What does the solid phase in a two-dimensional system with Lennard-Jones potential look like? Consider a system of two dimensional particles interacting via Lennard-Jones pair potential:
$$u(r) = 4[(\frac{1}{r^{12}})-(\frac{1}{r^{6}})]$$ where r is the distance between two particles.
What does the solid phase** look like? Is it a lattice of triangles? Of squares? Is there more than one solid phase? At what pressure/density/temperature do we get a solid phase?
** When I say solid phase I mean a soft solid phase, something that resembles a crystal. 
 A: You cannot have a solid phase in two dimensions, as the latter would break translational symmetry and thus violate the Mermin-Wagner theorem.
Rigorous proofs of this fact can be found in the following papers:


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*On the absence of spontaneous symmetry breaking and of crystalline ordering in two-dimensional systems, J. Fröhlich and C. Pfister, Comm. Math. Phys. 81(2) (1981), 277-298.

*Translation-Invariance of Two-Dimensional Gibbsian Point Processes, T. Richthammer, Comm. Math. Phys. 274(1) (2007), 81-122.


Both papers deal with very general interactions (not only Lennard-Jones); the second one even allow for additional hard-core interactions (which are tricky to deal with, from a Mermin-Wagner perspective).
Of course, you might then inquire what ground-states look like, but the latter would not be stable at positive temperatures. Rigorous results about the ground states of systems with a class of interactions resembling Lennard-Jones can be found in the paper


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*A Proof of Crystallization in Two Dimensions, F. Theil, Comm. Math. Phys. 262(1) (2006), 209-236.


It is proved there that, under suitable assumptions on the boundary conditions, the ground state indeed forms a triangular lattice. Even for ground-states in two dimensions, the problem seems still not fully understood. There is also the following nice (and recent) review paper on this topic:


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*The crystallization conjecture: a review,  X. Blanc and M. Lewin, EMS Surveys in Mathematical Sciences 2(2) (2015), 255-306.

