Suppression of superfluid stiffness near a Lifshitz-point instability to finite-momentum superconductivity

We derive the effective Ginzburg-Landau (GL) theory for finite-momentum [Fulde-Ferrell-Larkin-Ovchinnikov/pair-density wave (FFLO/PDW)] superconductivity without spin population imbalance from a model with local attraction and repulsive pair hopping. We find that the GL free energy must include up to sixth-order derivatives of the order parameter, providing a unified description of the interdependency of zero and finite-momentum superconductivity. For weak pair hopping the phase diagram contains a line of Lifshitz points where vanishing superfluid stiffness induces a continuous change to a long wavelength FF state. For larger pair hopping there is a bicritical region where the pair momentum changes discontinuously. Here the FF type state is near degenerate with the LO or PDW type state. At the intersection of these two regimes there is a “super-Lifshitz” point with extra soft fluctuations. The instability to finite-momentum superconductivity occurs for arbitrarily weak pair hopping for sufficiently large attraction suggesting that even a small repulsive pair hopping may be significant in a microscopic model of strongly correlated superconductivity. Several generic features of the model may have bearing on the cuprate superconductors, including the suppression of superfluid stiffness in proximity to a Lifshitz point as well as the existence of subleading FFLO order (or vice versa) in the bicritical regime.

Figure 1

Transition temperatures for $Q=0$ superconducting (SC, blue lines) and $Q\ne 0$ Fulde-Ferrell (FF, red lines) states of the model (1) and (2) as a function of local attraction strength, $g$, and relative pair-hopping strength, $\alpha =g_{\text{pair}}/g_0$. A line of Lifshitz points [given by ${\left(m_B\right)}^{-1}=0$ solutions of (11)], are shown in solid blue in the $g,\alpha$ plane (dashed for subleading transition). The black solid line indicates a bicritical line. The black dot marks the super-Lifshitz point at the intersection of the bicritical and Lifshitz line. The red and green arrows indicate similar paths as in Fig. 2.

Figure 2

Mean-field phase diagram of the free energy (6) (for $Q_{-}=0$) with SC, $Q_{+}=0$, and FF, $Q_{+}\ne 0$, phases. (a) Phases as a function of $a$ ($Q_{+}^2$) and $c$ ($Q_{+}^4$), and (inset) the corresponding functional form of $A\left(Q_{+}\right)$ for $r=0$. For $c>0,a=0$ there is a line of Lifshitz points with a continuous transition in $Q_{+}$, as shown in (b). (Note that when including fluctuations, the Lifshitz point $r=a=0$ is pushed to $T=0$, see Fig. 1). Solid black line $c<0,a=\frac{3c^2}{16}$ indicates a line of bicritical points where $Q_{+}$ jumps, as shown in (c). Black dashed in (c) marks a first-order transition [vertical assuming $u\left(\mathbf{Q}\right)=u]$, whereas other dashed lines in (a) and (c) are boundaries of metastable subleading phases (for $0<a<c^2/4$) that are not thermodynamic transitions. The black dot in (a) marks the “super-Lifshitz” point discussed in the text. The red and green arrows indicate similar paths as in Fig. 1.

Figure 3

Evolution of parameters as a function of $g$ for $\alpha =0;0.005;0.01;0.05$. For $\alpha =0.005;0.01$ there is a continuous evolution of $Q_{+}$ (a) starting at the Lifshitz point where the mass $m_B=\sqrt{m_B^{+}{m}_B^{-}}$ diverges (b). For $\alpha =0.05$ there is a jump in $Q_{+}$, the $Q_{+}\ne 0$ branch is dashed and the bicritical point is marked with a black dot. For higher $g$ the $\mu$ is pushed to the negative side (c), which corresponds to a high occupation of bosons, $n_B$ (d).

Figure 4

${\mathrm{\Gamma }}^{-1}(p_{+},0)$ (measured in $m$) for $\alpha =0.01$ (dashed green) and $\alpha =0.05$ (solid red) for three different interaction strengths (other variables are the same as in Fig. 3). For $\alpha =0.01$ we observe a transition through a Lifshitz point where the minimum shifts continuously from $Q=0$ to $|Q_{+}|>0$ roughly at $\frac{mg}{2\pi }\backsimeq 0.4$. For $\alpha =0.05$ we observe a transition through a bicritical point where the momentum changes discontinuously for $\frac{mg}{2\pi }\backsimeq 0.2488$.

Figure 5

Diagrammatic representation of (B2).