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Explore homotopical aspects of C*-algebraic quantum spin systems in this 58-minute conference talk from Harvard CMSA's Workshop on Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras. Discover a homotopy theoretic framework for studying state spaces of quantum lattice spin systems using C*-algebraic quantum mechanics language, developed through collaborative research with Beaudry, Hermele, Moreno, Qi and Spiegel. Learn about old and new results concerning the state space of quasi-local algebras in quantum lattice spin systems when equipped with natural metric topology or weak* topology. Examine the determination of homotopy groups of unitary groups in UHF algebras and understand why the pure state space of any UHF algebra in weak* topology is weakly contractible. Investigate non-commutative tori as examples demonstrating how state spaces with weak* topology can have non-trivial homotopy types that are neither deformation nor Morita invariant in non-commutative C*-algebras. Gain insights into how these tools, combined with higher homotopy theory methods such as E_infinity spaces, may provide a framework for constructing Kitaev's loop-spectrum of bosonic invertible gapped phases of matter.
