Exact and mean-field analysis of the role of Hubbard interactions on flux driven circular current in a quantum ring

TL;DR

This work addresses how electron-electron interactions and disorder govern the persistent current in flux-threaded Hubbard rings within a tight-binding framework. By combining exact diagonalization and Hartree-Fock mean-field methods, and introducing a LIN table formalism to efficiently build the full many-body Hamiltonian, the authors compute the ground-state energy $E_g(\phi)$ and the current $I(\phi) = -\partial E_g/\partial \phi$, along with localization via the inverse participation ratio $IPR$. The key finding is that the on-site interaction $U$ generally enhances current at both half- and low-filling regimes, while the nearest-neighbor interaction $V$ has a nontrivial, filling-dependent effect: it can suppress current at low filling but enhances it near half-filling up to a critical ratio $V \approx U/2$, with disorder further enriching these trends. An eigenstate localization analysis based on $IPR$ corroborates the interplay among $U$, $V$, filling, and disorder in shaping PC. The results identify parameter regimes where extended Hubbard interactions significantly boost PC, offering insights for reconciling theory with experiment and guiding extensions to longer-range hopping and finite-temperature effects.

Abstract

We investigate circular current in both ordered and disordered Hubbard quantum rings threaded by magnetic flux, employing exact diagonalization and the Hartree-Fock mean-field approach within the tight-binding framework. The influence of on-site and extended Hubbard interactions, disorder, and electron filling on the persistent current is systematically analyzed. To construct the full many-body Hamiltonian, we introduce a linear table formalism, which, to our knowledge, has been rarely used in this context. In ordered rings, the current decreases monotonically with increasing on-site repulsion, while the impact of the extended interaction depends strongly on the filling factor. At low filling, stronger extended interaction suppresses the current, whereas near half-filling, it enhances the current up to a critical ratio, half of the on-site strength, before reducing it. Disorder significantly modifies these behaviors, notably enhancing the current at less than quarter-filling with increasing extended interaction. The localization properties of eigenstates, examined via the inverse participation ratio, further support the crucial roles of filling and the interplay between on-site and extended interactions in governing persistent current.

Figures (7)

• Figure 1: (Color online). Schematic diagram of a quantum ring threaded by a magnetic flux $\phi$, where the red spheres represent atomic sites in the ring.
• Figure 2: (Color online). Variations of the ground-state energy and the corresponding persistent current for an ordered ring with $N=10$. The first three columns present the ground-state energy for the cases $U=V=0$, $U=4,\,V=0$, and $U=4,\,V=2$, respectively, while the associated persistent currents are displayed in the last column. Beginning with $N_\uparrow = N_\downarrow = 1$ in the top row, the numbers of both $N_\uparrow$ and $N_\downarrow$ electrons are increased by one in each subsequent row until the half-filled configuration is reached in the bottom row.
• Figure 3: (Color online). Variations of the ground-state energy and persistent current with magnetic flux for a $10$-site disordered ring (random disorder) with disorder strength $W=1$. The left and right columns display the flux dependence of $E_g$ and PC, respectively. The top row corresponds to the half-filled configuration ($N_{\uparrow} = N_{\downarrow} = 5$), while the bottom row represents the less-than-quarter-filled case ($N_{\uparrow} = 2$, $N_{\downarrow} = 5$). In sub-figure (a), the black, blue, and red curves depict the variation of $E_g$ for $U=V=0$, $U=0.8,\,V=0$, and $U=0.8,\,V=0.4$, respectively. Sub-figure (b) shows the corresponding PC variations using the same color scheme. In sub-figure (c), the black curve represents $U=V=0$, while the blue and red curves correspond to $U=2,\,V=0$ and $U=2,\,V=1$, respectively. Sub-figure (d) presents the associated PC variations following the same color codes.
• Figure 4: (Color online). Dependence of the persistent current with magnetic flux $\phi$ for a correlated disordered ring ($W=1$) of size $N=10$. In sub-figure (a), the black, red, and blue curves represent the PC for $U=V=0$, $U=0.75$ with $V=0$, and $U=0.75,\,V=0.375$, respectively, under the half-filled band condition. Sub-figure (b) displays the corresponding variations for $N_{\uparrow}=2$ and $N_{\downarrow}=1$, where the black curve denotes $U=V=0$, the red curve corresponds to $U=0.75,\,V=0$, and the blue curve represents the case $U=0.75,\,V=0.3$.
• Figure 5: (Color online). Current-flux characteristics for a quantum ring of size $N=40$ with random disorder of strength $W=1$. The top row corresponds to the half-filled case ($N_\uparrow = N_\downarrow = 20$), while the bottom row represents a less-than-quarter-filled configuration ($N_\uparrow = N_\downarrow = 7$). All other relevant parameters for each curve are indicated within the figure.