Realizing and adiabatically preparing bosonic integer and fractional quantum Hall states in optical lattices

We study the ground states of two-dimensional lattice bosons in an artificial gauge field. Using state-of-the-art density matrix renormalization group (DMRG) simulations we obtain the zero-temperature phase diagram for hard-core bosons at densities $n_b$ with flux $n_{\varphi }$ per unit cell, which determines a filling $\nu =n_b/n_{\varphi }$. We find the bosonic Jain sequence $[\nu =p/(p+1\left)\right]$ states, in particular, a bosonic integer quantum Hall phase at $\nu =2$, are fairly robust in the hard-core boson limit, In addition to identifying Hamiltonians whose ground states realize these phases, we discuss their preparation, beginning from independent chains, and ramping up interchain couplings. Using time-dependent DMRG simulations, these are shown to reliably produce states close to the ground state for experimentally relevant system sizes. Our proposal only utilizes existing experimental capabilities.

Figure 1

Harper-Hofstadter model on (a) a square lattice with flux $\varphi =2\pi n_{\varphi }$ on each square plaquette, and (b) a triangular lattice with flux $n_{\varphi }/2$ on each triangle plaquette.

Figure 2

Numerical diagnosis of quantum Hall states $\nu =2,1/2,2/3$; $L_c$ is the circumference. (a) Quantized Hall conductance measured from flux insertion on an infinite cylinder. Charge transfer as a function of flux: $\langle Q\rangle =-{\sigma }_{xy}\frac{\theta }{2\pi }$. (b) The correlation length $\xi$ of the quantum Hall state vs bond dimension $m$ in DMRG simulations showing convergence. The truncation error of DMRG simulation is around ${10}^{-8}–{10}^{-10}$. The correlation length is small compared with the circumference, indicating that our DMRG simulation is reliably producing 2D physics.

Figure 3

Nonequilibrium dynamics simulation of the preparation scheme with different ramp times $T$, and we define $J=J_x$. We show the results of the BIQH state at $n_{\varphi }=1/6,n_b=1/3$ of a square lattice placed on both the $L_c=6$ infinite cylinder (a), (c) and the $6\times 6$ square geometry (b), (d). (a), (b) show the time evolution of entanglement entropy, where the solid line represents the entanglement entropy of the ground state vs $J_y/J_x$, and the dots represent the entanglement entropy from the time evolution. (c),(d) shows the wave-function overlap per site between the ground state $\psi \left(J_y\right)$ and the state from time evolution $\tilde{\psi }\left[J_y\left(t\right)\right]$. Intriguingly, $TJ=20$ seems to give a better ground state than $TJ=100$; the physical reason is unclear.

Figure 4

Diagnosis of the quantum Hall state by measuring the two-point correlation function $|\langle a_0^{†}\left(x\right)a_y\left(x\right)\rangle |$. (a) The cartoon picture for a quantum Hall state on a finite $L_x\times L_y$ cluster. For a quantum Hall state, $x=0,1$ corresponds to the edge on which the correlation function decays algebraically. $x>1$ corresponds to the bulk where the correlation function decays exponentially, however, if $a_y\left(x\right)$ hits the edge $\left(y\sim L_y\right)$, the correlation function obeys the power law. Numerical results: (b) $9\times 12$ cluster, $n_{\varphi }=1/6,n_b=1/3,\nu =2$ quantum Hall state. (c) $6\times 6,n_{\varphi }=1/6,n_b=1/3,\nu =2$ quantum Hall state. (d) $6\times 6,n_{\varphi }=0,n_b=1/3$, superfluid. (e) $6\times 6,n_{\varphi }=1/6,n_b=1/2$, staggered potential $\mathrm{\Delta }=2$, charge density wave.