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arXiv:2109.15320

C-P-T Fractionalization

Juven Wang

Published Sep 30, 2021• 1 citation
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Abstract

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-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 $(-1)^{\rm F}$. 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-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ 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$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

DOI
10.48550/arxiv.2109.15320

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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.
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\mathbb{C}C. In this work, we incorporate anti-unitary symmetries, such as time-reversal symmetry Z2T\mathbb{Z}_2^TZ2T​, and show that the relevant physical structures are naturally described by fusion categories over R\mathbb{R}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\mathbb{Z}_2^TZ2T​ itself. We discuss the basic structures of real fusion categories and present a range of examples, including the group-theoretical categories (GT)ω(G^T)^ω(GT)ω and Rep(GT)\mathsf{Rep}(G^T)Rep(GT) associated to anti-linear groups GTG^TGT, 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\mathbb{Z}_2^TZ2T​-enriched SymTFT. Morita equivalent anti-linear symmetries are shown to arise as different boundaries of the same Z2T\mathbb{Z}_2^TZ2T​-enriched SymTFT.
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We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2{\mathbb{Z}}_{2}Z2​ topological order, that is, on gapped Z2{\mathbb{Z}}_{2}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){H}^{2}(G,{\mathbb{Z}}_{2})H2(G,Z2​). This result leads us to a symmetry classification of gapped Z2{\mathbb{Z}}_{2}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)\mathrm{SO}(3)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{\mathbb{Z}}_{2}Z2​ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.