Superconductivity from repulsion: Variational results for the two-dimensional Hubbard model in the limit of weak interaction

The two-dimensional Hubbard model is studied for small values of the interaction strength ($U$ of the order of the hopping amplitude $t$), using a variational ansatz well suited for this regime. The wave function, a refined Gutzwiller ansatz, has a BCS mean-field state with $d$-wave symmetry as its reference state. Superconducting order is found for densities $n<1$ (but not for $n=1$). This resolves a discrepancy between results obtained with the functional renormalization group, which do predict superconducting order for small values of $U$, and numerical simulations, which did not find superconductivity for $U\lesssim 4t$. Both the gap parameter and the order parameter have a domelike shape as a function of $n$ with a maximum for $n\approx 0.8$. Expectation values for the energy, the particle number, and the superconducting order parameter are calculated using a linked-cluster expansion up to second order in $U$. In this way large systems (millions of sites) can be readily treated and well converged results are obtained. A big size is indeed required to see that the gap becomes very small at half filling and probably tends to zero in the thermodynamic limit, whereas away from half filling a finite asymptotic limit is reached. For a lattice of a given size the order parameter becomes finite only above a minimal coupling strength $U_c$. This threshold value decreases steadily with increasing system size, which indicates that superconductivity exists for arbitrarily small $U$ for an infinite system. For moderately large systems the size dependence is quite irregular, due to variations in level spacings at the Fermi energy. The fluctuations die out if the gap parameter spans several level spacings.

Figure 1

First-order diagrams for the expansion (14). Dots represent two-particle vertices, squares single-particle vertices.

Figure 2

Second-order diagrams with symbols as in Fig. 1.

Figure 3

Energy gain due to superconductivity as a function of the gap parameter ${\mathrm{\Delta }}_0$ for $n=0.8$ and $U=1$. Symbols correspond to different system sizes, $L=1000$ (circles), $L=500$ (squares), $L=200$ (down-pointing triangles), and $L=100$ (up-pointing triangles). The dashed line represents results obtained with the Gutzwiller ansatz (for $L=1000$).

Figure 4

Condensation energy for $U=1$ and different system sizes. Down-pointing triangles: $L=500$; squares: $L=1000$; circles and solid line: $L=2000$; up-pointing triangle: $L=4000$. The dashed line represents the BCS prediction, Eq. (49).

Figure 5

Changes in kinetic (up-pointing triangles), potential (down-pointing triangles), and total energies (circles) for $U=1$ and $L=2000$.

Figure 6

Crossover from unconventional (“kinetic-energy lowering”) to conventional (“potential-energy lowering”) superconductivity for very small values of $U$. The numerical results (dots) have been calculated for $L_x=2000$ and are well converged. The line is a linear interpolation of the data points.

Figure 7

Gap parameter as a function of electron density for $U=1$ and different system sizes. Down-pointing triangles: $L=500$; squares: $L=1000$; circles and dashed line: $L=2000$; up-pointing triangle: $L=4000$.

Figure 8

Gap parameter as a function of $U$ for $n=0.7$ and $L=2000$. The solid line is a linear fit through the data points.

Figure 9

Order parameter for $U=1$ and $L=1000$ as a function of the density $n$. Circles and the dashed line represent the second-order expansion, squares the zeroth-order contribution.

Figure 10

Order parameter as a function of $U$ for $n=0.7$ and $L=2000$. The solid line is a linear fit through the data points.

Figure 11

Variation of the gap parameter with system size ($U=1$). Circles and solid line: $n=1$; squares: $n=0.85$; triangles: $n=0.6$.

Figure 12

Size dependence of the gap parameter for $U=1$ at half filling ($n=1$). The dashed line corresponds to the HOMO-LUMO gap ${\mathrm{\Delta }}_{\text{HL}}$, given by Eq. (56). The solid line is a linear fit through those data points for which ${\mathrm{\Delta }}_0$ exceeds ${\mathrm{\Delta }}_{\text{HL}}$.

Figure 13

$U$ dependence of the gap parameter for $n=0.85$ and small system sizes, $L=12$ (circles), $L=14$ (squares), and $L=20$ (triangles).

Figure 14

$U$ dependence of the gap parameter for $n=0.85$ and intermediate system sizes, $L=100$ (circles) and $L=200$ (squares). The arrows indicate the critical points above which ${\mathrm{\Delta }}_0$ is finite.

Figure 15

Size dependence of the critical value $U_c$, below which superconducting order disappears, for $n=0.85$.

Figure 16

Average gap parameter ${\mathrm{\Delta }}_0$ (dots) and standard deviation (error bars) for $n=0.85, U=1$, and various (average) system sizes. The circles indicate the average HOMO-LUMO gap ${\mathrm{\Delta }}_{\text{HL}}$. Its standard deviation is of the order of the average value.

Figure 17

Gap parameter as a function of electron density for $L=1000$, calculated with the Gutzwiller ansatz. Circles and solid line: $U=3$; dots and dashed line: $U=2$.

Figure 18

Pairing-induced changes in kinetic (up-pointing triangles), potential (down-pointing triangles), and total energies (circles) for $U=2$ and $L=1000$, obtained with the Gutzwiller ansatz.