of the noninteracting half-filled Hubbard chain. The more rapid increase ofMand the corresponding lowering of the critical field-strengthBcwith increasing values ofU/t can be understood by recalling that the energy difference∆E(Sz = ~Na/2)between the fully-polarized and the unpolarized states, which represents the band-width for spin-excitations, decreases withU/t.
The zero-field magnetic susceptibilityχ can be directly related to the curvatureα of the ground-state energy E(Sz) in the vicinity ofSz = 0. In fact, from ∆E(Sz) = αSz2/~2Na + O(Sz4)one readily obtains χ = 2µB2Na/α. In Fig.4.9 (b)we compare the zero-field susceptibilityχ obtained in the linearIFE-approximation with Takahashi’s exact results for the half-filled one-dimensional Hubbard model [50]. In the noninter-acting limit (U/t = 0) our approximation correctly reproduces the well known Pauli susceptibility χP = 2µB2Naρ(εF), where ρ(εF) = 1/(2πt) is the single-particle DOS per spin (4.28) of the Hubbard chain at the Fermi-level εF = 0. Furthermore, the IFEapproximation explains qualitatively the increase of χ with increasing Coulomb repulsion strength, as the electrons start to localize and the band width of spin exci-tations narrows. Moreover, theIFEapproximation reproduces qualitatively the linear increase of χ in the strongly-correlated limit. However, the corresponding asymp-totic behaviorχ ' (π χP/4)(U/t)(see AppendixE.2) differs from the exact asymptotic result χ ' (χP/π)(U/t)derived from the Bethe-ansatz solution [50]. Consequently, our approximation overestimates the zero-field magnetic susceptibility by a factor π2/4 ≈ 2.5 in the strongly-correlated regime. This is consistent with the previously observed underestimation of the curvature of E(Sz) in the vicinity of Sz = 0 [see Fig.4.8 (a)].
by Saubanère and Pastor [86], who used a simple but very effective scaling approx-imation to the interaction-energy functionalW[γ], which is based on exact results for the Hubbard dimer (Na = 2). In this way it was possible to obtain very accurate ground-state observables for the case of bipartite lattices in 1–3 dimensions having homogeneous and alternating local energy levels.
It is the purpose of this section to investigate the attractive Hubbard model from a delocalizedk-space perspective by generalizing theIFEapproximation to the case of attractive interactions. A rather straight-forward approach towards this goal would be to exploit the electron-hole transformation discussed in the context of Eq. (2.39), which maps the attractive model on a bipartite lattice to a (generally spin-polarized) repulsive model, and to subsequently apply the well-established IFE approxima-tion (4.17) to the resulting repulsive Hubbard model withNa−N↓down-spins. Notice, however, that this approach would be restricted to the case of bipartite lattices. Our goal is to derive a physically sound approximation to attractive correlations from an independent perspective in a general way, which is not restricted to bipartite lattice structures. Nevertheless, the existence of a mapping between the attractive and repul-sive Hubbard models suggests that the correlations caused by attractive and repulrepul-sive interactions are deeply related. Indeed, with increasing interaction-strength|U|/tthe fermions tend to localize for attractive as well as for repulsive interactions. However, the physical nature of the localization is quite different. Repulsive interactions lead to the formation and stabilization of local magnetic moments, whereas attractive in-teractions tend to suppress the local moments and stabilize localized fermion pairs with opposite spin polarization. In spite of these differences, we can expect that the competition between delocalization, driven by the hybridizations, and localization, resulting from the interactions between the particles, is quite similar in the repulsive and attractive cases. This encourages us to use theIFEdefined in Eq. (4.10) as an effec-tive measure for the degree of the correlations between the particles also in the case of attractive interactions. In order to verify that theIFEis indeed a measure for the correlations caused by attractive interactions, and to quantify its relation to the inter-action energyW, we have performed exact numerical Lanczos diagonalizations for the ground state of the half-filled attractive Hubbard model on multiple lattice structures.
To this aim, we have varied the hopping integrals fromtij = 0 totij |U|in order to scan the complete range of possible values 0≤S ≤ S∞of theIFE. In Fig.4.10the inter-action energyW is scaled between the uncorrelatedHF-limitWHF =U N↑N↓/Na and the strongly-correlated limitW∞ =U D∞, whereD∞ = min{N↑,N↓} is the maximum number of double occupations. The results presented in Fig.4.10show a remarkable one-to-one correspondence betweenW and S, which, after proper scaling with re-spect to the extreme valuesWHF,W∞, andS∞, becomes approximately independent of the size and dimension of the system under consideration. This one-to-one correspon-dence betweenW andS could thus be exploited in order to derive broadly applicable
0.0 0.2 0.4 0.6 0.8 1.0
S/S∞
0.0 0.2 0.4 0.6 0.8 1.0
(W−W∞)/(WHF−W∞)
Figure 4.10: Relation between the interaction energyW and the IFES in the ground state of the half-filled attractive Hubbard model, as obtained from exact numerical Lanczos diag-onalizations for a number of different periodic lattice structures. Results are shown for fi-nite 1D rings having Na = 6 (circles), Na = 10 (upright triangles) and Na = 14 (squares) lattice sites, as well as for 2D square-lattices having Na = 2×4 (downright triangles) and Na =3×4 (diamonds) lattice sites. The solid symbols correspond to the Hubbard model with onlyNNhoppingt, while the open symbols represent results for the non-bipartite case where second-NNhoppingst2=t/2 are included.
approximations to the interaction-energy functionalW[η]. In the case of a bipartite lattice it is straight forward to derive the relationW(a)−W∞ =W(r) between the in-teraction energy in the ground state of the attractive and repulsive Hubbard models, by applying the electron-hole transformation discussed in the context of Eq. (2.39) to the majority spins. Therefore, in the case of bipartite lattices, the scaled ground-state interaction energy shown in Fig.4.10follows the exact same dependence as a function ofS as the scaled interaction energy in the ground state of the previously considered repulsive Hubbard model (see Fig.4.1). Notice, however, that Fig.4.10 also includes results for non-bipartite lattices, such as the periodic 3×4 square-lattice cluster and all lattices with second-NNhoppings. This shows that the common, approximately linear one-to-one relation between the (scaled) interaction energyW and theIFES is not restricted to the bipartite case, but instead also holds for more general systems.
Therefore, we propose to generalize our previous approximation to the interaction-energy functionalW[η]of the Hubbard model in terms of theIFES[η]as
W[η] −W∞ WHF−W∞ = f
S[η] S∞
, (4.37)
where the function f : [0,1] → [0,1] accounts for the common relation between the properly scaled interaction energyW and theIFES in the attractive and repul-sive cases, shown in Figs.4.1and4.10. Furthermore,WHF =U N↑N↓/Na refers to the uncorrelatedHFinteraction-energy and
W∞ =U D∞ with D∞=
(min{N↑,N↓} forU < 0
max{N −Na,0} forU > 0 (4.38) to the interaction energy in the strongly correlated limit (|U|/t → ∞). Notice that Eq. (4.37) applies to the Hubbard model withbothattractive and repulsive interactions, and thus the notiongeneralizedIFE-approximation is justified. Since Eq. (4.37) approx-imatesW[η]in terms ofS[η], the consequences of the approximation (4.12) also apply to Eq. (4.37) and thus also to the case of attractive interactions. This applies in particu-lar to the result (4.16) which states that ground-state occupation-numbersηkσfollow a Fermi-Dirac distribution with an effectiveU-dependent temperatureTeff =−∂W/∂S.
Before we apply the generalizedIFE-approximation (4.37) to the attractive Hubbard model on finite clusters, it is useful to analyze its relation to the electron-hole trans-formation (2.39) which maps between attractive and repulsive interactions. It turns out that the relations between the attractive and repulsive models on bipartite lattices are not reproduced for arbitrary particle numbers Nσ. Only for N↓ = Na/2, i. e., if the number of down-spins is conserved under the electron-hole transformation, one can easily show that the generalized IFE-approximation (4.37) correctly reproduces the relations K(a) = K(r) andW(a) + |U|N↑ = W(r) between the kinetic and inter-action energies of the attractive and repulsive Hubbard models on bipartite lattices.
Notice, however, that the generalizedIFE-approximation predicts these relations in-dependent of the lattice structure, i. e., also for non-bipartite lattices. Furthermore, forN↓ = Na/2 it is easy to verify that the generalizedIFE-approximation yields the same gs-SPDMγ(a) = γ(r) for the attractive and repulsive Hubbard models on a bi-partite lattice, while the electron-hole transformation implies thatγij(a↓) = ±γij(r↓) for i , j, where the negative sign applies if the sites i and j belong to the same sub-lattice and the positive otherwise. Consequently, the generalizedIFE-approximation fails to reproduce the sign-change of thoseSPDMelements that do not contribute to the kinetic energy of the Hubbard model on a bipartite lattice. TheseSPDMelements are non-vanishing in general, however, in the special case where the ground state forN↑ = N↓ = Na/2 is non-degenerate they vanish altogether (see Section4.2.1). In this case the predictions of the generalized IFE-approximation (4.37) are consistent with the implications of the electron-hole transformation.
In order to apply the generalized IFE-approximation to the attractive Hubbard model on finite lattices, we follow the route taken in the repulsive case and propose to approximate the function f in Eq. (4.37) by the linear relation f(x) = 1−x, such
0 5 10 15 20
−E0/Nat
(a)
|U|/2t 0.0
0.2 0.4 0.6 0.8 1.0
ηkσ
(c)
k=(0,0)
±π2,0
, 0,±π2 εk =εF
±π2,π, π,±π2
(π,π)
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
0.25 0.30 0.35 0.40 0.45 0.50
D/Na
(b)
U < 0 4×4 square
LDFT exact
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
−0.05 0.00 0.05 0.10 0.15 0.20
γ0δσ
γ01σ (d)
γ04σ
0.00 0.05 0.10
t/|U|
0.0 0.2 0.4
EB/Nat
0.0 0.5 1.0 1.5
−K/Nat
01 1 1
2 12
2 2
3 3 4 44 4
5
Figure 4.11:Ground-state properties of the attractive 2D Hubbard model on a 4×4 square lat-tice cluster with N↑ = N↓ = 8 and periodic boundary conditions as functions of the attraction strength |U|/t. Results of LDFT combined with the generalized linear IFE-approximation (4.39) (blue full curves) are compared with exact numerical Lanczos diagonal-izations (red crosses): (a) ground-state energyE0, (b) average number of double occupationsD and kinetic energyK, (c) natural-orbital occupation numbersηk↑=ηk↓, and (d)gs-SPDM el-ementsγ0δ↑ =γ0δ↓between sitei = 0 and itsδthNN, as illustrated in the inset. The inset in subfigure (a) shows the binding energyEB =W∞−E0in the strongly-correlated limit (|U| t).
that the generalizedIFE-approximation becomes
W[η]=WHF+(W∞−WHF)S[η]
S∞ . (4.39)
In the repulsive case withN ≤ Na, i. e.,W∞ = 0, this coincides with the linear IFE-approximation (4.17) discussed in the previous sections. In Fig. 4.11we show results for the ground-state properties of the half-filled attractive 2D Hubbard model on a 4×4 square-lattice cluster with periodic boundary conditions andN↑ = N↓ = 8. The comparison with exact Lanczos diagonalizations shows that the quality of LDFT in combination with the generalizedIFE-approximation (4.39) is as good as in the repul-sive case considered in Section4.2.2. This is not surprising, since the 4×4 cluster with NNhopping is bipartite, and forN↑ =N↓= Na/2 the generalizedIFE-approximation reproduces the relations between the attractive and repulsive Hubbard models in-ferred from the electron-hole transformation (2.39). Also in this case, where the at-tractive and repulsive models are connected by an electron-hole transformation, it is very interesting to take a closer look at the ground-state properties of the attractive Hubbard model and to investigate the differences and similarities to the previously considered repulsive case.
Figure4.11 (a)demonstrates that the ground-state energyE0of the attractive Hub-bard model is accurately reproduced by the linear IFE-approximation in the com-plete range of the interaction strength |U|/t. In contrast to the repulsive case, we find a monotonously decreasing ground-state energy as the attraction strength|U|/t increases, since the pair-binding energy of the condensing fermions overcompen-sates the increase in kinetic energy caused by the gradual fermionic localization.
Most notably, in the strongly correlated limit |U|/t → ∞ the pair-binding energy of the condensing fermions dominates and gives rise to a diverging ground-state en-ergyE0 'W∞ = −|U|Na/2. Notice that first order hopping processes are prohibited in the strongly-correlated limit, since they would break a strongly bound fermion pair. The fermions can, however, lower their kinetic energy K ∝ t2/U by second order (virtual) hopping processes which break and subsequently reassemble a lo-cal fermion pair. Thus, one can expect that the binding energy EB = W∞ −E0 be-haves likeEB = αt2/|U|withα > 0 in the strongly correlated limit [see the inset in Fig.4.11 (a)]. In fact, one infers−EB =E0+|U|D∞ =E(0r)from the electron-hole trans-formation (2.39), i. e., the binding energy resulting from the virtual hopping processes coincides with the negative ground-state energyE(0r)of the repulsive model. Thus, the generalized linearIFE-approximation reproduces the binding energyEB =αt2/|U|in the strongly correlated limit of the attractive Hubbard model qualitatively correct, with a coefficientαIFE= 5.55 which is only 13% larger than the exact valueαex =4.81 for the 4×4 square-cluster. These results are consistent with the ones for the repulsive Hubbard model on the 4×4 square-cluster discussed in Section4.2.2.
Besides some discrepancies for|U|/t < 2, which are the result of degeneracies at the Fermi-level, we find in Fig.4.11 (b)that also the average numberDof ground-state dou-ble occupations is very well reproduced by the generalized linearIFE-approximation.
As expected, we find increasing values of the double occupationsD as the fermionic attraction strength |U|/t increases, which is due to the formation of local fermion-pairs, until the maximum possible valueD∞ =Na/2 is attained in the limit|U|/t → ∞. The increase of the average numberD of double occupations with increasing inter-action strength|U|/t is accompanied with an increase of the kinetic energyK, since the fermions condense into localized pairs and one-fermion hopping processes are gradually suppressed as the interaction energy starts to overcompensate the binding energy generated by the fermionic motion. Clearly, in the strongly correlated limit a fully localized state is attained and the kinetic energy vanishes likeK ∝ t2/U, since first-order hopping processes would break the local strongly bound fermion-pairs.
The dependence of thegs-SPDM elementsγijσ on the interaction strength |U|/t is shown in Fig. 4.11 (d). Due to the symmetry of the underlying lattice it is suffi-cient to focus on the matrix elementsγ0δσ corresponding to some lattice sitei = 0 and itsδth NN, as illustrated in the inset. The non-vanishing SPDM elementsγ01σ andγ04σ display the typical correlation-induced suppression of the charge fluctua-tions as |U|/t increases and the fermions condense into localized pairs. Like in the previous case of repulsive interactions, the long-range charge fluctuations|γ04σ|are suppressed faster than the fluctuationsγ01σ betweenNNs. In fact, the dependence of thegs-SPDM elementsγ0δσ on the strength |U|/t of the attractive interaction coin-cides with the one in the previously considered repulsive case. This is a result of the electron-hole symmetry (2.39), which implies that thegs-SPDMof the attractive and repulsive Hubbard models on bipartite lattices withN↑ = N↓ = Na/2 coincide if the ground state is non-degenerate. This implication is exactly reproduced by the gener-alized linearIFE-approximation (4.39), such that the results for thegs-SPDMshown in Fig.4.11 (d)coincide with the ones of Fig.4.3 (d)which accounts for repulsive inter-actions. Clearly, since thegs-SPDMof the attractive and repulsive Hubbard models coincide, the same must be true for its eigenvalues, i. e., for the Bloch-state occupation numbersηkσ, which are shown in Fig.4.11 (c). The fact that the occupation numbers andgs-SPDMelements of the attractive and repulsive Hubbard models coincide on a bipartite lattice withN↑ = N↓= Na/2 is a manifestation of the deep underlying rela-tion between the correlarela-tion induced localizarela-tion caused by attractive and repulsive interactions in half-filled band systems.
In order to investigate situations where the ground-state properties of the repul-sive and attractive Hubbard models are not related by an electron-hole transforma-tion, we have to go beyond bipartite lattices. This can be achieved by considering essentially non-bipartite structures, such as the 2D triangular lattice, or by includ-ing hoppinclud-ings beyond first-NNs. In order to address the second option we have
com-0 5 10 15 20
−E0/Nat
(a)
|U|/2t 0.0
0.2 0.4 0.6 0.8 1.0
ηkσ
−6t (c)
−2t 0
εk =2t
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
0.25 0.30 0.35 0.40 0.45 0.50
D/Na
(b)
LDFT exact
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
−0.1 0.0 0.1 0.2
γ0δσ
(d)
4×4 square U < 0,t2 =t/2
γ01σ
γ02σ
γ03σ
γ04σ
γ05σ
0.00 0.05 0.10
t/|U|
0.0 0.2 0.4 0.6
EB/Nat
0.0 0.5 1.0 1.5
−K/Nat
0 1
1 2
t t/2
Figure 4.12: Ground-state properties of the attractive 2D Hubbard model with second-NNhoppingt2=t/2 [as illustrated in subfigure (c)] on a 4×4 square lattice withN↑=N↓=8 and periodic boundary conditions as functions of the attraction strength |U|/t. Results of LDFTcombined with the generalized linearIFE-approximation (4.39) (blue full curves) are compared with exact numerical Lanczos diagonalizations (red crosses): (a) ground-state en-ergyE0, (b) average number of double occupationsDand kinetic energyK, (c) natural-orbital occupation numbersηk↑ = ηk↓, and (d)gs-SPDMelementsγ0δ↑ = γ0δ↓between sitei = 0 and itsδthNN, as illustrated in the inset of Fig.4.11 (d). The inset in subfigure (a) shows the binding energyEB =W∞−E0in the strongly-correlated limit (|U| t).
puted the ground-state properties of the half-filled attractive 2D Hubbard model on a 4×4 square-lattice with second-NNhoppingt2 = t/2 and N↑ = N↓ = 8. The ground-state energyE0shown in Fig.4.12 (a)is very accurately reproduced by the generalized linear IFE-approximation (4.39) in the complete range from weak to strong attrac-tions. This is most remarkable, especially in the range of intermediate interactions (1 . |U|/t . 10), since the interplay between delocalization driven by hybridiza-tions and localization due to the formation of local fermion-pairs is far from triv-ial within this regime. Moreover, the asymptotic behavior of the ground-state en-ergyE0 ' −|U|Na/2 in the strongly correlated limit |U|/t → ∞ is exactly obtained within the linearIFE-approximation. The qualitative behaviourEB = αt2/|U|of the binding energy in the strongly-correlated regime is also correctly reproduced, with a coefficientαIFE = 6.93 which is about 18.5% larger than the exact valueαex = 5.65 deduced from the Lanczos calculations [see the inset in Fig.4.12 (a)]. We conclude that the generalizedIFE-approximation overestimates the binding energy in the strongly correlated limit slightly more if second-NNhoppings are included, when compared to the previous case where only hoppings between first-NNwere implied. The depen-dence of the average number of ground-state double occupationsD and the kinetic energyK on the interaction strength|U|/t, shown in Fig.4.12 (b), is very similar to the previously considered case where onlyNN-hoppings were involved [see Fig.4.11 (b)].
However, the inclusion of second-NNhopping allows the system to increase the ab-solute kinetic energy|K|by about 16.7% in the weakly correlated regime and, at the same time, to reduce the average number of double occupations D by about 8.2%
[compare Figs.4.11 (b)and4.12 (b)]. The kinetic-energy gain in the weakly-correlated regime, resulting from the inclusion of second-NN hoppings, is exactly reproduced within the linear IFE-approximation, while the average number of double occupa-tions display the typical discrepancies resulting from degeneracies of the Fermi level.
Nevertheless, the linearIFE-approximation predicts a decrease of the average num-ber of double occupations by about 12.5% in the weakly-correlated regime due to the inclusion of second-NNhoppings, which is in qualitative agreement with the 8.2% de-crease found in the exact results. Moreover, the discrepancies inD essentially disap-pear for|U|/t & 2.
Regarding the dependence of the Bloch-state occupation numbersηkσ on the in-teraction strength|U|/t, shown in Fig.4.12 (c), we find the typical decrease (increase) ofηkσ for Bloch states having εk < εF (εk > εF) as the interaction strength |U|/t increases, which reflects the localization of the fermions. As in the repulsive case, a homogeneous occupation of all Bloch states, i. e.,ηkσ =1/2 for allk, is attained in the strongly correlated limit|U|/t → ∞. In contrast to the case which involves only hop-pings betweenNNs, we find decreasing values of the occupation numbersηkσ also for the Bloch states havingεk =εF =0. This is a consequence of the qualitative changes in the single-particle spectrumεkresulting from the inclusion of second-NNhoppings.
They lead to a fourfold degenerate Fermi level which is occupied by six fermions in the noninteracting limit |U|/t → 0, such thatηkσ = 3/4 forσ = ↑,↓ and all four Bloch states havingεk = εF. As shown in Fig.4.12 (c), the occupation numbersηkσ obtained from the generalized linearIFE-approximation (4.39) are very accurate in the complete interaction range from weak to strong correlations. Only the occupation of the lowest-lying Bloch state [k = (0,0)] having εk = −6t is slightly overestimated for intermediate|U|/t, which results in an overestimation of the absolute kinetic en-ergy |K| in this range. For example, at |U|/t = 15 the average occupation-number of the lowest-lying Bloch state is overestimated by 8.7%, which corresponds to an overestimation of |K|by about 17%.
Thegs-SPDMelementsγ0δσ corresponding to a lattice-sitei =0 and itsδthNNare shown in Fig.4.12 (d). The hoppings between secondNNslift the electron-hole sym-metry discussed in the context of Eq. (4.25), and as a result the matrix elementsγ0δσ
withδ = 2,3 and 5, which correspond to sites within the same sublattice in the ab-sence of second-NN hoppings, are no longer identically zero. An interesting non-monotonous behaviour of γ05σ is observed, which is qualitatively reproduced, al-though somewhat exaggerated, by the generalized linear IFE-approximation. The non-monotonous|U|/tdependence ofγ05σ indicates that the second-NNhoppings en-hance a charge transfer beyond 4th-NNsat intermediate interaction strength, which optimizes the kinetic energy and, at the same time, lowers the interaction energy due to the formation of local fermion-pairs, i e., due to an increase of the average number of double occupationsD.
As an example of an essentially bipartite lattice (i. e., a lattice which is non-bipartite even if onlyNNhoppings are taken into account), we consider in Fig. 4.13 the 4×4 triangular lattice with periodic boundary conditions andN↑ = N↓ = 8. The comparison with exact numerical Lanczos diagonalizations in Fig.4.13 (a)shows that the ground-state energyE0 is again very accurately obtained within the generalized linear IFE-approximation in the complete range from weak to strong interactions.
Also the qualitative behaviourEB = αt2/|U| of the binding energy in the strongly-correlated limit is correctly reproduced, and the corresponding coefficientαIFE =8.32 is about 19% larger than the exact valueαex =6.73 deduced from the Lanczos calcula-tions [see the inset in Fig.4.13 (a)]. We conclude that the strongly attracting limit on the 4×4 triangular-cluster is much better reproduced than the corresponding repul-sive case, where the coefficientα is overestimated by 38% (see Section4.2.3).
The average number of double occupationsDand the kinetic energyK are shown in Fig.4.13 (b), and a very similar dependence on|U|/tas in the previously considered attractive models is observed, which is in general very well reproduced by the present IFE approximation. Only in the weakly interacting regime |U|/t . 3 we find the already observed significant overestimation of the double occupations D, which is known to be a consequence of the degeneracies in the single-particle spectrum at the
0 5 10 15 20
−E0/Nat
(a)
|U|/2t
0.0 0.2 0.4 0.6 0.8 1.0
ηkσ
−6t (c)
−2t
εk =2t LDFT
exact
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
0.25 0.30 0.35 0.40 0.45 0.50
D/Na
(b)
U < 0 4×4 triangular
0.0 0.2 0.4 0.6 0.8 1.0
|U|/(|U|+8t)
−0.05 0.00 0.05 0.10 0.15
γ0δσ
(d)
γ01σ
γ02σ =γ03σ
0.00 0.05 0.10
t/|U|
0.0 0.5
EB/Nat
0.0 0.5 1.0 1.5 2.0
−K/Nat
0 1
1 1 1
1 1
2 22 2 22
3 3 3
Figure 4.13: Ground-state properties of the attractive 2D Hubbard model on a 4×4 cluster of the triangular lattice withN↑ = N↓ = 8 and periodic boundary conditions as functions of the attraction strength|U|/t. Results ofLDFTcombined with the generalized linear IFE-approximation (4.39) (blue full curves) are compared with exact numerical Lanczos diagonal-izations (red crosses): (a) ground-state energyE0, (b) average number of double occupationsD and kinetic energyK, (c) natural-orbital occupation numbersηk↑=ηk↓, and (d)gs-SPDM el-ementsγ0δ↑ =γ0δ↓between sitei = 0 and itsδthNN, as illustrated in the inset. The inset in subfigure (a) shows the binding energyEB =W∞−E0in the strongly-correlated limit (|U| t).
Fermi level.
Regarding the Bloch-state occupation numbersηkσ, shown in Fig.4.13 (c), one ob-serves the familiar fermionic localization as|U|/t increases: decreasing (increasing) occupationsηkσ for Bloch states having εk < εF (εk > εF), starting fromηkσ = 1 (ηkσ = 0) in the noninteracting limit, untilηkσ = 1/2 is reached for allkσ in the strongly correlated limit. Concerning the occupation numbersηkσ of the Bloch states havingεk =εF =2t, we findηkσ =1/9 for|U|/t →0, since the nine-fold degenerate Fermi-level is occupied by two fermions with opposite spin directions.
A very interesting difference to the previously considered repulsive case is the fact that all Bloch states havingεk = εF = 2t are equally occupied in the complete range from weak to strong interactions, while they split into two groups if repulsive interac-tions are considered [see the exact numerical Lanczos results shown in Fig.4.4 (c)]. As already discussed in Section4.2.3, this splitting of the Fermi-level occupation numbers in the repulsive case is a subtle finite-size effect, by which the local Hubbard interac-tion between the two electrons at the Fermi level is completely suppressed. Clearly, such a suppression of the mutual interaction between the two electrons at the Fermi level would be energetically unfavorable in the present attractive case, and thus the Fermi-level occupation numbers do not split. Figure4.13 (c)shows that the generalized IFE-approximation (4.39) reproduces the Bloch-state occupation numbersηkσ quite accurately in the complete range from weak to strong interactions. Only the occupa-tion of the lowest-lying Bloch state [k = (0,0)] havingεk = −6t is slightly overesti-mated for intermediate values of the interaction strength|U|/t. It is worth noting that forN↑=N↓=Na/2 the generalizedIFE-approximation always yields the same occu-pation numbers for attractive and repulsive interactions, although the exactηkσmight be different if non-bipartite lattices are considered. Comparing Figs.4.4 (c)and4.13 (c), one concludes that theLDFTresults are more accurate in the attractive case. This is not only due to the absence of the finite-size splitting of the Fermi-level occupation numbers in the attractive case, but also due to the higher accuracy ofηkσ forεk < εF
and intermediate values of the interaction strength. The very good accuracy ofηkσ also implies accurate results for thegs-SPDM elementsγ0δσ, as seen in Fig.4.13 (d).
In the considered attractive case we findγ02σ =γ03σ in contrast to the repulsive case, whereγ02σ ,γ03σ due to the finite-size correlations which manage to suppress the in-teraction of the two fermions at the Fermi level. Since this type of correlation is not at all favorable in the attractive case, we findγ02σ =γ03σ, as predicted by the generalized linearIFE-approximation (4.39).
In summary, the results presented in this section show that an appropriate general-ization of the linearIFE-approximation allows us to extend the scope of LDFTto the ground-state properties of the half-filled attractive Hubbard model. This is not only true for bipartite lattices, where such an extension can be formally inferred from the electron-hole transformation (2.39) which maps between the attractive and repulsive
models, but also in more general situations involving non-bipartite structures or hop-pings beyond first-NNs. We therefore conclude that theIFE(4.10) is a suitable measure of the degree of fermionic correlations in the Hubbard model for both attractive and repulsive interactions.