Controlled expansion for certain non-Fermi-liquid metals

The destruction of Fermi-liquid behavior when a gapless Fermi surface is coupled to a fluctuating gapless boson field is studied theoretically. This problem arises in a number of different contexts in quantum many-body physics. Examples include fermions coupled to a fluctuating transverse gauge field pertinent to quantum spin-liquid Mott insulators, and quantum critical metals near a Pomeranchuk transition. We develop a controlled theoretical approach to determine the low-energy physics. Our approach relies on combining an expansion in the inverse number $\left(N\right)$ of fermion species with a further expansion in the parameter $ϵ=z_b-2$, where $z_b$ is the dynamical critical exponent of the boson field. We show how this limit allows a systematic calculation of the universal low-energy physics of these problems. The method is illustrated by studying spinon Fermi-surface spin liquids, and a quantum critical metal at a second-order electronic nematic phase transition. We calculate the low-energy single-particle spectra, and various interesting two-particle correlation functions. In some cases, deviations from the popular random-phase approximation results are found. Some of the same universal singularities are also calculated to leading nonvanishing order using a perturbative renormalization-group calculation at small $N$ extending previous results of Nayak and Wilczek. Implications for quantum spin liquids and for Pomeranchuk transitions are discussed. For quantum critical metals at a nematic transition, we show that the tunneling density of states has a power-law suppression at low energies.

Figure 1

(Color online) Our suggested phase diagram. Above the indicated curve, the putative critical theory is likely preempted by some other broken-symmetry state. The behavior of the proposed phase boundary at small $N$ is one possible extrapolation.

Figure 2

(Color online) An alternate possible phase diagram. Here the interesting point $z_b=3,N=2$ is in the unstable regime; in this case, it would be best accessed starting from the correct mean-field theory for the new broken-symmetry state.

Figure 3

One-loop boson self-energy.

Figure 4

One-loop fermion self-energy.

Figure 5

Three-loop boson self-energy diagrams.

Figure 6

Two-loop fermion self-energy diagrams merely renormalize the coefficient of ${\left|\omega \right|}^{2/z_b}$.

Figure 7

Three-loop fermion self-energy diagrams involving fermions on both patches. Fermions on the right patch are denoted by solid lines and fermions on the left patch by dashed lines.

Figure 8

One-loop correction to the fermion-boson vertex.

Figure 9

The cooper-channel scattering amplitude.

Figure 10

The particle-hole channel scattering amplitude.

Figure 11

After a scattering event in the $2K_f$ channel, two fermions are no longer perfectly nested.

Figure 12

After any scattering event in the Cooper channel, two fermions remain perfectly nested.

Figure 13

Three-loop fermion self-energy containing fermions from a single patch.

Figure 14

Numerical integral determining the fermion anomalous dimension ${\eta }_f$.

Figure 15

One-loop correction to the $2k_{\text{F}}$ vertex. The dotted line denotes a $2k_{\text{F}}$ density fluctuation.