A bosonic perspective on the classical mapping of fermionic quantum dynamics (original) (raw)
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Physical Review B, 2010
We develop a new method that allows us to map models of interacting fermions onto bosonic models describing collective excitations in an arbitrary dimension. This mapping becomes exact in the thermodynamic limit in the presence of a bath. The boson models can be written either in the form of a model of non-interacting bosons in a fluctuating auxiliary field or in the form of a superfield theory of interacting bosons. We show how one can study the latter version using perturbation theory. Using the developed diagrammatic technique we compared the first two orders of perturbation theory with the corresponding results for the original fermion model and found a perfect agreement. As concerns the former representation, we suggest a scheme that may be suitable for Monte Carlo simulations and demonstrate that it is free of the fermionic sign problem. We discuss in details the properties of the bosonic representation and argue that there should not be any obstacles preventing from an efficient computation.
Mapping fermion and boson systems onto the Fock space of harmonic oscillators
Physical Review E, 2010
The fluctuation-dissipation theorem (FDT) is very general and applies to a broad variety of different physical phenomena in condensed matter physics. With the help of the FDT and following the famous work of Caldeira and Leggett, we show that, whenever linear response theory applies, any generic bosonic or fermionic system at finite temperature T can be mapped onto a fictitious system of free harmonic oscillators. To the best of our knowledge, this is the first time that such a mapping is explicitly worked out. This finding provides further theoretical support to the phenomenological harmonic oscillator models commonly used in condensed matter. Moreover, our result helps in clarifying an interpretation issue related to the presence and physical origin of the Bose-Einstein factor in the FDT.
Physics Today, 2005
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe.
Physical Review B
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.
A density-functional approach to fermionization in the 1D Bose gas
Journal of Physics B: Atomic, Molecular and Optical Physics, 2004
A time-dependent Kohn-Sham scheme for 1D bosons with contact interaction is derived based on a model of spinor fermions. This model is specifically designed for the study of the strong interaction regime close to the Tonks gas. It allows us to treat the transition from the strongly interacting Tonks-Girardeau to the weakly interacting quasicondensate regime and provides an intuitive picture of the extent of fermionization in the system. An adiabatic localdensity approximation is devised for the study of time-dependent processes. This scheme is shown to yield not only accurate ground-state properties but also overall features of the elementary excitation spectrum, which is described exactly in the Tonks-gas limit. S288 J Brand a Fermi system can be extended to arbitrary interaction strength at the expense of a highly singular interaction in the fermionic picture. For the homogeneous Bose gas, exact solutions for stationary states at arbitrary interaction strength can be found using the Bethe ansatz . Although the exact solutions show a continuous transition between the perturbative regime of weakly interacting bosons and the strongly correlated, fermionized regime, they do not prove very useful for the study of time-dependent processes or inhomogeneous situations. Neither do they provide us with a simple, intuitive picture of how strong the degree of fermionization is in a given system. The crossover from the quasicondensate to the fermionized Tonks gas has also been studied in Inspired by the works of Haldane [13], Sutherland and others on the concept of exclusion statistics describing a crossover between fermionic and bosonic statistics in 1D systems, we study a model which explicitly allows this transition and provides an intuitive picture of the degree of fermionization. In order to devise a practical scheme for treating time-dependent and inhomogeneous systems we employ DFT and develop a time-dependent Kohn-Sham formalism based on the auxiliary model system of N non-interacting spin-(ν−1)/2 fermions. This model is chosen because the spin degeneracy may simulate the level attraction or the bunching of single-particle quasi momenta in the interacting bosonic system . The interaction energy of the Bose system is simulated by the kinetic energy of the spin-degenerate fermions. In this work we will study the model in the simplest and most generic approximation: the adiabatic local-density approximation (ALDA). The limiting case of infinite interaction strength is obtained easily and is treated exactly with ν = 1. The opposite limit of weak interaction can also be treated accurately within the proposed formalism with ν = N where the perturbative Gross-Pitaevskii and Bogoliubov equations are recovered asymptotically. In the general case of arbitrary interaction strength, the spin degeneracy ν is fixed by requiring the correct low-energy asymptotics of the excitation spectrum, which is analysed in the framework of linear-response theory. The resulting model is suitable for the study of time-dependent processes in inhomogeneous 1D Bose gases close to the Tonks-gas limit. The properties of this approximate model are analysed and systematic improvements are suggested.
Boson-fermion model: An exact diagonalization study
Physical Review B, 2003
The main features of a generic boson-fermion scenario for electron pairing in a many-body correlated fermionic system are: i) a cross-over from a poor metal to an insulator and finally a superconductor as the temperature decreases, ii) the build-up of a finite amplitude of local electron pairing below a certain temperature T * , followed by the onset of long-range phase correlations among electron pairs below a second characteristic temperature T φ , iii) the opening of a pseudogap in the DOS of the electrons below T * , rendering these electrons poorer and poorer quasi-particles as the temperature decreases, with the electron transport becoming ensured by electron pairs rather than by individual electrons. A number of these features have been so far obtained on the basis of different many-body techniques, all of which have their built-in shortcomings in the intermediate coupling regime, which is of interest here. In order to substantiate these features, we investigate them on the basis of an exact diagonalization study on rings up to eight sites. Particular emphasis has been put on the possibility of having persistent currents in mesoscopic rings tracking the change-over from single-to two-particle transport as the temperature decreases and the superconducting state is approached.
Fermion-mediated long-range interactions of bosons in the one-dimensional Bose-Fermi-Hubbard model
Physical Review A, 2010
The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show besides bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a novel phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as the interplay between fermion-induced mean-field potential and long range interaction with alternating sign.
2021
The operators of localized spins within a magnetic material commute at different sites of its lattice and anticommute on the same site, so they are neither fermionic nor bosonic operators. Thus, to construct diagrammatic many-body perturbation theory, requiring the Wick theorem, the spin operators are usually mapped to the bosonic ones -- the most popular in magnonics and spintronics literature has been the Holstein-Primakoff (HP) transformation. However, the square root of operators in the HP transformation has to be expanded into an infinite series, which is then truncated to some low order to make calculations tractable. This poses a question on the {\em range of validity of truncated HP transformation} when describing nonequilibrium dynamics of localized spins interacting with conduction electron spins -- a problem frequently encountered in numerous transport phenomena in spintronics. Here we apply exact diagonalization techniques to Hamiltonian of fermions (i.e., electrons) int...
Fermion mediated long-range interactions of bosons in the 1D Bose-Fermi-Hubbard model
2009
The ground-state phase diagram of mixtures of spin polarized fermions and bosons in a 1D periodic lattice is discussed in the limit of large fermion hopping and half filling of the fermions. Numerical simulations performed with the density matrix renormalization group (DMRG) show besides bosonic Mott insulating (MI), superfluid (SF), and charge density-wave phases (CDW) a novel phase with spatial separation of MI and CDW regions. We derive an effective bosonic theory which allows for a complete understanding and quantitative prediction of the bosonic phase diagram. In particular the origin of CDW phase and the MI-CDW phase separation is revealed as the interplay between fermion-induced mean-field potential and long range interaction with alternating sign.