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Bulletin of the American Physical Society

Transitions between symmetry-protected topological phases from pivoting

Nathanan Tantivasadakarn, Ryan Thorngren, Ruben Verresen, Ashvin Vishwanath

Published Mar 16, 2021• 0 citations
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Progress in understanding symmetry-protected 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 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) U ( 1 ) pivot symmetry which helps to stabilize a direct SPT transition. The sign-problem 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) S O ( 5 ) symmetry. The known anomaly of the latter is made possible by the non-local nature of the U(1) U ( 1 ) pivot symmetry. Interestingly, the pivot Hamiltonian generating this symmetry is nothing other than the staggered Baxter-Wu three-spin interaction. This work illustrates the importance of U(1) U ( 1 ) pivot symmetries and proposes how to generally construct sign-problem-free lattice models of SPT transitions with such anomalous symmetry groups for other lattices and dimensions.

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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 U . Here, we consider obtaining the entangler from a local ‘pivot’ Hamiltonian H_\text{pivot} H pivot such that U = e^{i\pi H_\text{pivot}} U = e i π H 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) U ( 1 ) pivot symmetry generated by H_\text{pivot} H 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) U ( 1 ) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for H_\text{pivot} H pivot , including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U(1) 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) 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.

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Generalized charge pumps are topological obstructions to trivializing loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points that must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, “pivot loops”, defined by two Hamiltonians, where the first is unitarily evolved by the second “pivot” Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops that pump charge for various symmetry groups, often leading to SPT phases—including dipole SPTs. Intriguingly, we find examples where nontrivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated with the nontrivial pump to explain the “unnecessary” criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show are equivalent to the Hamiltonians satisfying the Dolan-Grady relation—known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labeled by spaces that are not manifolds.

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Progress in understanding symmetry-protected 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 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) U ( 1 ) pivot symmetry which helps to stabilize a direct SPT transition. The sign-problem 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) S O ( 5 ) symmetry. The known anomaly of the latter is made possible by the non-local nature of the U(1) U ( 1 ) pivot symmetry. Interestingly, the pivot Hamiltonian generating this symmetry is nothing other than the staggered Baxter-Wu three-spin interaction. This work illustrates the importance of U(1) U ( 1 ) pivot symmetries and proposes how to generally construct sign-problem-free lattice models of SPT transitions with such anomalous symmetry groups for other lattices and dimensions.
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 U . Here, we consider obtaining the entangler from a local ‘pivot’ Hamiltonian H_\text{pivot} H pivot such that U = e^{i\pi H_\text{pivot}} U = e i π H 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) U ( 1 ) pivot symmetry generated by H_\text{pivot} H 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) U ( 1 ) pivot symmetry. We illustrate this in a variety of examples, assuming various forms for H_\text{pivot} H pivot , including the Ising chain, and the toric code Hamiltonian. A remarkable property of such a U(1) 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) 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.
Generalized charge pumps are topological obstructions to trivializing loops in the space of symmetric gapped Hamiltonians. We show that given mild conditions on such pumps, the associated loop has high-symmetry points that must be in distinct symmetry-protected topological (SPT) phases. To further elucidate the connection between pumps and SPTs, we focus on closed paths, “pivot loops”, defined by two Hamiltonians, where the first is unitarily evolved by the second “pivot” Hamiltonian. While such pivot loops have been studied as entanglers for SPTs, here we explore their connection to pumps. We construct families of pivot loops that pump charge for various symmetry groups, often leading to SPT phases—including dipole SPTs. Intriguingly, we find examples where nontrivial pumps do not lead to genuine SPTs but still entangle representation-SPTs (RSPTs). We use the anomaly associated with the nontrivial pump to explain the “unnecessary” criticality between these RSPTs. We also find that particularly nice pivot families form circles in Hamiltonian space, which we show are equivalent to the Hamiltonians satisfying the Dolan-Grady relation—known from the study of integrable models. This additional structure allows us to derive more powerful constraints on the phase diagram. Natural examples of such circular loops arise from pivoting with the Onsager-integrable chiral clock models, containing the aforementioned RSPT example. In fact, we show that these Onsager pivots underlie general group cohomology-based pumps in one spatial dimension. Finally, we recast the above in the language of equivariant families of Hamiltonians and relate the invariants of the pump to the candidate SPTs. We also highlight how certain SPTs arise in cases where the equivariant family is labeled by spaces that are not manifolds.