The Grassmann time-evolving matrix product operator (GTEMPO) method, which represents the Feynman-Vernon influence functional as a temporal matrix product state, has been shown to be a flexible and potentially scalable solution for fermionic quantum impurity problems. In this work, we extend GTEMPO to solve fermionic impurity problems in the Nambu formalism, in which the impurity is coupled to a superconducting bath. A key insight is that by employing the Bogoliubov transformation for the superconducting bath, one could obtain the analytic expression of the Feynman-Vernon influence functional in a similar form to the case of a normal bath, after which the core algorithms of GTEMPO can be straightforwardly adapted. We demonstrate the accuracy of our method by benchmarking it against exact diagonalization in several exactly solvable cases, and against the continuous-time quantum Monte Carlo method using converged dynamical mean-field theory iterations on the imaginary contour in the nonintegrable case. In all cases, we perform both imaginary- and real-time calculations to illustrate the flexibility of our method. These results illustrate that our method, termed Nambu-GTEMPO, could be potentially useful as a quantum impurity solver to study the superconducting states in both equilibrium and nonequilibrium systems.

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• Impurities in superconductors
• Matrix product states
• Strongly correlated systems
• Tensor network methods