\section{Introduction}

Bloch oscillations (BOs), i.e. the oscillatory motion of a quantum particle in a periodic potential driven by a
constant force, are one of the most striking effects of coherent quantum transport in the matter. Originally studied in
the context of electrons in crystals \cite{r1,r2,r3,r3bis}, they have been observed  in a wide variety of different physical systems such as
semiconductor superlattices \cite{r4,r5}, ultracold atomic gases \cite{r6,r7,r8}, optical and acoustical waves \cite{r9,r10,r11,r12,r13}.
In the simplest semiclassical picture, BOs can be explained from the
Ehrenfest equations of motion taking into account the periodic band structure of the crystal \cite{r14,r15,r16}. In particular,
owing to the so-called 'acceleration theorem'  the momentum distribution of the particle moves in momentum space
at a constant speed while preserving its shape \cite{r2,r16,r17,r18}. As a result, for a wave packet with narrow momentum distribution
 the oscillation trajectory of the particle wave packet in physical space exactly reproduces the shape of the lattice band dispersion curve.
In the spectral domain, BOs are ascribed to the transition of
the energy spectrum from a continuous band with delocalized
Bloch eigenstates in absence of the external force to a
discrete ladder spectrum with localized Wannier-Stark (WS)
eigenstates \cite{r3bis} when the external force is applied and Zener tunneling
among different lattice bands is negligible \cite{r16,r18,r19}. Several studies have extended BOs to
include the role of lattice disorder \cite{r20,r21}, nonlinearities \cite{r22}, lattice defects
and quasicrystals \cite{r23,r24}, inhomogeneous fields \cite{r25}, particle interaction and correlation \cite{r26,r27,r28,r29,r30}, BOs of nonclassical states of light \cite{r31,r32}, long-range hoppings \cite{r32bis}, etc.  \par
More recently, the onset of BOs have been investigated in lattices with a complex periodic potential \cite{r33,r34}, so-called complex crystals \cite{r35,r36} including those with parity-time ($\mathcal{PT}$) symmetry \cite{r37,r38}. The observation of BOs in complex crystals, reported in the landmark experiment of Ref.\cite{r39} using light waves in $\mathcal{PT}$-symmetric synthetic lattices, has raised the interest into the coherent transport properties in non-Hermitian lattices.
Some peculiar properties of non-Hermitian BOs  have been disclosed, such as the occurrence of complex WS ladders \cite{r33}, unidirectional BOs at the symmetry breaking transition \cite{r33,r39} arising from one-way Bragg scattering \cite{r36,r40}, and periodic BOs at exceptional points in spite of lattice truncation \cite{r41}.  However, it is not clear whether the 'acceleration theorem' and semiclassical picture of BOs for wave packets in ordinary lattices can be extended (and to what extent) to complex crystals. For the case where the imaginary part of the potential can be regarded as a perturbation to the real potential or whenever the band spectrum remains real in spite of the complex potential, the wave packet in real space undergoes an oscillatory trajectory like in the Hermitian case, except that BOs  are damped or amplified depending on the sign of the force \cite{r33}.  However, in strongly non-Hermitian potentials the band structure can become imaginary, and in this case it is not clear whether and how
 the semiclassical picture of BOs can be extended to account for a complex energy lattice band. Several examples of tight-binding lattice models that show a non-vanishing imaginary part of the band dispersion curve have been discussed in many works, including the Hatano-Nelson model describing the hopping motion of a quantum particle in a linear tight-binding lattice with an imaginary vector potential \cite{r33_1,r33_2,r33_3,r33_4}, the non-Hermitian extension of the Su-Schrieffer-Heeger tight-binding model \cite{r33_5,r33_6,r33_6bis}, $\mathcal{PT}$-symmetric binary superlattices \cite{r33,r33_4bis}, and the $\mathcal{PT}$-symmetric Aubry-Andre model \cite{r33_7}. Lattice bands with a non vanishing imaginary part could be realized in synthetic temporal optical crystals,  waveguide lattices, coupled-resonator optical waveguides, microwave resonator chains, etc. as discussed in several recent works \cite{r39,r41,r33_4,r33_4bis,r33_8,r33_9,r33_10}.\\  In this work we aim to investigate the onset of BOs in non-Hermitian lattices  with a non vanishing imaginary part of the band dispersion curve. Our analysis discloses that the ordinary wisdom of BOs  that uses a semiclassical description can not be trivially extended to non-Hermitian lattices when the band dispersion curve has a non vanishing imaginary part. In particular, we show that the acceleration theorem (i.e. the uniform drift in the momentum space of the wave packet distribution) holds only on average, whereas in real space  a particle wave packet with narrow spectral distribution undergoes a periodic motion {\it but}  following a closed orbit  in {\it complex} plane.  As a result, strong wave packet reshaping is generally observed within each BO cycle, in addition to the transverse oscillatory motion.  We exemplify the concept of BOs involving complex trajectories by considering the onset of BOs in two nearest-neighbor tight-binding lattice models with a complex energy band dispersion curve, including the Hatano-Nelson model describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential. In such a case, it is shown that the BO motion of a particle wave packet corresponds to an elliptical orbit in the complex plane and the trajectory followed by a wave packet in real space can not be merely predicted by the real part solely (group velocity) of the energy dispersion relation.\\
 The paper is organized as follows. In Sec.II we briefly review the semiclassical approach to BOs in ordinary crystals, and highlight the difficulties to extend such a simple picture to BOs in lattices described by an effective non-Hermitian Hamiltonian. The exact analysis of BOs is presented in Sec.III, where an extension of the acceleration theorem is derived and the concept of complex trajectories is introduced. Two examples illustrating the onset of BOs in lattice models with a complex energy band are presented in Sec.IV, whereas the main conclusions are outlined in Sec.V.

\section{The semiclassical analysis of Bloch oscillations}
In this section we briefly review the simple semiclassical model that describes BOs as introduced for ordinary crystals, and show the limitations that arise when trying to extend such an analysis to complex crystals.

\subsection{Single-band lattice model and effective Hamiltonian}
We consider the motion of a quantum particle in a one-dimensional periodic potential with lattice period $a$ driven by an external constant force $F$. As is well-known, for a weak force Zener tunneling among different bands is negligible and a single-band model can be used to describe the onset of BOs and the formation of WS localized states within each band \cite{r3bis,r18,r19}. Indicating by $W(x-na)$ the Wannier  state of the lattice band that localizes the particle at site $n$ of the lattice \cite{r3bis}, after expanding the particle wave function $\psi(x,t)$ as $\psi(x,t)=\sum_n c_n(t) W(x-na)$, the evolution equations for the site occupation probabilities read (taking $\hbar=1$)
\begin{equation}
i \frac{d c_n}{dt}=\sum_l \kappa_{l}c_{n+l}+nFac_n,
\end{equation}
where $F$ is the external driving force and $\kappa_{l}$  ($l \neq 0$) are the hopping rates among different lattice sites.The dispersion relation $E=E(q)$ of the lattice band is given by the relation
\begin{equation}
E(q)=\sum_{n} \kappa_{n} \exp(iqan)
\end{equation}
where $- \pi/a \leq q < \pi/a$ is the Bloch wave number. Note that the constant $\kappa_0$ just provides an energy bias of the lattice band, and it will be taken to be zero in the following analysis for the sake of definiteness. In an Hermitian lattice, $\kappa_{-n}=\kappa_{n}^*$ and the energy dispersion curve $E(q)$ is a real function, whereas for a non-Hermitian lattice $E(q)$ can become complex. As is well known, the solution to the tight-binding equations (1) can be expressed as $c_n(t)=\phi(x=na,t)$, where the function $\phi(x,t)$ satisfies  the continuous Schr\"{o}dinger equation $i (\partial \phi / \partial t)=\hat{H}  \phi(x,t)$ with the {\it effective} Hamiltonian
\begin{equation}
\hat{H}=E \left( \hat{p}_x \right) +Fx
\end{equation}
where $\hat{p}_x= -i ( \partial / \partial x)$ is the momentum operator and $E(q)$ is the band dispersion relation defined by the Fourier expansion (2).
\subsection{Semiclassical analysis}
In an ordinary crystal, the onset of BOs for a wave packet with a narrow spectral distribution is generally explained on the basis of the Ehrenfest equations derived from  the effective Hamiltonian (3), that describe the evolution of mean position  (in units of the lattice period $a$) and momentum distributions of the particle wave function. Such equations read explicitly
\begin{eqnarray}
i \frac{d \langle x \rangle}{dt} & = & \langle [x, \hat{H}] \rangle= i \langle  \frac{\partial E}{\partial q} (\hat{p}_x) \rangle \\
i \frac{d \langle \hat{p}_x \rangle}{dt} & = & \langle [ \hat{p_x}, \hat{H}] \rangle=-i F.
\end{eqnarray}
From Eq.(5) it follows that the mean particle momentum $\langle \hat{p}_x(t) \rangle$ drifts in time with a constant speed $F$, i.e.
\begin{equation}
\langle \hat{p}_x(t) \rangle= \langle \hat{p}_x(0) \rangle-Ft
\end{equation}
a relation generally referred to as the 'acceleration theorem' in its simplest version \cite{r14}. Moreover, provided that the momentum distribution remains narrow in the evolution, the approximation $\langle  \frac{\partial E}{\partial q} (\hat{p}_x) \rangle \simeq  \frac{\partial E}{\partial q} ( \langle \hat{p}_x \rangle)$ in Eq.(4) can be introduced, yielding
\begin{equation}
\langle x(t) \rangle = \langle x(0) \rangle - \frac{1}{F} \left\{ E( \langle \hat{p}_x(t) \rangle)- E( \langle \hat{p}_x(0) \rangle) \right\}.
\end{equation}
 Since $E(q)$ is periodic with period $ 2 \pi/a$, i.e. $E(q+2 \pi/a)=E(q)$, it follows that in real space the particle wave packet undergoes an oscillatory motion, at the BO frequency $\omega_B=Fa$, along a path that reproduces the band dispersion curve of the lattice [according to Eq.(7)].\\
 The question is whether such a simple semiclassical analysis of BOs can be extended to the case of a complex lattice with a band dispersion relation $E(q)$ which is complex-valued, i.e. $E(q)=E_R(q)+iE_I(q)$ where  $E_R(q)$ and $E_I(q)$ are the real and imaginary parts, respectively, of the band dispersion curve. It should be noted that, as we will rigorously prove in the next section,  even though the energy spectrum $E(q)$ of the lattice is complex, as soon as $F \neq 0$ a real WS ladder energy spectrum is obtained, corresponding to a periodic motion at the BO frequency $\omega_B=Fa$ like in an ordinary crystal. However, the extension to the non-Hermitian case of the semiclassical analysis is a nontrivial issue and it is actually not much helpful. The reasons thereof are that, since the effective Hamiltonian $\hat{H}$ is now non-Hermitian, (i) the norm of the wave packet $ \langle \phi(t) | \phi(t) \rangle \equiv \int dx |\phi(x,t)|^2$ is not conserved, and (ii) the Ehrenfest equations are modified taking into account the commutation/anti-commutaion relations of the Hermitian and anti-Hermitian parts of $\hat{H}$. Namely, for any self-adjoint operator $\hat{A}$ corresponding to an observable $A$ that does not explicitly depend on time, after introduction of the expectation value of $\hat{A}$ defined as
\begin{equation}
\langle \hat{A}(t) \rangle \equiv \frac{\langle \phi | \hat{A} \phi \rangle}{\langle \phi | \phi \rangle}=\frac{\int dx \phi^*(x,t) \hat{A} \phi(x,t)}{\int dx |\phi(x,t)|^2}
\end{equation}
 it can be readily shown that
 \begin{equation}
 i \frac{d}{dt} \langle \hat{A}(t) \rangle = \langle [\hat{A}, \hat{H}_1] \rangle+ \langle [\hat{A},\hat{H}_2]_+ \rangle -2 \langle \hat{A}(t) \rangle \langle \hat{H}_2 \rangle.
 \end{equation}
 To derive Eq.(9), we introduced the following decomposition of the effective Hamiltonian $\hat{H}$
 \begin{equation}
 \hat{H}=\hat{H}_1+\hat{H}_2
 \end{equation}
 as a sum of the Hermitian ($\hat{H}_1$) and anti-Hermitian ($\hat{H}_2$) parts, with \cite{note1}
 \begin{equation}
 \hat{H}_1=E_R(\hat{p}_x)+Fx \; ,\;\;\; \hat{H}_2=i E_I(\hat{p}_x)
 \end{equation}
 and indicated by $[\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}$ and by $[\hat{A},\hat{B}]_+=\hat{A}\hat{B}+\hat{B}\hat{A}$ the commutator and anti-commutator of operators $\hat{A}$ and $\hat{B}$, respectively.  The extension of the Eherenfest equations (4) and (5) to the non-Hermitian crystal is obtained from Eq.(9) by letting either $\hat{A}=x$ and $\hat{A}=\hat{p}_x$.  The resulting equations are rather cumbersome and are actually not much useful to predict the dynamical evolution of the particle wave packet. In particular, as compared to the Hermitian limit $\hat{H}_2=0$, additional terms, arising from the anticommutator and non-energy conservation terms in Eq.(9), do arise in the analysis, which completely destroy the simple solutions (6) and (7) and prevent to extend them in  the non-Hermitian case. As a matter of fact, in a non-Hermitian lattice with complex energy dispersion both the acceleration theorem [as least as stated by Eq.(6)] and Eq.(7) are violated, as discussed in the next section.

\section{Bloch oscillations, acceleration theorem and complex trajectories}
A comprehensive analysis of BOs in non-Hermitian lattices with a complex energy dispersion curve should abandon the simple semiclassical model. The exact analysis, discussed in this section, shows that the following general results hold:\\
(i) For any applied force $F \neq 0$, the energy spectrum of the Hamiltonian is a WS ladder of equally-spaced levels (energy spacing $\omega_B=Fa$) like in an ordinary Hermitian crystal; thus the particle motion is periodic with a period
\begin{equation}
T_B= \frac{2 \pi}{Fa};
\end{equation}
(ii) The acceleration theorem holds only {\it on average} in time, namely one has
\begin{equation}
\langle q(t) \rangle=\langle q(0) \rangle-Ft+\theta(t)
\end{equation}
  where $\langle q(t) \rangle$ is the mean value of the wave packet distribution in momentum space at time $t$ and $\theta(t)$ is a periodic function of
  time with period $T_B$, $\theta(t+T_R)=\theta(t)$, and
  $\theta(0)=0$; \\
  (iii) For a wave packet with initial narrow distribution in momentum space at around $q=0$, one has under certain smooth conditions
  \begin{equation}
  |\phi(x,t)|^2 \simeq G(t) |\phi(x-x_0(t),0)|^2
  \end{equation}
 where $x_0(t)$ is a path in the complex $x$ plane, defined by the relation
 \begin{equation}
 x_0(t)=\frac{E(0)-E(-Ft)}{F}
 \end{equation}
 whereas $G(t)$ is a periodic function with period $T_B$, describing norm oscillation within each BO cycle and given by
 \begin{equation}
 G(t) = \exp \left[ \frac{2}{F} \int_{-Ft}^{0} d \xi E_I( \xi) \right].
 \end{equation}

 \subsection{Wannier-Stark energy spectrum}
The exact analysis of BOs in non-Hermitian lattices can be done starting from the tight-binding equations (1) in the Wannier basis extending the procedure used for ordinary lattices. Let us introduced the spectrum $S(q,t)$ defined by the relation
\begin{equation}
S(q,t)=\sum_{n=-\infty}^{\infty} c_n(t) \exp(-i q n a).
\end{equation}
 which is a periodic function of $q$ with period $2 \pi /a$. Once the evolution of the spectrum $S(q,t)$ has been determined, the amplitude probabilities $c_n(t)$ are obtained after inversion of Eq.(17),namely one has
 \begin{equation}
 c_n(t)=\frac{a}{2 \pi} \int_{-\pi/a}^{\pi/a} dq S(q,t) \exp(iqna).
 \end{equation}
 From Eqs.(1), (2) and (17) it readily follows that the spectrum $S(q,t)$ satisfies the following differential equation
 \begin{equation}
i \frac{\partial S}{ \partial t}=E(q) S(q,t)+i F \frac{\partial S}{\partial q}.
 \end{equation}
 To determine the energy spectrum $\epsilon$ of the driven Hamiltonian, let us assume $c_n(t)=C_n \exp(-i \epsilon t)$, i.e. $S(q,t)=s(q) \exp(-i \epsilon t)$. Then from Eq.(19) it follows that
 \begin{equation}
 i F \frac{\partial s}{\partial q}= \left[ \epsilon-E(q) \right]s
 \end{equation}
which can be solved for $s(q)$, yielding
\begin{equation}
s(q)=s(0) \exp \left[  -i \frac{ \epsilon q}{F} +\frac{i}{F} \int_{0}^{q} d \xi E( \xi) \right].
\end{equation}
Since the spectrum $S(q,t)$ is a periodic function with respect to $q$ with period $2 \pi /a$, one has $s(2 \pi /a)=s(0)$, and thus from Eq.(21) it follows that the allowed energies $\epsilon$ satisfy the condition
\begin{equation}
-\frac{2 \pi \epsilon}{aF}+\frac{1}{F} \int_0^{2 \pi /a} d \xi E(\xi)=-2 l \pi
\end{equation}
where $l=0, \pm 1, \pm 2, \pm 3, ...$. Since $\int_0^{2 \pi /a} d \xi E(\xi)= 2 \pi \kappa_0 /a=0$ \cite{note2}, it follows that the energy spectrum is given by a Wannier-Stark ladder, i.e.
\begin{equation}
\epsilon_l=lFa \;\;\; (l=0, \pm 1, \pm 2, \pm 3, ...).
\end{equation}
The corresponding eigenstates, $C^{(l)}_n$, are readily obtained from Eqs.(18) and (21), and read explicitly
\begin{equation}
C^{(l)}_n= \mathcal{N}_l  \int_{-\pi/a}^{\pi/a} dq \exp \left[ iqa(n-l) +\frac{i}{F} \int_0^q d \xi E( \xi)   \right]
\end{equation}
 where $\mathcal{N}_l$ is a normalization constant.

 \subsection{Acceleration theorem}
 The extension of the 'acceleration theorem' to non-Hermitian lattices is obtained by looking at the general solution to the spectral equation (19), which reads explicitly
 \begin{equation}
 S(q,t)=S_0(q+Ft) \exp \left[ -i \int_0^t d \xi E(q+Ft-F \xi) \right]
 \end{equation}
 where $S_0(q)$ is the spectrum of the particle wave packet at initial time $t=0$, i.e. $S_0(q)=S(q,0)$. Clearly, in an Hermitian lattice in which $E(q)$ is a real function from Eq.(25) one has $|S(q,t)|^2=|S_0(q+Ft)|^2$, i.e. the particle wave packet in the momentum space drifts undistorted with a constant speed equal to the force $F$ (acceleration theorem). In a non-Hermitian lattice with complex energy dispersion curve $E(q)$ such a result {\it does not hold}, since one has in this case
 \begin{equation}
 |S(q,t)|^2=|S_0(q+Ft)|^2 \exp \left[ 2 \int_0^t d \xi E_I(q+Ft-F \xi) \right]
 \end{equation}
It is worth determining the motion of the center of mass $\langle q(t) \rangle$ of the spectrum, defined by the relation
\begin{equation}
\langle q(t) \rangle = \frac{\int_{-\pi/a}^{\pi/a} dq q |S(q,t)|^2}{\int_{-\pi /a}^{\pi /a} dq |S(q,t)|^2}.
\end{equation}
From Eqs.(26) and (27) it readily follows that \cite{note3}
\begin{equation}
\langle q(t) \rangle=\langle q(0) \rangle-Ft+ \theta(t)
\end{equation}
where $\theta(t)$ is a periodic function of period $T_B$, i.e. $\theta(t+T_B)=\theta(t)$, with $\theta(0)=0$, given by
\begin{eqnarray}
\theta(t) & = &  \frac{ \int_{-\pi/a}^{\pi/a} dq q |S_0(q)|^2  \exp \left[  2 \int_0^t d \xi E_I(q-F \xi) \right]}{\int_{-\pi/a}^{\pi/a} dq |S_0(q)|^2   \exp \left[  2 \int_0^t d \xi E_I(q-F \xi) \right] } \nonumber \\
& - & \frac{ \int_{-\pi/a}^{\pi/a} dq q |S_0(q)|^2}{\int_{-\pi/a}^{\pi/a} dq |S_0(q)|^2}
\end{eqnarray}
Equation (28) shows that, in addition to the constant drift of the particle momentum $ \langle q(t) \rangle$ found in the Hermitian limit, an oscillatory motion is superimposed, i.e. the drift occurs only on average over each BO cycle.

 \subsection{Bloch oscillations and complex trajectories}
 The evolved amplitude probabilities $c_n(t)$ at time $t>0$ are determined from the initial spectral distribution $S_0(q)$ of the particle wave packet and can be determined from Eqs.(17), (18) and (25). One obtains
 \[
 c_n(t)=\sum_l U_{n,l}(t) c_l(0)
 \]
where the propagator is given by
\begin{eqnarray}
U_{n,l}(t) =  \frac{a}{2 \pi} \exp(-iFant) \times  \;\;\;\;\;\;\;\;\;\;\;\; \\
   \int_{-\pi/a}^{\pi/a} dq \exp \left[  -i \int_0^t d \xi E(q-F \xi) +i q (n-l)a \right] \;\; \nonumber
\end{eqnarray}
Let us now consider a particle wave packet with an initial narrow distribution in momentum space, i.e. let us assume that $|S_0(q)|^2=|S(q,0)|^2$ is a narrow function of $q$ peaked at around $q=q_0$, falling off sufficient rapidly far from $q=q_0$. For the sake of simplicity, in the following we will consider the case $q_0=0$. In this case, the energy dispersion relation $E(q)$ in the exponent on the integral on the right hand side of Eq.(30) can be expanded in series of $q$ at around $q=0$,
 \begin{equation}
 E(q-F \xi) \simeq E(-F \xi)+q \left( \frac{d E}{ d q} \right)_{(-F \xi)}+ ... \;\;,
 \end{equation}
  and the integral can be extended from $q= -\infty$ to $q= \infty$. If we limit the expansion in Eq.(31) up to first order in $q$, after setting $c_n(t)=\phi(x=na,t)$ one readily obtains \cite{note4}
  \begin{equation}
  \phi(x,t) \simeq  \exp \left[ -iFx -i \int_0^t d \xi E(-F \xi) \right] \phi(x-x_0(t),0)
  \end{equation}
  where the complex path $x_0(t)$ is defined by Eq.(15). From Eq.(32) it follows that
  \begin{equation}
  |\phi(x,t)|^2  \simeq  G(t) |\phi(x-x_0(t),0)|^2
  \end{equation}
  where $G(t)$ is defined by Eq.(16) \cite{note5}. Hence, apart from an oscillation of the amplitude over each BO cycle as defined by the periodic function $G(t)$, in real space the particle wave packet undergoes an oscillatory motion described by the orbit $x_0(t)$ in {\it complex} plane. Note that the trajectory of the wave packet center of mass in real space is not simply related to $x_0(t)$, and it generally depends on the specific profile $\phi(x,0)$. For some initial profiles, e.g. for a Gaussian wave packet distribution $\phi(x,0) = \mathcal {N} \exp(-\gamma x^2)$ with $\gamma$ real, from Eq.(33) it turns out that the trajectory in real space is simply given by the real part of $x_0(t)$, i.e. it is determined by the real part of the energy dispersion curve (like in the Hermitian lattices). However, this is not a general rule, as discussed in the next section for a specific non-Hermitian lattice model.

\begin{figure}[htbp]

  \includegraphics[width=84mm]{Fig1}\\
   \caption{(color online) (a) Schematic of a one-dimensional tight-binding chain with an imaginary gauge field (Hatano-Nelson model), and (b) corresponding band diagram (real and imaginary parts of the band dispersion curve). (c) Closed orbit in complex $x$ plane (ellipse) followed by a particle wave packet  under a forcing $F$.}
\end{figure}

\begin{figure}[htbp]

  \includegraphics[width=84mm]{Fig2}\\
   \caption{(color online) (a) Propagation of a Gaussian wave packet (snapshots of $|c_n(t)|^2 / \sum_n |c_n(t)|^2$) in the Hatano-Nelson lattice over one BO cycle for parameter values $Fa/ \kappa=0.2$ and $\mu=0.1$. In (b) and (c) the trajectories of the beam center of mass $\langle n(t) \rangle$ and   $\langle q(t) \rangle$ in real and momentum space, over one BO cycle, are shown by solid lines. The dashed curve in (b), almost overlapped with the solid one, is the beam trajectory in real space as predicted by Eqs.(14) and (15), which turns out to be given simply by ${\rm Re}(x_0)$, where $x_0(t)$ is the elliptical orbit in complex plane depicted in Fig.1(c). Breakdown of the acceleration theorem is clearly shown in panel (c), where the mean momentum $\langle q(t) \rangle$ is not a linear function of time owing to a small correction $\theta(t)$ according to Eqs.(28) and (29). }
\end{figure}

\begin{figure}[htbp]

  \includegraphics[width=84mm]{Fig3}\\
   \caption{(color online) Same as Fig.3, but in the Hermitian limit ($\mu=0$).}
\end{figure}

  \section{Examples of Bloch oscillations with complex trajectories}
  To exemplify the concept of BOs with complex trajectories, we consider two exactly-solvable examples of non-Hermitian lattices with a complex band dispersion relation.\par
  The first example is provided by the Hatano-Nelson non-Hermitian tight-binding lattice, which describes the hopping motion of a quantum particle on a one-dimensional tight-binding lattice in the presence of an imaginary vector field; see Fig.1(a).  In the nearest-neighbor approximation, it is described by the following set of equations for the site occupation probabilities $c_n(t)$
  \begin{equation}
  i \frac{dc_n}{dt}= \kappa \exp(\mu) c_{n+1}+ \kappa \exp(-\mu) c_{n-1}+V_n c_n
  \end{equation}
  where $\kappa>0$ is the hopping rate between adjacent lattice sites, $\mu>0$ is the imaginary vector potential, and $V_n$ are the on-site energies. The model (34) was originally introduced by Hatano and Nelson in a pioneering work to study the motion of magnetic
flux lines in disordered type-II superconductors \cite{r33_1}, showing that an 'imaginary' gauge field in a disordered one-dimensional lattice can induce a delocalization transition, i.e. it can prevent Anderson localization \cite{r33_1,r33_2,r33_3}. A possible implementation in an optical setting of the Hatano-Nelson model (34), based on light transport in coupled resonator optical waveguides, has been recently suggested in Ref.\cite{r33_4}. Here we consider the case of an ordered lattice with $V_n=Fan$, so that the lattice model (34) is obtained from the general model described by Eq.(1) after setting
  \begin{equation}
  \kappa_l= \kappa \exp( \mu) \delta_{l,1}+\kappa \exp(-\mu) \delta_{l,-1}.
  \end{equation}
  Note that non-Hermiticity in the Hatano Nelson model arises from the imaginary gauge field ($ \mu \neq 0$), the Hermitian case being reached in the limit $\mu=0$ \cite{note6}. The energy dispersion relation of the lattice band reads [Fig.1(b)]
  \begin{equation}
  E(q)=2 \Delta_R \cos(qa)+2 i \Delta_I \sin(qa)
  \end{equation}
  where we have set
  \begin{equation}
  \Delta_R= \kappa \cosh( \mu) \; ,\;\;\; \Delta_I= \kappa \sinh( \mu).
  \end{equation}
   An interesting feature of the Hatano-Nelson model is that it can be mapped into an Hermitian lattice problem after the substitution $c_n(t)=a_n(t) \exp( -\mu n)$ in Eq.(34). Such a property can be readily used to derive analytical expression of the WS eigenstates $C_n^{(l)}$, which are given by [according to Eq.(24)]
   \begin{equation}
   C_n^{(l)}= \mathcal{N}_l \exp[- \mu n+i \pi (n-l)] J_{n-l} \left( \frac{2 \kappa}{Fa} \right),
   \end{equation}
   whereas the propagator $U_{n,l}(t)$, as obtained from Eq.(30), reads explicitly
   \begin{eqnarray}
   U_{n,l}(t) & = & J_{n-l} \left(  \frac{4 \kappa}{Fa} \sin \left( \frac{Fat}{2} \right)  \right) \exp(-iFnat) \times \nonumber \\
   & \times & \exp \left[ i(n-l) (Fat-\pi)/2+ 	\mu(l-n) \right]
   \end{eqnarray}
   where $J_n$ is the Bessel function of first kind and order $n$.
   According to Eq.(15), a particle wave packet with narrow distribution in momentum space undergoes an oscillatory motion with a trajectory in complex plane described by the equation
 \begin{equation}
 x_0(t)=\frac{2 \Delta_R}{F} \left[ 1-\cos(Fat) \right]+i \frac{2 \Delta_I}{F} \sin(Fat)
 \end{equation}
   which describes the orbit of an ellipse [see Fig.1(c)]. As an example, Fig.2 shows the numerically-computed propagation of an initial Gaussian wave packet with distribution $c_n(0) \propto  \exp(-\gamma n^2)$ for parameter values $Fa/\kappa=0.2$, $\gamma=0.02$ and $\mu=0.1$. In the figure, the evolution of the wave packet center of mass $\langle n (t) \rangle$ and $\langle q(t) \rangle$ in real and momentum space, over one BO cycle, are also depicted. For comparison, the limit of an Hermitian lattice ($\mu=0$) is shown in Fig.3. Note that in the non-Hermitian case the mean momentum $\langle q(t) \rangle$ does not exactly varies linearly in time because of a non vanishing (thought small) variation $\theta(t)$, as shown in Fig.2(c). Note also that for a Gaussian wave packet, according to Eqs.(14) and (15) the behavior of $\langle n(t) \rangle$ is simply given by
   \begin{equation}
   \langle n(t) \rangle={\rm Re} \left( x_0(t) /a \right)
   \end{equation}
    i.e. the trajectory of the wave packet in the real space over one BO cycle basically maps the {\it real part} of the band dispersion curve $E_R(q)$. At first sight one might think that this should be a general result, since the real part of the band dispersion curve is associated to the group velocity of the wave packet. However, this is not the case since the semiclassical model can not be readily extended to the non-Hermitian case, as discussed in Sec.II. For example, let us consider a two-humped complex initial distribution defined by the relation
    \begin{equation}
    c_n(0) 	\propto \frac{1}{\cosh^2[(\alpha+i \beta) n]}
    \end{equation}
\begin{figure}[htbp]

  \includegraphics[width=84mm]{Fig4}\\
   \caption{(color online) Same as Fig.2, but for an initial two-humped wave packet distribution defined by Eq.(42). Parameter values are given in the text. In panel (b) the dashed curve, almost overlapped with the solid one, corresponds to the trajectory of the wave packet center of mass as predicted by Eqs.(14) and (15), whereas the tiny dotted curve shows the behavior of ${\rm Re}(x_0(t)/a)$.}
\end{figure}
\begin{figure}[htbp]
  \includegraphics[width=84mm]{Fig5}\\
   \caption{(color online) Same as Fig.4, but in the Hermitian limit ($\mu=0$).}
\end{figure}
\begin{figure}[htbp]
  \includegraphics[width=84mm]{Fig6}\\
   \caption{(color online) (a) Propagation of a Gaussian wave packet (snapshots of $|c_n(t)|^2 / \sum_n |c_n(t)|^2$) over one BO cycle in the non-Hermitian lattice defined by Eqs.(1) and (44)  for $Fa/ \kappa=0.2$. The initial condition is $c_n(0) \propto \exp(-\gamma n^2)$ with $\gamma=0.02$. In (b)  the behavior of the wave packet width $\Delta n(t)$, defined as $\Delta n(t)= \sqrt{\sum_n n^2 |c_n(t)|^2 / \sum_n  |c_{n}(t)|^2}$, is shown, indicating a breathing (rather than oscillatory) dynamics of the wave packet. The behavior of the centroid in momentum space,  $\langle q(t) \rangle$, is shown in panel (c).}
\end{figure}
     with $\alpha$ and $\beta$ real parameters. A narrow wave packet distribution at around $q=0$ in momentum space is ensured by assuming $|\alpha|, |\beta| \ll 1$.  Figure 4 shows the numerically-computed wave packet evolution for the initial condition (42) and for parameter values $Fa / \kappa=0.2$, $\alpha=0.02$, $\beta=0.04$, and $\mu=0.1$. The Hermitian limit $\mu=0$ is shown, for comparison, in Fig.5. While in the Hermitian limit the wave packet trajectory in the real space maps the energy band dispersion curve, in the non-Hermitian case the trajectory does not merely correspond to Eq.(41).\\
     As a second example, let us consider a non-Hermitian lattice with a completely imaginary dispersion relation, given by
     \begin{equation}
     E(q)=2 i \kappa \cos(qa)
     \end{equation}
     with $\kappa>0$. Such a lattice model  is obtained from the general model described by Eq.(1) after setting
  \begin{equation}
  \kappa_l= i \kappa  \delta_{l,1}+i \kappa \delta_{l,-1}
  \end{equation}
  i.e. by taking imaginary hopping rates for nearest neighbor sites. Such a model  can be realized, for example, in optical waveguide lattices with synthetic complex hopping rates obtained by longitudinal gain/loss management \cite{r33_8,r41} or in optical fiber loops with an amplitude modulator \cite{r41,r33_9}.
  In this case, the orbit $x_0(t)$ in the complex plane is simply given by
  \begin{equation}
  x_0(t)=(2 \kappa /F) i [1-\cos(Fat)],
  \end{equation}
  i.e. it is a straight line along the imaginary axis. The propagator $U_{n,l}(t)$, as obtained from Eq.(30), reads explicitly
   \begin{eqnarray}
   U_{n,l}(t) & = & I_{n-l} \left(  \frac{4 \kappa}{Fa} \sin \left( \frac{Fat}{2} \right)  \right) \exp(-iFnat) \times \nonumber \\
   & \times & \exp \left[ i(n-l) Fat /2 \right].
   \end{eqnarray}
where $I_n$ is the modified Bessel function of order $n$.

   Interestingly, for an initial Gaussian distribution $c_n(0) \propto \exp(-\gamma n^2)$ the center of mass of the wave packet in real space does not undergo any oscillation, i.e. $\langle n(t) \rangle=0$;  the purely imaginary trajectory $x_0(t)$ in complex plane corresponds this time to a breathing dynamics of the wave packet, as shown in Fig.6. Hence rather generally the fact that in non-Hermitian lattices with a complex dispersion curve the trajectory of a wave packet occurs in complex plane means that a combined oscillatory and breathing dynamics can occur, and that such effects may depend on the specific initial wave packet profile.

   \section{Conclusion}
   In this work we have theoretically investigated the onset of BOs in non-Hermitian lattices with a complex energy band, highlighting a few distinctive and novel features as compared to BOs in lattices (either Hermitian or non-Hermitian) with an entirely real energy band. Specifically, we have shown that the simple semiclassical model of BOs, based on the Ehrenfest equations of motion for an effective Hermitian Hamiltonian, can not be extended to non-Hermitian lattices with a complex energy band. In particular, we have shown that the so-called 'acceleration theorem' must be extended to account for reshaping of the  wave packet  distribution in momentum space, and that the particle wave packet in real space undergoes a periodic motion describing rather generally a complex trajectory, i.e. it generally corresponds to reshaping and breathing of the wave packet in addition to a transverse oscillatory motion. The concept of BOs with complex trajectories has been exemplified by considering two examples of non-Hermitian lattices with a complex band dispersion relation, namely the Hatano-Nelson tight-binding Hamiltonian describing the hopping motion of a quantum particle on a linear lattice with an imaginary vector potential and the nearest-neighbor tight-binding lattice Hamiltonian with imaginary hopping rates. Our results shed new light onto the coherent transport properties in driven non-Hermitian crystals, and the predictions of the analysis could be observed in synthetic temporal or spatial crystals realized with optical structures \cite{r39,r33_4bis,r33_9}. Here we have focused our study to one-dimensional non-Hermitian lattices driven by a dc force, however it would be interesting to extend the analysis to ac and dc-ac driven lattices, to bi-dimensional lattices, lattices with synthetic gauge fields, etc. (see, for instance, \cite{r42,r43,r44} and references therein). In such systems, it will be interesting to investigate the impact of a non-Hermitian effective lattice band to such important phenomena like dynamic localization, trapping and unidirectional transport.


\section{\label{sec::intro}Introduction}
\begin{figure}
    \includegraphics[width=.8\columnwidth]{fig01}
    \caption{\label{fig::shear_transformation}
    Horizontal shear transformation of a box.
    The maximum shear amplitude $\gamma_0$ is given by the ratio of horizontal displacement $h$ over box height $v$.
    The model of Ref.~\cite{Corte08} is shown on the left, the right shows the symmetric version considered here.
    By shearing in one cycle both, to the left and the right, by half the amplitude $\tilde\gamma_0$ one obtains the same behaviour.
    }
\end{figure}
\begin{figure}
    \includegraphics[width=.8\columnwidth]{fig02}
    \caption{\label{fig::critical_gamma_averages}
    Dynamical model: fraction of active (colliding) particles in the shear cycle as function of the number of shear cycles, for different maximum shear amplitudes $\gamma_0$, at volume fraction $\phi=0.2$, for $N=1000$ particles.
    As reported in Ref.~\cite{Corte08}, the critical shear amplitude is $\gamma_{0,c}(0.2) \approx 2.66$.
    For $\gamma_0 < \gamma_{0,c}$, the activity dies out and particles arrange in quiescent states.
    Above the threshold, $\gamma_0 > \gamma_{0,c}$, the activity appears to persist indefinitely.
    At the critical point, the fraction of active particles decays as a power law with the number of cycles \cite{Corte08, Milz13}.
    }
\end{figure}
Gibbsian Statistical Mechanics is based on the assumption that the time-averaged properties of a system are equal to the ensemble averages. Although this `ergodic' hypothesis is not valid for all classical many-body systems, it is generally believed that it is correct for thermal equilibrium systems.  However, for non-equilibrium systems, there is no reason to expect ergodicity to hold.  It is, however, not straightforward to test the breakdown of ergodicity. To carry out such a test, we need to know exactly what parts of the phase space are accessible to the non-equilibrium system, and then test whether those regions  are sampled with the same frequency both in and out of equilibrium.

Here we describe simulations of a simple model system where it is possible to test the ergodicity of a non-equilibrium system.  To be more precise, we consider both a simple non-equilibrium system and an equilibrium system that has exactly the same accessible states as the non-equilibrium system.
The non-equilibrium model that we consider was first introduced by Cort\'e \emph{et al.}~\cite{Corte08} to model the experiments by Pine \emph{et al.}~\cite{Pine05} on periodically sheared suspensions of large (non-Brownian) particles. These experiments explored the time evolution of a system of spherical plastic particles of diameter ca $0.23\,\text{mm}$, suspended in a viscous, density-matched fluid. The fluid was confined between coaxial cylinders that were sheared periodically.   Pine \emph{et al.} found that  below a critical maximum shear amplitude, after a transient period, the particle trajectories are reversible, i.e.~the particles return to their original positions at the end of each shear cycle.

The resulting `quiescent'  arrangements of particles lack obvious geometrical order, yet they are clearly organised in such a way that particles avoid collision during a shear cycle.  Cort\'e \emph{et al.}~\cite{Corte08} introduced a simple model of the shear experiment, which starts by placing $N$ disks of diameter $\sigma$ uniformly at random in a rectangular box of area $A$ with periodic boundary conditions, giving a volume fraction of $\phi = N\pi\sigma^2/(4A)$. Each cycle consists in applying a shear transformation with maximum amplitude $\gamma_0$ in, say, horizontal ($x$-) direction and recording for each particle the number of other particles it encounters during the cycle (see Fig.~\ref{fig::shear_transformation}).
Then the box is restored to its initial shape and each particle is displaced randomly once for each encounter recorded during the cycle.
This model reproduces the threshold behaviour observed in experiments \cite{Pine05, Corte08}. Below the threshold shear amplitude, the model system reaches a quiescent state. Cort\'e \emph{et al.}~ coined the phrase ``random organisation'' to describe such states~\cite{Corte08}.

For shear amplitudes above a certain volume-fraction-dependent threshold value, particles keep colliding and random organisation into quiescent states does not occur. As an illustration, we show a typical example of the simulation results obtained with Cort\'{e}'s model:
Figure~\ref{fig::critical_gamma_averages} shows the fraction of particles that are displaced during each cycle as function of the number of cycles. At a volume fraction $\phi = 0.2$, quiescent states are observed for $\gamma_0 \lesssim 2.66$ \cite{Corte08}.
The existence of such a threshold separating reversible from non-reversible behaviour has also been reproduced in  more sophisticated Stokesian dynamics simulations \cite{Sierou01, Pine05}. However, the key non-equilibrium dynamics is already contained in the model of Ref.~\cite{Corte08}.
Cort\'{e}'s model has been argued to be in the universality class of conserved directed percolation \cite{Menon09}.

The properties of the model of Cort\'{e} \emph{et al.}~have been studied extensively.
In particular, recent studies have shown that the quiescent states that emerge at sub-critical shear amplitudes are hyperuniform, meaning that low-wavevector density fluctuations vanish in the limit that the wave-vector goes to zero~\cite{Hexner15, Tjhung15, Weijs15}.
Whilst this observation is intriguing, it is not immediately obvious whether this behaviour is a property of the quiescent states, in the sense that it would show up if we were to sample uniformly over all quiescent states, or whether the non-equilibrium dynamics does not sample the quiescent states uniformly.
In order to elucidate this question, we construct an equilibrium equivalent of the non-equilibrium model of Cort\'{e} \emph{et al.}
As we argue below, our simulations show that  different quiescent states are not sampled with equal probability by the dynamical protocol and that this `non-ergodicity' is at the origin of the observed hyperuniformity.

The remainder of this paper is organised as follows.
Section~\ref{sec::butterfly} introduces the equilibrium model that we study. In this model, all allowed states are quiescent and all quiescent states are allowed.
In Sec.~\ref{sec::equilibrium} we report equilibrium Monte Carlo studies that sample quiescent states of the equilibrium model.
Section~\ref{sec::hyperu} considers the nature of density fluctuations in these states.
\section{\label{sec::butterfly}Equilibrium `butterfly' model}
\begin{figure}
    \includegraphics[width=.8\columnwidth]{fig03}
    \caption{\label{fig::butterfly_geometry_symmetric}
        Sketch of shear butterfly geometry for unit particle diameter, $\sigma = 1$.
        The central area (red, grey) indicates the particle, the surrounding ring (green, light grey) is excluded due to hard overlap at zero shear, and the surrounding butterfly domain (blue, dark grey) is excluded as a result of one full shear cycle of maximum shear amplitude $\tilde\gamma_0 = \gamma_0/2 = 2$.
    }
\end{figure}
\begin{figure}
    \includegraphics[width=.8\columnwidth]{fig04}
    \caption{\label{fig::butterfly_eos_quench_full_gammaH1p33}
    Equation of state of the equilibrium butterfly fluid: The compressibility factor $P/\rho$ is shown as function of the volume fraction $\phi$ at maximum shear amplitude $\gamma_0 = 2.66$, for different number of butterflies $N$. Note that the equilibrium equation of state is smooth at ${\phi=0.2}$, the density where the transition between the quiescent and diffusive states takes place in the dynamical model.     }
\end{figure}
\begin{figure}
    \includegraphics[width=\columnwidth]{fig05}
    \caption{\label{fig::butterfly_gxy_gamma1p33}
    Pair-distance distribution ($\gamma_0=2.66$ and $N=64$): (a) shows $g(x,y)$ vs the horizontal, $x$, and vertical, $y$, centre of mass distances (see Fig.~\ref{fig::butterfly_geometry_symmetric}), at volume fraction $\phi=0.4$.
    (b) shows the vertically averaged data $g(x, \phi)$ vs $x$, for different $\phi$.
    (c) shows the horizontal average $g(y, \phi)$ vs $y$, for the same $\phi$ as in (b).
    }
\end{figure}
The dynamical shear model of Ref.~\cite{Corte08} requires the information whether or not two given disks $A$ and $B$, at positions $(x_A, y_A)^T$ and $(x_B, y_B)^T$, will overlap during the next shear cycle.
This can be determined by a straightforward geometrical argument, since the relative shear displacement of disks $A$ and $B$ only depends on their vertical distance, $\Delta y \equiv y_A - y_B$.
Parametrising the trajectories which the particles trace out in one shear cycle by $t$, such that $t=0$ corresponds to no shear and $\vert t \vert = 1$ to the maximum shear amplitude, the vector pointing from the centre of $B$ to the centre of $A$ is
\begin{equation}
    \mathbf{r} =
    \left(\begin{array}{c} x_A(t) - x_B(t) \\ y_A(t) - y_B(t) \end{array}\right)
    = \left(\begin{array}{c} \Delta x + t \gamma_0 \Delta y \\ \Delta y \end{array}\right).
\end{equation}
We note that $\Delta x$ and $\Delta y$ are the distances before applying shear and they are therefore independent of $t$.
The particles, assumed to have the same diameter $\sigma$, collide during the shear cycle if $\vert \mathbf{r}(t) \vert \leq \sigma$ for some $-1 \leq t^* \leq 1$.
Setting $\mathbf{r}^2(t^*) = \sigma^2$ gives a quadratic equation for $t^*$ which has no solutions if $\vert \Delta y \vert > \sigma$.
However if the vertical distance $\vert \Delta y \vert \leq \sigma$, one finds
\begin{equation}
    \label{eqn::bf_t_star}
    t^* = \frac{-\Delta x \pm \sqrt{\sigma^2 - \Delta y^2}}{\gamma_0 \Delta y}.
\end{equation}
Given the requirement that $\vert t^* \vert \leq 1$, one can check for collision between disks $A$ and $B$ as follows:
\begin{itemize}
    \item If $\Delta x^2 + \Delta y^2 \leq \sigma^2$, at least one collision occurs (even without application of shear).
    \item If $\Delta y^2 > \sigma^2$, no collision occurs.
    \item If $\Delta y^2 \leq \sigma^2$, at least one collision occurs if
        $\left\vert \Delta x \pm \sqrt{\sigma^2 - \Delta y^2} \right\vert \leq \gamma_0\Delta y.$
\end{itemize}
Figure \ref{fig::butterfly_geometry_symmetric} visualises how this results in a butterfly-shaped domain around particle $A$ such that $A$ and $B$ will collide if and only if the centre of $B$ is in that domain.

One can then place a given number of `butterflies' in a box and randomly displace the overlapping ones to simulate the dynamical model of Cort\'e \emph{et al.}~\cite{Corte08} (see Sec.~\ref{sec::intro}). Note that the butterfly model is not a hard-core model in the conventional sense, as the butterfly around $A$ excludes the {\em centre} of $B$ and conversely. Hence, overlaps of butterfly `wings' are allowed, as long as the centre of a butterfly remains outside the body of the other butterfly.

We note that mapping from the asymmetric shear considered in Ref.~\cite{Corte08} onto the symmetric butterfly described here involves a transform of the maximum shear amplitude and of the displacement region (see Fig.~\ref{fig::shear_transformation}\cite{sm}).
\section{\label{sec::equilibrium}Equilibrium sampling of quiescent states}
Configurations in which no butterfly domain contains the centre of mass of any other particle  are by construction quiescent states.
Therefore, for fixed $N$, $\phi$, and $\gamma_0$, we can sample with equal probability among all quiescent states using a straightforward Monte Carlo algorithm \cite{Metropolis53}.
The Hamiltonian in this Monte Carlo algorithm is such that quiescent states have zero potential energy, whereas  configurations with at least one overlap, have infinite potential energy.

We measure the compressibility factor, $Z = P / \rho$ (in this paper, $k_BT=1$), where $P$ is the pressure and $\rho = N/A$ the number density, as function of the volume fraction $\phi$, for different values of the maximum shear amplitude $\gamma_0$.
The data is shown in Fig.~\ref{fig::butterfly_eos_quench_full_gammaH1p33} correspond to a system with $\gamma_0=2.66$.
The equation of state exhibits a flat region around $\phi \approx 0.4$, which seems correlated with the density-driven transition to a striped phase, see also Fig.~\ref{fig::butterfly_gxy_gamma1p33}.
Of course,  hard disks exhibit no such striped phase\cite{sm}.  The transition to the striped phase occurs at a substantially higher volume fraction than the value $\phi_c\approx0.2$ above which quiescent states disappear in the dynamical model  (see Fig.~\ref{fig::critical_gamma_averages}): in other words, the quiescent striped phase is never observed in the dynamical model.
We find the same behaviour for a range of $\gamma_0$ values (see summary in Fig.~\ref{fig::bf_percolation_threshold_comparison}).
Technical aspects of the pressure measurement are described in the Appendix \cite{sm}.

Figure~\ref{fig::butterfly_gxy_gamma1p33} shows the pair distribution function $g(x,y)$ at $\gamma_0=2.66$ for different volume fractions $\phi$.
We measure $g(x,y)$ by counting the number of particle pair distances, binning them according to displacement in $x$- and $y$-direction, and normalising by the number of pair distances expected if the particle distribution were homogeneous with density $\rho$.
Horizontal stripe formation is observed at $\phi=0.4$ in Fig.~\ref{fig::butterfly_gxy_gamma1p33}(a).
Vertical and horizontal averages of this data is shown in Fig.~\ref{fig::butterfly_gxy_gamma1p33}(b) and (c).
As the figure shows,  any horizontal ordering decays within a few particle diameters,  while periodic ordering persists in the vertical direction.
\section{\label{sec::hyperu}Hyperuniformity and non-ergodicity}
\begin{figure*}
    \includegraphics[width=2\columnwidth]{fig06}
    \caption{\label{fig::sk_eq_vs_neq}\label{fig::sk_neq_comp_sup_sub_c_Sk_to_zero}
    (a) Structure factor $S(k)$ as function of wave vector magnitude $k$ for the dynamical and the equilibrium model with $\gamma_0=2.66$ at $\phi=0.2$ (which is the critical volume fraction of the dynamical model \cite{Corte08}).
    The solid black line shows the result for $S(k\to0)$, obtained by computing $\partial\rho / \partial P$ from our independent measurement of the equation of state of the equilibrium butterfly fluid.
    (b) compares $S(k)$ for the dynamical model at different volume fractions, both, above and below $\phi_c$.
    The solid arrows indicate the values of $S(k\to0)$ in equilibrium, obtained from $\partial\rho / \partial P$.
    (c) Structure factor low-$k$ limit $S(k\to0)$ as function of the volume fraction $\phi$ for four different models.
    For the dynamical and the equilibrium models, we plot $S(k_\text{min})$, where $k_\text{min}$ is the smallest accessible wave-vector.
    The solid green line is computed from $\partial\rho/\partial P$ with the hard disk equation of state of Ref.~\cite{Kolafa06}.
    The dashed line indicates the ideal gas result $S(k)=1$.
    }
\end{figure*}
Anomalous density fluctuations of sheared suspensions have been studied theoretically, for  the model of Ref.~\cite{Corte08} and related ones \cite{Hexner15, Tjhung15}.  Experimentally, the density fluctuations have been studied in microfluidics experiments by Weijs \emph{et al.}~\cite{Weijs15}, who also performed simulations of their experimental system.
Over a narrow range of driving amplitudes, these studies all find evidence for hyperuniformity, i.e.~the structure factor $S(\mathbf{k})$ vanishes as $\vert\mathbf{k}\vert\to0$ \cite{Torquato03, Dreyfus15}, in some parameter regimes.
Explicitly, the structure factor $S(\mathbf{k})$, at wave vector $\mathbf{k}\neq0$, for an ensemble of $N$ points is defined by
    $S(\mathbf{k}) = \langle \rho(\mathbf{k}) \rangle / N,$
where
    $\rho(\mathbf{k}) = \left\vert \sum_{j=1}^{N}\exp({\mathrm{i} \mathbf{k}\cdot \mathbf{r}_j}) \right\vert^2,$
with $\mathbf{r}_j$ the point positions and the average is taken over different realisations of the pattern \cite{Rosenfeld90, Frenkel13b}.
Then hyperuniformity is equivalent to
    $\lim_{\vert\mathbf{k}\vert\to0} S(\mathbf{k}) = 0,$
see e.g.~Refs.~\cite{Torquato03, Dreyfus15}.

For equilibrium systems, the long-wavelength behaviour of $S(k)$ is related to the compressibility via
    $\lim_{\vert\mathbf{k}\vert\to0} S(\mathbf{k}) = \partial \rho / \partial P,$
which, away from close packing, is finite.
The results of Refs.~\cite{Hexner15, Tjhung15, Weijs15} are therefore indicative of the non-equilibrium nature of quiescent, randomly organised states.
However, what the existing tests cannot tell us is whether hyperuniformity is related to the average behaviour of quiescent states, or that it is due to the non-ergodic sampling of quiescent states.
As we argue below, our simulations show that it is  non-ergodicity that leads to hyperuniformity.

Figures~\ref{fig::sk_eq_vs_neq}(a)-(b) show $S(k)$ measured for the equilibrium and dynamical models, for $\gamma_0=2.66$, at the critical volume fraction of the dynamical model, $\phi=0.2$, as well as for smaller and larger $\phi$.
We also computed $S(k=0)$ from $\partial\rho / \partial P$ (see Fig.~\ref{fig::butterfly_eos_quench_full_gammaH1p33}\cite{sm}).
Clearly, $S(k=0)$ is non-zero, and hence the equilibrium model is not hyperuniform.
By contrast the critical absorbing states of the dynamical model seem to be hyperuniform, as reported in Ref.~\cite{Hexner15} (see also Appendix \cite{sm}).
However, in the high-$k$ region $S(k)$ is similar for both models.
It seems that these short-wavelength similarities are also present in $S(\mathbf{k})$, i.e.~without radial averaging\cite{sm}.
We note that for the dynamical model at high volume fractions $\phi \gtrsim \phi_c \approx0.2$, a finite fraction of particles remains colliding, such that non-quiescent states are also found\cite{sm} and enter the computation of $S(k)$.
Figure~\ref{fig::sk_neq_comp_sup_sub_c_Sk_to_zero}(c) compares the low-$k$ region for both models with the results for hard disks at different $\phi$ and for the ideal gas.
At low $\phi$, the dynamical model behaves similarly to the ideal gas, increasing $\phi$ towards $\phi_c\approx 0.2$, $S(k\to0)$ drops and it increases again for higher $\phi > \phi_c$.
\section{\label{sec::conclusion}Conclusions}
\begin{figure}
    \includegraphics[width=.8\columnwidth]{fig07}
    \caption{\label{fig::bf_percolation_threshold_comparison}
    Critical volume fractions $\phi_c$ of shear models as function of the maximum shear amplitude $\gamma_0$.
    Squares indicate the continuum percolation threshold of interpenetrable butterflies\cite{sm}.
    The location of the inflection point in the equation of state of the equilibrium butterfly fluid, as discussed in Sec.~\ref{sec::equilibrium} (see Fig.~\ref{fig::butterfly_eos_quench_full_gammaH1p33}), is indicated by circles.
    Triangles indicate the non-equilibrium critical point of the dynamical model of Ref.~\cite{Corte08} (see e.g.~Fig.~\ref{fig::critical_gamma_averages}).
    }
\end{figure}
In our study we have compared an equilibrium model that uniformly samples all quiescent states of a model for a periodically sheared suspensions of non-diffusing disks with the corresponding dynamical model. We find that, in  the dynamical model, quiescent states are sampled non-ergodically.  The differences between the dynamical and the equilibrium model are so pronounced that neither the phase transition, nor the continuum percolation threshold in the equilibrium model is related to the dynamical phase transition in the periodically sheared system.

The different critical volume fractions are summarised in Fig.~\ref{fig::bf_percolation_threshold_comparison}.

We found that critical quiescent states are only hyperuniform if sampled by the non-equilibrium protocol of Ref.~\cite{Corte08} and not if sampled ergodically.
We note that, in principle, quiescent states can `nucleate' from diffusing states, even above the dynamical transition threshold.
It would be interesting to study the pathway for such a nucleation process because our results indicate that the nucleation `barrier' is determined by non-ergodicity, rather than by the lack of availability of quiescent states.
\begin{acknowledgments}
This work was supported by ERC Advanced Grant 227758 (COLSTRUCTION),  EPSRC Programme Grant EP/I001352/1 and by the Swiss National Science Foundation (Grant No. P2EZP2-152188 and No. P300P2-161078).
K.J.S. acknowledges useful discussions with Nuno Ara\'ujo, Tristan Cragnolini, Daphne Klotsa, Erik Luijten, Stefano Martiniani, An{\dj}ela \v{S}ari\'c, Iskra Staneva, Jacob Stevenson, and Peter Wirnsberger. \end{acknowledgments}