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arXiv:2210.15596 · Physical review. B./Physical review. B

Classification and construction of interacting fractonic higher-order topological phases

Jian-Hao Zhang, Meng Cheng, Zhen Bi

Published Jul 19, 2023• 12 citations
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Abstract

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.

DOI
10.1103/physrevb.108.045133

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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.
Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: ``higher-order symmetry-protected topological (HOSPT) phases.'' While the usual (first-order) SPT phases in ddd spatial dimensions support anomalous $(d\ensuremath{-}1)$-dimensional surface states, HOSPT phases in ddd dimensions are characterized by topological boundary states of dimension $(d\ensuremath{-}2)$ or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the K\"unneth formula of group cohomology. Based on a decorated domain-wall picture of the K\"unneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.
The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space-group SPT phases were also proposed for interacting bosonic systems. In this Rapid Communication, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point-group SPT phases for two-dimensional (2D) interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these different classes of point-group SPT phases in 2D correlated superconductors is addressed.
The natural existence of crystalline symmetry in real materials manifests the importance of understanding crystalline symmetry-protected topological phases, especially for interacting systems. In this paper, we systematically construct and classify all the point-group crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of a topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover several intrinsically interacting topological phases that cannot be realized in any free-fermion systems. Moreover, the crystalline equivalence principle for 3D interacting fermionic systems is also verified, with an additional subtle “twist” on the spin of fermions.
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.
In this work, we introduce a new type of topological order that is protected by subsystem symmetries that act on lower-dimensional subsets of lattice many-body system, e.g., along lines or planes in a three-dimensional system. The symmetry groups for such systems exhibit a macroscopic number of generators in the infinite-volume limit. We construct a set of exactly solvable models in 2D2D2D and 3D3D3D, which exhibit such subsystem SPT (SSPT) phases with one-dimensional subsystem symmetries. These phases exhibit analogs of phenomena seen in SPTs protected by global symmetries: gapless edge modes, projective realizations of the symmetries at the edge, and nonlocal order parameters. Such SSPT phases are proximate, in theory space, to previously studied phases that break the subsystem symmetries and phases with fracton order, which result upon gauging them.