Quantum Geometry Driven Crystallization
Happy New Year!
Why VMC?
This model is in continuum so DMRG as well as other lattice-based methods are not applicable. For hall like problem, which breaks time-reversal symmetry, QMC suffers from the sign problem. Therefore VMC is a natural choice.
For any reason, Hall is extremmely hard!
-jellium model
Definition 1 ( -jellium model) -jellium model is a simple two-band generalization of the jellium model,
where is a large and positive constsant,
Fast check of hermicity of
since is hermitian operator.
Construction of -jellium
We start from the Dirac hamiltonian and add a mass term to open a gap.
I'm so naive so step by step
Massless dirac dispersion
In -space
Diagonalize it
Making a linear dispersion.
You could imagine adding a uniform mass term that gaps the dirac point out and creates two separate quadratic bands.
Flat band
Problem. Why do they say this flat band has identical interaction to plain jellium?
answer: because the single particle wavefunction is not changed with jellium kinetic term.
Adding a non-uniform mass term gives
gives
Thus
where
The band gap is tuned by , we would focus on the lower flat band and its topology from now on.
The wavefunction is
Adding the jellium term back we have
which gives the same jellium dispersion and same wavefunction.
Remark
Take-away lesson: Topology lies in the wavefunction which is kinetical part, not in the dynamical dispersion part.
This dispersion matching is essential to convince that the wigner crystallization is driven by quantum geometry not kinetical difference.
Berry phase of this single particle hamiltonian
Would be pretty straightforward for hamiltonian.
In this system
Thus we have
Theorem 2 For two-band model, the Berry curvature is given by
Thus
which peaks at and decays as for large .
That's very much the topological details in -jellium model.
Problem. But, why would berry curvature concentration drive crystallization?
Problem. I'm also interested in why there is a Quantum Critical Point.
Comparison to Landau level
Landau level has flat berry curvature distribution.
jellium model has a localized berry curvature distribution in space.