Frustration-free free fermions

Abstract

We develop a general theory of frustration-free free-fermion systems and derive the necessary and sufficient conditions for such Hamiltonians. Assuming locality and translation invariance, we find that any band touching between the valence and the conduction bands is always quadratic or softer, which rules out the possibility of describing Dirac and Weyl semimetals using frustration-free local Hamiltonians. We further construct a frustration-free free-fermion model on the honeycomb lattice and show that its density fluctuations acquire an anomalous gap originating from the diverging quantum metric associated with the quadratic band-touching points. Nevertheless, an $O(1/L^2)$ finite-size scaling of the charge-neutral excitation gap can be verified even in the presence of interactions, consistent with the more general results we derive in an accompanying work [arXiv:2503.12879].

Figures (3)

• Figure 1: Frustration-free free-fermion model on the honeycomb. (a) Lattice convention and the definition of the local orbital corresponding to the conduction band. (b) Local orbital for the valence band. (c) Band structure. (d) Non-zero eigenvalue of the quantum metric tensor in arbitrary units, which diverges as $1/\delta q^2$ around the K point. (e) Excitation energy of density fluctuations within the single-mode approximation, which acquires an anomalous gap $O(1/\log L)$ due to the singular quantum metric. We take $\delta \bm k = (\bm b_1 + \bm b_2)/L$. Crosses are computed values and the dashed line indicates an inverse-log fit with fitting parameters reported in Appendix \ref{['app:honeycomb_details']}.
• Figure 2: Energy of particle-hole excitations in $\hat{H}^{\rm int}$ at $U=2t$. (a) Energy spectrum of $\hat{H}^{\rm int}$ restricted to the subspace spanned by $|\bm k, \bm q\rangle$ for $\bm k$ along the M-$\Gamma$-K line computed for $L= 50$. The boundary of the particle-hole continuum can be estimated as $\min_{\bm q} (t e_{\bm q} + (t+U) e_{\bm q + \bm k})$ (green solid line), which becomes exact in the non-interacting limit with $U=0$ (grey dashed line). Note that interaction $U>0$ modifies the excitation energies although it keeps the ground states unchanged. (b) System size dependence of the particle-hole energy at $\bm k = \bm 0$. The dashed line shows the $O(1/L^2)$ bound in Eq. \ref{['eq:zero_momentum_ph']}.
• Figure 3: Fitting of numerical data. Red crosses indicated numerical data and dashed black lines are the linear fits. (a) The linear fit of $t/E^{\rm SMA}(\delta \bm k)$ against $\log L$ gives a slope of $1.04(2)$ and an intercept of $-1.179(7)$. (b) The linear fit of the $\log$ of the minimum particle-hole excitation energy at $\bm k = \bm 0$ against $\log L$ gives a slope of $-1.99(2)$ and an intercept of $2.84(8)$.