SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States

This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``SO (n) AKLT chains'' and the phase they occupy the ``SO (n) Haldane phase''. We present new results describing their ground state structure and, when n is even, their peculiar O (n) -to-SO (n) symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing n, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group G to an SPT index for a compact Lie group G. We then compute this index, which takes values in the second Borel group cohomology H² (SO (n), U (1) ), at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.