Snapshot-based detection of $\nu =\frac{1}{2}$ Laughlin states: Coupled chains and central charge

Experimental realizations of topologically ordered states of matter, such as fractional quantum Hall states, with cold atoms are now within reach. In particular, optical lattices provide a promising platform for the realization and characterization of such states, where novel detection schemes enable an unprecedented microscopic understanding. Here we show that the central charge can be directly measured in current cold atom experiments using the number entropy as a proxy for the entanglement entropy. We perform density-matrix renormalization-group simulations of Hubbard-interacting bosons on coupled chains subject to a magnetic field with $\alpha =1/4$ flux quanta per plaquette. Tuning the interchain hopping, we find a transition from a trivial quasi-one-dimensional phase to the topologically ordered Laughlin state at magnetic filling factor $\nu =1/2$ for systems of three or more chains. We resolve the transition using the central charge, on-site correlations, momentum distributions, and the many-body Chern number. Additionally, we propose a scheme to experimentally estimate the central charge from Fock basis snapshots. The model studied here is experimentally realizable with existing cold atom techniques and the proposed observables pave the way for the detection and classification of a larger class of interacting topological states of matter.

Figure 1

Central charge as obtained from the bipartite entanglement entropy $S\left(x\right)$ for different system sizes after extrapolation to $L_x\to \infty$. Faded dotted lines indicate values for finite $L_x$ with longer systems being less faded. We find a clear change of behavior around $t_y/t_x\approx 0.6$. Above this critical value, the central charge is in agreement with the prediction $c_{\mathrm{LN}}=1$ for the Laughlin state. The sketches below the main panel illustrate the origin of the different behavior in the decoupled ($t_y/t_x=0$) and the isotropic ($t_y/t_x=1$) limit by only showing gapless chiral modes.

Figure 2

(a) Number entropy $S_n\left(x\right)$ from 6000 (2000) snapshots of a single chain (three decoupled chains) of length $L_x=61$. While the number entropy provides an accurate proxy for the entanglement entropy $S(L_y=1)$ of a single chain, it is not additive in the number of chains. (b) In the isotropic limit, $t_y/t_x=1$, the proxy $S_n\left(x\right)$ from 2000 snapshots of the three-leg system is relatively accurate. (c) Central charges for three-leg systems extracted from the number entropy $S_n\left(x\right)$ (solid lines) compared to the prediction from the entanglement entropy $S\left(x\right)$ (faded, dotted lines).

Figure 3

On-site correlations $g^{\left(2\right)}\left(0\right)$ for (a) $L_y=3$ and (b) $L_y=5$ chains. The transition region at intermediate $t_y/t_x$ is clearly visible by a maximum of $g^{\left(2\right)}\left(0\right)$. (c) Momentum distribution $n_y\left(k_x\right)$ for $L_y=5,L_x=11,U/t_x=5.0$. We observe the emergence of a chiral mode in the outermost chains ($y=1,5$) around $t_y/t_x\approx 0.6$. In these chains, the momentum distribution is peaked around a finite momentum $k_x\ne 0$. In the remaining bulk chains, the momentum distribution is peaked around $k_x=0$ at all $t_y/t_x$.

Figure 4

Many-body Chern number as a function of $t_y/t_x$ for a cylinder of size $L_x\times L_y=25\times 3$. We find a transition towards a topologically nontrivial phase close to the isotropic limit.