Mathematics Colloquia and Seminars

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Two gapped Hamiltonians are said to exhibit the same quantum phase of matter iff there exists a path of gapped Hamiltonians connecting them. The topological phases of matter in 2D are believed to be classified by a modular tensor category and a number called the chiral central charge.  We remain a long way from proving this, but I will explain some related progress.  First we ask, (1) which gapped ground states should be considered "topological," and (2) how do we assign a tensor category to these states?  Here we use a variant of the entanglement bootstrap program.  Then I will sketch an argument that two gapped ground states with the same tensor category can always be connected by a constant-depth circuit, up to stacking with an invertible state.  These claims are not rigorous for states with nonzero correlation length; time permitting, I will mention ongoing work about "approximate algebras" that seeks to address this.