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Persistent Currents and Magnetization in Two-Dimensional Magnetic Quantum Systems

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Persistent Currents and Magnetization in Two-Dimensional Magnetic Quantum Systems FR9800312 PERSISTENT CURRENTS and MAGNETIZATION in TWO-DIMENSIONAL MAGNETIC QUANTUM SYSTEMS Jean DESBOIS, Stephane OUVRY and Christophe TEXIER Division de Physique Theorique 1, IPN, Orsay Fr-91406 Abstract Persistent currents and magnetization are considered for a two-dimensional electron (or gas of electrons) coupled to various magnetic fields. Thermodynamic formulae for the magnetization and the persistent current are established and the "classical" relationship between current and magnetization is shown to hold for systems invariant both by translation and rotation. Applications are given, including the point vortex superposed to aft homogeneous magnetic field, the quantum Hall geometry (an electric field and an homogeneous magnetic field) and the random magnetic impurity problem (a random distribution of point vortices). PACS numbers: 05.30.-d, 05.40.+J, ll.10.-z IPNO/TH 97-43 December 1997 1 Unite de Recherche des Universites Paris 11 et Paris 6 associee au CNRS 29-43 Introduction Since the pioneering work of Bloch [1], several questions concerning persistent currents have been answered. The conducting ring case has been largely discussed in the literature [2], as well as the persistent current due to a point-like vortex [3]. However, many questions remain open. For example, we might ask about standard relationships such as M = IV (2) where /, M, E, V, $ are, respectively, the persistent current, magnetization, energy, area (denoted by V as volume in two dimensions) and magnetic flux through the system. These relations are clearly understood in the case of a ring. Do they apply to more general, for instance infinite, systems? More precisely, are they still correct, and if yes, in which case? The aim of this paper is to clarify these points in the case of two-dimensional systems. We will show that (2) holds true as soon as the system is invariant by translation and rotation. We will also insist on the role of the spin coupling, which happens to be crucial in the case of point vortices. We will examplify these considerations for an electron (or gas of electrons) coupled to - a point vortex superposed to an homogeneous magnetic field, - an uniform electric field and an homogeneous magnetic field (quantum Hall geometry), - a distribution of point-like vortices randomly dropped onto the plane according to Poisson's law, modelising disordered magnetic systems (analytical and numerical). The case of electrons on a ring threaded by a flux, and electrons in a plane coupled to a point vortex will be revisited in Appendix A. Some technical details on harmonic regularization for computing partition functions and the so called "Landau counting rule" will be given in Appendix B. 2 Basic Definitions and Formalism Consider the two-dimensional quantum mechanical problem of an electron with Hamiltonian (we set the electron mass and the Planck constant me = h = 1) e + V- -czB (3) V and A are, respectively, the scalar and vector potential (B = V x A) and crz/2 is the electron spin. The local current is (v = p- eA) j(?)=e-(v\r)(r\ + \r)(r\v) (4) The total magnetization is by definition e M=\j dr{rx{]{r)))-k+ -(az) (5) where k is the unit vector perpendicular to the plane and ( ) means average over Boltzmann or Fermi-Dirac distributions. In the Boltzmann case, one obtains the thermal magnetization We want to relate the thermal magnetization to a thermodynamical quantity. To this aim, we add a fictitious uniform magnetic field B' perpendicular to the plane. In the symmetric gauge, its vector potential is eA' = (e/2)B' x r. H becomes H(B') with the partition function Zp{B') = Trje"^^')}. First order perturbation theory in B' gives f{( )} ... (7) and thus, the magnetization, 1 ,. d\nZp(B') Mp = - hm 5-£T—- (8) p 13 S'-vo dB' v ' In the sequel, we will consider the spin-up and spin-down as two distinct physical situations. It follows that in (8), the spin induced part of the magnetization is \ozs thus the orbital part of the magnetization is Note that in the particular case of H containing an homogeneous magnetic field B perpendicular to the plane, (8) narrows down to the usual formula and (9) to 6_ M& - Note also that one can compute directly M^ using (8) by simply dropping the coupling of the fictitious B' field to the spin in Zp(B'), and, accordingly, using (10) in the case of H containing a homgeneous B field, the coupling of B to the spin in Zp. Let us now turn to the persistent current and consider in the plane a semi infinite line V starting at f0 and making an angle 6O with the horizontal x-axis. The orbital persistent or6 current 7 (fb, 90) through the line is , r if- ^o) x (](f)) „ r*(?o,0o) = / d\r- fo| —A 1- • k (12) where 'r~£_?r*^ • fc is the orthoradial component of j(r) to the line. It obviously depends on r*o &nd OQ. Consider now systems rotationnally invariant around r*o. /or6(roi ^o) no longer depends on #o a-nd, without loss of generality, can be averaged over <?o- So \r — and for the Boltzmann distribution Again, we want to establish a link between (14) and a thermodynamical quantity. We add to the system a fictitious vortex <f>' located at fo the origin of the line, with potential vector and magnetic field eBv = 2na'8(r- ro)k (16) where a' = <j>'/<f>0 -4>o is the flux quantum- can always be chosen in the interval [0,1] by periodicity. The resulting Hamiltonian is e (-eA-eAv)\v- -at{B + Bv) (17) with partition function Zp(ar). Though being a well-established fact [4], we recall here that the spin coupling term Cz(B + Bv) in (3,17) is necessary to define in a non ambiguous way point vortices quantum systems. The behavior of the wavefunctions at the location fo of the fictitious point vortex (or of a physical point vortex contained in the original Hamiltonian (3)) has to be specified properly. This is precisely achieved in a scale invariant way with the spin coupling term, which materializes either a repulsive hard-core (spin-down) ira'8{r — ?Q) or attractive (spin-up) —no/5(r — f0) point contact interaction at f0. In the standard Aharonov-Bohm problem, the wavefunctions are chosen to vanish at ?o, i.e. the spin-down repulsive prescription. However, we might as well take the attractive spin-up prescription. Still, the persistent current should clearly be insensitive to the nature of the fictitious vortex introduced to define it, and in particular on the way it is regularized at short distance. First order perturbation theory in a' gives ) = Zp + a'p (TT |e~^ ^I^p * • *} + ™^(f0, r0)) + ... (18) where G/3(f,f') = {r\e~0H\r') is the thermal propagator for the Hamiltonian (3). We define the total current around ro as - (19) and, from (14), Note that in the particular case of H containing a vortex <f> at r*o, or consisting of a ring of center f*o threaded by a magnetic flux <f> = a<fo) the orbital persistent current generated 2 in the plane around f0, or in the ring , (20) narrows down to *' {ro) ~ 2^d^r rzj If the first term in (20) is usually discussed in the literature under the weaker form appearing in (21) valid for the point vortex and the ring [l]-[3], the second term in (20,21) 2 In the case of a ring, the propagator at the center of the ring is by definition nul. triggered by the spin coupling to the propagator has been so far ignored. However, both terms are crucial, and if the spin term is absent, some contradictions do arise in the computation of orbital persistent currents. Indeed, orbital persistent currents -as well as orbital magnetizations- should vanish when H becomes free, and this is not the case if the spin term is ignored, as we will see later for the point vortex, or for the point vortex superposed to an homogeneous field. We will also encounter in the attractive point vortex case a situation where the orbital current by itself is not defined (in fact infinite), and where only the total (orbital + spin) current has a sensible physical meaning. To summarize, orbital magnetizations and currents are respectively given by (9) and (20). They should both vanish when the original Hamiltonian H becomes free. Total magnetizations and currents are given by (8) and (19). They obviously depend, as well as their orbital counterparts, on the spin degree of freedom defined in the original Hamiltonian (3). If we consider systems that are both invariant by translation and rotation, we can proceed further by averaging (14) over f0. Taking advantage of the identity [5] fi£0*°-«x, (22) we can write, using (6,14), iorb=vl drV(r~o) = vMOrb (23) which generalizes the conducting ring situation to translation and rotation invariant two- dimensional systems. Furthermore, since for a translation invariant system Ga(fo,fo) = Zp/V, (23) holds also for the total current and magnetization / = ±M (24) Finally, (23,24) still hold for non rotational invariant systems if the persistent current Iorb is understood as averaged over all directions 8Q of the semi infinite straight line. To have access to the orbital magnetization and persistent currents of a gas of non- interacting electrons, Zp(B') or Zp(a') have to be replaced in (9,20), and Z@ in (11,21), by the corresponding grand-partition functions. For instance, (11) and (21) respectively become (26) where E@(n) is the grand-partition function In "Bpin) = Tr |ln(l + e~^H~^) | for chemical potential fi and f(E) is the Fermi-Dirac distribution.
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