Parity-dependent double degeneracy and spectral statistics in the projected dice lattice

Abstract

We investigate the spectral statistics of an interacting fermionic system derived by projecting the Hubbard interaction onto the two lowest-energy, degenerate flat bands of the dice lattice subjected to a $π$-flux. Surprisingly, the distributions of level spacings and gap ratios correspond to distinct Gaussian ensembles, depending on the parity of the particle number. For an even number of particles, the spectra conform to the Gaussian Orthogonal Ensemble, as expected for a time-reversal-symmetric Hamiltonian. In stark contrast, the odd-parity sector exhibits exact double degeneracy of all eigenstates even after resolving all known symmetries, and the Gaussian Unitary Ensemble accurately describes the spacing distribution between these doublets. The simultaneous emergence of two different random-matrix ensembles within a single physical system constitutes an unprecedented finding, opening new avenues for both random matrix theory and flat-band physics.

Figures (3)

• Figure 1: (a) Schematic of the dice lattice. The box is the magnetic unit cell comprising six lattice sites, while the thin black (thick red) bonds represent positive (negative) hopping amplitudes $t > 0$ ($-t <0$). Wannier functions $w_{n\vb{l}}$ ($n = 1, 2$) of the two lowest flat bands are also shown. (b)-(d): Interior eigenvalues of $\mathcal{\hat{H}}_{\lambda_2, \lambda_3}$\ref{['eq:H_main']} for system size $N_x, N_y =3, 2$, momentum $\mathbf{k} = 0$, and particle numbers $N_\uparrow$, $N_\downarrow$ shown in each panel. Different colors indicate different pairs $(S, B)$ of total spin and pseudospin quantum numbers. For even particle numbers (b), the eigenvalues $E_i$ are nondegenerate. For odd particle numbers (c), the spectrum consists of degenerate doublets. When $\lambda_2=\lambda_3=0$ (d), highly degenerate multiplets appear due to known LIOMs Swaminathan_2023_Phys.Rev.Res._Signaturesmanybodylocalizationsupplementary.
• Figure 2: Level spacing (top) and gap ratio (bottom) statistics of $\mathcal{\hat{H}}_{1,1}$ for even-$N$, size $N_x, N_y=3, 2$, and quantum numbers $(N_\uparrow, N_\downarrow,k_x,k_y,f, S, B) = (6,6,0,0,+,0,1)$ (Hilbert-space dimension $\mathcal{N}=16529$). In the top row, the solid line shows the fit to \ref{['eq:kNN_wigner']}. In both rows, the dashed curve is the closest matching Wigner surmise with effective order $k^*$ (ensemble $\beta^*$). The dotted curve shows the $k$NN Poisson distribution, and the dash-dot curve is the closest $m$-block GOE distribution for each $k$. Both level-spacing and gap-ratio distributions are consistent with the superposition of four GOE spectra for any $k$. Note that $P(s/r, 1) = P^2(s/r, 1, 4)$ as seen in (b) and (f), while $P^2(s/r, 1) = P^4(s/r, 1, 4)$ from (d) and (h).
• Figure 3: Same as Fig. \ref{['fig:kNNeven']}, but for odd-$N$ and quantum numbers $(N_\uparrow,N_\downarrow,k_x,k_y,S,B) = (5,6,0,0,1/2,1/2)$ (dimension $\mathcal{N}=25176$). For $k = 1$, the double degeneracy leads to a pronounced peak at $s=0$ in the level-spacing distribution and identically zero gap ratio \ref{['eq:gapratio']} (not depicted). The $k=2,4$ statistics are very well described by the GUE with $k^* = 1, 2$, equivalent to the superposition of two GOE blocks due to the known relation $P^{2k^*}(s/r, 1, 2)=P^{k^*}(s/r, 2)$. However, the $k = 3$ case clearly deviates from the superposition of two GOE blocks. Note the relation $P(s/r, 2) = P^3(s/r, 1, 5)$ in (c) and (f).