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arXiv:2007.12534 · Physical review. B./Physical review. B

Collective magnetic fluctuations in Hubbard plaquettes captured by fluctuating local field method

Alexey N. Rubtsov, E. A. Stepanov, Alexander I. Lichtenstein

Published Dec 21, 2020• 19 citations
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Abstract

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.

DOI
10.1103/physrevb.102.224423

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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.
Dynamical mean-field theory (DMFT) provides an optimal local approximation for correlated lattice systems by mapping the lattice onto a self-consistent effective impurity model. To account for the missing long-range correlations, we propose a novel extended approach, which we term fluctuating dynamical mean-field theory (fDMFT). It incorporates collective fluctuations of auxiliary impurity models across different sites via functional integration. Technically, this method involves obtaining a family of DMFT solutions on a grid for a self-consistent auxiliary classical field applied to the lattice. While the result can, in principle, be improved diagrammatically, we find that the minimal version of the theory already yields accurate results, with lowest-order diagrammatic corrections offering only minor improvements. This consistent framework, based on our fluctuating local field concept, demonstrates superior performance for the nearly half-filled Hubbard model compared to other known diagrammatic extensions of DMFT.
We introduce regular series expansion for weakly and moderately correlated fermionic systems based on the fluctuating local field approach. The method relies on explicitly accounting for the leading fluctuating mode(s) and is therefore suitable for medium-sized lattices. It can also be used as a solver for the cluster approximations for infinite-size lattices. We introduce a classical fluctuating field coupled to a fermionic collective mode(s). In this way, fluctuations in selected modes are treated in a nonperturbative way. Other degrees of freedom are accounted for in the diagram expansion performed for each value of the fluctuating field. The method is benchmarked for the U/t=1U/t=1U/t=1 and U/t=2U/t=2U/t=2 Hubbard lattices at half filling. Results for susceptibility in the antiferromagnetic channel along with the spectral function are compared with the numerically exact quantum Monte Carlo data. Calculations up to the third order of the series expansion are performed and show a uniform convergence to the reference result for the susceptibility. This convergence is observed well below the dynamical mean-field theory (DMFT) Ne\'el temperature and makes our practically simple method applicable in a much wider temperature range than DMFT-based diagrammatic schemes.
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.
The impact of leading collective electronic fluctuations on a free energy of a prototype 1D model for molecular systems is considered within the recently developed fluctuating local field (FLF) approach. The FLF method is a nonperturbative extension of a mean-field theory, where a self-consistent effective constant field is replaced by a fluctuating one. Integrating the fluctuating field out numerically exactly allows one to account for collective electronic fluctuations mediated by this field without any assumptions on their magnitude, degree of nonlinearity, etc. Using a half-filled Hubbard ring as a benchmark system, we find that the FLF method noticeably improves a mean-field estimation for the free energy, in particular below the mean-field Ne\'el temperature. We further demonstrate that the mean-field result can be even more improved introducing a multimode FLF scheme that additionally takes into account subleading fluctuations. Possible applications for the thermodynamics of real molecules are also discussed.
In this thesis, we tackle various problems in strongly correlated electron systems, which can be addressed properly via the non-perturbative dynamical mean field theory (DMFT) approach using the continuous time quantum Monte Carlo method as an impurity solver. First, we revisit the old Nagaoka ferromagnetism problem in the U=oo Hubbard model and study the stability of the ferromagnetic state as a function of the temperature, the doping level, and the next-nearest-neighbor hopping t'. We then address the nature of the Mott transition in the two-dimensional Hubbard model at half-filling using cluster DMFT. Cluster DMFT can incorporate the short-range correlations beyond DMFT by extending the spatial range in which correlations are treated exactly to a finite cluster size. The non-local correlations reduce substantially the critical interaction U and modify the shape of the transition lines in the phase diagram. We then concentrate on the calculation of two-particle response functions from the ab initio perspective by means of computing the one-particle excitation spectrum using the combination of the density functional theory (DFT) and DMFT and extracting the two-particle irreducible vertex function from a local two-particle Green's function computed within DMFT. In particular, we derive the equations for calculating the magnetic/charge susceptibility and the pairing susceptibility in superconductivity. This approach is applied to the Hubbard model and the periodic Anderson model and we determine the phase diagram of magnetism and superconductivity in these models. We show that the superconducting phase is indeed stable near the magnetic phase where the pairing interaction mediated by spin fluctuations is dominantly enhanced. The non-local correlation effect to superconductivity is also discussed using the dual fermion approach and the dynamical vertex approximation. We finally apply the vertex function approach within DFT+DMFT to a Fe-based superconductor, BaFe2As2, and compute the dynamical magnetic susceptibility in this material. Our calculation results show a good agreement with the magnetic excitation spectra observed in a neutron scattering experiment. The response function calculation method derived in this thesis can capture both a localized and an itinerant nature of collective excitations in strongly correlated electron systems.
In this paper, we revisit the antiferromagnetic (AF) phase diagram of the single-band three-dimensional half-filled Hubbard model on a simple cubic lattice studied within the dynamical mean field theory (DMFT). Although this problem has been investigated extensively in the literature, a comprehensive understanding of the impact of the different one- and, in particular, two-particle local correlation functions of DMFT on the AF transition temperature is still missing. We have, hence, performed a fluctuation diagnostics of TNT_NTN​ with respect to different local bosonic fluctuations (charge, spin, particle-particle) contained in the two-particle vertex of DMFT. Our results indicate that, beyond weak coupling, the screening of the DMFT vertex by local fluctuations leads to an enhancement of TNT_NTN​ with respect to a random phase approximation (RPA) like calculation where this vertex is replaced by the bare interaction. The overall suppression of TNT_NTN​ in DMFT with respect to RPA is then solely due to the incoherence introduced by the DMFT self-energy in the one-particle Green's functions. This illustrates the Janus faced role of the local moment formation in the DMFT solution of the Hubbard model which leads to completely opposite effects in the one- and two-particle correlations functions.