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Journal of Physics C Solid State Physics

Correlation functions for the Tomonaga model

Hans C. Fogedby

Published Oct 28, 1976• 52 citations
Publisher

Abstract

The density, spin density and pair correlation functions for the spin-half Tomonaga model are derived analytically within the framework of ordinary many body theory. The author extends and formalises the method of Dzyaloshinskii and Larkin (Sov. Phys. JETP., vol.38, p.202 (1974)). The dynamical constraint imposed by the conservation law is expressed in terms of Wick's theorem and the correlation functions are formulated as functional integrals by means of the Stratonovich-Hubbard transformation (1957-59). The associated one-body problem is explicitly soluble and the correlation functions assume the form of Gaussian integrals which are evaluated analytically.

DOI
10.1088/0022-3719/9/20/011

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The density, spin density and pair correlation functions for the spin-half Tomonaga model are derived analytically within the framework of ordinary many body theory. The author extends and formalises the method of Dzyaloshinskii and Larkin (Sov. Phys. JETP., vol.38, p.202 (1974)). The dynamical constraint imposed by the conservation law is expressed in terms of Wick's theorem and the correlation functions are formulated as functional integrals by means of the Stratonovich-Hubbard transformation (1957-59). The associated one-body problem is explicitly soluble and the correlation functions assume the form of Gaussian integrals which are evaluated analytically.
The longitudinal and transverse space and time dependent spin correlation functions for the one-dimensional Heisenberg-Ising model are derived asymptotically in the weak-coupling-low-frequency-long-wavelength limit within the framework of ordinary many-body theory (field theory). By means of the Jordan-Wigner transformation and using methods developed previously for the Tomonaga model (1950), the zero-temperature correlation functions is expressed as functional integrals. The associated one-body problem is explicitly soluble in the asymptotic regions and the correlation functions assume the form of Gaussian integrals which are evaluated analytically in the weak-coupling-low-frequency-long-wavelength limit. Apart from a linear correction to the transverse exchange constant agreement with Luther and Peschel (1975) is obtained.
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 method of the boson representation is applied to the Tomonaga-Luttinger model to obtain the one-particle temperature Green's function which is derived by Dzyaloshinskii and Larkin. In case T=0, the method and the result are essentially same as those of Luther and Peshel. The density of states D(ω) is calculated in cases both T=0 and T≠0. It is shown that the quantity D(ω) near ω=0 is proportional to |ω|A in case T=0 and that the quantity D(0) at low temperatures is proportional to TA, where the quantity A is related to the strength of the interaction. A sum rule for D(ω) is derived.
We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.
A new bosonisation procedure based on an exact method of functional integration is applied to the one-dimensional Tomonaga (1950)-Luttinger (1963) model with forward scattering. This method circumvents the conventional procedure where the fermion field operator is represented by (bosonic) charge and spin density fields. Fictitious auxiliary fields are introduced to deal with the interaction term so that the fermionic degrees of freedom can be integrated out exactly. In addition to the usual description of the system in terms of charge and spin fluctuations, an effective action in terms of the auxiliary fields can be used. The evaluation of the Green function is facilitated if the latter route is taken. The authors' results for the spinless and spin-1/2 cases agree with previous work.