Abstract
We study relativistic fermionic systems in 3 + 1 spacetime dimensions at finite chemical potential and zero temperature, from a path-integral point of view. We show how to properly account for the iε term that projects on the finite density ground state, and compute the path integral analytically for free fermions in homogeneous external backgrounds, using complex analysis techniques. As an application, we show that the U(1) symmetry is always linearly realized for free fermions at finite charge density, differently from scalars. We study various aspects of finite density QED in a homogeneous magnetic background. We compute the free energy density, non-perturbatively in the electromagnetic coupling and the external magnetic field, obtaining the finite density generalization of classic results of Euler-Heisenberg and Schwinger. We also obtain analytically the magnetic susceptibility of a relativistic Fermi gas at finite density, reproducing the de Haas-van Alphen effect. Finally, we consider a (generalized) Gross-Neveu model for N interacting fermions at finite density. We compute its non-perturbative effective potential in the large-N limit, and discuss the fate of the U(1) vector and \( {\mathbb{Z}}_2^A \) axial symmetries.
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Acknowledgments
It is a pleasure to thank Alberto Nicolis for collaboration at the early stages of this work and for useful discussions. We also thank Austin Joyce for discussions and collaboration on related topics. LS is supported by the French Centre National de la Recherche Scientifique (CNRS). AP is supported by the DOE grant DE-SC0011941.
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Podo, A., Santoni, L. Fermions at finite density in the path integral approach. J. High Energ. Phys. 2024, 182 (2024). https://doi.org/10.1007/JHEP02(2024)182
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DOI: https://doi.org/10.1007/JHEP02(2024)182