2.1. Non-Hermitian Hamiltonian matrix

The governing equations for rotating Rayleigh–Bénard convection include the Navier–Stokes, continuity and temperature equations. The dimensionless form can be written as (cf. Favier & Knobloch Reference Favier and Knobloch2020)

(2.1a)$$\begin{gather} \frac{\partial\boldsymbol{u}}{\partial t}+\left(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\boldsymbol{u} ={-}\boldsymbol{\nabla}p-\lambda\boldsymbol{e}_{z}\times\boldsymbol{u} +\alpha\theta\boldsymbol{e}_{z}+E\nabla^{2}\boldsymbol{u}, \end{gather}$$

(2.1b)$$\begin{gather}\boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u}=0, \end{gather}$$

(2.1c)$$\begin{gather}\frac{\partial\theta}{\partial t}+\left(\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla}\right)\theta=w+\frac{E}{Pr}\nabla^{2}\theta, \end{gather}$$

where $\boldsymbol {u}=(u,v,w)$ is the velocity vector, $\lambda =\pm 1$ indicates the direction of rotation, $p$ is the pressure, $\theta$ is the temperature fluctuation relative to the linear conduction profile, $\alpha =RaE^{2}/Pr$ is the square of the convective Rossby number. The Rayleigh number is $Ra=\beta g\Delta Th^{3}/\nu \kappa$, where $\beta$ is the thermal expansion coefficient, $g$ is the acceleration due to gravity, $\Delta T$ is the temperature difference between the top and bottom plates, $h$ is the height of the fluid layer and $\nu$ and $\kappa$ are the viscosity and thermal diffusivity, respectively. It is a measure of the strength of the buoyancy-driven flow relative to the viscous forces. The Ekman number, $E =\nu /(2\varOmega h^2)$, describes the balance of viscous forces to Coriolis forces, where $\varOmega$ is the angular velocity of rotation. The Prandtl number is $Pr=\nu /\kappa$ and is a measure of the relative importance of viscous and thermal diffusion. It is worth noting that $\alpha$ is independent of both $\nu$ and $\kappa$, so that we can take the inviscid limit $E \to 0$ when necessary at a fixed non-zero $\alpha$.

Considering a normal-mode ansatz $(u,v,w,p,\theta ) = {\rm Re}\{(\hat {u},\hat {v},\hat {w},\hat {p},\hat {\theta }) \exp ({{\rm i}(\boldsymbol {k}\boldsymbol {\cdot } \boldsymbol {r}\,{-}\,\omega t)})\}$ with the wavenumber $\boldsymbol {k}=(k_x,k_y,k_z)$, the linearized equations become

(2.2)\begin{gather} -{\rm i}\omega\boldsymbol{\hat{u}}={-}{\rm i}\hat{p}\boldsymbol{k}-\lambda\boldsymbol{e}_{z}\times\boldsymbol{\hat{u}}+\alpha\hat{\theta}\boldsymbol{e}_z-Ek^{2}\boldsymbol{\hat{u}},\end{gather}

(2.3)\begin{gather} \boldsymbol{k}\boldsymbol{\cdot}\boldsymbol{\hat{u}}=0, \end{gather}

(2.4)\begin{gather} -{\rm i}\omega\hat{\theta}=\hat{w}-\frac{E}{Pr}k^{2}\hat{\theta}, \end{gather}

where $k=\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$. Taking the divergence of both sides of (2.1a) and combining with (2.1b), we get

(2.5)\begin{equation} 0=\hat{p} k^{2}+{\rm i}\lambda(k_{x}\hat{v}-k_{y}\hat{u})+{\rm i}\alpha k_{z}\hat{\theta}. \end{equation}

Then

(2.6)\begin{equation} \hat{p}={\rm i}\frac{\lambda({-}k_{x}\hat{v}+k_{y}\hat{u})-\alpha k_{z}\hat{\theta}}{k^{2}}. \end{equation}

With the wave vector $\boldsymbol{\psi} \equiv (\hat {u},\hat {v},\hat {w},\hat {\theta })$, we get the eigenequation from (2.2) and (2.4) as

(2.7)\begin{equation} \boldsymbol{\mathsf{H}}\boldsymbol{\psi}=\omega\boldsymbol{\psi}, \end{equation}

with

(2.8) \begin{equation} \boldsymbol{\mathsf{H}}={\rm i}\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} E_{0}+\lambda\dfrac{k_{x}k_{y}}{k^{2}} & \lambda\left(1-\dfrac{k_{x}^{2}}{k^{2}}\right) & 0 & -\alpha\dfrac{k_{x}k_{z}}{k^{2}}\\ \lambda\left({-}1+\dfrac{k_{y}^{2}}{k^{2}}\right) & E_{0}-\lambda\dfrac{k_{x}k_{y}}{k^{2}} & 0 & -\alpha\dfrac{k_{y}k_{z}}{k^{2}}\\ \lambda\dfrac{k_{y}k_{z}}{k^{2}} & -\lambda\dfrac{k_{x}k_{z}}{k^{2}} & E_{0} & \alpha-\alpha\dfrac{k_{z}^{2}}{k^{2}}\\ 0 & 0 & 1 & E_{1} \end{array}\right], \end{equation}

where $E_{0}=-Ek^{2}$, $E_{1}=E_{0}/Pr$. This is a non-Hermitian Hamiltonian because $H\neq H^{{\dagger} }$, where ${\dagger}$ denotes the conjugate transpose. Because we define the growth rate as ${\rm i} \omega$ for convenience, the eigenvalue $\omega$ can be a complex number, unlike in the Hermitian case where $\omega$ is real. The eigenvalue problem of $H^{{\dagger} }$ is

(2.9)\begin{equation} \boldsymbol{\mathsf{H}}^{{{\dagger}}}\boldsymbol{\psi}^{\prime}=\omega^{{\ast}}\boldsymbol{\psi}^{\prime}, \end{equation}

where $\omega ^{\ast }$ is the complex conjugate of $\omega$, and $\boldsymbol{\psi}^{\prime }$ is generally different from $\boldsymbol{\psi}$.

In (2.1), we do not include the boundary conditions, and the main text of this paper focuses on demonstrating the idea that, to calculate a Chern number for rotating convection, a complex vertical wavenumber is required. A discussion on realistic boundary conditions can be found in Appendix A.

2.2. Chern number in a non-Hermitian system

The Chern number is a topological invariant that was originally defined for systems with a periodic structure, such as crystals, by integrating the Berry curvature over the Brillouin zone (Zak Reference Zak1989; Xiao et al. Reference Xiao, Chang and Niu2010). It is a quantity with integer values and is numerically equal to the Berry phase divided by $2{\rm \pi}$. The Berry curvature characterizes the local geometry of wave polarization and is recognized to manifest in the equations of motion of wave packets with multiple components (Berry Reference Berry1984; Perez, Delplace & Venaille Reference Perez, Delplace and Venaille2021).

As a topological invariant, the Chern number is insensitive to small perturbations of the Hamiltonian that do not change the topology of the system. A non-zero Chern number implies the existence of a non-trivial bulk topology and the presence of robust edge states (Hasan & Kane Reference Hasan and Kane2010). The bulk-boundary correspondence principle links the topological properties of the bulk and edge states (Essin & Gurarie Reference Essin and Gurarie2011; Venaille et al. Reference Venaille, Onuki, Perez and Leclerc2023; Onuki et al. Reference Onuki, Venaille and Delplace2024).

In the wavenumber space, the Berry connection (Berry Reference Berry1984) is defined as

(2.10)\begin{equation} \boldsymbol{A}_{n}(\boldsymbol{k})={\rm i}\langle \boldsymbol{\psi}_{n}(\boldsymbol{k})|\boldsymbol{\nabla}_{\boldsymbol{k}}|\boldsymbol{\psi}_{n}(\boldsymbol{k})\rangle, \end{equation}

where $\boldsymbol{\psi} _{n}$ is the eigenvector, and $\boldsymbol {\nabla }_{\boldsymbol {k}}$ is the gradient with respect to the wave vector $\boldsymbol {k}$. As with the mathematical symbols of quantum mechanics, the right vector $|\boldsymbol{\psi} _{n}\rangle$ represents the general eigenvector, the left vector $\langle \boldsymbol{\psi} _{n}|$ represents the conjugate transpose of $|\boldsymbol{\psi} _{n}\rangle$ and $\langle \cdots |\cdots \rangle$ represents the inner product. The Berry connection is a vector-valued function whose integral along a closed path gives the Berry phase. Then, the Berry curvature is calculated by the curl of the Berry connection

(2.11)\begin{equation} \boldsymbol{\varOmega}_{n}(\boldsymbol{k})=\boldsymbol{\nabla}_{\boldsymbol{k}}\times\boldsymbol{A}_{n}(\boldsymbol{k}). \end{equation}

The Chern number is then defined as the integral of the Berry curvature over the Brillouin zone

(2.12)\begin{equation} C_{n}=\frac{1}{2{\rm \pi}}\int_{BZ}\boldsymbol{\varOmega}_{n}(\boldsymbol{k})\boldsymbol{\cdot} d\boldsymbol{S}, \end{equation}

where $d\boldsymbol {S}$ is the surface element of the Brillouin zone. The Berry curvature $\boldsymbol {\varOmega }_{n}$ is a pseudovector in three dimensions. Due to the direction of rotation in this system, we only consider its component in the $z$ direction, $\varOmega _{n}^{z}$, and $d\boldsymbol {S}$ takes the horizontal plane.

In non-Hermitian systems, the classic Brillouin zone is no longer sufficient to describe the band topology due to the significant difference in frequency spectra between open (non-Bloch) and periodic (Bloch and classic) boundaries. Therefore, a generalized Brillouin zone (GBZ) defined on the complex wavenumber space is needed (Yokomizo & Murakami Reference Yokomizo and Murakami2019; Yang et al. Reference Yang, Zhang, Fang and Hu2020; Wu et al. Reference Wu, Xie, Zhou and An2022). As a result, a non-Bloch Chern number is defined as the integral of the Berry curvature over the GBZ

(2.13)\begin{equation} C_{n}=\frac{1}{2{\rm \pi}}\int_{GBZ}\boldsymbol{\varOmega}_{n}(\tilde{\boldsymbol{k}})\boldsymbol{\cdot} d\tilde{\boldsymbol{S}}, \end{equation}

where $\tilde {\boldsymbol {k}}$ is complex and $d\tilde {\boldsymbol {S}}$ is the surface element of the GBZ. The Berry curvature can take a complex number, as long as the Chern number obtained by integration remains an integer (Yao, Song & Wang Reference Yao, Song and Wang2018).

Due to the non-Hermitian nature of the system (2.7), obtaining the Berry curvature requires the consideration of a biorthogonal basis set, $\{|\boldsymbol{\psi} _{n}\rangle \}$ and $\{|\boldsymbol{\psi} ^{\prime }_{n}\rangle \}$, which satisfies

(2.14a)$$\begin{gather} \boldsymbol{\mathsf{H}}|\boldsymbol{\psi}_{n}\rangle=\omega_{n}|\boldsymbol{\psi}_{n}\rangle, \end{gather}$$

(2.14b)$$\begin{gather}\boldsymbol{\mathsf{H}}^{{{\dagger}}}|\psi^{\prime}_{m}\rangle=\omega_{m}^{{\ast}}|\psi^{\prime}_{m}\rangle, \end{gather}$$

(2.14c)$$\begin{gather}\langle\psi^{\prime}_{m}|\boldsymbol{\psi}_{n}\rangle=\delta_{m,n}, \end{gather}$$

where $\omega _m$ and $\omega _n$ are eigenvalues with band indices $m$ and $n$. Then we can define the non-Hermitian Berry curvature of band $\omega _{n}$ as (Yao et al. Reference Yao, Song and Wang2018)

(2.15)\begin{equation} \varOmega_{n}^{z}={\rm i}\left[\left\langle \frac{\partial\boldsymbol{\psi}_{n}^{\prime}}{\partial k_{x}}\left|\frac{\partial\boldsymbol{\psi}_{n}}{\partial k_{y}}\right.\right\rangle -\left\langle \frac{\partial\boldsymbol{\psi}_{n}^{\prime}}{\partial k_{y}}\left|\frac{\partial\boldsymbol{\psi}_{n}}{\partial k_{x}}\right.\right\rangle \right]. \end{equation}

3.1. Complex wavenumber $k_{z}$

When the eigenvalues of the linearized equations (2.2)–(2.4) are not real, which often happens in non-Hermitian systems, the convection is linearly unstable with oscillatory modes that can grow or decay over time, and exploring the topological properties gets complicated. For example, in the inviscid case where $E=0$, the eigenvalues and eigenvectors are degenerated at $k_{x}^{2}+k_{y}^{2}=k_{z}^{2}/\alpha$ (cf. (3.4)), and a larger $k_{x}$ or $k_{y}$ would make the eigenvalues purely imaginary. Shen, Zhen & Fu (Reference Shen, Zhen and Fu2018) proposed that the non-Hermitian system loses its well-defined Chern numbers unless $k_{z}$ becomes complex: with a non-zero imaginary part in $k_{z}$, the frequency bands will always remain gapped in the complex space, and the integration of the Berry curvature will give an integer Chern number. After such a procedure, we have obtained a GBZ, using which we can treat non-Hermitian problems under open boundary conditions as if they were under periodic boundary conditions (Yokomizo & Murakami Reference Yokomizo and Murakami2019; Yang et al. Reference Yang, Zhang, Fang and Hu2020; Wu et al. Reference Wu, Xie, Zhou and An2022).

For the rotating convection system, figure 1 illustrates the eigenfrequency gap opened by introducing a complex vertical wavenumber $k_z$ with $k_z^2=1+0.1{\rm i}$ and $\alpha =1$. We can see that both the real and imaginary parts of $\omega$ approach $0$, but its absolute value is bounded from below. When $k_z^2=1+\epsilon {\rm i}$ with $\epsilon \ll 1$, one can asymptotically find that

(3.1)\begin{equation} |\omega|_{gap} = \sqrt{\frac{\epsilon\alpha}{1+\alpha}} + {\rm h.o.t.}, \end{equation}

where ${\rm h.o.t.}$ denotes high-order terms.

Figure 1.

Illustration of the eigenvalue with complex $k_z$. Here, $k_z^2=1+0.1{\rm i}$, $\alpha =1$ and $E=0$.

Complex $k_z$ is also physically reasonable. The top and bottom no-slip boundaries introduce an exponential decay form to the asymptotic solutions along the $z$-direction (Zhang & Liao Reference Zhang and Liao2009), which agrees with our numerical results in Appendix A. The choice of a complex $k_z$ is due to the boundary conditions we have adopted: the top and bottom surfaces are no slip and the sidewalls are smooth (where both $k_x$ and $k_y$ are real). If we consider another scenario where the sidewalls are no slip but the upper and lower surfaces are smooth, $k_x$ and $k_y$ need a specific imaginary part. Mathematically speaking, $k_z$ can take real values in this case, and we can still obtain a well-defined Chern number. On the other hand, under open boundary conditions, i.e. with external energy injection, the bulk eigenstates of non-Hermitian systems exhibit a localized behaviour towards the boundary, the so-called non-Hermitian skin effect (Okuma et al. Reference Okuma, Kawabata, Shiozaki and Sato2020; Zhang, Yang & Fang Reference Zhang, Yang and Fang2022; Zhang et al. Reference Zhang, Zhang, Lu and Chen2022), which differs from the extended Bloch waves in Hermitian systems. In our system, energy exchanges with the outsiders in the $z$-direction due to the temperature difference, so we take a complex $k_{z}$.

3.2. Inviscid topological invariant ($E=0$)

With the GBZ where $k_{z}$ is complex, we can calculate the Chern number for open boundaries through a normal process, just as we do under periodic boundary conditions. The solutions satisfying realistic boundary conditions can be found in Appendix A. To obtain a single-valued function when dealing with the square root, we redefine the square root operation on the complex domain, making its value domain lie in the right half-plane that does not include the negative imaginary axis, i.e. $\sqrt {-1}=i$.

We start to discuss the topological properties of the system with the simplest case $E=0$. Since $k_{z}$ is complex, the singularity in the eigenvalues and eigenvectors will not exist. The Hamiltonian matrix becomes

(3.2) \begin{equation} \boldsymbol{\mathsf{H}}_{0}={\rm i}\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{}} \lambda\dfrac{k_{x}k_{y}}{k^{2}} & \lambda\left(1-\dfrac{k_{x}^{2}}{k^{2}}\right) & 0 & -\alpha\dfrac{k_{x}k_{z}}{k^{2}}\\ \lambda\left({-}1+\dfrac{k_{y}^{2}}{k^{2}}\right) & -\lambda\dfrac{k_{x}k_{y}}{k^{2}} & 0 & -\alpha\dfrac{k_{y}k_{z}}{k^{2}}\\ \lambda\dfrac{k_{y}k_{z}}{k^{2}} & -\lambda\dfrac{k_{x}k_{z}}{k^{2}} & 0 & \alpha-\alpha\dfrac{k_{z}^{2}}{k^{2}}\\ 0 & 0 & 1 & 0 \end{array}\right]. \end{equation}

For the eigenequation

(3.3)\begin{equation} \boldsymbol{\mathsf{H}}_{0}\boldsymbol{\psi}=\omega\boldsymbol{\psi}, \end{equation}

the non-zero eigenvalues and corresponding right eigenvectors are

(3.4)\begin{gather} \omega_{{\pm}}={\pm}\omega_{0},\quad \omega_{0}=\frac{\sqrt{k_{z}^{2}-\alpha(k_{x}^{2}+k_{y}^{2})}}{k}, \end{gather}

(3.5) \begin{gather} \left.\left|\psi_{{\pm}}\right.\right\rangle =\frac{1}{\sqrt{2\left(\dfrac{k_{z}^{2}}{k_{x}^{2}+k_{y}^{2}}-\alpha\right)}}\left[\begin{array}{@{}c@{}} \dfrac{k_{z}\left(-{\rm i}\lambda k_{y}\mp k_{x}\omega_{0}\right)}{k_{x}^{2}+k_{y}^{2}}\\ \dfrac{k_{z}\left({\rm i}\lambda k_{x}\mp k_{y}\omega_{0}\right)}{k_{x}^{2}+k_{y}^{2}}\\ \pm\omega_{0}\\ {\rm i} \end{array}\right], \end{gather}

where $k=\sqrt {k_x^2+k_y^2+k_z^2}$ and $k_x^2+k_y^2\neq 0$. When $\rho =\sqrt {k_{x}^{2}+k_{y}^{2}}\xrightarrow {}0$, the eigenvectors become

(3.6)\begin{equation} \left.\left|\psi_{{\pm}}\right.\right\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{@{}c@{}} \pm \lambda {\rm i}\\ 1\\ 0\\ 0 \end{array}\right]. \end{equation}

When $\rho \xrightarrow {}+\infty$, the eigenvectors are still single valued so that

(3.7)\begin{equation} \left.\left|\psi_{{\pm}}\right.\right\rangle =\frac{1}{\sqrt{2}}\left[\begin{array}{@{}c@{}} 0\\ 0\\ \pm 1\\ \dfrac{{\rm i}}{\sqrt{-\alpha}} \end{array}\right]. \end{equation}

Thus, different from the shallow water model, the eigenvectors here are regular on a compact manifold, which guarantees a well-defined Chern number without introducing unphysical items (Tauber, Delplace & Venaille Reference Tauber, Delplace and Venaille2019).

For the eigenequation

(3.8)\begin{equation} \boldsymbol{\mathsf{H}}_{0}^{{{\dagger}}}\boldsymbol{\psi}^{\prime}=\omega^{{\ast}}\boldsymbol{\psi}^{\prime}, \end{equation}

the non-zero eigenvalues and corresponding left eigenvectors are

(3.9)\begin{gather} \omega_{{\pm}}^{{\ast}}={\pm}\omega_{0}^{{\ast}}, \end{gather}

(3.10) \begin{gather} \left\langle \left.\psi_{{\pm}}^{\prime}\right|\right.=\frac{1}{\sqrt{2\left(\dfrac{k_{z}^{2}}{k_{x}^{2}+k_{y}^{2}}-\alpha\right)}}\left[\begin{array}{@{}c@{}} \dfrac{k_{z}\left({\rm i}\lambda k_{y}\mp\dfrac{k_{x}}{\omega_{0}}\right)}{k_{x}^{2}+k_{y}^{2}}\\ \dfrac{k_{z}\left(-{\rm i}\lambda k_{x}\mp\dfrac{k_{y}}{\omega_{0}}\right)}{k_{x}^{2}+k_{y}^{2}}\\ \mp\dfrac{\alpha}{\omega_{0}}\\ {\rm i}\alpha \end{array}\right]^{\mathrm{T}}, \end{gather}

when $k_x^2+k_y^2\neq 0$. When $\rho \xrightarrow {}0$, the eigenvectors become

(3.11)\begin{equation} \left\langle \left.\psi_{{\pm}}^{\prime}\right|\right. =\frac{1}{\sqrt{2}}\left[\begin{array}{@{}c@{}} \mp \lambda {\rm i}\\ 1\\ 0\\ 0 \end{array}\right]^{\mathrm{T}}. \end{equation}

When $\rho \xrightarrow {}+\infty$, the eigenvectors are single valued so that

(3.12)\begin{equation} \left\langle \left.\psi_{{\pm}}^{\prime}\right|\right. =\frac{1}{\sqrt{2}}\left[\begin{array}{@{}c@{}} 0\\ 0\\ \pm 1\\ -{\rm i}\sqrt{-\alpha} \end{array}\right]^{\mathrm{T}}. \end{equation}

Substituting the expressions of eigenvectors in (2.15), the Berry curvature of the positive band (associated with $\omega _+$ and $\omega _+^{\ast }$) becomes

(3.13)\begin{equation} \varOmega^{z}={-}\lambda\frac{k_{z}^{2}\left[\rho^{2}k_{z}^{2}+2\alpha\left(\rho^{4}+4\rho^{2}k_{z}^{2}+2k_{z}^{4}\right)-\alpha^{2}\rho^{2}\left(2\rho^{2}+k_{z}^{2}\right)\right]}{2k^{3}\left(k_{z}^{2}-\alpha\rho^{2}\right)^{5/2}}, \end{equation}

where $\rho =\sqrt {k_{x}^{2}+k_{y}^{2}}$. Then the Chern number is

(3.14a)\begin{align} C&=\frac{1}{2{\rm \pi}}\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\varOmega^{z}\,{\rm d}k_{x}\,{\rm d}k_{y} \end{align}

(3.14b)\begin{align} &=\int_{0}^{+\infty}\rho\varOmega^{z}\,{\rm d}\rho \end{align}

(3.14c)\begin{align} &=\lambda\left.\frac{-2k_{z}^{4}+k_{z}^{2}(\alpha-1)\rho^{2}}{2\sqrt{\rho^{2}+k_{z}^{2}}\left(k_{z}^{2}-\alpha\rho^{2}\right)^{3/2}}\right|_0^{+\infty} \end{align}

(3.14d)\begin{align} &=\lambda, \end{align}

which implies the existence of topologically protected edge states only related to the direction of fluid rotation. This is a three-dimensional version of the topologically protected equatorial waves (Delplace et al. Reference Delplace, Marston and Venaille2017). Under the hydrostatic approximation, we obtain an exact two-dimensional counterpart, shown in § 5.

3.3. Topological invariant when $E\neq 0$

This section considers $E\neq 0$, which does not change the Chern number we obtained in the previous subsection. Firstly, When $Pr=1$, the Hamiltonian matrix $H$ becomes

(3.15)\begin{equation} \boldsymbol{\mathsf{H}}=H_{0}+{\rm i}E_{0}\boldsymbol{\hat{1}}, \end{equation}

where $H_{0}$ is the simple Hamiltonian where $E=0$ and $\boldsymbol {\hat {1}}$ is the unit matrix of the same order as $H_{0}$. The eigenvalues are

(3.16)\begin{equation} \omega_{{\pm}}={\pm}\omega_{0}+{\rm i}E_{0}. \end{equation}

It is just a shift of $\pm \omega _{0}$, the eigenvalues of $H_{0}$ and then the eigenvectors are independent of $E$. Therefore, both the Berry curvature and Chern number are unchanged compared with the case where $E=0$.

When $Pr\neq 1$, there are no simple expressions for the eigenvalues and eigenvectors, and we need to calculate the Chern number with the help of numerical calculations, seen in figure 2. To avoid the derivation of the eigenvectors, we rewrite (2.15) as

(3.17) \begin{equation} \varOmega_{n}^{z}={\rm i}\underset{m\neq n}{\sum}\frac{\left\langle\boldsymbol{\psi}_{n}^{\prime}\left|\dfrac{\partial H}{\partial k_{x}}\right|\psi_{m}\right\rangle\left\langle\psi_{m}^{\prime}\left|\dfrac{\partial H}{\partial k_{y}}\right|\boldsymbol{\psi}_{n}\right\rangle-\{x\leftrightarrow y\}}{(\omega_{n}-\omega_{m})^{2}}. \end{equation}

It is similar to the Hermitian case (Xiao et al. Reference Xiao, Chang and Niu2010) except that the left vector takes the biorthogonal partner. This form is very useful for numerical calculations because it can be evaluated under an unsmooth phase choice of the eigenstates, which often occurs in the standard diagonalization algorithms.

Figure 2.

Numerical results of the non-Hermitian Chern number. (a) Chern number with different Prandtl numbers $Pr$ and rotation directions $\lambda =\pm 1$. The upper limit of integral for $\rho$ is $k_{cut}=6$, the $k$-point accuracy per unit length is 200 (radial) * 200 (axial) using the Gauss–Legendre integration scheme. (b) Demonstrating the numerical convergence degree with different $k$-point accuracies as $k_{cut}$ increases. Other parameters are $\alpha =1$, $E=0.01$, $k_{z}=1+0.1 {\rm i}$.