Complex Berry phase and imperfect non-Hermitian phase transitions

We consider a classical or quantum two-level system described by an effective $2\times 2$ non-Hermitian matrix Hamiltonian $\mathcal{H}=\mathcal{H}(k)$, which depends on a real parameter $k$ and is periodic in $k$ with a period of $2\pi$, i.e. $\mathcal{H}(k+2\pi)=\mathcal{H}(k)$. As we will discuss in the next section, in the Wannier-Stark ladder problem of a non-Hermitian lattice driven by a dc field the matrix Hamiltonian $\mathcal{H}(k)$ corresponds to the Bloch Hamiltonian of a two-band lattice and $k$ is the quasi momentum, that drifts to span the entire Brillouin zone in the presence of the dc field.
The temporal dynamics of the system is described by the Schrödinger equation

$$i\frac{d}{dt}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)=\left(\begin{array}[]{cc}\mathcal{H}_{11}&\mathcal{H}_{12}\\ \mathcal{H}_{21}&\mathcal{H}_{22}\end{array}\right)\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)=\mathcal{H}(k)\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right).$$

(1)

We assume that the Hamiltonian is $\mathcal{PT}$ symmetric with parity $\mathcal{P}$ and time reversal $\mathcal{T}$ operators defined by

$$\mathcal{P}=\sigma_{x}=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)\;,\;\;\mathcal{T}=\mathcal{K}$$

(2)

where $\sigma_{x}$ is the Pauli matrix and $\mathcal{K}$ the element-wise complex conjugation operator. $\mathcal{PT}$ symmetry, i.e. the condition $\mathcal{PTH}=\mathcal{HPT}$, is satisfied provided that

$$\mathcal{H}_{22}=\mathcal{H}^{*}_{11}\;,\;\;\mathcal{H}_{21}=\mathcal{H}^{*}_{12}$$

(3)

so that the non-Hermiticity in the system is embedded in a non-vanishing imaginary part of $\mathcal{H}_{11}$. The most general form of matrix elements that respect the $\mathcal{PT}$ symmetry is thus

	$\displaystyle\mathcal{H}_{11}=\mathcal{H}_{22}^{*}=G(k)+i\lambda W(k)$		(4)
	$\displaystyle\mathcal{H}_{12}=\mathcal{H}_{21}^{*}=R(k)\exp[i\varphi(k)]$		(5)

where $G(k),W(k)$, $R(k)$ are real and periodic functions of $k$ with period $2\pi$, $\varphi(k)$ is a real function with $\varphi(k+2\pi)=\varphi(k)$ mod $2\pi$, and $\lambda\geq 0$ is a real parameter that measures the strength of non-Hermiticity in the system, the case $\lambda=0$ corresponding to $\mathcal{H}(k)$ Hermitian. Further, we assume that $R(k)$ is nonvanishing over the entire range $0\leq k\leq 2\pi$.
When the parameter $k$ is kept constant, the eigenenergies of $\mathcal{H}(k)$ are given by

$$E_{\pm}(k)=G(k)\pm\sqrt{R^{2}(k)-\lambda^{2}W^{2}(k)}$$

(6)

with corresponding (right) eigenvectors

	$\displaystyle\mathbf{u}_{+}(k)$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c}\cos\left(\frac{\theta}{2}\right)\\ \sin\left(\frac{\theta}{2}\right)\exp(-i\varphi)\end{array}\right)$		(9)
	$\displaystyle\mathbf{u}_{-}(k)$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c}\sin\left(\frac{\theta}{2}\right)\\ -\cos\left(\frac{\theta}{2}\right)\exp(-i\varphi)\end{array}\right).$		(12)

In the previous equations, the complex angle $\theta=\theta(k)$ is defined by the relation

$$\tan\theta(k)=\frac{R(k)}{i\lambda W(k)}.$$

(13)

Note that the imaginary part of the angle $\theta(k)$ diverges when $R(k)=\pm\lambda W(k)$, corresponding to the simultaneous coalescence of the two energies and eigenstates, i.e. to the appearance of an exceptional point. The left eigenvectors of $\mathcal{H}(k)$, i.e. the (right) eigenvectors of the adjoint $\mathcal{H}^{{\dagger}}(k)$ with eigenvalues $E_{\pm}^{*}(k)$, read

	$\displaystyle\mathbf{v}_{+}(k)$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c}\cos^{*}\left(\frac{\theta}{2}\right)\\ \sin^{*}\left(\frac{\theta}{2}\right)\exp(-i\varphi)\end{array}\right)$		(16)
	$\displaystyle\mathbf{v}_{-}(k)$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{c}\sin^{*}\left(\frac{\theta}{2}\right)\\ -\cos^{*}\left(\frac{\theta}{2}\right)\exp(-i\varphi)\end{array}\right)$		(19)

and the biorthogonal conditions

$$\langle\mathbf{v}_{n}(k)|\mathbf{u}_{m}(k)\rangle=\delta_{n,m}$$

(20)

are satisfied for any $k$, with $n,m=+,-$. After letting

$$\lambda_{c}(k)\equiv|R(k)/W(k)|,$$

(21)

from Eq.(6) it readily follows that the energy spectrum is real for $\lambda<\lambda_{c}(k)$ (unbroken $\mathcal{PT}$ phase), and complex for $\lambda>\lambda_{c}(k)$ (broken $\mathcal{PT}$ phase), with the appearance of an exceptional point at the critical point $\lambda=\lambda_{c}(k)$.

Let us now consider the two-level system when the Hamiltonian $\mathcal{H}(k)$ is periodically and adiabatically cycled in time. We assume that the parameter $k$ in the Hamiltonian varies in time according to

$$k=\omega t$$

(22)

where $\omega$ is the cycling frequency. The temporal dynamics of the two-level system is thus described by the equation

$$i\frac{d}{dt}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)=\mathcal{H}(\omega t)\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right).$$

(23)

In most of our analysis, we will limit our attention considering the system dynamics in the slow-cycling regime $\omega\rightarrow 0$. As we will comment below with reference to some specific examples, the main motivation thereof is that to observe smooth spectral phase transitions the system evolution must be slow enough. Without loss of generality we can assume $G(k)=0$, i.e. $\mathcal{H}_{11}(k)=\mathcal{H}_{22}^{*}(k)=i\lambda W(k)$, so that the instantaneous eigenenergies of $\mathcal{H}(k)$ read

$$E_{\pm}(k)=\pm\sqrt{R^{2}(k)-\lambda^{2}W^{2}(k)}$$

(24)

with $k=\omega t$. In fact, a non-vanishing value of $G(k)$ can be eliminated from the dynamics after the gauge transformation

$$\left(\begin{array}[]{c}\psi_{1}(t)\\ \psi_{2}(t)\end{array}\right)\rightarrow\left(\begin{array}[]{c}\psi_{1}(t)\\ \psi_{2}(t)\end{array}\right)\exp\left\{-\frac{i}{\omega}\int_{0}^{\omega t}G(k)dk\right\}.$$

According to Floquet theory, the most general solution to the Schrödinger equation (15) is given by

$$\left(\begin{array}[]{c}\psi_{1}(t)\\ \psi_{2}(t)\end{array}\right)=\mathcal{U}(t)\exp(-i\mathcal{R}t)\left(\begin{array}[]{c}\psi_{1}(0)\\ \psi_{2}(0)\end{array}\right)$$

(25)

where $\mathcal{R}$ is a time-independent $2\times 2$ matrix while $\mathcal{U}(t)$ is a time-dependent and periodic $2\times 2$ matrix, $\mathcal{U}(t+2\pi/\omega)=\mathcal{U}(t)$, with $\mathcal{U}(0)=\mathcal{I}$ (the identity matrix). The exponential of the matrix $\mathcal{R}$ can be expressed in terms of the path-ordered integral

$$\exp(-i\mathcal{R})=\bar{\mathcal{T}}\exp\left[-i\frac{\omega}{2\pi}\int_{0}^{2\pi/\omega}dt\;\mathcal{H}(\omega t)\right]$$

where $\bar{\mathcal{T}}$ indicates the time ordering. The two eigenvalues $\mu_{\pm}=\mu_{\pm}(\lambda)$ of $\mathcal{R}$ are the quasi energies of the time-periodic cycled system. The real parts of the quasi energies are defined apart from integer multiples than $\omega$. Note that for $G(k)=0$ the trace of $\mathcal{H}(k)$ vanishes, so that $\mu_{-}=-\mu_{+}$, i.e. the two quasi energies can be assumed to be opposite one another. A non-vanishing value of $G(k)$ would just lead to a shift of the quasi energies by the amount $(1/2\pi)\int_{0}^{2\pi}dkG(k)$.
A natural question arises: akin to the non-cycled $\mathcal{PT}$ symmetric system, is there a spectral phase transition, from real to complex quasi energies, as the non-Hermitian parameter $\lambda$ in the system is increased above a critical value? To answer this question, let us indicate by $\bar{\lambda}_{c}$ the minimum value of $\lambda_{c}(k)$ as $k$ spans the range $0\leq k\leq 2\pi$, i.e.

$$\bar{\lambda}_{c}=\min_{0\leq k\leq 2\pi}\lambda_{c}(k)=\min_{0\leq k\leq 2\pi}\left|\frac{R(k)}{W(k)}\right|.$$

(26)

Intuitively, for a slowly-cycled system one would expect the following scenario: for $\lambda<\bar{\lambda}_{c}$, the instantaneous eigenenergies $E_{\pm}(k=\omega t)$ of $\mathcal{H}(k=\omega t)$ are real, and thus we expect the quasi energies $\mu_{\pm}$ to remain real as well. On the other hand, for $\lambda>\bar{\lambda}_{c}$ within the modulation cycle there are time intervals where the instantaneous eigenenergies $E_{\pm}(k=\omega t)$ become complex: in this case we expect the quasi energies to become complex too. Hence, according to such an intuitive picture, we expect a spectral phase transition of the cycled two-level system, from real to complex quasi energies, when the non-Hermitian parameter $\lambda$ is increased above the critical value $\bar{\lambda}_{c}$. This result is indeed what one observes from a numerical computation of the quasi energies in several examples of cycled two-level $\mathcal{PT}$ symmetric models, as shown in the next subsection. However, in some other models it turns out that, for a small but non-vanishing oscillation frequency $\omega$, the phase transition is smooth, i.e. imperfect: below the critical value $\bar{\lambda}_{c}$ the imaginary part of the quasi energy takes a small but non-vanishing value, which scales as $\sim\omega$, i.e. it exactly vanishes only in the limit $\omega\rightarrow 0$. What is the physical origin of such an imperfect phase transition, which is observed in some models but not in others?
The answer to this question is rooted in the appearance of a complex Berry phase in certain models (but not in others), and can be gained from an adiabatic analysis of the time evolution of the system in the $\omega\rightarrow 0$ limit, which is detailed in the Appendices A and B. In the adiabatic analysis, the slow evolution of the amplitudes of the instantaneous eigenstates $\mathbf{u}_{\pm}(k=\omega t)$ of the Hamiltonian $\mathcal{H}(k=\omega t)$ is governed by the non-Hermitian Berry connection

$$\mathcal{A}_{n,l}(k)=-i\langle\mathbf{v}_{n}|\partial_{k}\mathbf{u}_{l}\rangle$$

(27)

($n,l=+,-$), which is defined in the context of the biorthonormal inner product. The integrals of the diagonal terms of Berry connection, $\mathcal{A}_{+,+}(k)$ and $\mathcal{A}_{-,-}(k)$, over the interval $0\leq k\leq 2\pi$, i.e.

$$\gamma_{B_{+}}=\int_{0}^{2\pi}dk\mathcal{A}_{+,+}(k)\;,\;\;\gamma_{B_{-}}=\int_{0}^{2\pi}dk\mathcal{A}_{-,-}(k)$$

(28)

are the non-Hermitian Berry phases associated to the two instantaneous eigenstates $\mathbf{u}_{\pm}(k)$. The explicit form of the Berry connection and Berry phases are derived in Appendix A. In particular, one has

$$\gamma_{B_{\pm}}=\mp\frac{1}{2}\int_{0}^{2\pi}dk\frac{d\varphi}{dk}\pm\frac{i}{2}\int_{0}^{2\pi}dk\frac{d\varphi}{dk}\sinh\psi(k)$$

(29)

where the function $\psi(k)$ is defined by the relation

$$\tanh\psi(k)=\frac{\lambda W(k)}{R(k)}.$$

(30)

Note that the Berry phase vanishes in any $\mathcal{PT}$-symmetric two-level system with $(d\varphi/dk)\equiv 0$.
The adiabatic analysis shows some subtleties and limitations when applied to our model, owing to the appearance of instantaneous EP on the cycle when $\lambda>\bar{\lambda_{c}}$. Technical details are given in Appendix B. The main result of the adiabatic analysis is that, in the limit $\omega\rightarrow 0$ and for $\lambda\neq\bar{\lambda_{c}}$, the two quasi energies are given by

$$\mu_{\pm}=\frac{1}{2\pi}\int_{0}^{2\pi}dkE_{\pm}(k)+\frac{\omega}{2\pi}\gamma_{B_{\pm}}.$$

(31)

Note that, since $E_{-}(k)=-E_{+}(k)$ and $\gamma_{B_{-}}=-\gamma_{B_{+}}$, one has $\mu_{-}=-\mu_{+}$, as it should. The above result provides an approximate form of the quasi energies in the adiabatic limit $\omega\rightarrow 0$ for any strength $\lambda$ of the non-Hermitian parameter far from the critical value $\lambda=\bar{\lambda_{c}}$, at which the Berry phase term becomes singular and the adiabatic analysis fails; a discussion on this point is given in the Appendix B.
According to Eq.(23), each quasi energy is given by the sum of two terms. The first one is related to the dynamical phase accumulated by the adiabatic eigenstates in one cycle and equals the average of the instantaneous energies $E_{\pm}(k)$ over one cycle. The dynamical phase term is clearly independent of the modulation frequency $\omega$ and is real for $\lambda<\bar{\lambda}_{c}$, while its imaginary part is non-vanishing for $\lambda>\bar{\lambda}_{c}$. The second term on the right hand side in Eq.(23) is the non-Hermitian Berry phase contribution. This is a small term which vanishes like $\sim\omega$ as $\omega\rightarrow 0$. Interestingly, for $\lambda<\bar{\lambda}_{c}$ the imaginary part of the quasi energies is provided solely by the Berry phase term, and vanishes as $\omega\rightarrow 0$. This explains why in the cycled two-level $\mathcal{PT}$-symmetric system with a vanishing Berry phase the spectral phase transition of the quasi energies, at $\lambda=\bar{\lambda}_{c}$, is sharp (exact), while it becomes smooth (imperfect) when the non-Hermitian Berry phase along the cycle is non-vanishing.

The main result of the adiabatic analysis is that the spectral phase transition of the quasi energies in the slow-cycled $\mathcal{PT}$-symmetric two-level system turns out be be imperfect (smooth) whenever the Berry phase in the cycle is complex, which requires the derivative $(d\varphi/dk)$ not to identically vanish. On the other hand, the phase transition is sharp (exact) whenever the Berry phase is real. Here we illustrate and confirm the predictions of the adiabatic analysis by considering three examples of cycled two-level $\mathcal{PT}$-symmetric systems.

1. First example. The first example is a simple and exactly-solvable model, corresponding to $G(k)=0$, $W(k)=1$, $R(k)=R_{0}$, $\varphi(k)=k$, i.e. to the $\mathcal{PT}$-symmetric Hamiltonian

$$\mathcal{H}(k)=\left(\begin{array}[]{cc}i\lambda&R_{0}\exp(ik)\\ R_{0}\exp(-ik)&-i\lambda\end{array}\right)$$

(32)

where $R_{0}>0$ is a real parameter. Physically, this model describes a two-level system, in which the two states are Hermitian-coupled by an amplitude $R_{0}$ and a gauge (Peierls) phase $k$ and with gain ($\lambda$) and loss ($-\lambda$) rates in the two levels. The system displays a $\mathcal{PT}$ symmetry breaking at a critical value $\lambda_{c}(k)=\bar{\lambda}_{c}$, independent of $k$, given by

$$\bar{\lambda}_{c}=R_{0}.$$

(33)

In the cycled system with $k=\omega t$, we expect an imperfect phase transition of quasi energies because $(d\varphi/dk)=1\neq 0$ and the imaginary part of the Berry phase does not vanish. The quasi energies $\mu_{\pm}$ can be calculated in an exact form, given that the time dependence of the Hamiltonian $\mathcal{H}(k=\omega t)$ can be removed from the dynamics after the gauge transformation

$$\psi_{1}(t)=\bar{\psi}_{1}(t)\exp\left(i\frac{\omega t}{2}\right)\;,\;\psi_{2}(t)=\bar{\psi}_{2}(t)\exp\left(-i\frac{\omega t}{2}\right).$$

(34)

The exact expression of the quasi energies can be readily computed, yielding

$$\mu_{\pm}=\pm\sqrt{R_{0}^{2}+\left(\frac{\omega}{2}+i\lambda\right)^{2}}\mp\frac{\omega}{2}$$

(35)

A typical behavior of the imaginary parts of the quasi energies versus $\lambda$, in the adiabatic limit $\omega\ll R_{0}$, is depicted in Fig.1, clearly showing the appearance of an imperfect spectral phase transition near $\lambda=\bar{\lambda}_{c}$. Note that, when $\lambda$ is not too close to $\bar{\lambda}_{c}=R_{0}$, in the adiabatic limit $\omega\rightarrow 0$ we can expand the right hand side of Eq.(27) in power series of $\omega$ and, up to first order in $\omega$, the following approximate expression of quasi energies is obtained

$$\mu_{\pm}\simeq\pm\sqrt{R_{0}^{2}-\lambda^{2}}\mp\frac{\omega}{2}\pm i\frac{\lambda\omega}{2\sqrt{R_{0}^{2}-\lambda^{2}}}.$$

(36)

It can be readily shown that Eq.(28) precisely reproduces the result predicted by the adiabatic analysis [Eq.(23)]. In fact, the dynamical phase contribution to the quasi energy is given by

$$\frac{1}{2\pi}\int_{0}^{2\pi}dkE_{\pm}(k)=\pm\sqrt{R_{0}^{2}-\lambda^{2}}$$

while the Berry phase contribution reads

	$\displaystyle\frac{\omega}{2\pi}\gamma_{B_{\pm}}$	$\displaystyle=$	$\displaystyle\mp\frac{\omega}{4\pi}\int_{0}^{2\pi}dk\frac{d\varphi}{dk}\pm\frac{i\omega}{4\pi}\int_{0}^{2\pi}dk\frac{d\varphi}{dk}\sinh\psi(k)$		(37)
		$\displaystyle=$	$\displaystyle\mp\frac{\omega}{2}\pm i\frac{\omega}{2}\sinh\psi=\mp\frac{\omega}{2}\pm i\frac{\lambda\omega}{2\sqrt{R_{0}^{2}-\lambda^{2}}}.$

In deriving Eq.(29), we used the property that $\psi(k)$, defined by the relation ${\rm tanh}\psi(k)=\lambda/R_{0}$, is independent of $k$ and $(d\varphi/dk)=1$. Note that the behavior of the imaginary part of the quasi energies, predicted by the adiabatic analysis, well reproduces the exact curves, except near the phase transition point $\lambda=\bar{\lambda}_{c}$ where the Berry phase contribution displays a singularity.

2. Second example. As a second example, let us consider the $\mathcal{PT}$-symmetric two-level Hamiltonian

$$\mathcal{H}(k)=\left(\begin{array}[]{cc}i\lambda&t_{1}+t_{2}\cos k\\ t_{1}+t_{2}\cos k&-i\lambda\end{array}\right)$$

(38)

corresponding to $G(k)=0$, $W(k)=1$, $R(k)=t_{1}+t_{2}\cos k$, and $\varphi(k)=0$, where $t_{1}$ and $t_{2}$ are real and positive parameters with $t_{1}>t_{2}$. Since $(d\varphi/dk)=0$, the Berry phase vanishes and, according to the adiabatic analysis, when the system is slowly cycled with $k=\omega t$ the spectral phase transition of the quasi energies is sharp (exact) and occurs at the critical value $\bar{\lambda}_{c}=t_{1}-t_{2}$ of the non-Hermitian parameter $\lambda$. The numerical computation of the quasi energies $\mu_{\pm}$ versus $\lambda$, as obtained by a direct numerical integration of the Schrödinger equation (15) using an accurate variable-step fourth-order Runge-Kutta method, confirms that the phase transition is sharp and the curves ${\rm Im}(\mu_{\pm}(\lambda))$ are well approximated by the behavior predicted by the adiabatic analysis, as shown in Fig.2.

3. Third example. As a third example, let us consider the $\mathcal{PT}$-symmetric two-level Hamiltonian

$$\mathcal{H}(k)=\left(\begin{array}[]{cc}i\lambda+t_{0}\cos k&t_{1}+t_{2}\exp(ik)\\ t_{1}+t_{2}\exp(-ik)&-i\lambda+t_{0}\cos k\end{array}\right)$$

(39)

corresponding to $G(k)=t_{0}\cos k$, $W(k)=1$, $R(k)=\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos k}$, and $\varphi(k)={\rm atan}[t_{2}\sin k/(t_{1}+t_{2}\cos k]$, where $t_{0}$, $t_{1}$ and $t_{2}$ are real and positive parameters with $t_{1}\neq t_{2}$. Since $(d\varphi/dk)\neq 0$, the imaginary part of the Berry phase does not vanish and, according to the adiabatic analysis, when the system is slowly cycled with $k=\omega t$ the spectral phase transition of the quasi energies is imperfect (smooth). The phase transition occurs at the critical value $\bar{\lambda}_{c}=|t_{2}-t_{1}|$ of the non-Hermitian parameter $\lambda$. The numerical computation of the quasi energies $\mu_{\pm}$ versus $\lambda$, as obtained by a direct numerical integration of the Schrödinger equation (15), confirms that the phase transition is imperfect and the curves ${\rm Im}(\mu_{\pm}(\lambda))$ are well approximated by the behavior predicted by the adiabatic analysis for $\lambda\neq\bar{\lambda}_{c}$, as shown in Fig.3. Note that in the slow-cycling regime [Fig.3(a)] the curves ${\rm Im}(\mu_{\pm})$ versus $\lambda$ display a characteristic knee shape, indicating a smooth spectral phase transition. We remark that the terminology ”smooth” phase transition is meaningful in the adiabatic limit $\omega\rightarrow 0$ solely, while when we cycle the system faster, so as $\omega$ becomes comparable to the other characteristic frequencies of the Hamiltonian (such as the separation of adiabatic energies), the knee shape of the curves is continuously spoiled out and there is not any evident sharp transition of the imaginary part of the quasi energies as $\lambda$ is increased; see Fig.3(b).

Let us consider a two-band tight-binding lattice model driven by a dc force $F$. In physical space, the temporal evolution of the single-particle state of the system is described by the Schrödinger equation

	$\displaystyle i\frac{da_{n}}{dt}$	$\displaystyle=$	$\displaystyle\sum_{l}\rho_{n-l}a_{l}+\sum_{l}\sigma_{n-l}b_{l}-Fna_{n}$		(40)
	$\displaystyle i\frac{db_{n}}{dt}$	$\displaystyle=$	$\displaystyle\sum_{l}\theta_{n-l}a_{l}+\sum_{l}\eta_{n-l}b_{l}-Fnb_{n}$		(41)

for the amplitudes $a_{n}$ and $b_{n}$ in the two sublattices A and B of the $n$-th unit cell of the crystal. In the above equations, the coefficients $\rho_{0}$ and $\eta_{0}$ are the on-site energy potentials in the two sublattices A and B, respectively; $\rho_{l}$ and $\eta_{l}$ ($l\neq 0$) are the intra-dimer hopping amplitudes; finally, $\sigma_{l}$ and $\theta_{l}$ are the inter-dimer hopping amplitudes. In the absence of the dc force, i.e. for $F=0$, we can assume $a_{n}(t)=\psi_{1}(t)\exp(ikn)$ and $b_{n}(t)=\psi_{2}(t)\exp(ikn)$, where $k$ is the Bloch wave number that spans the Brillouin zone $0\leq k\leq 2\pi$. In this case, from Eqs.(32) and (33) one obtains

$$i\frac{d}{dt}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)=\mathcal{H}(k)\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)$$

(42)

where the elements of the $2\times 2$ Bloch Hamiltonian $\mathcal{H}(k)$ are given by

	$\displaystyle\mathcal{H}_{11}(k)=\sum_{l}\rho_{l}\exp(-ikl)$		(43)
	$\displaystyle\mathcal{H}_{12}(k)=\sum_{l}\sigma_{l}\exp(-ikl)$		(44)
	$\displaystyle\mathcal{H}_{21}(k)=\sum_{l}\theta_{l}\exp(-ikl)$		(45)
	$\displaystyle\mathcal{H}_{22}(k)=\sum_{l}\eta_{l}\exp(-ikl).$		(46)

The Bloch Hamiltonian is $\mathcal{PT}$-symmetric, with $\mathcal{P}=\sigma_{x}$ and $\mathcal{T}=\mathcal{K}$, provided that

$$\theta^{*}_{-l}=\sigma_{l}\;,\;\;\eta_{-l}^{*}=\rho_{l}.$$

(47)

Such conditions ensure that $\mathcal{H}_{22}(k)=\mathcal{H}^{*}_{11}(k)$ and $\mathcal{H}_{21}(k)=\mathcal{H}^{*}_{12}(k)$. In this case, the lattice does not display the non-Hermitian skin effect skin and the energy spectrum is absolutely continuous and composed by two energy bands, with the dispersion relation given by Eq.(6). A $\mathcal{PT}$ symmetry breaking phase transition of Bloch bands arises when the non-Hermitian parameter $\lambda$ in the system is increased above the critical value $\bar{\lambda}_{c}$, alike in the two-level system discussed in Sec.II.

When the external dc force is applied, i.e. for $F\neq 0$, the energy spectrum becomes pure point and composed by two Wannier-Stark ladders WS1 ; WS2 , with the allowed energies given by

$$E_{l}=lF\pm\Theta$$

(48)

where $l=0,\pm 1,\pm 2,\pm 3,..$ and $\Theta$ describes the energy shift of the two ladders. The corresponding eigenstates are normalizable (localized) with a higher-than-exponential localization.
In an Hermitian lattice, the energy shift $\Theta$ is real and the dynamics in the time domain is generally aperiodic and corresponds to a superposition of Bloch oscillations and Zener tunneling between the two bands WS1 ; WS2 ; WS3 ; WS4 ; WS5 ; WS5bis ; WS6 ; WS7 . The dynamics is characterized by two time periods: The first one, $T_{1}=2\pi/F$, is determined by the mode spacing of each WS ladder and is related to the Bloch oscillation dynamics, whereas the second one, $T_{2}=\pi/\Theta$, is determined by the shift of the two interleaved WS ladders.
In a non-Hermitian lattice the energy shift $\Theta$ can become complex and, as we show below, in the small-forcing limit $F\rightarrow 0$ it contains the complex Zak phase of the Bloch Hamiltonian $\mathcal{H}(k)$. More precisely, we will show below that $\Theta$ is the quasi energy $\mu_{+}$ of the Bloch Hamiltonian $\mathcal{H}(k)$, cycled over the Brillouin zone at a frequency $\omega=F$, i.e. with $k=Ft$. This means that the WS energy spectrum undergoes a phase transition as $\lambda$ is increased above $\bar{\lambda}_{c}$, from real to complex energies, and the phase transition can be either sharp or smooth, depending on whether the imaginary part of the Zak phase for $\lambda<\bar{\lambda}_{c}$ is vanishing or not.
To calculate the WS energy spectrum $E$, let us assume $a_{n}(t)=\bar{a}_{n}\exp(-iEt)$, $b_{n}(t)=\bar{b}_{n}\exp(-iEt)$ in Eqs.(32-33), and let us introduce the spectral variables

	$\displaystyle\psi_{1}(k)$	$\displaystyle=$	$\displaystyle\exp(-iEk/F)\sum_{n}\bar{a}_{n}\exp(-ikn)$		(49)
	$\displaystyle\psi_{2}(k)$	$\displaystyle=$	$\displaystyle\exp(-iEk/F)\sum_{n}\bar{b}_{n}\exp(-ikn).$		(50)

It readily follows that $\psi_{1,2}(k)$ satisfy the Sturm-Liouville problem

$$iF\frac{d}{dk}\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)=\mathcal{H}(k)\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)$$

(51)

on the interval $0\leq k\leq 2\pi$, with the boundary conditions

$$\psi_{1,2}(2\pi)=\psi_{1,2}(0)\exp\left(-\frac{2\pi iE}{F}\right).$$

(52)

Once the spectral amplitudes $\psi_{1,2}(k)$ and eigenenergies $E$ have been determined, the eigenvectors $(\bar{a}_{n},\bar{b}_{n})$, corresponding to the energy $E$, are determined using the inverse relations

	$\displaystyle\bar{a}_{n}$	$\displaystyle=$	$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}dk\psi_{1}(k)\exp(ikn+iEk/F)$		(53)
	$\displaystyle\bar{b}_{n}$	$\displaystyle=$	$\displaystyle\frac{1}{2\pi}\int_{0}^{2\pi}dk\psi_{2}(k)\exp(ikn+iEk/F).$		(54)

Interestingly, after letting $k=\omega t$ Eq.(43) indicates that $\psi_{1,2}(k)$ can be viewed as the amplitudes of a two-level $\mathcal{PT}$-symmetric system, with Hamiltonian $\mathcal{H}(k)$, which is slowly cycled in time at the frequency $\omega=F$. This basically corresponds to the fact that in Bloch space the external force introduces a uniform drift of the quasi-momentum $k$ to span the entire Brillouin zone. The Sturm-Liouville problem, defined by Eqs.(43) and (44), can be solved as follows. Let us indicate by $\bm{\psi}_{+}$ and $\bm{\psi}_{-}$ the eigenvectors of the Floquet matrix $\mathcal{R}$, introduced in Sec.II.B, with eigenvalues (quasi energies) $\mu_{\pm}$. Then Eq.(43) is satisfied by letting either

	$\displaystyle\left(\begin{array}[]{c}\psi_{1}(k)\\ \psi_{2}(k)\end{array}\right)=\mathcal{U}\left(\frac{k}{F}\right)\exp(-i\mathcal{R}k/F)\bm{\psi}_{+}$		(57)
	$\displaystyle=\exp(-i\mu_{+}k/F)\mathcal{U}\left(\frac{k}{R}\right)\bm{\psi}_{+}$

or

	$\displaystyle\left(\begin{array}[]{c}\psi_{1}(k)\\ \psi_{2}(k)\end{array}\right)=\mathcal{U}\left(\frac{k}{F}\right)\exp(-i\mathcal{R}k/F)\bm{\psi}_{-}$		(60)
	$\displaystyle=\exp(-i\mu_{-}k/F)\mathcal{U}\left(\frac{k}{F}\right)\bm{\psi}_{-}.$

Since $\mathcal{U}(0)=\mathcal{U}(2\pi/F)=\mathcal{I}$ (the $2\times 2$ identity matrix), to satisfy the boundary conditions Eq.(44) one should have

$$\frac{2\pi}{F}\mu_{\pm}=\frac{2\pi E}{F}-2l\pi$$

i.e.

$$E=lF+\mu_{\pm}$$

(61)

where $l=0,\pm 1,\pm 2,...$. Equation (47) provides the general form of the Wannier-Stark ladders of allowed energies in terms of the quasi energies $\mu_{\pm}$ of the cycled two-level Bloch Hamiltonian $\mathcal{H}(k)$. It is precisely Eq.(40) with the energy shift parameter $\Theta$ given by $\Theta=\mu_{+}$.
When the external dc force is weak, i.e. in the limit $F\rightarrow 0$, the Bloch wave number $k=Ft$ in the two-level Bloch Hamiltonian $\mathcal{H}(k)$ varies slowly with time, and thus the quasi energies can be approximated by Eq.(23). One obtains

$$E=lF+\frac{1}{2\pi}\int_{0}^{2\pi}dkE_{\pm}(k)+\frac{F}{2\pi}\gamma_{B_{\pm}}$$

(62)

corresponding to the energy shift

$$\Theta=\mu_{+}=\frac{1}{2\pi}\int_{0}^{2\pi}dkE_{+}(k)+\frac{F}{2\pi}\gamma_{B_{+}}.$$

(63)

Therefore, under a weak external driving the Wannier-Stark energy spectrum undergoes a phase transition at $\lambda=\bar{\lambda}_{c}$, which is either sharp or smooth depending on whether the imaginary part of the Zak phase $\gamma_{B{{}_{+}}}$ is vanishing or not when $\lambda<\bar{\lambda}_{c}$. We mention that, contrary to other non-Hermitian lattice models where the spectral ($\mathcal{PT}$ symmetry breaking) phase transition coincides with a localization/delocalization phase transition uffa1 ; uffa1bis ; uffa2 , in the Wannier-Stark ladder problem the spectral phase transition does not correspond to a localization/delocalization phase transition because the eigenstates of the Wannier-Stark Hamiltonian are always localized, for both $\lambda<\bar{\lambda}_{c}$ and $\lambda\geq\bar{\lambda}_{c}$. This very general result follows from the fact that the spectral amplitudes $\psi_{1,2}(k)\exp(iEk/F)$, with $\psi_{1,2}(k)$ solutions to the Sturm-Liouville problem [Eqs.(43,44)], are periodic and continuously differentiable functions of $k$ and thus their Fourier coefficients $\bar{a}_{n}$, $\bar{b}_{n}$ decay as $n\rightarrow\pm\infty$ at least like $\sim 1/n$, regardless of the value of $\lambda$.
In a non-Hermitian lattice below the Wannier-Stark phase transition point, i.e. for $\lambda<\bar{\lambda}_{c}$, the dynamical signature of a complex Zak phase can be probed looking at the temporal behavior of Bloch-Zener oscillations b81 . When the imaginary part of the Zak phase is vanishing, the temporal dynamics is rather generally aperiodic and characterized by the two periods $T_{1}$ and $T_{2}$, like in an ordinary Hermitian lattice under a dc field: only accidentally the dynamics can be periodic. On the other hand, when the imaginary part of the complex Zak phase does not vanish, after an initial transient the dynamics becomes periodic with period $T_{1}$. In fact, a rather arbitrary excitation of the system at initial time $t=0$ can be decomposed as a superposition of localized Wannier-Stark eigenstates belonging to the two ladders, and the dynamics at successive times is governed by the interference of such localized eigenstates. The localized Wannier-Stark eigenstates in one ladder, excited by the initial condition, decay in time with a damping rate $\sim F{\rm{Im}}(\gamma_{B_{+}})$, while the eigenstates in the other ladder are amplified in time with an amplification rate $\sim F{\rm{Im}}(\gamma_{B_{+}})$. Therefore, after a transient time of order $\sim 1/F{\rm{Im}}(\gamma_{B_{+}})$ only the Wannier-Stark eigenstates in the former ladder survive and the dynamics become periodic with the period $T_{1}$ b81 .

As illustrative examples, let us consider the binary lattices depicted in Figs.4(a) and 4(b). The non-Hermiticity in the lattices is introduced by assuming energy gain and loss terms $\pm\lambda$ in the two sub lattices A and B. The binary lattice of Fig.4(a) was introduced in a previous work b74 and its Bloch Hamiltonian $\mathcal{H}(k)$ is given by Eq.(30), previously introduced in Sec.II.C. Since $(d\varphi/dk)\equiv 0$, the Wannier-Stark ladder phase transition in this model is sharp. The model shown in Fig.4(b) is a non-Hermitian extension of the Rice-Mele model b54basta1 and its Bloch Hamiltonian is given by Eq.(31). For this model, the spectral phase transition of the Wannier-Stark energies is imperfect. We emphasize that our analysis is very general and could be applied to a generic $\mathcal{PT}$-symmetric binary lattice, also displaying long-range hopping.