Fractional periodicity and magnetism of extended quantum rings

The magnetic properties and nature of the persistent current in small flux-penetrated $t\text{−}{t}^{\prime }\text{−}U$ rings are investigated. An effective rigid-rotator description is formulated for this system, which coincides with a transition to a ferromagnetic state in the model. The criteria for the onset of effective rigid rotation is given. The model is used to understand continuum model ground-state solutions for a two-dimensional few-particle hard-wall quantum dot, where ferromagnetic solutions are found even without the Zeeman coupling to spin. After the onset of an effective rigid rotation, a 97%–98% correspondence can be determined between the lattice model and continuum model eigenstate results.

Figure 1

(Color online) Geometry of the $t\text{−}{t}^{\prime }\text{−}U$ ring with penetrating magnetic flux. In accordance with Ref. 12, the combination of the nearest- and next-nearest-neighbor hopping terms is shown to distort the 1D geometry, resulting in a system consisting two coupled 1D chains. To illustrate this feature, the two rings are displaced along the direction of the magnetic field. This is done for visualization purposes only and does not alter the physics of the model.

Figure 2

(Color online) (a) Ground-state energy as a function of flux for an eight-site Hubbard system at quarter filling (2↑, 2↓), $U∕t=20$, $U∕t=200$ and 1000. (b) Persistent current as a function of flux for the $U∕t=1000$ result.

Figure 3

(Color online) (a) Ground-state energy as a function of flux for an eight-site $t\text{−}{t}^{\prime }\text{−}U$ system at quarterfilling (2↑, 2↓), with $t^{\prime }∕t=-0.05$. The values of $U∕t$ starting from the bottom curve are 0, 1, 2, 4, 6, 8, 10, 20, 40.2, and 1000. $U∕t=40.2$ is specifically shown as this indicates the onset of effective rigid rotation and, hence, near-perfect fractional periodicity. (b) Ground-state energy as a function of flux, with focus on the $U∕t=20$, 40.2, and 1000 results.

Figure 4

(Color online) (a) The local moment $⟨S_i^2⟩$ as a function of $U∕t$ for an eight-site $t\text{−}{t}^{\prime }\text{−}U$ system at quarter filling, (2↑, 2↓). The two solutions are the $\Phi ∕{\Phi }_o=0.5$, $t^{\prime }∕t=0$ and $t^{\prime }∕t=-0.05$ results, as indicated in the legend. Note the discontinuity in the $t^{\prime }∕t=-0.05$ solution, which occurs at the onset of effective rigid rotation $(U∕t\simeq 40.2)$ and corresponds to the $S=0\to 2$ ferromagnetic transition.

Figure 5

(Color online) (a) Ground-state energy as a function of flux for an eight-site $t\text{−}{t}^{\prime }\text{−}U$ system at quarter filling, (2↑, 2↓), with $t^{\prime }∕t=-0.1$. The values of $U∕t$ starting from the bottom curve are 10, 16.9, 20, 30, 40, 50, 100, 200, and 1000.

Figure 6

(Color online) Persistent current as a function of flux, for the eight-site, quarter-filled (a) Hubbard chain with $U∕t=20$ and 1000 and (b) $t\text{−}{t}^{\prime }\text{−}U$ system at $t^{\prime }∕t=-0.05$ and $U∕t=20$, 40.2, and 1000, respectively. Note that the $I=0$ axis has been shifted in order to separate the results for comparison purposes.

Figure 7

(Color online) (a) Ground-state energies and (b) radial electron densities for the four-particle hard-wall quantum dot, with $M=6,\dots ,10$. The quantum numbers of the levels are shown in (a) as the pair $(M,S)$. The cyclotron energy, $\hslash {\omega }_c∕2$, of the zero-point motion has been subtracted from these graphs.

Figure 8

Ground-state phase diagram of a four-particle hard-wall quantum dot with $R=5a_0^{*}$ as a function of the external magnetic flux $\Phi$ and effective interaction strength $C$. The shade of the filling represents the spin quantum number from dark to light $S=0,1,2$, and the integers represent the angular momentum $M$ of the state. Fractional quasiperiodicity appears at strong $\Phi$ and $C$.

Figure 9

(Color online) Ground-state energy as a function of flux for an eight-site, quarter-filled $t\text{−}{t}^{\prime }\text{−}U$ chain with $t^{\prime }∕t=-0.17$. The values of $U∕t$ starting from the bottom curve are 3, 5, 7.28, 10, 23, 50, 100, and 1000. Note that the point of effective rigid rotation occurs at $\simeq U∕t=7.28$.

Figure 10

(Color online) Ground-state energy as a function of flux for the ten-site, quarter-filled $t\text{−}{t}^{\prime }\text{−}U$ chain with $t^{\prime }∕t=-0.19$. The values of $U∕t$ starting from the bottom curve are 5, 6.47, 10, 13, 20, 50, 100, and 1000. Note that the point of effective rigid rotation occurs at $\simeq U∕t=6.47$.