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arXiv:2307.05474 · Physical review. B./Physical review. B

Fractonic higher-order topological phases in open quantum systems

Jian-Hao Zhang, Ke Ding, Shuo Yang, Zhen Bi

Published Oct 16, 2023• 17 citations
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Abstract

In this work, we study the generalization of decohered average symmetry-protected topological phases to open quantum systems with a combination of subsystem symmetries and global symmetries. In particular, we provide examples of two types of intrinsic average higher-order topological phases with average subsystem symmetries. A classification scheme for these phases based on generalized anomaly cancellation criteria of average symmetry is also discussed.

DOI
10.1103/physrevb.108.155123

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In this work, we study the generalization of decohered average symmetry-protected topological phases to open quantum systems with a combination of subsystem symmetries and global symmetries. In particular, we provide examples of two types of intrinsic average higher-order topological phases with average subsystem symmetries. A classification scheme for these phases based on generalized anomaly cancellation criteria of average symmetry is also discussed.
The notion of higher-order topological phases can have interesting generalizations to systems with subsystem symmetries that exhibit fractonic dynamics for charged excitations. In this work, we systematically study the higher-order topological phases protected by a combination of subsystem symmetries and ordinary global symmetries in two and three-dimensional interacting boson systems, with some interacting fermionic examples. In particular, we discover some important results, including (i) an Abelian subsystem symmetry group that does not allow for a nontrivial higher-order symmetry-protected topological phase in 2D without global symmetry. (ii) inhomogeneous subsystem symmetries do not allow for that either.
Global symmetries greatly enrich the landscape of topological quantum phases, playing an essential role from topological insulators to fractional quantum Hall effect. Topological phases in mixed quantum states, originating from decoherence in open quantum systems or disorders in imperfect crystalline solids, have recently garnered significant interest. Unlike pure states, mixed quantum states can exhibit average symmetries—symmetries that keep the total ensemble invariant but not on each individual state. In this work, we present a systematic classification and characterization of average symmetry-protected topological (ASPT) phases applicable to generic symmetry groups, encompassing both average and exact symmetries, for bosonic and fermionic systems. Moreover, we formulate the theory of average symmetry-enriched topological (ASET) orders in disordered bosonic systems. Our systematic approach helps clarify nuanced issues in previous literature and uncovers compelling new physics. Notably, we discover that (1) the definition and classification of ASPT phases in decohered and disordered systems exhibit subtle differences, (2) despite these differences, ASPT phases in both settings can be classified and characterized under a unified framework of defect decoration and spectral sequence, (3) this systematic classification uncovers a plethora of ASPT phases that are intrinsically mixed, implying they can exclusively manifest in decohered or disordered systems where part of the symmetry is average, and (4) similarly for ASET, we find intrinsically disordered phases exhibiting exotic anyon behaviors—the ground states of such phases necessarily contain localized anyons, with gapless (yet still localized) excitation blue spectra.
Decoherence in realistic quantum platforms motivates a mixed-state notion of topological phases of matter, including average symmetry-protected topological (ASPT) phases. Alongside this progress, generalized symmetries--notably noninvertible and dipole symmetries--have become powerful organizing principles for exotic quantum phases, yet their implications for mixed states remain less explored. In this work, we bridge these directions through a gauging correspondence between mixed-state phases with generalized symmetries and mixed-state phases with ordinary group symmetries, recasting the classification of noninvertible and dipole ASPT phases into familiar classifications of symmetry breaking and ASPT phases with dual symmetries. Using this approach, we classify and construct a subclass of ASPT phases with non-invertible and dipole symmetries in (1+1)d(1+1)d(1+1)d, including phases that are intrinsic to mixed states, and characterize them via string order parameters and protected edge modes.
We investigate a novel class of topological superconducting phases protected by exact fermion-parity symmetry and average crystalline symmetries. These phases belong to the broader class of average crystalline symmetry-protected topological (ACSPT) states and include numerous examples of intrinsic ACSPTs -- topological phases that arise only in the presence of disorder or decoherence. Unlike conventional symmetry-protected topological (SPT) phases, which require exact symmetry protection, average SPT (ASPT) phases remain robust as long as the symmetry is restored on average across disorder realizations or mixed-state ensembles. To classify these phases, we extend the real-space block state construction framework to account for average crystalline symmetries. In this generalized setting, lower-dimensional cells are decorated with ASPT phases, and the obstruction-free conditions are reformulated to incorporate the constraints imposed by average symmetry at block intersections. This provides a physically transparent and systematic method for classifying ASPTs with spatial symmetries that are only preserved statistically. We further validate our classification using a generalized spectral sequence analysis, which serves as an independent consistency check. Our results demonstrate that many crystalline topological superconductors remain well defined under realistic imperfections, and they uncover a rich landscape of intrinsically average-symmetry-protected phases that have no analog in clean systems.
In this work, we present a collection of three-dimensional higher-order symmetry-protected topological phases (HOSPTs) with gapless hinge modes that exist only in strongly interacting systems subject to subsystem symmetry constraints. We use a coupled wire construction to generate three families of microscopic lattice models: insulators with helical hinge modes, superconductors with chiral Majorana hinge modes, and fractionalized insulators with helical hinge modes that carry fractional charge. In particular, these HOSPTs do not require spatial symmetry protection, but are instead protected by subsystem symmetries, and support ``fractonic'' quasiparticle excitations that move within only a low-dimensional submanifold of the system. We analyze the anomaly structure for the boundary theory and the entanglement Hamiltonian, and show that the side surfaces of these HOSPTs, despite being partially gapped, exhibit symmetry anomalies, and can only be realized as the boundary of three-dimensional HOSPT phases.
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+12+12+1 dimensions that hosts symmetry-protected corner modes protected by a noninvertible symmetry. This construction is then generalized to a dthd^{th}dth-order SPT phase in d+1d+1d+1 dimensions, featuring similarly protected corner modes. Additionally, we demonstrate a second-order SPT phase in 3+13+13+1 dimensions exhibiting hinge modes protected by a noninvertible symmetry.