# P6: Large scale simulation of edge magnetism

## Project Description

In graphene, the effect of short-ranged electronic interactions (such as the Hubbard interaction) is suppressed due to the vanishing density of states at the charge-neutrality point (Dirac point)^{(1)}. This is not true, however, if graphene's perfect honeycomb structure is disturbed; at certain edges or lattice defects one finds localized states right at the Fermi level, which, at charge neutrality, are half-filled. Naturally, in such a situation the electron-electron interaction becomes very important. The most striking consequence of this scenario is a spatially localized one-dimensional magnetism, known as edge magnetism^{(2)-(4)}. Most of the literature on edge magnetism, however, is based on mean-field methods (Hartree-Fock or density functional theory)^{(3),(5)-(8)}, or is restricted to relatively small lattices^{(9)-(12)}. In this project we will study edge magnetism with methods that allow us to go beyond these restrictions.

The first part of this project aims at improving the recently-developed effective Heisenberg theory for correlations at graphene edges^{(13),(14)}. The fundamental principles on the basis of which this effective theory is derived are similar to what was described in the project P5. However, the present project is especially targeted to realistic system sizes with millions of lattice sites and lattice defects at the edges and/or in the bulk. The first step in the derivation of the

effective theory is the identification of its optimal basis. For this, one needs to separate the low- and high-energy regime of the hopping Hamiltonian, which will be done with a Lanczos algorithm, supplemented by a heuristic predictor for the initial wave functions. The latter is required in order to facilitate the convergence of the Lanczos algorithm to the sparsely distributed localized wave functions. Once the low-energy subspace is identified, the maximally localized Wannier states must be found within this subspace. Due to the size and the missing translational invariance of the lattice, these two steps will be numerically demanding. The lowest order of the effective Heisenberg theory can be derived directly from the Wannier states: each localized Wannier state hosts essentially one electron, the spin of which is the relevant degree of freedom. In addition to this, we will calculate the corrections to the effective theory due to the high energy states in 2nd order in perturbation theory by systematic Schrieffer-Wolff transformations^{(15)}.

Finally, the effective Heisenberg theory will be solved with QMC simulations. Due to the expected bipartite nature of the effective theory (this is a direct consequence of the bipartite honeycomb lattice), which ensures the absence of frustrations, there is no sign problem in QMC simulations based on the stochastic series expansion (SSE) method^{(16)}. With this method we expect to be able to solve effective theories with thousands of individual spins with arbitrary (long-range) interactions. Thus, the combination of effective theories and SSE-QMC will allow us to study graphene

flakes with realistic sizes.

There is a strong connection to the project P5 where more general effective edge state theories that are not accessible in QMC are derived and studied. Furthermore, there are overlaps with projects P1 and P2.

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