Frustration-Induced Superconductivity in the $t$-$t'$ Hubbard Model

Abstract

The two-dimensional (2D) Hubbard model is widely believed to capture key ingredients of high-$T_c$ superconductivity in cuprate materials. However, compelling evidence remains elusive. In particular, various magnetic orders may emerge as strong competitors of superconducting orders. Here, we study the ground state properties of the doped 2D $t$-$t'$ Hubbard model on a square lattice via the infinite Projected Entangled-Pair State (iPEPS) method with $\mathrm{U}(1)$ or $\mathrm{SU}(2)$ spin symmetry. The former is compatible with antiferromagnetic orders, while the latter forbids them. Therefore, we obtain by comparison a detailed understanding of the magnetic impact on superconductivity. Moreover, an additional $t'$ term accommodates the particle-hole asymmetry, which facilitates studies on the discrepancies between electron- and hole-doped systems. We demonstrate that (i) a positive $t'/t$ significantly amplifies the strength of superconducting orders; (ii) at sufficiently large doping levels, the $t$-$t'$ Hubbard model favors a uniform state with superconducting orders instead of stripe states with charge and spin modulations; and (iii) the enhancement of magnetic frustration, by increasing either the strength of NNN interactions or the charge doping, impairs stripe orders and helps stabilize superconductivity.

Figures (4)

• Figure 1: The ground state energy per site (a,b) and singlet pairing (c,d) vs. doping $\delta$ of the $t$-$t'$ Hubbard model at $U/t\!=\!10$ and (a,c) $t'\!/t\!=\!-0.25$ or (b,d) $t'\!/t\!=\!0.25$, computed via $\mathrm{U}(1)$ iPEPS (red squares) on an 8×2 supercell at bond dimension $D\!=\!12$ and $\mathrm{SU}(2)$ iPEPS (blue circles) on a 4×2 supercell keeping $D^*\!=\!7$ multiplets (bond dimension $D\!=\!12$). Green and yellow arrows, respectively, indicate the NN (including on-site) and NNN contributions to the energy for several typical data points. Inset: zoom into the region near $1/8$ doping. (e-g): Details of the $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ symmetric ground states on 8×2, 4×2 and 2×2 supercells. Areas of red circles and lengths of black arrows are proportional to the charge density (top rows) and the local moments (bottom rows), respectively. Bond widths indicate NN singlet pairing amplitudes and two different colors indicate opposite signs. For (f-g), we used $D^*[D]\!=\!8[13]$ for reasons explained in the Supplemental Material.
• Figure 2: The contribution of (a) the NN (including on-site) and (b) the NNN terms to the total energy per site in the $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ ground states, respectively, as a function of doping. (c) The NN and (d) the NNN spin-spin correlators in the $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ ground states, respectively.
• Figure 3: (a) The long-range spin-spin and pair-pair correlators in the $\mathrm{U}(1)$ and $\mathrm{SU}(2)$ ground states, respectively. All these correlators exhibit an exponential decay behavior. (b) The corresponding correlation lengths as a function of doping.
• Figure 4: The ground state phase diagram of the $t$-$t'$ Hubbard model with respect to doping and $t'\!/t$. The color scale indicates $e_1 - e_2$, obtained via linear interpolation from a discrete set of scanning points (white). The grey dashed line marks $e_1 = e_2$.