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Journal of Physics F Metal Physics

Functional integral method in itinerant electron magnetism

S. Hirooka, Masao Shimizu

Published Oct 1, 1988• 8 citations
Publisher

Abstract

The authors investigate the functional integral method which is frequently used in the study of itinerant electron magnetism. It is pointed out that fluctuating fields consist of two parts with contrary natures and they should pay attention to their difference when some approximate treatments are introduced. So-called ambiguities in the functional integral formalism are removed by taking into proper consideration the local aspect in the time correlation of fluctuating fields. Dynamical susceptibilities, which reduce to those of the random phase approximation theory in the weak interaction limit, are derived within the gaussian approximation in the Hubbard model. Moreover, the optimum interaction form of spin and charge density operators in the Stratonovich-Hubbard transformation is determined by the variational method.

DOI
10.1088/0305-4608/18/10/013

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•Jan 1, 1994•4 citations
The authors investigate the functional integral method which is frequently used in the study of itinerant electron magnetism. It is pointed out that fluctuating fields consist of two parts with contrary natures and they should pay attention to their difference when some approximate treatments are introduced. So-called ambiguities in the functional integral formalism are removed by taking into proper consideration the local aspect in the time correlation of fluctuating fields. Dynamical susceptibilities, which reduce to those of the random phase approximation theory in the weak interaction limit, are derived within the gaussian approximation in the Hubbard model. Moreover, the optimum interaction form of spin and charge density operators in the Stratonovich-Hubbard transformation is determined by the variational method.
The static approximation in the functional integral theory of itinerant magnetism is critically investigated. For the two-site Hubbard model the local magnetic moment and the specific heat are calculated as functions of temperature and interaction strength. By comparison with the exact results the error in the static approximation is found to be large at low temperatures and for small interaction strengths. It is shown that the subdivision of the ground state into a phase without a local moment (small interaction) and a phase with local moment (large interaction) is induced by the static approximation. The specific heat is underestimated and the zero-point entropy is finite. The implications for the infinite Hubbard model are briefly discussed.
The spin susceptibility of the electron gas is studied using a functional integral formulation. A Hubbard-Stratonovich transformation is used to decouple the electron-electron interaction. A uniform Fock mean field is chosen, and effects of small fluctuations about this mean field are retained up to second order. The term that gives the usual spin random-phase approximation susceptibility is identified, with the ladder sum contained in the propagator for the field fluctuations. Computations of the static spin susceptibility outline the region of stability as a function of electron-gas coupling parameter and wave vector. These computations parallel those of Hamann and Overhauser [Phys. Rev. 143, 183 (1966)] for the Yukawa interaction. We use the analytic continuation method of Staunton et al. [Phys. Rev. Lett. 82, 3340 (1999)] to extend the calculations to nonzero frequency. The resulting spectral density shows the increase in low-energy spin fluctuations as the spin-density-wave transition is approached, and the development of a pseudogap in the spin fluctuation spectrum near the wave vector of the transition.
The application of functional integral methods and the Hubbard--Stratonovich transformation to the Hubbard model is discussed. For the attractive case, using a simple gauge transformation of the superconducting order parameter field, the effective action for the low-energy phase excitations is derived. Inclusion of the electromagnetic field is straightforward. For the repulsive case at half--filling use of a fluctuating spin reference frame allows a derivation of the effective nonlinear sigma model for arbitrary coupling strength U. Results for the spin-wave velocity and the spin stiffness are obtained for arbitrary U. Away from half--filling, at large U an effective action for doped fermions and the spins is found which is related to the slave fermion formulation of the correlated fermion problem. The formalism is generalized to include the full SU(2)xSU(2) symmetry of the Hubbard model. This allows the derivation of a Ginzburg-Landau like theory that includes both particle-hole and particle-particle excitations. (paper at the NATO workshop on the Hubbard model, San Sebastian, Spain, oct. 1993)
We establish a way to handle main collective fluctuations in correlated quantum systems based on a fluctuating local field concept. This technique goes beyond standard mean-field approaches, such as Hartree-Fock and dynamical mean-field theories (DMFT), as it includes a fluctuating classical field that acts on the leading order parameter of the system. Effective model parameters of this theory are determined from the variational principle, which allows one to resolve the Fierz ambiguity in decoupling of the local interaction term. In the saddle-point approximation for the fluctuating field our method reproduces the mean-field result. The exact numerical integration over this field allows one to consider nonlinear fluctuations of the global order parameter of the system while local correlations can be accounted for by solving the DMFT impurity problem. We apply our method to the magnetic susceptibility of finite Hubbard systems at half-filling and demonstrate that the introduced technique leads to a superior improvement of results with respect to parental mean-field approaches, without significant numerical complications. We show that the fluctuating local field method can be used in a very broad range of temperatures substantially below the N\'eel temperature of DMFT, which remains a major challenge for all existing theoretical approaches.
We describe a new microscopic approach for analyzing interacting electron systems with local moments or, in principle, any local order parameter. We specialize attention to the doped Mott insulator phase of the Hubbard model, where standard weak-coupling perturbation methods fail. A rotationally invariant Stratonovich-Hubbard field is introduced to decouple the static spin components of the interaction at each site. Charge degrees of freedom are then treated by ``slave'' Hartree-Fock in the presence of a spatially varying random spin field. The effective action reduces to a classical Heisenberg model at half filling, insuring that the system has (i) finite-range order at T>0 with an exponentially diverging correlation length and (ii) a one-electron Mott-Hubbard gap in the presence of disorder. Away from half-filling properties are determined by strongly non-Gaussian fluctuations in the amplitude and orientation of the local spin fields. The saddle point equations of the theory at zero temperature reduce to inhomogeneous Hartree-Fock, so that disordering of domain walls at finite temperature is in principle included. We present preliminary small system results for the intermediate and large interaction regimes obtained by Monte Carlo simulation of random spin field configurations.
Functional integral methods provide a way to define mean–field theories and to systematically improve them. For the Hubbard model and similar strong–correlation problems, methods based in particular on the Hubbard–Stratonovich transformation have however been plagued by difficulties to formulate the problem in a spin–rotation invariant way. Here a formalism circumventing this problem by using a space – and time–dependent spin reference axis is discussed. This formulation is then used to suggest a possible alternative to Nagaoka ferromagnetism in the strongly correlated Hubbard model in the vicinity of half–filling. Finally, some aspects of single–particle spectra in a simplified model for a short–range ordered antiferromagnet are discussed. PACS numbers: 71.28.+d, 75.10.Lp, 75.30.Kz 1.