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International Journal of Modern Physics B

RENORMALIZATION GROUP APPROACH TO QUASI-ONE-DIMENSIONAL CONDUCTORS

C. Bourbonnais, Laurent G. Caron

Published Apr 1, 1991• 164 citations
Publisher

Abstract

We review the recent progress made in the application of the renormalization group method to the interacting quasi-one-dimensional electron gas. From the functional integral formulation of the partition function as expressed in terms of anticommuting Grassmann variables, the effects of single-electron interchain hopping on the mechanisms of propagation of correlations leading to long range ordering is analyzed in full details for the non-half-filled band case. Within a unified formalism, the scaling features of the purely one-dimensional correlations, the dimensionality crossover of both single and composite particles, the critical temperatures, and the responses functions are described.

DOI
10.1142/s0217979291000547

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We review the recent progress made in the application of the renormalization group method to the interacting quasi-one-dimensional electron gas. From the functional integral formulation of the partition function as expressed in terms of anticommuting Grassmann variables, the effects of single-electron interchain hopping on the mechanisms of propagation of correlations leading to long range ordering is analyzed in full details for the non-half-filled band case. Within a unified formalism, the scaling features of the purely one-dimensional correlations, the dimensionality crossover of both single and composite particles, the critical temperatures, and the responses functions are described.
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We study the interplay of randomness, electron correlation, and dimensionality effects in weakly coupled half-filled Hubbard chains with weak quenched random potentials, based on the renormalization-group (RG) approach. We perform a two-loop RG analysis of an effective action derived by using the replica trick, and examine the following crossovers and phase transitions from an incoherent metal regime: (1) a crossover to the Anderson localization regime, (2) an antiferromagnetic phase transition, and (3) a crossover to the quasi-one-dimensional weak-localization regime. The case of $d=1+\ensuremath{\epsilon}$ $(\ensuremath{\epsilon}\ensuremath{\ll}1)$ dimensions is also mentioned.
We study the relation between the charge gap and the interchain one-body correlation function in quasi-one-dimensional dimerized organic conductors at quarter filling, by applying the density matrix renormalization group method to a three-chain extended Hubbard model. The charge gap increases with the degree of dimerization in the intrachain hopping integrals. When the charge gap is larger than the interchain hopping integral, the interchain hopping correlation is strongly suppressed, as observed in the (TMTTF)2X salts.
We compare two fermionic renormalization group (RG) methods which have been used to investigate the electronic transport properties of one-dimensional metals with two-particle interaction (Luttinger liquids) and local inhomogeneities. The first one is a poor man's method set-up to resum 'leading-log' divergences of the effective transmission at the Fermi momentum. Generically the resulting equations can be solved analytically. The second approach is based on the functional RG (fRG) method and leads to a set of differential equations which can only for certain set-ups and in limiting cases be solved analytically, while in general it must be integrated numerically. Both methods are claimed to be applicable for inhomogeneities of arbitrary strength and to capture effects of the two-particle interaction, such as interaction dependent exponents, up to leading order. We critically review this for the simplest case of a single impurity. While on first glance the poor man's approach seems to describe the crossover from the 'perfect' to the 'open chain fixed point' we collect evidence that difficulties may arise close to the 'perfect chain fixed point'. Due to a subtle relation between the scaling dimensions of the two fixed points this becomes apparent only in a detailed analysis. In the fRG method the coupling of the different scattering channels is kept which leads to a better description of the underlying physics.
Renormalization group analysis for multi-band many-electron systems at half-filling at positive temperature is presented. The analysis includes the Matsubara ultra-violet integration and the infrared integration around the zero set of the dispersion relation. The multi-scale integration schemes are implemented in a finite-dimensional Grassmann algebra indexed by discrete position-time variables. In order that the multi-scale integrations are justified inductively, various scale-dependent estimates on Grassmann polynomials are established. We apply these theories in practice to prove that for the half-filled Hubbard model with nearest-neighbor hopping on a square lattice the infinite-volume, zero-temperature limit of the free energy density exists as an analytic function of the coupling constant in a neighborhood of the origin if the system contains the magnetic flux pi (mod 2 pi) per plaquette and 0 (mod 2 pi) through the large circles around the periodic lattice. Combined with Lieb's result on the flux phase problem ([Lieb, E. H., Phys. Rev. Lett. 73 (1994), 2158]), this theorem implies that the minimum free energy density of the flux phase problem converges to an analytic function of the coupling constant in the infinite-volume, zero-temperature limit. The proof of the theorem is based on a four-band formulation of the model Hamiltonian and an extension of Giuliani-Mastropietro's renormalization designed for the half-filled Hubbard model on the honeycomb lattice ([Giuliani, A. and V. Mastropietro, Commun. Math. Phys. 293 (2010), 301-346]).