Abstract
A system with charge conservation and lattice translation symmetry has a well-defined filling $\nu$, which is a real number representing the average charge per unit cell. We show that if $\nu$ is fractional (i.e. not an integer), this imposes very strong constraints on the low-energy theory of the system and give a framework to understand such constraints in great generality, vastly generalizing the Luttinger and Lieb-Schultz-Mattis theorems. The most powerful constraint comes about if $\nu$ is continuously tunable (i.e. the system is charge-compressible), in which case we show that the low-energy theory must have a very large emergent symmetry group -- larger than any compact Lie group. An example is the Fermi surface of a Fermi liquid, where the charge at every point on the Fermi surface is conserved. We expect that in many, if not all, cases, even exotic non-Fermi liquids will have the same emergent symmetry group as a Fermi liquid, even though they could have very different dynamics. We call a system with this property an "ersatz Fermi liquid". We show that ersatz Fermi liquids share a number of properties in common with Fermi liquids, including Luttinger's theorem (which is thus extended to a large class of non-Fermi liquids) and periodic "quantum oscillations" in the response to an applied magnetic field. We also establish versions of Luttinger's theorem for the composite Fermi liquid in quantum Hall systems and for spinon Fermi surfaces in Mott insulators. Our work makes connection between filling constraints and the theory of symmetry-protected topological (SPT) phases, in particular through the concept of " 't Hooft anomalies".
Highlights
In condensed matter physics, systems with prescribed microscopic degrees of freedom can exhibit varied and exotic emergent behavior at low energies
Our approach leads to a very general perspective on what it means for a system to have a Fermi surface. We find that this Fermi surface must obey Luttinger’s theorem, and that, if the Fermi-surface geometry is such that a Fermi liquid with that geometry would exhibit quantum oscillations in the dependence of physical properties on magnetic field, any ersatz Fermi liquid (EFL) with the same Fermi-surface geometry is expected to display quantum oscillations with the same periodicity
We find that, assuming the translational symmetry is unbroken, such a scenario is inconsistent with the IR theory being an EFL, except when the system exists on the boundary of a gapless bulk
Summary
Systems with prescribed microscopic degrees of freedom (usually electrons) can exhibit varied and exotic emergent behavior at low energies. Since none of the kinematic properties that define an EFL require that the system be weakly coupled or have a description in terms of quasiparticles, we expect a wide variety of exotic non-Fermi-liquid phenomena to be realizable within the class of EFLs. we argue based on the general theory of filling constraints that any IR theory which describes a compressible metal, i.e., the filling can be continuously tuned, [21] must have a very nontrivial emergent symmetry group, larger than any compact Lie group. We argue based on the general theory of filling constraints that any IR theory which describes a compressible metal, i.e., the filling can be continuously tuned, [21] must have a very nontrivial emergent symmetry group, larger than any compact Lie group Such a property is satisfied by EFLs (due to the infinitely many conserved quantities associated with the Fermi surface); whether it could be satisfied in a different way that leads to fundamentally different kinematic properties is an important open problem. It is desirable to have a more general nonperturbative argument for Luttinger’s theorem in this context, which we provide in this paper
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