2.1. Berry phase

2.1.1. Discrete Berry phase

A Berry phase describes phase evolution of a complex vector as it changes around a closed loop (Berry Reference Berry1984; Hannay Reference Hannay1985; Berry & Hannay Reference Berry and Hannay1988). The Berry phase probes the underlying geometric structure. A non-zero Berry phase is analogous to the situation of a vector not returning to its original direction when it undergoes parallel transport around a loop on a curved surface. A standard example for where a Berry phase arises is the adiabatic evolution of a quantum mechanical wavefunction. Berry or geometrical phases have also found numerous applications in plasma physics (Littlejohn Reference Littlejohn1988; Liu & Qin Reference Liu and Qin2011, Reference Liu and Qin2012; Brizard & de Guillebon Reference Brizard and de Guillebon2012; Burby & Qin Reference Burby and Qin2013; Rax & Gueroult Reference Rax and Gueroult2019). To discuss Berry phase in a general way, our setting is a Hilbert space, and we use bra-ket notation, where the Hermitian product of two vectors ${\mathinner{|{u}\rangle}}$ and ${\mathinner{|{v}\rangle}}$ is denoted by ${\mathinner{\langle{u|v}\rangle}}$. If $a$ and $b$ are constants, then ${\mathinner{\langle{au|bv}\rangle}} = a^* b {\mathinner{\langle{u|v}\rangle}}$, and an asterisk denotes complex conjugation.

As is often the case, one can first gain intuition in a discrete setting. Suppose we have $N$ unit vectors, ${\mathinner{|{u_1}\rangle}}, \ldots , {\mathinner{|{{u_{N}}}\rangle}}$, as depicted in figure 1(a). The Berry phase of this sequence of vectors is defined as

(2.1)\begin{equation} \gamma ={-}\mbox{Im} \ln \left[ {\mathinner{\langle{u_1 | u_2}\rangle}} {\mathinner{\langle{u_2 | u_3}\rangle}} \cdots {\mathinner{\langle{u_{N}| u_1}\rangle}} \right], \end{equation}

where $\gamma$ is the Berry phase around a closed loop formed by the discrete sequence. For a complex number $z = |z|\ \textrm {e}^{\mathrm {i}\varphi }$, $\mbox {Im} \ln z = \mbox {Im}( \ln |z| + \mathrm {i} \varphi ) = \varphi$, so the $\mbox {Im} \ln (\cdot )$ operation yields the complex phase and discards the magnitude. The product of the $N$ inner products of the vectors has some complex phase, and the negative of that phase is the Berry phase. Since different branch choices of the complex logarithm leads to non-uniqueness of the phase up to integer multiples of $2{\rm \pi}$, the Berry phase is defined modulo $2{\rm \pi}$.

Figure 1. Discrete (a) and continuous (b) vectors for Berry phase, where c is a closed curve.

Typically, when working with complex unit vectors, their overall phase is arbitrary such that any physical result does not depend on the phase. The Berry phase is constructed such that it is invariant to these phases. To see this, consider a gauge transformation induced by phase factors $\beta_j$, where a new set of $N$ vectors is defined

(2.2)\begin{equation} {\mathinner{|{u_j}\rangle}} \to \textrm{e}^{-\mathrm{i} \beta_j} {\mathinner{|{u_j}\rangle}}. \end{equation}

The Berry phase computed from the transformed vectors is exactly $\gamma$ because all of the individual phases cancel out. The Berry phase is said to be invariant to the gauge transformation, or gauge invariant. The gauge invariance of Berry phase suggests it may be connected to a physically observable phenomenon.

2.1.2. Continuous formulation of Berry phase

Let us take the continuum limit of the Berry phase. We start from the expression

(2.3)\begin{equation} \gamma ={-}\sum_{j=0}^{N-1} \mbox{Im} \ln {\mathinner{\langle{u_j | u_{j+1}}\rangle}}. \end{equation}

which is equivalent to (2.1) modulo $2{\rm \pi}$. We suppose $j$ is an index that parameterizes some property, and we let $j$ pass to the continuous parameter $s$ and ${\mathinner{|{u_j}\rangle}} \to {\mathinner{|{u(s)}\rangle}}$ as shown in figure 1. An additional constraint is that we impose that ${\mathinner{|{u(s)}\rangle}}$ be continuous and differentiable.

In the continuum limit, the intuitive notion is to let ${\mathinner{\langle{u_j | u_{j+1}}\rangle}} \to {\mathinner{\langle{u(s) | u(s + \mathrm {d} s)}\rangle}}$, but we immediately replace this by setting

(2.4)\begin{equation} {\mathinner{|{{u(s + \mathrm{d} s)}}\rangle}} = {\mathinner{|{u(s)}\rangle}} + {\mathinner{|{u'(s)}\rangle}} \mathrm{d} s + O(\mathrm{d} s^2), \end{equation}

where ${\mathinner{|{u'(s)}\rangle}} = {\mathrm {d}}/{\mathrm {d} s} {\mathinner{|{u(s)}\rangle}}$ is the tangent vector to ${\mathinner{|{u(s)}\rangle}}$. Then,

(2.5)\begin{align} {\mathinner{\langle{u(s) | u(s + \mathrm{d} s)}\rangle}} &= {\mathinner{\langle{u(s) | u(s)}\rangle}} + {\mathinner{\langle{u(s) | u'(s)}\rangle}} \mathrm{d} s + O(\mathrm{d} s^2) \nonumber\\ & = 1 + {\mathinner{\langle{u(s) | u'(s)}\rangle}} \mathrm{d} s + O(\mathrm{d} s^2). \end{align}

Hence,

(2.6)\begin{equation} \ln {\mathinner{\langle{u(s) | u(s + \mathrm{d} s)}\rangle}} = {\mathinner{\langle{u(s) | u'(s)}\rangle}} \mathrm{d} s + O(\mathrm{d} s^2). \end{equation}

In the continuum limit of (2.3) and taking the sum to an integral, we obtain

(2.7)\begin{equation} \gamma ={-} \mbox{Im} \oint \mathrm{d} s {\mathinner{\langle{u(s) | u'(s)}\rangle}}, \end{equation}

where the integral is over the closed loop.

An important property is that ${\mathinner{\langle{u(s) | u'(s)}\rangle}} $ is pure imaginary, which follows from differentiating ${\mathinner{\langle{u(s) | u(s)}\rangle}} =1$ with respect to $s$. Thus, equivalent to (2.7) is

(2.8)\begin{equation} \gamma = \mathrm{i} \oint \mathrm{d} s {\mathinner{\langle{u(s) | u'(s)}\rangle}}. \end{equation}

2.2. Berry connection

Thus far, we have merely dealt with a parameterized loop of unit vectors, with vectors defined only on that loop. Now, suppose there is a two-dimensional (2-D) parameter space, with coordinates denoted by $\boldsymbol {k} = (k_x, k_y)$ as in figure 2. While this parameter space can be completely general, we will primarily be concerned with the situation where the parameters are wave vectors in Fourier space. Furthermore, we assume one can define vectors ${\mathinner{|{{u(\boldsymbol {k})}}\rangle}}$ which exist within some neighbourhood of $c$, not just $c$ itself. Along the path $c$, we can write ${\mathinner{|{u(s)}\rangle}} \to {\mathinner{|{{u(\boldsymbol {k}(s))}}\rangle}} $, and express

(2.9)\begin{equation} \frac{\mathrm{d} }{\mathrm{d} s} u(\boldsymbol{k}(s)) = \frac{\partial u}{\partial k_j} \frac{\mathrm{d} k_j}{\mathrm{d} s}, \end{equation}

where a sum over repeated indices is implied. The Berry phase can be written

(2.10)\begin{equation} \gamma = \mathrm{i} \oint \mathrm{d}\boldsymbol{k} \boldsymbol{\cdot} {\left\langle{u(\boldsymbol{k}) | \boldsymbol{\nabla}_{\boldsymbol{k}} u(\boldsymbol{k})}\right\rangle}. \end{equation}

We define

(2.11)\begin{equation} \boldsymbol{A}(\boldsymbol{k}) = \mathrm{i} {\left\langle{u(\boldsymbol{k}) | \boldsymbol{\nabla}_{\boldsymbol{k}} u(\boldsymbol{k})}\right\rangle}, \end{equation}

which is called the Berry connection or Berry potential. The terminology ‘connection’ comes from differential geometry, whereas the term ‘potential’ arises from an analogy with the vector potential of electromagnetism. The Berry connection is pure real, as can be seen by taking the gradient of ${\left\langle{u | u}\right\rangle} = 1$ with respect to $\boldsymbol {k}$.

Figure 2. Parameter space $\boldsymbol {k}$. Here, $S_1$ and $S_2$ denote the inside and outside of the closed loop $c$.

Let us consider how the Berry connection and Berry phase transform under a gauge transformation. Similar to the discrete case, define a gauge transformation to construct a new set of unit vectors differing from the original by a phase

(2.12)\begin{equation} {\mathinner{|{{u(\boldsymbol{k})}}\rangle}} \to \textrm{e}^{-\mathrm{i} \beta(\boldsymbol{k})} {\mathinner{|{{u(\boldsymbol{k})}}\rangle}}, \end{equation}

where the phase $\beta (\boldsymbol {k})$ is real and differentiable. Using (2.11), the Berry connection transforms as

(2.13)\begin{equation} \boldsymbol{A}(\boldsymbol{k}) \to \boldsymbol{A}(\boldsymbol{k}) + \boldsymbol{\nabla}_{\boldsymbol{k}} \beta(\boldsymbol{k}), \end{equation}

and therefore the Berry connection is not gauge invariant. Because the factor $\textrm {e}^{-\textrm {i} \beta (\boldsymbol {k})}$ must be single-valued, the Berry phase is gauge invariant modulo $2{\rm \pi}$.

2.3. Berry curvature

Define the Berry curvature

(2.14)\begin{equation} \boldsymbol{F}(\boldsymbol{k}) = \boldsymbol{\nabla}_{\boldsymbol{k}} \times \boldsymbol{A}(\boldsymbol{k}). \end{equation}

The curvature encodes information about the local geometric structure. The curl applies in a three-dimensional (3-D) (or 2-D restriction thereof) setting. One can formulate the curvature and other concepts here in higher dimensions using differential forms, but we have no need for that machinery at the moment. For a 2-D parameter space $(k_x, k_y)$, there is only one component of the Berry curvature:

(2.15)\begin{equation} F(\boldsymbol{k}) = \mathrm{i} \left( {\left\langle{ \frac{\partial u}{\partial k_x} | \frac{\partial u}{\partial k_y}}\right\rangle} - {\left\langle{ \frac{\partial u}{\partial k_y} | \frac{\partial u}{\partial k_x}}\right\rangle} \right) ={-}2\, \mbox{Im} {\left\langle{ \frac{\partial u}{\partial k_x} | \frac{\partial u}{\partial k_y}}\right\rangle}. \end{equation}

The Berry curvature is gauge invariant. This fact follows from (2.13) and that the curl of a gradient vanishes. Thus there is a suggestive analogy with the magnetic field. The Berry connection $\boldsymbol {A}$ is analogous to the vector potential, and is not invariant under a gauge transformation. The Berry curvature $\boldsymbol {F}$ is analogous to the magnetic field, and is invariant under a gauge transformation.

2.4. Chern theorem

A simple version of the Chern theorem states that the integral of the Berry curvature over a closed 2-D manifold is

(2.16)\begin{equation} C = \frac{1}{2{\rm \pi}} \int \mathrm{d} \boldsymbol{S} \boldsymbol{\cdot} \boldsymbol{F}(\boldsymbol{k}), \end{equation}

for some integer $C$. Here, $C$ is called the Chern number of the surface. It is a topological invariant associated with the manifold of states ${\mathinner{|{{u(\boldsymbol {k})}}\rangle}}$ defined on the surface. The Chern number is a property of the collection of complex vectors over the surface, not just the surface itself. As a topological invariant, the Chern number provides a topological quantization.

That the Chern number must be an integer can be understood intuitively as follows. Consider again figure 2. Let $\boldsymbol {A}_1$ be the Berry connection constructed with a gauge such that it is smooth on $S_1$, and similarly for $\boldsymbol {A}_2$ on $S_2$. The Chern number is given by

(2.17)\begin{align} C &= \frac{1}{2{\rm \pi}} \left( \int_{S_1} \mathrm{d}\boldsymbol{S} \boldsymbol{\cdot} \boldsymbol{F} + \int_{S_2} \mathrm{d} \boldsymbol{S} \boldsymbol{\cdot} \boldsymbol{F} \right) \end{align}

(2.18)\begin{align} &= \frac{1}{2{\rm \pi}} \left( \oint_c \mathrm{d}\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{A}_1 - \oint_c \mathrm{d}\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{A}_2 \right) \end{align}

(2.19)\begin{align} &= \frac{1}{2{\rm \pi}} \left( \gamma_1 - \gamma_2 \right). \end{align}

Since gauge invariance requires the Berry phases $\gamma _1$ and $\gamma _2$ be equal modulo $2{\rm \pi}$, the Chern number must be an integer.

When the Chern number is non-zero, one cannot construct a smooth, continuous gauge for ${\mathinner{|{{u(\boldsymbol {k})}}\rangle}}$ over the entire closed surface. A related concept is the hairy ball theorem, which states that due to the topology of the sphere, any vector field on the sphere must have singularities or vanishing points. While there is as yet no direct physical interpretation of the Chern number in continuum systems, the physical interpretation of the Chern number in photonic crystals has been advanced recently; it has been shown the photonic Chern number is related to the thermal fluctuation-induced angular momentum (Silveirinha Reference Silveirinha2019a,b).

The Chern theorem relates geometry and topology. This is analogous to the Gauss–Bonnet theorem, which relates an integral of a local geometric quantity, the Gaussian curvature, to a global topological quantity, the Euler characteristic. Here, the Berry curvature is a local geometric quantity, whereas the Chern number is a global topological property. The analogy is not perfect, however, as an important distinction is that Gaussian curvature reflects a property of the base manifold while the Berry curvature reflects a property of a vector field on the base manifold.

2.5. Alternative form of the Berry curvature

Given some unit vector as a function of two parameters $\boldsymbol {k} = (k_x, k_y)$, (2.14) provides a formula for the local Berry curvature $F(\boldsymbol {k})$. This standard form depends on derivatives of the vectors at different parameter values, which poses difficulties for numerical computations because great care is required to ensure a smooth gauge. An alternative form for the Berry curvature which is often useful can be given under the conditions that the parameterized vector arises from a non-degenerate Hermitian eigenvalue problem. The alternative form is manifestly gauge invariant.

Consider the eigenvalue problem:

(2.20)\begin{equation} H {\mathinner{|{n}\rangle}} = \omega_n {\mathinner{|{n}\rangle}}, \end{equation}

where $H$ is a Hermitian $N \times N$ matrix, which acts as an effective Hamiltonian. Thus the $\omega _n$ are real. Here we assume that there is no degeneracy, so that all $N$ eigenvalues are distinct. A discussion of the degenerate case can be found in Bernevig & Hughes (Reference Bernevig and Hughes2013). Let $\{ {\mathinner{|{n}\rangle}} \}$, $n=1, \ldots , N$ be an orthonormal eigenbasis. The Hamiltonian depends on $\boldsymbol {k}$, and therefore so do the eigenvalues and eigenvectors. From (2.15), the Berry curvature corresponding to eigenvector ${\mathinner{|{n}\rangle}}$ is

(2.21)\begin{equation} F_n(\boldsymbol{k}) ={-}2\, \mbox{Im} {\left\langle{\frac{\partial n}{\partial k_x} | \frac{\partial n}{\partial k_y}}\right\rangle}. \end{equation}

Apply $\partial / \partial k_i$ to (2.20), obtaining

(2.22)\begin{equation} \frac{\partial H}{\partial k_i} {\mathinner{|{n}\rangle}} + H {\left|{\frac{\partial n}{\partial k_i}}\right\rangle} = \frac{\partial \omega_n}{\partial k_i} {\mathinner{|{n}\rangle}} + \omega_n {\left|{\frac{\partial n}{\partial k_i}}\right\rangle} . \end{equation}

Act on (2.22) from the left with ${\mathinner{\langle{m}|}} $. For $n \neq m$, and assuming non-degenerate eigenvalues, we obtain

(2.23)\begin{equation} {\left\langle{m | \frac{\partial n}{\partial k_i}}\right\rangle} = \frac{ \displaystyle {\left\langle{m | \frac{\partial H}{\partial k_i} | n}\right\rangle}} {\omega_n - \omega_m}. \end{equation}

Use $I = \sum _m {\mathinner{|{m}\rangle}} {\mathinner{\langle{m}|}}$ in (2.21) to obtain

(2.24)\begin{equation} F_n(\boldsymbol{k}) ={-}2\, \mbox{Im} \left[ \sum_m {\left\langle{\frac{\partial n}{\partial k_x} | m}\right\rangle} {\left\langle{m | \frac{\partial n}{\partial k_y}}\right\rangle} \right]. \end{equation}

Consider the $m=n$ term in the sum. We have previously seen that when ${\mathinner{|{n}\rangle}}$ is a unit vector, $\langle n | \partial n/\partial k_i\rangle$ is purely imaginary. Therefore, $\langle \partial n/\partial k_{x} | n\rangle \langle n | \partial n/\partial k_{y}\rangle$ is real and does not contribute to the sum. Using (2.23) for the remaining terms, we obtain

(2.25)\begin{equation} F_n(\boldsymbol{k}) ={-}2\, \mbox{Im} \left[ \sum_{m \neq n} \frac{\displaystyle {\left\langle{n | \frac{\partial H}{\partial k_x} | m}\right\rangle} {\left\langle{m | \frac{\partial H}{\partial k_y} | n}\right\rangle} }{(\omega_n - \omega_m)^2} \right]. \end{equation}

This can also be written in the usual form without the explicit imaginary part:

(2.26)\begin{equation} F_n(\boldsymbol{k}) = \mathrm{i} \sum_{m \neq n} \frac{\displaystyle {\left\langle{n | \frac{\partial H}{\partial k_x} | m}\right\rangle} {\left\langle{m | \frac{\partial H}{\partial k_y} | n}\right\rangle} - {\left\langle{m | \frac{\partial H}{\partial k_x} | n}\right\rangle} {\left\langle{n | \frac{\partial H}{\partial k_y} | m}\right\rangle}}{(\omega_n - \omega_m)^2}. \end{equation}

This form of the Berry curvature is manifestly gauge invariant, because any phase on the eigenvectors from a gauge transformation cancels out. This form can be useful in practice, particularly for numerical computations. The original form of the Berry curvature is not manifestly gauge invariant. It contains derivatives of the eigenfunctions. In contrast, (2.26) places the derivative on the Hamiltonian rather than on the eigenfunction and eliminates issues of needing to numerically constrain to a smooth gauge.

2.6. Bulk-boundary correspondence

One of the most important reasons for the widespread interest in topological phases is the bulk-boundary correspondence, which states that the bulk properties and edge properties of systems are connected. While our discussion so far has been purely an abstract, mathematical discussion, we now turn to the physical manifestations of topological phase. As already mentioned, the abstract unit vectors ${\mathinner{|{{u(\boldsymbol {k})}}\rangle}}$ discussed previously can represent the eigenfunctions of a Hamiltonian, with dependence on wave vector $\boldsymbol {k}$. By use of a Fourier transform to $\boldsymbol {k}$ space, one is implicitly considering an infinite material, or that finite-size systems are sufficiently large. Chern numbers can be computed for each band in the bulk.

The bulk-boundary correspondence principle states that when two materials with differing topological phases and a common gapped spectrum are brought next to each other, modes localized to the interface and crossing the gap must appear at the interface (Hasan & Kane Reference Hasan and Kane2010). The bandgap Chern number for one material is $C_{\text {gap},1} = \sum _{n < n_{\text {gap}}} C_n^{(1)}$, summed over all bands below the bandgap in the first material. Similarly, for the second material, the gap Chern number is $C_{\text {gap},2} = \sum _{n < n_{\text {gap}}} C_n^{(2)}$. If $C_{\text {gap},1} - C_{\text {gap},2} \neq 0$, propagating surface modes are present in the gap. The standard heuristic argument for why modes at the interface must appear is that, for a gapped spectrum, the Chern number cannot change across the interface unless the gap closes somewhere at the interface. Closing the gap is accomplished by the surface mode. Moreover, the difference in Chern number dictates the number and direction of the propagating surface modes (Hassani Gangaraj, Silveirinha & Hanson Reference Hassani Gangaraj, Silveirinha and Hanson2017).

While the conventional understanding just given is often assumed to hold, it is typically proven only for specific model systems (Silveirinha Reference Silveirinha2019a). Additionally, in some cases of continuous-media systems, the bulk-boundary correspondence principle has been found to not apply straightforwardly (Hassani Gangaraj & Monticone Reference Hassani Gangaraj and Monticone2020). Tauber, Delplace & Venaille (Reference Tauber, Delplace and Venaille2020) found that the number of edge modes can be boundary-condition dependent, and restoration of the correspondence between Chern numbers and number of edge modes requires a more generalized accounting of possible ghost edge modes.

2.7. Compactness

The Chern theorem of (2.16) holds for a closed manifold (a manifold without boundary that is compact). In condensed matter systems or photonic crystals that have an underlying periodic lattice, the wave vector space of the first Brillouin zone is also periodic, topologically equivalent to a torus, and compact. The Chern theorem can therefore be applied directly.

However, in continuum models that are typically used in plasmas or fluid dynamics, non-compact wave vector manifolds arise naturally. The wave vector space extends to infinity; $|\boldsymbol {k}| = \infty$ can be thought of as the boundary. It is therefore important to delve at least a little into the issue of compactness to understand whether and how the Chern theorem is applicable. To be fully precise here, we distinguish between a Chern number, which is an integer-valued topological invariant, and the integral of the Berry curvature. For a compact manifold, the Chern theorem guarantees these two values are equal. For a non-compact manifold, that is not necessarily the case, and in fact one may find non-integer results for the integral of the Berry curvature. The effects of non-compactness are subtle, and may or may not cause difficulties in any given problem.

For example, one of the frequency bands in the cold plasma model discussed in § 3.2 has a non-integer integral of the Berry curvature, although the other bands have integer values (Parker et al. Reference Parker, Marston, Tobias and Zhu2020b). Moreover, in the shallow-water model discussed in § 3.1, all frequency bands result in integer-valued Berry curvature integrals. Interpretation in terms of the bulk-boundary correspondence is unclear when non-integer values are present.

There are various ways of dealing with the lack of compactness in continuum models (Silveirinha Reference Silveirinha2015; Souslov et al. Reference Souslov, Dasbiswas, Fruchart, Vaikuntanathan and Vitelli2019; Tauber, Delplace & Venaille Reference Tauber, Delplace and Venaille2019). If the problem stems from infinite wave vectors, one method is to introduce a regularization at small scales that enables compactification. For example, if the behaviour is regularized to decay sufficiently rapidly at large wave vectors, the infinite $\boldsymbol {k}$-plane can be mapped onto the Riemann sphere, which is a compact manifold and enables the Chern theorem to apply. Physically, such a regularization can be justified because the continuum model ceases to be valid at the microscopic scale of the interparticle spacing and the discreteness of the plasma becomes apparent. Regularization based on plasma discreteness for the cold plasma model was used by Parker et al. (Reference Parker, Marston, Tobias and Zhu2020b).

Instead of regularizing based on some physically motivated reason, one might try to tackle the lack of compactness directly. For non-compact manifolds, the index theorems relating an analytical index and topological index can be generalized, and there are additional boundary terms in the index formula (Eguchi, Gilkey & Hanson Reference Eguchi, Gilkey and Hanson1980). The boundary data arising from infinite wave vectors can be responsible for the non-integer integral of the Berry curvature.

3.1. Shallow-water equations and equatorial waves

Following Delplace et al. (Reference Delplace, Marston and Venaille2017), the non-dimensionalized, linearized fluid equations of motion of the shallow-water system are

(3.1a)\begin{gather} \partial_t \eta ={-}\partial_x u_x - \partial_y u_y, \end{gather}

(3.1b)\begin{gather}\partial_t u_x ={-}\partial_x \eta + f u_y, \end{gather}

(3.1c)\begin{gather}\partial_t u_y ={-}\partial_y \eta - f u_x, \end{gather}

where $u_x$ and $u_y$ are the fluid velocities and $\eta$ is the perturbation about the mean height. The $f$-plane model is used here, which is a local model of a rotating sphere using a constant value for the Coriolis parameter $f$ at a particular latitude, and $x$ and $y$ are the coordinates on the tangent plane. The sign of $f$ changes across the equator from the northern to southern hemisphere. To facilitate analysis, the $f$-plane is taken to be infinite and homogeneous. Note that $f$ appears in a manner similar to the cyclotron frequency of charged particles moving in a magnetic field.

Following standard Fourier analysis, we treat all perturbation quantities as having dependence ${\rm e}^{{\rm i}(k_{x} x + k_{y} y - \omega t)}$. The linearized system can then be written as the eigenvalue equation

(3.2)\begin{equation} H {\mathinner{|{\psi}\rangle}} = \omega {\mathinner{|{\psi}\rangle}}, \end{equation}

where the frequency $\omega$ is the eigenvalue,

(3.3)\begin{equation} {\mathinner{|{\psi}\rangle}} = \begin{pmatrix} \eta \\ u_x \\ u_y \end{pmatrix}, \end{equation}

and

(3.4)\begin{equation} H = \begin{bmatrix} 0 & k_x & k_y \\ k_x & 0 & \mathrm{i} f \\ k_y & -\mathrm{i} f & 0 \end{bmatrix}. \end{equation}

The effective Hamiltonian $H$ is Hermitian. The eigenvalues are $\omega _{\pm } = \pm \sqrt {k^2 + f^2}$ and $\omega _0 = 0$, where $k^2 = k_x^2 + k_y^2$. These modes are the Poincaré waves and a degenerate zero-frequency Rossby wave. The non-normalized eigenfunctions are

(3.5a,b)\begin{equation} {\mathinner{|{\psi_\pm}\rangle}} = \begin{pmatrix} k^2 \\ \pm k_x \sqrt{k^2 + f^2} + \mathrm{i} f k_y \\ \pm k_y \sqrt{k^2 + f^2} -\mathrm{i} f k_x \end{pmatrix}, \quad {\mathinner{|{\psi_0}\rangle}} = \begin{pmatrix} f \\ -\mathrm{i} k_y \\ \mathrm{i} k_x \end{pmatrix}. \end{equation}

The three frequency bands are shown in figure 3.

Figure 3. Dispersion relation for the three frequency bands in the shallow-water model. Colour shows the value of the Berry curvature, and the Chern number of each band is indicated. Here, $f=0.5$.

Using the concepts developed in § 2, we are now in a position to calculate the Berry connection, Berry curvature and Chern number of each band. We show the computation in detail for the $\omega _+$ band; the other two bands are analogous. The standard inner product is used.

First we compute the Berry connection for this band. An equivalent expression to (2.11) for a non-normalized eigenfunction is

(3.6)\begin{equation} \boldsymbol{A}_+(\boldsymbol{k}) ={-}\frac{\mbox{Im} {\left\langle{\psi_+ | \boldsymbol{\nabla}_{\boldsymbol{k}} \psi_+}\right\rangle}}{{\left\langle{\psi_+ | \psi_+}\right\rangle}}. \end{equation}

It is convenient to express $\boldsymbol {\nabla }_{\boldsymbol {k}}$ in polar coordinates, where $k_x = k \cos \varphi$ and $k_y = k \sin \varphi$. Thus,

(3.7)\begin{equation} {\mathinner{|{\psi_+}\rangle}} = \begin{pmatrix} k^2 \\ k\sqrt{k^2 + f^2} \cos \varphi + \mathrm{i} f k \sin \varphi \\ k \sqrt{k^2 + f^2} \sin \varphi -\mathrm{i} f k \cos \varphi \end{pmatrix}, \end{equation}

and

(3.8)\begin{align} \boldsymbol{\nabla}_{\boldsymbol{k}} {\mathinner{|{\psi_+}\rangle}} = {\boldsymbol{\hat{k}}} \begin{pmatrix} 2k \\ \dfrac{2k^2 + f^2}{\sqrt{k^2 + f^2}} \cos\varphi + \mathrm{i} f \sin \varphi \\ \dfrac{2k^2 + f^2}{\sqrt{k^2 + f^2}} \sin\varphi - \mathrm{i} f \cos \varphi \end{pmatrix} + \frac{\boldsymbol{\hat{\varphi}}}{k} \begin{pmatrix} 0 \\ -k \sqrt{k^2 + f^2} \sin \varphi + \mathrm{i} f k \cos \varphi \\ k \sqrt{k^2 + f^2} \cos \varphi + \mathrm{i} f k \sin \varphi \end{pmatrix}. \end{align}

The ${\boldsymbol {\hat {k}}}$ component of ${\left\langle{\psi _+ | \boldsymbol {\nabla }_{\boldsymbol {k}} \psi _+}\right\rangle} $ is real and hence does not contribute to $\boldsymbol {A}_+$. We obtain

(3.9)\begin{equation} \boldsymbol{A}_+(\boldsymbol{k}) ={-}\frac{f}{k \sqrt{k^2 + f^2}} \boldsymbol{\hat{\varphi}}. \end{equation}

The Berry curvature is

(3.10)\begin{equation} F_{+}(\boldsymbol{k}) = \frac{f}{(k^2 + f^2)^{3/2}}. \end{equation}

The Chern number of this band is

(3.11)\begin{equation} C_+{=} \frac{1}{2{\rm \pi}} \int \mathrm{d}^2 k\, F_{+} = \operatorname{sign}(\,f). \end{equation}

Topological quantization has emerged: the Chern number can only take on integer values. The breaking of time-reversal symmetry by the rotation of the Earth results in a topologically non-trivial bulk fluid in the $f$-plane model of the shallow-water system. One can similarly show that the Chern numbers for the other bands are $C_0 = 0$ and $C_- = -\operatorname {sign}(\,f)$. The Chern numbers are indicated in figure 3.

This example offers a clear demonstration of the bulk-boundary correspondence principle. The direct implication is that if $f$ in (3.1) is a function of $y$ rather than constant, then there must be a unidirectional wave localized to the spatial region around $f=0$ that spans the frequency gap. Because the change in Chern number across such an interface is $C_+(\,f > 0) - C_+(\,f < 0) = 2$, there are two localized waves.

The more physical case of interest is an actual spherical surface without the Cartesian approximation. The Coriolis parameter $f$ changes sign across the equator. Hence, the equator forms an interface dividing the topologically distinct northern and southern hemispheres. In fact, the two expected waves guaranteed by the bulk-boundary correspondence principle are the well-known equatorially trapped modes, the Kelvin wave and the Yanai wave (Delplace et al. Reference Delplace, Marston and Venaille2017). The dispersion relation for both the Kelvin and Yanai wave is monotonic, indicating group velocities of unidirectional, eastward travelling waves.

Despite the fact that the Cartesian $f$-plane neglects spherical curvature, which is an order-unity effect, analysis of the $f$-plane has yielded the key topological insight that the northern and southern hemisphere are topologically distinct. Kelvin waves have been clearly observed in the spectrum of fluctuations in the Earth's atmosphere (Wheeler & Kiladis Reference Wheeler and Kiladis1999). In simulations, Delplace et al. (Reference Delplace, Marston and Venaille2017) found that equatorially trapped Kelvin waves lying in the frequency gap experienced reduced scattering against static perturbations compared with modes not in the frequency gap, a signature of topological protection.

Although the infinite $\boldsymbol {k}$-plane is not compact, the behaviour at infinite $\boldsymbol {k}$ has not in this case spoiled the result of finding an integer Chern number by integrating the Berry curvature. The compactness issue was handled in an alternate way by Delplace et al. (Reference Delplace, Marston and Venaille2017), who considered a 2-D compact surface, a sphere, within the 3-D parameter space $(k_x, k_y, f)$. The Berry curvature within the 3-D parameter space is that of a monopole at the origin, and hence any closed surface containing the origin will yield the same Chern number. This calculation can be reconciled with the one presented above by considering a cylinder centred at the origin of finite height in $f$ and very large radius in the $(k_x, k_y)$-plane. The Berry flux through the side of the cylinder vanishes, and the flux through one end of the cylinder is equal to the flux through the infinite $\boldsymbol {k}$-plane at constant $f$ above. Yet another way of dealing with compactness is through the addition of odd viscosity (Souslov et al. Reference Souslov, Dasbiswas, Fruchart, Vaikuntanathan and Vitelli2019; Tauber et al. Reference Tauber, Delplace and Venaille2019).

3.2. Magnetized cold plasma and the gaseous plasmon polariton

In this section, we examine a simple magnetized cold plasma and show that it can host topological phases along with related interface modes. The magnetic field breaks time-reversal symmetry.

Consider an infinite, homogeneous, ion-electron plasma. When considering high-frequency electromagnetic waves, it is appropriate to treat the ions as a fixed neutralizing background and only consider electron motion. The mathematical description of a cold plasma consists of the electron equation of motion and Maxwell's equations:

(3.12a)\begin{gather} \frac{\partial \boldsymbol{v}}{\partial t} ={-}\frac{e}{m_e} (\boldsymbol{E} + \boldsymbol{v} \times \boldsymbol{B}_0), \end{gather}

(3.12b)\begin{gather}\frac{\partial \boldsymbol{E}}{\partial t} = c^2 \boldsymbol{\nabla} \times \boldsymbol{B} + \frac{e n_e}{\epsilon_0} \boldsymbol{v}, \end{gather}

(3.12c)\begin{gather}\frac{\partial \boldsymbol{B}}{\partial t} ={-}\boldsymbol{\nabla} \times \boldsymbol{E}, \end{gather}

where ${\boldsymbol{v}}$ is the electron velocity, ${\boldsymbol{E}}$ the electric field, ${\boldsymbol{B}}_0 = B_0 \hat{\boldsymbol{z}}$ the background magnetic field, ${\boldsymbol{B}}$ the perturbation magnetic field, $n_e$ the background electron density, $m_e$ the electron mass, c the speed of light, and $\epsilon_0$ the permittivity of free space.

We consider a fixed $k_z$ parallel to the background magnetic field and choose a two-dimensional parameter space $(k_x, k_y)$. After proper non-dimensionalization and Fourier analysis, one obtains the Hermitian eigenvalue problem $\omega {\mathinner{|{\psi }\rangle}} = H {\mathinner{|{\psi }\rangle}}$, where H is a $9\times 9$ matrix and ${\mathinner{|{\psi }\rangle}} = [\boldsymbol {v}, \boldsymbol {E}, \boldsymbol {B}]$.

Figure 4 shows the four positive-frequency bands. Non-trivial topology is found in multiple bands, as indicated by the non-zero Chern numbers (Parker et al. Reference Parker, Marston, Tobias and Zhu2020b; Fu & Qin Reference Fu and Qin2020). Unlike the shallow-water example, here the straightforward integration of the Berry curvature yields a non-integer result for one of the bands. As discussed in § 2.7, this stems from a lack of compactness. To obtain the integer Chern numbers shown in Figure 4, a large-wavenumber cutoff of the plasma response was introduced to regularize the small-scale behaviour, motivated by the physical fact that the continuum description breaks down at the scales of the interparticle spacing.

Figure 4. Spectrum of a magnetized, homogeneous cold plasma as a function of $k_y$ ($k_x$ set to zero, but the system is isotropic in the $xy$-plane), where only electron motion is retained. The two cases show (a) $k_z < k_z^*$ and (b) $k_z > k_z^*$, where $k_z^*$ is a critical point at which a topological transition occurs. The Chern numbers of the positive-frequency bands are shown (Parker et al. Reference Parker, Marston, Tobias and Zhu2020b).

When the topologically non-trivial plasma is placed next to the trivial vacuum, bulk-boundary correspondence suggests the existence of modes at the interface. One can consider a semi-infinite planar system, where the plasma and vacuum each occupies half of the space (Yang et al. Reference Yang, Lawrence, Gao, Guo and Zhang2016). A more physically realizable system is a confined cylindrical plasma with a radially decreasing density, transitioning to a low-density vacuum-like region. Parker et al. (Reference Parker, Marston, Tobias and Zhu2020b) investigated this system and demonstrated the existence of topological boundary waves. An important component of that study was accounting for a finite width of the density interface. A gaseous plasma cannot sustain a discontinuous density interface with vacuum, and the interface width is typically limited by classical or turbulent diffusion processes. A discontinuous step in density serves as a good first approximation but is quantitatively limited. Because the width of the density interface may be comparable in size to the wavelength of the wave, a quantitative treatment is necessary to accurately determine whether the wave can exist.

Figure 5 shows the surface mode at the plasma–vacuum interface. This mode is the gaseous plasmon polariton (GPP), named for its similarity to surface plasmon polaritons occurring at the surface of metals. The spectrum of the inhomogeneous plasma was computed by solving the differential eigenvalue equation in radius. In the figure, the dispersion relation of the GPP is unidirectional and crosses the bandgap. In Parker et al. (Reference Parker, Marston, Tobias and Zhu2020b), a typo led to the GPP being described as ‘undirectional’ rather than the correct ‘unidirectional’. The GPP can exist in planar as well as cylindrical geometries.

Figure 5. Spectrum of a cylindrical, inhomogeneous, magnetized plasma. The eigenvalue differential equation in radius was solved using the plasma density as a function of radius shown in panel (a). (b) Non-zero components of GPP electric field at azimuthal mode number $m=-8$. (c) Spectrum as a function of $m$. The GPP dispersion relation is indicated and crosses the frequency bandgap (Parker et al. Reference Parker, Marston, Tobias and Zhu2020b).

This study also showed that the GPP can be realized in plasma regimes achievable in laboratory experiments. The parameters used by Parker et al. (Reference Parker, Marston, Tobias and Zhu2020b) were directly motivated by the plasma parameters of the Large Plasma Device (Gekelman et al. Reference Gekelman, Pribyl, Lucky, Drandell, Leneman, Maggs, Vincena, Van Compernolle, Tripathi and Morales2016). In this case, a peak plasma density of $n=4 \times 10^{11}\ \textrm {m}^{-3}$, magnetic field $B=0.1$ T, and density scale length of $L_n \approx 5$ cm were used, and the GPP was calculated to have a frequency of ${\sim }2$ GHz. Hence, the GPP offers a window into the experimental study of topological phenomena in plasma systems.