Symmetry and causality constraints on Fermi liquids

TL;DR

The paper systematically derives how spacetime symmetries and causality constrain Fermi liquid data, revealing that scale invariance fixes $F_0$ in parallel with boost constraints that fix $F_1$. In conformal Fermi liquids, microcausality imposes strong bounds on the $(F_0,F_1)$ plane and restricts collective mode behavior, while nonlinear response probes require generalized Landau parameters $F^{(2,0)}_1$ and $F^{(3,0)}$, tied to symmetry constraints in a nonlinear EFT. The authors formulate a coadjoint-orbit EFT with a hierarchy of generalized Landau parameters, derive both linear and nonlinear symmetry constraints (Galilean, scale, Lorentz), and validate them across microscopic models including free Dirac fermions, weakly interacting Fermi liquids, and Chern-Simons-matter theories. These results illuminate the allowed landscape of conformal and non-conformal Fermi liquids and have potential implications for dense QCD, holography, and cold-atom experiments probing nonlinear density responses.

Abstract

We investigate symmetry and causality constraints on interacting Fermi liquids. Whereas Galilean or Lorentz boost symmetry leads to a well-known constraint on the first Landau parameter $F_1$, we show that scale invariance similarly constrains $F_0$. In the case of conformal Fermi liquids, we show that causality constraints on the particle-hole continuum and on zero sound strongly restrict the available parameter space for interacting Fermi liquids. We also consider nonlinear response, which we show is sensitive to additional ``generalized Landau parameters'' even at lowest orders in the long wavelength limit. We impose Galilean, Lorentz and scale symmetry on these generalized Landau parameters, obtaining further nonlinear constraints. We test our constraints in several microscopic models that enter a Fermi liquid phase.

Figures (10)

• Figure 1: (a) The particle-hole continuum of Fermi surfaces lead to (b) low energy spectral densities of currents $j^\mu$ at finite wavevector $k\leq 2k_F$ (dark gray, with multi-particle-hole continuum shown in lighter gray). (c) In contrast, QFT spectral densities in the vacuum only have support for $\omega\geq k$.
• Figure 2: (a) In Fermi liquids, the particle-hole continuum and collective excitations such as zero sound (blue) and shear sound (green) produce non-analyticities in $G^R_{\rho\rho}(\omega,q)$. (b) Causality constraints on conformal Fermi liquids in $d=3$, in the space of the first Landau parameters $F_0,F_1$. The dark gray region is excluded by Eq. \ref{['3d_vF_causality']}. Demanding the collective modes be causal leads to a stronger constraint, excluding the light gray region. The remaining allowed parameter space either features no collective excitation (white), a coherent zero sound mode (blue), or both zero sound and shear sound (green). See Sec. \ref{['sec_linear']} and App. \ref{['app_dimgen']} for details.
• Figure 3: Causality constraints on conformal Fermi liquids in $d=2$ spatial dimensions, in the space of the first Landau parameters $F_0,F_1$. The dark gray region is excluded by \ref{['eq_F0F1_CFTconstraint']}. Stronger constraints arise from demanding that collective excitations be causal; if higher Landau parameters are negligible, the constraint \ref{['eq_subluminal_ZS']} leads to the light-gray exclusion region. The remaining parameter space either features no coherent sound mode (white), a zero sound mode when \ref{['eq_ZS_exists']} is satisfied (blue) or both zero sound and shear sound when $F_1\geq 1$ (green). The red line $(F_0,F_1)\in 0\otimes \mathbb R_+$ shows the values realized in the class of CFTs considered in Sec. \ref{['ssec_CS']}.
• Figure 4: Density three-point function in a free Fermi gas.
• Figure 5: Density three-point function in a general Fermi liquid.