We analyze density wave and Pomeranchuk orders in twisted bilayer graphene. This complements our earlier analysis of the pairing instabilities. We assume that near half filling of either conduction or valence band, the Fermi level is close to Van Hove points, where the density of states diverges, and study potential instabilities in the particle-hole channel within a patch model with two valley degrees of freedom. The hexagonal symmetry of twisted bilayer graphene allows for either six or twelve Van Hove points. We consider both cases and find the same two leading candidates for particle-hole order. One is an SU(2)-breaking spin state with ferromagnetism within a valley. A subleading intervalley hopping induces antiferromagnetism between the valleys. The same state has also been obtained in strong-coupling approaches, indicating that this order is robust. The other is a mixed state with 120∘ complex spin order and orthogonal complex charge order. In addition, we find a weaker but still attractive interaction in nematic channels, and discuss the type of a nematic order.
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We thank V. Andreeva, E. Andrei, M. Christensen, R. Fernandes, L. Fu, D. Goldhaber-Gordon, P. Jarillo-Herrero, J. Kang, A. Klein, J. Schmalian, D. Shaffer, O. Vafek, and A. Vishwanath for fruitful discussions. The work was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0014402. D.C. gratefully acknowledges support from the Allen M. Goldman Fellowship of the University of Minnesota. L.C. was supported by the Humboldt Foundation, and work at BNL is supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No. DE-SC0012704.