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Fermi surface of heavy electron systems plays a fundamental role inunderstanding their variety of puzzling phenomena, for example, quantumcriticality, strange metal behavior, unconventional superconductivity and evenenigmatic phases with yet unknown order parameters. The spectroscopymeasurement of typical heavy fermion superconductor CeCoIn$_{5}$ hasdemonstrated multi-Fermi surface structure, which has not been in detailstudied theoretically in a model system like the Kondo-Heisenberg model. Inthis work, we make a step toward such an issue with revisiting theKondo-Heisenberg model. It is surprising to find that the usual self-consistentcalculation cannot reproduced the fermionology of the experimental observationof the system due to the unfounded sign binding between the hopping of theconduction electrons and the mean-field valence-bond order. To overcome suchinconsistency, we assume that the sign binding should be relaxed and themean-field valence-bond order can be considered as a free/fit parameter so asto meet with real-life experiments. Given the fermionology, the calculatedeffective mass enhancement, entropy, superfluid density and Knight shift areall in qualitative agreement with the experimental results of CeCoIn$_{5}$,which confirms our assumption. Our result supports a $d_{x^{2}-y^{2}}$-wavepairing structure in heavy fermion material CeCoIn$_{5}$. In addition, we havealso provided the scanning tunneling microscopy (STM) spectra of the system,which is able to be tested by the present STM experiments.

Fermionology in the Kondo-Heisenberg model: the case of CeCoIn$_{5}$