Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in $2+1$ dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a $d^{th}$ -order SPT phase in $d+1$ dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in $3+1$ dimensions exhibiting hinge modes protected by a noninvertible symmetry.Higher-order topological phases with invertible symmetries have been extensively studied in recent years, revealing gapless modes localized on boundaries of higher codimension. In this work, we extend the framework of higher-order symmetry-protected topological (SPT) phases to include noninvertible symmetries. We construct a concrete model of a second-order SPT phase in $2+1$ dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a $d^{th}$-order SPT phase in $d+1$ dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in $3+1$ dimensions exhibiting hinge modes protected by a noninvertible symmetry.

arXiv:2505.18119Physical review. B./Physical review. BApr 22, 20261 citation

Minimalist approach to the classification of symmetry protected topological phases

Zhaoxi Xiong

Bulletin of the American Physical SocietyMar 14, 20170 citations

Bulk-boundary correspondence for intrinsically gapless symmetry-protected topological phases from group cohomology

Rui Wen, Andrew C. Potter

Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.

arXiv:2208.09001Physical review. B./Physical review. BJun 20, 202358 citations

Classification of fermionic symmetry-protected topological phases

Qing-Rui Wang, Zheng‐Cheng Gu

Bulletin of the American Physical SocietyMar 4, 20190 citations

Bosonic symmetry-protected topological phases with reflection symmetry

Tsuneya Yoshida, Takahiro Morimoto, Akira Furusaki

We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer- $S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even $N$ and 0 (no SPT phase) for odd $N$ , while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are ${\mathbb{Z}}_{2}$ . We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a ${\mathbb{Z}}_{2}$ SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.We study two-dimensional bosonic symmetry-protected topological (SPT) phases that are protected by reflection symmetry and local symmetry [${Z}_{N}\ensuremath{\rtimes}R, {Z}_{N}\ifmmode\times\else\texttimes\fi{}R$, U(1)$\ensuremath{\rtimes}R$, or U(1)$\ifmmode\times\else\texttimes\fi{}R$], in the search for two-dimensional bosonic analogs of topological crystalline insulators in integer-$S$ spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even $N$ and 0 (no SPT phase) for odd $N$, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are ${\mathbb{Z}}_{2}$. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of $S=2$ spins on the square lattice is a ${\mathbb{Z}}_{2}$ SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.

arXiv:1509.04395Physical Review BDec 14, 201518 citations

2+1

dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a

d^{th}

-order SPT phase in

d+1

dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in

3+1

dimensions exhibiting hinge modes protected by a noninvertible symmetry.

S

spin systems with reflection and spin-rotation symmetries. To classify them, we employ a Chern-Simons approach and examine the stability of edge states against perturbations that preserve the assumed symmetries. We find that SPT phases protected by ${Z}_{N}\ensuremath{\rtimes}R$ symmetry are classified as ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ for even

N

and 0 (no SPT phase) for odd

N

, while those protected by U(1)$\ensuremath{\rtimes}R$ symmetry are

{\mathbb{Z}}_{2}

. We point out that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state of

S=2

spins on the square lattice is a

{\mathbb{Z}}_{2}

SPT phase protected by reflection and $\ensuremath{\pi}$-rotation symmetries.