Ground-state phase diagram of the repulsive fermionic $t-t^{\prime }$ Hubbard model on the square lattice from weak coupling

We obtain a complete and numerically exact in the weak-coupling limit $\left(U\to 0\right)$ ground-state phase diagram of the repulsive fermionic Hubbard model on the square lattice for filling factors $0<n<2$ and next-nearest-neighbor hopping amplitudes $0\le t^{\prime }\le 0.5$. Phases are distinguished by the symmetry and the number of nodes of the superfluid order parameter. The phase diagram is richer than may be expected and typically features states with a high—higher than that of the fundamental mode of the corresponding irreducible representation—number of nodes. The effective coupling strength in the Cooper channel $\lambda$, which determines the critical temperature $T_c$ of the superfluid transition, is calculated in the whole parameter space and regions with high values of $\lambda$ are identified. It is shown that besides the expected increase of $\lambda$ near the Van Hove singularity line, joining the ferromagnetic and antiferromagnetic points, another region with high values of $\lambda$ can be found at quarter filling and $t^{\prime }=0.5$ due to the presence of a line of nesting at $t^{\prime }\ge 0.5$. The results can serve as benchmarks for controlled nonperturbative methods and guide the ongoing search for high-$T_c$ superconductivity in the Hubbard model.

Figure 1

The Bethe-Salpeter equation for ${\mathrm{\Gamma }}^{\text{pp}}$ with $p_i\equiv \left({\xi }_i,{\mathbf{k}}_i\right)$. Summation over ${\xi }_i$ and integration over ${\mathbf{k}}_i$ is assumed.

Figure 2

The second-order diagram contributing to ${\widehat{\mathrm{\Gamma }}}^{\text{pp}}$. The wavy lines are the interaction vertices $U$, the straight lines with arrows are the noninteracting propagators $G_0$. Integration over internal momenta is assumed.

Figure 3

The form of basis functions in the Brillouin zone (red, blue, and white colors correspond to positive values, negative values, and nodes respectively) categorized by irreducible representation and the order of harmonic $m$. [See Eqs. (15, 16, 17, 18, 19)].

Figure 4

The change of Fermi surface topology in the range of parameters $0\le n\le 2$ and $0\le t^{\prime }\le 0.7$. The Van Hove line (dashed), second line of nesting (dot-dashed), and line beneath which a Fermi pocket exists (dotted) are plotted; see text for definitions of the lines. Characteristic Fermi surfaces in the first Brillouin zone $\left(k_x,k_y\in [-\pi ,\pi )\right)$ are shown for specific points in the parameter space: (a) the antiferromagnetic point $\{n=1,t^{\prime }=0\}$, (b) the ferromagnetic point $\{n=0,t^{\prime }=0.5\}$, (c) an arbitrary point along the Van Hove line $\{n=0.78,t^{\prime }=0.25\}$, (d) above the Van Hove line $\{n=1.60,t^{\prime }=0.4\}$, (e) below the Van Hove line $\{n=0.58,t^{\prime }=0.1\}$, (f) below the second line of nesting $\{n=0.27,t^{\prime }=0.65\}$, (g) above the second line of nesting $\{n=0.76,t^{\prime }=0.65\}$, in the region where a Fermi pocket exists around $(k_x,k_y)=(0,0)$, (h) on the second line of nesting $\{n=0.61,t^{\prime }=0.6\}$.

Figure 5

top: Phase diagram for the parameter range of $0<n<2$ and $0<t^{\prime }<0.5$. bottom: same parameter range by leading irreducible representation. The presence of a Van Hove singularity is portrayed by the black dashed line.

Figure 6

The transition between the first four waves of the $E_1$ irreducible representation are seen along the line of $0.05\le n\le 0.35$ at $t^{\prime }=0.375$. The type of wave is identified below each figure. It has to be noted that at the point at $n=0.30$ the leading wave is $g^{\left(8\right)}$, however we have decided to include the point for completeness reasons. Individual Fermi surfaces are drawn in black.

Figure 7

Plot of highest effective coupling constant ${\lambda }_{\max }$ irrespective of type of wave. Inset shows a 3D version of the same plot. Clear maxima can be seen at the FM and AFM point as well as an increase in coupling strength in the vicinity of the Van Hove line.

Figure 8

The leading eigenvalue for each of the irreducible representations is given as a function of $n$ at $t^{\prime }=\{0,0.1,0.2,0.3\}$. Dashed lines correspond to Van Hove densities at $n_{VH}=\{1,0.918,0.830,0.726\}$, respectively. The closest measurements to the Van Hove points were performed at $n_{VH}\pm 0.001$.

Figure 9

The leading effective coupling constant $\lambda$ for each of the irreducible representations. From left to right: $A_1,A_2,B_1,B_2,E_1$. One can see that the AFM point does not affect the effective coupling of $A_1$ and $B_1$ representations, while the FM point seems to affect all waves alike. A clear increase in $\lambda$ for $E_1$ can only be seen along the Van Hove line, while for $d_{x^2-y^2}^{\left(4\right)}$ the coupling strength dies out along the Van Hove line away from the AFM point.