Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, Ruben Verresen
SciPost Phys. 14, 013 (2023) ·
published 2 February 2023

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Progress in understanding symmetryprotected topological (SPT) phases has been greatly aided by our ability to construct lattice models realizing these states. In contrast, a systematic approach to constructing models that realize quantum critical points between SPT phases is lacking, particularly in dimension $d>1$. Here, we show how the recently introduced notion of the pivot Hamiltonian—generating rotations between SPT phases—facilitates such a construction. We demonstrate this approach by constructing a spin model on the triangular lattice, which is midway between a trivial and SPT phase. The pivot Hamiltonian generates a $U(1)$ pivot symmetry which helps to stabilize a direct SPT transition. The signproblem free nature of the model—with an additional Ising interaction preserving the pivot symmetry—allows us to obtain the phase diagram using quantum Monte Carlo simulations. We find evidence for a direct transition between trivial and SPT phases that is consistent with a deconfined quantum critical point with emergent $SO(5)$ symmetry. The known anomaly of the latter is made possible by the nonlocal nature of the $U(1)$ pivot symmetry. Interestingly, the pivot Hamiltonian generating this symmetry is nothing other than the staggered BaxterWu threespin interaction. This work illustrates the importance of $U(1)$ pivot symmetries and proposes how to generally construct signproblemfree lattice models of SPT transitions with such anomalous symmetry groups for other lattices and dimensions.
Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath, Ruben Verresen
SciPost Phys. 14, 012 (2023) ·
published 2 February 2023

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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 $H_\text{pivot}$ such that $U = e^{i\pi H_\text{pivot}}$. 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 $H_\text{pivot}$? 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 $H_\text{pivot}$, 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.
SciPost Phys. 10, 148 (2021) ·
published 18 June 2021

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We gauge the fermion parity symmetry of the Kitaev chain. While the bulk of the model becomes an Ising chain of gaugeinvariant spins in a tilted field, near the boundaries the global fermion parity symmetry survives gauging, leading to local gaugeinvariant Majorana operators. In the absence of vortices, the Higgs phase exhibits fermionic symmetryprotected topological (SPT) order distinct from the Kitaev chain. Moreover, the deconfined phase can be stable even in the presence of vortices. We also undertake a comprehensive study of a gently gauged model which interpolates between the ordinary and gauged Kitaev chains. This showcases rich quantum criticality and illuminates the topological nature of the Higgs phase. Even in the absence of superconducting terms, gauging leads to an SPT phase which is intrinsically gapless due to an emergent anomaly.