100872,China

Keywords: strong correlated superconductivity,exactly solvable model,mean-field theory

As one of the most important and challenging problems in condensed matter physics,electron correlations are responsible for a rich variety of properties of the many-body quantum systems. Among them, the correlated superconductivity is most extensively investigated in heavy-fermion materials, organic compounds, high-temperature cuprates[1]and nickelate,[2]as well as iron pnictides and chalcogenides.[3]Soon after,the early discovery of the high-Tccuprates by Bednorz and M¨uller[4]in 1986,the single-band Hubbard model[5]was suggested by Anderson[6]to describe the unusual properties in both the normal and superconducting states. The Hubbard Hamiltonian includes an on-site repulsiveUterm to account for the strong electron correlation. After decades of intensive research, the highly nontrivial problem of solving the Hubbard model for general cases is still far from being settled.

Recently,a correlated electron model formulated by Hatsugai and Kohmoto (HK)[7]was revisited as a toy model whose metallic phases exhibit non-Fermi liquid character with a superconducting instability.[8]Similar to the Hubbard model,the HK model also possesses aUterm reflecting the interaction between spin-up and spin-down electrons. However,theUterm in the HK model is local in momentum space instead of real space. The locality of interaction inkmakes the HK model exactly soluble.[9–12]Phillipset al.[8]demonstrated an analogous Cooper instability for the HK model of a doped Mott insulator. This intriguing observation motivates us to investigate the correlation effect on the superconductivity when a weak pairing interaction is present in the HK model.

To study how the electron pairing is influenced byU,we first elucidate some typical normal phases of the HK model.The Hamiltonian of the HK model is of the form

Whether the lowest-lying state of each subsystem is empty,singly occupied or full relies onξkandUas shown in Figs. 1(a) and 1(b). Consequently, the electron distribution in the energy band is changed due to the electron correlation,and the system undergoes metal–insulator transition with the variations inUand the band filling(or chemical potential).

Fig.1. The eigenenergies for empty(blue line),singly occupied(green line)and fully occupied(red line)states as functions of ξk,for the cases of(a)U <0 and(b)U >0.

For the three correlated metallic phases,(i),(ii)and(iii),there areksegments on which the energy levels are singly occupied by either spin-up or spin-down electrons,which results in highly degenerate ground states of the correlated metallic phases. This high degeneracy leads to possible instability of the normal phase in the presence of other electronelectron interactions towards the formation of magnetic or superconducting orders. Indeed,the superconducting instability is unveiled[8]after studying the Cooper problem in the correlated metals.

With the knowledge of the normal phases of the HK model,we then examine the electron pairing when an effective electron-electron attraction is present.The model Hamiltonian is written as

whereN(ε)=∑k δ(ε ?εk) is the single-particle density of states. To concentrate on the effect of electron correlation on the superconductivity,we adopt a simple model of the electron band,in which the single-particle density of state is constant,i.e.,N(ε)=N0,within the energy windowε ∈[?W/2,W/2].Furthermore,in the following we consider the the usual weakcoupling superconducting case,i.e.,N0V0?1.

To solve the self-consistent equation forΔ, we calculate firstly the eigen-wavefunctionxk,1,yk,1andzk,1by diagonalizing the 3×3 eigenequation (12) exactly, then the pairing amplitude according to Eq. (15), and finally the integral of Eq.(16).

Figure 2 illustrates theUdependence ofΔat fixed band fillingsn=1 by setting the chemical potentialμ=U/2. AsUvaries, the normal phase of the system evolves from Mott insulator for negativeUto correlated metal for 0W. In Fig.2,Δ0denotes the BCS value of the pair potential, i.e.,ΔatU=0.NearU=0,Δexhibits linear dependence onU. From Fig.2 we find thatΔis suppressed by negativeUand a rather weak negativeUwhich exceedsUc≈?Δ0can destroy the superconductivity. On the contrary,Δis enhanced by positiveUprovided thatUis less than but not too close toWas shown in Fig.2. ForU>Wat half filling,the normal state is a Mott insulator which cannot accommodate superconductivity.

Fig.2. Variation of the pair potential Δ as a function of U at half filling n=1. Here N0V0 =0.2,which gives rise to Δ0/W ˙=1/148. The black solid line shows the exact numerical results. The red and blue dashed lines draw Δ(U)according to the approximate formulas Eqs.(27)and(35),respectively. The inset shows the U dependence of Δ in the vicinity of U =0.

We then examine theUdependence ofΔat band fillingsn<1. AsUvaries, the normal phase of the system evolves from Mott insulator for negativeUto weakly correlated metal for 0nW.To fix the band filling whenUincreases, the variation of the chemical potential isμ=(U+n?1)/2 orμ=n?1/2 for the weakly or strongly correlated metallic phases,respectively.As shown in Fig. 3, whenUnW,Δ(U) is strengthened compared withΔ0for positiveUin then<1 case, similar to then=1 case. On the other hand, when the system is strongly correlated,Δis suppressed forUnWand approaches to a finite non-zero valueΔ∞whenU →∞.

The suppression ofΔby negativeUis attributed to the single-particle excitation gap of the Mott insulating phase,which is proportional to?U. To have a better interpretation of the numerical results for positiveUas shown in Figs.2 and 3,in the following we will tackle the eigen problem of Eq.(12)approximately.

Fig.3. Variation of the pair potential Δ as a function of U at band filling n=0.8. Here N0V0 =0.2,which gives rise to Δ0/W ˙=1/151. The black solid line shows the exact numerical results. The red, blue and green dashed lines draw Δ(U) according to Eqs. (27), (35) and (36),respectively.

The weak-coupling case:|U|?Δ. In the presence of a weakU, the superconductivity can be examined in the perturbed BCS framework by the first-order perturbation theory.The unperturbed system,i.e.,Eq.(8)forU=0,is a quadratic form whose eigen states can be obtained by Bogoliubov quasiparticle transformation[14]or by solving Eq.(12)exactly. The three eigenenergies are TheU-independent term on the right side of Eq. (26) is just the usual pairing amplitude of the BCS theory. Substituting Eq. (26) into Eq. (16), we have the first-order approximation ofΔ,

The intermediate-coupling case:Δ ?U ?W. The electron pairing is intimately related to the low-energy excitations of the normal state. As shown in Fig.1(b),forξk ≈0 the energies of the empty and spin-singlet states,2η0and 2η1,are close to each other, whereas the energy of the fully occupied state|f〉kis around 2Uand far above. Therefore, the electron pairing is effectively formed between|0〉kand|s〉k.To the leading order,the eigenvalue equation(12)reduces to

The above approximate formula exhibiting theUdependence ofΔfits quite well to the exact results as shown in Figs.2 and 3. ForU ?W,although the normal state is weakly correlated,the appearance ofUas well as a factor of 2 in the exponential function as shown in the formula ofΔ(35) indicates that the superconducting ground state is strongly correlated provided thatU ?Δ. Furthermore, the enhancement ofΔbyUis associated with the double-peak structure as shown in Fig.4 in comparison with the single peak for the BCS case.

Fig.4.The approximate and exact pairing amplitude g(ξ,Δ,U)as functions of ξ for U/Δ =10. Red solid line: approximate results calculated according to Eq. (34). Black dashed line: exact results calculated according to Eq.(15)utilizing the wavefunction obtained by exact diagonalization of Eq.(12).

The strong-coupling case:U ?W. At half filling, the pair potentialΔvanishes trivially forU>Was shown in Fig.2 since the normal phase is Mott insulating. Thus,it is interesting to examine the case of doped Mott insulators, i.e.,n/=1.Here, since the electron- and hole-doped cases are linked via electron-hole transformation, we only show the result for then<1 case. Distinct from the intermediate-coupling case,|f〉kis excluded from electron pairing forallenergy levelξk. Employing the perturbation theory,we have the following expression for the pairing amplitude up to the first order of 1/U,from which the asymptotic behavior ofΔ(U) is obtained as follows:

Fig.5. The U dependence of the overlap of the correlated and uncorrelated superconducting ground states. As shown in Eq.(37),the fidelity is equal to the product of N numbers with N being the number of k,and thus we plot the geometric mean of the fidelity at the thermodynamic limit,i.e.,limN→∞f(U)1/N.Black solid line:the half-filling case n=1.Red dashed line: below half filling,n=0.8.

Single-particle excitation gapΔgapWhen a single electron is added to or removed from the ground state, a singleparticle excitation is created. Since the ground state is of even parity,the single-particle excited state must live in the odd sector. Thus,Δgapis equal to

and the forms of the Hamiltonian in the other three odd subspaces is of similar structure.

Appendix A:Exact results

Table A1. Basis vectors built according to their quantum numbers. P,N,S and Sz denote the eigenvalues of of ?Pk, ?Nk, ?S and ?Sz,respectively.

Since 0≤θk ≤π/3 according to Eq. (A6), we haveEk,1≤Ek,2≤Ek,3from Eq. (A4).Ek,1=Ek,2orEk,2=Ek,3occur whenθk= 0 orπ/3, which is satisfied if and only ifΔ=0 andξk(ξk+U)(2ξk+U)=0. As a result, the three eigenenergies are arranged in strictly ascending order, i.e.,Ek,1

Furthermore, according to the variational principle, the minimum of the three eigenvalues, i.e.,Ek,1, is less than the minimum of the three diagonal elements of the Hermitian matrix in Eq. (12) whenΔ/= 0. Namely,Ek,1

For the other three 2D subspaces,the eigenenergies are exactly the same as Eq.(A8). Therefore,Ek,±are both four-fold degenerate,andEk,?is the lowest eigenvalue in the odd sector.

Making use of the above two inequalities,we have

where equality will occur if and only ifΔ=0 andξk(ξk+U)(2ξk+U)=0. Thus,whenΔ/=0,we always haveEk,1