Few-particle dynamics of fractional quantum Hall lattice models

Considering lattice Hamiltonians designed using conformal field theory to have fractional quantum Hall states as ground states, we study the dynamics of one or two particles on such lattices. Examining the eigenspectrum and dynamics of the single-particle sector, we demonstrate that these Hamiltonians cannot be regarded as describing interacting particles placed on a Chern band, so that the physics is fundamentally different from fractional Chern insulators. The single-particle spectrum is shown to consist of eigenstates localized in shells which have a larger radius for larger eigenenergies. This leads to chiral dynamics along reasonably well-defined orbits, in both the single-particle and the two-particle sectors.

Figure 1

(a) The potential $C_5$ for a $20\times 20$ lattice. The sites $i\equiv (i_x,i_y)$ are indexed by the horizontal and vertical coordinates $i_x$ and $i_y$, which are each integers running from 1 to 20. (b) Magnetic flux through each smallest plaquette for a $20\times 20$ lattice. The color at position $(i_x+0.5,i_y+0.5)$ in this plot represents the strength of the flux through the plaquette surrounded by the sites $(i_x,i_y)$, $(i_x+1,i_y)$, $(i_x,i_y+1)$, and $(i_x+1,i_y+1)$.

Figure 2

Eigenenergies of the single-particle system, indexed in ascending order.

Figure 3

Site occupancies $\langle E_{\alpha }\left|d_i^{†}{d}_i\right|E_{\alpha }\rangle$ for different eigenstates $\left|E_{\alpha }\right.\rangle$, where $\alpha$ indexes the eigenstates according to increasing energy. The number above each plot indicates $\alpha$ (e.g., $\alpha =398$ for the last plot). A different color scale has been used for each row, because in the higher eigenstates the weight is spread out among a larger number of sites, resulting in smaller nonzero site occupancies.

Figure 4

(a) We choose the initial state $\left|\psi \left(0\right)\right.\rangle$ to be a state with a single particle at a particular site. The site could, e.g., be one of the sites $a$, $b$, $c$, etc. (b) Norm square $P_{\alpha }={\left|〈E_{\alpha }|\psi \left(0\right)〉\right|}^2$ of the coefficients of the initial state in the basis of the energy eigenstates. For each plot, the legend gives the chosen initial position of the particle, and the lattice has $20\times 20$ sites. One can notice from these plots that if we start the dynamics with the particle initially at the edge, then the particle is more likely to be found in the higher excited states. As we move towards the bulk, the contributing states have lower and lower energy.

Figure 5

Propagation dynamics of a single particle initially placed at the corner of a $20\times 20$ lattice. The dynamics is illustrated through a series of snapshots of the occupancy profiles, i.e., values of $\langle \psi \left(t\right)\left|d_i^{†}{d}_i\right|\psi \left(t\right)\rangle$ at every site. The particle is seen to execute chiral motion around the edge, in addition to dispersion along the edge on a longer timescale. There is very little weight spreading into the bulk.

Figure 6

Propagation dynamics of a single particle in a $20\times 20$ lattice. The particle is initially placed some distance away from the edge, at position (6,6), referred to in the text as site $f$. The particle is seen to move chirally along an orbit roughly equidistant from the edge.

Figure 7

Energy eigenvalues of the two-particle systems, for a small ($5\times 5$) square lattice. The two-fermion and two-boson systems are seen to have very similar spectra (two lower datasets, barely distinguishable from each other). Also shown (curves with slightly higher energies) are the spectra of two-fermion and two-boson systems in which the interaction terms ($C_2$ and $C_3$ terms) have been suppressed from the Hamiltonian. These two datasets are also very close and hence barely distinguishable from each other.

Figure 8

Number density (site occupancy) profiles in different eigenstates of the two-fermion system. The eigenstate labels are indicated on top of the individual density plots. A different color (density) scale has been used for each row, because in the higher eigenstates the weight is spread out among a larger number of sites, resulting in smaller nonzero site occupancies. The two-boson case (not shown) shows the same overall features as one moves from low-energy to high-energy eigenstates.

Figure 9

Dynamics of the two-boson system on a $12\times 12$ lattice. In the initial state, the two bosons are at the positions $a$ and $d$, i.e., at the corner site (1,1) and at the site (4,4).