Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

Debray, Arun

doi:10.48550/arxiv.2102.02941

Cited by 5 publications

(11 citation statements)

References 54 publications

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“…Compare all classification results calculated in this paper to the classifications of 3D fSPT phases protected by corresponding internal symmetry groups, including results in Refs. [98,99] and that we compute for ω 2 = 0 using the formulas in Refs. [12,13] and the algorithm in Ref.…”

Section: Summary Of Main Resultsmentioning

confidence: 99%

“…In this section, we discuss how to generalize the crystalline equivalence principle that has been conjectured and justified in interacting bosonic systems [56] and 2D interacting fermionic systems [58,59,67]. By comparing the classification results of crystalline TSC and TI summarized in Tables I, II, III and IV with the classification results of 3D fSPT phases protected by corresponding internal symmetry groups [98], we verify the fermionic version of crystalline equivalence principle in 3D systems for all TSC and TI constructed in this paper, for both spinless and spin-1/2 fermions.…”

Section: Generalized Crystalline Equivalence Principlementioning

confidence: 94%

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Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems

Zhang¹,

Yang²,

Gu³

2022

Preprint

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The natural existences of crystalline symmetry in real materials manifest the importance of understanding crystalline symmetry protected topological(SPT) phases, especially for interacting systems. In this paper, we systematically construct and classify all the crystalline topological superconductors and insulators in three-dimensional (3D) interacting fermion systems using the novel concept of topological crystal. The corresponding higher-order topological surface theory can also be systematically studied via higher-order bulk-boundary correspondence. In particular, we discover an intriguing fact that almost all topological crystals with nontrivial 2D block-states are 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.

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Section: Summary Of Main Resultsmentioning

confidence: 99%

Section: Generalized Crystalline Equivalence Principlementioning

confidence: 94%

Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems

Zhang¹,

Yang²,

Gu³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…First, regarding classification of strongly correlated topological crystalline phases, there have been many works in the literature [37,41,[43][44][45][46][47][48][49][50][51][52]. The dimensional reduction approach proposed in Ref.…”

Section: B Relation To Prior Workmentioning

confidence: 99%

“…The two schemes are later shown to be equivalent [43][44][45][46]. There are also more mathematical approaches such as the cobordism theory [50,53] and invertible topological field theory [40,51,52]. For our purpose of defining bulk topological invariants using physical observables, we find the dimensional reduction approach more suitable and adopt it extensively in this work.…”

Section: B Relation To Prior Workmentioning

confidence: 99%

Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Ning,

Mao,

et al. 2021

Preprint

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We derive a series of quantitative bulk-boundary correspondences for 3D bosonic and fermionic symmetry-protected topological (SPT) phases under the assumption that the surface is gapped, symmetric and topologically ordered, i.e., a symmetry-enriched topological (SET) state. We consider those SPT phases that are protected by the mirror symmetry and continuous symmetries that form a group of U (1), SU (2) or SO(3). In particular, the fermionic cases correspond to a crystalline version of 3D topological insulators and topological superconductors in the famous ten-fold-way classification, with the time-reversal symmetry replaced by the mirror symmetry and with strong interaction taken into account. For surface SETs, the most general interplay between symmetries and anyon excitations is considered. Based on the previously proposed dimension reduction and folding approaches, we re-derive the classification of bulk SPT phases and define a complete set of bulk topological invariants for every symmetry group under consideration, and then derive explicit expressions of the bulk invariants in terms of surface topological properties (such as topological spin, quantum dimension) and symmetry properties (such as mirror fractionalization, fractional charge or spin). These expressions are our quantitative bulk-boundary correspondences. Meanwhile, the bulk topological invariants can be interpreted as anomaly indicators for the surface SETs which carry 't Hooft anomalies of the associated symmetries whenever the bulk is topologically non-trivial. Hence, the quantitative bulk-boundary correspondences provide an easy way to compute the 't Hooft anomalies of the surface SETs. Moreover, our anomaly indicators are complete. Our derivations of the bulk-boundary correspondences and anomaly indicators are explicit and physically transparent. The anomaly indicators obtained in this work can be straightforwardly translated to their timereversal counterparts that apply to the usual topological insulators and topological superconductors, due to a known correspondence between mirror and time-reversal topological phases.

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“…An important consequence in this framework is that the classification of cSPT phases with a crystalline symmetry group G is the same as the classification of SPT phases with internal symmetry group G, which is known as the "Crystalline Equivalence Principle" (See also Ref. [49][50][51]. The topological crystal and the smooth state approach are actually equivalent as shown in Ref.…”

Section: Introductionmentioning

confidence: 97%

Effective field theories of topological crystalline insulators and topological crystals

Huang,

Hsieh,

2021

Preprint

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We present a general approach to obtain effective field theories for topological crystalline insulators whose low-energy theories are described by massive Dirac fermions. We show that these phases are characterized by the responses to spatially dependent mass parameters with interfaces. These mass interfaces implement the dimensional reduction procedure such that the state of interest is smoothly deformed into a topological crystal, which serves as a representative state of a phase in the general classification. Effective field theories are obtained by integrating out the massive Dirac fermions, and various quantized topological terms are uncovered. Our approach can be generalized to other crystalline symmetry protected topological phases and provides a general strategy to derive effective field theories for such crystalline topological phases.

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Invertible phases for mixed spatial symmetries and the fermionic crystalline equivalence principle

Cited by 5 publications

References 54 publications

Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems

Construction and classification of crystalline topological superconductor and insulators in three-dimensional interacting fermion systems

Anomaly indicators and bulk-boundary correspondences for 3D interacting topological crystalline phases with mirror and continuous symmetries

Effective field theories of topological crystalline insulators and topological crystals

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