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

Ning, Shang-Qiang; Mao, Bin-Bin; Li, Zhengqiao; Wang, Chenjie

doi:10.1103/physrevb.104.075111

Cited by 5 publications

(8 citation statements)

References 94 publications

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“…For the Abelian topological order, the results shown below produce all of the anomaly indicator formulas proposed by Lapa and Levin in [17]. In the case with the time-reversal symmetry, the indicators were derived in [24] based on a completely different method where the time-reversal is replaced by a crystalline reflection symmetry.…”

Section: Construction Of Topological Path Integralsmentioning

confidence: 99%

“…10. On a + or − 4-simplex, the 15j symbol evaluates as Z + (01234) = e −2πi 42 q012(c+δσh)234−2πi 43 (δsσ q)0123h34 • U 34 (013, 123, 0123) s(34) U 34 (023, 32 012, 0123) s (34) η 012 (23, 34) s (24) • Z + 0 (01234), Z − (01234) = e 2πi 42 q012(c+δσ h)234+2πi 43 (δsσ q)0123h34 • U 34 (023, 32 012, 0123) s (34) η 012 (23, 34) s (24) U 34 (013, 123, 0123) s (34) • Z − 0 (01234),…”

Section: Bosonic State Sum Of (3+1)d Tqft With U(1) ⋊ H Symmetrymentioning

confidence: 99%

“…N = 2), this corresponds to the possibility of a spin-1/2 carried by the anyons. The path integral of the state sum interacting with the Z N 't Hooft line should allow derivation of the anomaly indicators involving SO(3) symmetry, which were studied in [24] using different methods. However, it is not clear how to generalize our Villain graphical calculus in the presence of non-zero SU(N )-valued h fields.…”

Section: Partition Function Onmentioning

confidence: 99%

“…On the left-hand-side (LHS) of Eq. (F1), the product of the symmetry factors involving η and U are evaluated as Symm H LHS = U 34 (013, 123, 0123) s(34) U 34 (023, 32 012, 0123) s (34) η 012 (23, 34) s (24) s (45) × U 45 (014, 124, 0124) s(45) U 45 (024, 42 012, 0124) s (45) η 012 (24, 45) s (25) × U 45 (024, 234, 0234) s(45) U 45 (034, 43 023, 0234) s (45) η 023 (34, 45) s (35) , (F2) where the exponent s(45) on the first row corresponds to the the orientation of 01234 . Meanwhile, the corresponding object on the right-hand-side (RHS) is given by Symm H RHS = U 35 (013, 123, 0123) s(35) U 35 (023, 32 012, 0123) s (35) η 012 (23,35) s (25) × U 45 (014, 134, 0134) s(45) U 45 (034, 42 013, 0134) s (45) η 012 (34, 45) s (35) × U 45 (124, 234, 1234) s(45) U 45 (134, 43 123, 0234) s (45) η 123 (34, 45) s (35) .…”

Section: Locality-respecting Natural Isomorphisms In the Fermionic Casementioning

confidence: 99%

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(3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases

Kobayashi¹,

Barkeshli²

2021

Preprint

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Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group G = U(1) ⋊ H, we construct a (3+1)D topologically invariant path integral in the presence of a curved background G gauge field, as an exact combinatorial state sum. Specifically, the U(1) component of the G gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat G bundles. Our construction expresses the U(1) gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form R gauge field and 2-form Z gauge field. We develop a new graphical calculus for anyons interacting with "Villain symmetry defects", associated with the 1-form and 2-form background gauge fields. This graphical calculus is used to define the (3+1)D path integral, which can describe either a bosonic or fermionic symmetry-protected topological (SPT) phase. For example, we can construct the topological path integral on curved U(1) bundles for the (3+1)D fermionic topological insulator in class AII and topological superconductor in class AIII given appropriate (2+1)D fermionic symmetry fractionalization data; these then give invariants of 4-manifolds with Spin c or Pin c structures and their generalizations. The (3+1)D path integrals define anomaly indicators for the (2+1)D topological orders; in the case of Abelian (2+1)D topological orders, we derive by explicit computation all of the mixed U(1) anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spin c generalization of the Gauss-Milgram sum, valid for super-modular categories.

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Section: Construction Of Topological Path Integralsmentioning

confidence: 99%

Section: Bosonic State Sum Of (3+1)d Tqft With U(1) ⋊ H Symmetrymentioning

confidence: 99%

Section: Partition Function Onmentioning

confidence: 99%

Section: Locality-respecting Natural Isomorphisms In the Fermionic Casementioning

confidence: 99%

See 2 more Smart Citations

(3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases

Kobayashi¹,

Barkeshli²

2021

Preprint

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“…In this work, we propose a general dimensional reduction method for the duality between the topological NLSM models with WZW terms and QED theory with generic dispersive fermions [66]. The application of the dimensional reduction method to condensed matter systems in the literature has been mainly focused on gapped systems [67][68][69][70][71][72][73]. In the seminal paper [67], it was proposed that lower dimensional topological insulators and their topological response terms can be derived from even dimensional Chern insulators by compacting the extra dimensions and viewing the momenta in these dimensions as parameters.…”

Section: Introductionmentioning

confidence: 99%

Dual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operators

Wu¹,

Li²

2022

Preprint

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Nonlinear σ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or QED theory. In this paper, we propose a dimensional reduction scheme to derive the U(1) gauged topological NLSM in n-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the N f = 4 QED 3 with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold U(4) U( 2)×U(2) coupled with a dynamical U(1) gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the U(1) monopole operators and the corresponding quantum number.

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Enforced symmetry breaking by invertible topological order

Ning

et al. 2023

Phys. Rev. Research

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

Cited by 5 publications

References 94 publications

(3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases

(3+1)D path integral state sums on curved U(1) bundles and U(1) anomalies of (2+1)D topological phases

Dual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operators

Enforced symmetry breaking by invertible topological order

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