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Topological phases protected by crystalline symmetries and internal symmetries are shown to enjoy fascinating one-to-one correspondence in classification. Here we investigate the physics content behind the abstract correspondence in three or higher-dimensional systems. We show correspondence between anomalous boundary states, which provides a new way to explore the quantum anomaly of symmetry from its crystalline equivalent counterpart. We show such correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical surface states. (1) First of all, for the surface SET correspondence, we demonstrate it by considering examples involving time-reversal symmetry and mirror symmetry. We show that one 2D topological order can carry the time reversal anomaly as long as it can carry the mirror anomaly and vice versa, by directly establishing the mapping of the time reversal anomaly indicators and mirror anomaly indicators. Besides, we also consider other cases involving continuous symmetry, which leads us to introduce some new anomaly indicators for symmetry from its counterpart. (2) Furthermore, we also build up direct correspondence for (near) critical boundaries. Again taking topological phases protected by time reversal and mirror symmetry as examples, the direct correspondence of their (near) critical boundaries can be built up by coupled chain construction that was first proposed by Senthil and Fisher. The examples of critical boundary correspondence we consider in this paper can be understood in a unified framework that is related to \textithierarchy structure of topological $O(n)$ nonlinear sigma model, that generalizes the Haldane's derivation of $O(3)$ sigma model from spin one-half system.

In the past decade, there has been a systematic investigation of symmetry-protected topological (SPT) phases in interacting fermion systems. Specifically, by utilizing the concept of equivalence classes of finite-depth fermionic symmetric local unitary (FSLU) transformations and the fluctuating decorated symmetry domain wall picture, a large class of fixed-point wave functions have been constructed for fermionic SPT (FSPT) phases. Remarkably, this construction coincides with the Atiyah-Hirzebruch spectral sequence, enabling a complete classification of FSPT phases. However, unlike bosonic SPT phases, the stacking group structure in fermion systems proves to be much more intricate. The construction of fixed-point wave functions does not explicitly provide this information. In this paper, we employ FSLU transformations to investigate the stacking group structure of FSPT phases. Specifically, we demonstrate how to compute stacking FSPT data from the input FSPT data in each layer, considering both unitary and anti-unitary symmetry, up to 2+1 dimensions. As concrete examples, we explicitly compute the stacking group structure for crystalline FSPT phases in all 17 wallpaper groups and the mixture of wallpaper groups with onsite time-reversal symmetry using the fermionic crystalline equivalence principle. Importantly, our approach can be readily extended to higher dimensions, offering a versatile method for exploring the stacking group structure of FSPT phases.

$\mathcal{N}=1$ superconformal minimal models are the first series of unitary conformal field theories (CFTs) extending beyond Virasoro algebra. Using coset constructions, we characterize CFTs in $\mathcal{N}=1$ superconformal minimal models using combinations of a parafermion theory, an Ising theory and a free boson theory. Supercurrent operators in the original theory also becomes sums of operators from each constituent theory. If we take our $\mathcal{N}=1$ superconformal theories as the neutral part of the edge theory of a fractional quantum Hall state, we present a systematic way of calculating its ground state wavefunction using free field methods. Each ground state wavefunction is known previously as a sum of polynomials with distinct clustering behaviours. Based on our decomposition, we find explicit expressions for each summand polynomial. A brief generalization to $S_3$ minimal models using coset construction is also included.

We construct a family of one-dimensional (1D) quantum lattice models based on $G$-graded unitary fusion category $\mathcal{C}_G$. This family realize an interpolation between the anyon-chain models and edge models of 2D symmetry-protected topological states, and can be thought of as edge models of 2D symmetry-enriched topological states. The models display a set of unconventional global symmetries that are characterized by the input category $\mathcal{C}_G$. While spontaneous symmetry breaking is also possible, our numerical evidence shows that the category symmetry constrains the the models to the extent that the low-energy physics has a large likelihood to be gapless.

The natural existence of crystalline symmetry in real materials manifests 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.

The boundary of topological phases of matter can manifest its topology nature, which leads to the so-called boundary-bulk correspondence (BBC) of topological phases. In this Letter, we construct a one-to-one correspondence between the boundary theories of fermionic SPT (fSPT) phases protected by crystalline symmetry and on-site symmetry in 2D fermionic systems, which follow the so-called crystalline equivalence principle. We dub such correspondence crystalline equivalent BBC. We illustrate this correspondence by two simple examples and as an application, we discover the topological boundary theory of 2D fSPT phase with spin-1/2 fermions, protected by a non-Abelian group $\mathbb{Z}_4\rtimes\mathbb{Z}_2^T$ with $A^4=\mathcal{T}^2=P_f$ where $A$ is the generator of $\mathbb{Z}_4$, from its crystalline equivalent partner -- 2D higher-order fSPT phase with spinless fermions, protected by $D_4$ symmetry.

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking by topological orders. In this work, we systematically study enforced breaking of a general finite group $G_f$ by a class of topological orders, namely 0D, 1D and 2D fermionic invertible topological orders. Mathematically, the symmetry group $G_f$ is a central extension of a bosonic group $G$ by the fermion parity group $Z_2^f$, characterized by a 2-cocycle $\lambda$ $\in H^2(G,Z_2)$. With some minor assumptions and for given $G$ and $\lambda$, we are able to obtain a series of criteria on the existence or non-existence of enforced symmetry breaking by the fermionic invertible topological orders. Using these criteria, we discover many examples that are not known previously. For 2D systems, we define the physical quantities to describe symmetry-enriched invertible topological orders and derive some obstruction functions using both fermionic and bosonic languages. In the latter case which is done via gauging the fermion parity, we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. We also study enforced breaking of the continuous group $SU_f(N)$ by 2D invertible topological orders through a different argument.

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 \emphcomplete 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 \emphanomaly 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.

In this work we investigate the decorated domain wall construction in bosonic group-cohomology symmetry-protected topological (SPT) phases and related quantum anomalies in bosonic topological phases. We first show that a general decorated domain wall construction can be described mathematically as an Atiyah-Hirzebruch spectral sequence, where the terms on the $E_2$ page correspond to decorations by lower-dimensional SPT states at domain wall junctions. For bosonic group-cohomology SPT phases, the spectral sequence becomes the Lyndon-Hochschild-Serre (LHS) spectral sequence for ordinary group cohomology. We then discuss the physical interpretations of the differentials in the spectral sequence, particularly in the context of anomalous SPT phases and symmetry-enriched gauge theories. As the main technical result, we obtain a full description of the LHS spectral sequence concretely at the cochain level. The explicit formulae are then applied to explain Lieb-Schultz-Mattis theorems for SPT phases, and also derive a new LSM theorem for easy-plane spin model in a $\pi$ flux lattice. We also revisit the classifications of symmetry-enriched 2D and 3D Abelian gauge theories using our results.

Abelian Chern-Simons theory, characterized by the so-called $K$ matrix, has been quite successful in characterizing and classifying Abelian fractional quantum hall effect (FQHE) as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT(fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry $G_f$. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry $G_f$ has three new features: (1) it may support gapless majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if $G_f$ is a nontrivial extention of bosonic symmetry $G_b$ over $\mathbb{Z}_2^f$, (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories (CFT) with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge $n-1$ for Type-III bosonic SPT phases protected by $(\mathbb{Z}_n)^3$ symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used here might be generalized to derive the edge theories of fSPT phases with arbitrary unitary finite Abelian total symmetry $G_f$.

We propose an exotic scenario that topological superconductivity can emerge by doping strongly interacting fermionic systems whose spin degrees of freedom form bosonic symmetry protected topological (SPT) state. Specifically, we study a 1-dimensional (1D) example where the spin degrees of freedom form a spin-1 Haldane phase. Before doping, the charge and spin degrees of freedom are both gapped. Upon doping, the charge channel becomes gapless and is described by a $c=1$ compactified bosonic conformal field theory (CFT), while the spin channel remains gapped and still form a bosonic SPT state. Interestingly, an instability toward $p$-wave topological superconductivity is induced coexisting with the symmetry protected spin edge modes that are inherited from the Haldane phase. This scenario is confirmed by density-matrix renormalization group simulation of a concrete lattice model, where we find that topological superconductivity is robust against interactions. We further show that by stacking doped Haldane phases an exotic 2D anisotropic superconductor can be realized, where the boundaries transverse to the chain-direction are either gapless or spontaneously symmetry-broken due to the Lieb-Schultz-Mattis (LSM) anomaly.

We classify symmetry fractionalization and anomalies in a (3+1)d U(1) gauge theory enriched by a global symmetry group $G$. We find that, in general, a symmetry-enrichment pattern is specified by 4 pieces of data: $\rho$, a map from $G$ to the duality symmetry group of this $\mathrm{U}(1)$ gauge theory which physically encodes how the symmetry permutes the fractional excitations, $\nu\in\mathcal{H}^2_{\rho}[G, \mathrm{U}_\mathsf{T}(1)]$, the symmetry actions on the electric charge, $p\in\mathcal{H}^1[G, \mathbb{Z}_\mathsf{T}]$, indication of certain domain wall decoration with bosonic integer quantum Hall (BIQH) states, and a torsor $n$ over $\mathcal{H}^3_{\rho}[G, \mathbb{Z}]$, the symmetry actions on the magnetic monopole. However, certain choices of $(\rho, \nu, p, n)$ are not physically realizable, i.e. they are anomalous. We find that there are two levels of anomalies. The first level of anomalies obstruct the fractional excitations being deconfined, thus are referred to as the deconfinement anomaly. States with these anomalies can be realized on the boundary of a (4+1)d long-range entangled state. If a state does not suffer from a deconfinement anomaly, there can still be the second level of anomaly, the more familiar 't Hooft anomaly, which forbids certain types of symmetry fractionalization patterns to be implemented in an on-site fashion. States with these anomalies can be realized on the boundary of a (4+1)d short-range entangled state. We apply these results to some interesting physical examples.

In this paper we attempt to understand Lorentzian tensor networks, as a preparation for constructing tensor networks that can describe more exotic backgrounds such as black holes. To define notions of reference frames and switching of reference frames on a tensor network, we will borrow ideas from the algebraic quantum field theory literature. With these definitions, we construct simple examples of Lorentzian tensor networks and solve the spectrum for a choice of ``inertial frame'' based on Gaussian models of fermions and integrable models. In particular, the tensor network can be viewed as a periodically driven Floquet system, that by-pass the ``doubling problem'' and gives rise to fermions with exactly linear dispersion relations. We will find that a boost operator connecting different inertial frames, and notions of ``Rindler observers'' can be defined, and that important physics in Lorentz invariant QFT, such as the Unruh effect, can be captured by such skeleton of spacetime. We find interesting subtleties when the same approach is directly applied to bosons -- the operator algebra contains commutators that take the wrong sign -- resembling bosons behind horizons.

Symmetry fractionalization (SF) on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in 2D, the understanding on SF in 3D is far from complete. A long-standing challenge is to understand SF on looplike topological excitations which are spatially extended objects. In this work, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of SF. For systems with Abelian gauge groups ($G_g$) and Abelian symmetry groups ($G_s$), we successfully establish equivalence classes that lead to a finite number of patterns of SF, although there are notoriously infinite number of Lagrangian-descriptions of the system. We compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of SF. We present detailed calculations on many concrete examples. As an example, we find that the SF in an untwisted $\mathbb{Z}_2\times \mathbb{Z}_2$ topological order with $\mathbb{Z}_2$ symmetry is classified by $ (\mathbb{Z}_2)^6\oplus (\mathbb{Z}_2)^2\oplus (\mathbb{Z}_2)^2\oplus (\mathbb{Z}_2)^2$. If the topological order is twisted, the classification reduces to $(\mathbb{Z}_2)^6$ in which particle excitations always carry integer charge. Pure algebraic formalism of the classification is given by: $ \bigoplus_{\nu_i} \mathcal H^4 ( G_g\leftthreetimes_{\nu_i}G_s, U (1)) /\Gamma_\omega ({\nu_i}) \,$. Several future directions are proposed.

In this paper, we present an exactly solvable model for two dimensional topological superconductor with helical Majorana edge modes protected by time reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two dimensional lattice, which was used for the construction of the symmetry protected fermion phase with $Z_2$ symmetry in Ref. \onlineciteTarantino2016,Ware2016. By decorating the time reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two dimensional topological superconductor. From our construction, it can be seen that the $T^2=-1$ transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with $T^2=1$, our construction does not work.

While two-dimensional symmetry-enriched topological phases ($\mathsf{SET}$s) have been studied intensively and systematically, three-dimensional ones are still open issues. We propose an algorithmic approach of imposing global symmetry $G_s$ on gauge theories (denoted by $\mathsf{GT}$) with gauge group $G_g$. The resulting symmetric gauge theories are dubbed "symmetry-enriched gauge theories" ($\mathsf{SEG}$), which may be served as low-energy effective theories of three-dimensional symmetric topological quantum spin liquids. We focus on $\mathsf{SEG}$s with gauge group $G_g=\mathbb{Z}_{N_1}\times\mathbb{Z}_{N_2}\times\cdots$ and on-site unitary symmetry group $G_s=\mathbb{Z}_{K_1}\times\mathbb{Z}_{K_2}\times\cdots$ or $G_s=\mathrm{U(1)}\times \mathbb{Z}_{K_1}\times\cdots$. Each $\mathsf{SEG}(G_g,G_s)$ is described in the path integral formalism associated with certain symmetry assignment. From the path-integral expression, we propose how to physically diagnose the ground state properties (i.e., $\mathsf{SET}$ orders) of $\mathsf{SEG}$s in experiments of charge-loop braidings (patterns of symmetry fractionalization) and the \emphmixed multi-loop braidings among deconfined loop excitations and confined symmetry fluxes. From these symmetry-enriched properties, one can obtain the map from $\mathsf{SEG}$s to $\mathsf{SET}$s. By giving full dynamics to background gauge fields, $\mathsf{SEG}$s may be eventually promoted to a set of new gauge theories (denoted by $\mathsf{GT}^*$). Based on their gauge groups, $\mathsf{GT}^*$s may be further regrouped into different classes each of which is labeled by a gauge group ${G}^*_g$. Finally, a web of gauge theories involving $\mathsf{GT}$, $\mathsf{SEG}$, $\mathsf{SET}$ and $\mathsf{GT}^*$ is achieved. We demonstrate the above symmetry-enrichment physics and the web of gauge theories through many concrete examples.