Charge conjugation (C), mirror reflection (R), and time reversal (T) symmetries, along with internal symmetries, are essential for massless Majorana and Dirac fermions. These symmetries are sufficient to rule out potential fermion bilinear mass terms, thereby establishing a gapless free fermion fixed point phase, pivotal for symmetric mass generation (SMG) transition. In this work, we systematically study the anomaly of C-R-T-internal symmetry in all spacetime dimensions by analyzing the projective representation (i.e. the fractionalization) of the C-R-T-internal symmetry group in the quantum many-body Hilbert space on the lattice. By discovering the fermion-flavor-number-dependent C-R-T-internal symmetry's anomaly structure, we demonstrate an alternative way to derive the minimal flavor number for SMG, which shows consistency with known results from Kähler-Dirac fermion or cobordism classification. Our findings reveal that, in general spatial dimensions, either 8 copies of staggered Majorana fermions or 4 copies of staggered Dirac fermions admit SMG. By directly searching for 4-fermion interactions that form commuting stabilizers respecting all symmetry constraints, we can prove the explicit SMG gapping retained a unique ground state in the codespace. Furthermore, we establish the correspondence between the symmetry operators of staggered fermions and free fermions, which is instrumental in facilitating the analysis of symmetry fractionalization at the field theory level.
Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as
Z2-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity
(−1)F. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-
(−1)F group structures, e.g., Dirac spinor is in a projective representation of
Z2C×Z2P×Z2T but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T
2=(−1)F. The general theme may be coined as C-P-T or C-R-T fractionalization.
Symmetric mass generation (SMG) is a mechanism to give gapless fermions a mass gap by nonperturbative interactions without generating any fermion bilinear condensation. The previous studies of SMG have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this paper, we generalize the concept of SMG to Fermi liquid (FL) with a finite Fermi volume and discuss how to gap out the Fermi surfaces (FSs) by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of FS SMG in both
(1+1)-dimensional
[(1+1)D] and
(2+1)-dimensional FL systems when several FSs together cancel the FS anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a nonperturbative interaction mechanism is the only way to gap out the FSs. This symmetric FS reconstruction is in contrast to the conventional symmetry-breaking gapping mechanism in the FL. As a side product, our model provides a pristine one-dimensional lattice regularization for the
(1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction.
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 classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with
Z2 topological order, that is, on gapped
Z2 spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group
H2(G,Z2). This result leads us to a symmetry classification of gapped
Z2 spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and
SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric
Z2 spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter
K) that reduces the scaling dimension of the 3-4-5-0 interaction from
5 to
5K, making it relevant for
K<2/5. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.
We present a fermionic lattice Hamiltonian that exactly realizes the
U(1)L×U(1)R global symmetry and the associated chiral anomalies of a massless Dirac fermion in 1+1 dimensions. The construction couples Villain bosons to a Kitaev chain of Majorana fermions, where the bosonic fields are essential for evading Nielsen-Ninomiya-type no-go theorems. For two copies of our model, we realize the anomaly-free
U(1)3450 global symmetry and construct symmetric boundary conditions. We analytically demonstrate symmetric mass generation by mapping the symmetry-preserving six-fermion interactions to fermion bilinear terms using fermionic T-duality. Finally, by gauging general anomaly-free global symmetries, we obtain a broad class of lattice chiral gauge theories. As nontrivial applications, we compute the mass spectra of the lattice Schwinger model and the 3450 gauge theory, finding agreement with the corresponding continuum results.