Benoit Douçot
LPTHE, Sorbonne University and CNRS, Paris
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Spatial patterns of spin-valley entanglement in skyrmion crystals
Two-dimensional electron gases under a strong magnetic field have tremendously expanded our understanding of many-body physics, with the discovery of integer and fractional quantum Hall effects, together with chiral edge states, fractional excitations, and anyons. Another striking effect is the strong coupling between charge and spin and valley degrees of freedom, which takes place near integer filling of the magnetic Landau levels. More precisely, because of the large energy gap associated to cyclotron motion, any slow spatial variation of the spin background induces a variation of the electronic density proportional to the topological density of the spin background. Minimizing Coulomb energy leads to an exotic class of two-dimensional crystals, which exhibit a periodic non-collinear spin texture called a skyrmion lattice. Recent tunnelling experiments have provided striking images of valley skyrmions bound to charged defects in a graphene layer.
We have analysed the spatial structure of spin and valley entanglement in a regime where the crystalline charge density pattern is fixed by electrostatic interactions. In the limit where we neglect anisotropic couplings in the 4-dimensional spin and valley internal space, these skyrmion lattices spontaneously break the underlying SU(4) symmetry. Using a variational approach in a space of holomorphic functions, we have shown that minimizing various anisotropic contributions (such as the Zeeman term acting on the spin and the difference between in plane and out of plane exchange energies in the valley sector), while keeping a fixed charge density, forces these skyrmion crystals to develop regions with maximal entanglement between spin and valley degrees of freedom. We have found that these spatial entanglement patterns often break the point group symmetry of the charge density crystal, leading to entanglement smectic and stripe phases. I will suggest that magnon scattering across such crystals could provide a way to experimentally probe these interesting phases.