Symmetry under time-reversal appears in the microscopic description of many physical systems. In a quantum mechanical setting it acts as an anti-unitary operator, so does not fall under general analyses based on unitary symmetries. In classifying zero temperature phases of matter in (1+1)d lattice models, the role of anti-unitary symmetries is, however, well-understood. In recent years, the Symmetry Topological Field Theory (SymTFT) approach to this classification has given a general framework to understand symmetries as topological defects, but does not naturally include anti-unitary symmetries. Following recent proposals in the literature, we adopt a symmetry-enriched SymTFT for a theory with both internal and time-reversal symmetry. In particular, we take a standard SymTFT associated with an internal unitary symmetry that is then enriched by a background time-reversal symmetry. A detailed analysis of the topological boundary conditions of this enriched SymTFT allows us to characterize the corresponding (1+1)d gapped phases that preserve the enriching symmetry (i.e. those that do not spontaneously break this symmetry in the ground state). Line operators in the SymTFT approach are related to non-local string-order parameters (with charged end-point operators) for SPT phases. These are subtle in the anti-unitary case and we explore them both on the lattice and in the continuum. We include an analysis of unitary string order parameters that reveal the Klein bottle SPT invariant. On the lattice, we show that the correct end-point charge coincides with the time-reversal-charge only when the end-point operator is hermitian.
Symmetry under time-reversal appears in the microscopic description of many physical systems. In a quantum mechanical setting it acts as an anti-unitary operator, so does not fall under general analyses based on unitary symmetries. In classifying zero temperature phases of matter in (1+1)d lattice models, the role of anti-unitary symmetries is, however, well-understood. In recent years, the Symmetry Topological Field Theory (SymTFT) approach to this classification has given a general framework to understand symmetries as topological defects, but does not naturally include anti-unitary symmetries. Following recent proposals in the literature, we adopt a symmetry-enriched SymTFT for a theory with both internal and time-reversal symmetry. In particular, we take a standard SymTFT associated with an internal unitary symmetry that is then enriched by a background time-reversal symmetry. A detailed analysis of the topological boundary conditions of this enriched SymTFT allows us to characterize the corresponding (1+1)d gapped phases that preserve the enriching symmetry (i.e. those that do not spontaneously break this symmetry in the ground state). Line operators in the SymTFT approach are related to non-local string-order parameters (with charged end-point operators) for SPT phases. These are subtle in the anti-unitary case and we explore them both on the lattice and in the continuum. We include an analysis of unitary string order parameters that reveal the Klein bottle SPT invariant. On the lattice, we show that the correct end-point charge coincides with the time-reversal-charge only when the end-point operator is hermitian.
We extend the Symmetry Topological Field Theory (SymTFT) framework beyond internal symmetries by including geometric data that encode spacetime symmetries. Concretely, we enrich the SymTFT of an internal symmetry by spacetime symmetries and study the resulting symmetry-enriched topological (SET) order, which captures the interplay between the spacetime and internal symmetries. We illustrate the framework by focusing on symmetries in {1+1} 1 + 1 D. To this end, we first analyze how gapped boundaries of {2+1} 2 + 1 D SETs affect the enriching symmetry, and apply this within the SymTFT framework to gauging and detecting anomalies of the {1+1} 1 + 1 D symmetry, as well as to classifying {1+1} 1 + 1 D symmetry-enriched phases. We then consider quantum spin chains and explicitly construct the SymTFTs for three prototypical spacetime symmetries: lattice translations, spatial reflections, and time reversal. For lattice translations, the interplay with internal symmetries is encoded in the SymTFT by translations permuting anyons, which causes the continuum description of the SymTFT to be a foliated field theory. Using this, we elucidate the relation between Lieb-Schultz-Mattis (LSM) anomalies and modulated symmetries and classify modulated symmetry-protected topological (SPT) phases. For reflection and time-reversal symmetries, the interplay can additionally be encoded by symmetry fractionalization data in the SymTFT, and we identify mixed anomalies and study gauging for such examples.
The classification of phases using categorical symmetries has greatly expanded the landscape of gapped and gapless phases. So far, however, these developments have largely been restricted to phases with unitary (higher-)categorical symmetries over C. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry Z2T, and show that the relevant physical structures are naturally described by fusion categories over R. A class of real fusion categories, which we call Galois-real fusion categories, provides the correct categorical model for anti-unitary symmetries. A simple example is the time-reversal symmetry Z2T itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories (GT)ω and Rep(GT) associated to anti-linear groups GT, as well as non-invertible time-reversal symmetries described by a real analogue of Tambara--Yamagami fusion categories. We then classify gapped phases enriched with anti-linear symmetries in terms of module categories over Galois-real fusion categories. We furthermore apply the categorical formulation to prove dualities (i.e. gauge or Morita equivalences) of anti-linear symmetries generated by gauging subgroups. Complementing this, we also develop a Symmetry Topological Field Theory (SymTFT) framework, in which Galois-real fusion categories arise as boundary conditions of a Z2T-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same Z2T-enriched SymTFT.
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.
Symmetry topological field theory (SymTFT) gives a holographic correspondence between systems with a global symmetry and a higher-dimensional topological field theory. In this framework, classification of gapped phases of matter in spacetime dimension 1+1D correspond to classifications of mechanisms to confine the SymTFT by condensing anyons. In this work, we extend these results to characterize gapless symmetry-protected topological states: symmetry-enriched gapless phases or critical points that exhibit edge modes protected by symmetry and topology. We establish a one-to-one correspondence between 1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From general physical considerations, we determine the set of data and consistency conditions required to define a 1+1D gSPT, and show that this data precisely matches that of symmetry-preserving partial confinement (or partially gapped boundaries) of 2+1D quantum double models. We illustrate this correspondence through a dimensional reduction (thin-slab) construction, which enables a physically-intuitive derivation of how properties of the gSPT such as edge modes, emergent anomalies, and stability to perturbations arise from the SymTFT perspective.ditions required to define a 1+1D gSPT and show that they fully determine the physics of the gSPT including edge modes and emergent anomaly.
Symmetry protected 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 a correspondence directly in two scenarios, including the anomalous symmetry-enriched topological orders (SET) and critical boundary 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. In this perspective, we first consider the edge-corner correspondence between edge theory as 1+1D conformal field theory of internal fermionic SPT and the 0+1D corner modes of (higher-order) crystalline fermionic SPT. By viewing the corner modes on 1D boundary as perturbed CFT is crucial insight for the correspondence, but also help to discover the boundary theory of some intrinsically interacting fermionic SPT, whcih are challenging. Next, we discuss this correspondence bosonic systems, 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 hierarchy structure of topological O(n) O ( n ) nonlinear sigma model (NL \sigma σ M), that generalizes the Haldane’s derivation of O(3) O ( 3 ) sigma model from spin one-half system. To our best of knowledge, this hierarchy structure of the O(n) O ( n ) NL \sigma σ M is first exposed, which inspires us to explore more interesting connection between different theories in the light of the boundary correspondence.
Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with time-reversal (TR) symmetry satisfying T2=1. Without extra symmetry, it has been found that Zp parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labeled by the divisors n of p. In the presence of TR symmetry, however, the enriched classification is characterized by three indices n, $\ensuremath{\kappa}$, and $\ensuremath{\mu}$, where $\ensuremath{\kappa}\ensuremath{\in}{\mathbb{Z}}_{2}$ denotes the linear or projective TR actions on the edges, and $\ensuremath{\mu}\ensuremath{\in}{\mathbb{Z}}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the Zr-symmetric parafermionic ground states, where r=p for trivial or topological phases, and r=p/n for SSB phases, each original gapped phase with odd r is divided into two phases, while each phase with even r is further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of these resulting topological phases, we can obtain the topologically protected degeneracies of their entanglement spectra.