Strong renormalization of the Fermi-surface topology close to the Mott transition

The underlying Fermi surface is a key concept for strongly interacting electron models and has been introduced to generalize the usual notion of the Fermi surface to generic (superconducting or insulating) systems. By using improved correlated wave functions that contain backflow and Jastrow terms, we examine the two-dimensional $t$-$t^{\prime }$ Hubbard model and find a nontrivial renormalization of the topology of the underlying Fermi surface close to the Mott insulator. Moreover, we observe a sharp crossover region, which arises from the metal-insulator transition, from a weakly interacting metal at small coupling to a resonating valence-bond superconductor at intermediate coupling. A violation of the Luttinger theorem is detected at low hole dopings.

Figure 1

Upper panel: variational hopping ratio ${\tilde{t}}^{\prime }/\tilde{t}$ as a function of $U/t$ for $t^{\prime }/t=-0.4$ and $-0.75$. Full (empty) symbols refer to the presence (absence) of backflow correlations in the variational wave function. Lower panel: variational hopping ratio ${\tilde{t}}^{\prime }/\tilde{t}$ as a function of the doping $\delta$ for $U/t=5$ and 10 with $t^{\prime }/t=-0.4$. Here, only results with backflow terms are shown. Data are shown for an $L=162$ lattice; in the lower panel a few points on an $L=242$ lattice are added close to half filling.

Figure 2

Variational hopping ratio ${\tilde{t}}^{\prime }/\tilde{t}$ as a function of doping $\delta$ and $U/t$ for the case with $t^{\prime }/t=-0.4$. The density plot is obtained by using results with $L=162$ and 242.

Figure 3

Static spin-spin correlations $S\left(q\right)$ as a function of $\left|q\right|/\pi$ along the $\Gamma =(0,0)$ to $M=(\pi ,\pi )$ direction, for $t^{\prime }/t=-0.4$. Data are presented for $U/t=5$ (squares), $U/t=5.6$ (triangles), $U/t=6$ (circles), and $U/t=7$ (diamonds). Full (empty) symbols refer to $L=242$ ($L=162$). The location of the point $Q=(\pi ,\pi )$ is marked with an arrow.

Figure 4

Left panels: The underlying Fermi surface defined by ${\xi }_k=0$ (solid blue lines) and the noninteracting Fermi surface ${\xi }_k^0=0$ (dashed black lines) for the same parameter values of the corresponding right panels (see text for more details). Right panels: Momentum distribution function $n_k$ along the path in the Brillouin zone connecting the points $\Gamma =(0,0)$, $M=(\pi ,\pi )$, and $X=(\pi ,0)$. Data are shown at $U/t=10$ and $t^{\prime }/t=-0.4$ for a $L=242$ lattice size at three dopings ($\delta =0$, $0.033$, and $0.174$). Arrows indicate the position of the underlying Fermi surface, as found in left panels.

Figure 5

Left panel: Ratio between the number of electrons enclosed by the renormalized Fermi surface $\tilde{n}$ and the number of electrons effectively present in the system $n$ as a function of doping. Data refer to the case $U/t=10$ and $L=242$. Right panel: Same quantity as in the left panel as a function of $U/t$ at half filling, for $L=242$.

Figure 6

Pairing amplitude ${\Delta }_{\mathrm{BCS}}/\tilde{t}$ as a function of doping for $t^{\prime }/t=-0.4$ and different values of $U/t$. Data are presented for an $L=162$ lattice size.

Figure 7

Superconducting order parameter squared ${\phi }^2$ as a function of doping for $t^{\prime }/t=-0.4$ and $U/t=7$ (left panel) and $U/t=10$ (right panel).

Figure 8

The underlying Fermi surface defined by ${\xi }_k=0$ (solid blue lines) and the noninteracting Fermi surface ${\xi }_k^0=0$ (dashed black lines) as a function of doping. Data are presented at $U/t=10$ for $L=242$.

Figure 9

The underlying Fermi surface defined by ${\xi }_k=0$ (solid blue lines) and the noninteracting Fermi surface ${\xi }_k^0=0$ (dashed black lines) as a function of doping. Data are presented at $U/t=5$ for $L=242$.