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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., Comment: Correct some errors in 1+1D case. Add more examples for 2+1D crystalline FSPT protected by symmetries being the mixture of wallpaper groups with onsite time-reversal symmetry

We study topological defect lines (TDLs) in two-dimensional $\mathbb Z_N$-parafermoinic CFTs. Different from the bosonic case, in the 2d parafermionic CFTs, there exist parafermionic defect operators that can live on the TDLs and satisfy interesting fractional statistics. We propose a categorical description for these TDLs, dubbed as ``para-fusion category", which contains various novel features, including $\mathbb Z_M$ $q$-type objects for $M\vert N$, and parafermoinic defect operators as a type of specialized 1-morphisms of the TDLs. The para-fusion category in parafermionic CFTs can be regarded as a natural generalization of the super-fusion category for the description of TDLs in 2d fermionic CFTs. We investigate these distinguishing features in para-fusion category from both a 2d pure CFT perspective, and also a 3d anyon condensation viewpoint. In the latter approach, we introduce a generalized parafermionic anyon condensation, and use it to establish a functor from the parent fusion category for TDLs in bosonic CFTs to the para-fusion category for TDLs in the parafermionized ones. At last, we provide many examples to illustrate the properties of the proposed para-fusion category, and also give a full classification for a universal para-fusion category obtained from parafermionic condensation of Tambara-Yamagami $\mathbb Z_N$ fusion category., Comment: 45+4 pages

Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed.

In this lecture note, we give a basic introduction to the rapidly developing concepts of generalized symmetries, from the perspectives of both high energy physics and condensed matter physics. In particular, we emphasize on the (invertible) higher-form and higher-group symmetries. For the physical applications, we discuss the geometric engineering of QFTs in string theory and the symmetry-protected topological (SPT) phases in condensed matter physics. The lecture note is based on a short course on generalized symmetries, jointly given by Yi-Nan Wang and Qing-Rui Wang in Feb. 2023, which took place at School of Physics, Peking University (https://indico.ihep.ac.cn/event/18796/)., Comment: 83 pages

Braiding and fusion rules of topological excitations are indispensable topological invariants in topological quantum computation and topological orders. While excitations in 2D are always particle-like anyons, those in 3D incorporate not only particles but also loops -- spatially nonlocal objects -- making it novel and challenging to study topological invariants in higher dimensions. While 2D fusion rules have been well understood from bulk Chern-Simons field theory and edge conformal field theory, it is yet to be thoroughly explored for 3D fusion rules from higher dimensional bulk topological field theory. Here, we perform a field-theoretical study on (i) how loops that carry Abelian gauge fluxes fuse and (ii) how loops are shrunk into particles in the path integral, which generates fusion rules, loop-shrinking rules, and descendent invariants, e.g., quantum dimensions. We first assign a gauge-invariant Wilson operator to each excitation and determine the number of distinct excitations through equivalence classes of Wilson operators. Then, we adiabatically shift two Wilson operators together to observe how they fuse and are split in the path integral; despite the Abelian nature of the gauge fluxes carried by loops, their fusions may be of non-Abelian nature. Meanwhile, we adiabatically deform world-sheets of unknotted loops into world-lines and examine the shrinking outcomes; we find that the resulting loop-shrinking rules are algebraically consistent to fusion rules. Interestingly, fusing a pair of loop and anti-loop may generate multiple vacua, but fusing a pair of anyon and anti-anyon in 2D has one vacuum only. By establishing a field-theoretical ground for fusion and shrinking in 3D, this work leaves intriguing directions, e.g., symmetry enrichment, quantum gates, and physics of braided monoidal 2-category of 2-group., Comment: Title adjusted. Abstract, Intro and Discussions revised. about 30 pages, 5 figures. 9 tables

In the past decade, tremendous efforts have been made towards understanding fermionic symmetry protected topological (FSPT) phases in interacting systems. Nevertheless, for systems with continuum symmetry, e.g., electronic insulators, it is still unclear how to construct an exactly solvable model with a finite dimensional Hilbert space in general. In this paper, we give a lattice model construction and classification for 2D interacting electronic insulators. Based on the physical picture of $\mathrm{U(1)}_f$-charge decorations, we illustrate the key idea by considering the well known 2D interacting topological insulator. Then we generalize our construction to an arbitrary 2D interacting electronic insulator with symmetry $G_f=\mathrm{U(1)}_f \rtimes_{\rho_1,\omega_2} G$, where $\mathrm{U(1)}_f$ is the charge conservation symmetry and $\rho_1, \omega_2$ are additional data which fully characterize the group structure of $G_f$. Finally we study more examples, including the full interacting classification of 2D crystalline topological insulators., Comment: 6+11 pages, 2 figures, 3 tables

In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of equivalence classes of fermionic local unitary(FLU) transformations was proposed to systematically understand non-chiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors $(N^{ij}_{k},F^{ij}_{k},F^{ijm,\alpha\beta}_{kln,\chi\delta},n_{i},d_{i})$, which satisfy a set of nonlinear algebraic equations parameterized by phase factors $\Xi^{ijm,\alpha\beta}_{kl}$, $\Xi^{ij}_{kln,\chi\delta}$, $\Omega^{kim,\alpha\beta}_{jl}$ and $\Omega^{ki}_{jln,\chi\delta}$. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-Yamagami category for $\mathbb{Z}_{2N}$, which can be regarded as the $\mathbb{Z}_{2N}$ parafermion generalization of Ising fermionic topological phase., Comment: 51 pages, 3 figures

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., Comment: 37 pages

We propose a general construction of commuting projector lattice models for 2D and 3D topological phases enriched by U(1) symmetry, with finite-dimensional Hilbert space per site. The construction starts from a commuting projector model of the topological phase and decorates U(1) charges to the state space in a consistent manner. We show that all 2D U(1) symmetry-enriched topological phases which allow gapped boundary without breaking symmetry, can be realized through our construction. We also construct a large class of 3D topological phases with U(1) symmetry fractionalized on particles or loop excitations., Comment: 16 pages, 2 figures

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. In this paper, based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups, Comment: 6+11 pages, 2 tables. As published in Chin. Phys. Lett

Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing universal topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for loop-like excitations for 3D fermionic topological phases. Most surprisingly, we discovered new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with fermionic particles). The simplest example of such non-Abelian braiding statistics can be realized in interacting fermionic systems with a gauge group $\mathbb{Z}_2 \times \mathbb{Z}_8$ or $\mathbb{Z}_4 \times \mathbb{Z}_4$, and the physical origin of non-Abelian statistics can be viewed as attaching an open Majorana chain onto a pair of linked loops, which will naturally reduce to the well known Ising non-Abelian statistics via the standard dimension reduction scheme. Moreover, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also give rise to an alternative way to classify FSPT phases with unitary symmetries. We further compare the classification results for FSPT phases with arbitrary Abelian total symmetry $G^f$ and find systematical agreement with previous studies using other methods. We believe that the proposed framework of understanding three-loop braiding statistics (including both Abelian and non-Abelian cases) in interacting fermion systems applies for generic fermonic topological phases in 3D., Comment: 36 pages, 8 figures, all comments and suggestions are very welcome

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 $G_f$., Comment: 31 pages, 1 table, 4 figures. Common and suggestion are welcome

The construction and classification of symmetry-protected topological (SPT) phases in interacting bosonic and fermionic systems have been intensively studied in the past few years. Very recently, a complete classification and construction of space group SPT phases were also proposed for interacting bosonic systems. In this paper, we attempt to generalize this classification and construction scheme systematically into interacting fermion systems. In particular, we construct and classify point group SPT phases for 2D interacting fermion systems via lower-dimensional block-state decorations. We discover several intriguing fermionic SPT states that can only be realized in interacting fermion systems (i.e., not in free-fermion or bosonic SPT systems). Moreover, we also verify the recently conjectured crystalline equivalence principle for 2D interacting fermion systems. Finally, the potential experimental realization of these new classes of point group SPT phases in 2D correlated superconductors is addressed., Comment: 4+epsilon pages, 4 figures, 2 tables; with supplementary material; Published version

We study a ground-state ansatz for the single-hole doped $t$-$J$ model in two dimensions via a variational Monte Carlo (VMC) method. Such a single-hole wave function possesses finite angular momenta generated by hidden spin currents, which give rise to a novel ground state degeneracy in agreement with recent exact diagonalization (ED) and density matrix renormalization group (DMGR) results. We further show that the wave function can be decomposed into a quasiparticle component and an incoherent momentum distribution in excellent agreement with the DMRG results up to an $8\times 8 $ lattice. Such a two-component structure indicates the breakdown of Landau's one-to-one correspondence principle, and in particular, the quasiparticle spectral weight vanishes by a power law in the large sample-size limit. By contrast, turning off the phase string induced by the hole hopping in the so-called $\sigma\cdot t\text{-}J$ model, a conventional Bloch-wave wave function with a finite quasiparticle spectral weight can be recovered, also in agreement with the ED and DMRG results. The present study shows that a singular effect already takes place in the single-hole-doped Mott insulator, by which the bare hole is turned into a non-Landau quasiparticle with translational symmetry breaking. Generalizations to pairing and finite doping are briefly discussed., Comment: 12pages, 8figures, resubmitted

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 $G_f=\mathbb{Z}_2^f \times_{\omega_2} G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb{Z}_2^f = \{1,P_f\}$. 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., Comment: 73 pages, 18 figures, 7 tables; added more examples and one more appendix

Topological qauntum field theory(TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In $2+1$D, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasi-particle braiding statistics, chiral central charge and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in $3+1$D and it has been shown that loop like excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and loop like excitation in $3+1$D. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases., Comment: 13 pages, 1 figure

The classification and construction of symmetry protected topological (SPT) phases have been intensively studied in interacting systems recently. To our surprise, in interacting fermion systems, there exists a new class of the so-called anomalous SPT (ASPT) states which are only well defined on the boundary of a trivial fermionic bulk system. We first demonstrate the essential idea by considering an anomalous topological superconductor with time reversal symmetry $T^2=1$ in 2D. The physical reason is that the fermion parity might be changed locally by certain symmetry action, but is conserved if we introduce a bulk. Then we discuss the layer structure and systematical construction of ASPT states in interacting fermion systems in 2D with a total symmetry $G_f=G_b\times\mathbb{Z}_2^f$. Finally, potential experimental realizations of ASPT states are also addressed., Comment: published version

Classification and construction of symmetry protected topological (SPT) phases in interacting boson and fermion systems have become a fascinating theoretical direction in recent years. It has been shown that the (generalized) group cohomology theory or cobordism theory can give rise to a complete classification of SPT phases in interacting boson/spin systems. Nevertheless, the construction and classification of SPT phases in interacting fermion systems are much more complicated, especially in 3D. In this work, we revisit this problem based on the equivalent class of fermionic symmetric local unitary (FSLU) transformations. We construct very general fixed point SPT wavefunctions for interacting fermion systems. We naturally reproduce the partial classifications given by special group super-cohomology theory, and we show that with an additional $\tilde{B}H^2(G_b, \mathbb Z_2)$ (the so-called obstruction free subgroup of $H^2(G_b, \mathbb Z_2)$) structure, a complete classification of SPT phases for three-dimensional interacting fermion systems with a total symmetry group $G_f=G_b\times \mathbb Z_2^f$ can be obtained for unitary symmetry group $G_b$. We also discuss the procedure of deriving a general group super-cohomology theory in arbitrary dimensions., Comment: 48 pages, 35 figures, published version

Injecting a single hole into a one-dimensional Heisenberg spin chain is probably the simplest case of doping a Mott insulator. The motion of such a single hole will generally induce a many-body phase shift, which can be identified by an exact sign structure of the model known as the phase string. We show that the sign structure is nontrivial even in this simplest problem, which is responsible for the essential properties of Mott physics. We find that the characteristic momentum structure, the Luttinger liquid behavior, and the quantum phase interference of the hole under a periodic boundary condition, can all be attributed to it. We use the density matrix renormalization group (DMRG) numerical simulation to make a comparative study of the $t$-$J$ chain and a model in which the sign structure is switched off. We further show that the key DMRG results can be reproduced by a variational wave function with incorporating the correct sign structure. Physical implications of the sign structure for doped Mott insulators in general are also discussed., Comment: 16+5 pages, 8 figures

Based on three general guiding principles, i.e., no double occupancy constraint, accurate description of antiferromagnetism at half-filling, and the precise sign structure of the $t$-$J$ model, a new ground state wave function has been constructed recently [Weng, New J. Phys. 13, 103039 (2011)]. In this paper, we specifically study such kind of variational ground state for the one-hole-doped anisotropic two-leg $t$-$J$ ladder using variational Monte Carlo (VMC) method. The results are then systematically compared with those recently obtained by density matrix renormalization group (DMRG) simulation. An excellent agreement is found between the VMC and DMRG results, including a "quantum critical point" at the anisotropy parameter $\alpha=\alpha_c\approx0.7$ (with the parameters $t/J=3$), and the emergence of charge modulation and momentum (Fermi point) reconstruction at $\alpha>\alpha_c$ due to the quantum interference of the sign structure. In particular, the wave function indicates that a Landau's quasiparticle description remains valid at $\alpha<\alpha_c$ but fails at $\alpha>\alpha_c$ due to the breakdown of the one-to-one correspondence of momentum and translational symmetry of the hole. The explicit form of the wave function provides a direct understanding on how the many-body strong correlation effect takes place non-perturbatively in a doped Mott insulator, which sheds interesting light on the two-dimensional case where the same type of wave function was proposed to describe the cuprate superconductor., Comment: 12+9 pages, 10 figures

The antiferromagnetic Heisenberg spin chain of odd spin $S$ is in the Haldane phase with several defining physical properties, such as thermodynamical ground-state degeneracy, symmetry-protected edge states, and nonzero string order parameter. If nonzero hole concentration $\delta$ and hole hopping energy $t$ are considered, the spin chain is replaced by a spin-$S$ $t$-$J$ chain. The motivation of this paper is to generalize the discussions of the Haldane phase to the doped spin chain. The \emph{first result} of this paper is that, for the model considered here, the $\mathbb{Z}_2$ sign structure in the usual Ising basis can be totally removed by two consecutive unitary transformations consisting of a spatially local one and a nonlocal one. Direct from the sign structure, the \emph{second result} of this paper is that the Marshall theorem and the Lieb-Mattis theorem for pure spin systems are generalized to the $t$-$J$ chain for arbitrary $S$ and $\delta$. A corollary of the theorem provides us with the ground-state degeneracy in the thermodynamic limit. The \emph{third result} of this paper is about the phase diagram. We show that the defining properties of the Haldane phase survive in the small $t/J$ limit. The large $t/J$ phase supports a gapped spin sector with similar properties (ground-state degeneracy, edge state, and string order parameter) of the Haldane chain, although the charge sector is gapless., Comment: 14 pages, 4 figures. Published version

We present a quantum field theoretic description on the t$-$J model on a square lattice with dilute holes (i.e. near half-filling), based on the compact mutual Chern-Simons gauge theory. We show that, due to the presence of non-perturbative monopole plasma configuration from the antiferromagnetic background, holons (carrying electric charge) are linearly confined and strongly localized even without extrinsic disorder taken into account. Accordingly, the translation symmetry is spontaneously broken at ground state. Such an exotic localization is distinct from Anderson localization and essentially rooted in intrinsic Mott physics of the t$-$J model. Finally, a finite-temperature phase diagram is proposed. The metal-insulator transition observed in in-plane resistivity measurement is identified to a confinement-deconfinement transition from the perspective of gauge theory. The transition is characterized by the order parameter "Polyakov-line"., Comment: 8 papges, 1 figure, accepted by Nucl. Phys. B