Competing Orders in a Nearly Antiferromagnetic Metal

We study the onset of spin-density wave order in itinerant electron systems via a two-dimensional lattice model amenable to numerically exact, sign-problem-free determinantal quantum Monte Carlo simulations. The finite-temperature phase diagram of the model reveals a dome-shaped $d$-wave superconducting phase near the magnetic quantum phase transition. Above the critical superconducting temperature, an extended fluctuation regime manifests itself in the opening of a gap in the electronic density of states and an enhanced diamagnetic response. While charge density wave fluctuations are moderately enhanced in the proximity of the magnetic quantum phase transition, they remain short ranged. The striking similarity of our results to the phenomenology of many unconventional superconductors points a way to a microscopic understanding of such strongly coupled systems in a controlled manner.

Figure 1

Phase diagram of model (1) showing the transition temperature $T_{\mathrm{SDW}}$ to magnetic spin density wave (SDW) order, the superconducting $T_c$, and the onset of diamagnetism at $T_{\text{dia}}$. Solid lines indicate a Berezinskii-Kosterlitz-Thouless transition. The SDW transition inside the SC dome, marked by a dashed line, possibly is a weakly first-order transition (see the main text). The shaded region indicates the coexistence region between SDW and SC quasi-long-range orders. The inset shows the Fermi surface, with different colors indicating the sign of the superconducting order parameter. The hot spots and the SDW ordering wave vector $\mathbf{Q}$ are marked.

Figure 2

(a) SDW susceptibility $\chi$ and (b) $d$-wave superconducting susceptibility $P_{-}(\mathbf{q}=0)$ at $T=1/30$ in the region of phase coexistence at $r=9.417$ and in the nonmagnetic superconducting phase at $r=10.026$.

Figure 3

(a) Inverse magnetic susceptibility across the phase diagram with the gray lines indicating contour lines. We show data for $L=14$ at those temperatures indicated by ticks on the left inside of the plot and interpolate linearly between them. (b) Magnetic susceptibility near the maximum of the superconducting dome as a function of temperature for different system sizes.

Figure 4

The integrated density of states $\tilde{N}(T)$, as defined in the text, versus temperature for various values of the tuning parameter $r$. The dashed (dotted) lines indicate the location of the SC (SDW) transition temperatures, respectively.

Figure 5

(a) $d$-wave CDW and (b) $d$-wave PDW susceptibilities, as defined in the text, across the Brillouin zone. Shown here is data for $L=14$, $T=0.083$, and $r=10.4$. The data point $P_{-}(\mathbf{q}=\mathbf{0})$ (i.e., the uniform superconducting susceptiblity) has been excluded from the data (white square).

Figure 6

(a) The $d$-wave CDW susceptibility versus momentum along the high-symmetry cut $\mathbf{q}=(\pi ,q_y)$ for various system sizes. The solid line is a guide to the eye. (b) Temperature dependence of the CDW susceptibility at the optimal $q$ (taken at $L=12$), for multiple values of $r$.

Figure 7

The finite-size superfluid density ${\rho }_s(L)$ within the superconducting phase versus $r$ at $T=0.025$.