Bosonization with a background U(1) gauge field

Abstract

Bosonization is one of the most significant frameworks to analyze fermionic systems. In this work, we propose a new bosonization of Dirac fermion coupled with U (1) background gauge field. Our new bosonization is consistent with gauge invariance, global chiral anomaly matching and fermion-boson operator correspondence, either of which is not satisfied by previously developed bosonizations. The bilinear Dirac-mass term condensation paradox and its generalized form are resolved by our bosonization. This new bosonization approach to interacting systems also correctly captures the conformal characters of a significant class of critical lattice models, such as conformal dimensions and the conformal anomaly of XXZ spin chain with twisted boundary condition. Our work clarifies the equivalence of fermionic flux insertion and bosonic background charge insertion for two-dimensional conformal field theory. arXiv:1902.06584v5 [cond-mat.str-el]

Anomaly inflow and $p$-form gauge theories

“…Thus ď and ě can only contribute to the anomaly of the type discussed in Sec. 3. By restricting the background fields to be of the form Č = ši ⋆ či , there is no contribution to the anomaly at all.…”

“…Let η(D (q) ) be the η-invariant of the Dirac operator of a fermion with Z 8 charge q. By using the values of the η-invariants in Appendix D, one can check that η(D (3) ) and η(D (1) ) + 9η(D (3) ) generate the dual of the bordism groups, Hom(Z 32 , U(1)) and Hom(Z 2 , U(1)), where Z 32 and Z 2 are the ones appearing in Ω spin-Z8 5 = Z 32 ⊕ Z 2 . A generator of Hom(Z 9 , U(1)) (where Z 9 = Ω spin 5 (BZ 3 )) is η(D (1) ) in a similar notation.…”

Parafermionization, bosonization, and critical parafermionic theories

“…However, the Majorana fermions and the Ising spins are significantly different in nature in that the Majorana fermions, which are local excitations in Majorana system and obey fermion statistics, are forbidden to exist in the local excitation content of the Ising chain, whose local excitations are bosonic. Thus, more precisely speaking, the massless Majorana fermion chain is actually equivalent to a proper stacking [24][25][26][27] of the critical Ising chain and a gapped Kitaev chain in its Z 2 -topologically nontrivial phase [28] providing the fermionic nature. Therefore, the critical theories of fermions (e.g., the critical theory of Majorana chains) are called "fermionic conformal field theories" to be distinguished from the critical bosonic theories, i.e., the so-called conformal field theories (CFTs).…”

“…As a main method of our study, we first develop a (1+1)-dimensional parafermionization together with a bosonization as its inverse to relate a parafermionic theory to a bosonic theory by a one-to-one correspondence. It can be also regarded as an attachment construction using a nontrivial topological phase of a parafermionic chain [12,[14][15][16][17][18] generalizing the Kitaev-chain attachment argument in k = 2 [25][26][27]. From this viewpoint, parafermionic chains and bosonic chains are expected to be indistinguishable locally since their differences result from this global topological factor.…”

“…When k = 2, z s 1 ,s 2 a 1 ,a 2 reduces to the partition function of the nontrivial topological phase of the Kitaev chain as the Z 2 -Arf invariant [25][26][27], coupled with a background Z 2 -gauge field (a 1 , a 2 ). Additionally, when both the parafermionic and the bosonic theories are coupled to dynamic Z k -gauge fields, the resultant gauged theories are the same modular invariant theory with the orbifold partition function…”