VOLUME 69, NUMBER 1 P H YSICAL R EV I E%' LETTERS 6 JULY 1992 Hall Conductance and Adiabatic Charge Transport of Leaky Tori Joseph E. Avron, Markus Klein, ' and Aye1et Pnueli Department of Physics, Technion Isr-ael Institute of Technology, Haifa, 320001, srael Lorenzo Sadun Department of Mathematics, Unioersity of Texas, AustinT, exas 78712 (Received 27 February 1992) Leaky tori are two-dimensional surfaces with g handles and r horns (punctures) that extend to infinity. They model certain features of mesoscopic systems with multiple Aharonov-Bohm-type geometries connected to infinitely long leads. For a large class of leaky tori with finite area, in the pres- ence of a constant magnetic field and threading fluxes, we calculate exactly the persistent currents, adia- batic charge transport, and the (appropriately defined) Hall conductances. PACS numbers: 72. 10. Bg, 03. 65. — w Leaky tori, a term coined by Gutzwiller [I], are two- dimensional analytic surfaces with g handles and r punc- tures. The metric near the punctures is such that it de- scribes infinitely long horns. An example with one handle and two horns is shown in Fig. 1. Leaky tori with finite area have horns (called cusps by mathematicians) that are effectively one dimensional near infinity and capture some of the features of mesoscopic systems which are multiply connected and connect to infinitely long, one- dimensional, leads. Unlike mesoscopic systems, leaky tori have no boundaries at finite distances. Aharonov-Bohm Auxes for leaky tori fall into three dis- tinct classes: r Auxes p~, j =1, ... , r, which thread the horns; 2g Auxes PJ, j =r+1, ... , r+2g, through the g handles; and Auxes that pierce the surface. %e shall take one piercing Aux IIio. We denote the Auxes collectively by III and with the piercing Aux excluded by I3). %ith such a surface we associate a Landau Hamiltoni- an (the Schrodinger operator in magnetic field 8) H(8, $). We shall describe exact results for (1) the low- lying eigenvalues E„(8, &) and their degeneracies; (2) the 2g+r persistent currents associated to the jth Aux and nth eigenvalue defined by — t)+E„; (3) the charges trans- ported from infinity along the ith horn and then back to infinity along the jth horn when some of the Auxes change adiabatically by a unit of quantum Aux; and (4) the Hall conductances at fixed Fermi energy EF, set in a gap [2]. The results hold for a large class of leaky tori in the presence of a constant magnetic field B, and parts extend IE z Z3 to more general surfaces and Hamiltonians. For reasons of space, we shall merely outline the basic strategies in- volved in the more mathematical issues. Detailed proofs shall be presented elsewhere [3]. Leaky tori with finite volume and constant Gaussian curvature E =— 1 can be represented as the quotient of the complex upper half plane H, with the Poincare metric ds =y (dx +dy ), by some discrete subgroup I of SL(2,IR) [4]. Thus the surface is represented by a polygon in H (the boundary of a fundamental domain) with appropriate identifications of the sides. Recall that the geodesics are semicircles centered on the boundary of H, |)H [y =0] U [~], and the sides of the polygon can be taken to be circular arcs. Each horn corresponds to a cusplike vertex of the polygon (with zero opening angle) on t)H. A fundamental polygon for the leaky torus of Fig. 1 is shown in Fig. 2. Specification of curvature, number of horns, and number of handles does not fix a unique surface, but a choice of I does (up to translations, scaling, and hyperbolic rotations — the isometrics of H). There is a [6(g — 1)+2r]-dimensional family of such leaky tori, known as the moduli space. Our results turn out to be completely independent of the choice of the point in the moduli space. By the Gauss-Bonnet theorem we have JK =2tr(2 — 2g r). Therefore, the — area is 2tt(2g+r — 2). We set — h K/2m =1, and choose the quantum Aux unit to be Zp Zp Zl Z~ Zp FIG. 1. Leaky torus with one handle and two horns. p3 is a Aux threading the handle and &0 is a piercing flux at zo. FIG. 2. The fundamental domain (polygon) for the leaky torus with one handle and two horns. The arrows show the identifications. All cusps except zi are identified. The points z3, z4 are base points used in the text to define the handle flux; see Eq. (4). 1992 The American Physical Society