Edge-localized states in quantum one-dimensional lattices

In one-dimensional quantum lattice models with open boundaries, we find and study localization at the lattice edge. We show that edge-localized eigenstates can be found in both bosonic and fermionic systems, specifically, in the Bose-Hubbard model with on-site interactions and in the spinless fermion model with nearest-neighbor interactions. We characterize the localization through spectral studies via numerical diagonalization and perturbation theory through considerations of the eigenfunctions and through the study of explicit time evolution. We concentrate on few-particle systems, showing how more complicated edge states appear as the number of particles is increased.

Figure 1

(Color online) Energy spectrum of ten-site Bose-Hubbard chain with $\gamma =10$. Left panels (a) and (c) show the case of two bosons $\left(n=2\right)$, right panels (b) and (d) show $n=3$. Top panels (a) and (b) plot-ordered energies against eigenvalue index $\nu$, for fixed hopping parameter $t=1$. Lower panels (c) and (d) plot energies against hopping strength $t$. Insets focus on the lowest (breather) band, showing that two edge-localized states separate out [(b) and (d)] for three bosons but not [(a) and (c)] for two bosons. Here $t$, $\gamma$, and $E_{\nu }$ are expressed in the same arbitrary energy units.

Figure 2

(Color online) Spatial profile of site occupancies for several eigenstates of the three-boson Bose-Hubbard chain. Inset shows the same plot in semilogarithmic scale, the linear behavior indicating exponential localization in the edge states. Here $L=10$ and $\gamma =10t$.

Figure 3

(Color online) Size-dependence of $\text{max}⟨n_j⟩$ for (a) all eigenstates of the two-boson Bose-Hubbard chain and (b) several eigenstates of the three-boson Bose-Hubbard chain labeled by $\nu$. Here $\gamma =10t$, and $L$ goes from $L=5$ to $L=17$.

Figure 4

(Color online) Spectrum and density profiles at smaller interactions for three-boson system in ten sites $\left(t=1\right)$. Here $t$, $\gamma$, and $E_{\nu }$ are expressed in the same arbitrary units. (a) The bands merge at small enough $\gamma$. (b) For this system the breather band loses its identity around $\gamma \sim 1.8$. We follow the edge-localized states at smaller $\gamma$; shown surrounded by a thick-lined dashed polygon. At smaller $\gamma$, the $\nu =\left(L-1\right)$ and $\nu =L$ states are no longer the edge states. (c) Site occupancies at $\gamma =1.4$. The almost exponentially localized curves are for the edge states, and the two nonlocalized curves are the $\nu =\left(L-1\right)$ and $\nu =L$ states, which are no longer the edge-localized states. (d) Size-dependence of $\text{max}⟨n_j⟩$.

Figure 5

(Color online) Construction of the effective single-particle chain for the Bose-Hubbard model with (a) two bosons and (b) three bosons. The sequence of hops of the two bosons (gray circles), initially at the same lattice site, are shown with the corresponding values of the matrix element of the Bose-Hubbard Hamiltonian. The energies ${\epsilon }_1$ and ${\epsilon }_2$ of the states after each hop are also shown. The resulting effective single-particle chain is shown in the lower part of the figures.

Figure 6

(Color online) Energies from second-order perturbation theory (dashed and dotted lines) compared with full spectrum (solid lines), for $\gamma =10$ and $L=10$. Here $t$, $\gamma$, and $E_{\nu }$ are expressed in the same arbitrary units.

Figure 7

(Color online) (a) Energy spectrum of the Bose-Hubbard chain with four bosons, plotted against eigenvalue label. The inset zooms onto the breather band, while (b) zooms onto the $\left(2+2\right)$-boson band, as indicated by the dotted red arrows. Here $\gamma =10$, $t=1$, and $L=10$. (c) Spectrum of same chain as a function of the hopping parameter $t$. The insets focus, from left to right, on the four-boson bound state band, the $2+2$-boson band, and the $2+2$-boson bound state subband, showing the separation of edge-localized states out of the former and the latter. Here $t$, $\gamma$, and $E_{\nu }$ are expressed in the same arbitrary units.

Figure 8

(Color online) Profile of the site occupancy in the Bose-Hubbard chain with four bosons for four eigenstates in (a) the four-boson bound state band and (b) the $2+2$-boson bound state band. The same plots in semilogarithmic scale are shown in the insets, showing exponential localization. Here $L=10$ and $\gamma =10t$.

Figure 9

(Color online) Energy spectrum of the spinless fermion model, for two fermions in left panels (a) and (c) and three fermions in right panels (b) and (d). Here $L=15$, $V=10$. Also, $t=1$ [(a) and (b)] for the top figures. Insets focus on the (topmost) breather band. [(b) and (d)] The three-fermion cases have two states separating from this band; [(a) and (c)] the two-fermion cases do not have this feature. Here $t$, $\gamma$, and $E_{\nu }$ are expressed in the same arbitrary units.

Figure 10

(Color online) Spatial profile of fermion occupancies, for several eigenstates of the three-fermion model with $L=15$ and $V=10t$. Same plot in semilogarithmic scale is shown in the inset, indicating exponential localization.

Figure 11

(Color online) $1/L$-dependence of $\text{max}⟨n_j⟩$ for (a) all eigenstates of the two-fermion chain and (b) some eigenstates of the three-fermion chain. Here $V=10t$, and $L$ goes from $L=9$ to $L=21$. Here $s_3$ is the Hilbert space size.

Figure 12

(Color online) Dynamics of three fermions in ten sites, shown through the evolution of site occupancies $n_j$. The three different initial conditions are shown on top. In each case, to avoid clutter we split the ten $n_j$ into two groups of five. Thus the left top and left bottom describe two parts of the same evolving chain. Here $V=20$ and $t=1$.