Abstract
In this work we develop EPOCH: Equilibrium Propagator by Orthogonal polynomial CHain, a computationally efficient method to calculate the time-dependent equilibrium Green's functions, including the anomalous Green's functions of superconductors, to capture the time-evolution in large inhomogeneous systems. The EPOCH method generalizes the Chebyshev wave-packet propagation method from quantum chemistry and efficiently incorporates the Fermi-Dirac statistics that is needed for equilibrium quantum condensed matter systems. The computational cost of EPOCH scales only linearly in the system degrees of freedom, generating an extremely efficient algorithm also for very large systems. We demonstrate the power of the EPOCH method by calculating the time-evolution of an excitation near a superconductor-normal metal interface in two and three dimensions, capturing transmission as well as normal and Andreev reflections.
Highlights
The study of time evolution in quantum mechanical systems provides fundamental insights into the system’s underlying structure, allowing for the development of applications harnessing dynamical quantum effects, with prominent examples in quantum information processing [1,2,3,4]
To provide a simple starting point and ensure that our approach is applicable to condensed matter systems with or without superconductivity, we review the standard Bogoliubov–de Gennes (BdG) formalism
We develop a computationally efficient method EPOCH to extract the equilibrium thermal time-dependent Green’s functions directly in the time domain that excels for any large inhomogeneous system described by a time-independent Hamiltonian
Summary
The study of time evolution in quantum mechanical systems provides fundamental insights into the system’s underlying structure, allowing for the development of applications harnessing dynamical quantum effects, with prominent examples in quantum information processing [1,2,3,4]. We develop the equilibrium propagator by orthogonal polynomial chain, or EPOCH, method, to efficiently calculate the time-domain equilibrium Green’s functions for both normal and anomalous correlations in generic quantum condensed matter systems. The computational cost in the EPOCH method is kept extremely low as the method scales linearly with the degrees of freedom (system size) and with the longest evolved time, i.e., running the time evolution for twice as long will only take twice as long to compute, all else being equal This is because each expansion order of Eq (2) requires only one additional multiplication by H (on very general grounds is a sparse matrix), which follows from the three-term recursion relationship of the Legendre polynomials. EPOCH enables calculation of the response to pulse probes directly in the time domain, predicting observables and material properties measured by, e.g., scattering, polarizability, and transport [36]
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