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Journal of Physics Condensed Matter

Persistent current and Drude weight for the one-dimensional Hubbard model from current lattice density functional theory

Akinlolu Akande, Stefano Sanvito

Published Jan 17, 2012• 12 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/24/5/055602

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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.
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.
We perform a numerical study of interacting one-dimensional Hubbard rings with a single impurity potential and pierced by a magnetic flux. Our calculations are carried out at the level of current lattice density functional theory (CLDFT) for the Hubbard model and compared to known results obtained in the thermodynamical limit from the Bethe ansatz. In particular, we investigate the effects of disorder and Coulomb interaction on the persistent current (PC) and the Drude weight. It is found that CLDFT is able to accurately describe qualitative and quantitative features of these ground state properties in the presence of disorder and electronic interaction. When the impurity potential is switched off, the CLDFT approach describes well the velocity of the Luttinger liquid excitations as a function of both interaction strength and electron filling. Then, when the impurity scattering potential is finite, we find the PC to vanish as [Formula: see text] for large L and independent on the strength of the scattering potential, in good agreement with Luttinger liquid theory.
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 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 Hubbard model in the presence of disorder. We compute the charge stiffness for a mesoscopic ring as a function of the size L, which is a measure of the persistent currents. We find that for finite disorder the persistent currents of the system with repulsive interactions are larger than those of the system with attractive interactions. This counterintuitive result is due to the fact that local-density fluctuations are reduced in the presence of repulsive interactions.
We study the ground state of two interacting bosonic particles confined in a ring-shaped lattice potential and subjected to a synthetic magnetic flux. The system is described by the Bose-Hubbard model and solved exactly through a plane-wave Ansatz of the wave function. We obtain energies and correlation functions of the system both for repulsive and attractive interactions. In contrast with the one-dimensional continuous theory described by the Lieb-Liniger model, in the lattice case we prove that the center of mass of the two particles is coupled with its relative coordinate. Distinctive features clearly emerge in the persistent current of the system. While for repulsive bosons the persistent current displays a periodicity given by the standard flux quantum for any interaction strength, in the attractive case the flux quantum becomes fractionalized in a manner that depends on the interaction. We also study the density after a long time expansion of the system. Our results can be used to benchmark approximate schemes for the many-body problem such as the density matrix renormalization group or other variational schemes.