Electron pairing by Coulomb repulsion in narrow band structures

We study analytically and numerically dynamics and eigenstates of two electrons with Coulomb repulsion on a tight-binding lattice in one and two dimensions. The total energy and momentum of electrons are conserved and we show that for a certain momentum range the dynamics is exactly reduced to an evolution in an effective narrow energy band where the energy conservation forces the two electrons to propagate together through the whole system at moderate or even weak repulsion strength. We argue that such a mechanism of electron pair formation by the repulsive Coulomb interaction is rather generic and that it can be at the origin of unconventional superconductivity in twisted bilayer graphene.

Figure 1

Plot of wave function amplitudes obtained from the exact quantum time evolution in one dimension at times $t=138\mathrm{\Delta }t$ and ${10}^5\mathrm{\Delta }t$ in left and right panels, respectively ($\mathrm{\Delta }t=1/B_1$ is an elementary time step of inverse bandwidth $B_1=8+U$). The top panels show the TIP wave function amplitude $\left|\psi \right(x_1,x_2,t\left)\right|$ at $x_1$ and $x_2$ for both axes. The bottom panels show $|\overline{\psi }(p_{+},\mathrm{\Delta }x,t)|$ with $\mathrm{\Delta }x=x_2-x_1$ (taken modulo $N$) the relative coordinate (horizontal axis) and $0\le p_{+}<2\pi$ the total momentum (vertical axis). The initial state at $t=0$ is a symmetrized state with one electron localized at $N/2$ and the other one at $N/2+1$, i.e., initial distance $\mathrm{\Delta }\overline{x}=1$. The other parameters are $U=1$ and $N=128$. The color bar represents (here and in certain subsequent figures) the ratio of the quantity shown (here the modulus of the wave function amplitude) to its maximal value. Related videos are available in [15, 16].

Figure 2

Plot of wave function amplitude $|\overline{\psi }(p_{+x},p_{+y},\mathrm{\Delta }x,\mathrm{\Delta }y)|$ in block representation obtained from the 2D quantum time evolution at iteration time $t={10}^5\mathrm{\Delta }t$ in the $\mathrm{\Delta }x\text{−}\mathrm{\Delta }y$ plane for certain $p_{+x}$ and $p_{+y}$ sectors and $U=2$ $[\mathrm{\Delta }t=1/B_2=1/(16+U)]$. The initial block state at $t=0$ is localized at $\mathrm{\Delta }x=\mathrm{\Delta }y=1$. All panels show a close-up of the region $(0\le \mathrm{\Delta }x,\mathrm{\Delta }y<32)$. The values of $p_{+}=p_{+x}=p_{+y}$ and $N$ are $p_{+}=0$ and $N=128$ (top left), $p_{+}=21\pi /32\approx 2\pi /3$ and $N=128$ (top right), $p_{+}=63\pi /64\approx \pi$ and $N=128$ (bottom left), and $k=85\pi /128\approx 2\pi /3$ and $N=512$ (bottom right). Related videos are available in [15, 16].

Figure 3

Two-dimensional wave function densities obtained from the time evolution shown at times $t=445\mathrm{\Delta }t$ and ${10}^4\mathrm{\Delta }t$ in left and right panels, respectively, for initial electron positions at approximately $(N/2,N/2)$, with $N=128$ and $U=2$ [$\mathrm{\Delta }t=1/B_2=1/(16+U)$ is the Trotter integration time step]. The top panels show a close-up of the density for $(0\le \mathrm{\Delta }x,\mathrm{\Delta }y<32)$ in the $\mathrm{\Delta }x\text{−}\mathrm{\Delta }y$ plane of relative coordinates obtained from a sum over $x_1$ and $y_1$. The bottom panels show the density in the $x_1\text{−}{x}_2$ plane obtained from a sum over $y_1$ and $y_2$. The corresponding values of the probability near the diagonal $w_{10}$ are $w_{10}=0.106$ and 0.133 for the left and right panels, respectively (see the text). Related videos are available in [15, 16].

Figure 4

Time dependence of the pair formation probability $w_{10}$ for $U=2$ and different cases of either the exact 2D time evolution in certain sectors of conserved momenta $p_{+}=p_{+x}=p_{+y}$ or the full space 2D time evolution using the Trotter formula approximation (initial states as in Figs. 2 and 3). The precise values of $p_{+}$ are $p_{+}=0$, $p_{+}=85\pi /128\approx 2\pi /3$, and $p_{+}=255\pi /256\approx \pi$ for $N=512$ and $p_{+}=21\pi /32\approx 2\pi /3$ for $N=128$. The top blue line at $w_{10}=1$ is accurate with a numerical error below ${10}^{-14}$ for all time values.

Figure 5

Dependence of the pair formation probability $w_{10}$ on the filling factor $\nu$ for $U=2$ (red curve) and 0.5 (blue curve) for $N=256$ in two dimensions. The dashed line shows the ergodic probability $w_{\mathrm{erg}}$ assuming a uniform wave function.