Physics – High Energy Physics – High Energy Physics - Phenomenology
Scientific paper
2012-01-19
Nucl. Phys. B, 860 (2012) 295-309
Physics
High Energy Physics
High Energy Physics - Phenomenology
Latex, 21 pages
Scientific paper
10.1016/j.nuclphysb.2012.03.002
Topological invariants for the 4D gapped system are discussed with application to the quantum vacua of relativistic quantum fields. Expression $\tilde{\cal N}_3$ for the 4D systems with mass gap defined in \cite{Volovik2010} is considered. It is demonstrated that $\tilde{\cal N}_3$ remains the topological invariant when the interacting theory in deep ultraviolet is effectively massless. We also consider the 5D systems and demonstrate how 4D invariants emerge as a result of the dimensional reduction. In particular, the new 4D invariant $\tilde{\cal N}_5$ is suggested. The index theorem is proved that defines the number of massless fermions $n_F$ in the intermediate vacuum, which exists at the transition line between the massive vacua with different values of $\tilde{\cal N}_5$. Namely, $ 2 n_F$ is equal to the jump $\Delta\tilde{\cal N}_5$ across the transition. The jump $\Delta\tilde{\cal N}_3$ at the transition determines the number of only those massless fermions, which live near the hypersurface $\omega=0$. The considered invariants are calculated for the lattice model with Wilson fermions.
Volovik Grigori E.
Zubkov M. . A.
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