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Journal de Physique I

Extension of the Popov-Fedotov method to arbitrary spin

O. Veits, R. Oppermann, M. Binderberger, J. Stein

Published Apr 1, 1994• 19 citations
Publisher

Abstract

The partition function and thermodynamic expectation values of a spin system with arbitrary spin quantum number are expressed by an exact fermionic representation on Fock space. Extending a suggestion of Popov and Fedotov, this is achieved by introducing a set of imaginary chemical potentials. Spin operators are replaced by usual bilinear combinations of Fermi operators satisfying the spin commutation relations.

DOI
10.1051/jp1:1994154

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The partition function and thermodynamic expectation values of a spin system with arbitrary spin quantum number are expressed by an exact fermionic representation on Fock space. Extending a suggestion of Popov and Fedotov, this is achieved by introducing a set of imaginary chemical potentials. Spin operators are replaced by usual bilinear combinations of Fermi operators satisfying the spin commutation relations.
We show that the Popov-Fedotov trick of mapping spin-1/2 lattice systems onto two-component fermions with imaginary chemical potential readily generalizes to bosons with a fixed (but not limited) maximal site occupation number as well as to fermionic Hamiltonians with various constraints on the site Fock states. In a general case, the mapping---fermionization---is on multicomponent fermions with many-body non-Hermitian interactions. Additionally, the fermionization approach allows one to convert large many-body couplings into single-particle energies, rendering the diagrammatic series free of large expansion parameters; the latter is essential for the efficiency and convergence of the diagrammatic Monte Carlo method.
I derive a mapping of the Hubbard and t-J models onto a description in terms of spins and spin-less fermions. This transformation can be combined with the Fedotov-Popov fermionisation technique, where spin-1/2 systems are mapped onto two-component lattice fermions with imaginary chemical potential. The result is a generalisation of the Fedotov-Popov formulation to the case of finite doping that makes it possible to treat the Hubbard and t-J models as spin-models in the presence of a finite density of holes and doublons. While strongly interacting lattice fermions near half-filling is an extremely challenging problem, spin-systems can be efficiently treated with current simulation techniques.
Clock-spin models are attracting great interest, due to both their rich phase diagram and their connection to parafermions. In this context, we derive an exact local mapping from clock-spin to fermionic partition functions. Such mapping, akin to techniques introduced by Fedotov and Popov for spin \frac{1}{2} 1 2 chains, grants access to well established numerical tools for the perturbative treatment of fermionic systems in the clock-spin framework. Moreover, in one dimension, it allows to use bosonization to access the low energy properties of clock-spin models. Finally, aside from the direct application in clock-spin models, this new mapping enables the conception of interesting fermionic models, based on the clock-spin counterparts.
I derive a dual description of lattice fermions, specifically focusing on the t-J and Hubbard models, that allow diagrammatic techniques to be employed efficiently in the strongly correlated regime, as well as for systems with a restricted Hilbert space. These constructions are based on spin-charge transformation, where the lattice fermions of the original model are mapped onto spins and spin-less fermions. This mapping can then be combined with Popov-Fedotov fermionisation, where the spins are mapped onto lattice fermions with imaginary chemical potential. The resulting models do not contain any large expansion parameters, even for strongly correlated systems. Also, they exhibit dramatically smaller corrections to the density matrix from nonlinear terms in the Hamiltonian. The combination of these two properties means that they can be addressed with diagrammatic methods, including simulation techniques based on stochastic sampling of diagrammatic expansions.
We present a general derivation of semi-fermionic representation for spin operators in terms of a bilinear combination of fermions in real and imaginary time formalisms. The constraint on fermionic occupation numbers is fulfilled by means of imaginary Lagrange multipliers resulting in special shape of quasiparticle distribution functions. We show how Schwinger–Keldysh technique for spin operators is constructed with the help of semi-fermions. We demonstrate how the idea of semi-fermionic representation might be extended to the groups possessing dynamic symmetries. We illustrate the application of semi-fermionic representations for various problems of strongly correlated and mesoscopic physics.
We present a general derivation of semi-fermionic representation for generators of SU(N) group as a bilinear combination of Fermi operators. The constraints are fulfilled by means of imaginary Lagrange multipliers. The important case of SU(2) group is discussed. We demonstrate how the idea of semi-fermionic representation might be extended to the groups possessing dynamic symmetries. As an example, SO(4) group is considered. We illustrate the application of semi-fermionic representations for various problems of strongly correlated physics.