Hubbard Bands and Spectral-Weight Transfer

Correlation-driven localization redistributes single-particle spectral weight between low-energy coherent states and incoherent Hubbard bands. The exact sum rule is fixed, so the diagnostic is not merely a two-peak line shape: one must follow integrated weight, quasiparticle residue, chemical potential, gap, and order as interaction or doping changes.

Required background. Use the Hubbard atomic and strong-coupling limits and spectral moments and sum rules. Helpful background. Dyson equations define $Z$ and self-energy poles.

Atomic spectral function

At half filling in the particle–hole-symmetric convention, the paramagnetic atomic Green function per spin is

$$G_{\mathrm{at}}(z)=\frac12\left(\frac1{z+U/2}+\frac1{z-U/2}\right) =\frac{z}{z^2-(U/2)^2}.$$

Thus

$$A_{\mathrm{at}}(\omega)=\pi\delta(\omega+U/2) +\pi\delta(\omega-U/2), \qquad \int\frac{\mathrm d\omega}{2\pi}\,A_{\mathrm{at}}(\omega)=1.$$

The poles are electron-removal and addition transitions; their separation is the atomic charge scale. Turning on hopping broadens them into lower and upper Hubbard bands. In a correlated metal, a narrow coherent feature can also develop near zero energy with residue

$$Z=\left[1-\left.\partial_\omega\operatorname{Re}\Sigma^R(\omega)\right|_{0}\right]^{-1}$$

when a local Fermi-liquid expansion exists.

Transfer with interaction and doping

Changing $U$ or carrier density moves weight over energy scales of order $U$, not only within a narrow quasiparticle band. Upon hole doping, new low-energy electron-addition states appear because empty sites become available. The integrated occupied and unoccupied weights follow exact anticommutator and moment sum rules; tracking only peak heights is unreliable because linewidth and matrix elements change.

A Mott transition can be approached through collapse of $Z$, opening of a paramagnetic gap, or coexistence of metallic and insulating solutions in a controlled DMFT setting. Magnetic order can instead fold bands and open a Slater gap before this paramagnetic localization is reached. Imada, Fujimori, and Tokura review these distinct spectral routes in Imada, Fujimori, and Tokura 1998, §§ III–IV.

Checks and limitations

Enforce positivity, normalization, and at least the first spectral moment. Report analytic-continuation uncertainty, temperature, momentum/orbital resolution, and matrix elements. A pair of broad peaks can arise from hybridization, disorder, phonon sidebands, or symmetry breaking; it is not a unique Mott diagnostic.

Exercises

Find the self-energy of the symmetric atomic Green function using a noninteracting propagator $G_0(z)=1/z$.

Solution

$G^{-1}=z-(U/2)^2/z$, while $G_0^{-1}=z$. Dyson’s equation gives $\Sigma(z)=U^2/(4z)$. Its zero-frequency pole removes spectral weight from the chemical potential and produces the atomic gap.

References

• Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal–Insulator Transitions,” Reviews of Modern Physics 70 (1998) 1039–1263, §§ III–IV, doi:10.1103/RevModPhys.70.1039.