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.
We prove a trade-off theorem for order and disorder parameters in one-dimensional quantum spin systems with quenched disorder. For a disordered ensemble with exact Ising symmetry and average translation symmetry, any gapped ensemble must have one and only one of the following: an
O(1) order parameter or an
O(1) disorder parameter with even parity, both of the Edwards-Anderson type. The result extends to nearly gapped ensembles that accommodate Griffiths-type rare-region effects. These results offer a powerful and rigorous framework to understand the disorder effects beyond perturbative approaches. As applications, we (1) establish the existence of string order parameters for SPT phases; (2) derive a Lieb-Schultz-Mattis-type constraint for disordered ensembles, which requires a nearly gapped ensemble to spontaneously break the symmetry; and (3) discuss similar trade-off relations for disordered fermion chains, leading to an improved understanding of certain "intrinsically disordered" topological phases.
We show that disordered flat phases in crystal surfaces are equivalent to valence-bond-type phases in integer and half-integer spin quantum chains. In the quantum spin representation the disordered flat phase represents a fluid-type phase with long-range antiferromagnetic spin order. This order is stabilized dynamically by the hopping of the particles and short-range spin-exchange interactions. The mass of N\'eel solitons is finite. Numerical finite-size-scaling results confirm this. We identify the order parameter of the valence-bond phase. The Haldane conjecture suggests a fundamental difference between half-integer and integer antiferromagnetic Heisenberg spin chains. We find that disordered flat phases are realized in both cases, have exactly the same type of long-range antiferromagnetic spin order, and are stabilized by exactly the same mechanism. They differ only in the mathematical formulation of broken symmetry in the spin representation. We suggest experimental methods of observing disordered flat phases in crystal surfaces.