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arXiv:2106.10703 · Physical review. B./Physical review. B

Lattice construction of exotic invertible topological phases

Ryohei Kobayashi

Published Jan 31, 2022• 5 citations
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Abstract

In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian [arXiv:2105.09454 [cond-mat.str-el]. The exotic phase has time-reversal $(T)$ symmetry, and depends on a choice of the space-time structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases and thus gives a novel class of invertible topological phases. When the $T$ symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with $T$ symmetry defects decorated with a fermionic phase that depends on a spin structure of the $T$ symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the ${\mathbb{Z}}_{8}$ classification of the $(3+1)$d invertible phase based on the Wu structure. This generalizes the ${\mathbb{Z}}_{8}$ classification of the $T$-symmetric $(1+1)$d topological superconductor proposed by Fidkowski and Kitaev. On oriented space-time, this $(3+1)$d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of $G$-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic space-time dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases.

DOI
10.1103/physrevb.105.035153

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In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian [arXiv:2105.09454 [cond-mat.str-el]. The exotic phase has time-reversal (T)(T)(T) symmetry, and depends on a choice of the space-time structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases and thus gives a novel class of invertible topological phases. When the TTT symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with TTT symmetry defects decorated with a fermionic phase that depends on a spin structure of the TTT symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the Z8{\mathbb{Z}}_{8}Z8​ classification of the (3+1)(3+1)(3+1)d invertible phase based on the Wu structure. This generalizes the Z8{\mathbb{Z}}_{8}Z8​ classification of the TTT-symmetric (1+1)(1+1)(1+1)d topological superconductor proposed by Fidkowski and Kitaev. On oriented space-time, this (3+1)(3+1)(3+1)d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of GGG-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic space-time dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases.
We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called “gauged Gu-Wen SPT defects” that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D \mathbb{Z}_2 ℤ 2 toric code stacked with a (2+1)D ancilla trivial atomic insulator. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the \mathbb{Z}_8 ℤ 8 classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry.
Abstract We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the Z8\mathbb{Z}_8Z8​ classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the Z16\mathbb{Z}_{16}Z16​ classification.
Abstract We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin ± case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin_ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the ℤ8 classification of (1+1)d topological superconductors. We also compute the indicator formula of ℤ16 valued time-reversal anomaly for (2+1)d pin+ TQFT based on our construction.
The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wave function construction of fermionic SPT (FSPT) states for generic fermionic symmetry group Gf=Z2f×ω2GbG_f=\mathbb{Z}_2^f \times_{\omega_2} G_bGf​=Z2f​×ω2​​Gb​ which is a central extension of bosonic symmetry group GbG_bGb​ (may contain time reversal symmetry) by the fermion parity symmetry group Z2f={1,Pf}\mathbb{Z}_2^f = \{1,P_f\}Z2f​={1,Pf​}. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction and classification of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction-free and trivialization-free constructions naturally lead to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.
Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group G=U(1)⋊HG = \mathrm{U}(1) \rtimes HG=U(1)⋊H, we construct a (3+1)D topologically invariant path integral in the presence of a curved background GGG gauge field, as an exact combinatorial state sum. Specifically, the U(1)\mathrm{U}(1)U(1) component of the GGG gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat GGG bundles. Our construction expresses the U(1)\mathrm{U}(1)U(1) gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form R\mathbb{R}R gauge field and 2-form Z\mathbb{Z}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)\mathrm{U}(1)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 Spinc^cc or Pinc^cc 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)\mathrm{U}(1)U(1) anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spinc^cc generalization of the Gauss-Milgram sum, valid for super-modular categories.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.