Abstract
We derive a rigorous, quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. We show how our scheme can be used to map a general many-fermion Hamiltonian and then consider two specific models that encode the fundamental physics of many fermionic systems, the Anderson impurity and Hubbard models. We use these models to demonstrate how efficient mappings of these Hamiltonians can be constructed using a judicious choice of index ordering of the fermions. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.
Highlights
The Meyer-Miller-Stock-Thoss (MMST) approach provides an exact prescription to map a Hamiltonian consisting of discrete states to one in terms of continuous Cartesian phase space variables
The Hilbert space constructed using M single-particle orbitals contains 2M many-body states, which, upon mapping, results in twice this number of phase space variables (2M+1). This exponential scaling with the number of single-particle orbitals is mildly ameliorated in cases involving a fixed number of fermions, N, where the 2M dimensional Hilbert space can be limited to the M!/N!(M−N)! dimensional Fock space, but in practice this still renders the MMST treatment infeasible in most cases
We show how our map can be applied to the Anderson and Hubbard models, which are representative of fermionic systems belonging to the impurity and lattice families, respectively
Summary
Quantum mechanical map of fermionic creation and annihilation operators to continuous Cartesian variables that exactly reproduces the matrix structure of the many-fermion problem. Dimensional Fock space, but in practice this still renders the MMST treatment infeasible in most cases To obviate this highly unfavorable scaling with the number of single-particle orbitals, one could instead consider directly mapping the fermionic creation and annihilation operators, which naturally encode the antisymmetry of the fermionic wavefunction by virtue of their anticommutivity. Our map provides an exact starting point for the application of the entire arsenal of quantum-classical and semiclassical techniques to investigate the statics and dynamics of problems involving many fermions These methods can be used to elucidate physical processes in systems ranging from electrochemical interfaces to nanojunctions and strongly correlated materials
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