$p$-Wave Superfluidity by Spin-Nematic Fermi Surface Deformation

We study attractively interacting fermions on a square lattice with dispersion relations exhibiting strong spin-dependent anisotropy. The resulting Fermi surface mismatch suppresses the $s$-wave BCS-type instability, clearing the way for unconventional types of order. Unbiased sampling of the Feynman diagrammatic series using diagrammatic Monte Carlo methods reveals a rich phase diagram in the regime of intermediate coupling strength. Instead of a proposed Cooper-pair Bose metal phase [A. E. Feiguin and M. P. A. Fisher, Phys. Rev. Lett. 103, 025303 (2009)], we find an incommensurate density wave at strong anisotropy and two different $p$-wave superfluid states with unconventional symmetry at intermediate anisotropy.

Figure 1

Schematic low-temperature phase diagram at density $n=1.2$. SSF (TSF) is a singlet (triplet) superfluid, and IDW is an incommensurate density wave; see the main text for an explanation of $s$ and $p$ symmetry classifications. Black triangles indicate phase transitions in the weak-coupling limit. Black circles mark the onset of instability in the $s\mathrm{SSF}$ channel (with dashed lines interpolating between them). Stars mark points where various ordering channels were compared to each other. Colored regions delimited by dotted lines indicate possible extents of the phases consistent with our data. Except for the data points represented by symbols, we do not claim precise location of the respective phase boundaries. At $\alpha \lesssim 0.2$, the $p_2\mathrm{TSF}$ and $p_3\mathrm{TSF}$ phases are nearly degenerate.

Figure 2

(top row) Momentum distributions (color plots) of free $\downarrow$ fermions (dashed lines indicate ${\mathrm{FS}}_{\uparrow }$) with hopping anisotropy $\alpha$ increasing from left to right at temperature $T/D=0.02$. Axes correspond to momenta $k_{x,y}\in [-\pi ,\pi ]$ in the first Brillouin zone of the square lattice. At ${\alpha }^{*}\approx 0.23$, the FS topology changes from a single closed contour (“2D-like”) to two disconnected lines that wind around the BZ boundaries in one direction (“1D-like”). (bottom row) Pair propagator ${\chi }_{\uparrow \downarrow }^{pp}$ for the same systems at zero frequency. The maximum value $D/4T$ (red color) is independent of $\alpha$ and diverges linearly with inverse temperature. However, for $\alpha \ne 0$ its support shrinks to four discrete points in the $T\to 0$ limit, rendering the integral over $\mathbf{k}$ finite.

Figure 3

(left) Weak-coupling eigenvalues for pairing of $\uparrow$ particles. The dashed vertical line marks the crossover at ${\alpha }^{*}$ from 2D to 1D topology of the FS. (right) Momentum-space structure of the leading pairing eigenvector obtained with DiagMC at finite attraction $|U|=D$. Axes are the same as in Fig. 2. As in the weak-coupling analysis, the leading instability at intermediate anisotropy $\alpha =0.375$ belongs to the representation $B_{3u}$ with horizontal nodal line $k_y=0$ (top) whereas at large anisotropy $\alpha =0.9$ the $B_{2u}$ configuration with vertical node $k_x=0$ dominates (bottom).

Figure 4

Temperature dependence of the leading eigenvalues at $|U|=D$ for strong anisotropy $\alpha =0.9$ (top) and $\alpha =1$ (bottom). Dashed lines are guides to the eye. The solid lines going through the $p$-wave data points show the $T$ dependence predicted by Fermi-liquid theory based on Fermi-liquid parameters and eigenvalues of the irreducible pairing vertex on the FS with $T\to 0$ extrapolation; only a constant offset accounting for high-temperature effects has been fitted to the finite $T$ data points. The solid line going through the $Q=(2k_F,2k_F)$ eigenvalues is a linear fit in $\ln T$. The inset shows the convergence of the transition temperature $T_{\mathrm{DW}}$ with diagram order $N_{*}$. The error bars on DW and $s\mathrm{SSF}$ data points at the lowest temperatures are systematic and dominated by extrapolation in the number of Matsubara frequencies.

Figure 5

Spatial structure of the irreducible vertex in the $p_2\mathrm{TSF}$ (left) and $p_3\mathrm{TSF}$ (right) channels for the anisotropies ${\alpha }_1=0.375$ and ${\alpha }_2=0.9$ calculated by DiagMC (for $|U|=D$) and within RPA approximation. RPA data have been calculated for weak interaction $|U|=1.8t_a$ and scaled with a constant factor for each channel.