4.4 Susceptibilities and magnetism in the Hubbard model
4.4.1 Sketch of the phase diagram
4 The Hubbard model
Magnetism depends strongly on the lattice structure and the number of electrons per lattice site (filling). For instance, in theories for the magnetism of localized magnetic moments, such as the Heisenberg model, the arrangement and coupling of the spins on some lattice is essential and geometric frustration has the effect of suppressing order. On the other hand, for metals the geometry of the hopping matrix elementstij, i.e. the resulting band structuretk= N1
0
P
ijeik(Ri−Rj)tij, and the Fermi surface play a crucial role.
If large, almost parallel regions of the Fermi surface with the appropriate electron- and hole-like character exist, a very large phase space volume for magnetic scattering processes at the Fermi surface results and an enhanced magnetic susceptibility is the consequence.
This is known as(perfect) nesting and the finite wave vectorq0 for which it occurs is known as thenesting vector, i.e.
tk+q
0 ≈tk (4.4.1)
for manyk on the Fermi surface. This is the case for the simple cubic and body centered cubic lattices with nearest neighbor hopping at half filling8.
As another point, the interplay and competition between strong correlations and the kinetic energy are believed to be the causes for itinerant ferromagnetism (FM) in transition metals. FM is thus very sensitive to the filling and the lattice structure, i.e. the distribution of spectral weight in non-interacting DOS.
As a starting point for the investigation, it is worthwhile to consider the Hubbard model on the three dimensional simple cubic (3d-SC) lattice. The SC lattice exhibits perfect nesting in any space dimensiondfor the antiferromagnetic (AFM) nesting vector,qAF M = π(1,1,1)T ford= 3, which leads to the presence of strong AFM spin fluctuations. Within a weak coupling (U/t1) Stoner (RPA) treatment a diverging static susceptibility for the AFM wave vector qAF M and thus a AFM phase transition is found with the critical N´eel temperature at half filling (cf. [Pen66, Hir87, Faz99])
TNRP A= 1.13W e−
1
ρ0(0)U , (4.4.2)
whereρ0(0) is the non-interacting tight-binding DOS at the Fermi level andW = 2dt= 3 the half bandwidth. The characteristict9 and U dependence of this result remains true for the SC lattice in any spacial dimension except for d= 2 where the logarithmic divergence of the non-interacting DOS at the Fermi level leads to a additional square root in the exponent10, i.e.
TNRP A(d= 2)∼W e−
√t
U. (4.4.3)
The AFM transition occurs at half-filling for any non-zero value of the Coulomb repulsion U >0, which is a drastic consequence of the perfect nesting property of the SC lattices.
The AFM phase transition is also present for fillings away from half filling nσ 6= 12 up to a critical value of doping which at zero temperatureT = 0 can be deduced within the weak coupling approximation
δRP AC = 1−2nσ ∼ 1 U e−
1
ρ0(0)U . (4.4.4)
8 For details on some lattice structures see appendix D.
9 Rememberρ0(0)∼1/t.
10The same is true for the three dimensional body centered cubic lattice.
4.4 Susceptibilities and magnetism in the Hubbard model
In the opposite strong coupling limit (U/t 1) and for half filling δ = 0, the critical N´eel temperature can be calculated from the high-temperature expansion for the antiferro-magnetic Heisenberg model [Rus74] with the appropriate coupling (see (4.2.7) and (4.2.8)),
TNstrong = 3.83 t2
U . (4.4.5)
As it can be already anticipated by the presence of an AFM exchange coupling for the large U limit in (4.2.7), AFM is expected to be a robust phenomenon for the Hubbard model. In situations, where additional next-nearest neighbor hopping amplitudes t0 are introduced, the perfect nesting property of the SC lattice is lost and the AFM region is expected to shrink [San93, Ari00, Zit04b, Pru05]. Only in situations, wheret0 is very large or on lattices with geometric frustration and without any sort of nesting, AFM is expected to vanish [Ulm98, Mer06, Kyu07].
At intermediate to large U ferromagnetism is expected away from half filling[Pen66, Bul90, Her97, Faz99]. In contrast to the robust occurrence of AFM, ferromagnetism is a very subtle phenomenon in the Hubbard model. A rigorous proof for fully polarized FM in the Hubbard model atU =∞ for various cubic lattices and a filling of one electron plus or minus half filling was presented by Nagaoka [Nag65, Nag66] (for a more general treatment see [Faz99, Han96, Han97]). The necessary restrictions on the lattice structure, signs of the hopping matrix elements and doping already suggests that FM is found in the Hubbard model only for very specific situations.
Quite generally, the conditions favoring FM in single-band Hubbard models include low densities [MH95, Pie96], situations where the Fermi level lies inside a flat part of a band or generally a flat band near one band edge resulting in a sharp structure in the non-interacting DOS [Tas92, Mie93, Obe97, Wah98, Ulm98, Vol99, Faz99, Ari00, Mer06, Pan07].
The common feature of all these scenarios is that the kinetic energy and especially its balance with the interaction energy is the key quantity deciding whether a tendency towards FM is developed or not. For certain band structures it could be beneficial to put electrons with the minority spin in higher energy band states and thereby avoid interactions in the majority band through the symmetry correlations, leading to a FM ground state with an essential weakly interacting band of majority spins. Or the quantum mechanical interference of hopping processes for electrons or holes on certain paths through the lattice produce effectively exchange interactions favoring FM, as it is found for the Nagaoka-type FM and e.g. three-site ring exchange processes on some special non-bipartite lattices [Pen96, Tas98a].
For intermediate values of the Coulomb repulsion U a transition from the antiferromag-netism of a weakly interacting electronic band to the phase of localized spins is expected at half filling. The transition temperature is thus an interpolation between the asymptotic forms (4.4.3) and (4.4.5) [Jar93, Dar00]. For systems off half filling antiferromagnetism is expected to extend to some value of doping, maybe transforming into some sort of in-commensurate (IC) magnetism before disappearing altogether [Pen66, Fre95, TZ97]. This is expected since for increasing doping δ 6= 0, the system is always metallic and particle-hole excitations are gradually suppressed due to the deviation from perfect nesting of the Fermi surface, but possibly other approximate nesting vectors appear. Additionally, sev-eral forms of a phase separated ground state are expected for largeU and away from half filling [Vis74, Eme90, Zit02, Pru03, Mac06a]
4 The Hubbard model
0 0.1
0.2 0.3
0.4 0 0.2
0.4 0.6
0.8 1 000
0.05 0.1 0.15 0.2 0.25 0.3
U/(1+U) δ
T/W
PM AFI
FM AF/PS
AF/PS PM
Figure 4.19: Schematic phase diagram of the single band Hubbard model on the simple cubic lattice with nearest neighbor hopping within the locally complete approximation taken from [Pru03]. PM denotes the paramagnetic metal, AFI the antiferromagnetic insulator, AF/PS the phase-separated antiferromagnetism and FM ferromagnetism. Note that in this picture W is chosen as the full bandwidth, while in the present work it is defined as the half bandwidth.
As a summary of the above remarks, a schematic phase diagram is shown in figure 4.19.
The diagram portrays the situation for the Hubbard model on the simple cubic lattice. As argued, at the edges of the AF/PS phase incommensurate magnetism is generally expected.
The regions of the various phases are enlarged or shrunk depending on the specific lattice structure and frustration.
Of special importance for the latter considerations is the different nature of the magnetism in various parts of the phase diagram. As it was already mentioned, at small Coulomb inter-actions the itinerant Stoner-type magnetism is encountered. Magnetic ordering is produced by an enlarged scattering for particle-hole pairs at the Fermi surface. At largeU and at (or very close to) half filling large local magnetic moments are present and the electrons are almost localized. The systems thus consists of localized spins and Heisenberg-type local mo-ment magnetism is encountered. In situations with intermediate interaction strength and close to half filling or for very large U and off half filling, considerable magnetic moments are formed, but the system is still metallic. These cases correspond to a correlated metal where both of the above archetypes mix and the details of the lattice might be essential.