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

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.

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:

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

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

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!

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.

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

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  

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.

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)

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

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.

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