Physics

Fermi Gamma-ray Space Telescope

Fermi surface

Pure mathematics

Discrete symmetry

Mathematics

Asymmetry

Renormalization group

Quartic function

Spontaneous symmetry breaking

Mathematical analysis

Symmetry breaking

Luttinger liquid

Geometry

Superconductivity

Fixed point

Quantum mechanics

Homogeneous space

Fermi liquid theory

Quantum

Nature Physics

Because Fermi liquids are inherently non-interacting states of matter, all electronic levels below the chemical potential are doubly occupied. Consequently, the simplest way of breaking the Fermi-liquid theory is to engineer a model in which some of those states are singly occupied, keeping time-reversal invariance intact. We show that breaking an overlooked1 local-in-momentum space                                                                   $${{\mathbb{Z}}}_{2}$$                               symmetry of a Fermi liquid does precisely this. As a result, although the Mott transition from a Fermi liquid is correctly believed to arise without breaking any continuous symmetry, a discrete symmetry is broken. This symmetry breaking serves as an organizing principle for Mott physics whether it arises from the tractable Hatsugai–Kohmoto model or the intractable Hubbard model. Through a renormalization-group analysis, we establish that both are controlled by the same fixed point. An experimental manifestation of this fixed point is the onset of particle–hole asymmetry, a widely observed2–10 phenomenon in strongly correlated systems. Theoretically, the singly occupied region of the spectrum gives rise to a surface of zeros of the single-particle Green function, denoted as the Luttinger surface. Using K-homology, we show that the Bott topological invariant guarantees the stability of this surface to local perturbations. Our proof demonstrates that the strongly coupled fixed point only corresponds to those Luttinger surfaces with co-dimension p + 1 with odd p. We conclude that both Hubbard and Hatsugai–Kohmoto models lie in the same high-temperature universality class and are controlled by a quartic fixed point with broken                                                                   $${{\mathbb{Z}}}_{2}$$                               symmetry. The Mott metal-to-insulator transition plays a key role in theoretical studies of high-temperature superconductors. A mathematical analysis of the theory of metals identifies a renormalization-group fixed point describing Mott physics.

Discrete symmetry breaking defines the Mott quartic fixed point