0.0 0.2 0.4 0.6 0.8 1.0
N/Na
0.0 0.2 0.4 0.6 0.8 1.0 1.2
−E0/tNa
(a)
0.0 0.2 0.4 0.6 0.8 1.0
N/Na
0.00 0.05 0.10 0.15 0.20 0.25
D/Na
(b) U/t
0.00.5 1.02.0 4.08.0
∞
Figure 2.2:Ground-state properties of the infinite one-dimensional Hubbard chain as func-tions of the electron density N/Na, obtained by solving Shiba’s integral equations (2.28) and (2.33). In (a) the ground-state energyE0 is shown for a number of representative val-ues of the Coulomb-repulsion strengthU/t, and the corresponding average number of double occupationsDis displayed in subfigure (b).
forN/Na = 1/2. This is in sharp contrast to the noninteracting caseU/t = 0, where the electrons fill the band from the bottom such that the minimal value of the ground-state energy is attained at half band-fillingN/Na = 1. The position of the minimum in the ground-state energyE0 shifts continuously with increasingU/t, starting from the noninteracting limit where the minimum is attained atN/Na = 1, to the strongly-interacting limit where the minimum inE0occurs atN/Na = 1/2. Furthermore, the electrons tend to reduce the average number of double occupationsDat the expense of a kinetic-energy gain asU/tincreases, and consequently one observes in Fig.2.2 (b) a gradual suppression ofDwith increasing values ofU/t, starting fromD = N2/(4Na) in the noninteracting case toD =0 in the strongly correlated limit.
book of Essleret al.[98]. Furthermore, we will consider the strongly interacting limit of the Hubbard model and demonstrate that the low-lying excitations in this limit are described by thet-J model, which reduces to the well known antiferromagnetic (AFM)Heisenberg model in the case of a half-filled band.
2.3.1 Selected symmetries of the Hubbard model
The fact, that the Hamiltonian of the Hubbard model (2.7) commutes with the number operator ˆNσ = P
icˆiσ†cˆiσ for electrons with spin polarizationσ is obvious, since each term in ˆH contains as many creation operators ˆciσ† as annihilation operators ˆcjσ. Con-sequently, the numberNσof electrons with spin polarizationσis a conserved quantity in the Hubbard model. It follows that the total number of electronsN = N↑+N↓and thez-component of the spinSz = (N↑− N↓)/2 are conserved quantities as well. In order to demonstrate that ˆH also commutes with the spin components ˆSx and ˆSy, one considers the ladder operators
Sˆ+ =Sˆx +i ˆSy =X
i
cˆi†↑cˆi↓ and ˆS− =Sˆx −i ˆSy =X
i
cˆi†↓cˆi↑. (2.37) Using the fundamental fermionic anticommutator relations{cˆα,cˆβ} =0 and{cˆα,cˆβ†} = δα β, it is straight forward to verify that both terms of the Hubbard-model Hamilto-nian (2.7), i. e., the kinetic energy ˆK and the local Coulomb repulsion ˆW, individually commute with the operators ˆS±. Consequently, both terms commute with all three components of the spin ˆSx = (Sˆ++Sˆ−)/2, ˆSy = −i(Sˆ+−Sˆ−)/2, and ˆSz, which means that they are invariant with respect to arbitrary rotations in spin-space.
In order to introduce the particle-hole symmetry, let us consider the Hubbard model on a bipartite lattice, i. e., a latticeT that can be divided into two sublatticesTAandTB
withT = TA∪ TBsuch that there is no hopping within the two sublattices, i. e.,tij = 0 ifi,j ∈ TAori,j ∈ TB. In this case, the unitary transformation which maps ˆciσ† → ±cˆiσ, where the positive sign applies ifi ∈ TA and the negative ifi ∈ TB, transforms the Hamiltonian of the Hubbard model (2.7) into
Hˆh =−X
ijσ
tijcˆiσcˆ†jσ+U X
i
cˆi↑cˆi†↑cˆi↓cˆi†↓=X
ijσ
tij∗cˆiσ†cˆjσ+U X
i
nˆi↑nˆi↓+U (Na−Nˆ). (2.38) Therefore, if the hopping matrixt is real, i. e.,tij ∈ Rfor alli,j, the Hamiltonian ˆHh differs from (2.7) only by the termU (Na −Nˆ), which is an irrelevant constant if one works in a basis with a fixed number of particles. Since the transformation under consideration maps electrons into holes and vice versa, it follows that one can infer theN-electron spectrum and eigenstates of the Hubbard model on a bipartite lattice from the corresponding problem with 2Na −N electrons. This means, for a bipartite
lattice it is sufficient to analyze the caseN ≤ Na, since the solution of the more than half-filled bandN > Na can be derived from it.
Let us now restrict the electron-hole transformation to the down spins, i. e., let us consider the transformation which maps ˆci†↓ → ±cˆi↓, where, again, the positive sign applies ifi ∈ TAand the negative ifi ∈ TB. This transformation maps the Hubbard-model Hamiltonian (2.7) into
Hˆa =X
ij
tij
cˆi†↑cˆj↑−cˆi↓cˆ†j↓
+U X
i
cˆi†↑cˆi↑cˆi↓cˆi†↓=
=X
ij
tijcˆi†↑cˆj↑+X
ij
tij∗cˆ†i↓cˆj↓−U X
i
nˆi↑nˆi↓+UNˆ↑.
(2.39)
If the hopping matrixt is real, the transformed Hamiltonian (2.39) differs from the Hubbard-model Hamiltonian (2.7) by the sign of the Coulomb integral U and the termUNˆ↑, which only contributes an irrelevant constant if one works in a basis with a fixed number of up spins. This means, the electron-hole transformation of the down spins effectively maps the Hubbard model with repulsive Coulomb interaction onto a corresponding model with attractive interaction. Therefore, on a bipartite lattice one can obtain the solution of the attractive Hubbard model with coupling constantU < 0 andN↓down spins from the corresponding solution of the repulsive Hubbard model withU > 0 andNa−N↓down spins.
2.3.2 Related models of strongly interacting electrons
In Section 2.1 we have already analyzed the noninteracting limitU =0, where the Hubbard model reduces to a simple tight-binding Hamiltonian. Furthermore, we have reviewed the atomic limittij = 0 for alli,j, where all states with minimal number of double occupations are ground states, such that the corresponding ground-state en-ergy is highly degenerate. Let us now consider the limit of strong Coulomb interac-tions, where the hopping integrals are non-vanishing but several orders of magnitude smaller than the Coulomb-repulsion strength, i. e.,U |tij| for alli,j. In this case, the large ground-state degeneracy encountered in the atomic limit is lifted by the kinetic-energy term ˆK, which can be treated like a small perturbation to the interac-tion energy ˆW. The ground state is then obtained by diagonalizing ˆK in the lower Hubbard subband D0, which is the subspace formed by all states having minimal double occupations, i. e.,D = 0 ifN ≤ Na andD = N −Na else. In the following we will derive an effective Hamiltonian which describes the low-lying excitations of the strongly correlated Hubbard model, i. e., the excitations within the low-energy sub-spaceD0. To this aim, let us split the kinetic-energy term ˆK defined in Eq. (2.7) into
the following projected hopping operators Kˆ0= X
ijσ
tij
h 1−nˆi,−σcˆiσ†cˆjσ 1−nˆj,−σ
+nˆi,−σcˆiσ†cˆjσnˆj,−σ
i, (2.40) Kˆ+ = X
ijσ
tijnˆi,−σcˆiσ†cˆjσ 1−nˆj,−σ
, (2.41)
Kˆ− = X
ijσ
tij 1−nˆi,−σcˆiσ†cˆjσnˆj,−σ. (2.42) It is easy to verify that ˆK = Kˆ0 + Kˆ+ +Kˆ−. The first term ˆK0 neither creates nor annihilates double occupations, but represents the hopping of double occupations and vacancies across the lattice. Consequently, the term ˆK0does not lead out of the lower subbandD0. In contrast to ˆK0, the operators ˆK±lead to transitions between the low-energy and the high-low-energy subspaces by creating or annihilating double occupations.
However, higher order hopping processes, such as ˆK−Kˆ+, generate and subsequently eliminate double occupations, and thus do not lead out of D0. These intermediate double occupations are calledvirtualand they are generated by hopping processes of second or higher order in the operators ˆK±. In order to find an effective Hamiltonian which describes the action of the Hubbard model (2.7) in the subspaceD0, we follow the work of Schrieffer and Wolff [99] and seek for a unitary transformation
Hˆeff =ei ˆSHˆ e−i ˆS =Hˆ +i[S,ˆ Hˆ] − 1
2[S,ˆ [Sˆ,Hˆ]]+· · · (2.43) which eliminates all terms in ˆH that give rise to transitions between the lower and upper Hubbard subbands. It turns out that this goal cannot be achieved in a finite number of steps, since, for example, all terms that are of odd order in the opera-tors ˆK±lead to transitions between the Hubbard subbands. Therefore, we will restrict ourselves to the elimination of the leading term ˆK1 =Kˆ++Kˆ− in the Hamiltonian ˆH which, regarding Eq. (2.43), can be achieved by choosing the generator ˆS such that
i[Wˆ,Sˆ]=Kˆ1. (2.44) Since the operator ˆK+( ˆK−) increases (decreases) the number of doubly occupied sites by one, it holds
[Wˆ,Kˆ±]=U
DˆKˆ±−Kˆ±Dˆ
=±U Kˆ±, (2.45)
and consequently one has
[Wˆ,Kˆ+−Kˆ−]=U Kˆ1, (2.46) such that by comparison with Eq. (2.44) one finds
Sˆ= − i U
Kˆ+−Kˆ−
. (2.47)
This means that ˆS is of the ordert/U 1 wheret = maxij |tij|. Using i[Sˆ,Kˆ1] = 2[Kˆ+,Kˆ−]/U, one obtains by substituting Eq. (2.47) into Eq. (2.43)
Hˆeff =Kˆ0+Wˆ +i[S,ˆ Kˆ0]+ 1
U [Kˆ+,Kˆ−]+O t3/U2
. (2.48)
Let us now focus on the casen =N/Na ≤ 1, where the lower Hubbard subbandD0is formed by the states without doubly occupied sites. Consequently, there is no contri-bution from ˆW in the effective Hamiltonian (2.48) in this case. Furthermore, the term in ˆK0which accounts for the hopping of double occupations does not contribute, such that withinD0one has
Kˆ0 =X
ijσ
tij 1−nˆi,−σcˆiσ†cˆjσ 1−nˆj,−σ
. (2.49)
Let us now consider the commutator [Kˆ+,Kˆ−]. If ˆK+(ij) and ˆK−(kl) denote the terms in Eqs. (2.41) and (2.42) which account for the site pairs (ij) and (kl), it is clear that[Kˆ+(ij),Kˆ−(kl)] = 0 if both pairs are disjoint. Therefore, only terms involving two or three sites contribute to[Kˆ+,Kˆ−]. The same is true for the commutator[Sˆ,Kˆ0]. It is easy to see that only three-site terms contribute to [Sˆ,Kˆ0]and, furthermore, one can show (see for example [100, Chapter 5]) that the two-site terms of the opera-tor[Kˆ+,Kˆ−]/U can be expressed as
Hˆspin= X
ij
Jij
σˆi·σˆj − nˆinˆj
4
with Jij = 2tij2
U , (2.50)
where ˆσi =Sˆi/~with the local spin operator ˆSi, and ˆni =nˆi↑+nˆi↓is the local electron-number operator. Consequently, if we neglect all three-site contributions, the effective Hamiltonian (2.48) reduces to the so-calledt-J model
Hˆt J =X
ijσ
tij 1−nˆi,−σcˆiσ†cˆjσ 1−nˆj,−σ +X
ij
Jij
σˆi ·σˆj − nˆinˆj
4
. (2.51) It is often argued that the omission of the three-site contributions is well justified close to half band-filling, however, a complete treatment of the effective Hamiltonian (2.48), including three-site terms up to ordert2/U, is desirable in many situations and can be found, for example, in the excellent book of Fazekas [100]. The first term in the t-J model (2.51) represents the correlated motion of the electrons through the lattice, which avoids the creation of double occupations altogether. The second term de-scribes anAFMinteraction between the spins (Jij =2tij2/U > 0), which is reduced by a density-density interaction. At exactly half band-filling (n = 1) the electrons can not
move without creating doubly occupied sites, such that in this case thet-Jmodel (2.51) reduces to the well-knownAFMHeisenberg model
HˆH=X
ij
Jijσˆi·σˆj . (2.52) Here we have omitted the density-density interaction term ˆninˆj which appears in Eq. (2.50), since at half band-filling it only contributes an irrelevant constant due to the fact that the electrons can not move such thathnˆinˆji= hnˆiihnˆji=1 for alli,j.