Adiabatic preparation of fractional Chern insulators from an effective thin-torus limit

We explore the quasi-one-dimensional (thin torus, or TT) limit of fractional Chern insulators (FCIs) as a starting point for their adiabatic preparation in quantum simulators. Our approach is based on tuning the hopping amplitude in one direction as an experimentally amenable knob to dynamically change the effective aspect ratio of the system. Similar to the TT limit of fractional quantum Hall systems in the continuum, we find that the hopping-induced TT limit adiabatically connects the FCI state to a trivial charge density wave (CDW) ground state. This adiabatic path may be harnessed for state preparation schemes relying on the initialization of a CDW state followed by the adiabatic decrease of a hopping anisotropy. Our findings are based on the calculation of the excitation gap in a number of FCI models, both on a lattice and consisting of coupled wires. By analytical calculation of the gap in the limit of strongly anisotropic hopping, we show that its scaling is compatible with the preparation of large-size FCIs for sufficiently large hopping anisotropy, where the amenable system sizes are only limited by the maximal hopping amplitude. Our numerical simulations in the framework of exact diagonalization explore the full anisotropy range to corroborate these results.

Figure 1

Illustration of the adiabatic path from an effectively one-dimensional charge density wave state (top) to a two-dimensional fractional Chern insulator state (bottom) in an array of quantum wires coupled by a tunable hopping $J_x(t,y)=J\left(t\right)e^{i\varphi y}$ (see Sec. 3b). In particular, a $\nu =\frac{1}{2}$ Laughlin phase may adiabatically form in the presence of contact interactions upon reducing $J$ from a large value towards a moderate value $J/E_{\mathrm{R}}\approx 1$. The magnetic recoil energy $E_{\mathrm{R}}$ and $J\left(t\right)$ represent the kinetic energy scale along the continuous and discrete directions, respectively.

Figure 2

Qualitative phase diagram for the coupled wire model [Eqs. (3) and (5)]. Tuning the amplitude $J$ of the interwire coupling between moderate values comparable to the magnetic recoil energy $E_{\mathrm{R}}$ and very large values $J\gg E_{\mathrm{R}}$ induces a phase transition between a topological FCI phase (red) and a trivial CDW (gray). $J$ and $E_{\mathrm{R}}$ represent the kinetic energy scale along the $x$ and $y$ directions, respectively. The Harper-Hofstadter model [Eqs. (8) and (9)] yields a similar phase diagram for certain geometries, where the hoppings $J_x$ and $J_y$ take the roles of $J$ and $E_{\mathrm{R}}$, respectively.

Figure 3

ED data for the coupled wire model as described by Eqs. (3) and (5) with $N_x=6, N_y=3, p=9$ particles, and interaction strength $U=E_{\mathrm{R}}{l}_B$. All energies are measured in units of the magnetic recoil energy $E_{\mathrm{R}}$. Top panel: Gaps ${\mathrm{\Delta }}_{1,2}$ and ${\mathrm{\Delta }}_{2,3}$ between the first and second eigenstates and between the second and third eigenstates, respectively, as a function of the interwire coupling $J$ along with the perturbative prediction ${\mathrm{\Delta }}^{\mathrm{CW}}$ [cf. Eq. (7)] for the excitation gap. The zoom-ins show the low-energy spectrum at $J=E_{\mathrm{R}}$ and $J=19E_{\mathrm{R}}$, respectively. Bottom panel: Particle entanglement spectrum of each twofold-degenerate ground state, as obtained by tracing out $N_{\mathrm{B}}=5$ particles. The colors indicate the number of states expected from Eqs. (20) and (21) for the FCI and CDW phases. The point at which the transition to the CDW is complete is marked by a red line in both plots, which corresponds to the critical value ${\tau }_{\mathrm{TT}}=ln\left(5\right)$ of the dimensionless anisotropy parameter.

Figure 4

The excitation gap ${\mathrm{\Delta }}_{2,3}$ above the twofold-degenerate ground state of the coupled wire model [Eqs. (3) and (5)] showing a collapse of different system sizes as a function of the dimensionless anisotropy parameter $\tau =2\sqrt{J/E_{\mathrm{R}}}{(\pi /N_x)}^2$. The legend entries indicate the system size as ${\mathrm{\Delta }}_{2,3}(N_x,N_y)$, and the particle number is $p=\frac{N_x{N}_y}{2}$. All data points fall onto the same curve and are almost undistinguishable, except for the smallest system size (red points, $p=6$ bosons). For comparison, the analytical prediction ${\mathrm{\Delta }}^{\mathrm{CW}}$ [Eq. (7)] is shown as a continuous line. The value ${\tau }_{\mathrm{TT}}=ln\left(5\right)$, which roughly marks the CDW transition for all studied system sizes, is indicated as a red line.

Figure 5

ED data for the interacting HH model at half filling described by Eqs. (8) and (9) with $\varphi =\frac{1}{18}, p=9$ particles, and a system size of $N_x=\frac{1}{\varphi }, N_y=1$ unit cells. Top panel: Gaps between the first and second eigenstates and between the second and third eigenstates as a function of the hopping ratio $\frac{J_y}{J_x}$ along with the perturbative prediction ${\mathrm{\Delta }}^{\mathrm{HH}}$ [cf. Eq. (14)] for the excitation gap. Energy is measured in units of $U$. Bottom panel: Particle entanglement spectrum of the twofold-degenerate ground state obtained by tracing out $N_{\mathrm{B}}=5$ particles. The colors indicate the number of states expected from Eqs. (20) and (21) for the FCI and CDW phases.

Figure 6

ED data for the interacting HH model at half filling described by Eqs. (8) and (9) with $\varphi =\frac{1}{6}, p=4$ particles, and a system size of $N_x=8, N_y=1$ sites. Top panel: Gaps between the first and second eigenstates and between the second and third eigenstates as a function of the hopping ratio $\frac{J_y}{J_x}$. The energy is measured in units of $U$. Bottom panel: Particle entanglement spectrum of the twofold-degenerate ground state obtained by tracing out $N_{\mathrm{B}}=2$ particles. The colors indicate the number of states expected from Eqs. (20) and (21) for the FCI and CDW phases.

Figure 7

ED data for the anisotropic KM model at half filling as described in Sec. 3d with $\varphi =\frac{1}{2}, p=4$ particles, a system size of $N_x=4, N_y=4$ sites, and hard-core interaction. The anisotropy parameter ${\alpha }^2$ is proportional to the geometric aspect ratio, such that the TT limit ($\alpha \ll 1$) appears on the left-hand side of the figure while the isotropic limit ($\alpha =1$) appears on the right-hand side of the figure. Top panel: Ground-state degeneracy lifting ${\mathrm{\Delta }}_{1,2}$ and excitation gap ${\mathrm{\Delta }}_{2,3}$. The energy is measured in units of $t$ [cf. Eq. (16)]. Bottom panel: Particle entanglement spectrum of the twofold-degenerate ground state obtained by tracing out $N_{\mathrm{B}}=2$ particles. The colors indicate the number of states expected from Eqs. (20) and (21) for the FCI and CDW phases.

Figure 8

Single-particle gap above the lowest band of the KM model for flux $\mathrm{\Phi }=\frac{1}{2}$ and system sizes of (a) $4\times 4$ sites and (b) $8\times 8$ sites with logarithmic scale.