arXiv subjects

Xiao-Gang Wen

Publications and source records attributed to Xiao-Gang Wen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness

Two global symmetries are holo-equivalent if their algebras of local symmetric operators are isomorphic. Holo-equivalent classes of global symmetries are classified by gappable-boundary topological orders (TO) in one higher dimension (called symmetry TO), which leads to a symmetry/topological-order (Symm/TO) correspondence. We establish that: (1) For systems with a symmetry described by symmetry TO $M$, their gapped and gapless states are classified by condensable algebras $A$, formed by elementary excitations in $M$ with trivial self/mutual statistics. Such classified states (called $A$-states) can describe symmetry breaking orders, symmetry protected topological orders, symmetry enriched topological orders, gapless critical points, etc., in a unified way. (2) The local low-energy properties of an $A$-state can be calculated from its reduced symmetry TO $M_{/A}$, using holographic modular bootstrap (holoMB) which takes $M_{/A}$ as an input. Here $M_{/A}$ is obtained from $M$ by condensing excitations in $A$. Notably, an $A$-state must be gapless if $M_{/A}$ is nontrivial. This provides a unified understanding of the emergence and symmetry protection of gaplessness that applies to symmetries that are anomalous, higher-form, and/or non-invertible. (3) The relations between condensable algebras constrain the structure of the global phase diagram. (4) 1+1D bosonic systems with $S_3$ symmetry have four gapped phases with unbroken symmetries $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\mathbb{Z}_1$. We find a duality between two transitions $S_3 \leftrightarrow \mathbb{Z}_1$ and $\mathbb{Z}_3 \leftrightarrow \mathbb{Z}_2$: they are either both first order or both (stably) continuous, and in the latter case, they are described by the same conformal field theory (CFT).

cond-mat.str-el↗

arXiv2023

A Lattice Chiral Boson Theory in $1+1$d

Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a $1+1$d chiral field theory as the edge of a $2+1$d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the $2+1$d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a $1+1$d spacetime lattice which carries an anomalous chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.

cond-mat.str-el↗

arXiv2023

Symmetry as a shadow of topological order and a derivation of topological holographic principle

Symmetry is usually defined via transformations described by a (higher) group. But a symmetry really corresponds to an algebra of local symmetric operators, which directly constrains the properties of the system. In this paper, we point out that the algebra of local symmetric operators contains a special class of extended operators -- transparent patch operators, which reveal the selection sectors and hence the corresponding symmetry. The algebra of those transparent patch operators in $n$-dimensional space gives rise to a non-degenerate braided fusion $n$-category, which happens to describe a topological order in one higher dimension (for finite symmetry). Such a holographic theory not only describes (higher) symmetries, it also describes anomalous (higher) symmetries, non-invertible (higher) symmetries (also known as algebraic higher symmetries), and non-invertible gravitational anomalies. Thus, topological order in one higher dimension, replacing group, provides a unified and systematic description of the above generalized symmetries. This is referred to symmetry/topological-order (Symm/TO) correspondence. Our approach also leads to a derivation of topological holographic principle: \emph{boundary uniquely determines the bulk}, or more precisely, the algebra of local boundary operators uniquely determines the bulk topological order. As an application of the Symm/TO correspondence, we show the equivalence between $\mathbb{Z}_2\times \mathbb{Z}_2$ symmetry with mixed anomaly and $\mathbb{Z}_4$ symmetry, as well as between many other symmetries, in 1-dimensional space.

cond-mat.str-el↗

arXiv2023

Emergent higher-symmetry protected topological orders in the confined phase of $U(1)$ gauge theory

We consider compact $U^κ(1)$ gauge theory in 3+1D with the $2π$-quantized topological term ${\sum_{I, J =1}^κ\frac{K_{IJ}}{4π}\int_{M^4}F^I\wedge F^J}$. At energies below the gauge charges' gaps but above the monopoles' gaps, this field theory has an emergent ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, where $k_i$ are the diagonal elements of the Smith normal form of $K$ and $\mathbb{Z}_{0}^{(1)}$ is regarded as $U(1)^{(1)}$. In the $U^κ(1)$ confined phase, the boundary's IR properties are described by Chern-Simons field theory and has a ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry that can be anomalous. To show these results, we develop a bosonic lattice model whose IR properties are described by this field theory, thus acting as its UV completion. The lattice model in the aforementioned limit has an exact ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We find that a gapped phase of the lattice model, corresponding to the confined phase of the $U^κ(1)$ gauge theory, is a symmetry protected topological (SPT) phase for the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry, whose SPT invariant is ${e^{iπ\sum_{I, J}K_{IJ}\int B_I\smile B_J+B_I\underset{1}{\smile} d B_J}e^{iπ\sum_{I< J}K_{IJ}\int d B_I\underset{2}{\smile}d B_J}}$. Here, the background 2-cochains $B_I$ satisfy ${d B_I=\sum_I B_{I}K_{IJ} = 0}$ mod $1$ and describe the symmetry twist of the ${\mathbb{Z}_{k_1}^{(1)}\times\mathbb{Z}_{k_2}^{(1)}\times\cdots}$ 1-symmetry. We apply this general result to a few examples with simple $K$ matrices. We find the non-trivial SPT order in the confined phases of these models and discuss its classifications using the fourth cohomology group of the corresponding 2-group.

cond-mat.str-el↗

arXiv2023

Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules

A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask "whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$?" In particular, we propose that the 3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffering from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. We show a topological non-perturbative proof of the equivalence between $G$-symmetric 't Hooft anomaly cancellation conditions and $G$-symmetric gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multi-U(1) symmetry. We expect our result holds universally regardless of spatial Hamiltonian or Lagrangian/spacetime path integral formulation. Numerical tests are demanding tasks but highly desirable for future work.

hep-lat↗

arXiv2022

Position-Dependent Excitations and UV/IR Mixing in the $\mathbb{Z}_{N}$ Rank-2 Toric Code and its Low-Energy Effective Field Theory

We investigate how symmetry and topological order are coupled in the ${2+1}$d $\mathbb{Z}_{N}$ rank-2 toric code for general $N$, which is an exactly solvable point in the Higgs phase of a symmetric rank-2 $U(1)$ gauge theory. The symmetry enriched topological order present has a non-trivial realization of square-lattice translation (and rotation/reflection) symmetry, where anyons on different lattice sites have different types and belong to different superselection sectors. We call such particles "position-dependent excitations." As a result, in the rank-2 toric code anyons can hop by one lattice site in some directions while only by $N$ lattice sites in others, reminiscent of fracton topological order in ${3+1}$d. We find that while there are $N^2$ flavors of $e$ charges and $2N$ flavors of $m$ fluxes, there are not $N^{N^{2} + 2N}$ anyon types. Instead, there are $N^{6}$ anyon types, and we can use Chern-Simons theory with six $U(1)$ gauge fields to describe all of them. While the lattice translations permute anyon types, we find that such permutations cannot be expressed as transformations on the six $U(1)$ gauge fields. Thus the realization of translation symmetry in the $U^6(1)$ Chern-Simons theory is not known. Despite this, we find a way to calculate the translation-dependent properties of the theory. In particular, we find that the ground state degeneracy on an ${L_{x}\times L_{y}}$ torus is ${N^{3}\gcd(L_{x},N) \gcd(L_{y},N) \gcd(L_{x},L_{y},N)}$, where $\gcd$ stands for "greatest common divisor." We argue that this is a manifestation of UV/IR mixing which arises from the interplay between lattice symmetries and topological order.

cond-mat.str-el↗

arXiv2022

3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

We study bosonic systems on a spacetime lattice defined by path integrals of commuting fields. We introduce branch-independent bosonic (BIB) systems, whose path integral is independent of the branch structure of the spacetime simplicial complex, even for a spacetime 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 (e.g., 4+1d w$_2$w$_3$) to be nontrivial for BIB systems, but this topological order and a trivial gapped tensor product state belong to the same phase for GLB systems. The invertible topological orders in GLB systems 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. 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 BIB systems to realize the w$_2$w$_3$ topological order, and its 3+1d gapped or gapless boundaries. A 3+1d $\mathbb{Z}_2$ gauge theory with (1) fermionic $\mathbb{Z}_2$ gauge charge particle trivializes w$_2$ and (2) fermionic $\mathbb{Z}_2$ gauge flux line trivializes w$_3$. In particular, if the flux loop's worldsheet is unorientable, then an orientation-reversal 1d worldline corresponds to a fermion worldline carrying no $\mathbb{Z}_2$ gauge charge. Spin and Spin$^c$ structures trivialize the w$_2$w$_3$ global pure gravitational anomaly to zero (which helps to construct 3+1d $\mathbb{Z}_2$ and all-fermion U(1) gauge theories), but the Spin$^h$ and Spin$\times_{\mathbb{Z}_2}$Spin$(n \geq 3)$ structures modify the w$_2$w$_3$ into a global mixed gauge-gravitational anomaly, which helps to constrain Grand Unifications (e.g., $n=10,18$) or construct new models.

cond-mat.str-el↗

arXiv2022

The boundaries of 2+1D fermionic topological orders

$2+1$D bosonic topological orders can be characterized by the $S,T$ matrices that encode the statistics of topological excitations. In particular, the $S,T$ matrices can be used to systematically obtain the gapped boundaries of bosonic topological orders. Such an approach, however, does not naively apply to fermionic topological orders (FTOs). In this work, we propose a systematic approach to obtain the gapped boundaries of $2+1$D abelian FTOs. The main trick is to construct a bosonic extension in which the fermionic excitation is "condensed" to form the associated FTOs. Here we choose the parent bosonic topological order to be the $\mathbb{Z}_2$ topological order, which indeed has a fermionic excitation. Such a construction allows us to find an explicit correspondence between abelian FTOs (described by odd $K$-matrix $K_F$) and the "fermion-" condensed $\mathbb{Z}_2$ topological orders (described by even $K$-matrix $K_B$). This provides a systematic algorithm to obtain the modular covariant boundary partition functions as well as the boundary topological excitations of abelian FTOs. For example, the $ν=1-\frac{1}{m}$ Laughlin's states have exactly one type of gapped boundary when $m$ is a square, whose boundary excitations form a $\mathbb{Z}_{2}\times\mathbb{Z}_{\sqrt{m}}$ fusion ring. Our approach can be easily generalized to obtain gapped and gapless boundaries of non-abelian fermionic topological orders.

cond-mat.str-el↗

arXiv2022

Reconstruction of modular data from $SL_2(\mathbb{Z})$ representations

Modular data is the most significant invariant of a modular tensor category. We pursue an approach to the classification of modular data of modular tensor categories by building the modular $S$ and $T$ matrices directly from irreducible representations of $SL_2(\mathbb{Z}/n \mathbb{Z})$. We discover and collect many conditions on the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations to identify those that correspond to some modular data. To arrive at concrete matrices from representations, we also develop methods that allow us to select the proper basis of the $SL_2(\mathbb{Z}/n \mathbb{Z})$ representations so that they have the form of modular data. We apply this technique to the classification of rank-$6$ modular tensor categories, obtaining a classification up to modular data. Most of the calculations can be automated using a computer algebraic system, which can be employed to classify modular data of higher rank modular tensor categories.

math.QA↗

arXiv2022

One dimensional gapped quantum phases and enriched fusion categories

In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and certain gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category. We also study the categorical descriptions of the boundaries of these models. In the end, we provide a classification of and the categorical descriptions of all 1-dimensional (the spatial dimension) gapped quantum phases with a finite onsite symmetry.

cond-mat.str-el↗

arXiv2022

A unified view on symmetry, anomalous symmetry and non-invertible gravitational anomaly

In this paper, using 1+1D models as examples, we study symmetries and anomalous symmetries via multi-component partition functions obtained through symmetry twists, and their transformations under the mapping class group of spacetime. This point of view allows us to treat symmetries and anomalous symmetries as non-invertible gravitational anomalies (which are also described by multi-component partition functions, transforming covariantly under the mapping group transformations). This allows us to directly see how symmetry and anomalous symmetry constraint the low energy dynamics of the systems, since the low energy dynamics is directly encoded in the partition functions. More generally, symmetries, anomalous symmetries, non-invertible gravitational anomalies, and their combinations, can all be viewed as constraints on low energy dynamics. In this paper, we demonstrate that they all can be viewed uniformally and systematically as pure (non-invertible) gravitational anomalies.

cond-mat.str-el↗

arXiv2021

Topological Mott Insulators and Discontinuous $U(1)$ $θ$-Terms

We introduce a lattice field theory that describes the transition between a superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a $U(1)$ symmetry protected topological phase labeled by an integer level $k$ and possessing an even integer $2k\frac{e^2}{h}$ quantized Hall conductance. Our model differs from the usual $2+1$d XY model by a topological term that vanishes on closed manifolds and in the absence of an applied gauge field, which implies that the critical exponents of the SF-tMI transition are identical to those of the well-studied $2+1$d XY transition. Our formalism predicts a "level-shift" symmetry: in the absence of an applied gauge field, the bulk correlation functions of all local operators are identical for models differing by the topological term. %, for example, near the SF-MI and SF-tMI transitions, and hence the extremely well-studied critical exponents of the $2+1$d XY model apply to the SF-tMI transition. In the presence of a background gauge field, the topological term leads to a quantized Hall response in the tMI phase, and we argue that this quantized Hall effect persists in the vicinity of the phase transition into the SF phase. Our formalism paves the way for other exact lattice descriptions of symmetry-protected-topological (SPT) phases, and mappings of critical exponents between transitions from symmetry-breaking to trivial states and transitions from symmetry-breaking to SPT states. A similar "level shift" symmetry should appear between all group cohomology SPT states protected by the same symmetry.

cond-mat.str-el↗

arXiv2021

A Commuting Projector Model with a Non-zero Quantized Hall conductance

By ungauging a recently discovered lattice rotor model for Chern-Simons theory, we create an exactly soluble path integral on spacetime lattice for $U^κ(1)$ Symmetry Protected Topological (SPT) phases in $2+1$ dimensions with a non-zero Hall conductance. We then convert the path integral on a $2+1$d spacetime lattice into a $2$d Hamiltonian lattice model, and show that the Hamiltonian consists of mutually commuting local projectors. We confirm the non-zero Hall conductance by calculating the Chern number of the exact ground state. It has recently been suggested that no commuting projector model can host a nonzero Hall conductance. We evade this no-go theorem by considering a rotor model, with a countably infinite number of states per site.

cond-mat.str-el↗

arXiv2021

Low energy effective field theories of fermion liquids and mixed $U(1)\times \mathbb{R}^d$ anomaly

In this paper we study gapless fermionic and bosonic systems in $d$-dimensional continuum space with $U(1)$ particle-number conservation and $\mathbb{R}^d$ translation symmetry. We write down low energy effective field theories for several gapless phases where $U(1)\times \mathbb{R}^d$ is viewed as internal symmetry. The $U(1)\times \mathbb{R}^d$ symmetry, when viewed as an internal symmetry, has a mixed anomaly, and the different effective field theories for different phases must have the same mixed anomaly. Such a mixed anomaly is proportional to the particle number density, and can be measured from the distribution of the total momentum $\boldsymbol{k}_\text{tot}$ for low energy many-body states (\ie how such a distribution is shifted by $U(1)$ symmetry twist $\boldsymbol{a}$), as well as some other low energy universal properties of the systems. In particular, we write down low energy effective field theory for Fermi liquid with infinite number of fields, in the presence of both real space magnetic field and $\boldsymbol{k}$-space "magnetic" field. The effective field theory also captures the mixed anomaly, which constraints the low energy dynamics, such as determine the volume of Fermi surface (which is another formulation of Luttinger-Ward-Oshikawa theorem).

cond-mat.str-el↗

arXiv2020

Metallic states beyond Tomonaga-Luttinger liquids in one dimension

In this paper, we propose some new strongly correlated gapless states (or critical states) of spin-1/2 electrons in 1+1-dimensions, such as doped anti-ferromagnetic spin-1/2 Ising chain. We find doped anti-ferromagnetic Ising chain to be a different metallic phase from the doped ferromagnetic Ising chain, despite the two have identical symmetry. The doped anti-ferromagnetic Ising chain has a finite energy gap for all charge-1 fermionic excitations even without pairing caused by attractive interactions, resembling the pseudo-gap phase of underdoped high Tc superconductors. Applying a transverse field to the ferromagnetic and anti-ferromagnetic metallic phases can restore the $Z_2$ symmetry, which gives rise to two distinct critical points despite that the two transitions have exactly the same symmetry breaking pattern. We also propose new chiral metallic states. All those new gapless states are strongly correlated in the sense that they do not belong to the usual Tomonaga-Luttinger phase of fermions, i.e., they cannot be smoothly deformed into the non-interacting fermion systems of the same symmetry. Our non-perturbative results are obtained by noticing that gapless quantum systems have emergent categorical symmetries, i.e., non-invertible gravitational anomalies), which are described by multi-component partition functions that are modular covariant. This allows us to calculate the scaling dimensions and quantum numbers of all the low energy operators for those strongly correlated gapless states. This demonstrates an application of emergent categorical symmetries in determining low energy properties of strongly correlated gapless states, which are hard to obtain otherwise.

cond-mat.str-el↗

arXiv2020

Categorical symmetry and non-invertible anomaly in symmetry-breaking and topological phase transitions

For a zero-temperature Landau symmetry breaking transition in $n$-dimensional space that completely breaks a finite symmetry $G$, the critical point at the transition has the symmetry $G$. In this paper, we show that the critical point also has a dual symmetry - a $(n-1)$-symmetry described by a higher group when $G$ is Abelian or an algebraic $(n-1)$-symmetry beyond higher group when $G$ is non-Abelian. In fact, any $G$-symmetric system can be viewed as a boundary of $G$-gauge theory in one higher dimension. The conservation of gauge charge and gauge flux in the bulk $G$-gauge theory gives rise to the symmetry and the dual symmetry respectively. So any $G$-symmetric system actually has a larger symmetry called categorical symmetry, which is a combination of the symmetry and the dual symmetry. However, part (and only part) of the categorical symmetry must be spontaneously broken in any gapped phase of the system, but there exists a gapless state where the categorical symmetry is not spontaneously broken. Such a gapless state corresponds to the usual critical point of Landau symmetry breaking transition. The above results remain valid even if we expand the notion of symmetry to include higher symmetries and algebraic higher symmetries. Thus our result also applies to critical points for transitions between topological phases of matter. In particular, we show that there can be several critical points for the transition from the 3+1D $Z_2$ gauge theory to a trivial phase. The critical point from Higgs condensation has a categorical symmetry formed by a $Z_2$ 0-symmetry and its dual - a $Z_2$ 2-symmetry, while the critical point of the confinement transition has a categorical symmetry formed by a $Z_2$ 1-symmetry and its dual - another $Z_2$ 1-symmetry.

cond-mat.str-el↗

arXiv2020

Algebraic higher symmetry and categorical symmetry -- a holographic and entanglement view of symmetry

We introduce the notion of algebraic higher symmetry, which generalizes higher symmetry and is beyond higher group. We show that an algebraic higher symmetry in a bosonic system in $n$-dimensional space is characterized and classified by a local fusion $n$-category. We find another way to describe algebraic higher symmetry by restricting to symmetric sub Hilbert space where symmetry transformations all become trivial. In this case, algebraic higher symmetry can be fully characterized by a non-invertible gravitational anomaly (i.e. an topological order in one higher dimension). Thus we also refer to non-invertible gravitational anomaly as categorical symmetry to stress its connection to symmetry. This provides a holographic and entanglement view of symmetries. For a system with a categorical symmetry, its gapped state must spontaneously break part (not all) of the symmetry, and the state with the full symmetry must be gapless. Using such a holographic point of view, we obtain (1) the gauging of the algebraic higher symmetry; (2) the classification of anomalies for an algebraic higher symmetry; (3) the equivalence between classes of systems, with different (potentially anomalous) algebraic higher symmetries or different sets of low energy excitations, as long as they have the same categorical symmetry; (4) the classification of gapped liquid phases for bosonic/fermionic systems with a categorical symmetry, as gapped boundaries of a topological order in one higher dimension (that corresponds to the categorical symmetry). This classification includes symmetry protected trivial (SPT) orders and symmetry enriched topological (SET) orders with an algebraic higher symmetry.

cond-mat.str-el↗

arXiv2020

Classification of topological phases with finite internal symmetries in all dimensions

We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

math-ph↗