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Journal of Physics C Solid State Physics

An improved method for computing correlation functions of one-dimensional fermion systems

Johannes Voit

Published Dec 20, 1988• 11 citations
Publisher

Abstract

Correlation functions of one-dimensional Tomonaga-Luttinger models with sine-Gordon-like extensions are computed using bosonisation and scaling techniques. The author shows that by expanding the full correlation functions instead of considering only their first cumulants, exponents for their asymptotic decay, logarithmic corrections as well as renormalised amplitudes can be obtained. Applied to the Hubbard model, the method produces for the first time correlation functions consistent with its symmetries, exact spectrum and numerical simulations.

DOI
10.1088/0022-3719/21/35/003

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Correlation functions of one-dimensional Tomonaga-Luttinger models with sine-Gordon-like extensions are computed using bosonisation and scaling techniques. The author shows that by expanding the full correlation functions instead of considering only their first cumulants, exponents for their asymptotic decay, logarithmic corrections as well as renormalised amplitudes can be obtained. Applied to the Hubbard model, the method produces for the first time correlation functions consistent with its symmetries, exact spectrum and numerical simulations.
A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.
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.
We study the correlation functions of one-dimensional Hubbard model in the presence of external magnetic field through the conformal field method. The long distance behaviour of the correlation functions and their unusual exponents for the model in the presence of a magnetic field are developed by solving the dressed charge matrix equations and setting the number of occupancies to one, as alternative to the usual zero used by authors in literatures. This work shows that the exponent of the correlation functions is a monotonous function of magnetic field and the correlation functions decay as powers of these unusual exponents. As the magnetic field goes to zero, we obtain the exponents as 8.125, 11.125, 17.125, 26.125 and 38.125 at kF, 3kF, 5kF, 7kF and 9kF. Our analytical results will provide insights into criticality in condensed matter physics.
We show that the prefactors of all terms of the one-dimensional (1D) Hubbard model correlation-function asymptotic expansions have a universal form, as the corresponding critical exponents. In addition to calculating such prefactors, our study clarifies the relation of the low-energy Tomonaga-Luttinger-liquid behavior to the scattering mechanisms which control the spectral properties of the model at all energy scales. Our results are of general nature for many integrable interacting models and provide a broader understanding of the unusual properties of quasi-1D nanostructures, organic conductors, and optical lattices of fermionic atoms.
The one-dimensional Hubbard model in a magnetic field is equivalent under renormalization-group transformation to a multicomponent Tomonaga-Luttinger model. The mapping between the Coulomb repulsion U of the Hubbard model and the couplings of the Tomonaga-Luttinger model is obtained for aribitrary magnetic field h and band filling n from a comparison of the correlation function exponents. These quantities are calculated for the Hubbard model from the finite-size corrections to the excitation energies, since the assumption of conformal invariance relates them to the critical exponents. On the other hand, the correlation functions of the multicomponent Tomonaga-Luttinger model are determined by solving the exact equation of motion derived by the use of generalized Ward identities.
Bosonisation is a technique for studying one-dimensional fermion systems by the means of transforming them to a boson system. S. Coleman showed that the correlation functions for the Sine Gordon field, a massless boson with a cosine interaction term, coincide with those of the massive Thirring model, which is a Dirac fermion with self-interactions. An explicit map can be found between these two systems, and it can in fact be generalised to an entire class of fermions in which the boson-fermion correspondence is a well-defined operator equality in Fock space. More generally, it is a widely used technique to transform a model into another whose nature simplifies the solution or allows for obtaining more information that appears more clearly in the new representation. Another example is the Jordan Wigner transformation, which associates quantum spin chains to one-dimensional fermions. Some aspects about the statistical properties of the chain can be inferred from the fermionic representation, and more properties can be studied by applying bosonisation and examine the system in this new representation.