Browsing by Subject "Representation Theory"

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Weak Order and Disorder of Quantum States Invariant Under Onsite Action by an Arbitrary Finite Group on a Spin Chain
(2019-12-13) Zelenko, Maxim; Freed, Daniel S.
In his paper “Constraints on order and disorder parameters in quantum spin chains,” Michael Levin derived general constraints on order and disorder parameters in Ising symmetric quantum spin chains. Levin’s main result in his paper was a theorem showing that in a circular spin chain, any local Hamiltonian that has a non-degenerate ground state and is translationally invariant and Ising symmetric must at least have either a nonzero order parameter or a nonzero disorder parameter. In the process of proving the theorem, he also proved a lemma that made a general statement about correlation and symmetry defect properties of any state in a circular spin chain. These properties namely had to do with notions of order and disorder that are weaker than long-range order and disorder. In this thesis, I prove an extension of the lemma to any simultaneous eigenstate of an onsite representation of an arbitrary finite group. Based on this generalization, we discuss some possible implications regarding how Levin’s theorem could be generalized to arbitrary finite symmetries as well. It is important to note that my generalization only applies to representations with one-dimensional invariant subspaces. While this condition is satisfied by any representation of an abelian group, it only holds for a subset of non-abelian group representations.