Abstract
A system with charge conservation and lattice translation symmetry has a well-defined filling , which is a real number representing the average charge per unit cell. We show that if 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 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 a connection between filling constraints and the theory of symmetry-protected topological phases, in particular through the concept of “’t Hooft anomalies.”
- Received 17 August 2020
- Revised 22 December 2020
- Accepted 21 January 2021
DOI:https://doi.org/10.1103/PhysRevX.11.021005
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In a solid material, the motion of electrons is influenced by the regular crystalline arrangement of the surrounding atoms. At low temperatures, there are many possible different behaviors that these electrons can exhibit collectively, some of which are very challenging to understand. In this work, we provide a general way to understand certain features of the collective behavior of electrons in a solid as consequences of the concentration of electrons.
In particular, the concentration of electrons allows one to define a number called the “filling,” which is the number of electrons per the fundamental repeating unit of the crystal lattice. Generally, one expects that at low temperatures and over large length scales, the physics should not be sensitive to the detailed microscopic properties of the system and, instead, can be described by an “effective field theory.” However, the fractional part of the filling must still have a manifestation in this field theory. We identify, in great generality, which properties of this effective field theory must be the manifestation of the fractionality of the filling.
Using this general framework, we obtain powerful and unexpected conclusions. We show, for example, that in a metal, where one expects the filling can be a generic real number (not required to be a rational number, for instance), there must be infinitely many conserved quantities that emerge at low temperatures. These results give insight into the physics of metals, even exotic ones not described by conventional theories.
