Cooper pairing in non-Fermi liquids

States of matter with a sharp Fermi surface but no well-defined Landau quasiparticles arise in a number of physical systems. Examples include (i) quantum critical points associated with the onset of order in metals; (ii) spinon Fermi-surface [U(1) spin-liquid] state of a Mott insulator; (iii) Halperin-Lee-Read composite fermion charge liquid state of a half-filled Landau level. In this work, we use renormalization group techniques to investigate possible instabilities of such non-Fermi liquids in two spatial dimensions to Cooper pairing. We consider the Ising-nematic quantum critical point as an example of an ordering phase transition in a metal, and demonstrate that the attractive interaction mediated by the order-parameter fluctuations always leads to a superconducting instability. Moreover, in the regime where our calculation is controlled, superconductivity preempts the destruction of electronic quasiparticles. On the other hand, the spinon Fermi surface and the Halperin-Lee-Read states are stable against Cooper pairing for a sufficiently weak attractive short-range interaction; however, once the strength of attraction exceeds a critical value, pairing sets in. We describe the ensuing quantum phase transition between (i) $\text{U}\left(1\right)$ and $Z_2$ spin-liquid states; (ii) Halperin-Lee-Read and Moore-Read states.

Figure 1

Top: conventional phase diagram of a quantum critical point (QCP) associated with an order parameter $\varphi$, with a superconducting dome (SC) partially overlapping the quantum critical region of the “bare” QCP of a metal. Bottom: the phase diagram obtained in this paper, with the SC dome fully overlapping the incipient regime of incoherent fermionic quasiparticles, while the quantum critical $\varphi$ fluctuations survive into higher temperatures in the normal state.

Figure 2

A pair of antipodal patches, labeled by a fixed $j$, on the Fermi surface. The values of ${\Lambda }_x$ and ${\Lambda }_y$ are constrained as in Eq. (2.1).

Figure 3

Top: our RG procedure. During each RG step, each patch of the Fermi surface is divided into two smaller patches. The relationship between the widths and heights of the patches remains as in Eq. (2.1). Bottom: single-boson exchange mediates a nonlocal intrapatch interaction (left). Here and below, solid (dashed) lines are fermion (boson) propagators. As high momentum boson modes are integrated out in the RG process, a local interpatch four-Fermi interaction in the BCS channel $\delta V({\vec{k}}_1,-{\vec{k}}_1;{\vec{k}}_2,-{\vec{k}}_2)$ is generated (right).

Figure 4

Renormalization of the BCS interaction $V({\vec{k}}_1,-{\vec{k}}_1;{\vec{k}}_2,-{\vec{k}}_2)$ in a FL.

Figure 5

RG flow of the interpatch BCS interaction ${\tilde{V}}_m^{s,a}$ in the spinon FS/HLR phase with $0<\epsilon \ll 1.$ The IR stable fixed point ${\tilde{V}}_{*}^{+}\approx \sqrt{\epsilon /2}$ controls the gapless FS phase, while the IR unstable fixed point ${\tilde{V}}_{*}^{-}\approx -\sqrt{\epsilon /2}$ controls the continuous transition to the paired phase.

Figure 6

RG flow of the intrapatch coupling constant $\tilde{\alpha }$ and the interpatch BCS interaction ${\tilde{V}}_m^{s,a}$ in the HLR phase with Coulomb interactions $(\epsilon =0)$. Note the attractor line $\tilde{V}\approx \sqrt{\tilde{\alpha }}$ (dashed red curve) and the separatrix $\tilde{V}\approx -\sqrt{\tilde{\alpha }}$ (solid red curve). The HLR phase is controlled by the logarithmic flow of the attractor line into the fixed point $(\tilde{V},\tilde{\alpha })=(0,0).$ The phase transition to the paired CF phase is controlled by the logarithmic flow of the separatrix into the same fixed point $(\tilde{V},\tilde{\alpha })=(0,0)$.

Figure 7

Determination of the pairing scale ${\ell }_p$ and the corresponding value ${\overline{\alpha }}_p$ in the regimes $\tilde{\alpha }\sim O\left({\epsilon }^2\right)$ and $\tilde{\alpha }\ll O\left({\epsilon }^2\right)$ [see Eqs. (A13) and (A15)].