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Functional Integrals for Correlated Fermions

Heinz J. Schulz

Published Jan 1, 1994• 4 citations
OpenAlex

Abstract

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.

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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.
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)
The Hubbard model is investigated using functional integrals and a Hubbard-Stratonovich decomposition of the interaction term. Using a spin-space reference frame that fluctuates in time and space, the effective action both is spin-rotation invariant and exhibits the correct Hartree-Fock saddle point. For strong correlation the effective action for free carriers and spins coupled via a gauge field is obtained as a systematic expansion in t/U. The spiral state existing for small doping is discussed. The formalism allows straightforward generalization to intermediate and weak coupling.
We review various ways to express the partition function of the single-band Hubard model as a path integral. The emphasis is made on the derivation of the action in the integrand of the path integral and the results obtained from this approach are discussed only briefly. Since the single-band Hubbard model is a pure fermionic model on the lattice and its Hamiltonian is a polynomial in creation and annihilation fermionic operators, with the help of the fermionic coherent states of holomorphic representions the partition function of the model can be straightforwardly expressed as a path integral with respect to Grassmann fields. Then the question arises how to calculate this path integral specially in the most interesting case of the strong coupling, which is described by the term quartic in the fermion fields. The standard way to get rid of this quartic term is to use so-called Hubbard-Stratonovich transformation by introducing auxilary boson field (this procedure is usually called bosonization). Since there are several equivalent forms of the interaction term in the single-band Hubbard model the bosonization procedure is not unique. In the first section of the article we discuss two types of bosoniztions one with two real scalar fields (charge and spin fields) and another with a real vector field. The second section is devoted to a spin-rotation-invariant bosonization procedure invented by Schulz and Weng, Ting and Lee, where an arbitary quantization axis is introduced at each point in space and time and an integration over all possible orientations enters the path integral measure. Spin rotation invariant formulation is necessary to study e.g. spiral magnetization. In the third section we present the contstruction of a path integral for the partition function when the single-band Hubbard Hamiltonian is written in terms of the Hubbard operators, which are elements of the doubly graded semi-simple algebra. In this case one should use generalized coherent states for superalgebra in order to get a path integral formulation. We follow the procedure proposed recently by T ngler and Kopp, to investigate the case of strong coupling.
We present a slave-boson formulation for the Hubbard model that preserves the spin-rotation and particle-hole symmetries of the paramagnetic phase at half filling. Fermion and boson fields are treated on an equal footing in a functional integral formulation. We show that, while in two and three dimensions there is a metal-insulator transition as the interaction U increases, in one dimension our approach always gives an insulating state. Short-range spin-spin correlations are included at the saddle-point level for all values of U, giving a good description of the insulating state. We show that the energy gap in the one-particle spectrum increases from zero as (U-Uc{\mathit{U}}_{\mathit{c}}Uc​${)}^{\mathrm{\ensuremath{\alpha}}}$, where Uc{\mathit{U}}_{\mathit{c}}Uc​ is the critical value of the on-site interaction and the exponent \ensuremath{\alpha} is 1 and 1/2 for 2 and 3 dimensions, respectively. \textcopyright{} 1996 The American Physical Society.
The Hubbard model was originally introduced as the simplest system which may exhibit an insulating (Mott) state. This state is the result of strong electron–electron interactions. In this chapter we consider the Hubbard model at half-filling. The main goal here is the study of the magnetic properties of its ground state. Apart from an exact solution in one dimension, no exact results are available for this problem. This leads to the use of several approximations. The most popular one, and the oldest, is the mean-field theory (MFT). In the MFT one has the bias that the ground state does have some sort of magnetic order (i.e. ferromagnetic, Néel antiferromagnetic, etc.). The problem is then usually solved by means of a variational ansatz. However, one is usually interested in more than just the ground-state energy, which, after all, is not directly measurable and depends very sensitively on the properties at short distances. Most often we wish to evaluate the long-distance, low-frequency, properties of the correlation and response functions of this theory. Moreover, in some cases, such as in one dimension, the fluctuations overwhelm the MFT predictions. In this chapter we will consider the standard MFT (i.e. Hartree–Fock), which is expected to become accurate at weak coupling. We will consider both ferromagnetic and antiferromagnetic states. We will also rederive these results using path integrals. As a byproduct, we will also have a theory of the fluctuations: the non-linear sigma model.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.