Modulated symmetries are internal symmetries which do not commute with spatial symmetries; dipolar symmetries are a prime example. We give a general recipe for constructing topological phases protected by modulated symmetries via a defect network construction, generalizing the crystalline equivalence principle to modulated symmetries. We demonstrate that modulated symmetries can be treated identically to unmodulated symmetries in the absence of spatial symmetries, but in the presence of spatial symmetries, some defect networks which are non-anomalous for unmodulated symmetries become anomalous for modulated symmetries. We apply this understanding to classify symmetry-protected topological phases protected by translation symmetry plus either discrete or continuous dipolar symmetries in (1+1)D and (2+1)D and obtain a number of other (1+1)D classification results for modulated symmetry-protected topological phases.
Modulated symmetries are internal symmetries which do not commute with spatial symmetries; dipolar symmetries are a prime example. We give a general recipe for constructing topological phases protected by modulated symmetries via a defect network construction, generalizing the crystalline equivalence principle to modulated symmetries. We demonstrate that modulated symmetries can be treated identically to unmodulated symmetries in the absence of spatial symmetries, but in the presence of spatial symmetries, some defect networks which are non-anomalous for unmodulated symmetries become anomalous for modulated symmetries. We apply this understanding to classify symmetry-protected topological phases protected by translation symmetry plus either discrete or continuous dipolar symmetries in (1+1)D and (2+1)D and obtain a number of other (1+1)D classification results for modulated symmetry-protected topological phases.
We investigate the physics of one-dimensional symmetry protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls to spatially modulated symmetry defects, and develop several tools for characterizing and classifying modulated SPT phases. A salient feature of modulated symmetries is that they are generically only present for open chains, and are broken upon the imposition of periodic boundary conditions. Nevertheless, we show that SPT order is present even with periodic boundary conditions, a phenomenon we understand within the context of an object we dub a ``bundle symmetry''. In addition, we show that modulated SPT phases can avoid a certain no-go theorem, leading to an unusual algebraic structure in their matrix product state descriptions.
Modulated symmetries are internal symmetries that act in a spatially non-uniform manner. Consequently, when a modulated symmetry Gint is combined with a spatial symmetry Gsp, the total symmetry group takes the form of a semidirect product G=Gint⋊Gsp. Using matrix product states, we classify topological phases protected by modulated symmetries together with spatial symmetries in one spatial dimension. We show that these modulated symmetry-protected topological (SPT) phases are classified by H2(G,U(1)s), in agreement with the crystalline equivalence principle, which states that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally. Furthermore, we provide a matrix product state derivation of the Lyndon-Hochschild-Serre spectral sequence for the corresponding internal SPT phases, which enables us to construct an explicit correspondence between modulated SPT phases and internal SPT phases. As applications of this classification, we prove a Lieb-Schultz-Mattis (LSM) theorem for modulated symmetries that forbids the existence of symmetric short-ranged entangled ground state, as well as an SPT-LSM constraint that enforces nontrivial entanglement in the SPT ground state. Finally, we use the classification to establish a similar LSM-type constraints for non-invertible Kramers-Wannier reflection symmetries.
We investigate the physics of one-dimensional symmetry-protected topological (SPT) phases protected by symmetries whose symmetry generators exhibit spatial modulation. We focus in particular on phases protected by symmetries with linear (i.e., dipolar), quadratic, and exponential modulations. We present a simple recipe for constructing modulated SPT models by generalizing the concept of decorated domain walls to spatially modulated symmetry defects, and develop several tools for characterizing and classifying modulated SPT phases. A salient feature of modulated symmetries is that they are generically only present for open chains, and are broken upon the imposition of periodic boundary conditions. Nevertheless, we show that SPT order is present even with periodic boundary conditions, a phenomenon we understand within the context of an object we dub a ``bundle symmetry.'' In addition, we show that modulated SPT phases can avoid a certain no-go theorem, leading to an unusual algebraic structure in their matrix product state descriptions.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
In order to understand topological phases of matter that exist in crystalline solids, it is necessary to take into account their spatial symmetries, which gives a notion of crystalline topological phases. Here, the authors give a general picture for understanding such phases. The idea is that ground states in such phases can be schematically represented by a crystalline network of defects imprinted on a topological substrate. Equivalence classes of such defect networks under symmetric deformations correspond precisely to crystalline topological phases.
Symmetry-protected topological (SPT) phases with either internal or spatial symmetries have been studied in different methods within this decade. In this paper, we study the classification and characterization of 3+1d topological phases protected simultaneously by both spatial and internal symmetries from the topological response theory. Physical observables derived from these response theories are discussed and compared with their analogs in 2+1d systems, some of which have been well-studied in quantum Hall (QH) systems, like the Wen-Zee (WZ) term. We then study the bulk topological gauge theories for the corresponding SPT phases. Generalizations to symmetry-enriched topological (SET) phases are also discussed at the end of this paper.