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Pivot Hamiltonians as generators of symmetry and entanglement
by Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath and Ruben Verresen
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Submission summary
Authors (as registered SciPost users):  Nathanan Tantivasadakarn · Ruben Verresen 
Submission information  

Preprint Link:  scipost_202204_00017v2 (pdf) 
Date accepted:  20221021 
Date submitted:  20221015 22:14 
Submitted by:  Tantivasadakarn, Nathanan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
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: (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 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.
Published as SciPost Phys. 14, 012 (2023)
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.