In principle, there is no obstacle to gapping fermions preserving any global symmetry that does not suffer a 't Hooft anomaly. In practice, preserving a symmetry that is realised on fermions in a chiral manner necessitates some dynamics beyond simple quadratic mass terms. We show how this can be achieved using familiar results about supersymmetric gauge theories and, in particular, the phenomenon of confinement without chiral symmetry breaking. We present simple models that gap fermions while preserving a symmetry group under which they transform in chiral representations. For example, we show how to gap a collection of 4d fermions that carry the quantum numbers of one generation of the Standard Model, but without breaking electroweak symmetry. We further show how to gap fermions in groups of 16 while preserving certain discrete symmetries that exhibit a mod 16 anomaly.
We prove that in $(3+1)d$ quantum field theories with $\mathbb{Z}_N$ symmetry, certain anomalies forbid a symmetry-preserving vacuum state with a gapped spectrum. In particular, this applies to discrete chiral symmetries which are frequently present in gauge theories as we illustrate in examples. Our results also constrain the long-distance behavior of certain condensed matter systems such as Weyl-semimetals and may have applications to crystallographic phases with symmetry protected topological order. These results may be viewed as analogs of the Lieb-Schultz-Mattis theorem for continuum fiel…
We study gapped boundaries characterized by "fermionic condensates" in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/ junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We also connect these results with topological defects in super modular invariant CFTs. To render our discussion …
Boundary topological orders of (4+1)d fermionic $\mathbb{Z}_{2N}^{\mathrm{F}}$ SPT states
Yale University（耶鲁大学）；London Institute for Mathematical Sciences, Royal Institution（伦敦数学科学研究所，皇家学会）；Center of Mathematical Sciences and Applications, Harvard University（哈佛大学数学科学及应用中心）
We investigate symmetry-preserving gapped boundary of (2+1)D topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)$^f$-symmetric (2+1)D fermionic topological phases, which are beyond chiral central charge $c_-$ and ele…
In arXiv:2110.06932, we argued that the chiral central charge -- a topologically protected quantity characterizing the edge theory of a gapped (2+1)-dimensional system -- can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap progr…
Competition and duality correspondence between chiral and superconducting channels in (2+1)-dimensional four-fermion models with fermion number and chiral chemical potentials
An analogous "Strong CP problem" is identified in a toy model in 2-dimensional spacetime: a general 1+1d abelian U(1) anomaly-free chiral fermion and chiral gauge theory with a generic theta instanton term $\fracθ{2 π} \int F$. The theta term alone violates the charge-conjugation-time-reversal CT and the parity P discrete symmetries. The analogous puzzle here is the CT or P problem in 1+1d: Why can the $\barθ$ angle (including the effect of $θ$ and the complex phase of a mass matrix) be zero or small for a natural reason? We show that this CT or P problem can be solved by a Symmetric Mass Gene…
We concisely derive chiral magnetic effect through Wigner function approach for chiral fermion system. Then we derive chiral magnetic effect through solving the Landau levels of chiral fermions in detail. The procedures of second quantization and ensemble average lead to the equation of chiral magnetic effect for righthand and lefthand fermion systems. Chiral magnetic effect only comes from the contribution of the lowest Landau level. We carefully analyze the lowest Landau level, and find that all righthand (chirality is +1) fermions move along positive z-direction and all lefthand (chirality …
This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unif…
I discretize axion string configuration coupled to a Dirac fermion, which in the continuum binds a massless chiral fermion in its core when the winding is one. I show that such a configuration can host one or more chiral fermions when regulated on the lattice. Realization of these chiral fermions relies on the presence of Wilson-like terms similar to the Wilson term used in lattice domain wall fermions. The number of chiral fermions on the string jumps as the Wilson-like parameter is varied with respect to the other mass scales in the problem. These jumps coincide with phase transitions along …
Using results on topological band theory of phases of matter and discrete symmetries, we study topological properties of band structure of physical systems involving spin $\frac{1}{2}$ and $\frac{3}{2}$ fermions. We apply this approach to study partial breaking in 4D $\mathcal{N}=2$ gauged supergravity in rigid limit and we describe the fermionic gapless mode in terms of chiral anomaly. We study as well the homologue of the usual spin-orbit coupling $\vec{L}.\vec{S}$, that opens the vanishing band gap for free $s=\frac{1}{2}$ fermions; and show that is precisely given by the central extension …
