A bstract In this work, we explore topological phases of matter obtained by effectively gauging or fermionizing a system, where the Gauss law constraint is only enforced energetically. In contrast to conventional gauging or fermionization, the symmetry that is effectively gauged at low energies still generates a global symmetry that acts on the whole Hilbert space faithfully. This symmetry turns out to protect a nontrivial topological phase together with other symmetries, or it can carry a nontrivial emergent ’t Hooft anomaly. We provide a precise formula for the topological response action involving these symmetries in a general setup, as well as a formula for ’t Hooft anomalies. As an application, we apply the general treatment of the procedure to gapless systems and find various new gapless SPT phases, such as the one carrying the Gu-Wen fermionic anomalies at low energy.
A bstract In this work, we explore topological phases of matter obtained by effectively gauging or fermionizing a system, where the Gauss law constraint is only enforced energetically. In contrast to conventional gauging or fermionization, the symmetry that is effectively gauged at low energies still generates a global symmetry that acts on the whole Hilbert space faithfully. This symmetry turns out to protect a nontrivial topological phase together with other symmetries, or it can carry a nontrivial emergent ’t Hooft anomaly. We provide a precise formula for the topological response action involving these symmetries in a general setup, as well as a formula for ’t Hooft anomalies. As an application, we apply the general treatment of the procedure to gapless systems and find various new gapless SPT phases, such as the one carrying the Gu-Wen fermionic anomalies at low energy.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.
Symmetry is foundational in theoretical physics. In this thesis, we show how recent advances in symmetry have reinforced its power to classify, characterize, and constrain quantum systems non-perturbatively. We study symmetries, ranging from spatial symmetries to generalized symmetries, in both quantum field theory and quantum lattice systems. In particular, this thesis develops five sets of results. First, we show that ordinary ordered phases generically host emergent generalized symmetries in the infrared, associated with homotopy defects of the order-parameter space. We use these emergent generalized symmetries to characterize and classify defect-suppressed disorder transitions to phases with topological order and emergent gauge structures. Second, we turn to lattice anomalies and show how the chiral anomaly of a massless Dirac fermion in two spacetime dimensions can be realized on the lattice. In particular, we identify lattice realizations of the vector and axial symmetries and demonstrate how their anomaly is matched by the non-abelian Onsager algebra in the staggered fermion Hamiltonian. Third, we give a microscopic lattice understanding of T-duality in quantum spin chains using non-invertible symmetry. This non-invertible symmetry, together with lattice momentum and winding symmetries, also enforces gaplessness, hence providing a novel way to realize perturbative anomalies in quantum spin-chain models. Fourth, we show that in local lattice models of spinless fermions, fermion-number conservation combined with a non-onsite Majorana translation symmetry enforces a Fermi surface. This symmetry-enforced Fermi surface is a strong form of symmetry-enforced gaplessness. Finally, we introduce a framework of stratified symmetries and stratified anomalies to understand and discover new Lieb–Schultz–Mattis constraints. We apply this framework to modulated symmetries and find an explicit criterion for when local anomaly data imply obstructions to a trivial gapped phase. Together, these results demonstrate the broad applicability and power of symmetry and its generalizations, both for organizing quantum phases and for deriving robust non-perturbative constraints.
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.
We analyze 2+1d and 3+1d bosonic symmetry protected topological (SPT) phases of matter protected by onsite symmetry group G by using dual bulk and boundary approaches. In the bulk, we study an effective field theory, which upon coupling to a background flat G gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Further, SPTs can be gauged by summing over all isomorphism classes of flat G gauge fields to obtain Dijkgraaf-Witten topological G gauge theories. These topological gauge theories can be ungauged by first introducing and then proliferating defects that spoil the gauge symmetry. This mechanism is related to anyon condensation in 2+1d and condensing bosonic gauge charges in 3+1d. In the dual boundary approach, we study 1+1d and 2+1d quantum field theories that have G 't-Hooft anomalies that can be precisely canceled by (the response theory of) the corresponding bulk SPT. We show how to construct/compute topological invariants for the bulk SPTs directly from the boundary theories. Further, we sum over boundary partition functions with different background gauge fields to construct G characters that generate topological data for the bulk topological gauge theory. Finally, we study a 2+1d quantum field theory with a mixed ${\mathbb{Z}}_{2}^{T/R}\ifmmode\times\else\texttimes\fi{}\text{U}(1)$ anomaly where Z2T/R is time-reversal/reflection symmetry, and the U(1) could be a 0-form or 1-form symmetry depending on the choice of time reversal/reflection action. We briefly discuss the bulk effective action and topological response for a theory in 3+1d that cancels this anomaly. This signals the existence of SPTs in 3+1d protected by 0,1-form $\text{U}(1)\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T,R}$.
We show how 1+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e., nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity ${(\ensuremath{-}1)}^{F}$, while the antiunitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., ``parity'' anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multilayers of 1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher space-time dimensions via a cobordism theory.
Symmetry-protected topological (SPT) states have boundary 't Hooft anomalies that obstruct the effective boundary theory realized in its own dimension with UV completion and with an on-site G-symmetry.In this work, yet we show that a certain anomalous non-on-site G-symmetry along the boundary becomes on-site when viewed as an extended H-symmetry, via a suitable group extension 1 → K → H → G → 1. Namely, a nonperturbative global (gauge or gravitational) anomaly in G becomes anomaly free in H.This guides us to construct an exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries applicable to any SPT state of any finite symmetry group, including on-site unitary and antiunitary time-reversal symmetries.The resulting symmetric gapped boundary can be described either by an H-symmetry extended boundary in any spacetime dimension or, more naturally, by a topological emergent K-gauge theory with a global symmetry G on a 3 þ 1D bulk or above.The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry G, described by representations of H. (Applying our approach to a 1 þ 1D boundary of 2 þ 1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order.The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.)In contrast to known gapped boundaries or interfaces obtained via symmetry breaking (either global symmetry breaking or the Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension.More generally, applying our approach to SPT states, topologically ordered gauge theories, and symmetry enriched topologically ordered (SET) states leads to generic boundaries or interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.