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Statistics of excitations play an essential role in understanding phases of matter. In this paper, we introduce a universal method for studying the generalized statistics of Abelian particles and extended excitations in lattices of any dimension. We compute the statistics using the Berry phase of a sequence of unitary operators that transports the excitations while canceling local ambiguities at each step. The sequence is derived from locality, using the Smith normal form. We show that the statistics are quantized invariants. Our method unifies the statistics for the braiding and fusion of particles and loops, and leads to the discovery of novel statistics for membrane excitations. The statistics can be interpreted as the quantum anomaly of a generalized global symmetry, which manifests as an obstruction to gauging the symmetry on lattices. Furthermore, we show that non-trivial statistics forbid short-range entangled states, establishing the dynamical consequence of anomalies in microscopic lattice models., Comment: 39 pages, 11 figures

We systematically construct and classify fault-tolerant logical gates implemented by constant-depth circuits for quantum codes using cohomology operations and symmetry. These logical gates are obtained from unitary operators given by symmetry-protected topological responses, which correspond to generators of group cohomology and can be expressed explicitly on the lattice using cohomology operations including cup product, Steenrod squares and new combinations of higher cup products called higher Pontryagin powers. Our study covers most types of the cohomology operations in the literature. This hence gives rise to several new classes of diagonal and non-diagonal logical gates in increasing Clifford hierarchies beyond the usual color code paradigm corresponding to an $n$-fold cup product, including the logical $R_k$ and multi-controlled $C^m R_k$ gates for codes defined in projective spaces. Implementing these gates could make it more efficient to compile specific types of quantum algorithms such as Shor's algorithm. We further extend the construction to quantum codes with boundaries, which generalizes the folding approach in color codes. We also present a formalism for addressable and parallelizable logical gates in LDPC codes via higher-form symmetries and apply it to construct addressable logical Clifford gates in expander-based codes including the asymptotically good LDPC codes and hypergraph-product codes. As a byproduct, we find new topological responses of finite higher-form symmetries using higher Pontryagin powers., Comment: 31 pages, 7 figures, 2 tables

We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a "fractional charge" under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enriched $S_3$ and $O(2)$ gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a "sandwich" construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified., Comment: 40 pages, 10 figures. added refs, minor edits

In the presence of crystalline symmetry, topologically ordered states can acquire a host of symmetry-protected invariants. These determine the patterns of crystalline symmetry fractionalization of the anyons in addition to fractionally quantized responses to lattice defects. Here we show how ground state expectation values of partial rotations centered at high symmetry points can be used to extract crystalline invariants. Using methods from conformal field theory and G-crossed braided tensor categories, we develop a theory of invariants obtained from partial rotations, which apply to both Abelian and non-Abelian topological orders. We then perform numerical Monte Carlo calculations for projected parton wave functions of fractional Chern insulators, demonstrating remarkable agreement between theory and numerics. For the topological orders we consider, we show that the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. Our results also yield invariants of continuum fractional quantum Hall states protected by spatial rotational symmetry., Comment: 17 pages, 7 figures. minor edits

We construct a family of infinitely many new candidate non-Abelian self-correcting topological quantum memories in $D\geq 5+1$ spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of $\mathbb{Z}_2^3$ higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group $\mathbb{D}_8$ gauge theory in (2+1)D. The simplest example includes a new non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability, and devise a probabilistic local cellular-automaton decoder., Comment: 20 pages, 5 figures

A (2+1)D topological ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It is recently discovered that there are "higher" versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states, which obstructs symmetry-preserving gapped edge state beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the "partial rotation" unitary which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with $\nu=1/3$, $1/5$, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determines if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented., Comment: 17 pages, 4 figures, minor edits

It is common in condensed matter systems for reflection ($R$) and time-reversal ($T$) symmetry to both be broken while the combination $RT$ is preserved. In this paper we study invariants that arise due to $RT$ symmetry. We consider many-body systems of interacting fermions with fermionic symmetry groups $G_f = \mathbb{Z}_2^f \times \mathbb{Z}_2^{RT}$, $U(1)^f \rtimes \mathbb{Z}_2^{RT}$, and $U(1)^f \times \mathbb{Z}_2^{RT}$. We show that (2+1)D invertible fermionic topological phases with these symmetries have a $\mathbb{Z} \times \mathbb{Z}_8$, $\mathbb{Z}^2 \times \mathbb{Z}_2$, and $\mathbb{Z}^2 \times \mathbb{Z}_4$ classification, respectively, which we compute using the framework of $G$-crossed braided tensor categories. We provide a many-body $RT$ invariant in terms of a tripartite entanglement measure, and which we show can be understood using an edge conformal field theory computation in terms of vertex states. For $G_f = U(1)^f \rtimes \mathbb{Z}_2^{RT}$, which applies to charged fermions in a magnetic field, the non-trivial value of the $\mathbb{Z}_2$ invariant requires strong interactions. For symmetry-preserving boundaries, the phases are distinguished by zero modes at the intersection of the reflection axis and the boundary. Additional invariants arise in the presence of translation or rotation symmetry., Comment: 7 + 8 pages, 3 + 1 figures

It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D $\mathbb{Z}_2$ gauge theory with an emergent fermion, and point out that pumping chiral $p+ip$ topological states gives rise to a $\mathbb{Z}_{8}$ 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization of $T^3$ (3-torus) and $T^2 \rtimes_{C_2} S^1$ (2-torus bundle over the circle) respectively, and pumping $p+ip$ states. Our considerations also imply the possibility of a logical $T$ gate by placing the code on $\mathbb{RP}^3$ and pumping a $p+ip$ topological state., Comment: 34 pages, 6 figures. minor revisions

We consider the $\mathbb{Z}_2$ toric code, surface code and Floquet code defined on a non-orientable surface, which can be considered as families of codes extending Shor's 9-qubit code. We investigate the fault-tolerant logical gates of the $\mathbb{Z}_2$ toric code in this setup, which corresponds to $e\leftrightarrow m$ exchanging symmetry of the underlying $\mathbb{Z}_2$ gauge theory. We find that non-orientable geometry provides a new way the emergent symmetry acts on the code space, and discover the new realization of the fault-tolerant Hadamard gate of 2d $\mathbb{Z}_2$ toric code on a surface with a single cross-cap, dubbed a non-orientable toric code. This Hadamard gate can be realized by a constant-depth local unitary circuit modulo non-locality caused by a cross-cap. Via folding, the non-orientable surface code can be turned into a bilayer local quantum code, where the folded cross-cap is equivalent to a bi-layer twist terminated on a gapped boundary and the logical Hadamard only contains local gates with intra-layer couplings. We further obtain the complete logical Clifford gate set for a stack of non-orientable surface codes. We then construct the honeycomb Floquet code in the presence of a single cross-cap, and find that the period of the sequential Pauli measurements acts as a $HZ$ logical gate on the single logical qubit, where the cross-cap enriches the dynamics compared with the orientable case. We find that the dynamics of the honeycomb Floquet code is precisely described by a condensation operator of the $\mathbb{Z}_2$ gauge theory, and illustrate the exotic dynamics of our code in terms of a condensation operator supported at a non-orientable surface., Comment: 26 pages, 19 figures. Added Sec. II D, major revision in Sec. III F, and numerous improvements. To appear in PRX Quantum

The theory of topological phases of matter predicts invariants protected only by crystalline symmetry, yet it has been unclear how to extract these from microscopic calculations in general. Here we show how to extract a set of many-body invariants $\{\Theta_{\text{o}}^{\pm}\}$, where ${\text{o}}$ is a high symmetry point, from partial rotations in (2+1)D invertible fermionic states. Our results apply in the presence of magnetic field and Chern number $C \neq 0$, in contrast to previous work. $\{\Theta_{\text{o}}^{\pm}\}$ together with $C$, chiral central charge $c_-$, and filling $\nu$ provide a complete many-body characterization of the topological state with symmetry group $G = \text{U}(1) \times_\phi [\mathbb{Z}^2 \rtimes \mathbb{Z}_M]$. Moreover, all these many-body invariants can be obtained from a single bulk ground state, without inserting additional defects. We perform numerical computations on the square lattice Hofstadter model. Remarkably, these match calculations from conformal and topological field theory, where $G$-crossed modular $S, T$ matrices of symmetry defects play a crucial role. Our results provide additional colorings of Hofstadter's butterfly, extending recently discovered colorings by the discrete shift and quantized charge polarization., Comment: 6+12 pages, 3+8 figures, minor edits to the main text and the appendix

A (2+1)D topologically ordered phase may or may not have a gappable edge, even if its chiral central charge $c_-$ is vanishing. Recently, it is discovered that a quantity regarded as a "higher" version of chiral central charge gives a further obstruction beyond $c_-$ to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the \textit{partial rotation} operator acting on the wavefunction of the topologically ordered state. This allows us to extract the higher central charge from a single wavefunction, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the $\nu=1/2$ bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to $\mathrm{U}(1)_2$ and Ising topological order respectively. The letter establishes a numerical method to obtain a set of obstructions to the gappable edge of (2+1)D bosonic topological order beyond $c_-$, which enables us to completely determine if a (2+1)D bosonic Abelian topological order has a gappable edge or not. We also point out that the expectation values of the partial rotation on a single wavefunction put a constraint on the low-energy spectrum of the bulk-boundary system of (2+1)D bosonic topological order, reminiscent of the Lieb-Schultz-Mattis type theorems., Comment: 21 pages, 13 figures. v4: added numerical simulations for the non-chiral \nu=2/3 FQH state. Accepted in Physical Review Letters

We discuss the codimension-1 defects of (2+1)D bosonic topological phases, where the defects can support fermionic degrees of freedom. We refer to such defects as fermionic defects, and introduce a certain subclass of invertible fermionic defects called "gauged Gu-Wen SPT defects" that can shift self-statistics of anyons. We derive a canonical form of a general fermionic invertible defect, in terms of the fusion of a gauged Gu-Wen SPT defect and a bosonic invertible defect decoupled from fermions on the defect. We then derive the fusion rule of generic invertible fermionic defects. The gauged Gu-Wen SPT defects give rise to interesting logical gates of stabilizer codes in the presence of additional ancilla fermions. For example, we find a realization of the CZ logical gate on the (2+1)D $\mathbb{Z}_2$ toric code stacked with a (2+1)D ancilla trivial atomic insulator, which is implemented by a finite depth circuit. We also investigate a gapped fermionic interface between (2+1)D bosonic topological phases realized on the boundary of the (3+1)D Walker-Wang model. In that case, the gapped interface can shift the chiral central charge of the (2+1)D phase. Among these fermionic interfaces, we study an interesting example where the (3+1)D phase has a spatial reflection symmetry, and the fermionic interface is supported on a reflection plane that interpolates a (2+1)D surface topological order and its orientation-reversal. We construct a (3+1)D exactly solvable Hamiltonian realizing this setup, and find that the model generates the $\mathbb{Z}_8$ classification of the (3+1)D invertible phase with spatial reflection symmetry and fermion parity on the reflection plane. We make contact with an effective field theory, known in literature as the exotic invertible phase with spacetime higher-group symmetry., Comment: 38 pages, 16 figures

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper we show how such gauge theories possess a higher-group global symmetry, which we study in detail. We derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] \in H^5(BG, U(1))$ obstruction that has appeared in prior work. We also show how the $d$-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We discover new logical gates in stabilizer codes using the $d$-group symmetry, such as a Controlled-Z gate in (3+1)D $\mathbb{Z}_2$ toric code., Comment: 42 pages, 6 figures; v3: added subsection 2.3

(3+1)D topological phases of matter can host a broad class of non-trivial topological defects of codimension-1, 2, and 3, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays a crucial role both in the classification of phases of matter and the possible fault-tolerant logical operations in topological quantum error correcting codes. In this paper, we study several examples of such higher codimension defects from distinct perspectives. We mainly study a class of invertible codimension-2 topological defects, which we refer to as twist strings. We provide a number of general constructions for twist strings, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects. We study some special examples in the context of $\mathbb{Z}_2$ gauge theory with fermionic charges, in $\mathbb{Z}_2 \times \mathbb{Z}_2$ gauge theory with bosonic charges, and also in non-Abelian discrete gauge theories based on dihedral ($D_n$) and alternating ($A_6$) groups. The intersection between twist strings and Abelian flux loops sources Abelian point charges, which defines an $H^4$ cohomology class that characterizes part of an underlying 3-group symmetry of the topological order. The equations involving background gauge fields for the 3-group symmetry have been explicitly written down for various cases. We also study examples of twist strings interacting with non-Abelian flux loops (defining part of a non-invertible higher symmetry), examples of non-invertible codimension-2 defects, and examples of interplay of codimension-2 defects with codimension-1 defects. We also find an example of geometric, not fully topological, twist strings in (3+1)D $A_6$ gauge theory., Comment: 70 pages, 42 figures

We investigate symmetry-preserving gapped boundary of (2+1)D topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)$^f$-symmetric (2+1)D fermionic topological phases, which are beyond chiral central charge $c_-$ and electric Hall conductivity $\sigma_H$. These obstructions are given by a simple Gauss-Milgram type formula valid for super-modular category, and regarded as a higher version of $c_-$ and $\sigma_H$., Comment: 44 pages, 13 figures

Given the algebraic data characterizing any (2+1)D bosonic or fermionic topological order with a global symmetry group $G = \mathrm{U}(1) \rtimes H$, we construct a (3+1)D topologically invariant path integral in the presence of a curved background $G$ gauge field, as an exact combinatorial state sum. Specifically, the $\mathrm{U}(1)$ component of the $G$ gauge field can have a non-trivial second Chern class, extending previous work that was restricted to flat $G$ bundles. Our construction expresses the $\mathrm{U}(1)$ gauge field in terms of a Villain formulation on the triangulation, which includes a 1-form $\mathbb{R}$ gauge field and 2-form $\mathbb{Z}$ gauge field. We develop a new graphical calculus for anyons interacting with "Villain symmetry defects", associated with the 1-form and 2-form background gauge fields. This graphical calculus is used to define the (3+1)D path integral, which can describe either a bosonic or fermionic symmetry-protected topological (SPT) phase. For example, we can construct the topological path integral on curved $\mathrm{U}(1)$ bundles for the (3+1)D fermionic topological insulator in class AII and topological superconductor in class AIII given appropriate (2+1)D fermionic symmetry fractionalization data; these then give invariants of 4-manifolds with Spin$^c$ or Pin$^c$ structures and their generalizations. The (3+1)D path integrals define anomaly indicators for the (2+1)D topological orders; in the case of Abelian (2+1)D topological orders, we derive by explicit computation all of the mixed $\mathrm{U}(1)$ anomaly indicator formulas proposed by Lapa and Levin. We also propose a Spin$^c$ generalization of the Gauss-Milgram sum, valid for super-modular categories., Comment: 34+21 pages, 11+7 figures

In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian. The exotic phase has time reversal ($T$) symmetry, and depends on a choice of the spacetime structure called the Wu structure. The exotic phase cannot be captured by the classification of any bosonic or fermionic topological phases, and thus gives a novel class of invertible topological phases. When the $T$ symmetry defect admits a spin structure, our construction reduces to a sort of the decorated domain wall construction, in terms of a bosonic theory with $T$ symmetry defects decorated with a fermionic phase that depends on a spin structure of the $T$ symmetry defect. By utilizing our path integral, we propose a lattice construction for the exotic phase that generates the $\mathbb{Z}_8$ classification of the (3+1)d invertible phase based on the Wu structure. This generalizes the $\mathbb{Z}_8$ classification of the $T$-symmetric (1+1)d topological superconductor proposed by Fidkowski and Kitaev. On oriented spacetime, this (3+1)d invertible phase with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT which has a topological ordered state with a semion on its boundary. Moreover, we propose a subclass of $G$-SPT phases based on the Wu structure labeled by a pair of cohomological data in generic spacetime dimensions. This generalizes the Gu-Wen subclass of fermionic SPT phases., Comment: 30 pages, 7 figures, typos corrected, minor revision made in appendix A, references added

Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group $G$, we show how to construct a (3+1)D topologically invariant path integral for a fermionic $G$ symmetry-protected topological state ($G$-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Equivalently, our construction provides an exact (3+1)D combinatorial state sum for a path integral of any FSPT that admits a symmetry-preserving gapped boundary, including the (3+1)D topological insulators and superconductors in class AII, AIII, DIII, and CII that arise in the free fermion classification. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form $\mathbb{Z}_2$ background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the $\mathbb{Z}_{16}$ anomaly indicator for time-reversal symmetric topological superconductors with ${\bf T}^2 = (-1)^F$. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all $\mathbb{Z}_{16}$ $\mathrm{Pin}^+$ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin$^+$ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure., Comment: 44+68 pages, 11+51 figures. v2: updated references, added Appendix F

We examine topological terms of $(2+1)$d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the $U(N)/U(1)^N$ flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group $\pi_3(U(N)/U(1)^N)=\mathbb{Z}$, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist ${N(N-1)\over 2}-1$ types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians., Comment: 29 pages; (v2) minor changes, refs updated

In this short paper, we argue that the chiral central charge $c_-$ of a (2+1)d topological ordered state is sometimes strongly constrained by 't Hooft anomaly of anti-unitary global symmetry. For example, if a (2+1)d fermionic TQFT has a time reversal anomaly with $T^2=(-1)^F$ labeled as $\nu\in\mathbb{Z}_{16}$, the TQFT must have $c_-=1/4$ mod $1/2$ for odd $\nu$, while $c_-=0$ mod $1/2$ for even $\nu$. This generalizes the fact that the bosonic TQFT with $T$ anomaly in a particular class must carry $c_-=4$ mod $8$ to fermionic cases. We also study such a constraint for fermionic TQFT with $U(1)\times CT$ symmetry, which is regarded as a gapped surface of the topological superconductor in class AIII., Comment: 16 pages, 2 figures, references added, accepted in Physical Review Research

We discuss a way to construct a commuting projector Hamiltonian model for a (3+1)d topological superconductor in class DIII. The wave function is given by a sort of string net of the Kitaev wire, decorated on the time reversal (T) domain wall. Our Hamiltonian is provided on a generic 3d manifold equipped with a discrete form of the spin structure. We will see how the 3d spin structure induces a 2d spin structure (called a Kasteleyn direction on a 2d lattice) on T domain walls, which makes possible to define fluctuating Kitaev wires on them. Upon breaking the T symmetry in our model, we find the unbroken remnant of the symmetry which is defined on the time reversal domain wall. The domain wall supports the 2d non-trivial SPT protected by the unbroken symmetry, which allows us to determine the SPT classification of our model, based on the recent QFT argument by Hason, Komargodski, and Thorngren., Comment: 32 pages, 15 figures. v2: minor changes

We study quantized non-local order parameters, constructed by using partial time-reversal and partial reflection, for fermionic topological phases of matter in one spatial dimension protected by an orientation reversing symmetry, using topological quantum field theories (TQFTs). By formulating the order parameters in the Hilbert space of state sum TQFT, we establish the connection between the quantized non-local order parameters and the underlying field theory, clarifying the nature of the order parameters as topological invariants. We also formulate several entanglement measures including the entanglement negativity on state sum spin TQFT, and describe the exact correspondence of the entanglement measures to path integrals on a closed surface equipped with a specific spin structure., Comment: 29 pages, 6 figures

We discuss a recipe to produce a lattice construction of fermionic phases of matter on unoriented manifolds. This is performed by extending the construction of spin TQFT via the Grassmann integral proposed by Gaiotto and Kapustin, to the unoriented pin$_\pm$ case. As an application, we construct gapped boundaries for time-reversal-invariant Gu-Wen fermionic SPT phases. In addition, we provide a lattice definition of (1+1)d pin$_-$ invertible theory whose partition function is the Arf-Brown-Kervaire invariant, which generates the $\mathbb{Z}_8$ classification of (1+1)d topological superconductors. We also compute the indicator formula of $\mathbb{Z}_{16}$ valued time-reversal anomaly for (2+1)d pin$_+$ TQFT based on our construction., Comment: 26 pages, 6 figures

We discuss a strategy to construct gapped boundaries for a large class of symmetry-protected topological phases (SPT phases) beyond group cohomology. This is done by a generalization of the symmetry extension method previously used for cohomological SPT phases. We find that this method allows us to construct gapped boundaries for time-reversal-invariant bosonic SPT phases and for fermionic Gu-Wen SPT phases for arbitrary finite internal symmetry groups., Comment: 22 pages, 4 figures

We examine (3+1)D topological ordered phases with $C_k$ rotation symmetry. We show that some rotation symmetric (3+1)D topological orders are anomalous, in the sense that they cannot exist in standalone (3+1)D systems, but only exist on the surface of (4+1)D SPT phases. For (3+1)D discrete gauge theories, we propose anomaly indicator that can diagnose the $\mathbb{Z}_k$ valued rotation anomaly. Since (3+1)D topological phases support both point-like and loop-like excitations, the indicator is expressed in terms of the symmetry properties of point and loop-like excitations, and topological data of (3+1)D discrete gauge theories., Comment: 21 pages, 7 figures

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U(1)$ charge conservation are both preserved. In this paper, we generalize the Lieb-Schultz-Mattis theorem to systems with higher-form symmetries, which act on extended objects of dimension $n > 0$. The prototypical lattice system with higher-form symmetry is the pure abelian lattice gauge theory whose action consists only of the field strength. We first construct the higher-form generalization of the Lieb-Schultz-Mattis theorem with a proof. We then apply it to the $U(1)$ lattice gauge theory description of the quantum dimer model on bipartite lattices. Finally, using the continuum field theory description in the vicinity of the Rokhsar-Kivelson point of the quantum dimer model, we diagnose and compute the mixed 't Hooft anomaly corresponding to the higher-form Lieb-Schultz-Mattis theorem., Comment: 20 pages, 4 figures

Resta proposed a definition of the electric polarization in one-dimensional systems in terms of the ground-state expectation value of the large gauge transformation operator. Vanishing of the expectation value in the thermodynamic limit implies that the system is a conductor. We study Resta's polarization amplitude (expectation value) in the $S=1/2$ XXZ chain and its several generalizations, in the gapless conducting Tomonaga-Luttinger Liquid phase. We obtain an analytical expression in the lowest-order perturbation theory about the free fermion point (XY chain), and an exact result for the Haldane-Shastry model with long-range interactions. We also obtain numerical results, mostly using the exact diagonalization method. We find that the amplitude exhibits a power-law scaling in the system size (chain length) and vanishes in the thermodynamic limit. On the other hand, the exponent depends on the model even when the low-energy limit is described by the Tomonaga-Luttinger Liquid with the same Luttinger parameter. We find that a change in the exponent occurs when the Umklapp term(s) are eliminated, suggesting the importance of the Umklapp terms., Comment: 30 pages, 3 figures

We study the Kitaev chain under generalized twisted boundary conditions, for which both the amplitudes and the phases of the boundary couplings can be tuned at will. We explicitly show the presence of exact zero modes for large chains belonging to the topological phase in the most general case, in spite of the absence of "edges" in the system. For specific values of the phase parameters, we rigorously obtain the condition for the presence of the exact zero modes in finite chains, and show that the zero modes obtained are indeed localized. The full spectrum of the twisted chains with zero chemical potential is analytically presented. Finally, we demonstrate the persistence of zero modes (level crossing) even in the presence of disorder or interactions., Comment: 15 pages, 7 figures