1. A. Ambrosetti, F. Pederiva and E. Lipparini
Quantum Monte Carlo study of two dimensional electron gas in presence of Rashba interaction
A. Ambrosetti, F. Pederiva and E. Lipparini

2. The Rashba Interaction
Rashba interaction has been proved to exist in semiconductor heterostructures, where electrons are subject to a quantum well confinement and therefore move in a 2D space (plane).
It is a spin-orbit-like interaction, coupling momentum with spin.
It can be tuned in strenght through gate voltage.

3. The Rashba Interaction
Due to the well asimmetry, electrons are subject to an electric field perpendicular to their plane of motion.
This causes electrons to sense an in-plane effective magnetic field because of relativistic effects
The electron spin couples to the magnetic field giving rise to the Rashba interaction:

4. Switching off Coulomb
In absence of Coulomb interaction the problem is exactly solvable
What we get is two different eigenstates for each wavevector k, consisting of different k-dependent spin states with two different energies
This generates two energy
bands, giving “quasi up –
quasi down” spin
polarization

5. Switching Coulomb on
When Coulomb interaction is introduced the solution to this problem is not known analytically
We need to use a numerical approach.
Diffusion Monte Carlo (DMC) is our method of choice:
Widely used for electrons
Very accurate
We know how to treat SO interactions

6. HOW DOES DMC WORK? All excited states will be multiplied by the factor
Take an initial wave function
Make it evolve in imaginary time
Expand over the Hamiltonian eigenstates:
Multiply by
where = ground state energy:
Let go to infinity
All excited states will be multiplied by the factor
Projection over the ground state is obtained!

7. DMC algorithm
Suppose that our Hamiltonian contains only a kinetic term
The solution is given by
where we used the free particle Green’s function
In terms of walkers, free propagation
means generating displacements
-> DIFFUSION.

8. DMC algorithm
Now suppose we have a kinetic term plus a central potential
From Trotter’s formula  
Take into account then the effect of the interaction term over the “renormalized” wavefunction
This can be seen as a weight,
i.e. the probability for the walker
in R to survive after a time

9. Implementation of DMC
A possibile implementation of the projection algorithm is:
Generate initial walkers distribution according to
Diffuse walkers due to free propagation
Kill or multiply walkers due to weight
Repeat steps until convergence is achieved

10. Spin-Orbit propagator
For small time steps
The idea is using Coulomb potential as a weight, and applying the Rashba term right after the free propagator
We can thus rewrite this as
Which means we will need to sample displacements with the free propagator, and then rotate spins according to the just sampled.

11. Checking DMC with SO propagator
In absence of Coulomb potential the problem is analytically solvable.
The exact ground state solution is a slater determinant of plane waves.
Modify it multiplying by a jastrow factor:
DMC must be able to project over ground state (red dot)

12. RESULTS
Ground state energies are shown at constant density for different values of Rashba strenght.
The minimum is shifted when interaction strenght increases

13. RESULTS
Hartree-Fock energy is known analytically, like the energy in absence of Coulomb interaction. In such cases solutions are made of plane waves Slater determinants.

14. CONCLUSIONS
We developed a functioning algorithm based on previous work in nuclear physics, using spin-orbit propagation
We have made some tests on method and trial wavefunction
We are calculating the equation of state for the 2D electron gas in presence of Rashba interaction
We expect to use this method for further research on other systems in presence of spin and momentum dependent interactions