Theory of the Luttinger surface in doped Mott insulators

We prove that the Mott insulating state is characterized by a divergence of the electron self-energy at well-defined values of momenta in the first Brillouin zone. When particle-hole symmetry is present, the divergence obtains at the momenta of the Fermi surface for the corresponding noninteracting system. Such a divergence gives rise to a surface of zeros (the Luttinger surface) of the single-particle Green function and offers a single unifying principle of Mottness from which pseudogap phenomena, spectral weight transfer, and broad spectral features emerge in doped Mott insulators. We also show that only when particle-hole symmetry is present does the volume of the zero surface equal the particle density. We identify that the general breakdown of Luttinger’s theorem in a Mott insulator arises from the breakdown of a perturbative expansion for the self-energy in the single-particle Green function around the noninteracting limit. A modified version of Luttinger’s theorem is derived for special cases.

Figure 1

Spectral function for a doped Mott insulator at a filling of $n=0.95$ at $T=0.07t$ for a path in momentum space from (0, 0) to $(\pi ,0)$ to $(\pi ,\pi )$ and then back to (0, 0). Equation (4) must undergo a sign change by passing through zero because at (0, 0) most of the spectral weight lies below the chemical potential, whereas at $(\pi ,\pi )$, it lies above. The temperature dependence of the density of states at the Fermi level is shown in the inset. That the spectral features are broad near the chemical potential is a direct consequence of the divergence of the self-energy at $\omega =0$ at the Luttinger surface. The spectral function was computed using the two-site self-consistent method of Stanescu and Phillips. (Ref. 17)

Figure 2

(Color online) Schematic depiction of two possibilities for the distribution of spectral weight in the single-particle density of states, $n\left(\omega \right)$, in a lightly doped Mott insulator. In both cases, the peak above the chemical potential represents the low-energy spectral weight. The weight of this peak increases as least as fast as $2x$, where $x$ is the number of holes. (A) The Luttinger surface lies along the zone diagonal and the transferred spectral weight must be asymmetrically located relative to the chemical potential to lead to a vanishing of $\mathrm{Re}G$. (B) The spectral weight redistribution is symmetrical and the Luttinger surface lies away from the zone diagonal, for example, close to $(\pi ,\pi )$.

Figure 3

(Color online) Evolution of the surface of zeros in the first quadrant of the FBZ. Yellow (light) indicates $\mathrm{Re}G>0$ while blue (dark) $\mathrm{Re}G<0$. The Hubbard model is constrained to have a surface of zeros as $n\to 1$ whereas the $t\text{−}J$ model is not. The two options upon doping represent weak coupling (a) or (c) to (d) and strong coupling (a) to (b). The transition from (a) to (b) requires a critical point at $n_c$ whereas (d) does not. Experiments (Refs. 33, 34) indicating an insulating state for $n>n_c$ are consistent with an abrupt transition from (a) to (b).