$\eta$-pairing ground states in the non-Hermitian Hubbard model

The introduction of non-Hermiticity has greatly enriched the research field of traditional condensed matter physics and eventually led to a series of discoveries of exotic phenomena. We investigate the effect of non-Hermitian imaginary hoppings on the Hubbard model. The exact bound-pair solution shows that the electron-electron correlation suppresses the non-Hermiticity, resulting in off-diagonal long-range order (ODLRO) ground state in the attractive Hubbard model. In a large $U$ limit, such non-Hermiticity contributes an extra minus sign in the virtual exchange of the particles. As a consequence, the energy of the effective spin model describing the behavior of the ground state and low energy excitations will be reversed. The corresponding ground state experiences a transition from antiferromagnetism to ferromagnetism characterized by the appearance of a non-decaying correlation function. The numerical result indicates that the $\eta$-pairing ground state exits in 1D and 2D systems and is insensitive to the disorder. We further propose a protocol to adiabatically generate the $\eta$-pairing state. Our results provide a promising approach for the non-Hermitian strongly correlated system.

Figure 1

Comparison of the two-particle spectrum within the subspace (0,0) between the non-Hermitian setting and its parent Hermitian system for (a) $U=-0.8t$, (b) $-2t$, and (c) $-4.5t$, respectively. The upper and lower panels present the spectrum of the Hermitian and non-Hermitian system, respectively. The red circle and gray shading denote the bound pair and scattering state. The parent Hermitian system can be obtained by assuming $it\to t$. For the Hermitian system, the bound pair with the lowest energy lies in the $K=0$ subspace while the ground state of two-particle non-Hermitian setting locates on the subspace indexed by $K=\pi$. It is shown that the presence of the imaginary hopping not only makes all scattering energy bands imaginary but also reverses the whole bound band. In the condition of small $U$, there can exist an EP characterized by the divergence of $\partial {\epsilon }_K/\partial K$. Such non-Hermiticity alters significantly the paring mechanism and hence favors superconductivity.

Figure 2

Plot of the fidelity $\mathcal{F}$ as a function of the interaction strength$U$ for the homogeneous 1D systems: (a) four-site system with filled particle pairs $m=2$, and 4; (b) six-site system with filled particle pairs $m=2$, 4, and 6; (c) eight-site system with filled particle pairs $m=2$, 4, 6, and 8; and (d) ten-site system with filled particle pairs $m=2$, 4, 6, 8, and 10. It is shown that the system possesses the $\eta$-pairing ground state as long as $U$ is switched on.

Figure 3

Plots of the overlap $\overline{\mathcal{F}}$ and correlator $\overline{C_j}$ as a function of the strength of interaction disorder $b$ for (a) $t=1$, $a=0.1t$, $U=-0.5t$ (b) $t=1$, $a=0.3t$, $U=-1.5t$ (c) $t=1$, $a=0$, $U=-4t$, and (d) $t=1$, $a=0.2t$, $U=-4t$. The numerical simulation is performed for the 6 site 1D Hubbard model at half filling and $s^z=0$. Here the strength of the hopping disorder $a$ is set to be constant for each subfigure and the correlator $\overline{C_j}$ is averaged over all sites separated by a distance $j$. When $b=0$, no matter what value $a$ takes, as long as $U$ is nonzero, one can always get a perfect $\eta$-pairing ground state. The variation of $\overline{C_j}$ indicates that the increase of $b$ will not result in the significant change of the $\eta$-pairing ground state; the coexistence of hopping and interaction disorders does not cause the ground state to deviate from the perfect $\eta$-pairing state, which can be seen from (c) and (d). Therefore the performance of the correlator is the consequence of the interplay between two such disorders, which provides a scheme to prepare $\eta$-pairing ground state in the experiment.

Figure 4

Numerical simulation of 2D corrugation pattern for (a1) eight-site Hubbard model with four filled particles, (b1) nine-site Hubbard model with 6 filled particles, and (c1) 13 site Hubbard model with four filled particles. The different sizes of solid circles and different lengths of red edges denote the disorder of the interaction strength $U_j$ and hopping strength $t_j$ of Eq. (24). (a2)–(c2) Plots of $\overline{\mathcal{F}}$ and $\overline{C_2}$ as function of disorder strength $b$. The system parameters are (a) $t=1$, $a=0$, $U=-0.8t$ (b) $t=1$, $a=0.2t$, $U=-0.8t$ and (c) $t=1$, $a=0.2t$, $U=-0.5t$. Similar with 1D system, the induced fluctuation of $\overline{C_2}$ is minor in comparison with disorder free case ($b=0$) indicating that the existing result of 1D can be extended to 2D or higher dimensional system.

Figure 5

Dynamical generation of the $\eta$-pairing superconducting ground state. [(a) and (b)] depict the variation of the instantaneous energy spectrum with respect to time $\tau$, and (c) presents the fidelity ${\mathcal{F}}_g$. The numerical simulation is performed for the six-site Hubbard model with four filled particles. The other system parameters are ${\lambda }_t=0.01$, ${\lambda }_{\mu }=0.05$, and ${\mu }_{1\left(2\right),\sigma }\left(0\right)=0.5$. The time $\tau$ is in units of $t^{-1}$, where $t$ is the energy scale of the Hamiltonian. The adiabatic path denoted by red line is chosen to avoid the level crossing and coalescing such that the evolved state stay at the ground state. The variation of ${\mathcal{F}}_g$ indicates that the ground state undergoes a transition from conventional paring to $\eta$ paring.