Interaction-induced Fermi-surface renormalization in the $t_1-t_2$ Hubbard model close to the Mott-Hubbard transition

We investigate the nature of the interaction-driven Mott-Hubbard transition of the half-filled $t_1-t_2$ Hubbard model in one dimension, using a full-fledged variational Monte Carlo approach including a distance-dependent Jastrow factor and backflow correlations. We present data for the evolution of the magnetic properties across the Mott-Hubbard transition and on the commensurate to incommensurate transition in the insulating state. Analyzing renormalized excitation spectra, we find that the Fermi surface renormalizes to perfect nesting right at the Mott-Hubbard transition in the insulating state, with a first-order reorganization when crossing into the conducting state.

Figure 1

(Color online) Phase diagram of the $t_1-t_2$ Hubbard model at half-filling with the metallic phase with gapped spin excitations (C1S0) and the insulating phase with gapless spin excitations (C0S1). The insulating phase with gapped spin excitations for larger $U$ and $t_2/t_1>1/2$ has regions with commensurate $\left(Q=\pi \right)$ and incommensurate ($Q$ incomm) spin-spin correlations. A crossover region separates the phase where the peak in $S\left(q\right)$ is incommensurate and the one with the peak commensurate to a doubled unit cell $\left(Q\sim \pi /2\right)$.

Figure 2

(Color online) For a chain with $L=120$ sites, the density-density correlations $N\left(q\right)$, divided by the momentum $q$, across the metal-insulator transition for $t_2/t_1=0.75$, 1.1, and 1.5. The metallic (insulating) state is characterized by a finite (vanishing) value of $N\left(q\right)/q$, in the limit $q\to 0$.

Figure 3

(Color online) Upper panel: spin-spin correlations $S\left(q\right)$ at $t_2/t_1=0.75$, for a $L=120$ chain. Data are shown for $U/t_1=0,4$ (metal) and for $U/t_1=5,6,12$ (insulator). Lower panel: single-particle spectrum $E_q/W$ at $t_2/t_1=0.75$ for the same values of the electron-electron repulsion $U$ and the same chain length.

Figure 4

(Color online) Evolution of $2k_1$ (distance between the minima of the single-particle spectrum $E_q$) in the insulating phase for $t_2/t_1=0.75$. Note the transition from an incommensurate to a commensurate value for $U/t_1\simeq 9$.

Figure 5

(Color online) Upper panel: spin-spin correlations $S\left(q\right)$ at $t_2/t_1=0.9$, for a $L=120$ chain. Data are shown for $U/t_1=5$ (metal) and $U/t_1=6,7,8,12$ (insulator). Lower panel: single-particle spectrum $E_q/W$ at $t_2/t_1=0.9$ for the same values of the electron-electron repulsion $U$ and the same chain length.

Figure 6

(Color online) The spin-spin correlations $S\left(q\right)$ at $U/t_1=12$ and for $L=120$ sites, for different values of the hopping ratio $t_2/t_1$. Arrows indicate the quantity $2k_1$, obtained from the single-particle spectrum $E_q$.

Figure 7

(Color online) The variationally optimized hopping ratio ${\tilde{t}}_2/{\tilde{t}}_1$ in $|\text{BCS}⟩$, see Eq. (5), as a function of $U/t_1$. The metal-insulator transition takes place for $U/t_1=4.5\pm 0.5$, $5.5\pm 0.5$, and $6.5\pm 0.5$ for $t_2/t_1=0.75$ (triangles), 0.9 (squares), and 1.0 (circles), respectively. The variational state $|\text{BCS}⟩$ contains only a single Fermi sea for ${\tilde{t}}_2/{\tilde{t}}_1\le 0.5$ (horizontal line).