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Physical Review B

Running coupling constant and possible bound state in a Luttinger liquid

Yu-Liang Liu

Published Nov 16, 2004• 9 citations
Publisher

Abstract

By studying a spinless fermion system, which under some conditions is equivalent to a XXZ spin-$1∕2$ Heisenberg chain coupled to a spin-$1∕2$ magnetic impurity, we introduce a new method which can be used systematically to find the stable infrared critical fixed point and corresponding effective action of the system. Moreover, near this stable infrared critical fixed point, in terms of the corresponding effective action, we show that the phase factor field appearing in the hybridization amplitude plays a key role in forming a possible bound state near the impurity site. This result can be tested in future numerical simulations.

DOI
10.1103/physrevb.70.195111

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We study a lattice realization of a Luttinger liquid interacting with quantum spins in terms of an antiferromagnetic S=1∕2S=1∕2S=1∕2 Heisenberg chain, where each spin is also coupled to a $\ensuremath{\sigma}=1∕2$ Kondo spin degree of freedom. This model describes the low-energy spin dynamics in quasi-one-dimensional materials, where the electronic spins of the magnetic ions interact with those of impurities, nuclei, and possibly other spin species present in their environment. For large ferromagnetic and antiferromagnetic Kondo interaction ${J}^{\ensuremath{'}}$, there are two phases corresponding to an effective spin-1 Heisenberg chain and a dimerized spin-1∕21∕21∕2 ladder, respectively. For weak Kondo interaction, we establish that the Kondo interaction drives the system to a strong coupling regime. This suggests that ${J}^{\ensuremath{'}}=0$ is the only critical point in the system.
We study the low-temperature properties of a spin-1/2 magnetic impurity coupled to a one-dimensional interacting electron system. Using a formalism by Affleck and Ludwig, with a scale invariant boundary condition replacing the impurity, we exploit boundary conformal field theory to deduce the impurity thermal and magnetic response. In the case of only forward electron scattering off the impurity, we predict the same critical scaling as for the two-channel Kondo effect for noninteracting electrons, but with a different Wilson ratio. Backward electron scattering off the impurity destabilizes this behavior and drives the system to a new fixed point. In the case of equal amplitudes for forward and backward scattering (Kondo interaction), we show that there are only two types of scaling behaviors consistent with the symmetries of the problem: either a local Fermi liquid or a critical theory with an anomalous specific heat. The latter case agrees with a recent poor-man's scaling result proposed by Furusaki and Nagaosa. \textcopyright{} 1996 The American Physical Society.
We have studied the zero and low-temperature behavior of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-1/2 chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a new order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.
We have studied the zero- and low-temperature behaviors of anisotropic alternating antiferromagnetic-ferromagnetic Heisenberg spin-12\frac{1}{2}21​ chains in a transverse magnetic field. Using the analytical spinless fermion approach, the thermodynamic behavior of the model has been studied. We have introduced a different order parameter to distinguish the gapless Luttinger liquid phase from the other gapped phases. The exact diagonalization Lanczos results are also used to compare with the spinless fermion ones. We have found a double peak structure in the specific-heat curves for the region between the two quantum critical points. Using the numerical full diagonalization results, the existence of the double peak structure is confirmed.
We study a system of an impurity spin coupled to a junction of several Tomonaga-Luttinger liquids using a renormalization group scheme. By starting close to the decoupled S-matrix at the junction, we find a new stable fixed point at a finite ferromagnetic value of the Kondo coupling, for repulsive inter-electron interactions; this leads to the possibility of spin-flip scatterings even at the lowest temperatures. If the junction is governed by the Gri_ths S-matrix, we find that the Kondo coupling flows to the strong coupling fixed point (Furusaki-Nagaosa point) where all the wires are decoupled and the conductance vanishes at zero temperature.
Using a functional renormalization group, we compute the flow of the renormalized impurity potential for a single impurity in a Luttinger liquid over the entire energy range from the microscopic scale of a lattice-fermion model down to the low-energy limit. The nonperturbative method provides a complete real-space picture of the effective impurity potential. We confirm the universality of the open chain fixed point, but it turns out that very large systems (104−−105{(10}^{4}--{10}^{5}(104−−105 sites) are required to reach the fixed point for realistic choices of the impurity and interaction parameters.