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Classification of Emergent Weyl Spinors in Multi-Fermion Systems

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In the fermionic systems with topologically stable Fermi points, the emergent two-component Weyl fermions appear. We propose the topological classification of these fermions based on the two invariants composed of the two-component Green’s function. We define these invariants using the Wigner–Weyl formalism also in case of essentially inhomogeneous systems. In the case where values of these invariants are minimal (±1), we deal with emergent relativistic symmetry. The emergent gravity appears, and our classification of Weyl fermions gives rise to the classification of vielbein. Transformations between emergent relativistic Weyl fermions of different types correspond to parity conjugation, time reversal, and charge conjugation.

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Notes

  1. In the case of lattice models, we require that all fields depending on coordinates almost do not vary at the distances of the order of lattice spacing. Under these conditions, the sum over the lattice points in the majority of expressions may be replaced by an integral.

  2. More precisely, the operator \(\hat {Q}\) is Hermitian, but its inverse \(\hat {G}\) is considered in space of generalized (rather than ordinary) operator-valued functions, and the mentioned would be infinitely small correction to \(\hat {Q}\) actually has the meaning of the proper definition of \(\mathop {\hat {Q}}\nolimits^{ - 1} \). In space of ordinary operators, the inverse to \(\hat {Q}\) does not exist.

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ACKNOWLEDGMENTS

I am grateful to G.E. Volovik for useful discussions.

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Authors and Affiliations

  1. Physics Department, Ariel University, 40700, Ariel, Israel

    M. A. Zubkov

  2. Alikhanov Institute for Theoretical and Experimental Physics, National Research Center Kurchatov Institute, 117259, Moscow, Russia

    M. A. Zubkov

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Zubkov, M.A. Classification of Emergent Weyl Spinors in Multi-Fermion Systems. Jetp Lett. 113, 445–453 (2021). https://doi.org/10.1134/S0021364021070031

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