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

• URL: http://arxiv.org/abs/2403.09951v1
• Date: Fri, 15 Mar 2024 01:22:49 GMT
• Title: SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States
• Authors: Michael Ragone,
• Abstract summary: This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems. We present new results describing their ground state structure and, when $n$ is even, their peculiar $O(n)$-to-$SO(n)$ symmetry breaking. 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 $G$.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $H^2(SO(n),U(1))$, 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.

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