On the concept of local invariance in the theory of spin glasses I.E. Dzyaloshinsk, G.E. Volovik To cite this version: I.E. Dzyaloshinsk, G.E. Volovik. On the concept of local invariance in the theory of spin glasses. Journal de Physique, 1978, 39 (6), pp.693-700. 10.1051/jphys:01978003906069300. jpa-00208802 HAL Id: jpa-00208802 https://hal.archives-ouvertes.fr/jpa-00208802 Submitted on 1 Jan 1978 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LE JOURNAL DE PHYSIQUE TOME 39, JUIN 1978, 693 Classification Physics Abstracts 75. 0H ON THE CONCEPT OF LOCAL INVARIANCE IN THE THEORY OF SPIN GLASSES I. E. DZYALOSHINSKII and G. E. VOLOVIK Landau Institute for Theoretical Physics, Vorobyevskoe shosse 2, Moscow 117334 U.S.S.R. (Reçu le 3 février 1978, accepté le 6 mars 1978) Résumé. 2014 Nous étudions la possibilité d’appliquer le concept d’invariance d’échange locale (champ de Yang-Mills [3]) à la description des verres de spin. Partant de l’invariance locale microsco- pique (discrète) et du concept de frustration étudié par Toulouse [8] et Villain [7], nous montrons qu’à l’échelle macroscopique un verre de spin est naturellement décrit par des champs de Yang-Mills continus. En particulier, les champs électriques et magnétiques s’interprètent comme des densités et des courants de disclinations. Nous analysons ainsi les différents types d’ordre à longue distance, le spectre d’oscillations et les éventuelles transitions de phase dans de tels systèmes. Abstract. 2014 We consider the possibility of applying the concept of local exchange invariance (Yang-Mills fields [3]) to describe spin glasses. We start with the microscopic discrete local invariance and frustration concept studied by Toulouse [8] and Villain [7] and show that on the macroscopic level a spin glass is naturally described by continuous Yang-Mills fields. In particular, magnetic and electric fields have the meaning of density and current of disclinations. We also study types of long-range order, oscillation spectra and possible phase transitions in such systems. 1. - While reading the works on spin glasses To make this functional local-invariant one has to (e.g. the brilliant survey by Anderson [1]) it is difficult introduce Yang-Mills fields to abstain from the temptation to use a concept of local exchange invariance (LEI) to describe this peculiar state. LEI leads to strong degeneracy of the which under infinitesimal rota- ground state of a spin glass. If the local symmetry coordinate-dependent tion in are transformed as were exact, the energy functional would remain isotopic space 0(x) invariant not only under simultaneous rotation of all the spins by the same angle, but also for rotation of only one spin with all the rest stationary. In fact however, LEI can only be approximate (see below). This means that with the rotation of one particular while under the same rotation the field S(x) is trans- its itself in such a way spin neighbourhood adjusts formed in the usual way : that in the leading approximation the energy of the system will not change. But even this truncated local invariance and the essentially weaker degeneracy which comes with it us to a permits reproduce majority The space derivatives of S may enter a local inva- of features of - absence of qualitative spin glasses riant functional only in a certain combination with conventional etc... long-range order, mictomagnetism, fields bi as so-called covariant derivatives The mathematical tools of any theory with conti- nuous local invariance, such as LEI, are widely known. They are the so-called Yang-Mills fields [3] and can be introduced as follows. We shall describe the spin glass by the vector field S(x) of unit length : S2 = 1 [2]. The energy of the system is some general functional of which under rotations (2), (3) are transformed as the the field S(x) and its derivatives ôS/ôx;, 02S/0Xi ôxk, ... field S itself. The Yang-Mills fields bi may appear in Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003906069300 694 the energy functional by themselves in S-independent S(a) being three-component classical vectors of unit combinations length. The exchange integral J(a, b) has - with a certain probability - only two values J = + 1. If the probability distribution is chaotic enough, one hopes that this model will display the properties of a spin glass. which are obviously,locàl-invariant too : The system described by the Hamiltonian (10) exhibits the local discrete invariance (LDI) : If besides this space local invariance one requires the time local invariance, we introduce the Yang-Mills field The group (11) is a discrete [5] analogue of the Yang-Mills group (2)-(3). This group is the only transforming as microscopic local symmetry in the physics of spin systems. The concept of LDI in the theory of spin glasses has been employed extensively by Mattis [6], Villain [7] and Toulouse [8]. In particular, Toulouse has provided under rotations The covariant time-depèndent 0(t). a complete description of the ground and excited time-derivative is states of the glass in terms of a frustration network. His results could be formulated as follows. Let us describe the spin orientation by rotation angle 0(a) depending on the lattice site, so that When this new field a is present another locally invariant quantity comes in to play : where So is a constant unit vector. According to Toulouse, the field 0(a) varies fairly smoothly almost everywhere except [9] in the vicinity of the network of singular lines and singular planes connecting them - Formulae (1)-(9) expressing LEI generalize the the frustration network. Therefore we may employ an regular gauge-invariance of Maxwell electrodynamic. intermediate semi-macroscopic picture : the field of The Yang-Mills fields bi and a are analogues of vector the unit vector S(x), the derivatives of which are given and scalar potentials, and fik and gi are analogues of by formula magnetic and electric fields [4]. However, in solid state physics LEI does not exist on the microscopic level (contrary to the situation in particle physics). It might emerge only as the symmetry of some macroscopic quantities averaged over suffi- Vector fields S and 0 are singular on frustration lines ciently large volumes. This averaging will automa- and have a discontinuity on planes stretched upon tically provide the physical meaning of Yang-Mills these lines. fields. Moreover to this definition local according Frustration lines thus defined are disclinations in invariance for will be violated macroscopic quantities the fields S and 0 similar to disclinations in liquid at small enough distances (at sufficiently large gra- crystals. This similarity, however, is superficial because That means that in the of in dients). expansion energy in liquid crystals planes stretched upon disclinations of the of, say, average power gradients (macroscopic) are not singular at all ; their actual arrangements and the first term may be invariant spin m(x) only locally shapes are completely irrelevant provided disclination while the next terms will (- (Vim)’), display ordinary lines are given. In our case these surfaces exhibit real derivatives instead of covariant ones. physical singulàrities. The discontinuity of S and 0 on such a surface is determined the distri- - by probability 2. To understand how to derive the macroscopic bution of the values of J(a, b). And in this respect our relations of this kind we shall consider a certain model disclination in spin glass is a twin brother of the of a spin glass : Heisenberg spins on a frustrated disclination studied by the theory of elasticity [lo]. lattice. This is a model with the standard Heisenberg Now we can take the last and our Hamiltonian with nearest-neighbour interaction : step average picture over scales larger than distances between disclinations. This procedure is well known in the elasticity theory and consists of going over to a conti- nuous distribution of disclinations. Let us define [11] 695 a mean angle of rotation cp(x) and tensor b,(x) similar To make the system complete one should add the to the distortion tensor of the elasticity theory (see [10]). equation which express the conservation of angular The latter is not generally speaking a gradient of any (spin) momentum i in the absence of an external function and has the same transformation properties magnetic field [13] : as the Yang-Mills fields bi in (1), (2). Further we introduce the density of disclinations (see [10]) where llk is current of spin-momentum. For the spin- momentum density it is natural to assume (Eikl is the antisymmetric unit tensor), which in a linear approximation coincides with fik of (5), and the current of disclinations Finally providing some phenomenological expres- sion for Uk (given for instance by (20)) we come to a problem which is formally equivalent to the standard which in the linear approximation is identical with gi problem in the theory of plastic flow (see [10]). of (9). The set of equations (13)-(19) describes a whole For the fourth Yang-Mills field a of (6) one can range of so-called mictomagnetic phenomena. How- usually assume (see [lo]) ever we shall not dwell on them further, but consider the simplest possible case when the mean disclination density in equilibrium is zero [14] and when disclina- tion motion is entirely dissipative.