Competition of Fermi surface symmetry breaking and superconductivity

We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a $d$-wave Pomeranchuk instability and a BCS pairing interaction driving $d$-wave superconductivity. By tuning the interaction parameters a rich variety of phase diagrams is obtained. If the BCS interaction is not too strong, Fermi surface symmetry breaking is stabilized around van Hove filling, and coexists with superconductivity at low temperatures. For pure forward scattering Fermi surface symmetry breaking occurs typically via a first order transition at low temperatures. The presence of superconductivity reduces the first order character of this transition and, if strong enough, can turn it into a continuous one. This gives rise to a quantum critical point within the superconducting phase. The superconducting gap tends to suppress Fermi surface symmetry breaking. For a relatively strong BCS interaction, Fermi surface symmetry breaking can be limited to intermediate temperatures, or can be suppressed completely by pairing.

Figure 1

Critical temperature $T_f\left(\mu \right)$ for Fermi surface symmetry breaking in the absence of superconductivity $(\Delta =0)$ for $g_f=1$ and critical temperature for superconductivity $T_c\left(\mu \right)$ in the absence of Fermi surface symmetry breaking $(\eta =0)$ for various choices of $g_c$. The superconducting transition is always of second order. Fermi surface symmetry breaking occurs via a first order transition at temperatures below the tricritical points $T_f^{\mathrm{tri}}$, and via a second order transition above.

Figure 2

Phase diagram in the $(\mu ,T)$-plane for $g_f=1$ and $g_c=0.7$. The symbols $\eta$ and $\Delta$ indicate which order parameters are finite in the various regions confined by the transition temperatures. Fermi surface symmetry breaking occurs via a first order transition in the temperature range shown in the plot. Continuing the (then metastable) phase with a symmetric Fermi surface beyond the first order transition line leads to a diverging $d$-wave compressibility at the ficticious second order transition line “$T_f^{2\mathrm{nd}}$” also shown in the plot.

Figure 3

Phase diagram in the $(\mu ,T)$-plane for $g_f=1$ and $g_c=0.9$. The first order transition line for Fermi surface symmetry breaking is very close to the ficticious second order line in this case.

Figure 4

Reduced Landau energy $\omega \left(\Delta \right)=\omega [{\eta }_{\min }\left(\Delta \right),\Delta ]-\omega [{\eta }_{\min }\left(0\right),0]$, minimized with respect to $\eta$, as a function of $\mid \Delta \mid$. The interaction parameters are $g_f=1$ and $g_c=0.9$ as in Fig. 3, the temperature is $T=0.09$. The two choices of $\mu$ correspond to two points close to but on opposite sides of the first order transition line between a superconducting state with a symmetric Fermi surface and a normal state with a $d$-wave deformed Fermi surface.

Figure 5

Order parameters $\eta$ and $\Delta$ as a function of $\mu$ for various temperatures, $T=0.05,0.07,0.09$. The interaction parameters are $g_f=1$ and $g_c=0.9$ as in Fig. 3.

Figure 6

Temperature dependence of the order parameters $\eta$ and $\Delta$ for $\mu =-0.7$ and interaction parameters as in Fig. 3.

Figure 7

Temperature dependence of the inverse Stoner enhancement of the $d$-wave compressibility along the right first order transition line between the phases with symmetric and symmetry-broken Fermi surface. Interaction parameters are $g_f=1$ and $g_c=0,0.7,0.9$. The tricritical point at the high temperature end of the transition line $(T_f^{\mathrm{tri}}=0.130)$ is the same in all cases.

Figure 8

Phase diagram in the $(\mu ,T)$-plane for $g_f=1$ and $g_c=1$. The ficticious second order transition line “$T_f^{2\mathrm{nd}}$” is so close to the first order line $T_f^{1\mathrm{st}}$ that it is hidden by the latter.

Figure 9

Phase diagram in the $(\mu ,T)$-plane for $g_f=1$ and $g_c=1.12$. All transitions are of second order.

Figure 10

Phase diagram in the $(\mu ,T)$-plane for $g_f=1$ and $g_c=1.2$. All transitions are of second order.