L lines, C points and Chern numbers: understanding band structure topology using polarization fields

Topology has appeared in different physical contexts. The most prominent application is topologically protected edge transport in condensed matter physics. The Chern number, the topological invariant of gapped Bloch Hamiltonians, is an important quantity in this field. Another example of topology, in polarization physics, are polarization singularities, called L lines and C points. By establishing a connection between these two theories, we develop a novel technique to visualize and potentially measure the Chern number: it can be expressed either as the winding of the polarization azimuth along L lines in reciprocal space, or in terms of the handedness and the index of the C points. For mechanical systems, this is directly connected to the visible motion patterns.


Introduction
Ever since the discovery of the transverse wave nature of electromagnetic waves, the study of the polarization properties of such vector waves has attracted a great deal of attention and led to the introduction of novel mathematical concepts. For a generic plane wave, the tip of the electric field vector traces an ellipse that defines the polarization state. In just the same way, we can describe the mechanical motion of a single pendulum that is free to move along two orthogonal directions.
For an arbitrary (monochromatic) field, its polarization becomes position-dependent, and the resulting polarizaton field can display complex spatial patterns. Again, there is a mechanical analogy, in the form of 2D arrays of coupled mechanical oscillators. Such coupled oscillator arrays have already been used as a platform to implement topologically protected transport of sound waves, using coupled pendula [1] and coupled gyroscopes [2]. Eventually, they could also be realized on the nanoscale, e. g. using nanopillar arrays [3,4]. The timeevolution of such an array under monochromatic driving also exhibits elliptical motion that has the same mathematical description as the electromagnetic polarization fields.
The study of the complex spatial polarization patterns in random electromagnetic waves has led to interesting topological concepts. The central objects of interest are lines where the polarization gets linear, and points with circular polarization. These 'L lines' [5,6] and 'C points' [6,7] have been studied thoroughly in random optical fields, but are still not widely known. However, they have been found in many different physical contexts, for example the sunlight in the sky [8] and speckle fields [9]. In recent years, unconventional C points in optical fields where the rotation of the electric field occurs in a plane containing the propagation direction of light have been both predicted theoretically and verified experimentally [10][11][12].
There is, of course, another branch of physics where topology has become very prominent recently: the analysis of band structures. According both to their phenomenology and their theoretical description, two categories can be identified. On the one hand, there are the Chern insulators associated with the quantum Hall effect [13,14] and the anomalous quantum Hall effect [15]; they exhibit chiral edge transport as a result of a nontrivial topological invariant, the Chern number [16][17][18]. On the other hand, topological insulators have been established by the discovery of the quantum spin Hall effect [19][20][21][22][23]; their edge channels are helical, and their topological properties are encoded in a binary ( 2  ) topological invariant [24]. The importance of those topological features on transport is by now well-documented [25].
In the present work, it is our aim to connect these two strands of topology, in a general way that is particularly useful for mechanical systems. Our approach helps to visualize (and, in principle, measure) the Chern numbers based on the polarization fields of bulk excitations. In fact, we offer two different approaches to extract the Chern numbers of the band structure, one based on L lines and the other based on C points.
We will explain how this generalizes the well-known Skyrmion configuration method [26,27]. Furthermore, we will discuss that our method is an alternative to other recently explored techniques to obtain Chern numbers for bosonic systems, both from bulk features [28,29], from dynamics at the boundary [30] and in more general settings [31,32].

Polarization fields
In this section, we will very briefly provide a qualitative description of polarization fields, L lines [5,6] and C points [6,7]. The prototypical example is a monochromatic electromagnetic wave. We observe a twodimensional cut through this field, and restrict to the projection on its in-plane components. For purely transverse polarization, this is naturally given for a plane perpendicular to the propagation direction. Using the common complex representation of the oscillating electric field, we are hence dealing with a (continuous) twocomponent complex vector field in two spatial dimensions.
This situation can be depicted as follows: for each point in space, we draw the orbit that the electric field vector traces during one cycle of oscillation. In general, this will be an ellipse (traversed with a certain handedness). A typical example for a random polarization pattern is shown in figure 1. Two particular features stand out: first, all the points with a linear polarization pattern (where the ellipse degenerates into a line) are arranged on closed curves; these are called the L lines. Second, there are isolated points of circular polarization, the so called C points.
Both L lines and C points are topologically robust in the sense that they are stable under smooth deformations of the polarization field. That is, they will only deform and move around, but they cannot suddenly appear or disappear. Destruction and creation of C points is only possible pairwise, i. e. two C points of the same handedness (same circular polarization) have to annihilate, and always two C points of the same handedness are created together. The only way for an L line to disappear is to contract down to a point, and new L lines can only emerge in the inverse process; in addition, L lines can also merge and split.

2.2.
Constructing an auxiliary polarization field in momentum space for Bloch bands Now, we will look at a different situation and relate it to the polarization picture subsequently. We consider a particle or excitation hopping around on a lattice in two dimensions. We will assume that there is some internal degree of freedom associated with the particle, or, equivalently, a certain number of basis states situated at each lattice site. The physical origin may vary: the spin of a particle, the atomic orbital for atoms in a crystal (s, p, d, ...), the different oscillation directions of a mechanical resonator in a phononic lattice, the sense of circulation of the whispering gallery modes in a lattice of coupled optical disk resonators [65], or the polarization of photonexciton polaritons in micropillars [66][67][68]. In addition, one unit cell might consist of multiple sites, depending on the lattice geometry. For all these cases, we will speak of the 'polarization degree of freedom'.
The total number N of basis states associated with a unit cell of the Bravais lattice determines the number of bands. In the following, we will denote the real-space basis states as j s where n is the index of the band (with n N 1   ) and r j  the position of the unit cell.
In order to make a connection with the polarization fields introduced before, we need to identify a complex twocomponent vector field which encodes the Bloch state. The projections of any Bloch wave function along two arbitrary, linearly independent polarization directions define such a complex vector field in momentum space: Without loss of generality, the two projections can always be chosen along orthogonal directions, which we will assume in the following. For particles with spin-1/2 or equivalent systems, like mechanical oscillators with two degrees of freedom (oscillation directions), s 1 and s 2 can refer to the two basis states of the internal degree of freedom. If the lattice is a non-Bravais lattice, the projection onto one sublattice is a natural choice. In the mechanical case, on the selected site. We note that, in the most general case, the orbit will be elliptical, relating directly to the polarization patterns observed in electromagnetic waves.

Geometrical interpretation of the polarization
In a mechanical setting, the angle k q  ( ) gives the direction of the major axis of the elliptical orbit and is therefore called the polarization azimuth, k c  ( ) represents a measure of its ellipticity and handnedness, and S 0 (the total intensity along the two projections) is the squared diagonal of the axis-aligned bounding box, see figures 2(a) and (b). In addition, j is the oscillation phase which is a gauge freedom for the eigenstate.
One can characterize the polarization of the field k n y    Figure 1. Polarization singularities in a random field. We imagine a monochromatic electric field with two components in the usual complex representation, x y , ) of the electric field vector; we can only draw a finite selection of points, here arranged on a square lattice, but note that the field itself is continuous and defined in every intermediate point as well. The polarization becomes exactly linear on the L lines (green) which separate regions with opposite handedness (white versus gray background), and perfectly circular in the C points (yellow).
The parameters S 1 , S 2 , and S 3 can be identified with the coordinates on the Poincaré sphere, while S 0 is the corresponding radius, see figure 2(c).

Chern number
By now, it is well-established [25,27] that Bloch wave functions defined on a two-dimensional momentum space can be viewed as a mapping from a torus (the Brillouin zone) to a complex vector space, and that close inspection of this mapping can be used to define topological invariants. We will now consider the situation of an arbitrary periodic Hamiltonian where time-reversal symmetry is broken (e. g. due to application of a magnetic field, in the case of electrons). Then, the topological invariant in question is the so-called Chern number (an integer). This quantity is defined for band insulators where the bands remain distinct, i. e. there is a gap between those bands. The Chern numbers of individual bands cannot change under smooth transformations of the Hamiltonian as long as the bands do not touch.
Chern numbers derive their importance from the so-called bulk-boundary correspondence: they can be used to predict whether there will be chiral, topologically protected edge channels located at the boundary of any finite sample (and the net number of clockwise versus counter-clockwise channels). In this sense, a bulk property (the Chern number associated to the infinitely extended periodic system) is connected to the boundary properties of a finite sample.
The Chern number of the band n can be calculated as is therefore sufficient to determine the Chern number of the nth band. This may even be of experimental relevance, for platforms like mechanical systems where direct measurement of the motion pattern in a given Bloch state is feasible.
It turns out that the Chern number can even be calculated by knowing the projections (in polarization space) of the Bloch state k n F ñ  | ( ) along just two linearly independent directions, as long as these projections do not both vanish simultaneously for the same quasimomentum k  (for a proof, see section 5). This fact is the basis for what ). (c) Representation on the Poincaré sphere. The three coordinate axes S 1 , S 2 and S 3 are the Stokes parameters. A half rotation of the major axis in the two-dimensional physical space is already sufficient for a full rotation around the S 3 axis since it only gives rise to a global minus sign which can be absorbed into the complex phase j. Note that two orthogonal states ( a b 0 á ñ = | ) are located on opposite spots of the Poincaré sphere.
we will discuss in the remainder of this article: the Chern number is directly related to the polarization pattern encoded in the two-component complex vector field k n y   ( ) ( ) introduced in equation (2).

L lines and Chern numbers
In the previous section, we have shown how to assign to each band n an auxiliary polarization field k n y   ( ) ( ) , see equation (2). In the remainder of the paper, we will make use of this field and of the geometrical interpretation of the polarization to provide a recipe to visualize the Chern number of the band.
A typical polarization pattern is shown in figure 3(a). There, it can be seen that the polarization gets perfectly linear at the interfaces between the regions with left-and right-handed polarization. These interfaces are therefore called L lines. They are a generic feature of continuous polarized fields. L lines have been studied especially in polarized random fields [5,6]. They are structurally stable upon small perturbations Their robustness can be motivated by a topological argument: because the mapping on the Poincaré sphere is continuous, any path between two points on different hemispheres, i. e. opposite handedness, has to inevitably cross the equator where the polarization is linear.
Below, we explain how to simply read off the Chern numbers by inspecting the L lines of the polarization field. This is possible under a basic assumption: there should be no amplitude vortex in the polarization field. Amplitude vortices are the points where S 0 vanishes. For a two-band model, this condition is equivalent to a degeneracy point, i. e. the band gap has to be closed there, which implies that in this case the Chern number is not well-defined anyway. Otherwise, it can always be argued that amplitude vortices are not topologically stable in two dimensions because they require that four parameters vanish simultaneously (the real and imaginary part of 1 y and 2 y ). Thus, they do not appear in generic random fields. While they could emerge as a consequence of some lattice point symmetry (for three or more bands), they will disappear in the presence of a small perturbation that breaks that symmetry (and does not change the Chern number). Alternatively, one could eliminate such structures even without modifying the underlying Hamiltonian, by choosing different projections to define the auxiliary polarization field k n y   ( ) ( ) . For the above reasons, the scenario analyzed here where no amplitude vortex is present is not a special case but rather a generic one. In view of establishing a connection between the L lines and the Chern numbers, we define the winding number z of the polarization azimuth along one L line. This is defined as follows: we observe how the linear polarization direction changes as we traverse the closed L line in a specified sense. If the polarization turns around the origin counterclockwise (clockwise), z acquires a positive (negative) sign, and its absolute value is determined by the number of full turns. This value can be half-integer, since the linear polarization direction is defined only modulo 180°. For our purposes, we specify that the traversal direction of the L line should always be chosen to match the handedness of the polarization field in the enclosed region (clockwise in the example of figure 3(a)).
We now state (and later prove) one of the main messages of this paper: the winding number z of the polarization azimuth along this L line is directly connected to the Chern number. In the simplest case with one L line per Brillouin zone, it is directly given by

C point classifications and Chern numbers
Next, we show that the Chern numbers can also be related to the properties of the so-called C points [6,7]. The C points are the points where the polarization gets perfectly circular. In other words, they correspond to the poles of the Poincaré sphere, and they are the nodes of the scalar field   figure 3(a). The color wheel in (b) tells how to translate a color into the corresponding direction; note that q is defined only up to p (not 2p). We can see that the q field in (a) has six singularities (vortices) per Brillouin zone; they correspond to the C points in k . The L line, drawn in white (with arrows indicating the handedness of the polarization field enclosed inside the L line), separates the one right-handed (S 0 3 < ) C point at K ¢ from the lefthanded (S 0 3 > ) C points, one at K and four near to Γ. In addition to the handedness, each C point is attributed with an index I It turns out that the C points can be classified according to four different criteria [7,72,73]. Below, we show that two of these criteria are relevant in determining the Chern number. The first relevant criterium is based on their handedness, i. e. whether the polarization is purely left-handed (S 0 3 > ) or right-handed (S 0 3 < ). The other relevant criterium is based on the so-called I classification: the polarization azimuth q is not well-defined at a C point, and the index I counts its winding around the corresponding singularity (in counter-clockwise direction) Its half-integer nature emerges from the fact that a half rotation of q corresponds to a full rotation around the S 3 axis, i. e. it is already sufficient to restore the original elliptical motion pattern. We note that a C point, by definition, emerges at any crossing of the contours S 0 1 = and S 0 2 = . Whenever any random perturbation is introduced, all the crossings with I 1 2 ¹  are splitted into several C points (the sum over all C points of the I indexes is conserved in this process). In contrast, those C points with I 1 2 =  are structurally stable, i. e. they will not split, and they cannot be created or destroyed spontaneously; pairwise creation and annihilation is however possible.
It turns out that also the sum of the I indexes weighted by the corresponding handedness (the sign of S k 3  ( )) is conserved for all these continuous transformations. In other words, such a weighted sum over all C points is a topological invariant. In the present setting where the polarization field is associated to the nth energy band of a translationally invariant system, the topological invariant discussed above can be identified with the Chern number

Connection to other methods and proof
We will now present a proof for our recipe in section 3 to obtain the Chern number from the L lines. In addition, we will relate the schemes presented in sections 3 and 4 with another, well-known technique to visualize the Chern number, the Skyrmion configuration method, and discuss common features and differences.

Kohmoto method
We will prove our main result formula, equation (6), for the Chern number by making a connection to the wellknown analytical method for calculating the Chern numbers which is due to Kohmoto [18]. In Kohmotoʼs method, one fixes a gauge by requiring that the overlap between the Bloch eigenstates of a particular band with a fixed state, for example j s , ñ | , is a real positive number. When this prescription is well defined in the whole Brillouin zone, the Chern number can be interpreted as the flux of a curl piercing a closed surface (the Brillouin zone which is a torus), see equation (5). In this case, Stokes theorem ensures that the Chern number will be zero. Thus, in all topologically non-trivial cases the prescription will be ill-defined for one or more values of the quasimomentum k  . These points are commonly referred to as obstructions. The reason why the gauge prescription is ill-defined at an obstruction is simply that the corresponding overlap matrix element vanishes there. In this situation, it is neccessary to divide the Brillouin zone into several regions where different gauge conditions are applied. Then, the Chern number is encoded in the phase mismatches AB j at the interface between the different regions, ( ) is the wavefunction in the gauge chosen in the regions A and B, respectively. If the subdivision into regions is chosen such that there are no trijunctions, i. e. all the interfaces are closed loops, the Chern number is where the sum is taken over interfaces between neighboring regions A, B (where A is enclosed by B), and the integral is along the respective interface line. In practice, an effective route to analytically calculate the Chern number is to choose one gauge k B F ñ  | ( ) , determine the corresponding obstructions, and, for each obstruction j, fix a suitable gauge k A j F ñ  | ( ) in a small region around it. This procedure reduces the task of calculating the Chern number to computing the phase mismatch in a few infinitesimally small regions. This often paves the way to an analytical treatment.
Our scheme discussed in section 3 can be viewed as a variation of Kohmotoʼs method where rather than trying to fix the same gauge in the whole Brillouin zone and inserting patches where this does not work, we focus from the very beginning on two different gauge choices. For this purpose, we define the left-and right-handed components of an eigenstate Fñ | (here, we omit the band index n) using the corresponding auxiliary polarization field from equation (2) . The crucial difference with respect to figure 4(a) is the presence of an amplitude vortex, located directly at the K point (cmp. c). This point can be taken into account by an additional contribution J to the Chern number (cmp. equation (10)), but it is impossible to determine its value if no additional component of the Bloch state k n F ñ  | ( ) is known. Alternatively, the value of J can be understood intuitively in the following way: the amplitude vortex is unstable and under a small perturbation it decays into C points. Since the amplitude vortex in (b) is protected by rotational symmetry, it can be destroyed only by some modification to the Hamiltonian which breaks this symmetry; (b′) and (d′) show the resulting q plots. Whereas nothing significant happens from (d) to (d′), subplot (b′) shows how the amplitude vortex in (b) decays into two C points with contributions ( ) is discontinuous at C points, so S 0 and 2q cannot simply be interpreted as the amplitude and the phase of a scalar field. Demanding either L Y or R Y to be real, we obtain two distinct gauge conditions for the state Fñ | This implicitely defines the gauge L F ( R F ) by demanding its component L Y ( R Y ) to be real-positive; starting from an arbitrary gauge k 0 F ñ  | ( ) , this can always be achieved by the choices We will fix the gauge using the condition for L Y when the state Fñ | is in the northern hemisphere of the Poincaré sphere (corresponding to L R Y > Y | | | |, i. e. left-handed polarization), and otherwise the one for R Y .
This construction is possible because we have assumed that there are no amplitude vortices, so the larger component can never become zero. By applying equation (11) to this situation, we see the remarkable fact that knowing two components of the Bloch band Fñ | , here L Y and R Y , is sufficient to determine its Chern number as they fix the gauges L F and R F , respectively. This already confirms our claim from section 2 that we need the projections along only two linearly independent polarization directions.
What remains is to show that the phase mismatch between these two gauges, defined by By inserting equation (12) into the numerator and comparing the result with the definition of σ in equation (7) rewritten in terms of s 1 Y and s 2 Y using equation (4), we find that ( )) | | ( )according to the last equality in equation (7). This directly leads to the result of this calculation which has been already stated in equation (6).

Skyrmion number
It has been discussed before [26,27] counts how many times the mapping d k   ( ) wraps the Bloch sphere, and is identical to the Chern number of the corresponding energy band. The skyrmion number has been developed in the context of magnetic skyrmions, quasiparticles which appear in special magnetic materials [74].
In that sense, our method generalizes this 'skyrmion method' to models with an arbitrary number of bands. In addition, it connects the Chern number to the theory of L lines and C points, and it provides an interpretation of the Chern number in terms of directly visible features.

Application to mechanics and optics
The illustrative examples in figures 3-5 have actually been calculated for a specific model which has already been implemented for gyroscopic pendula [2] and proposed for photon-exciton polaritons [66][67][68]75]. This tightbinding model features a 2  polarization degree of freedom on a honeycomb lattice. The on-site polarization degree of freedom represents two in-plane directions of motion of a two-dimensional mechanical (e. g. gyroscopic) oscillator or two directions of the field distributions in the case of photon-exciton polaritons.
Because of the two sublattices in a honeycomb lattice, there are in total four energy/frequency bands in such a tight-binding model. For a pair of sites, the longitudinal (transversal) mode is defined such that it is parallel (perpendicular) to the connecting line. The longitudinal (transversal) modes of nearest neighbor sites are coupled with coupling strength t L (t T ). The toy model for this interaction in mechanics are two-dimensional mass-spring networks. In this setting, additional on-site Zeeman splitting-breaking the time-reversal symmetry-gives rise to AQHE edge channels [38,75]. Because of its simple conception and the possibility to directly observe the polarization patterns in a mechanical system, such a system is a promising platform to demonstrate the determination and visualization of the Chern number with the technique described in this article. Whereas a macroscopic implementation of this model seems straightforward, a mesoscopic or microscopic one is more challenging due to the difficulties to fabricate springs at these lengthscales. However, there might be some more easily fabricated systems which intrinsically have this kind of interaction, e. g. arrays of nanopillars [3,4]. Unpublished experimental data [76] indicate that the motion of neighboring nanopillars is coupled, and the described coupling type is the only linear interaction model which respects all the present symmetries.
Conceptually, there are different options to implement Zeeman splitting including charged nanopillars in a homogeneous magnetic field, the conserved angular momentum [2] of magnetic tips in analogy to the Einsteinde Haas effect [77,78], and the Coriolis force [38] (cmp. figure 6). However, order-of-magnitude estimates indicate that in a nanopillar setup only the latter one can be strong enough to overcome the effect of dissipation and disorder.
Another possible application, in the context of optics, are dielectric materials with a discrete translational symmetry in two dimensions; chiral edge channels analogous to the QHE have already gained experimental verification in photonic crystals [48]. In these systems, the (complex) electric field E x y ,  ( )takes over the role of the electron wave function. If arbitrary quasimomentum states can be excited and the resultant polarization of the electric field E x y , 0 0  ( )can be measured at one single point x y , 0 0 ( )which is fixed previously, mapping these polarization patterns onto the Brillouin zone allows to directly read off the Chern number. Note that although the degree of polarization of the field E  is only two, the dimension of the Bloch space is infinite due to the dependency on the two (continuous) coordinates x and y. This is one example where the generalization to an arbitrary number of bands, compared to the Skyrmion method (see section 5), becomes important.

Measuring the Chern number
Due to its outstanding role in the description of topologically protected edge transport, the measurement of the Chern number may be a crucial point in an experimental analysis of a Chern insulator.
One possible approach to obtain the Chern numbers is to measure the quantized Hall conductance in the band gaps. For example, this measurement works with very high precision in the original setup, the quantum Hall effect [13,14]. However, there might be some situations in which wrong results are obtained due to technical limitations, e. g. if edge states in one band gap exist in principle, but they have a too large penetration length compared to the system size to be detectable. Or, if there is only a local, but not a global band gap, the Chern number is well-defined, but cannot be measured this way. More importantly, this method is by a fundamental reason restricted to fermionic systems as it relies on the quantized Hall conductance. Since there is no equivalent quantity in a bosonic system, different approaches to measure the Chern number for those had to be developed [28][29][30][31][32].
Our method could serve as an alternative to these techniques as it offers a very direct approach: experimental techniques like Fourier transform spectroscopy or the excitation with quasimomentum modes give full access to the Bloch states; this is sufficient input to identify the L lines and C points, and to determine their relevant properties. Therefore, straightforward application of the schemes in sections 3 and 4 can also be used to measure the Chern numbers for any type of Chern insulator.
Its universality and directness could be a great advantage in comparison to other strategies. We also emphasize that, as opposed to, for example, a numerical calculation using a band structure simulation (there are  [77,78]) whose magnetic moment corresponds to the one of the magnet indicated in the figure. (c) Global rotation of the nanopillar array. As observed in the co-rotating frame, there is the centrifugal force pointing directly away from the rotation axis (blue). Additionally, a relative movement of a pillar (indicated for one example) will induce a force perpendicular to its direction of motion, the Coriolis force, giving rise to Zeeman splitting. both analytical [18, section III] and numerical [79] methods), our method requires neither precise knowledge of the actual system parameters nor even the understanding of the underlying microscopic mechanisms.

Conclusion
The central aspect of this article is the connection between two topological concepts in modern physics: we have related the Chern number, one of the central quantities in the context of the QHE and the AQHE, to L lines and C points, the structurally stable objects known from random fields in polarized optics. What we obtain is a graphical interpretation of the Chern number which makes this abstract quantity a bit more tangible.
The term polarization is fundamental in the geometric description of oscillating electromagnetic fields. Even more, we can directly see what it means to a mechanical system by looking at the trajectories. This intuition makes these two classes of systems play an exclusive role for the scheme presented here.
Besides the visualization, we have also discussed the possibility to use this technique as a tool to measure the Chern number in an experiment.