Incommensurate nematic fluctuations in the two-dimensional Hubbard model

We analyze effective $d$-wave interactions in the two-dimensional extended Hubbard model at weak coupling and small to moderate doping. The interactions are computed from a renormalization group flow. Attractive $d$-wave interactions are generated via antiferromagnetic spin fluctuations in the pairing and charge channels. Above Van Hove filling, the $d$-wave charge interaction is maximal at incommensurate diagonal wave vectors, corresponding to nematic fluctuations with a diagonal modulation. Below Van Hove filling a modulation along the crystal axes can be favored. The nematic fluctuations are enhanced by the nearest-neighbor interaction in the extended Hubbard model, but they always remain smaller than the dominant antiferromagnetic, pairing, or charge density wave fluctuations.

Figure 1

Effective $d$-wave interactions at the scale ${\Lambda }_{*}$ in units of $t$ as a function of $t^{\prime }$, for $U=3t$ at Van Hove filling. The electron density ranges from $n=1$ at $t^{\prime }=0$ to $n=0.75$ at $t^{\prime }/t=-0.28$. The pairing interactions ${\gamma }_2^{\mathrm{SC},{\Lambda }_{*}}$ are plotted as circles and charge interactions ${\gamma }_2^{\mathrm{C},{\Lambda }_{*}}\left(\mathbf{0}\right)$ as squares. Solid lines correspond to $V=0$ and dashed lines to $V=0.74t$. In both cases the leading instability changes from antiferromagnetism to $d$-wave pairing at $t^{\prime }=-0.237t$ (corresponding to $n=0.79$).

Figure 2

Effective $d$-wave interactions ${\gamma }_2^{\mathrm{SC},{\Lambda }_{*}}$ (circles) and ${\gamma }_2^{\mathrm{C},{\Lambda }_{*}}\left(\mathbf{0}\right)$ (squares) in units of $t$ as a function of the nearest-neighbor interaction $V$, for $t^{\prime }=-0.15t$ and $U=3t$ at Van Hove filling ($n=0.88$). The leading instability changes from incommensurate antiferromagnetism to charge density wave at $V=0.758t$.

Figure 3

Effective $d$-wave interactions at the stopping scale ${\Lambda }_{*}$ of antiferromagnetism plotted as functions of $\mu$ for $t^{\prime }=-0.15t$, $U=3t$, and $V=0$. The density varies from $n=0.94$ at $\mu /t=0.1$ to $n=0.81$ at $\mu /t=-0.11$. Pairing interactions ${\gamma }_2^{\mathrm{SC},{\Lambda }_{*}}$ are plotted as circles connected by a solid line. The most attractive $d$-wave charge interactions ${\max }_{\mathbf{q}}\left|{\gamma }_2^{\mathrm{C},{\Lambda }_{*}}\left(\mathbf{q}\right)\right|$ are plotted as squares connected by a dashed line, while the points corresponding to ${\gamma }_2^{\mathrm{C},{\Lambda }_{*}}\left(\mathbf{0}\right)$ are connected by a solid line. Also shown are momenta $\mathbf{q}$ where $-{\gamma }_2^{\mathrm{C},{\Lambda }_{*}}\left(\mathbf{q}\right)$ becomes maximal, with components $q_x$ plotted as open diamonds and $q_y$ as filled diamonds, both connected by a dashed line. Momenta for which the bare $d$-wave particle-hole bubble ${\Phi }_{\mathrm{ph}}^{2,{\Lambda }_{*}}\left(\mathbf{q}\right)$ is extremal are shown similarly with components $q_x$ plotted as plus signs and $q_y$ as diagonal crosses. Stars are superpositions of plus signs and crosses for $q_x=q_y$.

Figure 4

Effective $d$-wave interactions and modulation vectors as in Fig. 3, but for a finite nearest-neighbor interaction $V=0.74t$.