Fidelity study of the superconducting phase diagram in the two-dimensional single-band Hubbard model

Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range $d$-wave pair field term, which precipitates a superconducting state in the $d$-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the $d$-wave pair/spin/charge-density matrices. The calculations show a $d$-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion $U$ and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping $t^{\prime }$. We also demonstrate that negative $t^{\prime }$ plays an important role in facilitating $d$-wave superconductivity.

Figure 1

The fidelity metric $g$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at half filling. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated. The small magnitude of $g$ indicates the incompatibility of $d$-wave order at half filling.

Figure 2

The ratio of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ and that of the spin-density matrix $R_{\text{spin}}$ as functions of $d$-wave pair field strength $\lambda$ (in units of $t$) at half filling. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 3

The fidelity metric $g$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at $12.5\%$ hole doping. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 4

The ratios of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ and that of the spin-density matrix $R_{\text{spin}}$ as functions of $d$-wave pair field strength $\lambda$ (in units of $t$) at $12.5\%$ hole doping. Coulomb $U=8t$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 5

The fidelity metric $g$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at $25\%$ hole doping. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated. A semilog scale is employed to highlight the behaviors around $\lambda =0$.

Figure 6

The ratio of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at 25% hole doping. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated. The behaviors near $\lambda =0$ are highlighted in the insets.

Figure 7

The fidelity metric $g$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at $37.5\%$ hole doping. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 8

The fidelity metric $g$ as a function of $d$-wave pair field strength $\lambda$ (in units of $t$) at $50\%$ hole doping. Coulomb $U(6t,8t,10t)$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 9

The CCFs (NNCCF, 2NNCCF, and 3NNCCF) and the ratio of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ as functions of $d$-wave pair field strength $\lambda$ (in units of $t$) at 37.5% hole doping. Coulomb $U=10t$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 10

The CCFs (NNCCF, 2NNCCF, and 3NNCCF) and the ratio of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ as functions of $d$-wave pair field strength $\lambda$ (in units of $t$) at 50% hole doping. Coulomb $U=10t$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 11

The CCFs (NN-CCF, 2NN-CCF, and 3NN-CCF) and the ratio of the largest to second-largest eigenvalue of the $d$-wave pair-density matrix $R_{d\text{-wave}}$ as functions of $d$-wave pair field strength $\lambda$ (in units of $t$) at 50% hole doping. Coulomb $U=6t$ and $t^{\prime }(0,-0.15t,-0.3t)$ are indicated.

Figure 12

The density of states for the 16-site single-band Hubbard model without $d$-wave pair field at 37.5% hole doping at $U=8t$ and $t^{\prime }(0,-0.15t,-0.3t)$; energy is in units of $t$. Energy zero represents the ground-state energy. The dashed line and solid line represent the occupied and unoccupied states, respectively.

Figure 13

Phase diagram of $\lambda$* (in units of $t$) vs hole-doping rate at $U=6t,8t,10t$ and $t^{\prime }=0,-0.15t,-0.3t$. The 2D single-band Hubbard model favors a $d$-wave SC phase at $25\%$ hole doping, an antiferromagnetic phase at low doping, and local checkerboard charge order at high doping. Strong antiferromagnetic order near half filling is mainly dependent on Coulomb $U$, while the strength of the local checkerboard charge order far from half filling is exclusively determined by $t^{\prime }$.