Relativistic Crystalline Symmetry Breaking and Anyonic States

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arXiv:cond-mat/0208004v1 [cond-mat.supr-con] 31 Jul 2002 une lo st li,fis,ta nosare anyons that first, claim, conse- to us These allow quences creates monopoles. breaking magnetic generating in This “effective” as bosons one Nambu–Goldstone 5]. PT the so-called the [4, break not theories but TM’s anyons symmetries the More- T that and notice [1]. P can computed one been with over, have compatible groups speak process magnetic can such The We ef- [3]. kineto-magnetoelectric fect kept. “inverse” of is kind a cell of interacting RCS the the which into givenof one, any “effective” of an frame to proper carrier Galilean initial the of RCS. initial pa- the varying, defining of continuous by rameters the cell case, given simplest the previous in this of breaks of kind can RCS cell this the magnetic car- elementary in charge an Indeed, a and rier between interaction . any [2] crystals, exist ef- into (ME) coupling [1] fects crystal magnetoelectric given non-trivial any which breaking to symmetry the associated crystalline (RCS) relativistic and one-cell (TM) the of moments creation the toroidal between of connection a exists There Abstract ANTEETI SUPERCONDUCTORS MAGNETOELECTRIC EAIITCCYTLIESYMMETRY CRYSTALLINE RELATIVISTIC hsbekn a eascae oachange a to associated be can breaking This ∗ † MgeolcrcItrcinPeoeai rsas Part Crystals, in Phenomena Interaction “Magnetoelectric EPC-2 oi# 04 Topic#: 2, - MEIPIC -al [email protected] E-mail: nacdadegihcretdupbihd20 eso fte19 the of version 2002 unpublished corrected english and Enhanced RAIGADAYNCSAE IN STATES ANYONIC AND BREAKING rceig fteitrainlcneec on conference international the of Proceedings M 2 NS-Uiested ie-Spi Antipolis, Sophia - Nice Université de - CNRS 129 UMR Ferroelectrics 31ruedslcoe,050Vlon,France Valbonne, 06560 lucioles, des route 1361 ntttd o-iár eNc (INLN), Nice Non-Linéaire de du Institut Rub h atrbigpbihdi the in published being latter the soa(wteln)Sp.93. Sept. (Switzerland) Ascona aqe .RUBIN L. Jacques o.1114 19) p 335-342. pp. (1994), 161(1-4) Vol. , coe 0 2018 30, October 1 1 breaking, symmetry Lagrangians. Chern-Simon anyons, symmetries, crystalline Words: Key supercon- in considered theory. highly second, ductors be and can TM’s, ME with that both mag- monopoles, “effective” netic with associated carriers charge edn nteapidmgei edinten- field magnetic applied the de- on tensor pending resistivity two- antisymetric a by dimensional together related are but proportionnal resistivity. longitudinal the (or of magneto-resistivity Hall values ularly, partic- a classical in changes, addition, so-called effect magneto-conductivity) In the is effect. That field magnetic appears. the lon- and the current to electric both gitudinal orthogonal field electric con- current, an electric a then an of by over direction crossed orthogonal layer ducting the mag- in a field applying that netic well-known is It [4]. theory high- the both with quan- link T the the and recall effect briefly Hall very tized we chapter, this In c hn urnsadeeti ed r olonger no are fields electric and currents Then, unie aleet high-T effect, Hall Quantized - uecnutvt hoyadteanyons the and theory superconductivity odciiyadayn theory anyons and conductivity † oodlmmns relativistic moments, toroidal 3one, 93 c super- I” , ∗ 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 Böhm 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 Schrödinger 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 electrons) of the gauge potential; The latter being

2 2 - Toroidal moments and broken relativistic 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 invariant 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 invariant a particular non-vanishing velocity vector. If such a vector exists and G′ =6 G, one strictly Figure 1 speaks about the relativistic crystalline symmetry 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. conductors or dielectric diamagnetic bodies con- Thus, if for example the system passes from taining densely packed atoms (agregates) [12]. the polarization vector P~ = ~0 to P~ =6 ~0, it in- volves a breaking, by group conjugaison, of the In such systems, in the presence of sponta- relativistic crystalline symmetry. Moreover, this neous currents, there may exist states for which conjugaison can be associated to a change of the configuration of the associated currents has Galilean frame, from an initialy fixed one to an a tore-shaped solenoid with a winding and so a “effective” moving one, in such a way that the toroidal moment. For a system with a dipolar initial G′ group is kept in the latter. toroidal moment density T~, the current density Then, phenomenologically, we can consider equals for instance, the following classical physical sys- ′ tem: a relativistic crystalline group G0 of an elementary polarized cell of a given crystal, with ~ ~ j = curl(curl(T )) . (1) an electron not in interaction with this magnetic cell in the initial state. We also assume that this electron has initially a Galilean motion with, for We can remark on Figure 1 below instance, a velocity ~v parallel to the invariant

3 ′ ′ axis of G0 (it is necessary to assume that this (for m and 2 we have two TM’s oriented in group is not invariant by smooth translations in an opposite direction; each one corresponding order to have only one such axis). Then, the to possible sublattices). All the corresponding electron interacts with the cell, inducing (by a normalizers N(G) contain, as a subgroup, the local magnetoelectric influence effect, by an ex- group of rotations in the plane: O(2), in order change interaction for instance or in fact what- to have the precession phenomena. To finish this ever is the interaction !) a change of the elec- chapter, let us give a supplementary important tric or magnetic polarization which breaks the remark. The electromagnetic and the “anyonic” initial relativistic crystalline symmetry group. potential 4-vectors A are toroidal moment den- During the interaction, the resulting symmetry sities, since they satisfy both the relation (1) in ′ group is Gi in the fixed laboratory frame. The a 3+1 dimensional anyons theory restricted in transformation from one symmetry to the other a 2+1 space, i.e. the fields do not depend on is realized, by group conjugaison, with an ele- the perpendicular coordinate. Then in fact, the ment S of N(G). It means that we pass with S problem is to work out a non-electromagnetic from one frame to another in which the motion potential vector in magnetoelectric materials as- of the electron is kept, i.e. Galilean. In some sociated to each charge carrier. We will see, as a way, we can speak of a kind of “inverse kineto- consequence, that CS terms appear and so any- magnetoelectric effect”, since usually the crystal onic statistics. moves whereas it is motionless in that present case. 3 - The relativistic effective Lagrangian Thus, the motion of the electron in the laboratory frame is determined by S , i.e. at most a In this chapter, we derive a density of free en- rotation around the invariant axis and a boost thalpy, or rather a relativistic Lagrangian den- along the same one. From the latter charac- sity, since first, we are not interest in ther- teristics of the electron motion and those of S, mal properties, and second, because our aim is the electron will get a solenoidal motion around to obtain a usable “microscopic” Lagrangian in the invariant axis, and if the electron is polar- anyons theory. Essentially for simplicity of no- ized, the Thomas precession of its spin along tations and to avoid discussions about mean- the trajectory would lead to a toroidal config- ing of a lot of particular tensorial coefficients, uration of spin currents [13]. Moreover, if the we will consider a system in a “nearly empty” electron has a fast solenoidal motion before the space since the reasoning we make, will be easily interaction with the cell, then with the condi- extended in matter without fundamental mod- tion that the interaction is adiabatic, one will ifications. The system we consider, is made of obtain, by the latter process, a slow solenoidal a free carrier and only one polarized elementary motion of the mean position of the electron, and magnetic cell freely moving in space. In fact, we so a tore-shaped motion during the interaction. consider a kind of local crystal field theory. Let us recall also that a non trivial Berry phase In some way, the carrier with the cell is a po- can occur in this process with two main conse- larized carrier, meaning that the particle gets a quences: an effective Yang-Mills field associated covariant (invariant under application of a boost to an anomaly such as a monopole and an any- and/or a rotation) tensorial property such as the onic statistic in cases of collective motions [14]. following scalar product (α =0, 1, 2, 3 and Ein- Then, to finish with this description, we stein convention): present the list of magnetic groups compatible α v . u = v uα = cste , (2) with such process of creation of toroidal moments [1] (let us remark that among the 16 com- where u is the velocity 4-vector of the carrier patible groups tabulated in [1], only 12 are as- (or of the mean position of the carrier in case sociated to a non-trivial O(2) action of the nor- of an adiabatic process) and v is a particular malizer; that is why four of them, namely the 4-vector associated to the magnetic cell. For in- groups 2, 3, 4, 6, are not indicated in the list stance, v can be the invariant velocity vector of below): the magnetic symmetry of the cell, or its magnetic or electric polarization vectors. We assume 1 , 2′ , m, m′ , 1¯′ , 2′/m , 3¯′ , that the interactions between the polarization vectors of the cell and the carrier are only de- 2/m′ , 4/m′ , 6/m′ , 4¯′ , 6¯′ , pending on the relative carrier-cell position. As

4 a fondamental result of the necessary covariance or pseudoscalar factor, one of the two polariza- of the relativistic symmetry, any kind of interaction vectors of the magnetic cell. Thus, if for tion is equivalent to a change of Galilean frame instance m′ is the magnetic symmetry, since the and this is related to the deep meaning of the two latter vectors are in the plane of the sym- concept of ”polarized carriers”. Here, we have a metry, then B~eff. is in the orthogonal direction. carrier polarized by a cell (!) not only by a spin. We have also to notice, as a consequence of the Then, the breaking occurs and the two 4- model, that there is such an effective magnetic vectors u and v are transformed by a Lorentz field (and so a toroidal moment) associated to transformation L ⊆ N(G), i.e. we observe in the each charge carrier. laboratory frame the primed vectors u′ = L.u At this step, unfortunately, we need very so- and v′ = L.v. Now, deriving relation (2) with phisticated mathematical tools such as exterior respect to the laboratory frame time, we obtain: differentiation d, co-differentiation δ, exact, co- exact and harmonic differential n-forms, coming ˙v . u + v . ˙u = 0 , from the de Rham cohomology theory of differential manifolds (see [9] for instance). where the dot indicates the time derivation. The fundamental consequence of the previ- Decomposing ˙u in a colinear and an orthogo- ous relation is that the total Faraday tensor nal part to v, we deduce (a × b = (a b − α β FTot. = F + Feff. is no longer a closed form, aβbα)Eα,β where Eα,β is the matrix with 1 on i.e. dF =6 0 (⇐⇒ divB~ =6 0) because of row α and column β and zero everywhere else): Tot. eff. the time t and the vector position ~r dependen- cies. It follows from the Hodge duality princi- ˙u = −( ˙v . u) v + ˙u1 = (v × ˙v) . u + ˙u1 . ple that we can define dFTot. (δFTot.) as a current of an “effective” magnetic (electric) charge: It can be shown that ˙u1 is only due to differ- ∗ ent interactions which don’t break the symme- dFTot. = jm (δFTot. = je). Let us remark try [15]. So, we cancel it out from the latter that the currents are not primed, because they expression. are currents in the laboratory frame, or equiva- This little computation leads us to write: lently, they have zero divergences in contradis- tinction with the primed currents. Moreover, ′ −1 ′ ′ ˙u = (L˙ L − Lv × L ˙v) . u ≡ Feff. . u . it means also that the electric charge-matter interaction is equivalent to an electric charge- This equation is not but the least, a kind of gen- magnetic charge interaction. eralized Thomas precession equation [15] which Hence, if the Lagrangian density for the “elec- can be associated, in this particular system, to tromagnetic” Faraday tensor F is (with Heavi- an effective Faraday tensor Feff. being only a side’s units): function of the relative cell-carrier position ~r 1 and time, and having not an electromagnetic − F . F − A . j , origin ! But, as a great surprise, the computa- 4 tion of Feff. shows, first, that its effective elec- we obtain for FTot. an other Lagrangian density tric field is vanishing, and second, its effective L such that: magnetic field is such that (v ≡ (γ, γ ~v),γ = 2 −1 2 1 (1 − ~v ) / ; θ being a function of t and ~r): L≡− F . F − A . j 4 Tot. Tot. e e 2 κ ∗ m γ ˙ − FTot. . FTot. − κ Am . jm , (4) B~eff. = ~v ∧ ~v ≃ θ ~v ∧ (~je . ∇~ ) ~v. 4 e (1 + γ) ∗ (3) where κ is a constant, FTot. the Hodge dual We think that it may be possible to interpret of FTot., and Ae and Am are respectively the this field as the effective magnetic field of the potential 4-vectors associated to the exact part ∗ flux tubes leading to the Aharanov–Böhm phase of FTot. and FTot.. As for the currents, the in the anyons theory. We can also notice from A’s are not primed. Let us remark that the the last term in (3) that B~eff. is a kind of vec- harmonic parts have no contributions to the dy- torial Lifshitz invariant. From this expression, namic of the charges, since they differentiations it follows also that ~v can’t be the invariant ve- and co-differentiations are vanishing, but they locity vector since the breaking doesn’t affect have one in the energy and consequently in the its direction. So ~v is, up to a suitable scalar Lagrangian density. These parts are associated

5 to the so-called instantons which are fields gen- [2] H. Schmid, Proceedings of the Symposium erated by the breaking and going away from the on Magnetoelectric Interaction Phenomena breaking area, carrying out a part of the en- in Crystals, Battelle Seattle Research Cen- ergy. Nevertheless, in the case of more than one ter, Seattle, Washington, U.S.A., May 21- electric charge, this field will interact with the 24, (1973), p. 47. others and can not be neglected. Thus, the total potential 4-vector A, interact- [3] E. Ascher, Proceedings of the Symposium ing with a one-charge current, will be the sum on Magnetoelectric Interaction Phenomena of the potentials of all the closed parts of the in Crystals, Battelle Seattle Research Cen- total corresponding Faraday tensor, for which ter, Seattle, Washington, U.S.A., May 21- the Poincaré’s lemma can be applied (indeed, 24, (1973), p. 69. from this lemma, in a suitable open subset not containing the interaction area, it always exists [4] F. Wilczek, Fractional Statistics And a potential since there is no more closed part Anyon Superconductivity, World Scientific inside). In fact, because FTot. is not only elec- (1990). tromagnetic and [5] V. M. Dubovik and V. V. Tugushev, ~ def. ~ Physics Reports, 187(4), (1990). δFTot. = je =⇒ curl(curl(Ae)) ≡ jAe ∗ ~ def. ~ dFTot. = jm =⇒ curl(curl(Am)) ≡ jAm , [6] K. Von Klitzing and al., Phys. Rev. Lett., 45, 494 (1980). then A~e and A~m are toroidal moment densities: ∗ A~e,m ≡ T~e,m. Moreover, FTot. . FTot. is the [7] D. C. Tsui and al., Phys. Rev. Lett., 48, expected Chern-Simon term of anyons theory 1559 (1982). and the harmonic part of FTot. will correspond to the famous singular potential of this theory, [8] B. B. Laughlin, Phys. Rev. Lett., 50, 1395 i.e. to magnetic monopoles. Consequently, at (1983). a macroscopic level, we must add in the free enthalpy, terms such as (E: electric field, H: [9] T. Eguchi, P. B. Gilkey and A. J. Hanson, magnetic field, J: current, T : dipolar toroidal Physics Reports, 66, No. 6, 213-393 (1980). moment): [10] S. Mandelstam, Monopoles in Quan- i j i j tum Field Theory, Proceedings of the γi,j E Je,m , µi,j H Je,m , Monopole Meeting, Trieste, Italy, Decem- i j i j ber 1981, World Scientific (1982) p.289. ρi,j Je,m Je,m , λi,j Je Jm , terms, up to order four, coming from relations [11] A. Janner and E. Ascher, Phys. Lett., (3) and (4), analogous terms substituting T i for 30A(4), 223 (1969). E. Ascher, J. Phys. J i, and crossing terms such as Soc. Japan, 28, Suppl., 7 (1970). E. As- cher, Phys. Stat. Sol. (b), 65, 677 (1974). i j i j A. Janner and E. Ascher, Physica, 48, τi,j Je,mTe,m , ωi,j Je,mTm,e , 425 (1978). W. Opechowski, Proceedings of where γ, µ, ρ, λ, τ and ω are the correspond- the Symposium on Magnetoelectric Interac- ing suitable susceptibility tensors. To finish, let tion Phenomena in Crystals, Battelle Seat- us add that it might allow permanent currents tle Research Center, Seattle, Washington, for particular choices of the latter tensors, and U.S.A., May 21-24, 69 (1973). so superconductivity with toroidal phase transitions [12]. [12] V. L. Ginsburg, Sol. State Comm., 39, 991 (1981). V. L. Ginsburg and al., Sol. State Comm., 50(4), 339 (1984). References [13] A. A. Gorbatsevitch and Yu. V. Kopaev, [1] J. L. Rubin, Il Nuovo Cimento D- JETP Lett., 39(12), 684 (1984). Yu. V. Cond. Matt. At., 15(1) (1993), p. 59. Kopaev and V. V. Tugushev, JETP Lett., http://arXiv.org/abs/cond-mat/0208003 41(8), 392 (1985).

6 [14] A. Shapere and F. Wilczek, Geometric Phases In Physics, Advanced Series in Mathematical Physics, Vol. 5, World Sci- entific (1989). [15] H. Bacry, Le¸cons sur la théorie des groupes et les symétries des particules élémentaires, Gordon and Breach, Paris (1967).