Fractional flux periodicity in tori made of square lattice

We present a study on fractional flux periodicity of the ground state in planar systems made from a square lattice whose boundary is compacted into a torus. The ground-state energy and persistent currents show a fractional period of the fundamental unit of magnetic flux depending on the twist around the torus axis. We discuss the possible relationship between the twist and genus of a torus. The Aharonov-Bohm effect 1) shows that a single electron wave function has a fundamental unit of magnetic flux Φ 0 = 2π/e, where −e is the electron charge * *). The electrical and magnetic properties of materials are governed by many electrons, and each constituent has the above-mentioned periodicity. However, the fundamental flux period of a material is not always Φ 0. For example, superconducting materials exhibit a period of Φ 0 /2, which can be understood by charge doubling due to the Cooper pair formation. This is clear if one imagines that a fundamental particle (or quasiparticle) has a −2e charge due to attractive interaction, and that the fundamental flux period of a material is equal to that of the quasiparticle (2π/2e) in the system. In Fig. 1, we depict schematic diagrams of typical experimental settings for the Aharonov-Bohm effect. Consider an electron going over and under a very long impenetrable cylinder, denoted as a black circle in Fig. 1(a). In the cylinder there is a magnetic field parallel to the cylinder axis, taken as normal to the plane of the figure. The probability of finding this electron in the interference region depends on the magnetic field Φ and the interference pattern having period Φ 0. When an electron is placed in a ring pierced by a magnetic flux Φ as shown in Fig. 1(b), a current I defined by differentiating the ground state energy with respect to the magnetic flux, appears and is expected to have the flux periodicity Φ 0. Actual mesoscopic rings contain many electrons, and non vanishing currents, known as persistent currents, 2)–4) are also expected. Although the ground state in mesoscopic rings consists of many electrons, because each electron has the flux periodicity Φ 0 one might think the flux periodicity of the systems is the same as the flux periodicity of the constituent electron. In this paper, we present an example in which the fundamental flux period of the ground-state does not coincide with the flux periodicity of …