Non-Fermi liquid at the FFLO quantum critical point

When a 2D superconductor is subjected to a strong in-plane magnetic field, Zeeman polarization of the Fermi surface can give rise to inhomogeneous FFLO order with a spatially modulated gap. Further increase of the magnetic field eventually drives the system into a normal metal state. Here, we perform a renormalization group analysis of this quantum phase transition, starting from an appropriate low-energy theory recently introduced in Phys. Rev. B 93, 085112 (2016). We compute one-loop flow equations within the controlled dimensional regularization scheme with fixed dimension of Fermi surface, expanding in $\epsilon =5/2-d$. We find a new stable non-Fermi-liquid fixed point and discuss its critical properties. One of the most interesting aspects of the FFLO non-Fermi-liquid scenario is that the quantum critical point is potentially naked, with the scaling regime observable down to arbitrary low temperatures. In order to study this possibility, we perform a general analysis of competing instabilities, which suggests that only charge density wave order is enhanced in the vicinity of the quantum critical point.

Figure 1

Typical temperature-magnetic field phase diagram of a superconductor susceptible to FFLO pairing. This picture was adapted from Ref. [14].

Figure 2

Typical Fermi surface of an anisotropic metal susceptible to FFLO pairing. Fluctuations of the pairing amplitude $\mathrm{\Delta }$ strongly couple left branch fermions with right branch fermions with opposite spin at the hot spots. This picture was adapted from [14].

Figure 3

Fermi surfaces in the FFLO phase at mean-field level. (a) Fermi surface without pairing, i.e., $g{\mathrm{\Delta }}_0=0$. (b) Fermi surface for electron/hole operators (see main text), at $\delta v\simeq 1.3$. Dashed lines: $g{\mathrm{\Delta }}_0=0$. Full lines: $g{\mathrm{\Delta }}_0=0.05$. (c) Same as (b), but $\delta v\simeq 0.2$.

Figure 4

One-loop diagrams. Dashed wavy lines (a) indicate bare boson propagators, while straight lines indicate electron propagators. Full wavy lines [(b) and (c)] represent bosons dressed with the self-energy of (a). External lines are amputated.

Figure 5

Zoom-in of Fig. 2, showing the two Fermi surface branches of fermions coupled by pairing fluctuations at $+{\mathbf{Q}}_{\text{FFLO}}$, shifted towards a common origin in momentum space. (a) Case of nonzero velocity detuning $\delta v\ne 0$, where only fermions with momenta close to $k=0$ are strongly entangled. (b) Case of vanishing velocity detuning $\delta v\to 0$, where the electrons with momenta $\pm {\mathbf{k}}_1$ and tangent vector ${\mathbf{q}}_1$ can be scattered to close-by momenta $\pm {\mathbf{k}}_2$ with similar tangent vector ${\mathbf{q}}_2$. (c) Same initial configuration as Fig. 5, but with different final momenta $\pm {\mathbf{k}}_2$; now, the final tangent vector ${\mathbf{q}}_2$ differs strongly from the initial one, and the phase space for the scattering is negligible.

Figure 6

Anomalous contributions to the fermion self-energy (in terms of the original fermion fields $\psi$). At one-loop, such contributions are impossible (a), but can arise at two loop (b).

Figure 7

Generic one-loop vertex correction in spinor space: (a) type-1 vertex (b) type-2 vertex.

Figure 8

${\mathrm{\Pi }}_0\left[h\right]$ numerically computed from Eq. (A8), with parameters $\mu =3.3,t=0.5$. The inset shows the same plot on a log-linear scale.