Paired Parton Trial States for the Superfluid-Fractional Chern Insulator Transition

Abstract

We consider a model of hard-core bosons on a lattice, half-filling a Chern band such that the system has a continuous transition between a fractional Chern insulator (FCI) and a superfluid state (SF) depending on the bandwidth to bandspacing ratio. We construct a parton-inspired trial wavefunction ansatz for the ground states that has remarkably high overlap with exact diagonalization in both phases and throughout the phase transition. Our ansatz is stable to adding some bosonic interactions beyond the on-site hard core constraint. We confirm that the transition is well described by a projective translation symmetry-protected multiple parton band gap closure, as has been previously predicted. However, unlike prior work, we find that our wavefunctions require anomalous (BCS-like) parton correlations to describe the phase transition and SF phase accurately.

Figures (7)

• Figure 1: Comparison of both paired (Pf) and uncorrelated (det prod.) parton wavefunction ansatz from VMC against exact diagonalization (ED) in a 9-boson system (total Hilbert space dimension $10^8$) as we move from a FCI phase to SF (shaded) by increasing bandwidth. (Top) Overlap of exact ground states with our paired ansatz for the $\mathbf{Q}=(0,0)$ state which is the lowest energy branch. (Bottom) energies of trial states versus ED. Both trial ground states are shown. Deep into the FCI phase these two are near degenerate, and the Det prod. wavefunction is also accurate. However, going into the SF phase, the two paired wavefunctions split with the $\mathbf{Q}=(\pi,0)$ and the Det. Prod. wavefunctions going up in energy. Note that the gap seen on the SF side is a finite-size effect, while the FCI gap remains finite for large systems. The horizontal axis does not extend down to zero bandwidth as we would need longer ranged hopping to achieve this.
• Figure 2: Energy and overlap comparison of our paired Pf VMC ansatz to exact diagonalization for a finite nearest neighbor repulsion $V_1$ in the FCI phase at Bandwidth 0.09, showing good overlap for both ground states for weak interaction. The overlap becomes worse as $V_1$ approaches unity, the scale of the hopping parameters and band gap.
• Figure S1: Hopping amplitudes for the checkerboard (a) and hexagonal (b) models. In (a), the curved lines appear between all diagonally separated pairs of like-colored points but are not shown for clarity. Similarly, in (b) both the $t'$ and $t"$ amplitudes appear in each plaquette.
• Figure S2: Comparison of the energies obtained from a short-range $U(1)$ invariant effective mean-field Hamiltonian (Eq. \ref{['eq:constrained_hmf']}) compared to the (strictly more general) unconstrained Pfaffian ansatz and exact diagonalization. The local $U(1)$ mean-field Hamiltonian correctly reproduces most of the energy variation, despite only short-range hopping and on-site pairing being allowed. The top shows the strength of the pairing $|\Delta_{12}^\text{MF}|^2$ in the effective parton mean-field Hamiltonian. The scale is chosen such that $t_{11,22}^\text{MF}$ have average magnitude unity -- this indicates that $|\Delta|\sim0.7|t|$ in the SF. The transition is not sharply defined, which we believe to be a finite system-size effect. The red dotted line indicates the expected generic behavior in the large-system thermodyanamic (TD) limit. The behaviour of our ansatz in the TD limit is analysed more carefully in Fig. \ref{['fig:tdlimscaling']}.
• Figure S3: The dependence of the bosonic band width of the FCI-SF transition in the checkerboard model (determined by the change in parton Chern numbers) on the number of bosons $N_b$ in the system for the paired and unpaired ansatz. Extrapolation to the thermodynamic limit gives $w_\text{unpaired}=0.69\pm0.05$ and $w_\text{paired}=0.58\pm0.04$. This analysis implies a difference $\delta w=w_\text{unpaired}-w_\text{paired}=0.11\pm0.06$ for large systems, demonstrating that mean-field parton pairing has a meaningful influence on the nature of the transition even in the thermodynamic limit.