# Dynamics of Ordered Active Columns: Flows, Twists, and Waves

S. J. Kole<EMAIL_ADDRESS>Centre for Condensed Matter Theory,
Department of Physics, Indian Institute of Science, Bangalore 560 012, India
Gareth P. Alexander<EMAIL_ADDRESS>Department of Physics, Gibbet
Hill Road, University of Warwick, Coventry CV4 7AL, United Kingdom  Ananyo
Maitra<EMAIL_ADDRESS>Laboratoire de Physique Théorique et
Modélisation, CNRS UMR 8089, CY Cergy Paris Université, F-95032 Cergy-Pontoise
Cedex, France  Sorbonne Université and CNRS, Laboratoire Jean Perrin, F-75005,
Paris, France  Sriram Ramaswamy<EMAIL_ADDRESS>Centre for Condensed Matter
Theory, Department of Physics, Indian Institute of Science, Bangalore 560 012,
India  Centre for Condensed Matter Theory, Department of Physics, Indian
Institute of Science, Bangalore 560 012, India  Department of Physics, Gibbet
Hill Road, University of Warwick, Coventry CV4 7AL, United Kingdom
Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, CY Cergy
Paris Université, F-95032 Cergy-Pontoise Cedex, France  Sorbonne Université
and CNRS, Laboratoire Jean Perrin, F-75005, Paris, France  Centre for
Condensed Matter Theory, Department of Physics, Indian Institute of Science,
Bangalore 560 012, India

###### Abstract

We formulate the hydrodynamics of active columnar phases, with two-dimensional
translational order in the plane perpendicular to the columns and no elastic
restoring force for relative sliding of the columns, using the general
formalism of an active model H∗. Our predictions include: two-dimensional odd
elasticity coming from three-dimensional plasmon-like oscillations of the
columns in chiral polar phases with a frequency that is independent of
wavenumber and non-analytic; a buckling instability coming from the generic
force-dipole active stress analogous to the mechanical Helfrich-Hurault
instability in passive materials; the selection of helical column undulations
by apolar chiral activity.

###### Abstract

In this supplement, we present detailed calculations corresponding to the
results of the main text. We derive the equations of motion of a apolar, polar
and chiral active columnar liquid crystals in Sec. A. We then show that effect
of the lowest order achiral active stress in a columnar material is _exactly_
equivalent to that of an externally imposed stress in Sec. B which allows us
to control active columnar systems. Sec. C describes a spontaneous Helfrich-
Hurault instability in active columnar materials. We describe the
hydrodynamics of apolar and polar (but achiral) active columnar systems in
Sec. D.1. We describe the hydrodynamics of chiral but apolar columnar
structures in Sec. D.2. In Sec. D.3, we describe the novel odd-elastic and
nonreciprocal response and wave propagation in polar and chiral columnar
materials. Finally, in Sec. E, we show that density fluctuations in active
columnar materials remain “normal”.

Living materials form a plethora of organised structures. Whereas many of
these have the spatial asymmetries seen in ordered phases at thermal
equilibrium, living matter is “active”, continually converting chemical energy
into work. A description of the mechanics and statistics of active systems
must either explicitly account for this chemistry or introduce a parameter
that breaks time-reversal symmetry at the microscopic level [1, 2, 3, 4, 5,
6]. The novel features of the resulting dynamics have been extensively
examined for orientationally ordered phases, but translational order has
received less attention [7, 8, 9, 10, 11, 12, 13, 14]. In particular, despite
the abundance of filamentous assemblies in soft matter and biology [15, 16,
17, 18], the active hydrodynamics of a two-dimensional crystalline arrangement
of fluid columns [19, 20, 21, 22] in three dimensions has not been explored.
Apart from the breaking of continuous rotational and translational invariance,
two discrete asymmetries play a central role in the present work. Chiral
symmetry [23] is broken when the elements composing the columns lack mirror
symmetry. Polar asymmetry implies that the columnar material lacks inversion
symmetry along the fluid direction – i.e., symmetry under the transformation
$z\to-z$ where $\hat{\bf z}$ is the mean direction of the column tangents,
orthogonal to the crystalline $\perp\equiv xy$ plane. Both of these
asymmetries are widely present in living and artificial columnar systems [24].
The most natural example of active columnar materials are axons, whose
subunits are both chiral and polar. Indeed, the microtubule bundles comprising
the axon in nerve cells are actively polarity-sorted and therefore possess
macroscopic polar order and chirality [25, 26]. In this article, we will
therefore consider four cases: apolar and achiral, polar and achiral, apolar
and chiral, and polar and chiral. We assume chirality leaves the columnar
structure intact and does not supplant it with an analogue of the TGB phases
[27].

Figure 1: (a) Column oscillations of an active polar, chiral phase for a plane
wave solution of (6) and (7). The horizontal plane shows a density plot of
$\psi$ and the background plane indicates the phase and direction of the plane
wave. Images are shown for every quarter of an oscillation period. (b) Polar
plot of $\text{Re}[\eta\omega(\theta)_{+}/|\zeta_{pc}|]$—the oscillation
frequency, non-dimensionalised by the active timescale $\eta/|\zeta_{pc}|$—of
the odd-elastic plasmon-like mode in the polar chiral columnar phase, for
$\lambda=\mu=\zeta$. $\theta=0$ corresponds to a perturbation purely along the
$\hat{{\bf z}}$ (polarity) direction while $\theta=\pi/2$ corresponds to a
perturbation purely along the planar, crystalline directions. Inset: Colour-
coded sectors (with different radii to ensure visibility) showing the range of
$\theta$—the angle between the wavevector of perturbation and $\hat{{\bf
z}}$—that elicits an oscillatory response for various values of
$\zeta/|\zeta_{pc}|$. This demonstrates that the angular range increases with
decreasing $\zeta/|\zeta_{pc}|$. While the angular range for
$\zeta/|\zeta_{pc}|\leq 1$ extends _almost_ to $\theta=0$, there is no
oscillatory response for a perturbation with wavevector along $\hat{{\bf z}}$.
Instead, $\text{Re}[\omega_{\pm}]\propto\theta^{2}$ at small $\theta$.

Our most interesting results concern polar, chiral columnar materials. We find
that such three-dimensional liquid crystals display odd _two-dimensional_
translational elasticity, with the two components of the displacement field
mimicking a position-momentum pair [12, 13]. Unlike other examples of planar
odd elasticity, for instance, in materials composed of two-dimensional
spinners [28] or two-dimensional crystals with “odd springs”, here odd
elasticity emerges naturally from the columnar organisation of three-
dimensional chiral active units. When the odd elastic modulus $\zeta_{pc}$ is
large enough, the interplay of oddness-induced non-reciprocity and Stokesian
hydrodynamics with viscosity $\eta$ leads to an _oscillatory, plasmon-like
mode_ for perturbations along most directions of the wavevector ${\bf
q}\equiv({\bf q}_{\perp},q_{z})$. The existence and frequency of this mode
depends on the angle between ${\bf q}$ and $\hat{\bf z}$ but, crucially, not
on the wavenumber. The squared vibrational frequency is
$(\zeta_{pc}/\eta)^{2}q_{\perp}^{4}q_{z}^{2}/q^{6}$. The three-dimensional
character of the columnar material is essential for this oscillation: the
Poisson-bracket-like coupling between the displacement fields vanishes for
distortions purely along the column (fluid) direction or purely transverse to
it (i.e., purely in the crystalline directions) and, therefore, there is no
oscillatory mode for such perturbations. With Lamé coefficients $\lambda$ and
$\mu$ of the two-dimensional elasticity, and an active achiral stress $\zeta$,
Fig. 1(a) displays the stroboscopic, three-dimensional conformations of the
columns every quarter of an oscillation period and (c) shows a polar plot of
the frequency of the oscillatory mode – non-dimensionalised by the active
frequency $|\zeta_{pc}|/\eta$ – as a function of the angle $\theta$ from
$\hat{{\bf z}}$. We also display the range of wavevector angles that elicit an
oscillatory response for various values of parameters in the inset of Fig.
1(b).

Another key result is an active column tension, positive or negative depending
on whether the uniaxial active achiral stress is contractile or extensile.
Contractile systems thus have displacement fluctuations with wavenumber $q$
have variance of order $1/q^{2}$ in all directions, while extensile systems
display a spontaneous column-undulation instability. We demonstrate that the
active achiral stress in columnars is functionally equivalent to an externally
imposed in-plane, isotropic stress, allowing us to calculate the extensile
instability threshold by a mapping to the columnar Helfrich-Hurault effect.

While polar and apolar achiral active columnar materials are hydrodynamically
equivalent, chirality breaks this degeneracy. In place of the oscillatory
modes predicted for polar systems, the apolar phases display active chiral
fluid flows that deform the columns into helices.

We now show how we obtain these results. We start from the coupled dynamics of
a conserved scalar field $\psi$ and the density ${\bf g}=\rho{\bf v}$ of a
conserved momentum, with overall incompressibility: a constant total mass
density $\rho$ and $\nabla\cdot{\bf v}=0$ for the total velocity field ${\bf
v}$. The equations of motion of the columnar phases emerge upon expanding
about a state in which $\psi$ has the form of a two-dimensional density wave.
We break detailed balance in Model H in the Hohenberg-Halperin [29]
classification by introducing active stresses and currents [30, 31, 14]. Of
these the simplest is
$\sigma^{a}_{ij}\propto\partial_{i}\psi\partial_{j}\psi$. If the material is
chiral, contributions proportional to the Levi-Civita tensor arise, such as
$\sigma^{ac}_{ij}\propto[\epsilon_{ilk}\partial_{l}(\partial_{k}\psi\partial_{j}\psi)]^{S}$,
and
$\sigma^{pc}_{ij}\propto[\epsilon_{ikl}P_{l}\partial_{k}\psi\partial_{j}\psi]^{S}$
if the material possesses a non-zero polar order parameter ${\bf P}$, where
the superscript $S$ denotes symmetrisation. In systems with only orientational
order, note that while $\sigma^{ac}_{ij}$ [14] has a nematic analogue
$[\epsilon_{ilk}\partial_{l}Q_{kj}]^{S}$, where Q is the nematic order
parameter [32, 33, 34, 35], $\sigma^{pc}_{ij}$ has no polar analogue.

In the supplement, we detail how we obtain the dynamical equations for active
columnar fluids from these extended active model H equations. The broken-
symmetry hydrodynamics of columnar materials requires the phases of the
$\psi$-density wave, that is the Nambu-Goldstone modes of spontaneously broken
translation symmetry, denoted by the two-component vector ${\bf
u}_{\perp}({\bf r},t)$, where ${\bf r}\equiv(x,y,z)$ is the three-dimensional
position vector.

Ignoring permeation, which we will show later to be irrelevant at long
wavelengths, the linearised equation of motion for ${\bf u}_{\perp}$ is
$\partial_{t}{\bf u}_{\perp}={\bf v}_{\perp}$. In the absence of activity, the
dynamics would be controlled by a free energy $F_{u}=(1/2)\int_{\bf
r}[\lambda(\mathrm{Tr}{\textsf{{E}}})^{2}+2\mu{\textsf{{E}}}:{\textsf{{E}}}+K\nabla^{2}{\bf
u}_{\perp}\cdot\nabla^{2}{\bf u}_{\perp}]$ [22], where
$E_{ij}=(1/2)(\partial_{i}u_{j}+\partial_{j}u_{i})$ is the strain, here
written in a linearised approximation, leading to a linearised force density
$-\delta F_{u}/\delta{\bf u}_{\perp}$. In active materials, extra non-
equilibrium force densities arise that cannot be obtained from a functional
derivative of a rotation-invariant functional. In this article, we will
examine the effects of three such force densities. The first, with a
coefficient $\zeta$, is present in all active columnar materials; the second,
with a coefficient $\zeta_{c}$, requires chiral asymmetry and the third, with
a coefficient $\zeta_{pc}$, requires both polar and chiral asymmetries. They
arise, respectively, from the stresses $\bm{\sigma}^{a}$, $\bm{\sigma}^{ac}$
and $\bm{\sigma}^{pc}$ in our active model H∗. Upon neglecting inertia, as
appropriate for the slow flows arising in most soft-matter and biological
contexts, the conservation of momentum reduces to balancing viscous force
densities against active and passive force densities involving ${\bf
u}_{\perp}$:

$\eta\nabla^{2}{\bf v}=\nabla\Pi-\mu\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu)\nabla_{\perp}\nabla_{\perp}\cdot{\bf u}_{\perp}\\\
-\zeta\nabla^{2}{\bf u}_{\perp}-\zeta_{c}\nabla^{2}\nabla\times{\bf
u}_{\perp}-\zeta_{pc}\nabla_{\perp}^{2}\bm{\epsilon}\cdot{\bf
u}_{\perp}+\bm{\xi}_{v},$ (1)

where the pressure $\Pi$ enforces incompressibility, $\bm{\xi}_{v}$ is a zero-
mean, Gaussian white noise whose correlator is $-2T\eta\nabla^{2}\delta({\bf
r}-{\bf r}^{\prime})\delta(t-t^{\prime})$ and $\bm{\epsilon}$ is the rank-2
Levi-Civita tensor. For polar materials there is also an achiral active force
density at $\mathcal{O}(\nabla^{3})$ $\propto\nabla^{2}\partial_{z}{\bf
u}_{\perp}$. We focus on the lower-order polar term but make some remarks
about this contribution later on. To the lowest order in wavenumbers, the
dynamics of achiral polar and apolar active columnar materials are exactly
equivalent.

We start by examining the response of achiral and apolar active columnar
liquid crystal to a perturbation with wavevector ${\bf
q}=q(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. Setting
$\zeta_{c}=\zeta_{pc}=0$ in (1), projecting it transverse to the wavevector,
eliminating the velocity field and defining longitudinal and transverse
displacement fields $u_{l}={\bf q}_{\perp}\cdot{\bf u}_{\perp}/|q_{\perp}|$
and $u_{t}=(q_{x}u_{y}-q_{y}u_{x})/|q_{\perp}|$, we obtain the decoupled
dynamics which is independent of the wavenumber $q$ for $q\to 0$:

$\dot{u}_{l}=-\frac{\cos^{2}\theta}{\eta}\left[{(2\mu+\lambda)\sin^{2}\theta}+{\zeta}\right]u_{l}+\xi_{l}$
(2)

and

$\dot{u}_{t}=-\frac{1}{\eta}\left[{\mu\sin^{2}\theta}+{\zeta}\right]u_{t}+\xi_{t}.$
(3)

The white noises $\xi_{l}$ and $\xi_{t}$ have variances $2T\cos^{2}\theta/\eta
q^{2}$ and $2T/\eta q^{2}$, so that the correlators for $u_{l}$ and $u_{t}$
both scales as $\sim 1/q^{2}$ (for $\zeta>0$) along _all_ spatial directions
(thereby cutting off the viscosity divergences that arise in the equilibrium
columnar phase [21]111Similarly, viscosity divergences of equilibrium smectics
[53, 54] are cut-off by the suppression of fluctuations in active layered
materials [7, 14, 8]) despite the relaxation rates scaling as $\sim q^{0}$.
This, however, crucially requires $\zeta>0$. In a passive columnar system
$\zeta=0$, the displacement fluctuations would diverge as $\sim 1/q^{4}$ for
fluctuations along the column direction. Activity leads to a column tension.
However, when $\zeta<0$, the column tension is destabilising and destroys the
columnar phase.

The effect of the active apolar stress $\zeta$ in an incompressible columnar
system is _exactly_ the same as that of an external stress, uniaxial along the
columns or, equivalently, isotropic in the plane normal to them, as we show in
the supplement. Thus, as in cholesterics and smectics [14], $\zeta<0$ leads to
a spontaneous Helfrich-Hurault instability with degeneracy in the polarisation
of the column undulation, which is shown in Fig. 2 for a planar undulation.
For a finite system of size $[-L/2,L/2]^{2}\times(-\infty,\infty)$ with a
helical modulation of the columns with displacement field ${\bf
u}(x,y,z)=\alpha x\,{\bf e}_{x}+\alpha y\,{\bf e}_{y}+u_{0}\cos(\pi
x/L)\cos(\pi y/L)\bigl{[}\cos qz\,{\bf e}_{x}+\sin qz\,{\bf e}_{y}\bigr{]}$
the mode that goes unstable first is $q^{2}=({\pi}/{L})\sqrt{{\mu}/{K}}$ for a
threshold strain of $\alpha=({\pi}/{L})\sqrt{{\mu}/{\lambda}}$. Equivalently,
the critical active stress for a finite system is
$\zeta_{th}=\pi{\sqrt{K\lambda}}/{L}$, details of the calculation can be found
in the supplement [37].

Figure 2: The columnar phase suffers a spontaneous Helfrich-Hurault
instability when $\zeta<0$ resulting in the buckling of columns. In chiral
systems, $\zeta_{c}\neq 0$ and even planar distortions produced by this
instability lead to flows (blue arrows) that tend to twist columns into
helices.

We now examine the chiral active force density $\propto\zeta_{c}$ in (1).
Though sub-leading in gradients, this term is physically interesting as it
couples longitudinal and transverse fluctuations:

$\dot{u}_{l}=-\frac{\cos^{2}\theta}{\eta}\left[{(2\mu+\lambda)\sin^{2}\theta}+{\zeta}\right]u_{l}-\frac{iq\zeta_{c}\cos\theta}{\eta}u_{t}+\xi_{l},$
(4)

and

$\dot{u}_{t}=-\frac{1}{\eta}\left[{\mu\sin^{2}\theta}+{\zeta}\right]u_{t}+\frac{iq\zeta_{c}\cos\theta}{\eta}u_{l}+\xi_{t},$
(5)

yielding, for perturbations purely along the column direction, $\hat{{\bf
z}}$, mode frequencies $\omega_{\pm}=-({i}/{\eta})[\zeta\pm\zeta_{c}q]$. That
is, chiral activity reduces the relaxation rate of one of the modes while
enhancing that of the other. Interestingly, unlike odd elasticity in two-
dimensional chiral solids [13], $\zeta_{c}$ here is a _dissipative_ Onsager-
like coupling between $u_{t}$ and $u_{l}$. Note that this coupling originates
from the same stress in active model H∗ that led to an odd elastic force
density in active layered materials directed primarily along contours of
constant mean curvature of the layers [14]. For the columnar phase, in
contrast, these transverse chiral flows generate a deformation of the columns
into helices even at linear order (see Fig. 2).

We now turn to phases with macroscopic polarity along $\hat{{\bf z}}$. As
discussed earlier, polar and apolar achiral columnar materials are
distinguishable only at sub-leading order in wavenumber. An achiral but polar
active force density $\zeta_{p}\partial_{z}\nabla^{2}{\bf u}_{\perp}$ leads to
a simple modification of (2), (3): $\zeta\to\zeta+i\zeta_{p}q\cos\theta$.
Chirality, by contrast, breaks the degeneracy of polar and apolar active
columnar liquid crystals at leading order in wavenumber. Intriguingly, the
polar, chiral active force density
$\zeta_{pc}\nabla_{\perp}^{2}\bm{\epsilon}\cdot{\bf u}_{\perp}$ in (1) is
exactly the odd elasticity discussed in _two-dimensional_ active solids [13,
12], but here realised in a fully three-dimensional active material. The
resulting coupling of longitudinal and transverse displacements

$\dot{u}_{l}=-{(2\mu+\lambda)q_{\perp}^{2}q_{z}^{2}+\zeta
q^{2}q_{z}^{2}\over\eta q^{4}}u_{l}-{\zeta_{pc}q_{z}^{2}q_{\perp}^{2}\over\eta
q^{4}}u_{t}+\xi_{l}$ (6)

and

$\dot{u}_{t}=-{\mu q_{\perp}^{2}+\zeta q^{2}\over\eta
q^{2}}u_{t}+{\zeta_{pc}q_{\perp}^{2}\over\eta q^{2}}u_{l}+\xi_{t},$ (7)

enters at zeroth order in wavenumber and has a Poisson-bracket-like structure.
This coupling is truly three-dimensional, vanishing for $\theta=0$ or $\pi/2$.
Its equivalent is ruled out in an incompressible, two-dimensional odd
poroelastic gel, where to $\mathcal{O}(q^{0})$, $\dot{u}_{l}=v_{l}=0$. Here,
however, incompressibility constrains the three-dimensional velocity field
and, therefore, $v_{l}\neq 0$ for a perturbation in a general direction; of
course, $\dot{u}_{l}$ vanishes (at this order in wavenumbers) for perturbation
purely in the solid directions. As discussed in the introduction, (6) and (7)
can lead to an oscillatory response for perturbations with $\theta\neq
0,\pi/2$ for large-enough $\zeta_{pc}$. While the general expression for the
eigenfrequency displayed in [37] is complicated, for $\theta=\pi/4$ it assumes
the form

$\omega_{\pm}(\theta=\pi/4)=\pm\frac{\mid{\zeta_{pc}}\mid}{2\sqrt{2}\eta}\sqrt{1-\left(\frac{\lambda-2\zeta}{2\sqrt{2}\zeta_{pc}}\right)^{2}}-i\frac{6\zeta+4\mu+\lambda}{8\eta}.$
(8)

Notice that (8) describes an overdamped mode when
$|\zeta_{pc}|<|\lambda-2\zeta|/2\sqrt{2}$. In general, perturbations with
wavevector making an angle $\theta\neq 0,\pi/2$ with $\hat{{\bf z}}$ will
elicit an oscillatory response when
$|\zeta_{pc}|>|\zeta+\mu-(2\mu+\lambda)\cos^{2}\theta|/2|\cos\theta|$. This
implies that when $\zeta_{pc}$ is larger than the other active and elastic
terms, a perturbation with a wavevector _almost_ fully along $\hat{{\bf z}}$
also leads to an oscillatory response with $\lim_{\theta\to
0}\mathrm{Re}[\omega_{\pm}]\propto\theta^{2}$. We display the angular ranges
in which we predict an oscillatory response in Fig. 1(b) for various values of
$\zeta_{pc}$.

An instructive comparison is with the dispersive wave induced by non-
reciprocity in two-dimensional compressible solids on frictional substrates
[13], where odd elasticity leads to the dynamics $\dot{u}_{l}\propto-
q_{\perp}^{2}u_{t}$ and $\dot{u}_{t}\propto q_{\perp}^{2}u_{l}$. In active
polar and chiral columnar phases, the presence of momentum conservation and
three-dimensional incompressibility leads to a radically different mode
structure. Momentum conservation induces a long-range interaction, replacing
the dispersive wave by a plasmon-like oscillatory mode that is non-analytic in
the $q\to 0$ limit. In the presence of a momentum sink, the damping in the
momentum equation would ultimately imply a dispersive wave at small
wavenumbers. Indeed, if the columnar material is confined in a geometry which
has a finite extent $\ell$ in the $z$ direction, with porous walls such that
the fluid can flow out of the system, then one can replace $q_{z}^{2}$ with
$1/\ell^{2}$ to obtain the displacement dynamics in [13] quoted above. Note
that the three-dimensional character of the active liquid crystal is still
crucial for this: it requires a $z$-variation of the $z$-component of the
velocity field at the scale of the system.

If instead the confinement is on a scale $\ell$ in the $\perp$ plane, the odd
elastic coupling for $q_{z}\ell\ll 1$ takes the form
$\dot{u}_{l}\propto-(\zeta_{pc}/\eta)\ell^{2}q_{z}^{2}u_{t}$ and
$\dot{u}_{t}\propto(\zeta_{pc}/\eta)u_{l}$. The possibility of active non-
dispersive waves à la [38, 39] is, however, ruled out by the effective damping
rates $(\mu+\zeta)/\eta$ and $(2\mu+\lambda+\zeta)\ell^{2}q_{z}^{2}/\eta$ of
$u_{t}$ and $u_{l}$ respectively. The resulting eigenfrequencies
$\omega_{+}=-i(\zeta+\mu)/\eta$ and
$\omega_{-}=-i\ell^{2}q_{z}^{2}[(\zeta+\lambda+2\mu)/\eta+\zeta_{pc}^{2}/\eta(\zeta+\mu)]$
have no real part.

We have described the hydrodynamics of active columnar phases, including both
chiral and polar materials. The polar chiral activity realises a form of two-
dimensional odd elasticity, with a plasmon-like oscillatory response of the
columns whose frequency scales as $q^{0}$ and is non-analytic as $q\to 0$.
Three-dimensionality and viscous hydrodynamic interaction are essential to the
mechanism. Active columnar phases are an ideal prospect for both natural and
synthetic odd elastic materials. Biological tissues, such as axons and
epithelia, have a columnar structure and a natural polarity, so chiral
activity in these tissues should generate odd elastic effects. Columnar liquid
crystals with macroscopic polarity [24], if suffused with chiral
microswimmers, could realize a material in which to test our predictions.

The achiral active stress is analogous to an applied mechanical stress and
produces a Helfrich-Hurault instability of the columns with degeneracy in the
polarisation of the undulations. As in active cholesterics, an imposed stress
can be used to control the instability. The bend instability in three-
dimensional extensile nematics [40, 3] is generically accompanied by twist to
satisfy geometric compatibility [41, 42, 43, 44]. Analogously, the active
columnar instability could favour spontaneous chiral-symmetry breaking through
a coherent twisting of the column undulations. In chiral active columnar
materials, $\zeta_{c}$ explicitly breaks parity, favouring one sign of helical
column undulation. This highlights the emergence of qualitatively different
effects from the same chiral stress in distinct broken-symmetry phases of
active model H∗.

An interesting extension of our results would be to determine the effects of
confinement and boundary conditions on the column undulations and their
associated odd elasticity. Finally, our predictions are based largely on
linear stability analysis. An understanding of the evolution beyond the linear
regime and the final state of the system requires a direct numerical solution
of the nonlinear equations of motion.

###### Acknowledgements.

We thank Jacques Prost for valuable discussions. SJK acknowledges support
through a Raman-Charpak Fellowship of CEFIPRA, hosted by Cesare Nardini, SPEC,
CEA-Saclay. AM was supported in part by a TALENT fellowship awarded by CY
Cergy Paris Université. SR was supported in part by a J C Bose Fellowship of
the SERB, India. Dynamics of Ordered Active Columns: Flows, Twists, and Waves
Supplementary Material

##  Appendix A Derivation of hydrodynamic equations of columnar materials
starting from a theory in terms of phase fields

The dynamics of columnar fluids can be consistently derived from that of a
conserved or non-conserved scalar field coupled to a velocity field. In an
earlier paper, we discussed an analogous construction of the hydrodynamic
equations for a layered system [14] starting with a scalar field theory and
examining fluctuations about a state with a one-dimensional density modulation
and now repeat the same construction for a columnar one. We will only “derive”
the dynamical equations starting from a conserved scalar field coupled to a
velocity field, but an exactly equivalent calculation can be performed for a
non-conserved scalar field.

We consider a bulk, three-dimensional system with three-dimensional density
$\psi({\bf r},t)$ and velocity ${\bf v}({\bf r},t)$ fields where ${\bf
r}=(x,y,z)$. A columnar state is realised in this system when the density
field has a two-dimensional, periodic density modulation. We will further
examine the breaking of two discrete symmetries: chirality and polarity. The
latter implies breaking the up-down symmetry in the fluid direction, i.e., the
direction transverse to the density modulations. We will therefore examine
four distinct kinds of active columnar liquid crystal, listed here from the
most to the least symmetric: i. apolar and achiral; ii. polar and achiral;
iii. apolar and chiral; and iv. polar and chiral. Note that the hydrodynamics
of active, polar but achiral columnar liquid-crystalline phases is exactly
equivalent to their apolar counterparts. Therefore, we will not need to
consider i. and ii. separately.

Before embarking on deriving the hydrodynamic equations, it is useful to
comment on the distinct ways in which polar asymmetry appears in columnar
materials and lamellar phases. Here, we discuss systems that have polar
asymmetry in the fluid direction, i.e., orthogonal to the periodic directions.
Polar lamellar or smectic phases [45] usually refer to systems in which the
polar asymmetry is _along_ the normal to the layers i.e., along the direction
of density modulation. A closer analogue to polar columnar phases are what are
known as smectic-P phases [46], which have a direction in the fluid plane of
the layers. However, since in three-dimensional lamellar phases, there are two
fluid directions, a polar asymmetry in the fluid directions requires breaking
a continuous symmetry; two-dimensional smectic-P states do break only a
discrete symmetry in the fluid direction like three-dimensional polar columnar
liquid crystals we will discuss. This distinction between polar lamellar and
columnar materials has certain interesting consequences. For instance, while
the hydrodynamics of polar and apolar smectic phases in Stokesian fluids are
equivalent, just like their columnar counterparts, the hydrodynamics of “dry”
polar and active smectics [47] – i.e. compressible polar and active smectics
moving through a frictional media (in arbitrary dimensions) – is distinct from
dry, active apolar lamellar phases [7]. There is no such distinction between a
dry polar and apolar columnar system. Since the polar drive in columnar
systems is _orthogonal_ to the plane in which translation symmetry is broken,
the displacement field dynamics is unaffected, to the lowest order in
wavenumbers, by this drive.

With this preamble, we now construct the hydrodynamic equations for active
columnar liquid crystals.

###  A.1 Apolar and achiral active columnar phase

We start with the general dynamics of a conserved scalar field coupled to a
velocity field – an active version of model H [30, 31, 29]

$\partial_{t}\psi=-\nabla\cdot({\psi}{\bf
v})+M\nabla^{2}\frac{\delta{F}[\psi]}{\delta\psi}+\nabla\cdot(\psi{\bf
V}_{an})+{\xi}_{\psi},$ (1)

where $\langle\xi_{\psi}({\bf r},t)\xi_{\psi}({\bf
r}^{\prime},t^{\prime})\rangle=-2D\nabla^{2}\delta({\bf r}-{\bf
r}^{\prime})\delta(t-t^{\prime})$ and the active “particle-phase” velocity is

${\bf
V}_{an}=\mu_{1}(\nabla\psi)\nabla^{2}\psi+\mu_{2}\nabla\cdot(\nabla\psi\nabla\psi)+\mu_{3}\nabla(\psi\nabla^{2}\psi).$
(2)

The velocity field, in the limit of small Reynolds number relevant for most
biological and soft matter systems, has an overdamped, Stokesian dynamics:

$\eta\nabla^{2}{v}_{i}=\psi\partial_{i}\frac{\delta
F[\psi]}{\delta\psi}+\partial_{i}p-{\zeta}_{H}\partial_{j}(\partial_{i}\psi\partial_{j}\psi)+\xi_{v_{i}},$
(3)

where $p$ is the pressure that enforces the incompressibility constraint
$\nabla\cdot{\bf v}=0$ and $\langle{{\xi}}_{v_{i}}({\bf
r},t){{\xi}}_{v_{j}}({\bf
r}^{\prime},t^{\prime})\rangle=-2\delta_{ij}{D}_{v}{\nabla}^{2}\delta({\bf
r}-{\bf r}^{\prime})\delta(t-t^{\prime})$. The force density with a
coefficient $\zeta_{H}$ is one of the active terms in this model. The
constraint of detailed balance would also introduce a relation between the
coefficients $\mu_{i}$ in (2). Further, in equilibrium, the noise strengths
are related via ${D}/{M}={D}_{v}/{\eta}$.

We now separate the field $\psi$ into a part $\psi_{0}$ that is uniform on
average and a part $\psi_{1}$ that is modulated on average and use a Swift-
Hohenberg free energy to describe the latter. The Swift-Hohenberg free energy
is given by

$F=\frac{\Upsilon}{2}\int_{\bf
x}\left[\frac{\alpha}{2}\psi_{1}^{2}+\frac{\beta}{4}\psi_{1}^{4}+\left|\\{\nabla^{2}+q^{2}_{s}\\}\psi_{1}\right|^{2}\right],$
(4)

where $q_{s}$ sets the lengthscale of the spatial modulation. To obtain the
dynamics of a columnar material from this, we assume a density profile that is
periodic in two-dimensions.

${\psi}_{1}|_{s.s}=\sum_{j}\psi^{0}_{j}[e^{i\phi_{j}}+e^{-i\phi_{j}}],$ (5)

where

$\phi_{j}=q_{s}({\bf q}_{j}\cdot{\bf x}-\bar{{\bf q}}_{j}\cdot{\bf
u}_{\perp})$ (6)

with ${\bf q}_{j}$ being the lattice vectors of the planar structure and ${\bf
u}_{\perp}$ is the two-dimensional displacement field about the periodically
ordered density wave state. We can take the lattice vectors to be
$(-\sqrt{3}/2,-1/2,0)$, $(\sqrt{3}/2,-1/2,0)$ and $(0,1,0)$ for a triangular
structure without loss of generality. We write the corresponding two-
dimensional vectors as $\bar{{\bf q}}_{1}=(-\sqrt{3}/2,-1/2)$, $\bar{{\bf
q}}_{2}=(\sqrt{3}/2,-1/2)$ and $\bar{{\bf q}}_{3}=(0,1)$. Taking all
$\psi^{0}_{j}$ to be equal to $\psi^{0}_{1}=\sqrt{|\alpha|/\beta}$, we obtain
a free energy density in terms of the displacement field alone:

$f_{{\bf
u}_{\perp}}=\frac{1}{2}\left[\lambda\text{Tr}[{\textsf{{E}}}]^{2}+2\mu{\textsf{{E}}}:{\textsf{{E}}}+K\nabla^{2}u_{k}\nabla^{2}u_{k}\right]$
(7)

where we have defined a rotation-invariant strain tensor

$E_{ij}=\frac{1}{2}[\partial_{i}u_{j}+\partial_{j}u_{i}-\partial_{i}u_{k}\partial_{j}u_{k}-\partial_{z}u_{i}\partial_{z}u_{j}]\\\
=\frac{1}{2}\begin{pmatrix}2\partial_{x}u_{x}-(\partial_{x}u_{x})^{2}-(\partial_{x}u_{y})^{2}-(\partial_{z}u_{x})^{2}&\partial_{x}u_{y}+\partial_{y}u_{x}-\partial_{x}u_{x}\partial_{y}u_{x}-\partial_{x}u_{y}\partial_{y}u_{y}-\partial_{z}u_{x}\partial_{z}u_{y}\\\
\partial_{x}u_{y}+\partial_{y}u_{x}-\partial_{x}u_{x}\partial_{y}u_{x}-\partial_{x}u_{y}\partial_{y}u_{y}-\partial_{z}u_{x}\partial_{z}u_{y}&2\partial_{y}u_{y}-(\partial_{y}u_{x})^{2}-(\partial_{y}u_{y})^{2}-(\partial_{z}u_{y})^{2}\end{pmatrix},$
(8)

and $\lambda=3\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{4}$,
$\mu=3\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{4}$ and
$K=3\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{2}$. Note that, as pointed out by Kamien
in the context of smectics [48], the _nonlinear_ strain tensor is _not
unique_. For instance, we could have equivalently defined another strain
tensor

$\bar{E}_{ij}=\frac{1}{2}[\partial_{i}u_{j}+\partial_{j}u_{i}-\partial_{k}u_{i}\partial_{k}u_{j}]$
(9)

in terms of which, the free energy would have the form

$\bar{f}_{{\bf
u}_{\perp}}=\frac{1}{2}\left[\bar{\lambda}\text{Tr}[\bar{{\textsf{{E}}}}]^{2}+\bar{\mu}\tilde{\bar{{\textsf{{E}}}}}:\tilde{\bar{{\textsf{{E}}}}}+\bar{K}\nabla^{2}u_{k}\nabla^{2}u_{k}\right]$
(10)

where $\tilde{\bar{{\textsf{{E}}}}}$ is the deviatoric part of the strain
tensor, $\bar{\lambda}=6\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{4}$,
$\bar{\mu}=6\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{4}$ and
$\bar{K}=6\Upsilon{\psi^{0}_{1}}^{2}q_{s}^{2}$.

With the definition of free energy density $f_{{\bf u}_{\perp}}$ and the
strain tensor $E_{ij}$ in hand, we are ready to construct the equations of
motion of the columnar phase in terms of the displacement field about a
periodic, two-dimensional density profile. For this, we will use the following
relations:

$\nabla\psi_{1}\nabla\psi_{1}=2{\psi^{0}_{1}}^{2}\sum_{j}\nabla\phi_{j}\nabla\phi_{j}$
(11)

Then, using the definition of $\phi_{j}$,

$\sum_{j}\nabla_{l}\phi_{j}\nabla_{k}\phi_{j}=q_{s}^{2}\sum_{j}q_{j_{i}}q_{j_{m}}[\delta_{ki}-\nabla_{k}u_{i}][\delta_{ml}-\nabla_{l}u_{m}]=\frac{3q_{s}^{2}}{2}[\delta_{ki}-\nabla_{k}u_{i}][\delta_{li}-\nabla_{l}u_{i}]$
(12)

Using the dynamical equations for $\psi$, we get a dynamical equation for
${\bf u}_{\perp}$:

$\partial_{t}{\bf u}_{\perp}={\bf v}_{\perp}-{\bf v}\cdot\nabla{\bf
u}_{\perp}-\Gamma\frac{\delta F_{{\bf u}}}{\delta{\bf u}_{\perp}}+{{\bf
V}_{a}}_{\perp},$ (13)

where $\Gamma\propto Mq_{s}^{2}$ and ${{\bf V}_{a}}_{\perp}$ is the active
permeation which, for simplicity, we only write to linear order:

${{\bf V}_{a}}_{\perp}=\mu_{1}\nabla^{2}{\bf
u}_{\perp}+\mu_{2}\nabla_{\perp}(\nabla_{\perp}\cdot{\bf u}_{\perp}),$ (14)

with $\mu_{1}=-3q_{s}^{2}\psi_{0}^{2}(\mu_{2}+\mu_{1})$ and
$\mu_{2}=-3q_{s}^{2}\psi_{0}^{2}(\mu_{2}-2\mu_{3})$.

The velocity of the mass-density wave – described by the dynamics of the
displacement field – is not the same as the joint velocity of the fluid and
the mass-density wave. This difference is captured by the active and passive
permeation term ${\delta F_{{\bf u}}}/{\delta{\bf u}_{\perp}}$ and ${\bf
V}_{a\perp}$ in the ${\bf u}_{\perp}$ equation. However, for fluctuations
along most directions of the wavevector space (but not all!), the effect of
these permeative terms will turn out to be subdominant.

Now, we construct the dynamical equation for the Stokesian velocity field.
This has the form

$\eta\nabla^{2}{v}_{i}=\partial_{i}\Pi+(\delta_{ij}-\partial_{i}{u_{\perp}}_{j})\frac{\delta
F_{{\bf u}}}{\delta{u_{\perp}}_{j}}+\nabla\cdot\bm{\sigma}^{a},$ (15)

where $\bm{\sigma}^{a}$ is the active stress and $\Pi$ is the pressure
enforcing the incompressibility constraint. The full, nonlinear active stress
is

$\zeta_{H}\partial_{i}\psi\partial_{j}\psi=3\zeta_{H}{\psi_{1}^{0}}^{2}q_{s}^{2}\begin{pmatrix}(\partial_{x}u_{x}-1)^{2}+(\partial_{x}u_{y})^{2}&\partial_{y}u_{x}(\partial_{x}u_{x}-1)+(\partial_{y}u_{y}-1)\partial_{x}u_{y}&\partial_{z}u_{x}(\partial_{x}u_{x}-1)+\partial_{z}u_{y}\partial_{x}u_{y}\\\
\partial_{y}u_{x}(\partial_{x}u_{x}-1)+\partial_{x}u_{y}(\partial_{y}u_{y}-1)&(\partial_{y}u_{x})^{2}+(\partial_{y}u_{y}-1)^{2}&\partial_{z}u_{x}\partial_{y}u_{x}+\partial_{z}u_{y}(\partial_{y}u_{y}-1)\\\
\partial_{z}u_{x}(\partial_{x}u_{x}-1)+\partial_{z}u_{y}\partial_{x}u_{y}&\partial_{z}u_{x}\partial_{y}u_{x}+\partial_{z}u_{y}(\partial_{y}u_{y}-1)&(\partial_{z}u_{x})^{2}+(\partial_{z}u_{y})^{2}\end{pmatrix}$
(16)

Note that this stress is invariant under arbitrary rotations, by definition.
The linearised force balance equation can, therefore, be explicitly written as

$\eta\nabla^{2}{\bf v}=\nabla\Pi-(\mu+\zeta)\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu+\zeta)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta\partial_{z}^{2}{\bf
u}_{\perp}-\zeta\partial_{z}(\nabla_{\perp}\cdot{\bf u}_{\perp})\hat{{\bf
z}},$ (17)

where $\zeta=-3\zeta_{H}{\psi_{1}^{0}}^{2}q_{s}^{2}$. Further, noting that the
joint velocity field is incompressible and accordingly redefining the
pressure, which is merely a Lagrange multiplier to enforce the
incompressibility constraint, the active force is seen to be simply
$\zeta\nabla^{2}{\bf u}_{\perp}$ i.e.,

$\eta\nabla^{2}{\bf v}=\nabla\Pi-\mu\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta\nabla^{2}{\bf u}_{\perp}.$ (18)

However, this is not the most general hydrodynamic theory of an active
columnar system. In particular, the coefficient of the active stress can be
anisotropic since the columnar state itself is anisotropic. The easiest way to
account for this anisotropy is to retain an explicit director field $\hat{n}$
and start with the columnar and active variant of the Chen-Lubensky model
[49]. For this, we replace the Swift-Hohenberg free energy with $F=\int_{\bf
r}[f_{\psi_{0}}+f_{n}+f_{\psi_{1},n}]$, where $f_{n}$ is the free energy
density for the director field, and

$f_{\psi_{1},n}=\frac{\Upsilon}{2}\left[\frac{\alpha}{2}\psi_{1}^{2}+\frac{\beta}{4}\psi_{1}^{4}+C_{\parallel}(\hat{n}\cdot\nabla\psi_{1})^{2}+\left|\\{(\nabla-\hat{n}\hat{n}\cdot\nabla)^{2}+q^{2}_{s}\\}\psi_{1}\right|^{2}\right].$
(19)

Taking the mean director direction to be $\hat{n}=\hat{{\bf z}}$ and using the
same $\psi_{1}$ as earlier, we obtain a free energy density containing both
${\bf u}_{\perp}$ and $\hat{n}$

$f_{{\bf u}_{\perp},\delta{\bf
n}_{\perp}}=\frac{1}{2}\left[\lambda\text{Tr}[{\textsf{{E}}}]^{2}+2\mu{\textsf{{E}}}:{\textsf{{E}}}+K\nabla^{2}u_{k}\nabla^{2}u_{k}+C(\delta{n}_{\perp}-\partial_{z}{\bf
u}_{\perp})^{2}\right]$ (20)

where $C=3\Upsilon C_{\parallel}{\psi^{0}_{1}}^{2}q_{s}^{2}$. This free energy
implies that the _fluctuations_ of the director are slaved to that of the
displacement field. That is, the director field is not a hydrodynamic quantity
and relaxes fast to

$\hat{n}=\frac{(\partial_{z}{\bf u}_{\perp},1)}{\sqrt{1+\partial_{z}{\bf
u}_{\perp}\cdot\partial_{z}{\bf u}_{\perp}}}\approx(\partial_{z}{\bf
u}_{\perp},1)$ (21)

where the approximate equality ignores higher order nonlinear contributions.
We can integrate out the director fluctuations from the free energy to obtain
an expression purely in terms of the displacement which has the form (7).

In general, the active stress also depends on $\hat{n}$; i.e., it is
anisotropic. This can be seen by, for instance, writing the stress as

$\sigma^{a}_{ij}=\zeta_{ijkl}\partial_{k}\psi\partial_{l}\psi$ (22)

where

$\zeta_{ijkl}=\zeta_{1}n_{i}n_{j}n_{k}n_{l}+\zeta_{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+\zeta_{3}(n_{i}n_{k}\delta_{jl}+n_{j}n_{k}\delta_{il}+n_{i}n_{l}\delta_{jk}+n_{j}n_{l}\delta_{ik})+\zeta_{4}\delta_{ij}\delta_{kl}+\zeta_{5}(\delta_{ij}n_{k}n_{l}+n_{i}n_{j}\delta_{kl}).$
(23)

To calculate the linearised active force density, we first obtain the
$\sigma_{ij}$ using (22), (23), (21) and (16) and expand to first order in the
displacement fields. The divergence of this yields the achiral active force
density as

${\bf f}^{a}=\zeta_{2}\nabla^{2}{\bf u}_{\perp}-2\zeta_{5}\partial_{z}^{2}{\bf
u}_{\perp},$ (24)

where we have dropped a term $\propto\nabla\nabla_{\perp}\cdot{\bf u}_{\perp}$
since it can be absorbed in a redefinition of the pressure. Defining
$\zeta_{\perp}=\zeta_{2}$ and $\zeta_{\parallel}=(\zeta_{2}-2\zeta_{5})$, the
linearised force balance equation, after eliminating terms that can be
absorbed into a redefinition of the pressure, becomes

$\eta\nabla^{2}{\bf v}=\nabla\Pi-(\mu+\zeta_{\perp})\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta_{\parallel}\partial_{z}^{2}{\bf u}_{\perp}.$ (25)

The careful reader may notice that we could have introduced another active
stress $\propto n_{i}n_{j}$ which would lead to a force density
$\partial_{z}^{2}{\bf u}_{\perp}+\hat{{\bf
z}}\partial_{z}\nabla_{\perp}\cdot{\bf u}_{\perp}$. Using incompressibility,
to redefine the pressure, we see that this force density renormalises the
coefficient of $\nabla_{\perp}\nabla_{\perp}\cdot{\bf u}_{\perp}$ and
$\partial_{z}^{2}{\bf u}_{\perp}$; i.e., it also leads to an active
modification of the bulk modulus. In general, in an incompressible columnar
system, the _only_ active force-density that cannot be reabsorbed by an
activity-dependent renormalisation of the shear and the bulk moduli, at this
order in gradients, is $\partial_{z}^{2}{\bf u}_{\perp}$. In compressible
materials, one would have a further purely active contribution to the force
density $\propto\partial_{z}\nabla_{\perp}\cdot{\bf u}_{\perp}$.

Similarly, the active permeation terms can also be anisotropic. However, as we
shall show, they become important only along one specific direction.
Therefore, we do not examine their anisotropy here.

###  A.2 Apolar and chiral active columnar phase

The dynamical equations for $\psi$ and the velocity field for a _chiral_
columnar phase have the following additional _chiral_ terms: i. Eq. (1) is
supplemented by the divergence of a _passive_ chiral current $\nabla\cdot{\bf
J}_{c}$ and ii. Eq. (3) is supplement both by a passive chiral stress which is
the Onsager counterpart to ${\bf J}_{c}$ [14, 50] and an active chiral stress
$\bm{\sigma}_{c}$. The chiral current is

${\bf J}_{c}=\Omega_{v}\psi\nabla^{2}(\nabla\times{\bf v}).$ (26)

This term distinguishes the dynamics of _passive_ achiral and chiral columnar
phases, which are indistinguishable at the hydrodynamic level, at higher order
in wavenumbers. However, this term yields velocity couplings
$\propto\nabla^{2}(\partial_{z}v_{y},-\partial_{z}v_{x})$ in the displacement
field dynamics which do not affect the long-time, large-scale dynamics.
Therefore, we will ignore both this term and its Onsager (anti)symmetric
counterpart in the force balance equation. The chiral active stress in (3) has
the form

$\sigma^{ac}_{ij}=\bar{z}_{c}\partial_{l}(\epsilon_{ijk}\partial_{k}\psi\partial_{l}\psi)$
(27)

where $\epsilon_{ijk}$ is the _three-dimensional_ Levi-Civita tensor. The
chiral active force-density, with a coefficient $\bar{z}_{c}$, appears at a
_higher_ order in gradients than the achiral active force. This force has a
form similar to the one used in theories of three-dimensional chiral nematic
liquid crystals [51, 34, 33, 35] and the velocity field resulting from it can
be shown to be divergence-free. Note that though this stress superficially
seems to be antisymmetric, it is allowed in angular momentum-conserved
systems; in fact, an equivalent (up to a Belinfante-Rosenfeld tensor)
explicitly _symmetric_ stress yielding the same velocity field can be
constructed [52, 35]:

$\bar{z}_{c}\partial_{l}(\epsilon_{ijk}\partial_{k}\psi\partial_{l}\psi)\equiv\bar{z}_{c}[\epsilon_{ilk}\partial_{l}(\partial_{k}\psi\partial_{j}\psi)+\epsilon_{jlk}\partial_{l}(\partial_{k}\psi\partial_{i}\psi)].$
(28)

In terms of the displacement field, the linearised chiral active stress
becomes

$\bar{z}_{c}\epsilon_{ijk}\partial_{l}(\partial_{l}\psi\partial_{k}\psi)=-3\bar{z}_{c}\psi_{0}^{2}q_{s}^{2}\begin{pmatrix}0&\partial_{z}\nabla_{\perp}\cdot{\bf
u}_{\perp}&-(\partial_{z}^{2}u_{y}+\nabla_{\perp}^{2}u_{y}+\partial_{y}\nabla_{\perp}\cdot{\bf
u}_{\perp})\\\ -\partial_{z}\nabla_{\perp}\cdot{\bf
u}_{\perp}&0&\partial_{z}^{2}u_{x}+\nabla_{\perp}^{2}u_{x}+\partial_{x}\nabla_{\perp}\cdot{\bf
u}_{\perp}\\\
(\partial_{z}^{2}u_{y}+\nabla_{\perp}^{2}u_{y}+\partial_{y}\nabla_{\perp}\cdot{\bf
u}_{\perp})&-(\partial_{z}^{2}u_{x}+\nabla_{\perp}^{2}u_{x}+\partial_{x}\nabla_{\perp}\cdot{\bf
u}_{\perp})&0\end{pmatrix}.$ (29)

Defining $\zeta_{c}=-3\bar{z}_{c}{\psi^{0}_{1}}^{2}q_{s}^{2}$, this implies
that the force balance equation for a chiral, apolar columnar material is

$\eta\nabla^{2}{\bf v}=\nabla\Pi-(\mu+\zeta)\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\mu+\lambda+\zeta)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta\partial_{z}^{2}{\bf
u}_{\perp}-\zeta\partial_{z}(\nabla_{\perp}\cdot{\bf
u}_{\perp})-\zeta_{c}\nabla^{2}\partial_{z}\bm{\epsilon}\cdot{\bf
u}_{\perp}+\zeta_{c}\nabla^{2}\nabla_{\perp}\times{\bf u}_{\perp},$ (30)

where $\bm{\epsilon}$ is the _rank-two_ Levi-Civita tensor. More compactly,

$\eta\nabla^{2}{\bf v}=\nabla\Pi-\mu\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\mu+\lambda)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta\nabla^{2}{\bf u}_{\perp}+\zeta_{c}\nabla^{2}\nabla\times{\bf
u}_{\perp}.$ (31)

Of course, as in the last section, the chiral active stress may also be
anisotropic. One (non-unique) version of this is obtained by writing

$\sigma^{ac}_{ij}=({{\bar{z}_{c}}})_{ijkl}[\epsilon_{kmn}\partial_{m}(\partial_{n}\psi\partial_{l}\psi)+\epsilon_{lmn}\partial_{m}(\partial_{n}\psi\partial_{k}\psi)],$
(32)

where we have used an explicitly symmetrised version of the chiral stress and

$-3{\psi^{0}_{1}}^{2}q_{s}^{2}({{\bar{z}_{c}}})_{ijkl}={\zeta_{c}}_{1}n_{i}n_{j}n_{k}n_{l}+{\zeta_{c}}_{2}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})+{\zeta_{c}}_{3}(n_{i}n_{k}\delta_{jl}+n_{j}n_{k}\delta_{il}+n_{i}n_{l}\delta_{jk}+n_{j}n_{l}\delta_{ik})+{\zeta_{c}}_{4}\delta_{ij}\delta_{kl}+{\zeta_{c}}_{5}(\delta_{ij}n_{k}n_{l}+n_{i}n_{j}\delta_{kl}).$
(33)

Again taking $\hat{n}\approx(\partial_{z}{\bf u}_{\perp},1)$, we get the
anisotropic force balance equation for a chiral columnar fluid:

$\eta\nabla^{2}{\bf v}=\nabla\Pi-(\mu+\zeta_{\perp})\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta_{\parallel}\partial_{z}^{2}{\bf
u}_{\perp}-[\zeta^{c}_{1}\partial_{z}^{2}+\zeta^{c}_{2}\nabla_{\perp}^{2}][\partial_{z}(\bm{\epsilon}\cdot{\bf
u}_{\perp})]+[\zeta^{c}_{3}\partial_{z}^{2}+\zeta^{c}_{4}\nabla_{\perp}^{2})](\nabla_{\perp}\times{\bf
u}_{\perp}),$ (34)

where we have defined $\zeta^{c}_{1}=(2\zeta_{c2}+\zeta_{c3})$,
$\zeta^{c}_{2}=2\zeta_{c2}$,
$\zeta^{c}_{3}=(4\zeta_{c1}+2\zeta_{c2}+2\zeta_{c3})$ and
$\zeta^{c}_{4}=(2\zeta_{c2}+\zeta_{c3})$.

###  A.3 Polar and chiral active columnar phase

In this section, we will construct the hydrodynamic equations for an active,
polar and chiral columnar material. It is tempting to obtain the equations of
motion of a chiral and polar columnar phase starting from a variant of model
$H$ which lacks $\psi\to-\psi$ symmetry. However, this is not possible since
the polar asymmetry is along the fluid direction. We will obtain the dynamics
by starting with a model analogous to the one which we used to obtain the
anisotropic apolar columnar dynamics in Sec. A.1 but with a polar vector ${\bf
P}$ instead of a director. The free energy is $F=\int_{\bf
r}[f_{\psi_{0}}+f_{P}+f_{\psi_{1},P}]$, where $F_{P}$ is the Landau-de Gennes
free energy density for the polar order parameter, and

$f_{\psi_{1},P}=\frac{\Upsilon}{2}\left[\frac{\alpha}{2}\psi_{1}^{2}+\frac{\beta}{4}\psi_{1}^{4}+C_{\parallel}(\hat{P}\cdot\nabla\psi_{1})^{2}+\left|\\{(\nabla-\hat{P}\hat{P}\cdot\nabla)^{2}+q^{2}_{s}\\}\psi_{1}\right|^{2}\right].$
(35)

The density wave state is assumed to arise in this case in a phase with a mean
${\bf P}$. Taking the mean ${\bf P}=P_{0}\hat{{\bf z}}$ and assuming a two-
dimensional density wave state as in Sec. A.1, we get from $f_{\psi_{1},P}$

$f_{{\bf u}_{\perp},\delta{\bf
P}_{\perp}}=\frac{1}{2}\left[\lambda\text{Tr}[{\textsf{{E}}}]^{2}+2\mu{\textsf{{E}}}:{\textsf{{E}}}+K\nabla^{2}u_{k}\nabla^{2}u_{k}+C(\delta{\bf
P}_{\perp}-\partial_{z}{\bf u}_{\perp})^{2}\right]$ (36)

where $C=3\Upsilon C_{\parallel}{\psi^{0}_{1}}^{2}q_{s}^{2}$. This free energy
implies that the fluctuations of the polarisation vector are slaved to that of
the displacement field. Therefore, in the following, we construct the
dynamical equations of active and chiral polar columnar phases in terms of the
displacement field ${\bf u}_{\perp}$ and the velocity field ${\bf v}$ just as
in the earlier sections. The polarisation vector is assumed to relax fast to
${\bf P}=P_{0}(\partial_{z}{\bf u}_{\perp},1)$. Note that the free energy of
polar columnar liquid crystals is exactly equivalent to their apolar
counterparts.

We now construct an active stress that is both _polar_ and _chiral_ ,

$\sigma^{pc}_{ij}=\bar{\zeta}_{pc}[\epsilon_{ikl}P_{l}\partial_{k}\psi\partial_{j}\psi+\epsilon_{jkl}P_{l}\partial_{k}\psi\partial_{i}\psi],$
(37)

which is symmetric by construction. Defining
$\zeta_{pc}=-3q_{s}^{2}{\psi^{0}_{1}}^{2}P_{0}\bar{\zeta}_{pc}$, we obtain the
simplest chiral and polar active stress contribution in columnar materials in
terms of the displacement fields from this; the linear part of this stress is

$\left[\bm{\sigma}^{pc}\right]_{lin}=\zeta_{pc}\begin{pmatrix}\partial_{x}u_{y}+\partial_{y}u_{x}&\partial_{y}u_{y}-\partial_{x}u_{x}&0\\\
\partial_{y}u_{y}-\partial_{x}u_{x}&-(\partial_{x}u_{y}+\partial_{y}u_{x})&0\\\
0&0&0\end{pmatrix}.$ (38)

This _linearised_ stress is _exactly_ equivalent to the odd elastic stress of
momentum-conserved two-dimensional chiral solids [12, 13]. The difference
between the polar, chiral stress in columnar materials and the odd elastic
stress in two-dimensional chiral solids appears only at the nonlinear level;
these are irrelevant for large-scale, long-time dynamics of the system and,
therefore, we do not explicitly display them. The stress $\bm{\sigma}^{pc}$
has this form because of a combination of rotation invariance and momentum
conservation. It has to be constructed out of the strain tensor, the three-
dimensional Levi-Civita tensor and the polarisation. Since the strain tensor
in a columnar system is two-dimensional, the only (linearised) odd elastic
stress that one can construct is equivalent to that in [12, 13].

The general force balance equation for a polar and chiral active columnar
system is

$\eta\nabla^{2}{\bf v}=\nabla\Pi-(\mu+\zeta_{\perp})\nabla_{\perp}^{2}{\bf
u}_{\perp}-(\lambda+\mu)\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\zeta_{\parallel}\partial_{z}^{2}{\bf
u}_{\perp}-\zeta_{pc}\nabla_{\perp}^{2}\bm{\epsilon}\cdot{\bf u}_{\perp},$
(39)

where we have discarded higher order in gradient contributions.

##  Appendix B Active columnars: Effect of an isotropic stress

In this section, we demonstrate that the effect of the active achiral and
apolar stress in an incompressible columnar system can be _exactly_
compensated by an external stress. To show this, we note that the nonlinear
strain tensor in a columnar is

${\textsf{{E}}}=\frac{1}{2}\begin{pmatrix}2\partial_{x}u_{x}-(\partial_{x}u_{x})^{2}-(\partial_{x}u_{y})^{2}-(\partial_{z}u_{x})^{2}&\partial_{x}u_{y}+\partial_{y}u_{x}-\partial_{x}u_{x}\partial_{y}u_{x}-\partial_{x}u_{y}\partial_{y}u_{y}-\partial_{z}u_{x}\partial_{z}u_{y}\\\
\partial_{x}u_{y}+\partial_{y}u_{x}-\partial_{x}u_{x}\partial_{y}u_{x}-\partial_{x}u_{y}\partial_{y}u_{y}-\partial_{z}u_{x}\partial_{z}u_{y}&2\partial_{y}u_{y}-(\partial_{y}u_{x})^{2}-(\partial_{y}u_{y})^{2}-(\partial_{z}u_{y})^{2}\end{pmatrix}$
(40)

and the free energy without an external stress is

$F=\frac{1}{2}\int[\lambda E_{ii}^{2}+2\mu E_{ij}E_{ij}].$ (41)

A purely compressive stress leads to the extra free energy term

$F_{\text{ext}}=\int\sigma_{0}E_{ii}=\int\sigma_{0}E$ (42)

where we have denoted $\text{Tr}[{\textsf{{E}}}]$ by

$E=\nabla_{l}u_{l}-\frac{1}{2}\nabla_{m}u_{l}\nabla_{m}u_{l}.$ (43)

(Note that exactly the same calculation (with the same result) can also be
performed in terms of the strain tensor $\bar{\textsf{{E}}}$.) The force due
to this external stress is

${\bf f}_{ext}=-(\delta_{ij}-\nabla_{i}u_{j})\frac{\delta
F_{\text{ext}}}{\delta
u_{j}}=\sigma_{0}(\delta_{ij}-\nabla_{i}u_{j})\nabla_{k}\frac{\partial
E}{\partial\nabla_{k}u_{j}}$ (44)

Using the definition of $E$,

$\frac{\partial E}{\partial\nabla_{k}u_{j}}=\delta_{kj}-\nabla_{k}u_{j},$ (45)

which implies

$(\delta_{ij}-\nabla_{i}u_{j})\nabla_{k}\frac{\partial
E}{\partial\nabla_{k}u_{j}}=\nabla_{k}[(\delta_{ij}-\nabla_{i}u_{j})(\delta_{kj}-\nabla_{k}u_{j})]-(\delta_{kj}-\nabla_{k}u_{j})\nabla_{k}(\delta_{ij}-\nabla_{i}u_{j})\\\
=\nabla_{k}\left[\frac{\partial E}{\partial\nabla_{i}u_{j}}\frac{\partial
E}{\partial\nabla_{k}u_{j}}\right]-(\delta_{kj}-\nabla_{k}u_{j})\nabla_{k}(\delta_{ij}-\nabla_{i}u_{j})$
(46)
$(\delta_{kj}-\nabla_{k}u_{j})\nabla_{k}(\delta_{ij}-\nabla_{i}u_{j})=-[\nabla_{i}\nabla_{j}u_{j}-(\nabla_{k}u_{j})\nabla_{i}(\nabla_{k}u_{j})]=-\nabla_{i}\left[\nabla_{j}u_{j}-\nabla_{k}u_{j}\nabla_{k}u_{j}\right]=-\nabla
E$ (47)

This implies that

${\bf f}_{ext}=\sigma_{0}\nabla_{k}\left[\frac{\partial
E}{\partial\nabla_{i}u_{j}}\frac{\partial
E}{\partial\nabla_{k}u_{j}}+E\delta_{ik}\right]$ (48)

The active, achiral stress is

$\bm{\sigma}^{a}\propto\partial_{k}\psi\partial_{l}\psi=3q_{s}^{2}{\psi^{0}_{1}}^{2}[\delta_{ki}-\nabla_{k}u_{i}][\delta_{li}-\nabla_{l}u_{i}].$
(49)

This implies that

$\bm{\sigma}^{a}=\zeta_{H}\partial_{k}\psi\partial_{l}\psi=3\zeta_{H}q_{s}^{2}{\psi^{0}_{1}}^{2}\left[\frac{\partial
E}{\partial\nabla_{k}u_{m}}\frac{\partial E}{\partial\nabla_{l}u_{m}}\right].$
(50)

That is, apart from a purely isotropic part which can be absorbed in a
redefinition of the pressure, the active stress in a columnar system can be
fully compensated by imposing an external stress of strength
$\sigma_{0}=-3\zeta_{H}q_{s}^{2}{\psi^{0}_{1}}^{2}=\zeta$.

##  Appendix C Active Helfrich-Hurault instability

In this section, we use the mapping discussed in the last section between an
active stress and an externally-imposed stress, to examine the active
Helfrich-Hurault instability of apolar columnar liquid crystals. For this, we
start with the well-known _passive_ Helfrich-Hurault under an external stress.
The rotation-invariant free energy for a columnar liquid crystal is

$F=\int\frac{\lambda}{2}\biggl{(}\nabla\cdot{\bf
u}-\frac{1}{2}\bigl{|}\nabla{\bf
u}\bigr{|}^{2}\biggr{)}^{2}+\frac{K}{2}\bigl{|}\nabla^{2}{\bf
u}\bigr{|}^{2}\,dV.$ (51)

As shown in the previous section, an active stress is equivalent to imposing
an isotropic dilatational stress in a passive columnar phase. Such a
dilatational stress corresponds leads to an Eulerian displacement field ${\bf
u}=[\alpha x,\alpha y]$ where $\alpha$ is some positive constant. For this
displacement field, the isotropic part of the strain field is

$\nabla\cdot{\bf u}-\frac{1}{2}\bigl{|}\nabla{\bf
u}\bigr{|}^{2}=2\alpha-\alpha^{2},$ (52)

and the curvature is zero $\nabla^{2}{\bf u}=0$. We consider a helical
modulation of the columns about this strained state with the displacement
field

${\bf u}=\alpha x\,{\bf e}_{x}+\alpha y\,{\bf e}_{y}+u_{0}(x,y)\bigl{[}\cos
qz\,{\bf e}_{x}+\sin qz\,{\bf e}_{y}\bigr{]}.$ (53)

The strain and curvature are

$\displaystyle\nabla\cdot{\bf u}-\frac{1}{2}\bigl{|}\nabla{\bf
u}\bigr{|}^{2}=2\alpha-\alpha^{2}+(1-\alpha)\bigl{[}\cos
qz\,\partial_{x}u_{0}+\sin
qz\,\partial_{y}u_{0}\bigr{]}-\frac{1}{2}\bigl{|}\nabla
u_{0}\bigr{|}^{2}-\frac{1}{2}q^{2}u_{0}^{2},$ (54)
$\displaystyle\nabla^{2}{\bf
u}=\Bigl{(}\nabla^{2}u_{0}-q^{2}u_{0}\Bigr{)}\bigl{[}\cos qz\,{\bf e}_{x}+\sin
qz\,{\bf e}_{y}\bigr{]}.$ (55)

Substituting into the free energy and gathering terms quadratic in $u_{0}$ we
find

$F_{2}=\frac{\lambda}{2}\int\biggl{\\{}-\bigl{(}2\alpha-\alpha^{2}\bigr{)}\Bigl{[}\bigl{|}\nabla
u_{0}\bigr{|}^{2}+q^{2}u_{0}^{2}\Bigr{]}+\frac{1}{2}(1-\alpha)^{2}\bigl{|}\nabla
u_{0}\bigr{|}^{2}+\gamma^{2}\Bigl{(}\nabla^{2}u_{0}-q^{2}u_{0}\Bigr{)}^{2}\biggr{\\}}dV,$
(56)

where $\gamma=\sqrt{K/\lambda}$ is the ‘penetration depth’.

We assume $u_{0}(x,y)=u_{0}\cos(\pi x/L)\cos(\pi y/L)$ in the domain
$[-L/2,L/2]^{2}\times H$ with a square cross-section and Dirichlet boundary
conditions in the plane. With this choice, the quadratic free energy is

$\begin{split}F_{2}&=\frac{\lambda
L^{2}Hu_{0}^{2}}{8}\biggl{\\{}-\bigl{(}2\alpha-\alpha^{2}\bigr{)}\Bigl{(}2\bigl{(}\pi^{2}/L^{2}\bigr{)}+q^{2}\Bigr{)}+(1-\alpha)^{2}\bigl{(}\pi^{2}/L^{2}\bigr{)}+\gamma^{2}\Bigl{(}2\bigl{(}\pi^{2}/L^{2}\bigr{)}+q^{2}\Bigr{)}^{2}\biggr{\\}},\\\
&=\frac{\lambda
L^{2}Hu_{0}^{2}}{8}\biggl{\\{}\gamma^{2}\biggl{(}2\bigl{(}\pi^{2}/L^{2}\bigr{)}+q^{2}-\frac{\alpha-\frac{1}{2}\alpha^{2}}{\gamma^{2}}\biggr{)}^{2}-\biggl{(}\frac{\alpha-\frac{1}{2}\alpha^{2}}{\gamma}\biggr{)}^{2}+(1-\alpha)^{2}\bigl{(}\pi^{2}/L^{2}\bigr{)}\biggr{\\}},\end{split}$
(57)

The instability threshold is obtained by examining when this quadratic energy
first vanishes; this yields both the threshold strain for the _passive_
Helfrich-Hurault instability and the selected length scale of the helical
modulation just beyond the instability:

$\displaystyle\alpha_{\mathrm{th}}$
$\displaystyle=1+\frac{\gamma\pi}{L}-\biggl{[}1+\frac{\gamma^{2}\pi^{2}}{L^{2}}\biggr{]}^{1/2}\approx\frac{\gamma\pi}{L},$
(58) $\displaystyle q_{\mathrm{th}}^{2}$
$\displaystyle=\frac{\alpha-\frac{1}{2}\alpha^{2}}{\gamma^{2}}-\frac{2\pi^{2}}{L^{2}}\approx\frac{\pi}{\gamma
L}.$ (59)

We now use the stress-strain relation to convert the threshold strain
$\alpha_{\mathrm{th}}$ to a threshold stress which must be applied on a
passive columnar liquid crystal for it suffer a Helfrich-Hurault instability:

$\sigma_{0_{th}}=\dfrac{\sqrt{K\lambda}\pi}{L}.$ (60)

Given the mapping discussed in the last section, this is also the threshold
_active_ stress $\zeta_{th}=-{\sqrt{K\lambda}\pi}/{L}$ at which an _active_
columnar liquid crystal undergoes a spontaneous Helfrich-Hurault instability.

##  Appendix D Linear dynamics of columnar systems

In this section, we will discuss the linearised dynamics of active columnar
systems. To construct the linear equations, we decompose the displacement
fields into in-plane longitudinal and transverse displacement fields
$u_{l}=(q_{x}u_{x}+q_{y}u_{y})/\sqrt{q_{x}^{2}+q_{y}^{2}}$ and
$u_{t}=(q_{x}u_{y}-q_{y}u_{x})/\sqrt{q_{x}^{2}+q_{y}^{2}}$. Note that in a
_pure_ two-dimensional poroelastic solid, two-dimensional incompressibility
would imply that $\partial_{t}u_{l}\sim\mathcal{O}(q^{2})$. However, since in
this columnar system, the incompressibility constraint is three-dimensional
while the displacement field is two-dimensional, $u_{l}$ has a
$\mathcal{O}(q^{0})$ relaxation rate except for _purely_ in-plane
perturbations. We also write the wavevectors as $q_{x}=q\cos\phi\sin\theta$,
$q_{y}=q\sin\phi\sin\theta$ and $q_{z}=q\cos\theta$. The white noises in the
displacement dynamics appear from two sources i. from the velocity coupling
and ii. the noise directly in the displacement field equation. Assuming the
noises are such that they would lead to an equilibrium dynamics in the absence
of active forces, the noise correlators for the longitudinal and transverse
displacement fields are, respectively,

$\langle\xi_{l}({\bf q},\omega)\xi_{l}({\bf
q}^{\prime},\omega^{\prime})\rangle=2T\left[\frac{\cos^{2}\theta}{\eta
q^{2}}+\Gamma\right]\delta({\bf q}+{\bf
q}^{\prime})\delta(\omega+\omega^{\prime}),$ (61) $\langle\xi_{t}({\bf
q},\omega)\xi_{t}({\bf
q}^{\prime},\omega^{\prime})\rangle=2T\left[\frac{1}{\eta
q^{2}}+\Gamma\right]\delta({\bf q}+{\bf
q}^{\prime})\delta(\omega+\omega^{\prime})$ (62)

and

$\langle\xi_{l}({\bf q},\omega)\xi_{t}({\bf
q}^{\prime},\omega^{\prime})\rangle=0.$ (63)

###  D.1 Achiral and apolar active columnars

The linearised dynamics of the displacement fields, in this case, turn out to
be

$\partial_{t}u_{l}=-\cos^{2}\theta\left[\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{l}+\xi_{l}$
(64)

and

$\partial_{t}u_{t}=-\left[\frac{\mu}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{t}+\xi_{t}.$
(65)

The longitudinal and the transverse displacement fluctuations are decoupled.
Note that, as advertised, the $\mathcal{O}(q^{0})$ relaxation rate in (64)
vanishes for a fluctuation with wavevector with $\theta=0$, i.e., for a
fluctuation purely in the crystalline plane. The relaxation rate of
longitudinal displacement fluctuations for fluctuations with wavectors with
$\theta=\pi/2$ will therefore be $\mathcal{O}(q^{2})$ and be given by the
permeative terms. From Sec. A.1, for a wavevector with $\theta=\pi/2$, the
relaxation rate of $u_{l}$ is, therefore,
$-[(2\mu+\lambda)\Gamma+2\mu_{1}+\mu_{2}]q^{2}$. For stability, therefore, we
also require $[(2\mu+\lambda)\Gamma+2\mu_{1}+\mu_{2}]>0$ in addition to
$\zeta_{\parallel}>0$ and $\zeta_{\perp}>-\mu$.

When the signs of the active coefficients are such that the columnar phase is
stable, the equal-time correlators of the displacement fluctuations are

$\langle u_{l}({\bf q},t)u_{l}({\bf q}^{\prime},t)\rangle=\frac{T\delta({\bf
q}+{\bf
q}^{\prime})}{q^{2}[(2\mu+\lambda+\zeta_{\perp})\sin^{2}\theta+\zeta_{\parallel}\cos^{2}\theta]},$
(66)

and

$\langle u_{t}({\bf q},t)u_{t}({\bf q}^{\prime},t)\rangle=\frac{T\delta({\bf
q}+{\bf
q}^{\prime})}{q^{2}[(\mu+\zeta_{\perp})\sin^{2}\theta+\zeta_{\parallel}\cos^{2}\theta]}.$
(67)

The displacement fluctuations thus diverge as $1/q^{2}$ along _all_ directions
of the wavevector space unlike in passive columnar liquid crystals (i.e. in
the absence of $\zeta$s) where displacement fluctuations for wavevectors with
$\theta=0$ diverges as $1/q^{4}$. This further implies that the linear
roughness exponent for displacement fluctuations is $\chi=-1/2$ (for a three-
dimensional columnar phase). Since the fluctuations scale as $1/q^{2}$ for all
directions of the wavevector space, the linear anisotropy exponent is $1$.
Finally, since the relaxation rate of displacement fluctuations is
$\mathcal{O}(q^{0})$ (as a result of Stokesian hydrodynamics), the linear
dynamical exponent is $z=0$. This last implies that the hydrodynamic behaviour
of active columnar systems in three dimensions are completely described by the
linear theory and the linear exponents discussed here are _exact_. That is,
there is no relevant nonlinearity. To see this, note that any nonlinear term
must at least be of the order $\mathcal{O}(q^{0}u^{2})$ which would have a
scaling dimension of $z+\chi$. Since, within the linear theory, $z=0$ and
$\chi<0$, all such nonlinear terms would be irrelevant. Furthermore, active
_suppression_ of displacement fluctuations which render all nonlinearities
irrelevant also cuts-off the low-frequency viscosity divergences known to
affect _passive_ columnar liquid crystals [21]; these now go to a constant
value at small frequencies. Since the displacement fluctuations in active
columnar and lamellar materials have the same low-frequency, small wavenumber
scaling, viscosities are non-divergent at small frequencies even in those
systems.

###  D.2 Chiral apolar active columnars

Chirality couples longitudinal and transverse displacement fluctuations at
$\mathcal{O}(q)$:

$\partial_{t}u_{l}=-\cos^{2}\theta\left[\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{l}\\\
-\frac{iq\cos\theta}{\eta}\left[\zeta^{c}_{1}\cos^{2}\theta(1+\sin^{2}\theta)+\zeta^{c}_{4}\sin^{2}\theta(1+\cos^{2}\theta)-(\zeta^{c}_{2}+\zeta^{c}_{3})\cos^{2}\theta\sin^{2}\theta\right]u_{t}+\xi_{l}$
(68)

and

$\partial_{t}u_{t}=-\left[\frac{\mu}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{t}+\frac{iq\cos\theta}{\eta}(\zeta^{c}_{1}\cos^{2}\theta+\zeta^{c}_{2}\sin^{2}\theta)u_{l}+\xi_{t}.$
(69)

These expressions are fairly complicated and difficult to read. However,
taking a one activity constant approximation, the dynamics become
significantly simpler.

$\partial_{t}u_{l}=-\cos^{2}\theta\left[\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right]u_{l}-\frac{i\zeta_{c}q\cos\theta}{\eta}u_{t}+\xi_{l}$
(70)

and

$\partial_{t}u_{t}=-\left[\frac{\mu}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right]u_{t}+\frac{iq\zeta_{c}\cos\theta}{\eta}u_{l}+\xi_{t}.$
(71)

The chiral terms couple the longitudinal and transverse displacement
fluctuations. Note that, despite appearances, the $\zeta_{c}$ term here is not
like a Poisson-bracket (i.e. reactive) coupling but a dissipative one. The
eigenfrequencies are

$\omega=-\frac{i}{2\eta}\left[[\zeta(1+\cos^{2}\theta)+\sin^{2}\theta\\{(2\mu+\lambda)\cos^{2}\theta+\mu\\}]\pm\sqrt{[\zeta(1-\cos^{2}\theta)-\sin^{2}\theta\\{(2\mu+\lambda)\cos^{2}\theta-\mu\\}]^{2}+4\zeta_{c}^{2}q^{2}\cos^{2}\theta}\right].$
(72)

The effect of the chiral active stress is most prominent for a wavevector
purely along the column direction i.e., when $\theta=0$. In this case, the
eigenfrequency becomes

$\omega=-\frac{i}{\eta}[\zeta\pm\zeta_{c}q].$ (73)

That is, chirality yields higher order in wavenumber corrections to growth and
decay rates of displacement fluctuations.

###  D.3 Chiral and polar active columnars

Using the dynamical equations obtained in Sec. A.3, we get the linearised
equations for the dynamics of longitudinal and transverse displacement fields
of chiral and polar active columnar liquid crystals:

$\partial_{t}u_{l}=-\cos^{2}\theta\left[\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{l}-\cos^{2}\theta\frac{\zeta_{pc}\sin^{2}\theta}{\eta}u_{t}+\xi_{l}$
(74)

and

$\partial_{t}u_{t}=-\left[\frac{\mu}{\eta}\sin^{2}\theta+\frac{\zeta_{\perp}}{\eta}\sin^{2}\theta+\frac{\zeta_{\parallel}}{\eta}\cos^{2}\theta\right]u_{t}+\frac{\zeta_{pc}\sin^{2}\theta}{\eta}u_{l}+\xi_{t}.$
(75)

Again, the chiral polar term couples longitudinal and transverse fluctuations.
Notice that $\zeta_{pc}$ looks almost like a _reactive_ Onsager coupling
(except for the $\cos^{2}\theta$ required by incompressibility in the $u_{l}$
equation) between $u_{l}$ and $u_{t}$. This excitatory-inhibitory coupling in
_non-reciprocal_ : Poisson bracket or antisymmetric couplings relate
quantities that have _opposite_ signs under time-reversal in detailed balance
obeying dynamics. Here, they relate two displacement fields, which obviously
have the same sign under time-reversal.

The odd-elastic coupling vanishes for fluctuations transverse to the
crystalline directions. While, as we discussed in Sec. A.3, the odd elastic
stress in polar columnar materials is analogous to that in two-dimensional
chiral active solids, we will now demonstrate that the mode structure of polar
columnar liquid crystals is very different from a two-dimensional active
solid. This is because the columnar is a _three-dimensional_ material with a
three-dimensional incompressibility constraint. In two-dimensional poroelastic
solids (with or without odd elasticity), the relaxation rate of $u_{l}$
fluctuations scale as $\sim q^{2}$ for all wavevector directions due to the
two-dimensional incompressibility constraint (these would appear from
permeative dynamics of poroelastic solid). Further, any coupling to $u_{t}$
would also appear at $\mathcal{O}(q^{2})$.

In contrast, in columnar materials, for wavevectors which have components both
in the crystalline plane and along the $\hat{{\bf z}}$ directions, i.e., for
$\theta\neq 0$ and $\theta\neq\pi/2$, odd-elasticity leads to a wavenumber-
independent excitatory-inhibitory coupling between $u_{l}$ and $u_{t}$. This
leads to an optical oscillatory mode in this overdamped system. Taking
$\zeta_{\perp}=\zeta_{\parallel}=\zeta$ for simplicity, we get the
eigenfrequencies

$\omega=-\frac{i}{2\eta}\left[[\zeta(1+\cos^{2}\theta)+\sin^{2}\theta\\{(2\mu+\lambda)\cos^{2}\theta+\mu\\}]\pm\sin^{2}\theta\sqrt{[\zeta-\\{(2\mu+\lambda)\cos^{2}\theta-\mu\\}]^{2}-4\zeta_{pc}^{2}\cos^{2}\theta}\right].$
(76)

The eigenfrequencies are wavenumber-independent because of the overdapmed
Stokesian hydrodynamics, yet, as discussed in the main text, they can have a
real part for large enough $\zeta_{pc}$. Indeed, such a real part exists as
long as

$|\zeta-(2\mu+\lambda)\cos^{2}\theta+\mu|<2|\zeta_{pc}\cos\theta|$ (77)

for _some_ $\theta\neq 0$. For instance, for a perturbation with a wavevector
making an angle $\pi/4$ with $\hat{{\bf z}}$, the eigenfrequency is

$\omega(\theta=\pi/4)=-\frac{i}{8\eta}\left[6\zeta+4\mu+\lambda\pm\sqrt{(\lambda-2\zeta)^{2}-8\zeta_{pc}^{2}}\right],$
(78)

which leads to oscillations when $\zeta_{pc}>|2\zeta-\lambda|/2\sqrt{2}$.

##  Appendix E Density fluctuations in a columnar system

We have, till now, been silent about the fluctuations of the density of the
active species $\delta c=c-c_{0}$ about a mean value $c_{0}$ in columnar
systems (the _joint_ density of the active species and the fluid is also
conserved and accounted for by the incompressibility constraint). We will now
demonstrate that like all translation symmetry-broken systems – such as
smectics or solids – the static structure factor of density fluctuations
$\langle|\delta c(q,t)|^{2}\rangle$ goes to a finite value as $q\to 0$. The
displacement field equation has a coupling to the density field due to a free-
energetic term $w\delta c\nabla_{\perp}\cdot{\bf u}_{\perp}$ (a free energetic
coupling between the polarisation and density $\propto\delta c\nabla\cdot{\bf
P}$ can be discounted since it is higher order in gradients compared to this;
since $\delta{\bf P}_{\perp}\sim\partial_{z}{\bf u}_{\perp}$, this term is
$\propto\delta c\partial_{z}\nabla_{\perp}\cdot{\bf u}_{\perp}$). However, a
lower order in wavenumber coupling to the density field arises from the active
flows. The force balance equation for _all_ columnar systems contains
gradients of density. This arises from multiple sources: i. There is an active
stress $\propto g(c)\hat{n}\hat{n}$ in apolar columnar systems or $\propto
g(c)\hat{P}\hat{P}$ in polar columnar system. This leads to a linearised,
force density $\propto\partial_{z}\delta c\hat{{\bf z}}$. ii. A force density
$\propto\nabla_{\perp}\delta c$ appears both from the free energy coupling
between density and the displacement fields and from active forces. However, a
term $\propto\nabla\delta c$ can be absorbed into the pressure which, in
incompressible fluids, is just a Lagrange multiplier that enforces the
incompressibility constraint. Therefore, it is enough to retain either the
$\partial_{z}\delta c\hat{{\bf z}}$ or the $\nabla_{\perp}\delta c$ force
density; we choose to retain the former. A force density
$\zeta_{c}\partial_{z}\delta c$ leads to the in-plane velocity field (in
Fourier space)

${\bf v}_{\perp}=-i\zeta_{c}\frac{q_{z}^{2}}{\eta q^{4}}q_{\perp}\delta c,$
(79)

which implies that the $u_{t}$ dynamics doesn’t couple to the density field
while the $u_{l}$ dynamics is augmented by a term
$-i\zeta_{c}\cos^{2}\theta|\sin\theta|\delta c/q$.

In polar columnars, there is a mass current ${\bf J}_{P}=\tilde{g}(c){\bf P}$
which accounts for the propulsion of the columns. This implies that the
density equation becomes

$\partial_{t}\delta c=-g^{\prime}\partial_{z}\delta
c-g_{0}\partial_{z}\nabla_{\perp}\cdot{\bf u}_{\perp}+D_{c}\nabla^{2}\delta
c+\xi_{c}=-ig^{\prime}q\cos\theta\delta c+g_{0}q^{2}|\sin\theta|\cos\theta
u_{l}-D_{c}q^{2}\delta c+\xi_{c},$ (80)

where $g^{\prime}=\partial_{c}g(c)|_{c=c_{0}}$, $g_{0}=g(c_{0})$ and $\xi_{c}$
is a conserving noise. All other couplings to the displacement field appear at
higher orders in gradients. These include the currents
$\partial_{z}\nabla_{\perp}\cdot{\bf u}_{\perp}\hat{{\bf z}}$,
$\partial_{z}^{2}{\bf u}_{\perp}$, $\nabla_{\perp}\nabla_{\perp}\cdot{\bf
u}_{\perp}$. In chiral materials, there is also a current
$\propto\partial_{z}(\nabla_{\perp}\times{\bf u}_{\perp})$. The displacement
field dynamics for polar columnars is

$\partial_{t}u_{l}=-\cos^{2}\theta\left[\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right]u_{l}-\cos^{2}\theta\frac{\zeta_{pc}\sin^{2}\theta}{\eta}u_{t}-i\zeta_{c}\frac{\cos^{2}\theta|\sin\theta|}{\eta
q}\delta c+\xi_{l}$ (81)

and

$\partial_{t}u_{t}=-\left[\frac{\mu}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right]u_{t}+\frac{\zeta_{pc}\sin^{2}\theta}{\eta}u_{l}+\xi_{t}.$
(82)

The density correlator is particularly simple to calculate in the limit
$\zeta_{pc}\to 0$ i.e., for a polar but achiral columnar system. The scaling
of the correlator remains unchanged even for $\zeta_{pc}\neq 0$. For
$\zeta_{pc}=0$, the $u_{t}$ dynamics decouple and the coupled $c$ and $u_{l}$
dynamics have the eigenfrequencies

$\omega_{\pm}=-\frac{i}{2}\Bigg{[}\cos^{2}\theta\left(\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right)+Dq^{2}+ig^{\prime}q\cos\theta\\\
\pm\sqrt{\left\\{\cos^{2}\theta\left(\frac{2\mu+\lambda}{\eta}\sin^{2}\theta+\frac{\zeta}{\eta}\right)-Dq^{2}-ig^{\prime}q\cos\theta\right\\}^{2}-4i\frac{\zeta_{c}}{\eta}g_{0}q\sin^{2}\theta\cos^{3}\theta}\Bigg{]}.$
(83)

These can be written in the form $\omega_{\pm}=c_{\pm}q\cos\theta-i\Xi_{\pm}$,
where $\Xi_{+}\sim\mathcal{O}(q^{0})$ and $\Xi_{-}\sim\mathcal{O}(q^{2})$ .
With this, the density correlator becomes

$\langle|\delta c({\bf q},\omega)|^{2}\rangle\approx\frac{\langle\xi_{l}({\bf
q},\omega)\xi_{l}(-{\bf q},-\omega)\rangle
g_{0}^{2}q^{4}\sin^{2}\theta\cos^{2}\theta}{[(\omega-
c_{+}q\cos\theta)^{2}+\Xi_{+}^{2}][(\omega-
c_{-}q\cos\theta)^{2}+\Xi_{-}^{2}]}=\frac{2Tg_{0}^{2}q^{2}\sin^{2}\theta\cos^{4}\theta}{\eta[(\omega-
c_{+}q\cos\theta)^{2}+\Xi_{+}^{2}][(\omega-
c_{-}q\cos\theta)^{2}+\Xi_{-}^{2}]}$ (84)

Therefore, the equal-time density correlator becomes

$\langle|\delta c({\bf
q},t)|^{2}\rangle=\frac{Tg_{0}^{2}q^{2}\sin^{2}\theta\cos^{4}\theta|\Xi_{+}+\Xi_{-}|}{\eta\Xi_{+}\Xi_{-}[(c_{+}-c_{-})q^{2}\cos^{2}\theta+(\Xi_{+}+\Xi_{-})^{2}]}.$
(85)

Since $\Xi_{+}\sim q^{0}$ and $\Xi_{-}\sim q^{2}$, $\Xi_{+}\Xi_{-}\sim q^{2}$
and $\Xi_{+}+\Xi_{-}\sim q^{0}$. Therefore, as $q\to 0$, $\langle|\delta
c({\bf q},t)|^{2}\rangle$ goes to a $q$-independent constant value. While the
coefficient of the $q^{0}$ part of $\Xi_{+}$ vanishes in some directions,
specifically for $\cos\theta=0$, the density and displacement fields decouple
along those directions to lowest order in wavenumbers. It is easy to see that
even in those directions $\langle|\delta c({\bf q},t)|^{2}\rangle\sim q^{0}$
using the conserving noise in the density equation.

The density correlator goes to a constant at small wavenumbers in apolar
systems as well (irrespective of whether it is chiral or not). In this case,
the $u_{l}$ equation is the same as (81) with $\zeta_{pc}=0$ and the density
equation, to lowest order in gradients, is

$\partial_{t}\delta c=-\gamma_{1}\partial_{z}^{2}\nabla_{\perp}\cdot{\bf
u}_{\perp}-\gamma_{2}\nabla_{\perp}^{2}\nabla_{\perp}\cdot{\bf
u}_{\perp}+D_{c}\nabla^{2}\delta
c+\xi_{c}=iq^{3}(\gamma_{1}\cos^{2}\theta+\gamma_{2}\sin^{2}\theta)\sin\theta
u_{l}-D_{c}q^{2}\delta c+\xi_{c},$ (86)

where we have ignored the chiral current which appears at the same order in
wavenumbers and doesn’t change the scaling of the static structure factor of
density fluctuations at small wavenumbers. Further, both the eigenvalues are
now purely relaxational $\omega_{\pm}=-i\Xi_{\pm}$ with $\Xi_{+}\sim q^{0}$
and $\Xi_{-}\sim q^{2}$. Therefore, $\langle|\delta c({\bf
q},t)|^{2}\rangle\sim q^{0}$ again at small wavenumbers. In fact, here the
couplings $\gamma_{i}$ to the displacement field don’t affect the small
wavenumber density structure factor.

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