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Published February 2023 | Published
Journal Article Open

Pivot Hamiltonians as generators of symmetry and entanglement

Abstract

It is well-known that symmetry-protected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finite-depth unitary operator U. Here, we consider obtaining the entangler from a local 'pivot' Hamiltonian H_(piv) such that U = e^(iπH_(piv)). This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions which we explore here. (i) Since SPT Hamiltonians and entanglers are now on the same footing, can we iterate this process to create other interesting states? (ii) Since entanglers are known to arise as discrete symmetries at SPT transitions, under what conditions can this be enhanced to U(1) 'pivot' symmetry generated by H_(piv)? In this work we explore both of these questions. With regard to the first, we give examples of a rich web of dualities obtained by iteratively using an SPT model as a pivot to generate the next one. For the second question, we derive a simple criterion guaranteeing that the direct interpolation between the trivial and SPT Hamiltonian has a U(1) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for H_(piv), including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U(1) pivot symmetry is that it shares a mutual anomaly with the symmetry protecting the nearby SPT phase. We discuss how such anomalous and non-onsite U(1) symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixed-point toric code state.

Additional Information

© 2023 N. Tantivasadakarn et al. This work is licensed under the Creative Commons Attribution 4.0 International License. Published by the SciPost Foundation. We thank Shu-Heng Shao for helpful discussions. NT is supported by NSERC. RV and AV are supported by the Simons Collaboration on Ultra-Quantum Matter, which is a grant from the Simons Foundation (651440, A.V.). RV is supported by the Harvard Quantum Initiative Postdoctoral Fellowship in Science and Engineering.

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Additional details

Created:
August 22, 2023
Modified:
October 25, 2023