Magnetic Circular Dichroism Versus Orbital Magnetization

PHYSICAL REVIEW RESEARCH 2, 023139 (2020) Magnetic circular dichroism versus orbital magnetization Raffaele Resta * Istituto Officina dei Materiali IOM-CNR, Strada Costiera 11, 34151 Trieste, Italy and Donostia International Physics Center, 20018 San Sebastián, Spain (Received 13 January 2020; accepted 13 April 2020; published 8 May 2020) The magnetic circular dichroism sum rule yields an extremely useful ground-state observable, which provides a quantitative measure of spontaneous time-reversal symmetry breaking (T-breaking) in a given material. Here I derive its explicit expression within band structure theory, in the general case: trivial insulators, topological insulators, and metals. Orbital magnetization provides a different measure of T-breaking in the electronic ground state. The two observables belong to the class of “geometrical” observables; both are local and admit a “density” in coordinate space. In both of them, one could include or exclude selected groups of bands in order to acquire element-specific information about the T-breaking material. Only in the case of an isolated flat band do the contributions to the two observables coincide. Finally, I provide the corresponding geometrical formula—in a different Hilbert space—for a many-body interacting system. DOI: 10.1103/PhysRevResearch.2.023139 I. INTRODUCTION electronic properties of crystalline materials. Unfortunately, PBCs are incompatible with the presence of a generic mag- Since the very popular 1992 paper by Thole et al. [1], netic field: for these reasons, the theory of orbital magneti- magnetic circular dichroism (MCD) has been widely regarded zation was only established in 2006 [3,4]. Therein, M is ex- as an approximate probe of orbital magnetization in bulk pressed as a reciprocal-space integral; it is worth stressing that solids. Some years later, it was clearly recognized that the no use is made of the angular momentum Lz, a “forbidden” MCD sum rule Iαβ (defined below) provides insight into operator within PBCs. The operator L is only legitimate when the magnetic properties of solids, although such “magnetic z addressing a bounded crystallite. properties” do not coincide with orbital magnetization ex- Here, I thoroughly investigate the analogies and differences cept in the extreme atomic limit [2]. It must be mentioned between the two observables, also providing three significant that at the time, no sound theory of orbital magnetization advances. (i) I give a microscopic expression for Iαβ, gauge in bulk solids was available. Orbital magnetization M is, invariant in form, for any crystalline material (either metal by definition, the derivative of the free-energy density with or insulator) within band structure theory. I also show that respect to magnetic field (orbital term thereof, and with a in a Chern insulator, Iαβ is not affected by the topologically minus sign). After Ref. [2], it is then pretty clear that Iαβ protected edge states (while M is affected). (ii) I show that is the free-energy derivative with respect to a different T- Iαβ is a local observable, in full analogy to M [5,6]. It must breaking probe: circularly polarized light, integrated over the be stressed that other geometrical observables are strongly whole spectrum. Owing to a fluctuation-dissipation theorem, nonlocal (most notably electrical polarization [5]). (iii) I go a frequency-integrated dynamical probe becomes effectively a beyond band structure theory and I show that even in a corre- static one; the said probe has the virtue of coupling to orbital lated many-electron system, Iαβ is a geometrical ground-state degrees of freedom only. The two observables M and Iαβ observable, although in a different Hilbert space. When PBCs provide two quantitatively different measures of spontaneous are abandoned, the very same geometrical formula for Iαβ T-breaking in the orbital degrees of freedom of a given mate- goes seamlessly into the center-of-mass angular momentum rial. On the experimental side, Iαβ is naturally endowed with formula of Ref. [2] (which only makes sense for a bounded core chemical specificity, at the root of its great success; while crystallite). instead only the total M value is experimentally accessible. Condensed-matter physics adopts Born–von-Kàrmàn periodic boundary conditions (PBCs), at the root of Bloch’s theorem, which in turn allows one to address the intensive II. THEORY Shortly after the theory of orbital magnetization was fully established [3], Souza and Vanderbilt derived an explicit *[email protected] expression for Iαβ in the special case of a topologically trivial insulator [7]. It was shown that both M and Iαβ are geometrical Published by the American Physical Society under the terms of the properties of the electronic ground state; an explicit expres- Creative Commons Attribution 4.0 International license. Further sion for their difference was also provided. distribution of this work must maintain attribution to the author(s) The MCD sum rule concerns the frequency integral of the and the published article’s title, journal citation, and DOI. imaginary part of the antisymmetric term in the conductivity 2643-1564/2020/2(2)/023139(6) 023139-1 Published by the American Physical Society RAFFAELE RESTA PHYSICAL REVIEW RESEARCH 2, 023139 (2020) −ik·r tensor, the periodic Bloch orbitals |u jk=e |ψ jk, eigenvectors −ik·r ik·r ∞ of Hk = e He : (−) Iαβ = Im dωσαβ (ω); (1) dk 0 r| P |r=V eik·(r−r )r| P |r, cell d k (7) BZ (2π ) a kind of fluctuation-dissipation theorem relates Iαβ to a ground-state property, both for a bounded sample (e.g., a Pk = |u jku jk|. (8) μ crystallite) within the so-called open boundary conditions jk (OBCs), and for an unbounded solid within PBCs. In both In order to establish a differential geometry in the space frameworks, all ground-state properties—at the independent- of the |u jk state vectors, we choose a gauge which makes particle level—can be expressed in terms of the relevant ∞ the |u smooth (i.e., C ) throughout the whole BZ. This ground-state projector P. jk is always possible, even in topologically nontrivial materials In the OBCs case, the projector (per spin channel) is k [10]. The integrand in Eq. (7) is periodical in , and hence the P = |ϕ ϕ |, BZ integral of its k derivative vanishes: j j (2) μ dk j 0 = i(r − r)r| P |r+V eik·(r−r )r| ∂ P |r, cell π d k k μ |ϕ BZ (2 ) where is the Fermi level and j are the single-particle (9) H eigenstates of the Hamiltonian with eigenvalues j.Inthe band structure case, the projector is instead dk i[r, P] =−V eik·r∂ P e−ik·r. (10) cell π d k k dk BZ (2 ) P = V |ψ ψ |, (3) cell π d jk jk BZ (2 ) μ We are now ready to replace this into Eq. (6), together with jk dk |ψ H = V eik·rH e−ik·r. where BZ is the Brillouin zone, jk are the Bloch orbitals cell d k (11) BZ (2π ) normalized to one in the crystal cell of volume Vcell, jk are the band energies, and d is the dimension; Vcell must be The three reciprocal-space integrals in the product contract to understood as the area for d = 2. The reason for adopting one (see the Appendix), and we arrive at the same symbol P in Eqs. (2) and (3) lies in the “nearsight- 2 P iπe dk edness” principle [8]. If one evaluates from Eq. (2)fora Iαβ = Tr (H − μ) ∂ P ,∂ P . 2 π d cell k kα k kβ k large bounded crystallite, and then further projects this P onto 2¯h BZ (2 ) the inner region of the crystallite, the result asymptotically (12) converges to the P value provided by Eq. (3) for the same The sum rule Iαβ, defined in Eq. (1), is the frequency in- material; the convergence is exponential in insulators and tegral of a linear response function; Eq. (12) shows that it power law in metals. coincides with a ground-state material property. Equalities of We start with a bounded sample within OBCs: the sum this kind belong to the general class of fluctuation-dissipation rule for Iαβ has a relatively straightforward expression [7]. A theorems [7]. tedious calculation (see the Appendix) shows that it can be Equation (12) is one of the major results of the present equivalently expressed as work. It applies on the same ground to trivial insulators, topological insulators, and metals. In the metallic case, the 2 iπe k derivative of Pk includes a δ-like singularity at the Fermi Iαβ =− Tr {(H − μ)[[rα, P], [rβ , P]]}. (4) 2¯h2V level, which is annihilated by antisymmetrization. When spe- cialized to the so-called Hamiltonian gauge [4], Eq. (12) The virtue of this expression becomes clear when switching to yields the same formula as in Ref. [7] (derived therein in a P PBCs and band structure theory. In the latter case, is lattice completely different way for trivial insulators only). periodical, Intensive material properties in crystalline materials are ex- r| P |r=r + R| P |r + R, (5) pressed as reciprocal-space integrals, such as Iαβ in Eq. (12); the analogous expression for orbital magnetization M was where R is a lattice vector. The position r is instead a forbid- found in 2006 [3,4]. While the observables have obviously den operator, incompatible with PBCs [9]. Notwithstanding, different dimensions, the two reciprocal-space integrals are the commutator [r, P] is an honest lattice-periodical operator similar—when expressed within the present formalism—but (like H and P), and hence Eq. (4) can be adopted as it is, not equal. Preliminarily, we observe that custom dictates the adoption of the field H in the free energy.

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