Topology Obstructing Anderson Localization of Light

Introduction:—Electromagnetic wave propagation in random media is a problem with many facets, including the prediction of Anderson localization Anderson (1958) with promising applications in photocatalysis Gomard et al. (2013); Cernuto et al. (2011), lasing Cao (2005); Wiersma (2008); Andreasen et al. (2011), and sensing Oliver et al. (2018). However, despite an extensive body of theoretical work, its experimental observation remains ‘stubbornly elusive’ Skipetrov and Page (2016); Yamilov et al. (2023): unlike with acoustic Condat and Kirkpatrick (1987); Hu et al. (2008), elastic Flores et al. (2013); Ángel et al. (2019), quantum matter Chabé et al. (2008); Lemarié et al. (2010); Billy et al. (2008); Roati et al. (2008); White et al. (2020), or optical fiber Schwartz et al. (2007); De Raedt et al. (1989) waves, the unambiguous observation of Anderson localization of light in three dimensions ($3d$) is pending. Recently, it has been proposed that the elusiveness of localization may originate in light’s vector nature Skipetrov and Sokolov (2014); Cherroret et al. (2016); van Tiggelen and Skipetrov (2021). We here bring an independent alternative into play, delocalization protected by topology.

Conceptually, Anderson localization is a highly universal phenomenon, backed by powerful concepts of statistical mechanics, and described in terms of effective field theories Efetov (1997); Evers and Mirlin (2008). The one principle powerful enough to trump localization in low spatial dimensions $d\leq 3$ is topology. A case in point are the $d\leq 4$-dimensional surfaces of topological insulators Ryu et al. (2010), for which the one-parameter scaling paradigm would require localization by strong disorder, while topological principles demand otherwise. Topology wins, and the ways in which it does can be described in different manners. One option starts with the observation that the band structure of a topological insulator is governed by a single (more generally, an odd number) of Dirac cones. The non-vanishing helicity of the latter requires that surface states remain extended. A more explicit approach reconsiders the universal field theories of localization to observe that in the presence of single Dirac cones they pick up topological Chern-Simons or $\theta$-terms blocking the renormalization group flow into a localized phase Evers and Mirlin (2008).

We here argue that similar principles are at work for light scattering. Our starting point will be a representation of the light scattering problem in terms of a Dirac equation, linear in space and time derivatives, as a direct descendant of Maxwell’s equations. (For our present discussion, this formulation is more natural than the effective second order differential equations more commonly employed in light-localization theory.) On general grounds, the Dirac problem splits into two sectors (‘Weyl-fermions’) distinguished by their helicity. Within the topological insulator analogy, these would correspond to opposite surfaces, separated in real space. Presently, the separation is in momentum space, a scenario otherwise realized in the Weyl-semimetal Armitage et al. (2018) Either way, as long as the two sectors are not coupled by helicity non-conserving scattering, delocalization is granted. The concrete question then is to what degree they remain isolated in the light scattering context, and how efficiently this principle extends localization lengths. Naturally, the answer depends on the microscopic realization of scattering channels, where the vectorial nature of light implies a rich spectrum of scenarios: from helicity conserving dual-scattering enjoying perfect protection to the irrelevance of the topological principle in the case of strong helicity-mixing. In the following, we combine elements of the theory of disordered Weyl quantum matter Altland and Bagrets (2015); Altland et al. (2002) with that of light scattering to a first principle analysis of the situation.

Dirac approach to light scattering:—Our starting point are Maxwell’s equations in absence of external sources (summation convention)

where the presence of a scattering medium enters through the constitutive relations $B_{i}=\mu H_{i}$, $D_{i}=\epsilon E_{i}$ with randomly fluctuating magnetic permeability $\mu=\mu_{0}\mu_{\mathbb{x}}$ and permittivity $\epsilon=\epsilon_{0}\epsilon_{\mathbb{x}}$, respectively ($c_{0}=1/\sqrt{\epsilon_{0}\mu_{0}}=1$ throughout). As an alternative to the standard strategy John (1984); Lagendijk and van Tiggelen (1996) of combining Eqs. (1) and (2) to a single random wave equation for the electric field, we here follow Ref. Bialynicki-Birula (1996) and introduce field components ${\cal E}_{i}=\sqrt{\epsilon}E_{i}$ and ${\cal H}_{i}=\sqrt{\mu}H_{i}$, to define the six-component ‘single photon wave function’ $\Phi=(\Phi^{+},\Phi^{-})$, comprising the complex linear combinations $\Phi^{\pm}=({\cal E}\pm i{\cal H})/\sqrt{2n}$ associated to vacuum solutions of opposite circular polarization, or helicity, with $n=\sqrt{\epsilon\mu}$ the (local) refractive index. Expressed in terms of this ‘single photon wave function’ Bialynicki-Birula (1996), Maxwell’s equations for solutions with monochromatic time dependence of frequency $\omega_{0}$ assume the form (see supplemental material)

Here, the matrices $\tau_{i}$ act between the helical subspaces of the $\Phi$-vector and we use a variant of the Feynman slash notation, $\not{v}=v_{i}S_{i}$, where $(S_{j})_{\alpha\beta}=-i\epsilon_{j\alpha\beta}$ are three-dimensional matrices representing the components of a spin-1 operator, with commutation relations $[S_{i},S_{j}]=i\epsilon_{ijk}S_{k}$ and $S_{i}S_{i}=2\openone_{3}$. Randomness enters this equation through a scalar potential $V=\omega_{0}(1-n)$, and a vector potential, $\not{a}_{Z}=\frac{1}{2}\not{\partial}\ln Z$, involving the local refractive index $n_{\mathbb{x}}$ and impedance, $Z_{\mathbb{x}}=\sqrt{\mu_{\mathbb{x}}/\epsilon_{\mathbb{x}}}$, respectively. We assume a short range correlated Gaussian distribution of the former $\langle V_{\mathbb{x}}\rangle=0$, $\langle V_{\mathbb{x}}V_{\mathbb{x}^{\prime}}\rangle=w_{0}\delta(\mathbb{x}-\mathbb{x}^{\prime})$, and leave the latter unspecified for now. Finally, the linear operator in Eq. (3) possesses time-reversal symmetry, $H=\textrm{T}H\equiv\tau_{3}H^{T}\tau_{3}$, squaring to unity, $\textrm{T}^{2}=\openone$, placing the problem into the orthogonal Wigner-Dyson class-${\rm AI}$.

Before turning to the analysis of the disorder, we note two key algebraic structures of Eq. (3). Helicity: in the absence of impedance variations, $\not{a}_{Z}=0$, the equation decouples into equations $(\chi\not{p}-\omega_{0})\Phi_{\chi}=0$, for the two sectors $\chi=\pm$. These play a role analogous to the states of opposite chirality in the Weyl-semimetal, and enjoy individual topological protection against Anderson localization. The second structure, which has no analog in the condensed matter context, is transversality: Unlike vacuum, inhomogeneous media admit longitudinal field components, which turn out to be key to the delocalization problem.

To organize the field solutions into transverse and longitudinal components, we consider the retarded vacuum field propagators of definite helicity, defined as $\left(\omega_{0}+i\delta-\chi\not{p}\right)G^{0}_{\chi,p}=\openone_{3}$, where $p=(\omega_{0}+i\delta,\mathbb{p})$ is the four-momentum and $\delta\searrow 0$. Introducing the projection operators $\bar{\pi}_{\mathbb{p}}=\mathbb{p}\mathbb{p}^{t}/(p_{i}p_{i})$, $\pi_{\mathbb{p}}=1-\bar{\pi}_{\mathbb{p}}$, a straightforward computation shows that the propagator assumes the form $G^{0}_{p}=G^{0}_{t,p}\pi_{\mathbb{p}}+G^{0}_{l,p}\bar{\pi}_{\mathbb{p}}$, with a propagating transverse ($t$) and non-propagating longitudinal ($l$) contribution given by

As a first step towards the inclusion of disorder, we consider these propagators averaged over realizations of $V$, $G_{0}\to\langle G_{0}\rangle\equiv G$. Referring to the supplemental material for details, we assume weak disorder and compute the propagator within a self-consistent Born approximation (SCBA), leading to $G_{s}=(G_{s,0}^{-1}-\Sigma)^{-1}$, with $s\equiv(\chi,t/l)$. For the simple disorder model used here, and at length scales $\gtrsim\omega_{0}^{-1}$ exceeding the wave length ¹¹1For a detailed discussion of the averaged propagator at short length scales, $<\omega_{0}$, see Ref. van Tiggelen and Skipetrov (2021)., the self energy $\Sigma=-i\kappa$ becomes a constant $\kappa=\kappa(w_{0},\omega_{0},\Lambda)\ll\omega_{0}$ depending on frequency, disorder strength, and an ultraviolett cutoff, $\Lambda$, set by the inverse width of the effective $\delta$-function determining the disorder correlation length. Physically, $\kappa$, describes the damping of the averaged propagator at length scales exceeding the mean free path $\sim\kappa^{-1}$.

Anderson localization shows in observables involving disorder averaged products of advanced and retarded propagators, such as the inverse participation ratio of the electromagnetic energy density Mirlin (2000); Efetov (1997). It is the result of multiple constructive self interference of diffusion modes formed by co-propagating amplitudes of opposite causality, at length scales parametrically exceeding the mean free path. While ‘weak localization’ precursors of the phenomenon can be computed in diagrammatic perturbation theory, the understanding of strong localization (or the absence thereof) requires field theoretical analysis in terms of nonlinear $\sigma$-models Mirlin (2000); Efetov (1997).

Field theory:—The application of the replica field theory formalism to light localization goes back to Refs. John (1987, 1984, 1985). Referring to the original reference and the supplemental material for details, we note that prior to the inclusion of helicity mixing, this theory is described by a functional integral ${\cal Z}=\int D(T_{+},T_{-})e^{-S_{+}[T_{+}]-S_{-}[T_{-}]}$, split into two helicity sectors $\chi=\pm$ with

Treating retarded and advanced propagators in a unified fashion, the action contains a Pauli matrix $\sigma_{3}^{\textrm{ra}}$ multiplying the causal symmetry breaking imaginary self energy $i\kappa$. The fact that prior to averaging the retarded and advanced action were identical (except of the infinitesimal $i\delta$), reflects in the presence of Goldstone modes, acting on the self energy operator $i\kappa\sigma_{3}^{\mathrm{ra}}$ as slowly varying rotation matrices $T(\textbf{x})\in\textrm{Sp}(4R)$, where $R$ is the number of replicas, the symplectic structure is a consequence Efetov (1997) of the class AI time reversal symmetry, and the trace in Eq. (5) extends over both, the $4R$-dimensional matrix space of the Goldstone modes, and Hilbert space. With the definition $Q_{\chi}=T_{\chi}\sigma_{3}^{\mathrm{ra}}T_{\chi}^{-1}$, we have identified the effective degrees of freedom of our theory, whose long range fluctuation behavior will decide over its localization behavior.

Eq. (5) is almost identical to the action of a single node in the Brillouin zone of a Weyl-semimetal in the presence of smoothly varying disorder (i.e., disorder excluding the scattering between nodes) Altland and Bagrets (2016). The differences — the matrices $S_{i}$ acting in a spin-1 representation of $SU(2)$, instead of spin-1/2 Pauli matrices in the Weyl context, and the Weyl Goldstone mode matrices $Q$ lacking the symplectic structure representing symmetry class AI — take no influence on the next step in the construction of the theory: expansion of the ‘Tr ln’ to leading order in derivatives acting on the slow differences. Referring to Refs. Altland et al. (2002); Altland and Bagrets (2016) and the supplemental material for details, this exercise leads to the effective action

where

is the action of the nonlinear $\sigma$ model for three-dimensional disordered media. In Eq. (7), ‘${\rm tr}$’ denotes the trace over internal degrees of freedom, and the coupling constant $\sigma=(\omega_{0}^{2}+3\kappa^{2})\left(\omega_{0}^{2}-\frac{1}{3}\kappa^{2}\right)/\left(\pi\kappa(\omega_{0}^{2}+\kappa^{2})\right)$ plays the role of a bare conductivity characterizing the transport properties of the medium at length scales of the order of the scattering mean free path $\kappa^{-1}$. Depending on whether $\sigma\kappa^{-1}$ is large or small compared to unity, this action describes the flow into a diffusive or a localized phase, the two scenarios being separated by the Anderson localization transition Evers and Mirlin (2008).

The reason why the effective action $S_{\chi}$ predicts different behavior has to do with the presence of the second term, $S_{\textrm{CS}}[T_{\chi}]\equiv S_{\textrm{CS}}[A_{\chi}^{+}]-S_{\textrm{CS}}[A_{\chi}^{-}]$, where

is a Chern-Simons action for the ‘vector potentials’ $A^{s}_{\chi}=\chi T^{-1}_{\chi}\nabla T_{\chi}P^{s}$, and $P^{s}=\frac{1}{2}(\mathds{1}^{\textrm{ar}}+s\sigma_{3}^{\textrm{ar}})$ projects onto retarded or advanced space. The presence of this term in the effective action follows from high-level reasons, namely the equivalence of Eq. (5) to the action of three-dimensional linearly dispersive fermions coupled to ‘gauge fluctuations’ $T_{\chi}$. For this system, the parity anomaly Redlich (1984) requires the presence of a Chern-Simons term. In the specific case, where the fields $T_{\chi}$ are generated by disorder averaging, this term assumes the specific form Eq. (8) Altland and Bagrets (2016).

The absence of localization in a system governed by the full action $S=S_{\textrm{d}}+S_{\textrm{CS}}$ follows from another high-level argument: the prototypical action Eq. (5) describes a three-dimensional linearly dispersive and time reversal invariant system, coupled to disorder. In condensed matter physics, such systems are conceptualized as surface states of four dimensional topological insulators in symmetry class AI, and topological obstructions prevent the localization of their microscopic quantum states. This statement includes strongly disordered cases, where the conductance $\sigma\kappa^{-1}$ becomes of order unity and the derivation of the effective action $S$ is no longer under parametric control.

Supplemental Material for “Topology Obstructing Anderson Localization of Light”

Tobias Micklitz

Alexander Altland

March 10, 2025