Non-extensive quantum statistics with particle–hole symmetry

• We generalize the KMS relation for non-exponential density matrices.
• We explore the requirement of the hole–antiparticle reinterpretation on smooth deformed exponential functions.
• We establish a connection between the mathematical forms of qq-exponential and κκ-exponential type approaches.
• We emphasize that the Sommerfeld expansion of a generalized Fermi distribution should contain only odd terms, beyond the leading 1/2.

Based on Tsallis entropy (1988) and the corresponding deformed exponential function, generalized distribution functions for bosons and fermions have been used since a while Teweldeberhan et al. (2003) and Silva et al. (2010). However, aiming at a non-extensive quantum statistics further requirements arise from the symmetric handling of particles and holes (excitations above and below the Fermi level). Naive replacements of the exponential function or “cut and paste” solutions fail to satisfy this symmetry and to be smooth at the Fermi level at the same time. We solve this problem by a general ansatz dividing the deformed exponential to odd and even terms and demonstrate that how earlier suggestions, like the κκ- and qq-exponential behave in this respect.