4.3 Breakdown of the Fermi liquid
4.3.3 Cubic lattices
After having explored the non-Fermi liquid properties of some model DOS, it is interesting to study real lattices with similar features. Almost all of the cubic DOS have noticeable van Hove singularities in not too large space dimensions, typically logarithmically divergences or square-root like cusps. For an overview and examples consult appendix D.
Figure 4.14 shows the different solutions for ˜T (upper rows) and ΣU (lower rows) of the Hubbard model for various cubic lattices in different space dimensions. The non-interacting DOS, which are also plotted in the insets of each graph, their parameter values and the positions of their van Hove singularities are best describe in the following list:
(a) Simple-cubic lattice with nearest neighbor hopping in two space dimensions (2d-SC).
This DOS has a logarithmic divergence in the band center.
(b) Simple-cubic lattice with nearest and next-nearest neighbor hopping (t0/t = −0.2) in two space dimensions (2d-SCNNN). This DOS has a logarithmic divergence at ω=−0.4.
(c)+(d) Body-centered cubic lattice with nearest neighbor hopping in three space dimensions (3d-BCC). This DOS has a logarithmic divergence in the band center.
(e) Face-centered cubic lattice with nearest neighbor hopping in three space dimensions (3d-FCC). This DOS has a logarithmic divergence atω =−2 and a square-root van Hove singularity atω = 0.
4.3 Breakdown of the Fermi liquid
(f) Simple-cubic lattice with nearest and next-nearest neighbor hopping (t0/t=−0.46) in three space dimensions (3d-SCNNN). This DOS has square-root van Hove singularities atω≈ −1,−0.82,−0.24 and 0.08.
It can be clearly seen in the graphs, that all effective media develop the minimum in the immediate vicinity or right at the Fermi level for low temperatures, except for case (f). In case (b), where the logarithmic divergence is not situated at the Fermi level but at ω ≈ −0.4, the minimum in Im ˜T is shifted to energies slightly below zero. For the three dimensional simple cubic lattice with considerable next-nearest neighbor hopping (f), the medium develops a very steep edge right at the Fermi level rather than a minimum.
All correlation self-energies show rather unusual behavior. In the cases (a), (b), (c) and (e) a maximum is developed at or near the Fermi level, as it is expected from the model-DOS studies of figure 4.13. For the asymmetric case (b), where the singularity is moved to lower energies, the maximum is also moved away from the Fermi level. For case (e), which represents also an asymmetric situation but the non-analyticity is still situated at the Fermi level, the maximum in ImΣU is also found at ω = 0, but the two minima next to it are asymmetric. In the 3d-BCC lattice with larger Coulomb repulsion, case (d), the formation of the maximum is not clearly visible and might be suppressed in favor of a linearly rather than quadratic dependence of the imaginary part around the Fermi level. The tendency to increase linearly away from the Fermi level can also be observed for the smaller value of U = 2.75 in case (c). For the 3d-SCNNN shown in part (f) a quadratic minimum is found in ImΣU around the Fermi level, but it is formed very asymmetric and remains so to very low temperatures, as it can be seen in the upper right inset. The increase in ImΣU(ω−iδ) from ω = 0 to negative energies is considerably slower and less steep than from zero to positive ω. For very low temperatures (inset) a quadratic fit to the minimum can be justified only up to energies of the size of the temperature, e.g. |ω−ωmin| ≈T. The question whether the quadratic dependence remains asT →0 or if it is replaced by an asymmetric functional dependence cannot be answered with this approach and thus remains unanswered.
While for the situations (a), (c) and (d) the destruction of the Fermi liquid is directly observable, Fermi liquid ground states might still be possible for cases (b), (e) and (f).
However, even if the Fermi liquid turns out to be stable in the latter cases, the finite temperature behavior is nonetheless expected to be rather peculiar, e.g. non-Fermi liquid like, due to the strong renormalizations and unusual energy dependencies of ImΣU in the vicinity of the Fermi level.
As it should be clear from the studies shown so far, the key ingredient for the suppression of the Fermi liquid formation is the non-analyticity in the non-interacting DOS at or near the Fermi level.
In order to understand the encountered behavior, notice that the origin of the van Hove singularities are flat parts in the dispersion relation. As examples, figure 4.15 shows the dispersion relations for the 2d-SC (left graph) and the 3d-FCC (right graph) lattice. The corresponding points in the Fermi surface leading to the van Hove singularities atω= 0 for half filling are for example k= (0,±π)T (2d-SC) and k= (±π/2,±π/2,3π/2)T (3d-FCC).
As it can be seen from the figures, all these points are associated with saddle points in the dispersion.
4 The Hubbard model
0 2
-4 -2 0 2 4
ImΣU(ω−iδ)
ω
U= 3 nσ= 0.49 0
0.5 1
Im˜T(ω−iδ) (a) T=0.4
0.26 0.125
0 0.5 1 1.5
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
ImΣU(ω−iδ)
ω U= 4 nσ= 0.5 0
0.5 1
Im˜T(ω−iδ) (b) T=0.12 0.06 0.02
0 0.2 0.4 0.6 0.8
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
ImΣU(ω−iδ)
ω
U= 2.75 nσ= 0.5 0.6
0.8
Im˜T(ω−iδ) (c)
T=0.3 0.14 0.05 0.03
0 0.5 1 1.5 2
-1 -0.5 0 0.5 1
ImΣU(ω−iδ)
ω U= 4 nσ= 0.5 0.4
0.6 0.8
Im˜T(ω−iδ) (d)
T=0.15 0.05 0.018 0.005
0 0.5 1 1.5
-8 -6 -4 -2 0 2 4 6 8 10
ImΣU(ω−iδ)
ω U= 3
=−1.9
0 0.5 1 1.5
Im˜T(ω−iδ) (e) T=1
0.7 0.5
0 5 10 15
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
ImΣU(ω−iδ)
ω U= 4 nσ= 0.5 0
0.5 1
Im˜T(ω−iδ) (f)T=0.04
0.01 0.009
0.008 0
0.5
-0.03 0 0.03
Figure 4.14: The effective medium (upper graphs) and correlation self-energy (lower graphs) of the Hubbard for different cubic lattices. The non-interacting DOS are shown in the insets. The parameter sets are: (a) 2d-SC,U = 3, nσ = 0.49; (b) 2d-SCNNN t0/t=−0.2,U = 4, nσ = 0.5;
(c) 3d-BCC,U = 2.75, nσ = 0.5; (d) 3d-BCC,U = 4, nσ = 0.5; (e) 3d-FCCU = 3, =−1.9; (f) 3d-SCNNN U = 4, t0/t =−0.46, nσ = 0.5 and the temperatures as indicated. The right inset in figure (f) shows ImΣU in the low energy region for even lower temperaturesT = 0.007, 0.006 and 0.004.
4.3 Breakdown of the Fermi liquid
π 0 -π
π 0
-π -2
-1 0 1 2
tSCk 2d-SC
kx
ky 0
-π -2π
2π π
0 -2
0 2
tF CCk 3d-FCC
kz= 3π/2
kx
ky
Figure 4.15: The dispersion relation for the 2d-SC (left) and 3d-FCC (right) lattice. In the latter case the z-component set to 3π/2 and only one quadrant is shown for clarity. The lines on the bottom of each graph indicate the position of the Fermi surface for half filling.
If these saddle points occur right at the Fermi surface, which is the case at half filling for these two examples, the dispersion relation cannot be approximated by a linear function around the Fermi surface, i.e.
tk 6≈ kF
m∗|∆k| , (4.3.17)
with ∆k=k−kF and kF a Fermi wave vector. The expansion around these points rather yield hyperbolic functional dependencies
tk∼
∆k2x−∆k2y 2d−SC
∆kx∆ky 3d−FCC . (4.3.18)
Therefore the usual argument leading to a quadratic dependence of the scattering amplitude on energy and temperature, i.e. ImΣU(ω −iδ) ∝ ω2 +π2T2, is not applicable at these special k-points. Instead the quadratic momentum dependence leads to an enhancement of low energy excitations due to the very small energy denominators, and consequently an enhanced scattering. Averaged over the whole Fermi surface in order to obtain the full (local) scattering amplitude, an energy and temperature independent additional constant contributed from these points seems possible.
The physical origin of the increased scattering could either be the removal of certain band states from the vicinity of the Fermi surface, i.e. the formation of a band gap, or a partial localization of those electrons. The difference between these two scenarios cannot be directly resolved on the basis of one-particle properties. Even though only the calculation of the conductivity or the resistivity could directly reveal the nature of the situation encountered
4 The Hubbard model
here, an indirect argument can be provided nonetheless. The spectral intensities of the effective media for all situations are reduced, when the increased scattering is encountered.
Since the effective medium characterizes the hybridization amplitude of a local lattice site, this indicates decreased hopping to neighboring sites within this energy interval. The reason can either be a a reduced mobility of the electrons (localization) or the lack of states within that energy region at the surrounding sites (band gap). In first order, the number of states available on the surrounding sites is given by the local density of states of these sites, i.e.
the local one-particle DOS. The latter function does not show any sign of a (pseudo) gap formation, which could be taken as a sign of some sort of localization taking place.
However, at this point care has to be taken, since the present approximation only accounts for momentum independent self-energies and effective media. Theses quantities represent momentum averages over iso-energy surfaces of the whole Brillouin zone. Thus details of possible band gaps developing only in some points of the Brillouin zone are averaged and maybe even smeared out. Therefore a band gap might still be there, but just not visible in the local DOS.
The physical picture of the scenario leading to the breakdown or at least strong modifi-cation of the Fermi liquid can be summarized: at high temperatures, thermal fluctuations are large and the details and flat parts of the Fermi surface are not noticeable. In that case, the system behaves regular and the tendency of the spin-12 SIAM to form a local Fermi liquid dominates. Lowering the temperature leads to an increased screening of the developing local moments and the band states can be mixed coherently to form the low energy Fermi liquid, i.e. the quasiparticles start to emerge. This is observed in the graphs, as the quadratic minimum in the self-energy and the new quasiparticle band around the Fermi level are formed, which both become more pronounced for lower temperatures. Since the self-energy is momentum independent, the quasiparticles attain the same Fermi surface as in the non-interacting case [MH89b]. Upon further lowering the temperature, the flat parts of the Fermi surface become influential and lead to a large scattering amplitude for the quasiparticles and the Fermi liquid formation is disturbed.
The physical origin of the increased scattering is speculated to be a consequence of a partial localization of the quasiparticles at the flat points on the Fermi surface. In order to bring more light into this subject, the calculation of the conductivity and approximations involving momentum dependent self-energies are desirable.