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arXiv:2112.12148 · Physical review. B./Physical review. B

(3+1)d boundaries with gravitational anomaly of (4+1)d invertible topological order for branch-independent bosonic systems

Zheyan Wan, Juven Wang, Xiao-Gang Wen

Published Jul 20, 2022• 12 citations
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Abstract

We study bosonic systems on a space-time lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the space-time simplicial complex, even for a space-time with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle [such as $(4+1)\mathrm{d}$ ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$] to be nontrivial for branch-independent bosonic systems, but this topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic lattice bosonic systems on a space-time lattice are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. In general, the branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct branch-independent bosonic systems to realize the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ topological order, and its $(3+1)\mathrm{d}$ gapped or gapless boundaries. One of the gapped boundaries is a $(3+1)\mathrm{d}$ ${\mathbb{Z}}_{2}$ gauge theory with (1) fermionic ${\mathbb{Z}}_{2}$ gauge charge particle which trivializes ${\mathrm{w}}_{2}$ and (2) ``fermionic'' ${\mathbb{Z}}_{2}$ gauge flux line trivializes ${\mathrm{w}}_{3}$. In particular, if the flux loop's world sheet is unorientable, then the orientation-reversal one-dimensional world line must correspond to a fermion world line that does not carry the ${\mathbb{Z}}_{2}$ gauge charge. We also explain why Spin and Spin${}^{c}$ structures trivialize the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ nonperturbative global pure gravitational anomaly to zero [which helps to construct the anomalous $(3+1)\mathrm{d}$ gapped ${\mathbb{Z}}_{2}$ and gapless all-fermion U(1) gauge theories], but the Spin${}^{h}$ and Spin${\ifmmode\times\else\texttimes\fi{}}_{{\mathbb{Z}}_{2}}$Spin$(n\ensuremath{\ge}3)$ structures modify the ${\mathrm{w}}_{2}{\mathrm{w}}_{3}$ into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain grand unifications (e.g., $n=10,18$) or construct new models.

DOI
10.1103/physrevb.106.045127

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We study bosonic systems on a space-time lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the space-time simplicial complex, even for a space-time with boundaries. In contrast, a generic lattice bosonic (GLB) system's path integral may depend on the branch structure. We find the invertible topological order characterized by the Stiefel-Whitney cocycle [such as (4+1)d(4+1)\mathrm{d}(4+1)d w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​] to be nontrivial for branch-independent bosonic systems, but this topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic lattice bosonic systems on a space-time lattice are not classified by the oriented cobordism. The branch dependence on a lattice may be related to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. In general, the branch structure on a discretized lattice may be related to a frame structure on a smooth manifold that trivializes any Stiefel-Whitney classes. We construct branch-independent bosonic systems to realize the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ topological order, and its (3+1)d(3+1)\mathrm{d}(3+1)d gapped or gapless boundaries. One of the gapped boundaries is a (3+1)d(3+1)\mathrm{d}(3+1)d Z2{\mathbb{Z}}_{2}Z2​ gauge theory with (1) fermionic Z2{\mathbb{Z}}_{2}Z2​ gauge charge particle which trivializes w2{\mathrm{w}}_{2}w2​ and (2) ``fermionic'' Z2{\mathbb{Z}}_{2}Z2​ gauge flux line trivializes w3{\mathrm{w}}_{3}w3​. In particular, if the flux loop's world sheet is unorientable, then the orientation-reversal one-dimensional world line must correspond to a fermion world line that does not carry the Z2{\mathbb{Z}}_{2}Z2​ gauge charge. We also explain why Spin and Spinc{}^{c}c structures trivialize the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ nonperturbative global pure gravitational anomaly to zero [which helps to construct the anomalous (3+1)d(3+1)\mathrm{d}(3+1)d gapped Z2{\mathbb{Z}}_{2}Z2​ and gapless all-fermion U(1) gauge theories], but the Spinh{}^{h}h and Spin\ifmmode×\else\texttimes\fiZ2{\ifmmode\times\else\texttimes\fi{}}_{{\mathbb{Z}}_{2}}\ifmmode×\else\texttimes\fiZ2​​Spin(n\ensuremath≥3)(n\ensuremath{\ge}3)(n\ensuremath≥3) structures modify the w2w3{\mathrm{w}}_{2}{\mathrm{w}}_{3}w2​w3​ into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain grand unifications (e.g., n=10,18n=10,18n=10,18) or construct new models.
We study gapped systems with anomalous time-reversal symmetry and global gravitational anomaly in three and four spacetime dimensions. These systems describe topological order on the boundary of bosonic Symmetry Protected Topological (SPT) Phases. Our description of these phases is via the recent cobordism proposal for their classification. In particular, the behavior of these systems is determined by the geometry of Stiefel-Whitney classes. We discuss electric and magnetic operators defined by these classes, and new types of Wilson lines and surfaces that sit on their boundary. The lines describe fermionic particles, while the surfaces describe a sort of fermionic string. We show that QED with a fermionic monopole exhibits the 4d global gravitational anomaly and has a fermionic π-flux.
Quasiparticle excitations in 3+13+13+1 dimensions can be either bosons or fermions. In this work, we introduce the notion of fermionic loop excitations in (3+1)(3+1)(3+1)-dimensional (3+1d)(3+1\text{d})(3+1d) topological phases. Specifically, we construct a new many-body lattice invariant of gapped Hamiltonians, the loop self-statistics $\ensuremath{\mu}=\ifmmode\pm\else\textpm\fi{}1$, that distinguishes two bosonic topological orders that both superficially resemble 3+1dZ23+1\mathrm{d}\phantom{\rule{4pt}{0ex}}{\mathbb{Z}}_{2}3+1dZ2​ gauge theory coupled to fermionic charged matter. The first has fermionic charges and bosonic Z2{\mathbb{Z}}_{2}Z2​ gauge flux loops (FcBl) and is just the ordinary fermionic toric code. The second has fermionic charges and fermionic loops (FcFl) and, as we argue, can only exist at the boundary of a nontrivial (4+1)(4+1)(4+1)-dimensional (4+1d)(4+1\mathrm{d})(4+1d) invertible phase, stable without any symmetries, i.e., it possesses a gravitational anomaly. We substantiate these claims by constructing an explicit exactly solvable 4+1d4+1\mathrm{d}4+1d model using a method that bootstraps a boundary theory into a bulk Hamiltonian, analogous to that of Walker and Wang, and computing the loop self-statistics in the fermionic Z2{\mathbb{Z}}_{2}Z2​ gauge theory hosted at its boundary. We also show that the FcFl phase has the same gravitational anomaly as all-fermion quantum electrodynamics. Our results are in agreement with the recent classification of nondegenerate braided fusion 2-categories by Johnson-Freyd, and with the cobordism prediction of a nontrivial Z2{\mathbb{Z}}_{2}Z2​-classified 4+1d4+1\mathrm{d}4+1d invertible phase with action $S=\frac{1}{2}\ensuremath{\int}{w}_{2}{w}_{3}$.
We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary \mathbb{Z}_2 ℤ 2 topological order with fermionic particle and fermionic loop excitations that have mutual \pi π statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary \mathbb{Z}_2 ℤ 2 symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.
We review a many-body formulation of gravitational anomalies as obstructions to realizing the complete low-energy data by a local {\it bosonic} lattice quantum system of the same dimension. In this formulation, an anomalous theory can be realized at the boundary of a gapped {\it bosonic} lattice bulk, whose topological order gives a one-to-one classification of the anomaly. An invertible bulk reproduces conventional anomaly inflow; a bulk with non-trivial topological excitations yields a non-invertible generalization. To describe the holographic theory of anomalies, we explain the distinction between uniquely defined topological order and many versions of topological quantum field theory and discuss the boundary-to-bulk map. As an application of the holographic theory of anomalies, we describe morphisms between local quantum theories (such as quantum field theories). These morphisms are built from a bulk domain wall and an invertible boundary junction, which give rise to a sandwich picture. Chiral boson/fermion theory, Laughlin edges, and boundaries of Z2\mathbb{Z}_2Z2​ higher gauge theory with bosonic or fermionic point-like excitations illustrate the different kinds of information about non-invertible anomalies contained in transport coefficients, superselection sectors, and condensation data. In particular, a fermionic theory can be viewed as a theory with a particular non-invertible gravitational anomaly. Those examples also reveal that {\it symmetry ∼\sim∼ gravitational anomaly ∼\sim∼ topological order in one higher dimension}.
Gravitational anomalies can be realized on the boundary of topologically ordered states in one higher dimension and are described by topological orders in one higher dimension. In this paper, we try to develop a general theory for both topological order and gravitational anomaly in any dimensions. (1) We introduce the notion of BF category to describe the braiding and fusion properties of topological excitations that can be point-like, string-like, etc. A subset of BF categories -- closed BF categories -- classify topological orders in any dimensions, while generic BF categories classify (potentially) anomalous topological orders that can appear at a boundary of a gapped quantum liquid in one higher dimension. (2) We introduce topological path integral based on tensor network to realize those topological orders. (3) Bosonic topological orders have an important topological invariant: the vector bundles of the degenerate ground states over the moduli spaces of closed spaces with different metrics. They may fully characterize topological orders. (4) We conjecture that a topological order has a gappable boundary iff the above mentioned vector bundles are flat. (5) We find a holographic phenomenon that every topological order with a gappable boundary can be uniquely determined by the knowledge of the boundary. As a consequence, BF categories in different dimensions form a (monoid) cochain complex, that reveals the structure and relation of topological orders and gravitational anomalies in different dimensions. We also studied the simplest kind of bosonic topological orders that have no non-trivial topological excitations. We find that this kind of topological orders form a Z\mathbb{Z}Z class in 2+1D (with gapless edge), a Z2\mathbb{Z}_2Z2​ class in 4+1D (with gappable boundary), and a Z⊕Z\mathbb{Z}\oplus \mathbb{Z}Z⊕Z class in 6+1D (with gapless boundary).
By definition, the physics of the ddd-dimensional (dim) boundary of a (d+1)(d+1)(d+1)-dim symmetry protected topological (SPT) state cannot be realized as itself on a ddd-dim lattice. If the symmetry of the system is unitary, then a formal way to determine whether a ddd-dim theory must be a boundary or not, is to couple this theory to a gauge field (or to ``gauge'' its symmetry), and check if there is a gauge anomaly. In this paper we discuss the following question: Can the boundary of a SPT state be driven into a fully gapped topological order which preserves all the symmetries? We argue (conjecture) that if the gauge anomaly of the boundary is ``perturbative,'' then the boundary must remain gapless; while if the boundary only has global gauge anomaly but no perturbative anomaly, then it is possible to gap out the boundary by driving it into a topological state, when $d\ensuremath{\ge}2$. We will demonstrate this conjecture with two examples: (1) the 3d3d3d spin-1/2 chiral fermion with the well-known Witten's global anomaly [Phys. Lett. 117, 324 (1982)], which can be realized on the boundary of a 4d4d4d topological superconductor with SU(2) or $U(1)\ensuremath{\rtimes}{Z}_{2}$ symmetry; and (2) the 4d4d4d boundary of a 5d5d5d topological superconductor with the same symmetry. We show that these boundary systems can be driven into a fully gapped Z2N{\mathbb{Z}}_{2N}Z2N​ topological order with topological degeneracy, but this Z2N{\mathbb{Z}}_{2N}Z2N​ topological order cannot be future driven into a trivial confined phase that preserves all the symmetries due to some special properties of its topological defects. Our study also leads to exotic states of matter in pure 3d3d3d space.