Can One Determine the Underlying Fermi Surface in the Superconducting State of Strongly Correlated Systems?

The question of determining the underlying Fermi surface (FS) that is gapped by superconductivity (SC) is of central importance in strongly correlated systems, particularly in view of angle-resolved photoemission experiments. Here we explore various definitions of the FS in the superconducting state using the zero-energy Green’s function, the excitation spectrum, and the momentum distribution. We examine (a) $d$-wave SC in high-$T_c$ cuprates, and (b) the $s$-wave superfluid in the BCS–Bose-Einstein condensation (BEC) crossover. In each case we show that the various definitions agree, to a large extent, but all of them violate the Luttinger count and do not enclose the total electron density. We discuss the important role of chemical potential renormalization and incoherent spectral weight in this violation.

Figure 1

(a) Momentum distribution $n(\mathbf{k})$ along $(0,0)\to (\pi ,\pi )\to (\pi ,0)\to (0,0)$ for three different doping levels. Comparison of renormalized mean field theory with variational Monte Carlo (MC) results of Ref. 15: (b) $n(\mathbf{k})$, and (c) nodal $Z(\mathbf{k})$ as function of $x$.

Figure 2

(a),(b),(c) Various “Fermi surface contours” as a function of hole doping $x$. The contours plotted are (1) ${\xi }_{\mathbf{k}}=0$, (2) $n(\mathbf{k})=1/2$, and (3) the noninteracting Fermi surface ${ϵ}_{\mathbf{k}}={\mu }^0$. Note that the “minimum gap locus” (see text) is indistinguishable from ${\xi }_{\mathbf{k}}=0$. (d) The fractional difference $\delta n/n$ between the area enclosed by ${\xi }_{\mathbf{k}}=0$ and $n$ plotted as a function of hole doping $x$.

Figure 3

ARPES intensity maps at zero energy $A(\mathbf{k},0)$ for $x=0.15$ (left) and $x=0.35$ (right).