Coexistence of $s$-Wave Superconductivity and Antiferromagnetism

We study the phase diagram of a new model that exhibits a first order transition between $s$-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard $U$ from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.

Figure 1

The phase diagram for the full Hamiltonian. The shaded region indicates the parameters accessible to sign-free simulations. For every region in the phase diagram the full symmetry of the Hamiltonian is indicated. Broken symmetries are marked by bars.

Figure 2

(a) Staggered susceptibility ${\chi }_{n,n}^{\prime }(\overrightarrow{Q},0)\propto {\xi }^2{\Delta }^{-1}$; (b) $⟨n_{\overrightarrow{Q}}{n}_{-\overrightarrow{Q}}⟩\propto {\xi }^2$. The dotted line is proportional to results from LSWT.

Figure 3

Density-density correlations $2⟨n_{\overrightarrow{i}}{n}_{\overrightarrow{i}+\overrightarrow{x}}⟩=-1/N(\partial F/\partial V)$, where $n_{\overrightarrow{i}}=n_{i,\uparrow }+n_{i,\downarrow }$. The derivative of the free energy is continuous.

Figure 4

(a) Density-density correlation function $⟨n_{(L/2,L/2)}{n}_{\overrightarrow{0}}⟩$ as a function of $U/t$. The correlation function jumps by a factor of 2 once a finite Hubbard $U$ term breaks the $Z_{2,\mathrm{p}\mathrm{h}}$ symmetry. (b),(c) The staggered susceptibility ${\chi }_{S^z,S^z}^{\prime }(\overrightarrow{k}=\overrightarrow{Q},\omega =0)\propto {\xi }^2{\xi }^z$ is well below the correlation function $⟨S_{\overrightarrow{k}}^z{S}_{-\overrightarrow{k}}^z⟩(\overrightarrow{k}=\overrightarrow{Q})\propto {\xi }^2$.

Figure 5

The jump in $⟨n_{\uparrow }{n}_{\downarrow }⟩=1/N(\partial F/\partial U)$ at $U=0$ signals a first order phase transition.

Figure 6

The staggered moment $m_s=\sqrt{3⟨S^z{S}^z⟩(Q)/N}$ which coexists with the superconducting order parameter $\Delta =m_s\sqrt{2}$ plotted as a function of $t_p$ ($U/t=V/t=0$). The inset shows finite size extrapolation for $t_p=4$ and $t_p=0.5$.

Figure 7

The $1/N⟨{\mathcal{D}}_{\overrightarrow{k}}{\mathcal{D}}_{\overrightarrow{k}}⟩$ ($\overrightarrow{k}=0$) correlations show that at least up to $T\le 1.8t$ particle-hole symmetry is broken.