Coherent-state path integrals in quantum thermodynamics

In these notes, we elucidate some subtle aspects of coherent-state path integrals, focusing on their application to the equilibrium thermodynamics of quantum many-particle systems. These subtleties emerge when evaluating path integrals in the continuum, either in imaginary time or in Matsubara-frequency space. Our central message is that, when handled with due care, the path integral yields results identical to those obtained from the canonical Hamiltonian approach. We illustrate this through a pedagogical treatment of several paradigmatic systems: the bosonic and fermionic harmonic oscillators, the single-site Bose-Hubbard and Hubbard models, the weakly-interacting Bose gas with finite-range interactions, and the BCS superconductor with finite-range interactions.

💡 Research Summary

The manuscript provides a thorough and pedagogical treatment of coherent‑state path integrals (CSPI) as a tool for calculating equilibrium thermodynamic quantities of quantum many‑body systems. The authors focus on the subtle technical issues that arise when one passes to the continuum limit, either in imaginary time or in Matsubara‑frequency space, and they demonstrate that, when these issues are handled correctly, the CSPI reproduces exactly the results obtained from the canonical Hamiltonian formalism.

The paper begins with a concise introduction that stresses the central role of the partition function (or grand partition function) in determining the Helmholtz free energy or grand potential. While the operator‑based trace evaluation is straightforward for non‑interacting systems, interacting Hamiltonians contain non‑commuting operators and usually require perturbative diagrammatic expansions. The CSPI offers an alternative by expressing the trace as a functional integral over complex (bosonic) or Grassmann (fermionic) fields, thereby bridging the gap between operator methods and field‑theoretic techniques.

In Section 2 the authors construct the CSPI for bosons and fermions. For bosons they introduce complex coherent variables (a(\tau), a^{*}(\tau)) and derive the discretized action by inserting (M) resolutions of the identity. By normal‑ordering the Hamiltonian and expanding the short‑time propagator to first order in (\delta\tau), they obtain the familiar Euclidean action
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