Splitting of electrons and violation of the Luttinger sum rule

We obtain a controlled description of a strongly correlated regime of electronic behavior. We begin by arguing that there are two ways to characterize the electronic degree of freedom, either by the canonical fermion algebra or the graded Lie algebra $\mathfrak{su}\left(2\right|2)$. The first underlies the Fermi liquid description of correlated matter, and we identify a regime governed by the latter. We exploit an exceptional central extension of $\mathfrak{su}\left(2\right|2)$ to employ a perturbative scheme recently developed by Shastry and obtain a series of successive approximations for the electronic Green's function. We then focus on the leading approximation, which reveals a splitting in two of the electronic dispersion. The Luttinger sum rule is violated, and a Mott metal-insulator transition is exhibited. We offer a perspective.

Figure 1

Correlated hopping is when the hopping amplitude depends on how the two sites are occupied by electrons of the opposite spin. Here we illustrate the four possibilities for a hopping spin-up electron (the final two of which are hermitian conjugate). We argue that decoupling these amplitudes from the uncorrelated limit may induce a splitting of the electron.

Figure 2

The splitting of the electronic dispersion, exemplified on a square lattice. We focus on the symmetric case $\tilde{\lambda }=0$, and $J$ does not contribute at this order of approximation. The left panel shows the electronic density of states (DOS), ${\sum }_{\sigma }{\int }_{BZ}\frac{d^{2}p}{{\left(2\pi \right)}^2}{A}_{p\sigma }\left(\omega \right)$, with the contributions of the $\circ$ (blue, lower) and $•$ (red, upper) bands distinguished. The central panel shows the band structure, an intensity plot of the electronic spectral function (with Lorentzian broadening of ${10}^{-3}$), along the $\mathrm{\Gamma }\text{−}X\text{−}M\text{−}\mathrm{\Gamma }$ high-symmetry path in the Brillouin zone. The right panel shows the spectral weights $a_{p\circ }$ and $a_{p•}$. The horizontal lines (a)–(d) indicate slices along which the spectral function is plotted in Fig. 3. Four examples (i)–(iv) of couplings $\kappa$ and $\tilde{U}$ are presented: (i) on leaving the noninteracting point the band structure splits with the introduction of weak and flat dispersion near $\omega =0$, and the 2D Van Hove singularity of the DOS also splits in two; (ii) when $\tilde{U}$ is increased above ${\tilde{U}}_M=\frac{8}{1+{\kappa }^2}$ the two bands separate and a Mott gap opens; (iii) as the strength of correlated hopping is amplified the two bands overlap significantly for $\tilde{U}<{\tilde{U}}_M$; (iv) as $\kappa$ approaches 1 the two bands decouple, each with half the weight of an electron, and $\tilde{U}$ behaves as an additional chemical potential that shifts the bands oppositely in $\omega$.

Figure 3

Plots of the electronic spectral function throughout the 2d Brillouin zone on the slices (a)-(d) indicated in Fig. 2. The violation of the Luttinger sum rule can be seen by contrasting between (a) and (b), both of which correspond to below half-filling: while less than half the Brillouin zone is enclosed in (a), this is clearly not the case in (b). In (c) and (d) the appearance of two surfaces is in sharp contrast to conventional band theory.

Figure 4

A schematic depiction of how metallic behavior may be governed by either canonical or $\mathfrak{su}\left(2\right|2)$ degrees of freedom in different regions of parameter space. Here correlated hopping, controlled by $\kappa$, and onsite repulsion, controlled by $U$, compete to organise the electronic degree of freedom in distinct ways. While the $\mathfrak{su}\left(2\right|2)$ regime is restricted to small $U$ for small $\kappa$, it may extend to large $U$ when $\kappa$ is $\mathcal{O}\left(1\right)$. We speculate on the nature of the ‘transition’ between the two regimes toward the end of the Discussion.