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Relativistic Crystalline Symmetry Breaking and Anyonic States

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Relativistic Crystalline Symmetry Breaking and Anyonic States RELATIVISTIC CRYSTALLINE SYMMETRY BREAKING AND ANYONIC STATES IN MAGNETOELECTRIC SUPERCONDUCTORS∗ Jacques L. RUBIN† Institut du Non-Lin´eaire de Nice (INLN), UMR 129 CNRS - Universit´ede Nice - Sophia Antipolis, 1361 route des lucioles, 06560 Valbonne, France Enhanced and english corrected unpublished 2002 version of the 1993 one, the latter being published in the Proceedings of the international conference on “Magnetoelectric Interaction Phenomena in Crystals, Part I”, Ascona (Switzerland) Sept. 93. Ferroelectrics, Vol. 161(1-4) (1994), pp. 335-342. October 30, 2018 Abstract charge carriers associated with “effective” mag- netic monopoles, both with TM’s, and second, There exists a connection between the creation that ME can be highly considered in supercon- of toroidal moments (TM) and the breaking ductors theory. of the one-cell relativistic crystalline symmetry (RCS) associated to any given crystal [1] into Key Words: toroidal moments, relativistic which non-trivial magnetoelectric coupling ef- crystalline symmetries, symmetry breaking, fects (ME) exist [2] . Indeed, in this kind of anyons, Chern-Simon Lagrangians. crystals, any interaction between a charge car- rier and an elementary magnetic cell can breaks 1 the RCS of this previous given cell by varying, - Quantized Hall effect, high-Tc super- in the simplest case, the continuous defining pa- conductivity and anyons theory rameters of the initial RCS. In this chapter, we very briefly recall the quan- This breaking can be associated to a change tized Hall effect and the link with both the high- of the initial Galilean proper frame of any given T superconductivity theory and the anyons carrier to an “effective” one, into which the RCS c theory [4]. It is well-known that applying a mag- of the interacting cell is kept. We can speak arXiv:cond-mat/0208004v1 [cond-mat.supr-con] 31 Jul 2002 netic field in the orthogonal direction of a con- of a kind of “inverse” kineto-magnetoelectric ef- ducting layer crossed over by an electric current, fect [3]. The magnetic groups compatible with then an electric field orthogonal both to the lon- such process have been computed [1]. More- gitudinal electric current and the magnetic field over, one can notice that the TM’s break the appears. That is the so-called classical Hall P and T symmetries but not the PT one as in effect. In addition, a magneto-resistivity (or anyons theories [4, 5]. This breaking creates magneto-conductivity) effect changes, in partic- so-called Nambu–Goldstone bosons generating ularly, values of the longitudinal resistivity. “effective” magnetic monopoles. These conse- Then, currents and electric fields are no longer quences allow us to claim, first, that anyons are proportionnal but are related together by a two- ∗MEIPIC - 2, Topic#: 04 Rub dimensional antisymetric resistivity tensor de- †E-mail: [email protected] pending on the applied magnetic field inten- 1 sity B. Each diagonal coefficient equals the lon- viewed as effective magnetic singular flux tubes. gitudinal resistance, whereas the non-diagonal The electric charge is shared by q of these flux coefficients are precisely plus or minus the so- tubes. It is from these considerations that the called Hall resistance RH = B/(nec), where n anyons theory and the anyonic superconductiv- is the surface density of electric charges. The ity are built. Indeed, the gauge choice made latter is linearly depending on the filling factor by Laughlin, generates a non-trivial Aharonov- ν = n/nB. This previous relation links the Hall B¨ohm phase when each electron moves on its resistance to the number of filled so-called Lan- cyclotron orbit. In particularly, it follows that dau levels in a bounded conducting layer. These the permutation or exchange operator has no levels are defined by the stationary eigenfunc- longer the eigenvalues ±1 only, but more gener- tions of a one-particle Schr¨odinger equation in ally, unitary complex numbers. That is the main a two-dimensional space with a magnetic inter- property of anyons together with the P and T action. It is a harmonic oscillator-like equation violations. ~ with eigenvalues EN = ωC(N +1/2), where ωC Actually, the litterature about anyons the- N is the cyclotron frequency (N ∈ ). The bound- ory and its mathematical formalisms and tools, ary conditions make each Landau level highly is highly increased. The basic mathematical degenerate with a nB surface density of degen- tool of all of these formalisms, is to take into eracies. account the so-called Chern-Simon (CS) La- So, a linear variation of the Hall resistance grangian term [4, 9]. The latter breaks, as ex- would have been experimentally observed, but pected, the P and T symmetries and is associ- as K. Von Klitzing and al. [6] have shown in ated to the singularities of the gauge fields. It 1980, the Hall resistance is quantized and the is necessary to point out a common confusion evolution with the magnetic intensity presents about the CS term. It is absolutely not the CS plateaus for integer values of ν. More, on the term in a 2+1 dimension space (it doesn’t ex- plateaus and when the temperature goes to zero, ist !), but the one of a curvature field in the the longitudinal resistance tends also towards 3+1 Minkowski space ! After integrating on a zero. The system becomes non-dissipative and 2+1 dimensional hypersurface, bounded by the permanent longitudinal currents appear. plane of the conducting layer, we obtain the In 1982, D.C. Tsui and al. [7] observed the “CS term” of the two-dimensional anyons the- fractional quantized Hall effect for rational val- ory. Moreover, the 3+1 CS term is ∗F . F , where ues of ν ≤ 1 (ν = p/q with q odd), meaning F is the Faraday tensor and ∗F its Hodge dual. that only the first Landau level is excited and Hence, in this paper, we consider anyons theory that electron-electron correlations produce con- in 3+1 dimensions and then the restriction onto densation in a lower energy level. Such results the conducting layer. have been explained in 1983 by B. B. Laugh- Generally, this term is added up in order to lin [8] considering a model of free fermions in a consider monopoles such as the Dirac one. Let plane, interacting with a very particular gauge us recall that the Dirac monopole is associated potential (the “anyonic” vector potential) not a with a singular gauge potential. Hence peo- priori electromagnetic. The main caracteristic ple have made models of anyons considering in- of this potential is the existence of singularities teraction between free electrons and magnetic in the plane at each locations of the fermions. monopoles. As S. Mandelstam showed [10], this Solving the corresponding Hamiltonian leads to leads to a confinement of the Fermi gas and the so-called Laughlin function, which points may generate superconductor states. Anyway, out strong correlations at a finite distance de- statistical calculus based on the equivalence be- pending on the gauge potential intensity. More- tween anyons gas and free electrons embeded in over, the computation of the total energy of the a static magnetic field, show that the anyons system effectively shows singularities (“cusps”) gas generates a Meissner effect. That is the for rational values (with an odd denominator) main reason, with the existence of permanent of ν. currents in the quantized Hall effect, for using One can display this model in considering ex- anyons theory in order to explain high-T su- cited fractional electric charges e/q, i.e. each c perconductivity in two dimensions. cyclotron orbit contains into its associated cir- cle, an odd number q of singularities (or elec- trons) of the gauge potential; The latter being 2 2 - Toroidal moments and broken rela- tivistic crystalline symmetries T Electric and magnetic crystals are characterized by their magnetic groups which are subgroups of the so-called Shubnikov group O(3)1′ (the time inversion is indicated by the “ ′ ” sym- bol). Among the 122 magnetic groups, only 106 are compatible with the existence of a linear or quadratic magnetoelectric effect. In the present paper, we consider relativistic symmetry group theory in crystals [11]. There- fore, we need, first, an extension from the Shub- nikov group O(3)1′ to the group O(1, 3) in the Minkowski space, and second and more partic- j ularly, transformations of O(1, 3) leaving invari- ant polarization and magnetization vectors, and generating a subgroup of the normalizer N(G) of G in O(1, 3). This subgroup G′ may not be iden- tified with the magnetic group if G leaves invari- ant a particular non-vanishing velocity vector. If such a vector exists and G′ =6 G, one strictly Figure 1 speaks about the relativistic crystalline symme- try G′. Only 31 magnetic groups are compati- that this kind of configuration of currents can be ble with the existence of a relativistic crystalline generated from a solenoidal one. The creation symmetry. or the closure to a tore-shaped solenoid from the The invariant non-vanishing velocity vectors latter, can be obtained adiabatically, applying can be linked with toroidal moments from the for instance, a homogeneous external magnetic point of view of magnetic symmetries, as it has field slowly rotating at a more lower frequency been already shown in previous papers (see [12] than the cyclotron one. But we present another for instance). The toroidal moments T are polar possibility in connection with the breaking of tensors which change sign under time inversion, the relativistic crystalline symmetries. Before, like velocity vectors or current vectors of electric it is absolutly necessary to understand that the ′ charges. There are refered to the order parame- groups G are strongly depending on the orien- ter in toroidal phase transitions, and involved, in tations of the polarization and/or magnetization particularly, in the superdiamagnetism of super- vectors.
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