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The exact ground state of the reduced BCS Hamiltonian is investigatednumerically for large system sizes and compared with the BCS ansatz. A"canonical'' order parameter is found to be equal to the largest eigenvalue ofYang's reduced density matrix in the thermodynamic limit. Moreover, thelimiting values of the exact analysis agree with those obtained for the BCSground state. Exact results for the ground state energy, level occupations anda pseudospin-pseudospin correlation function are also found to converge to theBCS values already for relatively small system sizes. However, discrepanciespersist for a pair-pair correlation function, for inter-level correlations ofoccupancies and for the fidelity susceptibility, even for large system sizeswhere these quantities have visibly converged to well-defined limits. Ourresults indicate that there exist non-perturbative corrections to the BCSpredictions in the thermodynamic limit.

Order parameter, correlation functions and fidelity susceptibility for the BCS model in the thermodynamic limit