4.3 Breakdown of the Fermi liquid
4.3.4 Relevance for the investigation of pseudogap behavior
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.
4.3 Breakdown of the Fermi liquid
Within the locally complete approximation, pseudogap behavior can only be introduced via ΣU. The local Green function is given by
G(loc)(z) = 1 N0
X
k
1 z−ΣU(z)−tk
= Z
dx ρ0(x)
z−ΣU(z)−x (4.3.19)
=GU=0(z−ΣU(z)) (4.3.20)
and is thus completely determined by the functional form of the non-interacting Green function, but evaluated for arguments in the complex plane atξ =z−ΣU(z). This implies for a Fermi liquid at T = 0, where the imaginary part of the self-energy vanishes at the Fermi level Imξ = 0, that the interacting DOS at ω = 0 is given by the value of the non-interacting one5. From this it is obvious, that a maximum growing in an otherwise small ImΣU at the Fermi level indeed indicates a tendency to produce a pseudogap in the local DOS.
Pseudogap behavior is found and studied in the two dimensional Hubbard and tJ models. The key ingredient to capture these effects is believed to be the inclusion of non-local correlations into the approximation. Therefore these models are usually stud-ied with the help of more advanced techniques, such as cluster extensions of the locally complete approximation [Mai05, Mac06b] or by explicitly accounting for non-local fluctua-tions [Vil97, Hau03, Sad05]. The occurrence of pseudogaps and the suppression of coherent quasiparticle formation for some parts of the Fermi surface, both effects are found within these calculations, are attributed to the presence of strong antiferromagnetic spin fluctua-tions. The self-energy, which is momentum dependent in these approximations, indicates a strongly enhanced scattering for some wave vectors, due to the presence of spin fluctuations.
It is very tempting to view the findings described in this study as precursors to the characteristic spin-fluctuation induced pseudogap behavior. One could argue, that due to the assumption of a local self-energy in the locally complete approximation, the non-local spin fluctuations are included only in a mean-field manner and the details of the k-dependence of the self-energy are averaged to yield a mean effect. The remnants of the pseudogap physics is then only seen in the slight maximum of ImΣU, which results from the average over the whole Fermi surface, but the effects are too much smeared out and less pronounced to leave any trace in the local DOS.
However, this is not in accord with the explanation presented here. The argument for the increased scattering rate of the quasiparticles did solely rely on the non-analytic structure of the non-interacting DOS and the concomitant flat parts of the Fermi surface! This is sufficient to violate some basic assumptions of a Fermi liquid description, leading to an increased scattering amplitude at some points in the Brillouin zone.
It could be certainly suspected, that both scenarios might be connected, since the flat parts of the Fermi surface in the 2d-SC and 3d-BCC lattice, which are responsible for the van Hove singularities, also cause a strongly enhanced AFM magnetic susceptibility, indicating the presence of strong AFM fluctuations. But the enhanced magnetic susceptibility in both these lattices results from (perfect) nesting with an antiferromagnetic nesting vector.
5 Notice, this would imply a diverging DOS for the Fermi liquid state at half-filling in the 2d-SC and 3d-BCC lattices! Since this contradicts the assumption of analyticity of the Fermi liquid, the breakdown of the latter can already be anticipated.
4 The Hubbard model
-0.5 0 0.5 1 1.5 2 2.5 3
0 0.2 0.4 0.6 0.8 1
T 1/χmag(q,0)
2d-SC U= 4 nσ= 0.5
q= 0 (FM) π(1/2,1/2) π(1,1) (AFM) π(1,1/3) π(1,0) π(1/3,0)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0 0.2 0.4 0.6 0.8 1
T 1/χmag(q,0)
3d-FCC U= 3 =−1.9
q= 0 (FM) π(1,1,1) (9/8)π(1,1,1) 2π(1,1,0) π(2,1,0)
Figure 4.16: The inverse static magnetic susceptibility for the 2d-SC lattice for U = 4, nσ = 0.5 (left graph) and the 3d-FCC lattice forU = 3,=−1.9 (right graph) as functions of temperature.
The different curves correspond to variousq-modes as indicated. The dashed grey line in the right graph is the linear extrapolation 3(T+ 0.1).
Such nesting is for example lacking in the 3d-FCC lattices and the presence of strong spin fluctuations is therefore questionable in this case.
To examine this issue, the magnetic susceptibility is calculated and shown in figure 4.16.
The graphs show the temperature dependency of the inverse susceptibility χ 1
mag(q,0) for some selected wave vectors on the 2d-SC (left graph) and 3d-FCC (right graph) lattice.
Lowering the temperature indeed strongly enhances the susceptibility in the 2d-SC case and even leads to a phase transition at T ≈ 0.14. The q = (π, π)T-component of the susceptibility is always largest, and thus diverges at the highest temperature, indicating the AFM instability. In this case, enhanced AFM fluctuations are indeed present in the system and might play a role in the destruction of the Fermi liquid. However, even though the magnetic susceptibility of the 3d-FCC lattice is also increasing with lower temperatures, the absolute value is by far lower than in the 2d-SC case. The extrapolation6 of the curves to lower temperatures even indicates the absence of a transition to an ordered state, since the critical temperature for the transition comes out to be below zero (dashed grey line).
Consequently, the magnetic fluctuations are considerably less and cannot be the cause of the observed strong anomalies, especially at the temperatures where the latter occur.
An argument usually presented in favor of a AFM spin-fluctuation driven pseudogap is, that introducing geometric frustration into the system by means of next-nearest neighbor hopping, suppresses AFM fluctuations and consequently leads to a vanishing of the pseudo-gap, which is indeed observed. However, the effects described above may lead to the same qualitative behavior: frustration changes the non-interacting DOSρ0 and moves the non-analyticity in ρ0 away from the Fermi level (either to higher or lower energies, depending
6 Data points below temperatures of aboutT ≤0.5 could not be obtained due to the pathology described in section 4.3.2.
4.3 Breakdown of the Fermi liquid
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
-4 -2 0 2 4 6
ω ρ(ω)
T=1 0.35 0.07 0.03 0.01 0.007
0 0.1 0.2 0.3
-0.1 0 0.1 0.2 0.3
0 1 2 3 4 5 6 7 8 9
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
ImΣ(ω−iδ)
T=0.35 0.07 0.03 0.01 0.007
0.1 1 10 100 1000
-4 -2 0 2 4 6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Im ˜T(ω−iδ)
T=0.35 0.07 0.03 0.01 0.007
-4 -2 0 2 4 6
Figure 4.17: Local DOS (upper graph), correlation self-energy (lower left) and effective medium (lower tight) for the asymmetric Hubbard model on a 2d-SC lattice withU = 6 and fillingnσ= 0.49 for various temperatures. The insets show the low energy (local DOS) and high energy (self-energy and medium) region.
on the sign of the next-nearest neighbor matrix element), as it can be seen in graph (b) of 4.14 or in appendix D. Consequently, the quasiparticle band structure is now disturbed at energies off the Fermi surface and the maximum in ImΣU is shifted to finite energies. The low energy excitations at the Fermi level do not have to respond to non-analyticities at low temperatures and thus, with increased frustration, the Fermi liquid ground state becomes stable again.
The same characteristic behavior can also be observed, if the unfrustrated lattice is doped away from half filling. There, the same reasoning applies as in the frustrated case. The flat parts of the iso-energy surface are shifted away from zero energy and consequently, the Fermi
4 The Hubbard model
(0,0) (π,0) (π, π) (0,0)
-4 -2 0 2 4 6
0 0.2 0.4 0.6
ρ(k, ω)
T=0.08
ω ρ(k, ω)
(0,0) (π,0) (π, π) (0,0)
-4 -2 0 2 4 6
0 0.5 1 1.5 2 2.5
ρ(k, ω)
T=0.007
ω ρ(k, ω)
(0,0) (π,0) (π, π) (0,0)
-0.4 -0.2 0 0.2 0.4 0
0.2 0.4 0.6
ρ(k, ω)
T=0.08
ω ρ(k, ω)
(0,0) (π,0) (π, π) (0,0)
-0.1 0 0.1 0.2 0.3 0
0.5 1 1.5 2 2.5
ρ(k, ω)
T=0.007
ω ρ(k, ω)
Figure 4.18: Band structure for the asymmetric Hubbard model on a 2d-SC lattice with U = 6 and fillingnσ = 0.49 for two low temperatures. The upper row shows the complete energy region, while the lower row show the low energy region around the Fermi level. The turquoise lines mark the specialkvectors as indicated on the axis, while the blue lines mark the approximate position of the Fermi wave-vectorskF.
surface becomes regular. The van Hove singularity, and correspondingly the maximum in ImΣU is also shifted to finite energy and the formation of a Fermi liquid becomes possible again. This is the case for the 2d-SC lattice at filling nσ = 0.49 and U = 6 as it is shown in figure 4.17. The upper graph of figure 4.17 shows the local spectral function with the lower and upper Hubbard bands7 and the many-body resonance at the Fermi level. The inset shows the latter for an enlarged interval around ω = 0. In can be seen, that while
7 The origin of the sharp spike in the local DOS at the lower edge of the upper Hubbard band forT = 0.1 is not clear yet. Within the numerical simulations, it behaves in a meta-stable manner, as it is not consistently observed for lower temperatures. It has been speculated, that it results from a bound state