Incommensurate antiferromagnetic fluctuations in the two-dimensional Hubbard model

Commensurate and incommensurate antiferromagnetic fluctuations in the two-dimensional repulsive $t\text{−}{t}^{\prime }$-Hubbard model are investigated using functional renormalization group equations. For a sufficient deviation from half-filling we establish the existence of local incommensurate order below a pseudocritical temperature $T_{pc}$. Fluctuations not accounted for in the mean-field approximation are important—they lower $T_{pc}$ by a factor $\approx 2.5$.

Figure 1

(Color online) Mean-field kinetic term $P_a\left(0,\mathbf{q}\right)/t$ of the $\mathbf{a}$-boson as a function of spacelike momenta for $U/t=3$ and $t^{\prime }=0$. In Fig. 1a $\mu =0$ and $T/t=0.205$, in Fig. 1b $\mu /t=-0.27$ and $T/t=0.0435$. Both temperatures are mean-field critical temperatures.

Figure 2

(Color online) Renormalization flow of the gradient coefficient $A_{a,k}$ for $U/t=3$ and $\mu /t=-0.12$ according to Eq. (18) in the parameter regime where we have commensurate antiferromagnetism and therefore ${\partial }_k\widehat{q}=0$. The solid line corresponds to $T/t=0.08$, the long-dashed line to $T/t=0.07$, and the short-dashed line to $T/t=0.058$.

Figure 3

(Color online) In (a) the mean-field approximation for bosonic kinetic term $P_a\left(0,q_x,0\right)/t$ is shown as a function of the $x$ component of spatial momenta. Parameters are $U/t=3$, $\mu /t=-0.35$, $t^{\prime }=0$, and $T/t=0.1$. Figure 3b shows the same quantity according to our approximation given by Eqs. (11, 16) with the values of $A_a$, $\widehat{q}$, and $D$ drawn from the mean-field computation.

Figure 4

(Color online) Renormalization flow of the gradient coefficient $A_{a,k}$ and the incommensurability $\widehat{q}$ according to Eqs. (18, 20) at $\mu /t=-0.12$ and $T/t=0.05$, where incommensurate antiferromagnetism dominates. The solid line shows $A_a$ decreasing to zero at $-\text{ln}k/t=3.09$ and increasing again when the incommensurability $\widehat{q}$ (short-dashed line) sets in.

Figure 5

(Color online) Renormalization group results for the pseudocritical temperature $T_{pc}/t$ as a function of $\mu /t$, given by the solid (commensurate) and dashed (incommensurate) lines. Results are displayed for $U/t=3$, $t^{\prime }=0$ (upper panel) and $t^{\prime }/t=-0.05$ (lower panel).

Figure 6

(Color online) Renormalization group results for the incommensurability $\widehat{q}$ as a function of $T$ for $U/t=3$, $t^{\prime }=0$, and $\mu /t=-0.12$ (solid line) and $\mu /t=-0.105$ (dashed line).

Figure 7

(Color online) Figure 7a shows the spin susceptibility ${\left[P_a\left(0,q_x,0\right)/t\right]}^{-1}$ and Fig. 7b the bosonic occupation number $n_a\left(q_x,0\right)$ for $\mu /t=-0.12$, $T/t=0.01$, $U/t=3$, and $t^{\prime }=0$ according to the renormalization group computations. Both curves are given as a function of spatial momentum in $x$ direction. A peak at $q_x=0$ would signal the dominance of commensurate antiferromagnetism. The actual peaks, located at $\widehat{q}=\pm 0.132$, indicate incommensurate antiferromagnetism.

Figure 8

(Color online) Mean-field effective Fermi surfaces for $\mu /t=-0.6$, $t^{\prime }/t=-0.2$, and gap parameter $A/t=0.1$. Figure 8a shows the commensurate case $\widehat{q}=0$ where the Fermi surface exhibits hole- and particle pockets at the magnetic Brillouin zone boundary. In Fig. 8b, the remainders of the effective Fermi surface are shown for a nonzero incommensurability $\widehat{q}=0.3$ along the $x$ axis.