Cumulative Geometric Frustration and the Intrinsic Approach to Physical Assemblies

Geometric frustration arises whenever the constituents of a physical assembly locally favor an arrangement that is incompatible with the geometry or topology of the space in which it resides.  It naturally arises in a variety of fields ranging from macromolecular assemblies to liquid-crystals to spin models, yet in distinct systems, geometric frustration may be associated with different phenomena. For example, in liquid crystals frustration may lead to spontaneous size limitation and to a unique ground state whose energy grows super-extensively, while for the Ising antiferromagnet on triangular lattice frustration leads to a highly degenerate ground state of extensive energy.

In this talk, I will discuss how the intrinsic approach, in which matter is described only through local properties available to an observer residing within the material, overcomes the lack of a stress-free rest state and leads to a general framework in which the geometric compatibility conditions assume a central role. This framework, in particular, allows predicting the super-extensive energy exponent for sufficiently small systems and explains the origin of the large variety of phenomena attributed to geometric frustration. I will discuss its application to several specific systems exhibiting geometric frustration including growing elastic bodies, frustrated liquid crystals, and frustrated spin systems. Time permitting, I will discuss how the discretization of the degrees of freedom affects a system’s response to geometric frustration. 



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