Pivot Hamiltonians as generators of symmetry and entanglement
Abstract
It is wellknown that symmetryprotected topological (SPT) phases can be obtained from the trivial phase by an entangler, a finitedepth unitary operator $U$. Here, we consider obtaining the entangler from a local 'pivot' Hamiltonian $H_{piv}$ such that $U = e^{i\pi H_{piv}}$. 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 nononsite $U(1)$ symmetries explain the exotic phase diagrams that can appear, including an SPT multicritical point where the gapless ground state is given by the fixedpoint toric code state.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.07599
 arXiv:
 arXiv:2110.07599
 Bibcode:
 2021arXiv211007599T
 Keywords:

 Condensed Matter  Strongly Correlated Electrons;
 Condensed Matter  Statistical Mechanics;
 Quantum Physics
 EPrint:
 22+6 pages, 6 figures. v2 published version