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arXiv:cond-mat/0612267 · Physical Review B

Persistent currents through a quantum impurity: Protection through integrability

Johan Nilsson, Hans-Peter Eckle, Henrik Johannesson

Published Aug 15, 2007• 28 citations
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Abstract

We consider an integrable model of a one-dimensional mesoscopic ring, with the conduction electrons coupled by a spin exchange to a magnetic impurity. A symmetry analysis based on a Bethe ansatz solution of the model reveals that the current is insensitive to the presence of the impurity. We argue that this is true for any integrable impurity-electron interaction, independent of the choice of physical parameters or couplings. We propose a simple physical picture of how the persistent current gets protected by integrability.

DOI
10.1103/physrevb.76.073408

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We consider an integrable model of a one-dimensional mesoscopic ring, with the conduction electrons coupled by a spin exchange to a magnetic impurity. A symmetry analysis based on a Bethe ansatz solution of the model reveals that the current is insensitive to the presence of the impurity. We argue that this is true for any integrable impurity-electron interaction, independent of the choice of physical parameters or couplings. We propose a simple physical picture of how the persistent current gets protected by integrability.
We consider a one-dimensional mesoscopic quantum ring filled with spinless electrons and threaded by a magnetic flux, which carries a persistent current at zero temperature. The interplay of Coulomb interactions and a single on-site impurity yields a nontrivial dependence of the persistent current on the size of the ring. We determine numerically the asymptotic power law for systems up to 32 00032\text{ }00032 000 sites for various impurity strengths and compare with predictions from Bethe ansatz solutions combined with bosonization. The numerical results are obtained using an improved functional renormalization-group (fRG) method. We apply the density-matrix renormalization-group (DMRG) and exact diagonalization methods to benchmark the fRG calculations. We use DMRG to study the persistent current at low electron concentrations in order to extend the validity of our results to quasicontinuous systems. We briefly comment on the quality of calculated fRG ground-state energies by comparison with exact DMRG data.
The persistent current in a lattice model of a one-dimensional interacting electron system is systematically studied using a complex version of the density-matrix renormalization group algorithm and the functional renormalization group method. We mainly focus on the situation where a single impurity is included in the ring penetrated by a magnetic flux. Due to the interplay of the electron-electron interaction and the impurity the persistent current in a system of N lattice sites vanishes faster than 1/N.1/N.1/N. Only for very large systems and large impurities our results are consistent with the Bosonization prediction obtained for an effective field theory. The results from the density-matrix renormalization group and the functional renormalization group agree well for interactions as large as half the bandwidth, even though as an approximation in the latter method the flow of the two-particle vertex is neglected. This confirms that the functional renormalization group method is a very powerful tool to investigate correlated electron systems. The method will become very useful for the theoretical description of the electronic properties of small conducting ring molecules.
We consider the one-dimensional t-J model, which consists of spin-1/2 electrons on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J=2t. We extend the model by introducing an impurity of arbitrary spin S that interacts with the electrons on the neighboring sites without destroying the integrability. The lattice model is defined by the scattering matrices via the quantum inverse scattering method. The interaction Hamiltonian between the impurity and the itinerant electrons is only explicitly constructed in the continuum limit. The discrete Bethe ansatz equations diagonalizing the model are derived and the solutions are classified according to the string hypothesis. The thermodynamic Bethe ansatz equations are derived and the impurity free energy is obtained for arbitrary bandfilling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter. The impurity can localize up to one itinerant electron and has in general mixed valent properties. The integer valent low T small H fixed point of the impurity corresponds to an asymptotically free spin S, while if either T or H (or both) become large the impurity behaves like an asymptotically free spin (S+).