Topological invariants of electronic currents in magnetic fields
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Topological invariants of electronic currents in magnetic fields
S. Selenu (Dated: February 9, 2021)In this article it is reported a formulation of the solenoidal nature of quantum electronic currents atthe nanoscale whose divergence is expressed as the coupling of a magnetic field, interacting with aquantum body, and a weighted Cern invariant vector making then a direct topological interpretationof this quantum magnetic phenomenon. Also a Fourier analysis of the signaling of electronic wavesis reported in an ab initio formalism from first pinciples[1].
INTRODUCTION
This article may be considered as a first attempt on re-gards the understanding of electronic current motion ofa quantum system interacting with magnetic fields andis physical properties of being solenoidal as it is well re-ported by experiments in physical labs[3–5]. The reach-ing of formulas making us able understanding the verynature of quantum magnetic electronic currents allowsthen to perform their engeneering, lead by the knowledgeof their solenoidal properties, either at the nano scale ormicroscopic scale via their control by magnetic fields. Itwill be shown also how latter properties are related tothe very nature of electronic waves nowaday accessibleon physical labs them related to a topological invariantcalled Berry curvature[6, 7] whose physical meaning aslong being evaded since the discovery of the linear con-nection by M. Berry[8]. It will be shown either the ap-pearence of a Cern invariant vector here thought of be-ing as the counterpart of any measurable classical vectorfield should we take into account during the engeneeringprocess of electronic current control across quantum mat-ter. Moreover appears useful to underline that the modelof the topological current density is associated to a wellmeasurable vector field in strict analogy to the classicalformulation of classical electromagnetism[2], where herethe leading hypothesis made on classical quantum me-chanics is considered, as it is already experimented onelectrons at the macroscopic scale[9], them behaving likewaves either at the nanoscale. A formulation of main re-sults therefore obtained will be expressed via an ab initioformalism from first principles in order to obtain a set offormulas, reached in order, it making us able performinga Fourier analysis of the signal data of electronic waveseither on experimental labs. The latter will be derivedand reported in the next section where conclusions areleft at the end of the article.
A TOPOLOGY OF ELECTRONIC CURRENTS INMAGNETIC FIELDS
In this part of the article it will be focused on thedeveloping of a new formulation of magnetism of quan- tum matter at the nanoscale while concerning the use ofmagnetic transformations[10] of a quantum eigenstate ofmatter due to the interaction of the latter with magneticfields. We may notice how the behaviour of quantumelectronic waves due to the scattering of their associatedquantum momenta with the classical potential vectorsenforces us to take into account a new formalism on thesearch of solutions of the Shroedinger equation, as foundin [10], with the aim to find the electronic waves across asample interacting with a magnetic field it shall be usedin order evaluate the quantum electronic current via thetransformation of the linear momentum in the Hilbertspace spanned by quantum eigenstates | Ψ i . Let us startour evaluation of the expectation value of the linear mo-mentum by considering the unitary transformation act-ing on eigenstates of the system making changing theirwave vector k : U = e i e ~ c ( B × r ) · i ∇ k (1)where e is the elementary charge, c the speed of light,and B a uniform magnetic field interacting with thequantum body. On a wave formalism the quantum mo-mentum appears being: p n = h Ψ n, k − ec B × r | ˆ p | Ψ n, k − ec B × r i (2)it evaluated in the n -th state of the electronic wave. Theassociate electronic current per state is directly calcu-lated as j n = ep n , can be studied in an eigenstate ref-erence Hilbert space transforming wave form of the elec-tronic wave signals across the sample body by mean ofthe quantum magnetic transformations reported in1. Infact, we can directly write: p n = h Ψ n, k | ˆ p − ec B × i ∇ k | Ψ n, k i (3)showing that with respect to the reference Hilbertspace spanned by eigenstates | Ψ n, k i the macroscopic av-erage of the quantum linear momentum is: P = X n f n p n = X n f n h Ψ n, k | ˆ p − ec B × i ∇ k | Ψ n, k i (4)= X n f n h Ψ n, k | ˆ p | Ψ n, k i − ec X n f n h Ψ n, k | B × i ∇ k | Ψ n, k i = p − ec B × [ X n f n h Ψ n, k | i ∇ k | Ψ n, k i ]= p + ec B × ¯r = p + ec A corresponding to a measurable classical potential vec-tor in classical electromagnetism, where f n are the oc-cupation numbers of the quantum electronic wave. Theassociate electric current J = eP carried by the quantalstate is then: J = e p + e c A (5)At this stage of our modelling it is not yet exploitedthe topolical invariance of the quantum divergence of theelectronic current it shall be shown appearing as the cou-pling of the Cern invariant with the magnetic vector mak-ing us able recognizing the averaged electronic currentdensity being a quantum topological invariant of the sys-tem. Let us firstly evaluate divergence of the electroniccurrent ∇ · J by directly calculating: ∇ · J = e ∇ · p + e V c ∇ A (6)= ∇ · j − e V c B · i [ X n f n h i ∇ Ψ n, k | × | i ∇ Ψ n, k i ]= ∇ · j − e V c B · i rot [ X n f n h Ψ n, k | i ∇ k | Ψ n, k i ]= ∇ · j + e V c B · i rot [¯ r ]being ¯ r the macroscopic position[11, 12] of the quantalbody expressed as a function of wave vectors k . in viewof the fact that expressing the Berry curvature of theelectronic waves, with constant transport numbers[13],implie: c = rot [ X n f n h Ψ n, k | i ∇ k | Ψ n, k i ] (7)= − i [ X n f n h i ∇ k Ψ n, k | × | i ∇ k Ψ n, k i ]= − i [ X n f n h i ∇ k Ψ n, k | Ψ n, k i × h Ψ n, k | i ∇ k Ψ n, k i ]= i X n f n [ h Ψ n, k | i ∇ k Ψ n, k i × h Ψ n, k | i ∇ k Ψ n, k i ] h Ψ n | Ψ n i = 0by making use of |∇ k Ψ i = h Ψ |∇ k Ψ i| Ψ i , being also eval-uated the vector product of the linear connection withrespect itself being equal to zero. Let us call C = V R c the weigthed Cern invariant, reducing to the Cern invari-ant for uniformly filled eigenstates. Because of eq.(7), theBerry curvature or either its weighted counterpart it iszero for any set of eigenstates, then the volume integralof the divergence of the electronic current is :1 V Z ∇ · j = 0 (8)1 V Z ∇ · J = e V c B · i C · C = 0where it has been considered a set of electronic waveswith constant transport number[13] on a sample not cur-ring macroscopic electronic currents j in absence of amagnetic field crossing the sample itself. Our demon-stration of formulas reported in eq. (8) put on a basethe topological invarinacy of the quantum divergence ofthe electronic current allowing us to notice how a nullCern invariant gives rise to a null divergence it corre-sponding to a solenoidal electronic current. The quan-tum electronic structure[1] of the quantum system willthen permit to lead and control the electronic currenton samples crossed by external magnetic fields, allow-ing then to an engeneering of quantum samples at thenanoscale underposed to electronic wave signaling. Auseful formula based on Fourier analysis[13] for the elec-tronic wave signaling, makes vary the magnitude of theaverage electronic currents across the quantum samples,can be evaluated by an ab initio modelling[13] as alsoreported as a conclusion of our article in what is the fol-lowing formulas:1 V Z ∇ · J = 0 (9)1 V Z J = e [ X G f G | c G | ~ G ] + ec B × [ X G f G c ∗ G i ∇ c G ]Considering this final result to put on a new base quan-tum magnetism by the finding of formulas involving topo-logical Cern invariants coupling with the magnetic fieldas a counterpart to classical magnetism still on use onclassical physics, we report conclusions on the next partof the article. CONCLUSIONS