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arXiv:cond-mat/0209588 · Physical review. B, Condensed matter

Persistent currents in mesoscopic rings: A numerical and renormalization group study

V. Meden, Ulrich Schollwöck

Published Jan 15, 2003• 49 citations
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Abstract

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 $1/N.$ 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.

DOI
10.1103/physrevb.67.035106

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Effect of electronic interactions on the persistent current in one-dimensional disordered rings

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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 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.
We study the persistent current and the Drude weight of a system of spinless fermions, with repulsive interactions and a hopping impurity, on a mesoscopic ring pierced by a magnetic flux, using a density matrix renormalization group (DMRG) algorithm for complex fields. Both the Luttinger liquid (LL) and the charge density wave (CDW) phases of the system are considered. Under a Jordan-Wigner transformation, the system is equivalent to a spin-1∕21∕21∕2 XXZXXZXXZ chain with a weakened exchange coupling. We find that the persistent current changes from an algebraic to an exponential decay with the system size, as the system crosses from the LL to the CDW phase with increasing interaction UUU. We also find that in the interacting system, the persistent current is invariant under the impurity transformation $\ensuremath{\rho}\ensuremath{\rightarrow}1∕\ensuremath{\rho}$, for large system sizes, where $\ensuremath{\rho}$ is the defect strength. The persistent current exhibits a decay that is in agreement with the behavior obtained for the Drude weight. We find that in the LL phase the Drude weight decreases algebraically with the number of lattice sites NNN, due to the interplay of the electron interaction with the impurity, while in the CDW phase, it decreases exponentially, defining a localization length which decreases with increasing interaction and impurity strength. Our results show that the impurity and the interactions always decrease the persistent current, and imply that the Drude weight vanishes in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ in both phases.
The Bethe ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compare favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates the value of CLDFT in describing the transport properties of one-dimensional correlated electron systems. As its computational overheads are rather modest, we propose this method as a tool for studying problems where both disorder and interaction are present.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study the persistent current of the interacting electron gas in a one-dimensional continuous ring containing a single δδδ barrier. We calculate the persistent current as a function of the ring circumference, magnetic flux threading the ring, barrier strength, etc. We compare our results with the results of correlated models like the Luttinger liquid model and the Hubbard model solved by means of the renormalization group. A good agreement is found. First, the persistent current decays with the increasing ring circumference (LLL) faster than L−1L^{-1}L−1 and eventually like L−α−1L^{-α-1}L−α−1, where α>0α>0α>0 depends only on the electron-electron interaction. Second, the persistent current is a sine-shaped function of magnetic flux. This sine-like dependence and in particular the universal power law L−α−1L^{-α-1}L−α−1 have sofar been believed to arise only in the correlated many-body models. Observation of these features within the Hartree-Fock model is a surprising result.
The Bethe-Ansatz local density approximation (LDA) to lattice density functional theory (LDFT) for the one-dimensional repulsive Hubbard model is extended to current-LDFT (CLDFT). The transport properties of mesoscopic Hubbard rings threaded by a magnetic flux are then systematically investigated by this scheme. In particular we present calculations of ground state energies, persistent currents and Drude weights for both a repulsive homogeneous and a single impurity Hubbard model. Our results for the ground state energies in the metallic phase compares favorably well with those obtained with numerically accurate many-body techniques. Also the dependence of the persistent currents on the Coulomb and the impurity interaction strength, and on the ring size are all well captured by LDA-CLDFT. Our study demonstrates that CLDFT is a powerful tool for studying one-dimensional correlated electron systems with high accuracy and low computational costs.
The persistent current is here studied in one-dimensional disordered rings that contain interacting electrons. We used the density matrix renormalization group algorithms in order to compute the stiffness, a measure that gives the magnitude of the persistent currents as a function of the boundary conditions for different sets of both interaction and disorder characteristics. In contrast to its noninteracting value, an increase in the stiffness parameter was observed for systems at and off half-filling for weak interactions and nonzero disorders. Within the strong interaction limit, the decrease in stiffness depends on the filling and an analytical approach is developed to recover the observed behaviors. This is required in order to understand its mechanisms. Finally, the study of the localization length confirms the enhancement of the persistent current for moderate interactions when disorders are present at half-filling. Our results reveal two different regimes, one for weak and one for strong interactions at and off half-filling.