# Pivot Hamiltonians as generators of symmetry and entanglement

### Submission summary

 Authors (as Contributors): Nathanan Tantivasadakarn · Ruben Verresen
Submission information
Date accepted: 2022-10-21
Date submitted: 2022-10-15 22:14
Submitted to: SciPost Physics
Ontological classification
Specialties:
• Condensed Matter Physics - Theory
• Quantum Physics
Approach: Theoretical

### 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 $U = e^{i\pi H_{piv}}$. This perspective of Hamiltonians pivoting between the trivial and SPT phase opens up two new directions: (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 for when 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.

###### Current status:
Accepted in target Journal

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### List of changes

1. Exposition of Sec. 4.1 has been improved, clarifying the distinction of the local pivot Hamiltonian vs the total pivot Hamiltonian.
2. In Sec. 5.4 a description of a lattice model that is likely to realize the $O(2)/\mathbb Z_2$ transition has been added, which can be numerically explored in future work.