EVOLUTION EQUATIONS FOR FERMION SYSTEMS IN THE CONTINUAL REPRESENTATION (original) (raw)
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MICROSCOPIC QUANTUM HYDRODYNAMICS OF 8YSTEl\1S OF FERMIONS: PART 1
The iundementel equanons of the microscopic quantum hydrodytuunics of Iermions i11 an extotuul elecrromagnerlc field (i.c., tlie pnrticlc balance cqllatioll, the II101IIcntlllll bnltusco cquat.iol1, tlu: CJWl'g,1'bnlnnco equatfon, and the magl1etic IllOlllellt beleuce cqua/.Íon) are derived IIsilJg tho Schrikliuger CC¡IlR,t.;OI/. TI/(' form of t-1wspin-spin illl-c~ractiol/ Hnmiltonien is spocilicá. 'Ib clase UIC sys(;()rJJ o! t-//() lmlance (~:II/,3./.ions Ior a lllll/Upartide
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It is well-known that operators of localized spins within a magnetic material satisfy neither fermionic nor bosonic commutation relations. Thus, to construct diagrammatic many-body perturbation theory requiring the Wick theorem, the spin operators are usually mapped to the bosonic ones with Holstein-Primakoff (HP) transformation being the most widely used in magnonics and spintronics literature. However, to make calculations tractable, the square root of operators in the HP transformation is expanded into a Taylor series truncated to some low order. This poses a question on the range of validity of the truncated HP transformation when describing nonequilibrium dynamics of localized spins interacting with each other or with conduction electron spins-a problem frequently encountered in numerous transport phenomena in magnonics and spintronics. Here we apply exact diagonalization techniques to a Hamiltonian of fermions (i.e., electrons) interacting with HP bosons versus a Hamiltonian of fermions interacting with the original localized spin operators to compare their many-body states and one-particle equilibrium and nonequilibrium Green's functions (GFs). We employ as a test bed a one-dimensional quantum Heisenberg ferromagnetic spinS XXX chain of N 7 sites, where S = 1 or S = 5/2, and the ferromagnet can be made metallic by allowing electrons to hop between the sites while interacting with the localized spins via sd exchange interaction. For these two different versions of the Hamiltonian of this model, we compare the structure of their ground states, time evolution of excited states, spectral functions computed from the retarded GF in equilibrium, and matrix elements of the lesser GF out of equilibrium. Interestingly, magnonic spectral function can be substantially modified by acquiring additional peaks due to quasibound states of electrons and magnons once the interaction between these subsystems is turned on. The Hamiltonian of fermions interacting with HP bosons gives an incorrect ground state and electronic spectral function unless a large number of terms are retained in the truncated HP transformation. Furthermore, tracking the nonequilibrium dynamics of localized spins over longer time intervals requires a progressively larger number of terms in truncated HP transformation, even if a small magnon density is excited initially, but the required number of terms is reduced when interaction with conduction electrons is turned on. Finally, we show that recently proposed [M. Vogl et al., Phys. Rev. Res. 2, 043243 (2020); J. König et al., SciPost Phys. 10, 007 (2021)] resummed HP transformation, where spin operators are expressed as polynomials in bosonic operators, resolves the trouble with truncated HP transformation while allowing us to derive an exact quantum many-body (manifestly Hermitian) Hamiltonian consisting of a finite and fixed number of boson-boson and electron-boson interacting terms.
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