Symmetry-enforced Fermi surfaces

TL;DR

The paper identifies a strong symmetry-enforced mechanism that guarantees a Fermi surface in any local, symmetric lattice-fermion model with one spinless fermion per unit cell, via a UV symmetry generated by onsite $U(1)$ fermion-number conservation and non-onsite $b$-Majorana translations leading to the non-compact group $\,\mathrm{Ons}_{d} \\rtimes \\prod_{i=1}^d \\mathbb{Z}_{L_i}$. It shows that the resulting most general local, symmetric quadratic Hamiltonian decouples into $a$- and $b$-Majorana sectors and maps to a free-fermion model with dispersion $\\epsilon_{\\mathbf{k}} = -2\\sum_{\\mathbf{v}} g_{\\mathbf{v}}\\sin(\\mathbf{k}\\cdot\\mathbf{v})$, with Fermi surface defined by $\\epsilon_{\\mathbf{k}} = 0$ and $\\epsilon_{-\\mathbf{k}} = -\\epsilon_{\\mathbf{k}}$, guaranteeing a nonempty Fermi surface occupying half the Brillouin zone. Topologically, any symmetry-enforced Fermi surface must have at least two non-contractible components, and a generic Fermi surface cannot pass through inversion-invariant points in a way that would trivialize its topology. In the IR, the UV Onsager-type symmetry yields an emanant symmetry containing a compact subgroup $\\prod_{\\mathbf{k} \\\in \\mathcal{F}^+} O_{\\mathbf{k}}$, leading to an emergent, though non-abelian, $L^e_{\\mathcal{F}} U(1)$-type structure acting on the Fermi surface; this provides a concrete link between lattice UV symmetries and IR gapless physics. Overall, the work establishes a lattice realization of strong SEG enforcing a Fermi surface and outlines rich avenues for exploring higher-codimension Fermi surfaces, topology, and background-gauge-field effects.

Abstract

We identify a symmetry that enforces every symmetric model to have a Fermi surface. These symmetry-enforced Fermi surfaces are realizations of a powerful form of symmetry-enforced gaplessness. The symmetry we construct exists in quantum lattice fermion models on a $d$-dimensional Bravais lattice, and is generated by the onsite U(1) fermion number symmetry and non-onsite Majorana translation symmetry. The resulting symmetry group is a non-compact Lie group closely related to the Onsager algebra. For a symmetry-enforced Fermi surface $\cal{F}$, we show that this UV symmetry group always includes the subgroup of the ersatz Fermi liquid L$_{\cal{F}}$U(1) symmetry group formed by even functions ${f(\mathbf{k})\in\mathrm{U}(1)}$ with ${\mathbf{k}\in \cal{F}}$. Furthermore, we comment on the topology of these symmetry-enforced Fermi surfaces, proving they generically exhibit at least two non-contractible components (i.e., open orbits).

Figures (1)

• Figure 1: Examples of the Fermi seas and Fermi surfaces enforced by our microscopic symmetry. Specifically, here we consider a special class of Hamiltonians of the form \ref{['Hg Hamiltonian']} on a ${d=2}$ square lattice with ${\epsilon_\mathbf{k} = \sin(k_x) + \alpha \sin(k_y) + \beta (\sin(3k_x)\!+\sin(3k_y))}$. Each panel shows the Brillouin zone with horizontal axis ${-\pi\leq k_x< \pi}$ and vertical axis ${-\pi\leq k_y< \pi}$. The top row shows, from left to right, ${\left(\alpha,\beta\right) = \left(\frac{3}{4},0\right), \left(1,0\right), \left(\frac{5}{4},0\right)}$. The bottom row shows, from left to right, ${\left(\alpha,\beta\right) = \left(\frac{3}{4},\frac{1}{4}\right), \left(\frac{3}{4},\frac{1}{2}\right), \left(\frac{3}{4},1\right)}$. Note that the Fermi surface at ${\left(\alpha,\beta\right) = \left(1,0\right)}$ is non-generic.