Interacting topological quantum chemistry of Mott atomic limits

Topological quantum chemistry (TQC) is a successful framework for identifying (noninteracting) topological materials. Based on the symmetry eigenvalues of Bloch eigenstates at maximal momenta, which are attainable from first principles calculations, a band structure can either be classified as an atomic limit, in other words adiabatically connected to independent electronic orbitals on the respective crystal lattice, or it is topological. For interacting systems, there is no single-particle band structure and hence, the TQC machinery grinds to a halt. We develop a framework analogous to TQC, but employing $n$-particle Green's function to classify interacting systems. Fundamentally, we define a class of interacting reference states that generalize the notion of atomic limits, which we call Mott atomic limits, and are symmetry protected topological states. Our formalism allows to fully classify these reference states (with $n=2$), which can themselves represent symmetry protected topological states. We present a comprehensive classification of such states in one dimension and provide numerical results on model systems. With this, we establish Mott atomic limit states as a generalization of the atomic limits to interacting systems.

Figure 1

Mott atomic limits and induced bands representations. (a) Unit cell and first Brillouin zone of the square lattice with the Wyckoff positions and high-symmetry points marked, respectively. (b) Two sets of single-particle irreps compatible with the site symmetry group $C_4^D$ of Wyckoff position $1a$ : The two blocks represent each two TRS-related states (Kramers pairs) and the colors indicate nontrivial rotation eigenvalues under $C_4$. (c) Examples of time-reversal symmetric two-particle irreps constructed out of the single-particle irreps in (b). In the first row we show two example of ALs: in this case, the two states of a Kramers pair from a Slater determinant and consequently, the state transforms in the $A$ representation of the point group. In the second row, we show an example of an MAL, where the state transforms in the $B$ representation of the point group. [(d),(f)] Show examples of two-particle irreps placed in each unit cell at Wyckoff position $1a$ with (d) an AL and (f) an MAL state on the square lattice. [(e),(g)] Schematically show the respective inverse spectra of the single-particle and two-particle Green's functions, marked by ${\lambda }_1^{-1}$ and ${\lambda }_2^{-1}$.

Figure 2

Crystalline SPTs. (a) Classification of 0D-block cSPTs in a $C_4$ symmetric unit cell with TRS ${\mathcal{T}}^2=+1$. (b) Illustration of the action of a partial symmetry operation applied to a subsystem in the square lattice. The pink region indicates the subsystem and the blue lines indicate the new lattice connections after the partial $C_4$ rotation is applied to the subsystem.

Figure 3

Green's functions spectra with a spectrally flattened Hamiltonian. [(a),(b)] One- and [(c),(d)] two-particle Green's functions inverse spectra for a AL or MAL ground state, evaluated in the flattened Hamiltonian limit. We plot $-{\lambda }_1^{-1}$, instead of ${\lambda }_1^{-1}$ to maintain the analogy with the topological Hamiltonian. Thick lines indicate eigenvalues with multiplicity higher than one, while the thin line in the MAL ${\underline{g}}^{\left(2\right)}$ inverted spectrum indicates a singly degenerate eigenvalue. In (b), there is a band of eigenvalues corresponding to the empty orbitals $3,\overline{3}$ at $-{\lambda }_1^{-1}/\mathrm{\Delta }=1$, whereas, for comparison, in Fig. 1 this line is missing since there are no empty orbitals remaining in the local Hilbert space.

Figure 4

One-dimensional lattice. Wyckoff positions of a 1D lattice. The inversion center coincides with the Wyckoff position $1a$.

Figure 5

Schematics of the models considered. Schematics of (a) the Hubbard square (Sec. 6a), (b) the Hubbard diamond chain (Sec. 6b), (c) the checkerboard lattice of Hubbard squares (Sec. 6c), and (d) the star of David cluster (Sec. 6d). In (b) and (c) some Wyckoff positions are explicitly marked for the two lattices, and the unit cell is highlighted by the yellow rectangle. The tunneling amplitudes discussed in the various models are highlighted in each schematic.

Figure 6

Hubbard square. Inverse eigenvalues of the two-particle Green's function for (a) the trivial phase ($t_2>t_1$) and (b) the nontrivial phase ($t_2<t_1$) of the Hubbard square (44). The boundary of the region shaded in yellow marks the two-particle gap ${\mathrm{\Delta }}_{{\underline{g}}^{\left(2\right)}}$ at each value of the Hubbard interaction strength $U$. The eigenvalues of ${\underline{g}}^{\left(2\right)}$ are colored according to the legend in (a), with the color code distinguishing the $C_4^D$ representation in which their eigenstates transform, either $A, B$ or the corepresentation ${}^{1}E{}^{2}E$, see Table S3 within the SM [55]. Solid (dashed) lines indicate singly (doubly) degenerate eigenvalues. The state in (b) below ${\mathrm{\Delta }}_{{\underline{g}}^{\left(2\right)}}$ transforming in the $B$ irrep of $C_4$ indicates the nontrivial MAL character of the ground state. All quantities are expressed in units of $t_1$.

Figure 7

Hubbard diamond chain. (a) Phase diagram of the diamond chain model in the $(U/t_1,t_3/t_1)$ plane. In the upper part, the system is in an MAL phase, while in the lower it is in an AFM phase. The phase transition is marked using three approaches: (i) ${\mathrm{\Delta }}_{\text{AFM}}$, by the vanishing gap between the ground state and the $S=1, k=\pi$ excited state computed within $L\to \infty$ extrapolation of the QMC data; (ii) ${\chi }_{\text{AFM}}$, inflection point in susceptibility towards the respective Néel order computed within QMC on $L=10$ unit cells; and (iii) ${\underline{g}}^{\left(2\right)}$, by the vanishing gap between the lowest-lying band and the rest of the inverted spectrum of ${\underline{g}}^{\left(2\right)}$, computed within QMC on the $L=8$ chain (see the SM [55]). The two crosses mark the points in the phase diagram corresponding to the plots (b)–(e). [(b),(c)] Inverted band structure of ${\underline{g}}^{\left(1\right)}$ of the diamond chain obtained using QMC simulations, computed at $U/t_1=2$ and $t_3/t_1=0.1,1.0$, for (b) and (c) respectively. The spectrum is doubly degenerate at each value of $q$ due to the spin degree of freedom. [(d),(e)] Inverted band structure of ${\underline{g}}^{\left(2\right)}$, for $U/t_1=2, u_{max}=6, L=8$ and $t_3/t_1=0.1,1.0$, for (c) and (d), respectively. The dashed line indicates the two-particle gap ${\mathrm{\Delta }}_{{\underline{g}}^{\left(2\right)}}$ estimated from the spectral gap of ${\left[{\underline{g}}^{\left(1\right)}\right]}^{-1}$, meaning ${\mathrm{\Delta }}_{{\underline{g}}^{\left(2\right)}}\approx 2{\mathrm{\Delta }}_1$. (f) Spectral gap ${\mathrm{\Delta }}_2$ between the lowest-lying and the remaining eigenvalues of ${\left[{\underline{g}}^{\left(2\right)}\right]}^{-1}$, as a function of $1/u_{max}$, computed at fixed $U/t_1=2$, and $t_3/t_1=0.0,0.1,0.5$, and 1.0.

Figure 8

Checkerboard lattice of Hubbard squares. (a) Phase diagram as a function of $\phi /(\pi /2)$ and $U/t$ ($t_1=tcos\phi ,t_3=tsin\phi$). The phase boundaries are evaluated using (i) ${\mathrm{\Delta }}_{\text{AFM}}$, the vanishing gap between the ground state and the $S=1$ and $\mathit{k}=(\pi ,\pi )$ excited state, extrapolated for $L\to \infty$ from the QMC data; (ii) ${\chi }_{\text{AFM}}$, the inflection point in susceptibility towards the Néel order, obtained within QMC on the $L=5$ lattice (corresponding to $L\times L$ unit cells); and (iii) ${\underline{g}}^{\left(2\right)}$, the gap closing in the inverse spectra of ${\underline{g}}^{\left(2\right)}$, obtained in QMC for a system with $L=3$ (see the SM [55]). The three crosses mark the points in the phase diagram corresponding to the (b)–(d) plots. [(b)–(d)] Inverted spectrum of ${\underline{g}}^{\left(2\right)}$ in the three regimes (b) $t_1\gg t_3$ ($\phi =0.08$), (c) $t_1\approx t_3$ ($\phi =0.74$), and (d) $t_1\ll t_3$ ($\phi =1.49$), for $U/t=2$ and $\phi =arctan(t_1/t_3)$. The spectra in (b)–(d) are evaluated with QMC, for a system of size $L=4$. (e) Convergence of the spectral gap in the inverted spectrum of ${\underline{g}}^{\left(2\right)}$ as a function of inverse cluster size $1/u_{max}$, for systems with $L=4$ and $\phi =0.00,0.17,1.41,1.57$.

Figure 9

Star of David. (a) Single-particle spectrum of the star of David at $U/t=0, {\mu }_{*}/t=1.5$. Each eigenvalue is doubly degenerate due to the spin degree of freedom. The two levels closest to zero energy correspond to the $s$ and $f$ states. Their wave function weights are shown in the insets. (b) Phase diagram obtained within ED of the star of David as a function of ${\mu }_{*}/t$ and $U/t$ in the Hilbert space of $N=12$ electrons, at fixed $t^{\prime }/t=1$ and $t^{\prime \prime }/t=0.4$. Phases are labeled by ${}^{2S+1}R$, where $R$ indicates the $C_6$ representation in which the ground state transforms and $S$ its total spin. (The representation labeled by $A$ corresponds to $A_1$ for $C_{6v}$, while $A^{\prime }$ corresponds to $A_2$.) The pink asterisk marks the parameters of (a). (c) Low-energy spectrum at $N=12$ as a function of $U$, plotted with respect to the lowest energy eigenvalue (i. e., the ground-state energy). (d) Inverted ${\underline{g}}^{\left(2\right)}$ spectrum obtained in ED with a cutoff of $m_{max}=300$ in each symmetry sector as a function of $U$ (see the SM [55]). The lowest eigenvalues are colored according to the physical representation of $C_6$ in which their eigenstates transform. The continuous (dashed) lines mark the $N-2$ ($N+2$) many-body gap, ${\mathrm{\Delta }}_{N\pm 2}=E_{min,N\pm 2}-E_{\mathrm{GS},N}$ with $N=12$. The gray (black) lines refer to excitations with $S_z=\pm 1$ ($S_z=0$). In (c)–(d), the parameters are ${\mu }_{*}/t=1.5, t^{\prime }/t=1$, and $t^{\prime \prime }/t=0.4$, corresponding to the dashed line in (b). The legend in (c) applies to both (c) and (d). In (d), only the lowest eigenvalues are filled with color, for visibility.