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Quantized toroidal dipole eigenvalues in nano-systems

The parity violation in nuclear reactions led to the discovery of the new class of toroidal multipoles. Since then, it was observed that toroidal multipoles are present in the electromagnetic structure of systems at all scales, from elementary particles, to solid state systems and metamaterials. The toroidal dipole T (the lowest order multipole) is the most common. This corresponds to the toroidal dipole operator T^ in quantum systems, with the projections T^i (i = 1, 2, 3) on the coordinate axes. These operators are observables if they are self-adjoint, but, although it is commonly discussed of toroidal dipoles of both, classical and quantum systems, up to now no system has been identified in which the operators are self-adjoint. Therefore, in this paper we use what are called the “natural coordinates” of the T^3 operator to give a general procedure to construct operators that commute with T^3 . Using this method, we introduce the operators p^(k),p^(k1) , and p^(k2) , which, together with T^3 and L^3 , form sets of commuting operators: (p^(k),T^3,L^3) and (p^(k1),p^(k2),T^3) . All these theoretical considerations open up the possibility to design metamaterials that could exploit the quantization and the general quantum properties of the toroidal dipoles.


Introduction
Toroidal moments first emerged as a new field of research in the context of nuclear physics, when Zeldovich introduced in 1957 a new type of electromagnetic interaction to explain parity nonconservation in β-decays [1]:Ĥ (S is the spin of the particle, J ext is the external current, and H ext is the external magnetic field). Since neither electric nor magnetic multipoles lead to an interaction of the type (1),this novel charge configuration came to be known as "anapole" and it was visualized as a toroidal solenoid with a current flowing along the poloidal lines (as in Fig. 1), which produces a magnetic field localized inside the torus, while the external electromagnetic field of the configuration is zero.In 1997, decades after Zeldovich published his pioneering article [1], the anapole moment was measured by Wood et al by determining the amplitude of the transition between 6S and 7S states in 133 Cs [35].
In the late 1970s, Dubovik et al expanded de concept of the anapole to classical electrodynamics and introduced a new family of multipoles, that he called "polar toroidal multipole moments" [2,3].Like the electric and magnetic moments, toroidal moments form an  Figure 1: A toroidal dipole T generated poloidal currents j on a toroidal solenoid independent multipole family that appears in the multipole expansion of the potentials [36,37] Fundamental symmetry considerations explain the need for this new family of multipoles.For example, the electric dipole is odd under spatial inversion and even under time reversal, while the magnetic dipole is odd under time reversal and even under spatial inversion. For a complete picture, one needs to include dipole moments which are even or odd under both transformations and these are the axial and polar toroidal dipoles [4].
Although overlooked for a long time, it has been realized that the toroidal moments and anapole configurations represent an important tool to describe the properties of systems at all scales, from particle physics to the physics of macroscopic systems and this new field expanded rapidly [5,6]. Nuclear anapole moments were explored thoroughly in a series of articles [9,6,7,8,12,10], focusing on both the theoretical implications, as well as the experimental techniques. Probably one of the most remarkable discoveries in this field is that massive Majorana particles can possess only toroidal moments, which in turn lead to the idea that dark matter particles could be characterized by the anapole [9,10,11,12] In solid state physics, toroidal ordering was first studied theoretically by Charles Kittel [13] and it indicates the existence of a new type of magnetoelectric effect. In the framework of condensed matter, toroidal moments are linked to another kind of order parameter known as toroidization (toroidal polarization). Media that exhibit macroscopic toroidization are called ferrotoroids and since they are expected to have promising technological applications (for example in data storage), it is crucial to investigate whether ferrotoroidicity is on an equal footing with the other feroic states (ferroelectric, ferromagnetic) [17,18,19,20,21,22,23,24]. On top of this, the toroidal moments were studied in relation to the antisymmetric magnetoelectric effect and it was shown that any bulk toroidization results in the emergence of an antisymmetric magnetoelectric tensor [14,15,16]. Moreover, a magnetoelectric effect was predicted in skyrmion crystals and it was quantified with the aid of spin toroidization [15].
Recently, anapole states have generated tremendous interest in the field of nanophotonics, optics and metamaterials and have great potential for a wide range of applications like lasers, sensing and nonscattering objects that may be used for their cloaking behaviour [25,26,27,28]. It has even been proposed that since anapole states interact weakly with electomagnetic fields, they could be used to protect qubits from environmental disturbance [29].
The lowest order toroidal moment corresponding to a current distribution j(r) is a polar vector of components where r ≡ |r|, j = |j|, T ≡ |T|, and i = 1, 2, 3 number the components of a vector on the three axes x, y, and z: T ≡ (T 1 , T 2 , T 3 ), j ≡ (j 1 , j 2 , j 3 ), r ≡ (r 1 , r 2 , r 3 ) ≡ (x, y, z) [34]. The may be decomposed into multipoles interactions, leading to the term characterizing the interaction of the toroidal moment with the external fields [34] H To correctly predict the toroidal moments of quantum systems (particles, nuclei, or macroscopic systems), one has first to study the quantum operators corresponding to them and their basic mathematical properties.
The quantum operators which correspond to the classical toroial moments (2) are [30,33] wherep j ≡ −i ∂/∂x j is the momentum operator along the j axis and m is the mass of the particle. We observe that the commutation relations [33] [ are satisfied, whereL j is the projection of the angular momentum operator on the j axis. In [33] it was shown that in R 3 space,T 3 (and, equivalently,T 1 andT 2 ) is a hypermaximal operator (admits several self-adjoint extensions).
The operatorT 3 has a specific "natural coordinate", denoted by u, such that (notice that in [33] u had a different scale, namelyT 3 = −[i /(10mc)]∂/∂u). Starting from u, one can form the so called "natural systems of coordinates" for the operatorT 3 , like (k, u, φ) and (k 1 , k 2 , u), as it will be explained in Section 2. In the natural systems of coordinates it is easy to construct operators that commute withT 3 , so one can form sets of commuting operators and find their eigenvectors and eigenvalues. Furthermore, we can translate the operators defined in the natural system of coordinates into operators acting on the (x, y, z) coordinates and vice-versa.

Method
The main tools used in this paper, as in Ref. [33], are two natural sets of coordinates of the toroidal moment operator, which we shall present in this section to introduce the notations and make the paper easier to read.

The natural coordinates for the toroidal moment operator
In the following we shall focus on the operatorT 3 . The analysis ofT 1 andT 2 may be done in a similar fashion. According to Eq. (6), [T 3 ,L 3 ] = 0, so it is more convenient to write in cylindrical coordinates [33]T  Figure 2: (a) The directions of the vector field T 3 (i.e., t 3 ≡ T 3 /T 3 ) in a plane that contains the z axis (the units on the axes are arbitrary). The vector field is tangent in every point to the continuous lines that represent the lines along which the "natural variable" u vary. In (b) we show the range of u, as a function of k (in arbitrary units): u ∈ (−a(k), a(k)).
where ρ ≡ x 2 + y 2 , r = ρ 2 + z 2 . We define the vector field T 3 by the relation We plot the directions of Fig. 2, in a plane which contains the z axis. We observe thatT 3 may be written as a derivative along the lines of this vector field, that is, along the continuous curves shown in the figure. This observation led to the introduction of the "natural" set of coordinates (k, u, φ) [33], in whichT 3 takes the form (7), whereas the curves in Fig. 2 correspond to k(ρ, z) = const. The variable u takes values in a finite interval (−a(k), a(k)), with [33] a(k) = 10mc 4k and C a ≈ 1.31103. Notice that all the end points −a(k) and a(k) (for all k) correspond to the points (z → ∞, ρ = 0) and (z → −∞, ρ = 0), respectively. Beside (k, u, φ), we may define the set of coordinates (k 1 , k 2 , u) by in which the relation (7) is still valid. In these variables, k 1 and k 2 take values in the interval (−∞, ∞), whereas u takes values in the interval (−a(k 2 1 + k 2 2 ), a(k 2 1 + k 2 2 )) (Eq. 11).

The Hilbert space
On the space of integrable single valued complex functions defined on R 3 , we define the scalar product of two functions f and g by We work with the Hilbert space H ≡ L 2 R 3 , which consists of the functions integrable in modulus square. The space R 3 minus the z axis, , and a ∈ (−a(k), a(k)). Then, the functions f (r), g(r), from H, are mapped into the functionsf (k, u, φ),g(k, u, φ) defined on the set M. The volume element is transformed as [33] so the scalar product and the norm (Eqs 15 and 13) are respectively. Then, byH ≡ L 2 (M) we denote the Hilbert space of functions of finite norms defined on M, which is the image of H(R 3 ).
In the coordinates (k 1 , k 2 , u) (12) we have the relation together with the scalar product and the norm, respectively. The domain of definition, k 1 , k 2 ∈ (−∞, ∞) and u ∈ −a( k 2 1 + k 2 2 ), a( k 2 1 + k 2 2 ) , will be denoted by M 1 and the Hilbert space of functions of finite norms on M 1 is denoted bỹ

Boundary conditions and eigenfunctions ofT 3
In both sets of coordinates, (k, u, φ) and (k 1 , k 2 , u), the operatorT 3 is defined as −i ∂/∂u. The sets M (in (k, u, φ)) and M 1 (in (k 1 , k 2 , u)) correspond to the whole R 3 (x, y, z) space and no boundary conditions are imposed on their frontiers-that is, when u = ±a(k) in (k, u, φ) and u = ±a( k 2 1 + k 2 2 ) in (k 1 , k 2 , u). Then, in the coordinates (k, u, φ),T 3 is defined on the set of smooth functions on M and we can impose the boundary conditions [33] f where θ(k, φ) can be any function of k and φ. Of course, the same conclusion is reached working in the (k 1 , k 2 , u) space, where the boundary conditions read A set of orthonormal eigenfunctions for theT 3 operator in the (k, u, φ) space, which satisfy periodic boundary conditions in the u direction, was proposed in Ref. [33]: where k 0 ∈ (0, ∞), t k 0 ≡ nπ /a(k 0 ), whereas m and n are integers. We see that the functions (20) are localized on the k axis of the space M.
IfT 3 is not defined on the whole space R 3 , then different boundary conditions may be imposed. For example, the definition domain may be a finite region D of R 3 , with Dirichlet boundary conditions on the frontier (e.g. a nucleus, a region of a condensed matter system, or an element of a metamaterial).

Commuting set of operators
In order to find other operators that commute withT 3 , we notice that in the space (k, u, φ) any operator which is a function of k, ∂/∂k, ∂/∂u, and ∂/∂φ commutes with both,T 3 and L 3 . Similarly, in the space (k 1 , k 2 , u), any operator that is a function of k 1 , k 2 , ∂/∂k 1 , ∂/∂k 2 , and ∂/∂u commutes withT 3 . In this section, we calculate the operators ∂/∂k (in the (k, u, φ) space), ∂/∂k 1 , and ∂/∂k 2 (in the (k 1 , k 2 , u) space). Having these, we can form complete sets of commuting operators involving partial derivatives. For example, in the coordinates (k 1 , k 2 , u), the operatorsp (k1) ≡ −i ∂/∂k 1 ,p (k2) ≡ −i ∂/∂k 2 , andT 3 ≡ −i ∂/∂u resemble the three components of a "momentum operator" and form a complete set of commuting operators. In the coordinates (k, u, φ), the operatorsT 3 ≡ −i ∂/∂u andL 3 ≡ −i ∂/∂φ correspond to the projections on the z axis of the toroidal moment and of the angular momentum, whereas the third operator, defined asp (k) ≡ −i ∂/∂k, represents the projection of the "momentum operator" in the (k 1 , k 2 , u) coordinates, on the radial direction k ≡ k 2 1 + k 2 2 .
3.1. The operatorp (k) , in the (k, u, φ) space From Eqs. (10) we obtain where To obtain the derivative of ρ with respect to z at constant u, we plug Eq. (23) into Eq. (21b) and, using also Eqs. (22), we obtain Combining (24) with (23), we write Then, the partial derivative in the (k, u, ϕ) space of an arbitrary functionF (k, u, ϕ) ≡ F (ρ, z, ϕ) may be written in cylindrical coordinates as Combining Eqs. (22) and (25) we obtain the expression forp (k) : On the other hand, if we invert Eqs. (10), by first formally writing from the second equation z as a function of k and u (say, z(k, u), since we do not have an analytical expression), then, from the first equation, In these notations, which may be plugged into (26), to obtainp (k) . Therefore, the operatorp (k) = −i ∂/∂k in the (k, u, φ) space may be expressed in terms of the (x, y, z) coordinates and has the form and all the partial derivatives are given in equations (22), whereasp ≡ −i ∇ is the momentum operator in the (x, y, z) coordinates.  Figure 3: The directions of the vector field P (k) (i.e., p (k) ≡ P (k) /P (k) ) and the curves of constant u(ρ, z). The vector field is tangent to the curves in every point.
The vector field P (k) in cylindrical coordinates is The directions of P (k) (that is, p (k) ≡ P (k) /P (k) ) are drawn in Fig. 3, together with the curves of constant u(ρ, z). These curves correspond to lines parallel to the k axis in the (k, u, φ) space (fixed u and φ), so the vector field P (k) is tangent to them in every point-similarly to T 3 in Fig. 2.
Equations (8) and (29) may be joined into a system from which we can calculate the partial derivatives in the (x, y, z) coordinates in terms of the partial derivatives in the (k, u, φ) coordinates: where By inverting the system (31), we write ∂/∂ρ and ∂/∂z in terms of ∂/∂k and ∂/∂u, in the natural set of coordinates forT 3 .

Conclusions
We analyzed the quantum properties of the toroidal dipole operator, by studying its projection T 3 on the z axis. For this, we changed from the cylindrical and Cartesian coordinates to what we called the "natural coordinates" of the operator, (k, u, φ) and (k 1 , k 2 , u), respectively, where k 1 = k cos φ and k 2 = k sin φ. The advantage of the natural coordinates is thatT 3 gets the much simpler formT 3 ≡ −i ∂/∂u and some of its properties become more apparent. For example, in the coordinates (k, u, φ) and (k 1 , k 2 , u), the variable u takes values in a finite interval (−a(k), a(k)) for any finite value of k ≡ k 2 1 + k 2 2 , where a(k) = 10mcC a /k (Eq. 11). This property ensures the integrability of the modulus square of the wavefunctions without the necessity to impose zero boundary condition on the border, which implies thatT 3 (as well as all the other projections of the toroidal dipole operator) is hypermaximal.
We also briefly discussed the eigenfunctions and eigenvectors ofT 3 , noticing that in physical systems (in nuclear physics [1,31,6], solid state physics [34], and metamaterials [25,26,32]) toroidal moments are not generated by currents corresponding to these eigenfunctions, that is, currents along the continuous lines in Fig. 2(a), but they are associated, in general, with current distributions along the poloidal lines of tori, as depicted in Fig. 1. Therefore, one can use the sets of commuting operators introduced here to construct new operators acting on closed lines of currents, which are more appropriate for physical (finite) systems, are self-adjoint, and correspond to observables. Such operators may be used to identify physical systems or design metamaterials in which their eigenfunctions (which are also eigenfunctions ofT 3 ) may be formed, eventually with quantized eigenvalues. The observation of the quantization of the toroidal momentum eigenvalues would constitute a signature for the toroidal properties of a system as well as an indication for its quantum properties. This subject will be analyzed elsewhere.