Quantum phase transitions of metals in two spatial dimensions: I. Ising-nematic order

TL;DR

The paper develops a renormalization-group framework for the onset of Ising-nematic order in two-dimensional metals, describing the critical point with an infinite set of 2+1D local field theories labeled by patches on the Fermi surface. It derives an anisotropic scaling with $k_y$ and $k_x$ ($k_y \to s k_y$, $k_x \to s^2 k_x$, $\omega \to s^z \omega$) and identifies the dynamic exponent $z$ and fermion anomalous dimension $\eta_\psi$, showing $z=3$ at three-loop order while $\eta_\psi$ is nonzero and patch-dependent. One finds Landau-damped boson dynamics, non-Fermi-liquid fermions with $G^{-1}(\vec{k},\omega) \sim \omega^{(2-\eta_\psi)/z}$, and a Fermi-surface shift $\Delta k$ that couples to the boson sector via a nonzero $\alpha$, with implications for compressibility. The results illuminate how Ising-nematic and related transitions in 2D metals manifest nontrivial scaling and non-Fermi-liquid behavior, and they connect to broader contexts such as spin liquids and gauge-field coupled Fermi surfaces.

Abstract

We present a renormalization group theory for the onset of Ising-nematic order in a Fermi liquid in two spatial dimensions. This is a quantum phase transition, driven by electron interactions, which spontaneously reduces the point-group symmetry from square to rectangular. The critical point is described by an infinite set of 2+1 dimensional local field theories, labeled by points on the Fermi surface. Each field theory contains a real scalar field representing the Ising order parameter, and fermionic fields representing a time-reversed pair of patches on the Fermi surface. We demonstrate that the field theories obey compatibility constraints required by our redundant representation of the underlying degrees of freedom. Scaling forms for the response functions are proposed, and supported by computations by up to three loops. Extensions of our results to other transitions of two-dimensional Fermi liquids with broken point-group and/or time-reversal symmetry are noted. Our results extend also to the problem of a Fermi surface coupled to a U(1) gauge field.

Figures (11)

• Figure 1: The shaded region represents the occupied states inside a Fermi surface. Fluctuations of the order parameter $\phi$ at wavevectors parallel to $\vec{q}$ couple most strongly to fermions near the Fermi surface points $\pm \vec{k}_0$. These fermions are denoted $\psi_\pm$.
• Figure 2: One loop contributions to the (a) boson, and (a) fermion self-energies.
• Figure 3: The momentum of the fermion at point $P$ can be measured with respect to either the co-ordinate system at $\vec{k}_0$, or that at $\vec{k}_1$.
• Figure 4: Two loop corrections to the polarization.
• Figure 5: Three loop corrections to the boson self-energy with one fermion loop.