# Broken living layers: dislocations in active smectics

Frank Julicher<EMAIL_ADDRESS>Max-Planck Institut für Physik Komplexer
Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany  Jacques Prost
<EMAIL_ADDRESS>Mechanobiology Institute and Department of Biological
Sciences, National University of Singapore, Singapore 117411  3 Laboratoire
Physico Chimie Curie, Institut Curie, PSL Research University, CNRS UMR168,
75005 Paris, France  John Toner<EMAIL_ADDRESS>Department of Physics and
Institute for Fundamental Science, University of Oregon, Eugene, OR $97403$

###### Abstract

We show that dislocations in active 2d smectics with underlying rotational
symmetry are always unbound in the presence of noise, meaning the active
smectic phase does not exist for non-zero noise in $d=2$. The active smectic
phase can, like equilibrium smectics in 2d, be stabilized by applying
rotational symmetry breaking fields; however, even in the presence of such
fields, active smectics are still much less stable against noise than
equilibrium ones, when the symmetry breaking field(s) are weak.

##  I Introduction

Much of the richness of condensed matter physics is due to the great variety
of possible different phases of matter. Each distinct phase breaks different
symmetries of the underlying physical laws of the universechaikin .

One of the most interesting equilibrium phases of matter is the smectic A
phasedegennes . This liquid crystalline phase is, as the term “liquid crystal”
suggests, a hybrid of a liquid and a crystalline solid. Specifically, a
$d$-dimensional smectic A can be thought of as a one dimensional stack of
$d-1$-dimensional isotropic fluids. In three dimensions, this is a stack of
two dimensional fluid layers.

These fascinating phases exhibit a number of unique properties, including
quasi-long-ranged order -i.e., algebraicly decaying translational
correlations- in three dimensionscaille , and a breakdown of linearized
hydrodynamicsMRT .

A priori, all of the phases found in equilibrium could also be exhibited by
active matter Vicsek ; TT1 ; TT2 ; TT3 ; TT4 ; TT5 ; Active4 ; Active1 ;
Active2 ; Active3 ; JF1 ; JF2  systems, in which the building blocks are kept
out of equilibrium by constant transduction of energy. Examples of such
systems are living organisms, molecular motors, and robots, to name just a
few. In this paper, we consider active smectics A.

Whether or not a particular active phase is stable, and robust against noise,
is, obviously, the first question one must ask about any potential phase of
active matter. Some phases of active matter - e.g., polar ordered active
matter, of which a uniformly moving flock is the most obvious example, are
actually more robust than their equilibrium counterparts. Polar ordered
flocks, for example, can exhibit long-ranged order in two dimensions, while
their nearest equilibrium analog, ferromagnets, can exhibit only quasi-long
ranged order in two dimensions in the presence of noise - i.e., at finite
temperature.

Other active phases, on the other hand, are less stable than their equilibrium
counterparts. “Wet” active nematics - that is, active nematics with momentum
conservation - are actually unstablesimha .

So in this paper, we ask the question of whether or not active smectics are
robust against noise. These systems have already been shownapolarsm ; polarsm
to be stable at zero noise, and to be stable against noise if topological
defects are ignored. These papers also asserted that these systems can be
stable in two spatial dimensions in the presence of noise against topological
defects as well. Here, we show that this is not the case, for the two simplest
possible types of active smectic: dry Malthusian active smectics, and dry
incompressible active smectics. We will now define these phases.

A dry Malthusian active smectic is one in which nothing - not energy, not
momentum, not even particle numberTT5  \- is conservedapolarsm ; polarsm . As
a further simplification, we will consider only apolar smectics; that is,
smectics with up-down symmetry (that is, symmetry between the two directions
normal to the smectic layers).

A dry incompressible active smectic is simply an active smectic with no
momentum conservation whose mean density is fixed. By “mean” density, we mean
the small wavelength components of the density; the components at
$q_{0}\equiv{2n\pi\over a}$, with $a$ the smectic layer spacing, are non-zero
for all integer $n$ as a result of the spontaneous density modulation that
defines the smectic.

Such completely non-conservative systems are by no means of purely academic
interest. Tissues growing or developing on a substrate tissue1 ; tissue , for
example, lack all conservation laws: cell number is not conserved due to cell
division and death. Momentum is lost due to friction with the substrate, and
can be gained due to active forces between the cells and that substrate.
Finally, energy is not conserved, both due to friction with the substrate, and
because living cells have a fuel source (i.e., food) which enables them to
consume energy. Hence, if such a system was thin compared to its lateral
extent, and spontaneously formed a layered structure, it would be a two
dimensional dry Malthusian active smectic of exactly the type we describe
here. However, smectic order has not been observed experimentally yet in two-
dimensional tissues.

Interestingly, the cell actomyosin system may provide an example of a
Malthusian active smectic. Cell contractility is known to result mainly from
the action of the myosin II molecular motors on the cortical actin gel. Myosin
II motors have a long tail, and two active heads. The tails of several motors
tend to bundle in a way similar to the tail bundling of phospholipids, but at
a different scale: the resulting head to head distance is on the order of 300
nmmyosins . Often the picture of the bundle is that of a symmetric flower
bouquet and their distribution in the actin gel is essentially random. More
ordered structures exist: there is well defined 3d crystalline order in
muscles, and 1d periodic patterns occur in stress fibers. An intermediate
arrangement is observed in lamellipodia of fibroblasts myosins ; myosins1 ;
myosins2 : the myosin tails arrange in lines separated from the heads,
building a clear 2d smectic order. The actin filaments, above this structure,
are on average orthogonal to the myosin lines with no specific translational
order. The bundling/unbundling process does not conserve myosin number and
interaction with the substrate exchange momentum. Hence, this is a ”dry”
system in the sense defined earlier. Such systems of stacked myosin II
filaments in cells have thus the characteristics of dry active Malthusian
smectics.

In addition, we think our results shed considerable light on the dislocation
behavior one might expect in more complex active smectic systems in which one
or more quantities are conserved. For instance, our results are valid in 2d
incompressible systems with boundary conditions allowing for layer number
variation as shown in the last section of this manuscript. This is for
instance the case for the roll structures in Rayleigh-Benard instabilities1st
.

We find that, despite being more stable in spin-wave theory - that is, when
topological defects (i.e., dislocations), are neglected, active smectics in
rotation invariant environments (which we will hereafter refer to by the
shorthand “rotation invariant active smectics”) are unstable against
dislocation unbinding in the presence of any noise, no matter how small.
Furthermore, although they can be stabilized by rotational symmetry breaking
fields, they are still less stable than equilibrium smectics in symmetry
breaking fields of the same strength. Specifically, in the active smectics we
study here, we’ll define $\Delta_{c}$ as the critical value of the noise
strength $\Delta$ above which dislocations unbind, causing the smectic to melt
into an active nematic. This critical value $\Delta_{c}$ grows linearly with
the applied symmetry breaking field strength $g$ for small $g$; that is

$\Delta_{c}\propto g\,\,\,{\rm as}\,\,\,\,g\to 0\,.$ (I.1)

This result should be contrasted with the equilibrium resultKT  for the
transition temperature $T_{c}$, (temperature is the equilibrium analog of the
noise strength in active smectics):

$T_{c}^{\rm eq}\propto\sqrt{g}\,\,\,{\rm as}\,\,\,\,g\to 0\,,$ (I.2)

whose derivation we’ll review in section II.2. We therefore see that, for
small symmetry breaking fields $g$, the critical noise strength $\Delta_{c}$
for dislocation unbinding and the melting of the smectic phase is much smaller
for active smectics than for equilibrium smectics. That is, active smectics
are less robust against melting, even in the presence of symmetry breaking
fields, than their equilibrium counterparts.

Like equilibrium smectics in the presence of a rotational symmetry breaking
field, active smectics in such a symmetry breaking field exhibit quasi-long-
ranged translational correlations for noise strength smaller than the critical
value. This is most clearly manifest in the Fourier transformed density-
density correlation function (which is also the X-ray structure factor in
scattering experiments). This exhibits peaks at wavevectors
$\mathbf{q}_{n}=nq_{0}\hat{y}+{\bf\delta q}$, where $q_{0}={2\pi\over a}$,
with $a$ the smectic layer spacing. Near these peaks, we have

$\langle|\rho(\mathbf{q},t)|^{2}\rangle\propto|{\bf\delta
q}|^{-2+n^{2}\eta(g,\Delta)}\,,$ (I.3)

with $\eta(g,\Delta)$ a non-universal exponent that depends on the symmetry
breaking field strength $g$ and the noise strength $\Delta$ (as well as other
smectic parameters).

Another consequence of our result (I.1) is that the critical value $\eta_{c}$
of the exponent $\eta$ vanishes linearly with symmetry breaking field:

$\eta_{c}\propto g\,\,\,{\rm as}\,\,\,\,g\to 0\,.$ (I.4)

in contrast to equilibrium smectics, for which $\eta_{c}=1/4$, universally,
independent of the applied symmetry breaking field.

The remainder of this paper is organized as follows: in section II, we review
the theory of equilibrium smectics, both in rotation-invariant systems II.1
and with rotational symmetry breaking fields II.2. In section III, we review
the spin-wave theory of active smectics (that is, the theory in which
dislocations are neglected). Section IV presents the calculation of the fields
due to dislocations, which prove to be identical in form to those found in
equilibrium smectics in the presence of a rotational symmetry breaking field.
Section V derives the equation of motion for dislocations in an active,
rotation invariant smectic. In section VI.1, we show that this equation of
motion leads, for fixed boundary conditions, to the achievement of a type of
“homeostasis,”, in which isolated dislocations do not spontaneously move. For
“constant stress” boundary conditions, on the other hand, the smectic will
either collapse, or grow without bound. In section VII, we show that even in
the homeostatic state, in the presence of noise, dislocations are always
unbound in rotation invariant smectics. Rotational symmetry breaking fields
can stabilize smectic quasi-long-ranged order in active smectics, as we show
in VIII. We also show in that section that although symmetry breaking fields
do stabilize smectic order, the resultant order is still very weak for small
fields, in the sense that it can be much more easily destroyed by noise than
in equilibrium smectics. We then generalize these results to the case of
incompressible dry active smectics. Finally, in section X, we summarize our
results, speculate about the behavior of more complicated smectic systems with
conservation laws, and suggest avenues for further work.

##  II Review of Equilibrium 2d smectics

###  II.1 Rotation invariant 2d equilibrium smectics

####  II.1.1 ”Spin wave” (Phonon) theory

Any smectic A phase (either equilibrium or active) is characterized by the
spontaneous breaking of translational invariance in one direction (in contrast
to a crystal, in which translational invariance is broken in all $d$
dimensions, where $d$ is the dimension of space). This is equivalent to saying
that the system spontaneously layers, with the additional requirement that the
layers are ”liquid-like”, in the sense of being homogeneous along the layers.
We will choose our co-ordinates throughout this paper so that the direction in
which translational invariance is broken is $y$. This means the layers, in the
absence of fluctuations, run parallel to $x$ (see figure [1]).

Figure 1: Schematic of the ideal smectic state, in which the layers are
parallel, and uniformly spaced. We choose our coordinates $(x,y)$ so that the
$x$-axis runs parallel to the layers, as shown.

Since the smectic breaks the continuous translational symmetry of free space,
such systems (again whether equilibrium or active) have a ”Goldstone mode”
associated with the breaking of this symmetry. In smectics, we usually take
this Goldstone mode to be the local displacement $u(\mathbf{r},t)$ of the
layers in the vicinity of the spatial point $\mathbf{r}$ away from some
reference set of parallel, regularly spaced layer positions (see figure [2]).

Figure 2: Definition of the layer displacement field $u(\mathbf{r},t)$. The
straight parallel line are the reference positions of the layers, while the
curved lines depict a fluctuation in the layer positions. The layer
displacement field $u(\mathbf{r},t)$ is the distance from the reference
position to the fluctuating position of the layers in the vicinity of the
spatial point $\mathbf{r}$, as illustrated.

To describe such systems in equilibrium, one introducescaille ; chaikin ;
degennes  a phenomenological elastic Hamiltonian (sometimes called the elastic
free energy). This is constructed as an expansion in powers of spatial
gradients of the displacement field $u(\mathbf{r},t)$, keeping all terms to
leading order in spatial gradients allowed by the symmetries of the system.
Translational invariance requires that all terms involve at least one spatial
derivative of $u(\mathbf{r},t)$, since a spatially uniform displacement (i.e.,
$u(\mathbf{r},t)={\rm constant}$) must cost no energy.

Rotation invariance is somewhat more subtle. Its implications can be
understood by recognizing that a uniform rotation of the layers by a small
angle $\phi\ll 1$ can be represented in our coordinate system by a non-uniform
displacement field

$u(\mathbf{r},t)=\phi x\,.$ (II.1)

From this expression, we see that

$\partial_{x}u=\phi\,,$ (II.2)

that is, the $x$-derivative of the displacement field $u$ gives the rotation
angle of the layers away from their reference orientation.

The relation (II.2) continues to apply for arbitrary layer distortions; that
is, the $x$-derivative of $u$ locally gives the local tilt of the layers away
from their reference orientation (provided that tilt is small, of course). We
will make much use of this relationship throughout this paper.

Rotation invariance therefore forbids the inclusion of terms that depend on
$\partial_{x}u$ in the elastic Hamiltonian, since such terms will be non-zero
for the uniform rotation (II.1). Therefore, the leading order term involving
$x$-derivatives of $u(\mathbf{r},t)$ in $H$ is a term proportional to
$(\partial_{x}^{2}u)^{2}$, which represents the energy cost of bending  the
layers.

There is no such prohibition against terms involving $\partial_{y}u$. Indeed,
a term proportional to $(\partial_{y}u)^{2}$ can easily be seen to represent
the energy cost of compressing the layers closer together (for
$\partial_{y}u<0$) or stretching them further apart (for $\partial_{y}u>0$).
It is straightforward to show that

$\delta a=a\partial_{y}u\,,$ (II.3)

where $\delta a$ is the departure of the local layer spacing from its
energetically optimal value $a$. This is another relation which we will use
repeatedly throughout this paper.

These considerations lead, to quadratic order in $u$, to the elastic
Hamiltoniancaille ; chaikin ; degennes :

$H_{\rm sm}=\frac{1}{2}\int{\rm
d}^{2}r\left[B(\partial_{y}u)^{2}+K(\partial_{x}^{2}u)^{2}\right]\,.$ (II.4)

While higher than quadratic order in $u$ terms are actually important in 2d
equilibrium smectics Golub , we will not consider them here, since they do not
play any role in active apolar smecticspolarsm ; polar smectic .

The simplest, purely relaxational, equilibrium dynamics associated with this
Hamiltonian is the time-dependent Landau-Ginsburg-Wilson equation of motion:

$\partial_{t}u=-\Gamma{\delta H_{\rm sm}\over\delta u}+f_{\rm u}\,.$ (II.5)

where $f_{\rm u}(\mathbf{r},t)$ is a Gaussian, zero mean white noise that
drives the smectic to thermal equilibrium, governed by the Hamiltonian $H_{\rm
sm}$ at temperature $T$. To do this, its variance must obey the “fluctuation-
dissipation theorem”chaikin , which requires

$\langle f_{u}({\bf r},t)f_{u}({\bf r}^{\prime},t^{\prime})\rangle=2\Gamma
k_{B}T\delta({\bf r}-{\bf r}^{\prime})\delta(t-t^{\prime})\,.$ (II.6)

Using the Hamiltonian (II.4) in the equation of motion (II.5) gives:

$\displaystyle\partial_{t}u=\Gamma B\partial_{z}^{2}u-\Gamma
K\partial_{x}^{4}u+f_{u}\,.$ (II.7)

Note that this equation exhibits subdiffusive behavior in the $x$-direction.
That is, relaxation along the $x$-direction is even slower than diffusive;
specifically, the lifetime of a plane wave $u$ field running along $x$ with
wavelength $\lambda$ grows like $\lambda^{4}$ for large $\lambda$, in contrast
to the $\lambda^{2}$ scaling of simple diffusion.

This slowness of response to distortions in the $x$-direction, like the
corresponding ”softness” in the $x$-direction of the elastic Hamiltonian
(II.4), is a direct consequence of the rotation invariance (i.e., the zero
energy cost of pure rotations (II.1)) discussed earlier.

This rotation invariance can be removed by applying an external symmetry
breaking field, as we’ll discuss in subsection B.

####  II.1.2 Dislocation effects: There are no 2d equilibrium smectics at
$T\neq 0$

The preceding discussion constitutes what is normally called ”spin-wave”
theory. It assumes that it is possible to define throughout the entire system
a unique, single valued displacement field $u(\mathbf{r},t)$ throughout the
system. This is in fact not the case if dislocations (see figure
[LABEL:Burgers]) are present in the system.

We will now review the theory of these defects in equilibrium, as first
developed by Pershanpershan .

Figure 3: Illustration of the Burgers’ construction for dislocations described
in the text.

As can easily be seen from figure [3], one can think of a dislocation in a
smectic as a place where a layer ends. That these ”topological defects” make
it impossible to define a single valued $u$ field can be seen by the well-
known ”Burgers’ construction”, which is also illustrated in figure 3. In this
construction, one ”walks” in a closed loop around the dislocation, crossing
equal numbers of layers while moving up and moving down. In the example of
figure 3, one crosses two layers while going up from the starting point S to
the corner A, plus another two while moving from the lower right corner E to
the final point F, for a total of four going up, and crosses four going down
between the upper left corner B and the lower left corner C. Clearly, if the
dislocation (at point D in figure 3) were not there, or if the path did not
enclose the dislocation (i.e., the point D), this path would have returned to
the starting point S. Equally clearly, when it does enclose the dislocation,
the path does not return to $S$, but, rather, overshoots closure by the length
of the thick blue arrow between S and F, whose length is clearly the layer
spacing $a$. This failure to close is known as the ”Burgers’ number” $b$ for
the dislocation, and in this case is $b=+1$.

It is straightforward to see that in general, the Burgers’ number will be an
integer, the integer being simply the difference between the number of layers
coming in from the left that end inside the loop, and the number coming in
from the right that do so.

The Burgers’ number, defined in this way, is the analog for smectics (which,
we remind the reader, only translationally order in one direction) to the
better-known ”Burgers’ vector” defined by an almost identical construction in
crystalline solids (which translationally order in all directions).

One way of thinking about this result is that, if we had defined the
displacement field at the starting point S to be $u(S)=0$, we would, after
having completed the loop, found that the layer had been displaced up by an
amount $na$ for $b=n$. That is, $u$ is no longer single valued, but instead
increases by $ba$ every time one moves around a loop enclosing the
dislocation. Mathematically, the contour integral $\int_{C}m\hat{\bf
n}\cdot{\bf dl}$ counts the number of layers traversed along the integration
path $C$, where $m=1/a(\mathbf{r})$ is the local layer density at the point
$\mathbf{r}$, and $\hat{n}(\mathbf{r})$ is the unit normal to the layers at
the point $\mathbf{r}$. The statement that $u$ is not single valued, but
changes when moving around a loop enclosing dislocations, is equivalent to the
statement that a nonvanishing number of layers is encountered by a closed loop

$\oint m\hat{\bf n}\cdot{\bf dl}=\sum_{\alpha}b_{\alpha}\,.$ (II.8)

Here we have now generalized to the case of many dislocations, labeled by
$\alpha$, with Burgers’ numbers $b_{\alpha}$, at positions
$\mathbf{r}_{\alpha}$ that are enclosed by the loop over which the contour
integral on the left-hand side of (II.8) is done. We reiterate that the
Burgers’ number $b_{\alpha}$ of each dislocation must be an integer that is,
$b_{\alpha}=n$, with $n$ an integer.

Applying Stokes’ theorem to (II.8) gives

$\nabla\times m\hat{\bf
n}=\sum_{\alpha}b_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha})\,,$ (II.9)

In (II.9), we have defined the curl in two dimensions in the usual way; that
is, as a scalar given, for any vector $\mathbf{v}$, by
$\nabla\times\mathbf{v}\equiv\partial_{x}v_{y}-\partial_{y}v_{x}$.

It is convenient to define ${\bf w}=(a_{0}m-1)\hat{\bf n}$, where $a_{0}$ is
the layer spacing in a reference state. For small displacements it can be
written as

$\mathbf{w}(\mathbf{r},t)\simeq\phi(\mathbf{r},t)\hat{x}+\left({\delta
a(\mathbf{r},t)\over a_{0}}\right)\hat{y}\,.$ (II.10)

with $\phi(\mathbf{r},t)$ and $\delta a(\mathbf{r},t)$ respectively the local
tilt of the layers at $\mathbf{r}$ at time $t$, and the local change in the
layer spacing.

Keeping in mind our earlier discussion of the relationships (II.2) and (II.3)
which hold between these two quantities $\phi(\mathbf{r},t)$ and $\delta
a(\mathbf{r},t)$ and the layer displacement field $u(\mathbf{r},t)$ in the
absence of dislocations, we see that $\mathbf{w}$ is clearly simply the
generalization of $\nabla u$ to situations in which dislocations are present.
To say this another way, when dislocations are absent,

$\mathbf{w}=\nabla u\ \ \ ,\ \ \ {\rm no}\,{\rm dislocations}\,.$ (II.11)

Thus, $\mathbf{w}$ is the natural generalization of the vector field $\nabla
u$ in the presence of dislocations, see Appendix A.

We can use this idea to calculate the $\mathbf{w}$ field of a dislocation.
That will be the field that minimizes the energy of the system for a given
configuration of dislocations. That is, we wish to find the field
$\mathbf{w}(\mathbf{r})$ that minimizes the energy of the system subject to
the constraint

$\nabla\times\mathbf{w}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(II.12)

which is where the dislocation configuration
$\\{b_{\alpha},\mathbf{r}_{\alpha}\\}$ enters the calculation.

In the absence of dislocations, the minimum energy configuration
$u(\mathbf{r})$ can be obtained from the Euler-Lagrange equation associated
with the smectic elastic Hamiltonian (II.4). That equation is easily seen to
be:

$B\partial^{2}_{y}u(\mathbf{r})-K\partial^{4}_{x}u(\mathbf{r})=0\,.$ (II.13)

We can obviously rewrite this as

$B\partial_{y}(\partial_{y}u(\mathbf{r}))-K\partial^{3}_{x}(\partial_{x}u(\mathbf{r}))=0\,.$
(II.14)

Now generalizing this equation to situations in which dislocations are present
by replacing $\nabla u$ with $\mathbf{w}$ gives

$B\partial_{y}w_{y}(\mathbf{r})-K\partial^{3}_{x}w_{x}(\mathbf{r})=0\,.$
(II.15)

The other condition on dislocations is the Burgers’ condition (II.12).

These two simultaneous linear equations (II.15) and (II.12) can be easily
solved by Fourier transforming in space; this gives

$Bq_{y}w_{y}(\mathbf{q})+Kq_{x}^{3}w_{x}(\mathbf{q})=0\,,$ (II.16)

and

$q_{x}w_{y}(\mathbf{q})-q_{y}w_{x}(\mathbf{q})=-i\sum_{\alpha}a_{0}b_{\alpha}e^{-i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\,,$
(II.17)

Solving these simple linear equations gives

$\displaystyle w_{x}(\mathbf{q})={iBq_{y}\over
Bq_{y}^{2}+Kq_{x}^{4}}\sum_{\alpha}a_{0}b_{\alpha}e^{-i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\,,$
(II.18) $\displaystyle w_{y}(\mathbf{q})={-iKq_{x}^{3}\over
Bq_{y}^{2}+Kq_{x}^{4}}\sum_{\alpha}a_{0}b_{\alpha}e^{-i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\,.$
(II.19)

Fourier transforming these solutions back to real space gives

$\displaystyle
w_{x}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{x}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(II.20) $\displaystyle
w_{y}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{y}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(II.21)

where the Green’s functions $G_{x,y}$ are given by

$\displaystyle G_{x}(\mathbf{r})=-{1\over
4\sqrt{\pi\lambda|y|}}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}{\rm sgn}(y)\,,$ (II.22) $\displaystyle
G_{y}(\mathbf{r})={x\over
8\sqrt{\pi\lambda|y|^{3}}}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}\,,$ (II.23)

where we’ve defined $\lambda\equiv(K/B)^{1/2}$.

We can also obtain the energy of interaction between dislocations by inserting
the solution (II.19) for $\mathbf{w}$ into our elastic Hamiltonian (II.4). To
do so, we must first rewrite (II.4) in terms of $\mathbf{w}$ using the same
replacement $\nabla u\to\mathbf{w}$ we’ve been using. This gives:

$H_{\rm sm}=\frac{1}{2}\int{\rm
d}^{2}r\left[Bw_{y}^{2}+K(\partial_{x}w_{x})^{2}\right]\,.$ (II.24)

Fourier transforming this, inserting the solution (II.19) for $\mathbf{w}$
into the result, and Fourier transforming back to real space gives

$H(\\{b_{\alpha},\mathbf{r}_{\alpha}\\})=\sum_{\alpha,\beta}a_{0}^{2}b_{\alpha}b_{\beta}U(\mathbf{r}_{\alpha}-\mathbf{r}_{\beta})$
(II.25)

where the pairwise interaction potential

$U(\mathbf{r})={B\over
4}\sqrt{\lambda\over\pi|y|}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}\,.$ (II.26)

Because this vanishes as the separation between dislocations goes to infinity,
dislocations will always be unbound in smectics at any non-zero temperature,
thereby destroying the smectic order. This means that 2d smectics do not
actually exist in rotation invariant systems at non-zero temperature, as first
shown (using exactly this argument) by 1st .

While we do not need the equation of motion for the dislocations in this
equilibrium system, since the statistical mechanics is determined entirely by
the Hamiltonian (II.4), it is instructive to formulate those equations of
motion. This will allow us later to compare and contrast them with the
equations of motion for dislocations in active smectics, for which the
equation of motion is the only information we have about the behavior of
dislocations in those non-equilibrium systems.

To obtain the equations of motion, we first calculate the forces arising from
the potential (II.26).

Consider a system with only two dislocations $\alpha=(1,2)$ of Burgers’
charges: $b_{1}$ and $b_{2}$, at positions $\mathbf{r}_{1}={\bf 0}$,
$\mathbf{r}_{2}=\mathbf{r}=x\hat{x}+y\hat{y}$. The dislocation Hamiltonian
(II.25) then implies that the energy of this pair will be

$V(\mathbf{r})={Bb_{1}b_{2}a_{0}^{2}\over
4}\sqrt{\lambda\over\pi|y|}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}\,.$ (II.27)

The force experienced by dislocation $\alpha=2$ will therefore be ${\bf
F}=F_{x}\hat{x}+F_{y}\hat{y}$, with its Cartesian components $F_{x,y}$ given
by

$\displaystyle F_{x}$ $\displaystyle=$
$\displaystyle-\partial_{x}V(\mathbf{r})={Bb_{1}b_{2}a_{0}^{2}x\over
8\sqrt{\pi\lambda|y|^{3}}}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}=Bb_{2}a_{0}w_{y}^{(1)}(\mathbf{r})$ $\displaystyle
F_{y}$ $\displaystyle=$
$\displaystyle-\partial_{y}V(\mathbf{r})=-{Bb_{1}b_{2}a_{0}^{2}{\rm
sgn}(y)\over
16\sqrt{\pi\lambda|y|^{5}}}\bigg{\\{}x^{2}-2\lambda|y|\bigg{\\}}\exp\bigg{[}-\left({x^{2}\over
4\lambda|y|}\right)\bigg{]}=Kb_{2}a_{0}\partial_{x}^{2}w_{x}^{(1)}(\mathbf{r})$
(II.28)

where by $\mathbf{w}^{1}(\mathbf{r})$, we mean the contribution to the field
$\mathbf{w}$ at the position $\mathbf{r}$ of the $\alpha=2$ dislocation coming
from the $\alpha=1$ dislocation (i.e., neglecting the the field created by
dislocation $\alpha=2$ itself). It is straightforward to show that the
generalization of these forces to configurations with more than two
dislocations is:

$\displaystyle F_{x}^{\alpha}$ $\displaystyle=$ $\displaystyle
Bb_{\alpha}a_{0}w_{y}(\mathbf{r}_{\alpha})$ $\displaystyle F_{y}^{\alpha}$
$\displaystyle=$ $\displaystyle
Kb_{\alpha}a_{0}\partial_{x}^{2}w_{x}(\mathbf{r}_{\alpha})$ (II.29)

where by $\mathbf{w}(\mathbf{r}_{\alpha})$, we mean the contribution to the
field $\mathbf{w}$ at the position $\mathbf{r}_{\alpha}$ of the $\alpha$’th
dislocation coming from all of the other dislocations (i.e., neglecting the
field created by dislocation $\alpha$ itself).

Since the dislocation cannot ”tell” whether the local field $\mathbf{w}$ is
created by other dislocations, by spin waves, or by externally applied
stresses, we expect (II.29) to hold more generally if we take
$\mathbf{w}_{\alpha}$ on the right-hand side of those equations to be the
entire $\mathbf{w}$ field, excluding the part due to dislocation $\alpha$
itself. This proves to be important when we consider the effect of stresses at
the boundary on dislocation motion.

There are two important features of the result (II.29) that should be noted:

1) the force on a given dislocation $\alpha$ is determined entirely by the
local value of the field $\mathbf{w}(\mathbf{r}_{\alpha})$ (and its
derivatives) at the location $\mathbf{r}_{\alpha}$ of that dislocation
(excluding the part of that field due to the given dislocation itself).

2) The dependence of the force on the $x$ component $w_{x}$ involves spatial
derivatives of that component; a uniform $w_{x}$ does not generate any force
on the dislocation. This is a consequence of rotation invariance: as shown by
equation (II.29), a spatially uniform $w_{x}$ corresponds to a uniform
rotation of the layers, which clearly cannot lead to any force on the
dislocation in a rotation-invariant system. This consideration will continue
to apply in active smectics, and will forbid certain terms in the force in
those systems which one might otherwise expect, as we will see in section [V]
below.

Since there will be friction between the dislocations and the underlying
substrate, we expect the velocity $\mathbf{v}$ (rather than the acceleration
$\dot{\mathbf{v}}$) of the dislocations to be linear in the force
$\mathbf{F}$. That is,

$\mathbf{v}_{\alpha}=\bm{\mu}\mathbf{F}_{\alpha}\,,$ (II.30)

where $\bm{\mu}$ is a constant ”mobility tensor”. On symmetry grounds, we
expect this tensor to be diagonal in our $(x,y)$ coordinate system (i.e., with
the $x$ and $y$ axes respectively parallel and perpendicular to the mean
smectic layers); hence

$v_{x}^{\alpha}=\mu_{x}F_{x}^{\alpha}\,,\,v_{y}^{\alpha}=\mu_{y}F_{y}^{\alpha}\,.$
(II.31)

Using our earlier results (II.29) for the forces on the dislocations, we can
rewrite these as

$\displaystyle v_{x}^{\alpha}$ $\displaystyle=$
$\displaystyle\mu_{x}Bb_{\alpha}a_{0}w_{y}(\mathbf{r}_{\alpha})$
$\displaystyle v_{y}^{\alpha}$ $\displaystyle=$ $\displaystyle
Kb_{\alpha}a_{0}\partial_{x}^{2}w_{x}(\mathbf{r}_{\alpha})\,.$ (II.32)

We see that, like the component $F_{y}$ of the force, the $y$-component of the
dislocation velocity (which is, after all, proportional to $F_{y}$) vanishes
for spatially uniform $w_{x}$. And it does so for the same reason: a spatially
uniform $w_{x}$ corresponds to a uniform rotation, which cannot lead to
dislocation motion in a rotation invariant system.

This result, based as it is purely on symmetry, proves to continue to apply
even in active, rotation invariant smectics, as we will see in section [V].

###  II.2 Non-rotation invariant 2d equilibrium smectics: effects of a
symmetry breaking field

####  II.2.1 ”Spin wave” (Phonon) theory with a symmetry breaking field:
Quasi-long-ranged order

As we have seen, rotation invariance plays a crucial role in the behavior of
equilibrium two-dimensional smectics - indeed, in some sense, it makes the
smectic phase impossible(at non-zero temperature) in $d=2$. One can, however,
make a 2d smectic non-rotation invariant. This can be done in a number of
ways. Two of the simplest are:

1) applying a magnetic field (${\bf H}$)

2) preparing the 2d surface on which the smectic lives in some non-rotation
invariant way. For example, one could rub the surface in one direction with an
abrasive cloth, or etch a set of parallel grooves along it.

Magnetic fields break rotation invariance by picking out a preferred direction
for the layer normal

$\hat{{\bf
n}}(\mathbf{r})=-\sin[\phi(\mathbf{r})]\,\hat{x}+\cos[\phi(\mathbf{r})]\,\hat{y}\,,$
(II.33)

where $\phi(\mathbf{r})$ is the angle between the local layers at $\mathbf{r}$
and the $y$-axis. They do this due to the fact that the magnetic
susceptibility tensor $\chi^{H}_{ij}$ must, by symmetry, have the layer normal
$\hat{{\bf n}}$ as one of their principal axes. This implies that the
susceptibility tensor can be written in the form degennes ; chaikin

$\chi^{H}_{ij}=\chi^{H}_{0}\delta_{ij}-\Delta\chi^{H}n_{i}n_{j}\,,$ (II.34)

where the material parameters $\chi^{H}_{0}$ and $\Delta\chi^{H}$ are
respectively the isotropic and anisotropic parts of the susceptibility.

The expression II.34 in turn implies that the magnetic energy of the smectic
are given by (note that the $H$ on the left-hand side of the following
expressions stands for “Hamiltonian”, while the $H$ on the right-hand side of
the first of them stands for magnetic field):

$\displaystyle H_{mag}$ $\displaystyle={1\over 2}\int
d^{2}r\chi^{H}_{ij}H_{i}H_{j}$ (II.35) $\displaystyle=-{1\over 2}\int
d^{2}r\Delta\chi^{H}({\bf H}\cdot\hat{{\bf n}})^{2}+{\rm constant}\,$

where the “constant” includes those parts of the energy independent of the
layer normal $\hat{{\bf n}}$.

Inserting our expression II.33 for $\hat{{\bf n}}$, and choosing our $y$-axis
to be along the magnetic field, gives

$H_{mag}=-{1\over 2}\int
d^{2}r\Delta\chi^{H}H^{2}\cos^{2}[\phi(\mathbf{r})]+{\rm constant}\,.$ (II.36)

Clearly, the magnetic energy favors (for positive $\Delta\chi^{H}$) alignment
of $\hat{{\bf n}}$ along the magnetic fields (i.e., $\phi=0$). If, e.g.,
$\Delta\chi^{H}$) is negative, then the lowest energy configuration will have
the layer normal $\hat{{\bf n}}$ perpendicular to ${\bf H}$. In these cases,
we can still arrive at the expression II.36 simply by choosing the $y$-axis to
be perpendicular to the applied field. So the result II.36 can easily be made
to hold in general.

Now assuming that fluctuations away from this minimum energy state are small,
we can expand II.36 for small $\phi$, obtaining

$H_{mag}={1\over 2}\int d^{2}r\Delta\chi^{H}H^{2}[\phi(\mathbf{r})]^{2}+{\rm
constant}\,$ (II.37)

Finally, using the relation $\partial_{x}u=\phi$ (i.e., (II.2)) between $\phi$
and the layer displacement field $u$, we can rewrite this as

$H_{mag}={1\over 2}\int d^{2}r\,g(\partial_{x}u)^{2}+{\rm constant}\,,$
(II.38)

where we’ve defined the “symmetry breaking field strength” $g$ via

$g\equiv\Delta\chi^{H}H^{2}$ (II.39)

for the case of an applied magnetic field.

Note that the symmetry breaking field strength $g$ is actually proportional to
the square of the applied field. This is a consequence of the fact that the
smectic is apolar; in a polar smectic, this symmetry breaking would be linear
in the applied field, since the constituents of the smectic would then have
spontaneous magnetic and electric dipole moments.

The second approach mentioned above, of breaking symmetry by preparing the
surface in a non-rotation invariant way, can be shown by similar arguments to
lead to a symmetry breaking contribution to the Hamiltonian of the form II.38
as well. The dependence of $g$ on whatever quantity one uses to characterize
the strength of the symmetry breaking of the surface preparation need not be
quadratic, however. For example, if the preparation consisted of etching or
rubbing a set of grooves onto the substrate, we would expect $g$ in II.38 to
be linear, not quadratic, in the density of such grooves, at least when that
density is small.

Adding this additional symmetry breaking energy II.39 to the terms already
present in our smectic energy gives the total smectic Hamiltonian for the non-
rotation invariant case:

$H_{\rm sm}=\frac{1}{2}\int{\rm
d}^{2}r\left[B(\partial_{y}u)^{2}+g(\partial_{x}u)^{2}+K(\partial_{x}^{2}u)^{2}\right]\,,$
(II.40)

The bend elasticity term $K(\partial_{x}^{2}u)^{2}$ is clearly negligible, at
long wavelengths (i.e., for small spatial gradients) relative to the symmetry
breaking term $g(\partial_{x}u)^{2}$. We will therefore henceforth drop it,
which leaves our Hamiltonian in the form:

$H_{\rm sm}=\frac{1}{2}\int{\rm
d}^{2}r\left[B(\partial_{y}u)^{2}+g(\partial_{x}u)^{2}\right]\,,$ (II.41)

This non-rotation invariant smectic problem can readily be seen to be
equivalent to an XY model . To see this, make a simple linear change of
variables from the layer spacing $u(\mathbf{r},t)$ to an ”angle field”
$\theta(\mathbf{r},t)$ defined via

$\theta\equiv q_{0}u$ (II.42)

where $q_{0}\equiv{2\pi\over a_{0}}$ is the wavevector of the smectic
layering. This has the effect of converting the invariance of the smectic
system under the translation $u(\mathbf{r},t)\to u(\mathbf{r},t)+a_{0}$ (which
is a symmetry since the smectic structure is periodic with period $a_{0}$ in
the $y$-direction) to invariance under $\theta\to\theta+2\pi$. The latter
symmetry implies that $\theta$ can be interpreted as the angle between a unit
spin and some reference direction, or, equivalently, as the phase of a complex
scalar. Both of these systems are XY models.

With the change of variables (II.42), the Hamiltonian (II.40) becomes

$H_{XY}=\frac{1}{2}\int{\rm
d}^{2}r\left[K_{y}(\partial_{y}\theta)^{2}+K_{x}(\partial_{x}\theta)^{2}\right]\,,$
(II.43)

with

$K_{y}=Bq_{0}^{-2}\ \ \ ,\ \ \ K_{x}=gq_{0}^{-2}\,.$ (II.44)

We can convert this into the most familiar, isotropic form of the XY model by
rescaling lengths anisotropically so as to make the coefficients of the two
terms in (II.46) equal. It is easy to show that the change of variables

$x=x^{\prime}\sqrt{K_{x}\over K_{y}}=x^{\prime}\sqrt{g\over B}$ (II.45)

accomplishes this, leading to an isotropic model

$H_{XYiso}=\frac{\bar{K}}{2}\int{\rm d}x^{\prime}{\rm
d}y|\nabla^{\prime}\theta|^{2}\,,$ (II.46)

where
$\nabla^{\prime}\equiv\partial_{x^{\prime}}\hat{x}^{\prime}+\partial_{y}\hat{y}$,
and the spin wave stiffness $\bar{K}$ is just the geometric mean of $K_{x}$
and $K_{y}$:

$\bar{K}=\sqrt{K_{x}K_{y}}\,.$ (II.47)

The model (II.46) with the “compactness condition” that the operation
$\theta\to\theta+2\pi$ takes one to the same physical state is the extremely
well-studied ”XY model”KT . It describes spin systems, with the local spin
$\mathbf{S}(\mathbf{r}^{\prime})=(\cos(\theta(\mathbf{r}^{\prime})),\sin(\theta(\mathbf{r}^{\prime})))$,
w here

$\mathbf{r}^{\prime}\equiv(x^{\prime},y)=\left(x\sqrt{B\over g},y\right)\,.$
(II.48)

It exhibits quasi-long-ranged order, that is, algebraically decaying spin
correlationsKT :

$\langle\mathbf{S}(\mathbf{r}^{\prime}_{1})\cdot\mathbf{S}(\mathbf{r}^{\prime}_{2})\rangle\propto|\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}|^{-\eta(T)}\,,$
(II.49)

with the non-universal, temperature-dependent exponentKT

$\eta(T)={k_{B}T\over 2\pi\bar{K}}={k_{B}T\over
2\pi\sqrt{K_{x}K_{y}}}={k_{B}Tq_{0}^{2}\over 2\pi\sqrt{gB}}\,.$ (II.50)

The correlation function in smectics most closely analogous to the spin-spin
correlation function (II.49) is the density-density correlation function. This
follows from a standard result of the scattering theory of smectics degennes ,
which states that the Fourier transformed equal-time correlations of the
density near the $n$’th Bragg peak (i.e., near wavevector
$\mathbf{q}=nq_{0}\hat{y}$ with integer $n$) are given by

$\langle|\rho(\mathbf{q},t)|^{2}\rangle\propto\mathbf{FT}\bigg{\\{}\langle\exp\\{inq_{0}[u(\mathbf{r}^{\prime}_{1})-u(\mathbf{r}^{\prime}_{2})]\\}\rangle\bigg{\\}}\bigg{|}_{{\bf\delta
q}}\,,$ (II.51)

where $\mathbf{FT}$ denotes a Fourier transform and ${\bf\delta
q}\equiv\mathbf{q}-nq_{0}\hat{y}$. Mapping this to the XY problem by using the
change of fields from $\theta$ to $u$ (II.42), and the change of coordinates
from $x$ to $x^{\prime}$ (II.45), we obtain power law correlations for the
complex exponential in (II.51):

$\langle\exp\\{inq_{0}[u(\mathbf{r}^{\prime}_{1})-u(\mathbf{r}^{\prime}_{2})]\\}\rangle\propto|\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}|^{-n^{2}\eta(T)}\,.$
(II.52)

This is easily shown to imply that the Bragg peaks become power law
singularities near the $n$’th Bragg peak:

$\langle|\rho(\mathbf{q},t)|^{2}\rangle\propto|{\bf\delta
q}|^{-2+n^{2}\eta(T)}\,.$ (II.53)

This can be measured experimentally by various scattering techniques (either
X-ray or light scattering, depending on the layer spacing $a$), or, in
experiments in which the constituent particles can actually be imaged, by
simply constructing the spatially Fourier transformed density correlations
directly from particle positions.

####  II.2.2 Equation of motion for the non-rotation invariant case

Again taking the simplest, purely relaxational, equilibrium dynamics
associated with this Hamiltonian, which is the time-dependent Landau-Ginsburg-
Wilson EOM (II.5), we obtain

$\displaystyle\partial_{t}u=D_{y}\partial_{y}^{2}u+D_{x}\partial_{x}^{2}u+f_{u}$
(II.54)

where we’ve defined $D_{y}\equiv\Gamma B$ and $D_{x}\equiv\Gamma g$.

In (II.54), we have, as in the rotation-invariant case, added a Gaussian white
noise $f_{u}$ with correlations:

$\langle f_{u}({\bf r},t)f_{u}({\bf r}^{\prime},t^{\prime})\rangle=2\Gamma
k_{B}T\delta({\bf r}-{\bf r}^{\prime})\delta(t-t^{\prime})\,,$ (II.55)

where the coefficient $\Gamma k_{B}T$ is required by the fluctuation-
dissipation theoremchaikin .

One somewhat surprising feature of the equation of motion (II.54) is that,
although it was derived from the non-rotation invariant free energy (II.40),
the equation of motion (II.54) itself is rotation invariant, as can be seen by
noting that the equation remains unchanged under the substitution
$u(\mathbf{r},t)\to u(\mathbf{r},t)+\phi x$, which corresponds to a uniform
rotation of all the smectic layers by an angle $\phi\ll 1$. We will use this
observation later to argue that (II.54) is therefore the spin wave equation of
motion we would expect even for rotation invariant active smectics, for which
there is no free energy.

####  II.2.3 Dislocation effects: Kosterlitz-Thouless transition

We can now treat dislocations in smectics without rotation invariance exactly
as we treated those with rotation invariance in the previous section. The
definition of the field $\mathbf{w}$ as the generalization of $\nabla u$ in
the presence of dislocations is unchanged, as is the Burgers’ condition
(II.12). All that changes is the Hamiltonian, which is now (II.40) rather than
(II.24). As a result, the Euler-Lagrange equation now becomes

$g\partial_{x}w_{x}(\mathbf{r})+B\partial_{y}w_{y}(\mathbf{r})=0\,,$ (II.56)

which can be rewritten in Fourier space as

$gq_{x}w_{x}(\mathbf{q})+Bq_{y}w_{y}(\mathbf{q})=0\,.$ (II.57)

Solving this simultaneously with the unchanged Burgers’ condition (II.12)
gives

$\displaystyle w_{x}(\mathbf{q})={iBq_{y}\over
gq_{x}^{2}+Bq_{y}^{2}}\sum_{\alpha}a_{0}b_{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\,,$
(II.58) $\displaystyle w_{y}(\mathbf{q})={-igq_{x}\over
gq_{x}^{2}+Bq_{y}^{2}}\sum_{\alpha}a_{0}b_{\alpha}e^{i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\,.$
(II.59)

Fourier transforming these solutions back to real space gives

$\displaystyle
w_{x}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{x}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(II.60) $\displaystyle
w_{y}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{y}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(II.61)

where the Green’s functions $G_{x,y}$ are now given by

$\displaystyle G_{x}(\mathbf{r})=-{y\sqrt{gB}\over 2\pi(gy^{2}+Bx^{2})}\,,$
(II.62) $\displaystyle G_{y}(\mathbf{r})={x\sqrt{gB}\over
2\pi(gy^{2}+Bx^{2})}\,.$ (II.63)

As we did for the rotation invariant smectic, we can calculate the energy of a
dislocation configuration by inserting these results into the elastic
Hamiltonian (II.40), which can be rewritten in terms of the components of
$\mathbf{w}$ as

$H_{\rm sm}=\frac{1}{2}\int{\rm d}^{2}r\left[gw_{x}^{2}+Bw_{y}^{2}\right]\,.$
(II.64)

Inserting our results (II.21) and (II.63) into this expression, and performing
the integral over $\mathbf{r}$ gives our dislocation Hamiltonian $H_{\rm
disl}$ for non-rotation invariant smectics:

$H_{\rm disl}=-{\sqrt{gB}\over
2\pi}\sum_{\langle\alpha\neq\beta\rangle}a_{0}^{2}b_{\alpha}b_{\beta}\ln\left({|\mathbf{r}^{\prime}_{\alpha}-\mathbf{r}^{\prime}_{\beta}|\over
a}\right)\,,$ (II.65)

where the sum is over pairs $\alpha$, $\beta$ of dislocations, with each pair
counted once, and $\alpha\neq\beta$. We remind the reader that
$\mathbf{r}\equiv(x,y)$ and $\mathbf{r}^{\prime}\equiv(x\sqrt{B\over g},y)$.
Note that the sign of this expression implies that the potential between two
oppositely charged dislocations is attractive.

From the form of this Hamiltonian, it is possible see why a Kosterlitz-
Thouless defect unbinding transition must occur, and to even determine its
temperature, by the following very simple argument, originally given (in a
slightly different form) by Kosterlitz and ThoulessKT .

Consider a minimal neutral pair of dislocations $\alpha=(1,2)$ of Burgers’
charges: $b_{1}=1$ and $b_{2}=-1$. From the dislocation Hamiltonian (II.65),
we see that the energy of this pair will be

$V(\mathbf{R})={\sqrt{gB}a_{0}^{2}\over
2\pi}\ln\left({|\mathbf{r}^{\prime}_{1}-\mathbf{r}^{\prime}_{2}|\over
a}\right)+2E_{c}\,,$ (II.66)

where $\mathbf{R}\equiv\mathbf{r}_{1}-\mathbf{r}_{2}$ is the separation of the
pair, and $E_{c}$ is a“core energy” that we have added to the energy of the
pair to take into account the energy coming from distortions within a distance
$a$ of the cores of the two dislocations, which are not accurately captured by
our elastic theory, because that theory is only valid at long distances. This
core energy $E_{c}$ also contains some constant ($\mathbf{R}$-independent)
contributions to the energy of the pair coming from the elastic energy outside
this region. Hence, by simple Boltzmann statistics, the probability density
$p(\mathbf{R})$ for this pair is

$\displaystyle p(\mathbf{R})=\exp\left(-{V(\mathbf{R})\over
k_{B}T}\right)=\kappa^{2}|R^{\prime}|^{-\nu}\,,$ (II.67)

where we’ve defined the “dislocation fugacity”
$\kappa\equiv\exp(-E_{c}/k_{B}T)$,
$\mathbf{R}^{\prime}\equiv(R_{x}\sqrt{B\over g},R_{y})$, and

$\nu\equiv{\sqrt{gB}a_{0}^{2}\over 2\pi k_{B}T}\,.$ (II.68)

The mean squared size of this dipole is

$\langle|\mathbf{R}|^{2}\rangle=\int
d^{2}R\,p(\mathbf{R})|\mathbf{R}|^{2}=\kappa^{2}\sqrt{g\over B}\int
d^{2}R^{\prime}\,|\mathbf{R}^{\prime}|^{2-\nu}\left({R\over
R^{\prime}}\right)^{2}\,.$ (II.69)

Since ${R\over R^{\prime}}$ is bounded between $1$ and $\sqrt{B\over g}$, as
the reader can easily convince herself, this mean squared value clearly
diverges if $\nu\leq 4$. This signals dislocation unbinding; i.e., the
Kosterlitz-Thouless transition, which corresponds to the loss of quasi-long-
ranged translational order, or equivalently, the melting of the smectic into a
nematic. This transition clearly occurs at the temperature $T_{KT}$ at which
$\nu=4$; hence, using (II.68), we have

${\sqrt{gB}a_{0}^{2}\over 2\pi k_{B}T_{KT}}=4\,.$ (II.70)

Using this result, and setting $T=T_{KT}$ in our expression (II.50) for the
exponent for algebraic decay of density correlations, we recover the famous
resultKT ; NK

$\eta(T_{KT})=1/4\,.$ (II.71)

The crucial point the reader should take away from this discussion: first, is
that, when the probability density for the separation of a dislocation pair
falls off like a power law $R^{-\nu}$ with distance, dislocations will be
bound if $\nu>4$, and unbound if $\nu<4$. We will later apply this criterion
to active, non-rotation invariant smectics to determine the critical noise
strength at which dislocations unbind, melting the smectic phase.

Solving our expression II.70 for $T_{KT}$ gives

$T_{KT}={\sqrt{gB}a_{0}^{2}\over 8\pi k_{B}}\propto\sqrt{g}\,.$ (II.72)

Thus, we see that the smectic melting temperature grows quite rapidly with the
symmetry breaking field strength $g$; specifically, like $\sqrt{g}$. We will
see in section [VIII] that, while active smectics can also be stabilized
against dislocation unbinding by rotational symmetry breaking fields, they are
much more delicate. Specifically, the critical noise strength (the analog in
those non-equilibrium systems of the critical temperature) grows only linearly
with the symmetry breaking field strength $g$.

Although we do not need to explicitly consider the dynamics of dislocations in
order to understand their unbinding in this equilibrium problem, it is
instructive to do so, in order to facilitate comparison with the active cases
that we will consider next, for which the dynamics is all we have.

To study the dynamics, consider first a general pair of dislocations
$\alpha=(1,2)$ of Burgers’ charges: $b_{1}$ and $b_{2}$, at positions
$\mathbf{r}_{1}={\bf 0}$, $\mathbf{r}_{2}=\mathbf{r}=x\hat{x}+y\hat{y}$. The
dislocation Hamiltonian (II.65) then implies that the energy of this pair will
be

$V(\mathbf{r})=-{\sqrt{gB}b_{1}b_{2}a_{0}^{2}\over
2\pi}\ln\left({|\mathbf{r}^{\prime}|\over a}\right)\,.$ (II.73)

The force experienced by dislocation $\alpha=2$ will therefore be ${\bf
F}=F_{x}\hat{x}+F_{y}\hat{y}$, with its Cartesian components $F_{x,y}$ given
by

$\displaystyle F_{x}$ $\displaystyle=$
$\displaystyle-\partial_{x}V(\mathbf{r})={b_{1}b_{2}a_{0}^{2}xB\sqrt{gB}\over
2\pi(gy^{2}+Bx^{2})}=Ba_{0}b_{2}w_{y}^{(1)}(\mathbf{r})$ $\displaystyle F_{y}$
$\displaystyle=$
$\displaystyle-\partial_{y}V(\mathbf{r})={b_{1}b_{2}a_{0}^{2}yg\sqrt{gB}\over
2\pi(gy^{2}+Bx^{2})}=-ga_{0}b_{2}w_{x}^{(1)}(\mathbf{r})$ (II.74)

where by $\mathbf{w}^{1}(\mathbf{r})$, we mean the contribution to the field
$\mathbf{w}$ at the position $\mathbf{r}$ of the $\alpha=2$ dislocation coming
from the $\alpha=1$ dislocation (i.e., neglecting the field created by
dislocation $\alpha=2$ itself). It is straightforward to show that the
generalization of these forces to configurations with more than two
dislocations is:

$\displaystyle F_{x}^{\alpha}$ $\displaystyle=$ $\displaystyle-
Ba_{0}b_{\alpha}w_{y}(\mathbf{r}_{\alpha})$ $\displaystyle F_{y}^{\alpha}$
$\displaystyle=$ $\displaystyle ga_{0}b_{\alpha}w_{x}(\mathbf{r}_{\alpha})$
(II.75)

where by $\mathbf{w}(\mathbf{r})$, we mean the contribution to the field
$\mathbf{w}$ at the position $\mathbf{r}_{\alpha}$ of the $\alpha$’th
dislocation coming from all of the other dislocations (i.e., neglecting the
field created by dislocation $\alpha$ itself).

Since the dislocation cannot ”tell” whether the local field $\mathbf{w}$ is
created by other dislocations, by spin waves, or by externally applied
stresses, we expect (II.29) to hold more generally if we take
$\mathbf{w}_{\alpha}$ on the right-hand side of those equations to be the
entire $\mathbf{w}$ field, excluding the part due to dislocation $\alpha$
itself. This proves to be important when we consider the effect of stresses at
the boundary on dislocation motion.

There are two important features of the result (II.29) that should be noted:

1) the force on a given dislocation $\alpha$ is determined entirely by the
local value of the field $\mathbf{w}(\mathbf{r}_{\alpha})$ (and its
derivatives) at the location $\mathbf{r}_{\alpha}$ of that dislocation
(excluding the part of that field due to the given dislocation itself).

2) The force in this non-rotation invariant case now depends directly on the
$x$ component $w_{x}$; no spatial derivatives are required. This means in
particular that a uniform $w_{x}$ does generate a force on the dislocation.
This is a consequence of the lack of rotation invariance: as shown by equation
(II.2), a spatially uniform $w_{x}$ corresponds to a uniform rotation of the
layers, which now can lead to a force on the dislocation, since the system is
not rotation-invariant. This consideration will continue to apply in active
smectics, and will allow certain terms in the force in those systems in the
rotation non-invariant case which are absent in the rotation invariant case.

As in the rotation-invariant case, we expect the velocity $\mathbf{v}$ of the
dislocations to be linear in the force $\mathbf{F}$. That is,

$\mathbf{v}_{\alpha}=\bm{\mu}\mathbf{F}_{\alpha}\,,$ (II.76)

where $\bm{\mu}$ is a constant ”mobility tensor”. On the same symmetry grounds
as before, we expect this tensor to be diagonal in our $(x,y)$ coordinate
system (i.e., with the $x$ and $y$ axes respectively parallel and
perpendicular to the mean smectic layers); hence

$v_{x}^{\alpha}=\mu_{x}F_{x}^{\alpha}\,,\,v_{y}^{\alpha}=\mu_{y}F_{y}^{\alpha}\,.$
(II.77)

Using our earlier results (II.75) for the forces on the dislocations, we can
rewrite these as

$\displaystyle v_{x}^{\alpha}$ $\displaystyle=$
$\displaystyle-\mu_{x}Ba_{0}b_{\alpha}w_{y}(\mathbf{r})$ $\displaystyle
v_{y}^{\alpha}$ $\displaystyle=$
$\displaystyle\mu_{y}ga_{0}b_{\alpha}w_{x}(\mathbf{r})\,.$ (II.78)

We see that, like the component $F_{x}$ of the force, the $x$-component of the
dislocation velocity (which is, after all, proportional to $F_{x}$) no longer
vanishes for spatially uniform $w_{x}$. And it need not, since, although a
spatially uniform $w_{x}$ still corresponds to a uniform rotation, in a non-
rotation invariant system, no symmetry forbids a uniform rotation from causing
dislocation motion.

###  II.3 Summary of the equilibrium cases

We have seen that, in equilibrium, rotation invariant 2d smectics,
translational order at non-zero temperature is always short ranged, even in
spin wave theory (i.,e., when dislocations are ignored). Furthermore,
dislocations are always unbound at any non-zero temperature, if the smectic is
rotation invariant. Effectively, this means that 2d smectics melt as soon as
the temperature becomes non-zero. Another way to say this is that 2d smectics
do not exist at temperatures $T>0$.

Breaking rotation invariance by, e.g., applying a rotational symmetry breaking
magnetic field, or breaking the underlying rotation invariance in other ways,
can stabilize quasi-long-ranged translational order (i.e., power law decay of
translational correlations) in spin wave theory. Rotational symmetry breaking
also stabilizes two-dimensional equilibrium smectics against dislocation
unbinding. The temperature $T_{m}$ at which these systems melt vanishes as the
strength $g$ of the applied symmetry breaking field vanishes, according to the
law II.72.

We will see that, while the presence or absence of rotation invariance has no
important effect on the spin wave dynamics of active smectics, nor on the
fields created by dislocations, it has a profound effect on the motion of
dislocations. In fact, like equilibrium 2d smectics, active 2d smectics are
only stable against dislocations if rotation invariance has been explicitly
broken by an externally applied symmetry breaking field. Furthermore, the
field required to stabilize active 2d smectics against dislocations is much
higher than the field needed in equilibrium 2d smectics.

##  III ”Spin wave” (Phonon) theory of active smectics

###  III.1 Rotation invariant 2d active smectics: spin wave theory

In this section, we will review the hydrodynamic theory of active, rotation-
invariant, apolar smectics. For more details, the interested reader is
referred to apolarsm . We will first limit our discussion to “spin-wave
theory”; that is, dislocation-free smectics.

Because there are no conserved quantities, the only hydrodynamic field in our
problem is the layer displacement $u$, which is the ”Goldstone mode”
associated with the breaking of translational symmetry by the layering. The
long-wavelength hydrodynamics of this field is therefore simply the most
general equation of motion, to leading order in space and time derivatives,
that respects the symmetries of this system. These symmetries are rotation and
translation invariance. As we noted earlier in our discussion of the non-
rotation invariant equilibrium smectic, the equation of motion (II.54) for
that system, oddly, is rotation invariant. Therefore, that equation, which we
repeat here for the readers’ convenience, also describes active, rotation
invariant smectics:

$\displaystyle\partial_{t}u=D_{y}\partial_{y}^{2}u+D_{x}\partial_{x}^{2}u+f,$
(III.1)

where $f^{u}$ is a Gaussian, zero-mean spatiotemporally white noise with

$\langle f_{u}({\bf r},t)f_{u}({\bf
r}^{\prime},t^{\prime})\rangle=2\Delta\delta({\bf r}-{\bf
r}^{\prime})\delta(t-t^{\prime})\,.$ (III.2)

Because this is a non-equilibrium system, there is no longer a fluctuation-
dissipation theoremchaikin  relating the noise variance $\Delta$ to the
dissipative terms in (III.1). However, the equation of motion (III.1), with
the noise correlations (III.2) is, as noted, identical to that of an
equilibrium, non-rotation invariant smectic with

$\Gamma k_{B}T=\Delta\,\ \ \ ,\ \ \ \Gamma B=D_{y}\,\ \ \ ,\ \ \ \Gamma
g=D_{x}\,.$ (III.3)

We can therefore use these relations to obtain any spin-wave correlation
function in the active, rotation-invariant smectic from the corresponding
correlation function in an equilibrium, non-rotation invariant smectic. In
particular, this reasoning predicts , in the absence of dislocations, that
active, rotation invariant smectics will exhibit power law singularities near
the $n$’th Bragg peak (i.e., for wavevector
$\mathbf{q}=nq_{0}\hat{y}+\delta\mathbf{q}$ with integer $n$ and
$|\delta\mathbf{q}|\ll q_{0})$:

$\langle|\rho(\mathbf{q},t)|^{2}\rangle\propto|{\bf\delta
q}|^{-2+n^{2}\eta(\Delta)}\,,$ (III.4)

with the non-universal exponentKT

$\eta(D_{x},D_{y},\Delta)={\Delta q_{0}^{2}\over 2\pi\sqrt{D_{x}D_{y}}}\,.$
(III.5)

As in equilibrium systems, this can be measured experimentally by various
scattering techniques (either X-ray or light scattering, depending on the
layer spacing $a$), or, in experiments in which the constituent particles can
actually be imaged, by simply constructing the spatially Fourier transformed
density correlations directly from particle positions. It can be determined
from simulations by the latter approach as well.

Interestingly, as we shall see, in rotation invariant active smectics, this
quasi-long-ranged order is destroyed by unbound dislocations, and they
therefore do not exhibit any singular behavior at the Bragg spots at all. In
rotation non-invariant active smectics, however, the results III.4 and III.6
will hold for sufficiently small noise.

Clearly, the non-universal exponent $\eta(D_{x},D_{y},\Delta)$ for algebraic
decay of translational correlations, when expressed in terms of $D_{x}$ and
$D_{y}$ will continue to be given in terms of $D_{x}$, $D_{y}$, and the noise
strength $\Delta$ (which we also expect to be independent of the symmetry
breaking field $g$ for small $g$) by the result found earlier for rotation
invariant active smectics; i.e.,

$\eta(T)={k_{B}Tq_{0}^{2}\over 2\pi\sqrt{gB}}={\Delta q_{0}^{2}\over
2\pi\sqrt{D_{x}D_{y}}}\,.$ (III.6)

Thus, it would appear, from this spin wave theory, that active smectics are
more robust against fluctuations that equilibrium ones: it looks like we can
get quasi-long-ranged translational order in these systems even without
breaking rotation invariance. As we’ll see in section (VII), this conclusion
is actually wrong, due to the unbinding of dislocations.

One last comment about our equation of motion (III.1) for active, rotation-
invariant smectics is in order. The term $D_{x}\partial_{x}^{2}u$ actten ;
Ramaswamy2000  is, as we discussed in section [II.1], forbidden in equilibrium
rotation-invariant smectic, where it would correspond to a term
$\propto(\partial_{x}u)^{2}$ in the Hamiltonian (II.4) , which is forbidden by
rotation invariance. In our out-of-equilibrium system, however, only the
equation of motion itself must be rotation-invariant, which we have already
shown equation III.1 is. Physically, the origin of this term is that local
vectorial asymmetry of a curved layer must inevitably lead to directed motion
in a self-driven system.

###  III.2 Non-rotation invariant 2d active smectics: effects of a symmetry
breaking field (spin wave theory)

We now turn to the spin-wave theory for active smectics in the presence of a
symmetry breaking field. Even when rotational symmetry is broken, the lowest
order in derivative terms allowed by $x\to-x$, $y\to-y$ symmetry are still
second order derivatives. Therefore, the equation of motion is still:

$\displaystyle\partial_{t}u=D_{y}\partial_{y}^{2}u+D_{x}\partial_{x}^{2}u+f,$
(III.7)

where $f^{u}$ is a Gaussian, zero-mean spatiotemporally white noise with

$\langle f_{u}({\bf r},t)f_{u}({\bf
r}^{\prime},t^{\prime})\rangle=2\Delta\delta({\bf r}-{\bf
r}^{\prime})\delta(t-t^{\prime})\,,$ (III.8)

as in the rotation-invariant case. The only difference between this and the
equilibrium case is that, in equilibrium, $D_{x}$ can only be non-zero due to
the presence of the symmetry breaking field. Indeed, we expect $D_{x}\propto
g$. In active smectics, on the other hand, as we’ve just seen, $D_{x}$ can be
non-zero simply due to the activity. Thus, for small symmetry breaking field
$g$ (the limit we will consider later), we expect $D_{x}$ to be essentially
independent of the symmetry breaking field $g$.

##  IV Dislocations in active smectics: configurations

In our non-equilibrium system, we can no longer determine the fields of the
dislocations by minimizing the free energy, since there is no free energy for
a non-equilibrium system. However, we can readily obtain the fields of static
dislocations simply by looking for steady state solutions of the equations of
motion (III.1), once those equations are suitably rewritten to take into
account the presence of dislocations. As has already been discussed, this
amounts to making the replacements $\partial_{x}u\to w_{x}$ and
$\partial_{y}u\to w_{y}$. Doing so in the equation of motion (III.1), and
setting all time derivatives to zero, gives

$D_{x}\partial_{x}w_{x}(\mathbf{r})+D_{y}\partial_{y}w_{y}(\mathbf{r})=0\,.$
(IV.1)

The other condition on dislocations is the Burger’s condition, which can be
written

$\partial_{x}w_{y}(\mathbf{r})-\partial_{y}w_{x}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha})\,.$
(IV.2)

These two simultaneous linear equations (II.16) and (II.17) are exactly the
same as those we obtained for non-rotation-invariant, equilibrium smectics if
we make the identifications (III.3). Therefore, we can simply transcribe the
solutions we obtained for that problem here. This gives

$\displaystyle
w_{x}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{x}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(IV.3) $\displaystyle
w_{y}(\mathbf{r})=\sum_{\alpha}a_{0}b_{\alpha}G_{y}(\mathbf{r}-\mathbf{r}_{\alpha})\,,$
(IV.4)

where the Green’s functions $G_{x,y}$ are now given by

$\displaystyle G_{x}(\mathbf{r})=-{y\sqrt{D_{x}D_{y}}\over
2\pi(D_{x}y^{2}+D_{y}x^{2})}\,,$ (IV.5) $\displaystyle
G_{y}(\mathbf{r})={x\sqrt{D_{x}D_{y}}\over 2\pi(D_{x}y^{2}+D_{y}x^{2})}\,.$
(IV.6)

Note that we can write these Greens functions in terms of the gradient of a
potential:

$\displaystyle G_{x}(\mathbf{r})=-\varpi\partial_{y}V(\mathbf{r})\,,$
$\displaystyle G_{y}(\mathbf{r})={1\over\varpi}\partial_{x}V(\mathbf{r})\,,$
(IV.7)

where the potential

$V(\mathbf{r})={1\over 2\pi}\ln\left({|\mathbf{r}^{\prime}|\over
a_{0}}\right)={1\over 4\pi}\ln\left({D_{y}x^{2}+D_{x}y^{2}\over
D_{x}a_{0}^{2}}\right)\,,$ (IV.8)

and we’ve defined $\varpi\equiv\sqrt{D_{y}\over D_{x}}$ and
$\mathbf{r}^{\prime}\equiv(x\sqrt{D_{y}\over D_{x}},y)$.

These dislocation fields are essentially identical in form to those we found
for equilibrium, non-rotation invariant smectics in section [II.2]. However,
we will see in the next section that the motion of dislocations in response to
these fields is very different from that case.

##  V Dislocation equation of motion for rotation invariant active smectics

We have established that both the spin wave theory of active, rotation-
invariant smectics, and the field $\mathbf{w}(\mathbf{r})$ generated by
dislocations, are the same as those found for a non-rotation invariant
equilibrium smectic. It might therefore seem reasonable to assume that the
motion of dislocations, and, as a result, the dislocation unbinding transition
in these active, rotation invariant systems would be the same as those of the
equilibrium, non-rotation invariant system. Indeed, precisely this argument
was made in earlier publications by one of usapolarsm ; polarsm .

However, this conclusion proves to be wrong. The reason is that, despite the
just noted similarities between active rotation invariant systems and
equilibrium, non-rotation invariant systems, active, rotation invariant
smectics are still rotation invariant. This fairly obvious (indeed,
tautological!) statement makes the motion of dislocations in an active,
rotation-invariant smectic very different from that in an equilibrium, non-
rotation invariant one. The motion is so different, in fact, that although
dislocations are always bound at sufficiently low temperatures in equilibrium,
non-rotation invariant smectics at sufficiently low temperature, they are
never bound in an active, rotation-invariant smectic with any non-zero noise,
no matter how small.

We will now demonstrate this. We begin by deriving the equation of motion for
dislocations in an active, rotation-invariant smectic.

We restrict ourselves to “unit” dislocations, by which we mean a dislocation
whose Burgers’ number $b$ has the smallest possible magnitude; i.e., $b=\pm
1$.

As can be seen from figure [3], and from our analytic solutions (IV.4) for the
dislocation fields, a dislocation in a smectic is an inherently polar object;
it breaks the left-right symmetry along the layers. This means that, in an
active system, there is no symmetry forbidding spontaneous motion of
dislocations either left or right along the layers. Therefore, by the Curie
principle curie , that “anything that’s not forbidden is compulsory”, we
expect such motion to occur.

Since a unit dislocation with $b=-1$ is just the mirror image of one with
$b=1$, if $b=+1$ dislocations spontaneously move to the left, $b=-1$ must
spontaneously move to the right, and visa-versa. The motion should be along
the local layer direction, since spontaneous motion perpendicular to the
layers is forbidden by the fact that dislocations do not break up-down
symmetry.

Of course, a local curvature of the layers will break up-down symmetry.
Indeed, this is the origin of the force in equilibrium smectics in the
$y$-direction given in equation (II.29). Note, however, that because this
effect involves curvature, it involves at least two derivatives of the
displacement field, or, equivalently, one derivative of $\mathbf{w}$. Hence,
to leading (i.e., zeroth) order in derivatives of $\mathbf{w}$, the motion of
a dislocation must be along the layers. This has profound implications for the
stability of the active smectic state in rotation invariant systems, as we
will see.

These considerations imply that, to zeroth order in gradients of $\mathbf{w}$,
the velocity $\mathbf{v}_{\alpha}$ of the $\alpha$’th dislocation must take
the form:

$\mathbf{v}_{\alpha}=v_{s}{\rm sgn}(b_{\alpha})\hat{z}\times\hat{{\bf
n}}(\mathbf{r}_{\alpha})$ (V.1)

where

$\hat{{\bf n}}\approx\hat{y}-\phi(\mathbf{r})\hat{x}$ (V.2)

is the local normal to the smectic layers. The characteristic speed $v_{s}$
appearing in (V.1) is a system-dependent parameter. It will depend,
importantly, on local properties like the mean layer spacing $a$ of the active
smectic. It is this dependence that makes it possible for the active smectic
to reach homeostasis, as we’ll argue below.

The direction $\hat{z}\times\hat{{\bf n}}$ of $\mathbf{v}$ is dictated by the
requirement that this spontaneous motion be along the layers, which, as we
just discussed, is required by up-down symmetry. The factor of ${\rm
sgn}(b_{\alpha})$ simply reflects the fact noted above that oppositely charged
dislocations must move in opposite directions.

Since our definition (II.10) of $\mathbf{w}$ implies that the local layer
spacing is simply

$a(\mathbf{r})=a_{0}(1+w_{y}(\mathbf{r}))\,,$ (V.3)

where $a_{0}$ is the “reference” layer spacing , relative to which we measure
$\mathbf{w}$, the dependence of the spontaneous speed $v_{s}$ on the layer
spacing $a$ is equivalent to dependence on $w_{y}$. That is, we can (and
will!) take $v_{s}$ to be a local function of $w_{y}$.

Note that rotation invariance forbids any dependence of $v_{s}$ on $w_{x}$,
since a uniform $w_{x}$ corresponds to a pure rotation, which cannot change
the spontaneous speed (or, indeed, any local scalar) in a rotation invariant
system.

These arguments imply that the spontaneous speed $v_{s}(\mathbf{r})$ at a
point $\mathbf{r}$ can be written as

$v_{s}(\mathbf{r})=v_{s}(a(\mathbf{r}))=v_{s}(a_{0}(1+w_{y}(\mathbf{r}))\equiv
v_{s}(w_{y}(\mathbf{r}))\,.$ (V.4)

Using (V.4) in our expression (V.1) for the dislocation velocity gives

$\displaystyle v_{x}^{\alpha}$ $\displaystyle=$ $\displaystyle v_{s}{\rm
sgn}(b_{\alpha})+\mu_{x}b_{\alpha}w_{y}(\mathbf{r}_{\alpha})$ $\displaystyle
v_{y}^{\alpha}$ $\displaystyle=$ $\displaystyle v_{s}{\rm
sgn}(b_{\alpha})w_{x}(\mathbf{r}_{\alpha})\,.$ (V.5)

We will argue in the next section that, for an active smectic confined between
fixed boundaries, $v_{s}$ vanishes in the steady state. We will refer to this
state as the state of “homeostasis”. Note that this implies that there is no
motion in the $y$-direction (i.e., normal to the smectic layers) in the
homeostatic state. This in turn implies that, in the presence of noise,
dislocations in an active smectic are always unbound. As a result, the active
smectic state is, at any non-zero noise, always destroyed by dislocation
unbinding in rotation invariant active smectics.

##  VI Dislocations: Self-propulsion and the approach to homeostasis

The dominant term in the equations of motion (V.5) is the “self-propulsion”
term $v_{s}{\rm sgn}(b_{\alpha})$. For $v_{s}>0$, this term will make positive
dislocations move to the right, and negative dislocations move to the left.
Obviously, this switches for $v_{s}<0$.

Because $w_{x,y}(\mathbf{r}_{\alpha})\to 0$ as $|\mathbf{r}|\to\infty$ (as can
be seen from equation (IV.4)), the interactions between dislocations cannot
compete with this constant “external force” -like motion. Hence, even tightly
bound pairs of dislocations will eventually be rent asunder by the spontaneous
motion , with all of the positive dislocations moving to one side, and all of
the negative dislocations moving to the other.

We therefore expect dislocation pairs to constantly nucleate, be ripped apart
by this spontaneous velocity, and traverse the system.

Consider first the case $v_{s}(w_{y}=0)>0$. In this case, all positive
dislocations will eventually traverse the system from left to right, while all
negative dislocations will eventually traverse the system from right to left.
It is easy to see, both by inspection of figure 3, and from our expressions
for the dislocation fields, that each time one of these happens (i.e., each
time either a positive dislocation traverses the system from left to right, or
a negative dislocation traverses the system from right to left), the number of
layers in the system is increased by one. Therefore, if the boundaries of the
system at the top and bottom remain fixed, the strain $w_{y}$ will decrease
(i.e., become more negative), since the mean layer spacing is reduced.

The crucial point here is that this self-propelled dislocation motion changes
the mean strain $w_{y}$ in the system. But since the dislocation self-
propulsion speed $v_{s}(w_{y})$ is itself a function of $w_{y}$, this implies
that, as this process of pair nucleation, separation, and motion across the
system continues, the strain $w_{y}$ will continue to evolve.

Will it ever stop? Yes, it will, provided that the speed $v_{s}(w_{y})$
vanishes at some negative $w_{y}$. We would expect it to do so: as shown by
(II.78), in an equilibrium smectic, a more negative $w_{y}$ causes positive
dislocations to move to the left dislocation with a speed proportional to
$w_{y}$. Stability requires that the negative strain induced by the process
just described will oppose the motion of positive dislocations to the right.
This opposition should get stronger as $w_{y}$ becomes more negative, so it is
very plausible that a type of “homeostasis” will eventually be reached, at
which the strain $w_{y}$ takes on a value $w_{y,{\rm h}}$ such that

$v_{s}(w_{y,{\rm h}})=0\,.$ (VI.1)

This will almost always happen whenever the direction of spontaneous motion of
the dislocations causes them to move in a direction that makes the strain
evolve in such a way as to oppose the spontaneous motion of the dislocations.
As just discussed, this is most likely to occur if $v_{s}(w_{y}=0)>0$, and
${dv_{s}\over dw_{y}}>0$ giving rise to a steady state extensile stress normal
to the layers. The opposite case of $v_{s}(w_{y}=0)<0$, and ${dv_{s}\over
dw_{y}}<0$ will also reach homeostasis, and give rise to a contractile steady
state stress normal to the layers.

So we expect stable active smectic systems to reach homeostasis, as defined by
VI.1. In such systems, the homeostatic layer spacing $a_{h}$ will be

$a_{h}=a_{0}(1+w_{y,{\rm h}})\,.$ (VI.2)

It clearly makes sense to define this homeostatic layer spacing as our
reference layer spacing, and measure our displacement field $u$ relative to
that. It is easy to relate the $u$ field $u_{h}(\mathbf{r})$ defined relative
to this homeostatic state to that $u_{0}(\mathbf{r})$ defined relative to the
initial state with layer spacing $a_{0}$; the required transformation is just
a linear function of the Cartesian coordinate $y$:

$u_{h}(\mathbf{r})=u_{0}(\mathbf{r})-w_{y,{\rm h}}y\,.$ (VI.3)

Henceforth, we will drop the subscript ${\rm h}$, and implicitly assume that
our $u$ field is measured relative to the homeostatic state. With this choice
of variables, the homeostatic condition VI.1 becomes simply

$v_{s}(w_{y}=0)=0\,.$ (VI.4)

Note that the argument just presented for the approach of active smectics to
the homeostatic state depended on the presence of fixed boundaries without
layer flux at the top and bottom of the system. For active smectics confined
under constant stress between movable boundaries, however, a homeostatic state
is never reached. Instead, the active smectic either grows arbitrarily large,
pushing the boundaries ever further out, or shrinks and disappears. The
unbounded growth scenario occurs if the applied normal stress $\sigma_{n}$ at
the boundaries is less than some homeostatic value $\sigma_{c}$, while
shrinkage and disappearance occurs if $\sigma_{n}>\sigma_{c}$.

The reason for this behavior is clear: like all elastic systems, fixing the
stress is equivalent to fixing the strain. In smectics, the “strain” is just
$w_{y}$. Hence, by fixing the stress at the boundary, we fix $w_{y}$. If
$w_{y}$ is fixed at a value such that $v_{s}(w_{y})>0$ (this corresponds to
$\sigma_{n}<\sigma_{c}$), then positive dislocations will move to the right,
and negative ones to the left, thereby adding layers to the system. The only
way to keep $w_{y}$ fixed, therefore, is for the top and bottom surfaces to
move out, to accommodate the extra layers. This process will continue
indefinitely.

On the other hand, if $w_{y}$ is fixed at a value such that $v_{s}(w_{y})<0$
(this corresponds to $\sigma_{n}>\sigma_{c}$), then positive dislocations will
move to the right, and negative ones to the left, thereby adding layers to the
system. The only way to keep $w_{y}$ fixed, therefore, is for the top and
bottom surfaces to move in, to accommodate the loss of layers. This process
will continue until the active smectic disappears completely. This behavior is
reminiscent of that predicted for tissues and observed in epithelia
homeoTissue ,homeoTissuetwo , homeoTissueexp . The addition and removal of
layers corresponds to the addition of cells by cell division and the removal
of cells by cell death. In a tissue under homeostatic conditions, the cell
division rate is exactly balanced by cell death rate. Since both cell division
and cell death depend on tissue pressure, in general there will be only one
tissue pressure value for which these rates balance exactly. This defines the
homeostatic pressure, corresponding to the stress $\sigma_{c}$ in the
homeostatic state of the smectic. If the tissue is given a prescribed volume,
keeping constant biochemical conditions, cells will divide and the tissue will
grow to occupy all space and settle to steady state, ie homeostasis, when the
tissue pressure reaches the homeostatic pressure. Alternatively if the tissue
is kept at a constant pressure larger than the homeostatic one, cell death
will win over cell division and the tissue will disappear. If the pressure is
kept at a lower value the tissue will invade all space. Tissue invasion of one
type of tissue by another one, will occur if the homeostatic pressure of the
invading tissue is larger than the homeostatic pressure of the invaded tissue.

While the above discussion has been very physical, biochemistry nonetheless
plays an important role, since the homeostatic pressure values depend on local
biochemical conditions.

Returning now to the case of fixed boundaries, and the resultant homeostatic
state, we can now ask , since the spontaneous velocity tearing dislocation
pairs apart vanishes in the homeostatic state, whether or not it is possible
to achieve a state free of unbound dislocations. Only if such a state is
possible can we have a true smectic.

In the presence of noise, the answer to this question proves to be no:
dislocations in a rotation invariant active smectic will always be unbound. We
will demonstrate this in the next section.

##  VII Motion at homeostasis and the destruction of the active smectic phase

As shown in the last section, at homeostasis,

$v_{s}(\mathbf{r})=v_{s}(a(\mathbf{r}))=v_{s}(a_{\rm
h}(1+w_{y}(\mathbf{r}))\approx\mu_{x}w_{y}(\mathbf{r})\,,$ (VII.1)

where in the last, approximate, equality, we have expanded for small
$w_{y}(\mathbf{r})$, and defined the “mobility” $\mu_{x}\equiv{1\over
a}\left({dv_{s}(a)\over da}\right)_{a=a_{\rm h}}$. We have also used the fact
that

$v_{s}(a_{\rm h})=0\,,$ (VII.2)

since, as discussed above, isolated dislocations do not spontaneously move in
the homeostatic state.

Inserting VII.1 into our general equation of motion (V.5) for the
dislocations, and linearizing those equations in the strain field $\mathbf{w}$
gives

$\displaystyle v_{x}^{\alpha}(t)$ $\displaystyle=$
$\displaystyle\mu_{x}b_{\alpha}w_{y}(\mathbf{r}_{\alpha},t)+f^{\alpha}_{x}(t)\,,$
$\displaystyle v_{y}^{\alpha}$ $\displaystyle=$ $\displaystyle
f^{\alpha}_{y}(t)\,,$ (VII.3)

where we have added a “Langevin force” - i.e., a random white noise
$\mathbf{f}^{\alpha}$ \- to the equation of motion. In equilibrium, these
would simply be thermal noises, with variances proportional to temperature. In
our non-equilibrium system, they can have non-thermal, active contributions as
well. We will assume, as seems reasonable, that these forces are white,
Gaussian, zero-mean, and decorrelated between different dislocations. Taking
these conditions together with the $x\to-x$, $y\to-y$ symmetries of the apolar
smectic state we’re considering in this paper, implies that these forces have
correlations

$\displaystyle\langle\mathbf{f}^{\alpha}(t)\rangle$ $\displaystyle={\bf 0}\,,$
$\displaystyle\langle f^{\alpha}_{i}(t)f^{\beta}_{j}(t^{\prime})\rangle$
$\displaystyle=\Bigg{(}\Delta_{x}\delta_{ij}^{x}+\Delta_{y}\delta_{ij}^{y}\Bigg{)}\delta(t-t^{\prime})\delta_{\alpha\beta}\,.$

Here $\delta_{ij}^{k}=1$, when $i=j=k$ and $0$ otherwise. Since the forces are
assumed to be Gaussian random variables (as suggested by the central limit
theorem), these correlations completely specify the distribution of the forces
$\mathbf{f}^{\alpha}(t)$.

This implies that, to linear order in the field $\mathbf{w}$, dislocation
motion in the $y$-direction is a perfectly random walk. Therefore, any pair of
dislocations will eventually wander arbitrarily far apart in the
$y$-direction. Hence, dislocations in active, rotation invariant smectics are
always unbound. Another way to say this is that a true active smectic phase
cannot exist in a noisy, rotation invariant system.

##  VIII Binding dislocations with a symmetry breaking field

As in equilibrium smectics, it is possible to make dislocations bind in 2d
active smectics by applying symmetry breaking fields. Once we do so, there is
no longer any symmetry argument forcing the dislocation velocity
$\mathbf{v}_{s}$ to be independent of the “rotational” component of the strain
$w_{x}$. Nor must this velocity be directed along the layers. Therefore, we
can write in general:

$\mathbf{v}_{s}^{\alpha}=\bigg{(}v_{s\parallel}\hat{z}\times\hat{{\bf
n}}(\mathbf{r}_{\alpha})+v_{s\perp}\hat{{\bf
n}}(\mathbf{r}_{\alpha})\bigg{)}{\rm sgn}(b_{\alpha})$ (VIII.1)

By up-down symmetry, $v_{s\perp}(w_{x}=0)=0$. Therefore, the leading order
term in the expansion of $v_{s\perp}(w_{x})$ in powers of $w_{x}$ is the
linear term:

$v_{s\perp}(w_{x})=-\mu_{y}w_{x}\,,$ (VIII.2)

where we’ve defined

$\mu_{y}\equiv-\left({dv_{d\perp}(w_{x})\over dw_{x}}\right)_{w_{x}=0}\,.$
(VIII.3)

By the same reasoning about homeostasis that we applied to the rotation
invariant case, we expect that, at homeostasis,

$v_{s\parallel}(a_{\rm h})=0\,.$ (VIII.4)

We can therefore again choose the state with $a=a_{\rm h}$ to be our reference
state, and expand for small strain $w_{y}$, obtaining, just as we did for the
rotation invariant case,

$v_{s\parallel}(\mathbf{r})=v_{s}(a(\mathbf{r}))=v_{s}(a_{\rm
h}(1+w_{y}(\mathbf{r}))\approx\mu_{x}w_{y}(\mathbf{r})\,,$ (VIII.5)

where again, we have excluded a term proportional to $w_{x}$ by up-down
symmetry.

Inserting the results (VIII.2) and (VIII.5) into (VIII.1), using again the
relation V.2 $\hat{{\bf n}}\approx\hat{y}-\phi(\mathbf{r})\hat{x}$ between the
layer normal $\hat{{\bf n}}$ and the strain $w_{x}$, and expanding to linear
order in the strain $\mathbf{w}$, we obtain

$v_{x}=\mu_{x}w_{y}\,,\,v_{y}=-\mu_{y}w_{x}\,.$ (VIII.6)

Since the $\mu_{y}$ term in (VIII.6) only appears due to the rotational
symmetry breaking field, $\mu_{y}$ must vanish as the strength $g$ of that
symmetry breaking field goes to zero. We therefore expect

$\mu_{y}\propto g\,$ (VIII.7)

for small symmetry breaking field strength $g$. This is the only parameter in
the dislocation dynamics that we expect to exhibit any strong dependence on
$g$ at small $g$. We will use this fact later to determine the behavior of the
critical noise strength at which dislocation unbinding occurs with $g$.

Note also that, as we saw in our discussion in section [II.2.1] of symmetry
breaking fields in equilibrium smectics, $g$ could scale non-linearly with
experimentally tunable parameters like magnetic field $H$. Indeed, we expect
that, as in equilibrium, $g\propto H^{2}$ for magnetic symmetry breaking
fields. On the other hand, for symmetry breaking induced by etching grooves
into the 2d surface, we expect $g$ to scale linearly with the density of those
grooves.

Consider now an isolated neutral pair of fundamental dislocations, one with
Burgers number $b_{+}=+a$ located at $\mathbf{r}_{+}\equiv(x_{+},y_{+})$, the
other with Burgers number $b_{-}=-a$ located at
$\mathbf{r}_{-}\equiv(x_{-},y_{-})$.

Since, as discussed earlier, the spin-wave equation of motion (III.7) is
unchanged in form by the presence of the symmetry breaking fields, and since,
furthermore, the Burgers’ condition is also unchanged, we can use the
expressions (IV.4) for the dislocation fields for active, rotation invariant
active smectics for this non-rotation invariant case as well. Therefore, the
strain field $\mathbf{w}(\mathbf{r}_{+})$ due to the $-$ dislocation at the
position $\mathbf{r}_{+}$ of the $+$ dislocation is, from equation (IV.4),

$w_{x}(\mathbf{r}_{+})=-aG_{x}(\mathbf{r})\,\ \ \ ,\ \ \
w_{y}(\mathbf{r}_{+})=-aG_{y}(\mathbf{r})\,,$ (VIII.8)

where we’ve defined the relative displacement
$\mathbf{r}\equiv(x_{+}-x_{-},y_{+}-y_{-})\equiv(x,y)$. As always, we exclude
the field of the $+$ dislocation at $\mathbf{r}_{+}$ from the field
$\mathbf{w}$ above, since the $+$ dislocation only responds to the field of
the other dislocation. Likewise, the strain field $\mathbf{w}(\mathbf{r}_{-})$
due to the $+$ dislocation at the position $\mathbf{r}_{-}$ of the $-$
dislocation is, from equation (IV.4),

$w_{x}=aG_{x}(\mathbf{r})\,\ \ \ ,\ \ \ w_{y}=aG_{y}(\mathbf{r})\,,$ (VIII.9)

Using the dislocation equation of motion (VIII.6), together with our
expressions (VIII.8) and (VIII.9) for the fields at the $+$ and $-$
dislocations, , we obtain the equations of motion for the dislocations:

$\displaystyle{dx_{\pm}\over
dt}=\mp\mu_{x}aG_{y}(\mathbf{r})+f^{\pm}_{x}(t)\,,$
$\displaystyle{dy_{\pm}\over
dt}=\pm\mu_{y}aG_{x}(\mathbf{r})+f^{\pm}_{y}(t)\,,$ (VIII.10)

where, as we did for the rotation invariant case, we have added random noises
$\mathbf{f}^{\pm}$ to the equation of motion for each dislocation, to take
into account random microscopic processes (including, but not limited to,
thermal fluctuations) that move the dislocations. We will continue to take
these noises to have correlations given by LABEL:disforcecorr, with the
indices $\alpha$ and $\beta$ running over the two values $+$ and $-$.

From VIII.10, it follows that the relative displacement $\mathbf{r}$ obeys

$\displaystyle{dx\over dt}$ $\displaystyle={dx_{+}\over dt}-{dx_{-}\over
dt}=-2\mu_{x}aG_{y}(\mathbf{r})+f_{x}(t)\,,$ $\displaystyle{dy\over dt}$
$\displaystyle={dy_{+}\over dt}-{dy_{-}\over
dt}=2\mu_{y}aG_{x}(\mathbf{r})+f_{y}(t)\,,$ (VIII.11)

where the “relative force” $\mathbf{f}=\mathbf{f}^{+}-\mathbf{f}^{-}$. From
this, it follows that the relative force is also Gaussian, with mean and
variance given by:

$\displaystyle\langle\mathbf{f}(t)\rangle$ $\displaystyle={\bf 0}\,,$
$\displaystyle\langle f_{i}(t)f_{j}(t^{\prime})\rangle$
$\displaystyle=2\Bigg{(}\Delta_{x}\delta_{ij}^{x}+\Delta_{y}\delta_{ij}^{y}\Bigg{)}\delta(t-t^{\prime})\delta_{\alpha\beta}\,.$

Using our earlier expression (IV.7) relating the Greens functions $G_{x,y}$ to
the gradient of the potential (IV.8), we can rewrite this as

$\displaystyle{dx\over
dt}=-2{\mu_{x}\over\varpi}a\partial_{x}V(\mathbf{r})+f_{x}(t)\,,$
$\displaystyle{dy\over dt}=-2\mu_{y}\varpi
a\partial_{y}V(\mathbf{r})+f_{y}(t)\,.$ (VIII.13)

Note that if the noise variances $\Delta_{x,y}$ and the effective “relative
mobilities”

$\mu^{\rm rel}_{x}\equiv{2a\mu_{x}\over\varpi}\ \ \ ,\ \ \ \mu^{\rm
rel}_{y}\equiv 2a{\mu_{y}\varpi}$ (VIII.14)

satisfied

${\Delta_{x}\over\Delta_{y}}={\mu^{\rm rel}_{x}\over\mu^{\rm rel}_{y}}\,,$
(VIII.15)

then the equations of motion VIII.13 could be written in the form

$\displaystyle{dr_{i}\over
dt}=-\tilde{\Gamma}_{ij}\partial_{j}U(\mathbf{r})+f_{i}(t)\,,$ (VIII.16)

with $U(\mathbf{r})=KV(\mathbf{r})$, a diagonal kinetic coefficient tensor

$\displaystyle\tilde{\Gamma}_{ij}=\Bigg{(}\tilde{\Gamma}_{x}\delta_{ij}^{x}+\tilde{\Gamma}_{y}\delta_{ij}^{y}\Bigg{)}\,,$
(VIII.17)

and the noise correlations given by

$\displaystyle\langle f^{\alpha}_{i}(t)f^{\beta}_{j}(t^{\prime})\rangle$
$\displaystyle=2k_{B}T\tilde{\Gamma}_{ij}\delta(t-t^{\prime})\delta_{\alpha\beta}\,.$

The reader can easily show for herself that this works provided the
temperature $T$, kinetic coefficient tensor components $\tilde{\Gamma}_{x,y}$,
and effective spin wave stiffness $K$ obey

${k_{B}T\over K}={\Delta_{x}\over\mu^{\rm rel}_{x}}={\Delta_{y}\over\mu^{\rm
rel}_{y}}\ \ \ ,\ \ \ \tilde{\Gamma}_{x,y}={\mu^{\rm rel}_{x,y}\over K}\,.$
(VIII.19)

The relation LABEL:disforcecorreqeq between these noise correlations and the
kinetic coefficient tensor $\tilde{\Gamma}_{ij}$ is exactly that required by
the fluctuation-dissipation relationschaikin . Therefore, the relative motion
of the dislocation pair is exactly that of a pair moving in equilibrium in the
potential $U(\mathbf{r})$. This is precisely the equilibrium model for the
Kosterlitz-Thouless transition discussed in section [II.2.3]. Therefore, if
this system did happen to satisfy the relation (VIII.15), it would undergo a
Kosterlitz-Thouless unbinding transition as noise was increased. Most
importantly, at sufficiently small, but non-zero, noise (i.e., sufficiently
low temperature in the equilibrium model), dislocations would remain bound,
and the active smectic phase would be stable.

However, in a weak symmetry breaking field, the system will always be far from
satisfying the relation (VIII.15). This can be seen from limiting behaviors of
the $\Delta$’s and $\mu$”s in the limit of weak symmetry breaking field $g$:
they all go to finite constants as $g\to 0$ except $\mu_{y}$, which, as
discussed earlier, vanishes according to $\mu_{y}\propto g$ as $g\to 0$.

Even if the symmetry breaking field is not weak, there is no reason our non-
equilibrium active smectic should obey (VIII.15). Therefore, the connection to
the equilibrium Kosterlitz-Thouless transition just made for the special case
in which equation (VIII.15) is satisfied will not, in general, hold.

Nonetheless, we expect this system to undergo something very like a
Kosterlitz-Thouless transition, at least for very weak symmetry breaking
fields. To establish this, however, we cannot use equilibrium arguments.
Instead, we must use the more general Fokker-Planck equation for the generic
non-equilibrium system, to which we now turn.

Using standard techniqueschaikin , we can show that the stochastic equation of
motion (VIII.13) for the relative displacement $\mathbf{r}$ of the two
dislocations implies that the probability density $\rho(x,y,t)$ for the
relative displacement vector $\mathbf{r}$ obeys the Fokker-Planck equation:

$\partial_{t}\rho+{\bf\nabla}\cdot(\mathbf{u}\rho)-(\Delta_{x}\partial_{x}^{2}+\Delta_{y}\partial_{y}^{2})\rho=0\,,$
(VIII.20)

where

$u_{i}(\mathbf{r})\equiv-\mu_{ij}\partial_{j}V(\mathbf{r})\,,$ (VIII.21)

with the effective relative mobility tensor

$\mu_{ij}=\Bigg{(}\mu^{\rm rel}_{x}\delta_{ij}^{x}+\mu^{\rm
rel}_{y}\delta_{ij}^{y}\Bigg{)}\,,$ (VIII.22)

is the deterministic part of the relative dislocation velocity in equation
(VIII.13).

Looking for steady-state solutions to this equation, we can set the time
derivative to zero, and drop the time dependence of $\rho(x,y,t)$ (that is,
set $\rho(x,y,t)=\rho(x,y)$). Doing so, and using our expression (IV.8) for
$V(\mathbf{r})$, leads to the steady state equation:

$(\Delta_{x}\partial_{x}^{2}+\Delta_{y}\partial_{y}^{2})\rho+\partial_{x}\left({\gamma_{x}x\rho\over
x^{2}+\alpha y^{2}}\right)+\partial_{y}\left({\gamma_{y}y\rho\over
x^{2}+\alpha y^{2}}\right)=0\,,$ (VIII.23)

where we’ve defined

$\alpha\equiv{D_{x}\over D_{y}}={1\over\varpi^{2}}$ (VIII.24)

and

$\gamma_{i}={\mu^{\rm rel}_{i}\over 2\pi}\ \ \ ,\ \ \ \,\,i=(x,y)\,.$
(VIII.25)

Since we expect that, for small symmetry breaking field $g$, $\mu_{y}\propto
g$, while all other parameters should go to finite, non-zero constants as
$g\to 0$, we therefore expect

$\gamma_{y}\propto g\,\,\,{\rm as}\,\,\,\,g\to 0\,.$ (VIII.26)

We will use this scaling law later to determine how the critical noise
strength $\Delta_{y}^{c}(\zeta,g)$ at which noise destroys smectic order
depends on activity $\zeta$ and symmetry breaking field strength $g$.

Our experience with the equilibrium case suggests that we seek a solution of
this equation which falls off like a power law; i.e., $\rho(\mathbf{r})\propto
r^{-\nu}$ for large $r$. We will therefore insert the scaling ansatz

$\rho(x,y)=y^{-\nu}\Phi(x/y)\,$ (VIII.27)

into VIII.23.

Doing so, we find that such a solution will indeed work, provided that the
scaling function $\Phi(z)$ obeys the ODE:

$\Delta_{x}\Phi^{\prime\prime}(z)+\Delta_{y}\bigg{[}z^{2}\Phi^{\prime\prime}(z)+2(\nu+1)z\Phi^{\prime}(z)+\nu(\nu+1)\Phi(z)\bigg{]}=-\gamma_{x}{d\over
dz}\left({z\Phi(z)\over
z^{2}+\alpha}\right)+\gamma_{y}\left[{(1+\nu)\Phi(z)\over
z^{2}+\alpha}+z{d\over dz}\left({\Phi(z)\over z^{2}+\alpha}\right)\right]\,.$
(VIII.28)

We will not need to solve this equation (fortunately!); rather, we will use it
simply to establish that the scaling function $\Phi(z)$ does nothing singular
as $\gamma_{y}\to 0$; that is, as the symmetry breaking field strength goes to
zero. In particular, the range of $\Phi(z)$ \- that is, the value of $z$ at
which $\Phi(z)$ starts falling off fast enough that its integral converges -
does not diverge as $\gamma_{y}\to 0$. To see this, consider (VIII.29) at
$z=0$, where it reads

$\Phi^{\prime\prime}(z=0)=-\bigg{[}\left({\Delta_{y}\over\Delta_{x}}\right)\nu(1+\nu)+{(\gamma_{x}-\gamma_{y}(1+\nu))\over\alpha\Delta_{x}}\bigg{]}\Phi(z=0)\,.$
(VIII.29)

Since $\Delta_{x}$ and $\Delta_{y}$ are of the same order of magnitude, the
coefficient of $\Phi(z=0)$ in (VIII.29) is at least $O(1)$. Therefore,
$\Phi^{\prime\prime}(z=0)$ is a negative number whose magnitude is $\gtrsim
O(1)$ times $\Phi(z=0)$. This means that we would expect $\Phi(z)$ to drop
from $\Phi(z=0)$ to small values on a scale no larger than $O(1)$. For large
$z$ the behavior of $\Phi(z)$ obeys
$z^{2}\Phi^{\prime\prime}(z)+2(\nu+1)z\Phi^{\prime}(z)+\nu(\nu+1)\Phi(z)=0$,
which implies $\Phi(z)\sim z^{-\nu}$ for large $z$.

Our scaling ansatz (VIII.27) implies that

$f(y)\equiv\int_{-\infty}^{\infty}\rho(x,y)dx=\int_{-\infty}^{\infty}y^{-\nu}\Phi(x/y)dx=y^{1-\nu}\Upsilon_{1}\,,$
(VIII.30)

where we’ve defined

$\Upsilon_{1}\equiv\int_{-\infty}^{\infty}\Phi(z)dz\,.$ (VIII.31)

Our above observation that $\Phi(z)$ becomes small for $z\gtrsim{\cal O}(1)$
regardless of the value of $\gamma_{y}$ implies that $\Upsilon_{1}={\cal
O}(1)$ regardless of the value of $\gamma_{y}$ as well. Note also that
$\Upsilon_{1}$ is independent of $y$.

Now, integrating equation (VIII.23) over $x$ from $-\infty$ to $\infty$ gives

$\Delta_{y}f^{\prime\prime}(y)=-\gamma_{y}{d\over
dy}\bigg{[}y\int_{-\infty}^{\infty}{\rho(x,y)dx\over x^{2}+\alpha
y^{2}}\bigg{]}\,.$ (VIII.32)

In deriving this equation, we have dropped surface terms that must vanish
since the scaling function $\Phi(z)$ vanishes sufficiently rapidly to be
integrable.

Using our scaling ansatz (VIII.27) enables us to rewrite

$\int_{-\infty}^{\infty}{\rho(x,y)dx\over x^{2}+\alpha
y^{2}}=\int_{-\infty}^{\infty}{y^{-\nu}\Phi(x/y)dx\over x^{2}+\alpha
y^{2}}=y^{-1-\nu}\Upsilon_{2}$ (VIII.33)

where we’ve defined

$\Upsilon_{2}\equiv\int_{-\infty}^{\infty}{\Phi(z)dz\over z^{2}+\alpha}\,.$
(VIII.34)

Like $\Upsilon_{1}$, $\Upsilon_{2}$ can be readily seen to be ${\cal O}(1)$,
even when $\gamma_{y}\to 0$. Thus VIII.32 can be rewritten as

$\Delta_{y}{d^{2}\over dy^{2}}(\Upsilon_{1}y^{1-\nu})=-\gamma_{y}{d\over
dy}(\Upsilon_{2}y^{-\nu})\,,$ (VIII.35)

which can readily be seen to work (because both sides scale the same way with
$y$; specifically, like $y^{-(\nu+1)}$) provided

$\Delta_{y}\Upsilon_{1}\nu(\nu-1)=\gamma_{y}\Upsilon_{2}\nu\,.$ (VIII.36)

Since $\nu=4$ at the dislocation unbinding transition, (VIII.36) implies that
the value $\Delta_{y}^{c}$ of $\Delta_{y}$ at the transition obeys

$\Delta_{y}^{c}=\gamma_{y}\left({\Upsilon_{2}\over 3\Upsilon_{1}}\right)\,.$
(VIII.37)

Since $\Upsilon_{1}$ and $\Upsilon_{2}$ are of ${\cal O}(1)$, even when
$\gamma_{y}\to 0$, (VIII.37) implies that

$\Delta_{y}^{c}=\gamma_{y}\times{\cal O}(1)\,.$ (VIII.38)

Given our earlier argument that $\gamma_{y}$ should vanish linearly with the
symmetry breaking field $g$ as $g\to 0$, this result implies that the noise
strength $\Delta_{y}^{c}$ at the transition should also vanish linearly with
$g$ as $g\to 0$.

Since we also expect that the noise strengths $\Delta_{x,y}$ in the
dislocation equation of motion noise correlations (LABEL:reldisforcecorr) are
both proportional to the spin wave noise strength $\Delta$, this implies that
the value $\Delta_{c}$ of the spin wave noise strength $\Delta$ at the
transition should also scale linearly with the symmetry breaking field
strength $g$ for small $g$; that is

$\Delta_{c}\propto g\,\,\,{\rm as}\,\,\,\,g\to 0\,.$ (VIII.39)

This result should be contrasted with the equilibrium result

$\Delta_{c}^{\rm eq}\propto\sqrt{g}\,\,\,{\rm as}\,\,\,\,g\to 0\,,$ (VIII.40)

which follows from equation (II.72) if we interpret $T_{KT}$ as the critical
noise correlation strength. We therefore see that, for small symmetry breaking
fields $g$, the critical noise strength $\Delta_{c}$ for dislocation unbinding
and the melting of the smectic phase is much weaker for active smectics than
for equilibrium smectics. Active smectics are less robust against melting,
even in the presence of symmetry breaking fields, than their equilibrium
counterparts.

Another consequence of this result VIII.39 is that the critical value
$\eta_{c}$ of the exponent $\eta$ for algebraic decay of smectic translational
order becomes non-universal. Indeed, since the diffusion constants $D_{x}$ and
$D_{y}$ go to finite, non-zero constants as the symmetry breaking field $g\to
0$, while the critical noise strength $\Delta_{c}$ vanishes according to
VIII.39, the value of $\eta_{c}$ also vanishes linearly with symmetry breaking
field:

$\eta_{c}\propto{\gamma_{y}\over\sqrt{D_{x}D_{y}}}\propto g\,\,\,{\rm
as}\,\,\,\,g\to 0\,.$ (VIII.41)

##  IX From Malthusian to incompressible active smectics

In the preceding paragraphs, for the sake of simplicity we have ignored the
existence of a local velocity field $\mathbf{v}(\mathbf{r})$. This field has
two effects: it can advect the dislocations, and it can modify the expressions
of $w_{x}(\mathbf{r})$ and $w_{y}(\mathbf{r})$. The equations of motion for
the dislocation pair separation now read

$\displaystyle{dx\over dt}$ $\displaystyle=2\psi
v_{x}(\mathbf{r})-2\mu_{x}w_{y}(\mathbf{r})+f_{x}(t)\,,$
$\displaystyle{dy\over dt}$ $\displaystyle=2\psi
v_{y}(\mathbf{r})+2\mu_{y}w_{x}(\mathbf{r})+f_{y}(t)\,,$ (IX.1)

where by $\mathbf{v}(\mathbf{r})$ we mean the velocity field at the point
$\mathbf{r}$ generated by a $+1$ dislocation, and the factor of $2$ comes from
the fact that an equal and opposite velocity field is generated by the $-1$
dislocation at the position of the $+1$ dislocation, and the two effects on
the relative motion add. Note that the prefactors of $v_{x}(\mathbf{r})$ and
$v_{y}(\mathbf{r})$ are not $2$, as one might naively expect, but $2\psi$,
where the factor $\psi$ captures the effects of drag between the dislocations
and the substrate, as discussed in the appendix. However, these effects do not
modify our analysis since we will show that the velocity contribution is
subdominant compared to that of $w_{x},w_{y}$. The equations for the velocity
field involve the force balance, the layer dynamics equation and the density
non conservation  equation. At steady state the layer dynamical equation
(III.1) becomes, see Appendix A:

$\displaystyle
0=v_{y}(\mathbf{r})+D_{y}\partial_{y}w_{y}+D_{x}\partial_{x}w_{x}+f,$ (IX.2)

This equation shows that $\mathbf{v}(\mathbf{r})$ is of the order of
derivatives of $\partial_{x}w_{x},\partial_{y}w_{y}$ which lead to terms
subdominant compared to $w_{x},w_{y}$ in the long distance limit, hence our
claim that the advection term in (IX.1) is subdominant. The force balance
expresses the fact that the momentum extracted from the substrate is balanced
by the divergence of the stress. The momentum exchange with the substrate
involves not only the usual friction term, but because we are dealing with an
active system, also involves gradients of layer spacing plus bend and splay of
the layer normal. The stress tensor involves the layer compression term as in
conventional smectics, the pressure term and the usual active stress of
anisotropic systems. The viscous term which is of higher order in gradients
than substrate friction may be omitted. The density balance equation involves
a source term which expresses that whenever the pressure departs from its
homeostatic value, the elements building the active smectic are either created
or destroyed. It has been shown that under these conditions on long time
scales one can replace the pressure term by a bulk viscous term homeoTissuetwo
. As already pointed out viscous terms can be omitted in the long wavelength
limit and it is then straightforward to show that the equation we found for
$w_{x},w_{y}$ given in Eq. (IV.1) is valid and the effect of flow is simply to
renormalize the coefficients. Thus our conclusions concerning the Malthusian
case are valid even when one takes into account a momentum exchange with the
substrate more complex than just friction.

We now turn to the incompressible active smectic case. The pressure becomes a
Lagrange multiplier which can be calculated with the condition that the
divergence of the velocity field vanishes. Other equations are identical to
the one we just discussed. They can be solved in a straightforward way. All
together, this adds up to eight parameters in the problem. Following the same
logic as before we obtain:

$\displaystyle w_{x}(\mathbf{r})=-{y\gamma^{(1)}_{x}\over
2\pi(\alpha_{1}y^{2}+x^{2})}+{y\gamma^{(2)}_{x}\over
2\pi(\alpha_{2}y^{2}+x^{2})}\,,$ (IX.3) $\displaystyle
w_{y}(\mathbf{r})={x\gamma^{(1)}_{y}\over
2\pi(\alpha_{1}y^{2}+D_{y}x^{2})}-{x\gamma^{(2)}_{y}\over
2\pi(\alpha_{2}y^{2}+D_{y}x^{2})}\,.$ (IX.4)

We have obtained expressions for
$\gamma^{(1)}_{x},\gamma^{(2)}_{x},\gamma^{(1)}_{y},\gamma^{(2)}_{y},\alpha_{1},\alpha_{2}$
in terms of the aforementioned 8 parameters, but they are not very
illuminating and so we will not give them here. In the large friction regime
one recovers the results of the Malthusian case, with
$\gamma^{(2)}_{x},\gamma^{(2)}_{y}$ going to zero,
$\gamma^{(1)}_{x},\gamma^{(1)}_{y}$ going to $\gamma_{x},\gamma_{y}$ and
$\alpha_{1},\alpha_{2}$ going to $\alpha$. The important point is that the
expressions IX.4 appear as the difference between two functions having the
same structure as the one appearing in the Malthusian case, with the same
scaling. Thus there is an important parameter space for which the sign of
$w_{x}(\mathbf{r}),w_{y}(\mathbf{r})$ and the scaling are the same as in the
Malthusian case. This means that the same procedure as in the preceding
section can be followed and that the conclusions detailed in the Malthusian
case are valid in general in a broad range of parameters even in the
incompressible case.

##  X Summary, conclusions, and suggestions for future work

We have shown that dislocations in dry active Malthusian smectics - that is,
smectics lacking all conservation laws, including that of particle number-
behave very differently from those in equilibrium smectics. Specifically:

1) they can move spontaneously, even in isolation.

2) Because of this, active smectics with “constant stress” boundary conditions
can never reach a steady state. Instead, they either grow forever, or shrink
and disappear. This behavior is similar to that of tissuestissue .

3) When their boundaries are fixed, active smectics reach a state of
“homeostasis”, in which the spontaneous motion of isolated dislocations
ceases.

4) However, even in the state of homeostasis, dislocations are always unbound
in rotation invariant active smectics, if there is any noise, however small.
This means that the active smectic phase does not, in fact, exist at finite
noise.

5) By applying rotational symmetry breaking fields, active smectics can be
stabilized against dislocation unbinding for sufficiently small noise.
However, for weak symmetry breaking fields, active smectics are less robust
against noise than their equilibrium counterparts. Specifically, the critical
noise strength $\Delta_{c}$ above which dislocations unbind and smectic order
is lost scales linearly with the symmetry breaking field strength $g$, in
contrast to the $\sqrt{g}$ scaling of the critical temperature in an
equilibrium smectic.

6) As a result, the exponent $\eta$ for the algebraic decay of smectic
correlations (given by equation (III.6)) becomes non-universal at melting, and
in fact vanishes linearly with symmetry breaking field strength $g$ as $g\to
0$. This should be contrasted with the universal value $\eta(T_{c})=1/4$ of
this exponent in equilibrium smectics with a symmetry breaking field, a result
that is completely independent of the symmetry breaking field $g$.

Our work here has, of course, focused mainly on a very particular type of
smectic, namely dry Malthusian apolar smectics. In contrast to equilibrium
systemschaikin , for which the exact nature of the dynamics - in particular,
what conservation laws the dynamics respects - has no effect on the equal time
correlations of smectic fluctuations, in active systems, because they are non-
equilibrium, all we have is dynamics, and, so, a priori, results could change
if we consider smectics with conservation laws. The most obvious of these is
number conservation, which our Malthusian smectics lack, due to “birth and
death” of the constituent active particles. One can of course imagine many
situations in which birth and death are absent (at last on the time scale of
an experiment). Such systems will have very different hydrodynamic equations,
because the conserved particle density will now become a slow, hydrodynamic
variableMPP , thereby completely changing the dynamics. The spin wave theory
for this case has already been worked out apolarsm , and our analysis of the
incompressible case suggests our results are quite general.

Likewise, momentum conservation, which will hold for freely suspended 2d
systems (i.e., those not in contact with a substrate to which they can lose
momentum) always radically changes the dynamics, for similar reasons.

Finally, polarity (ie., the absence of head-tail symmetry) is also
knownpolarsm  to change the nature of the spin-wave theory of active smectics.
So its role in dislocation behavior should also be investigated.

We strongly suspect that in all of these more complicated systems, our
fundamental conclusion - namely, that dislocations in a rotation invariant 2d
active smectic will always be unbound in the presence of noise, meaning that
the active smectic phase cannot exist at finite noise in two dimensions. We
suspect this because, whatever the conserved quantities, rotation invariance
forbids motion of dislocations, other than Brownian motion driven by noise ,
or motion driven by curvature of the layers, in the direction perpendicular to
the layers in a rotation invariant system. Because the curvature field induced
by dislocations will always fall off quite rapidly with distance from the
dislocation, the motion induced by them will always be insufficient to bind a
dislocation pair. Instead, dislocations can always unbind by diffusing apart
in the direction normal to the layers, as we found here for dry Malthusian
active smectics.

But, obviously, this conclusion is at best speculative until those other
systems enumerated above are explicitly investigated.

###### Acknowledgements.

JT thanks The Higgs Centre for Theoretical Physics at the University of
Edinburgh for their hospitality and support while this work was in progress.
He likewise thanks the Max Planck Institute for the Physics of Complex
Systems, Dresden, Germany, for their support through the Martin Gutzwiller
Fellowship. JP thanks A. Bershadsky for drawing his attention on smectic order
in fibroblast cells.

##  Appendix A Theory of active smectics

We describe the smectic configuration by the vector field $m\hat{\bf n}$ which
also defines ${\bf w}(\mathbf{r},t)=(a_{0}m-1)\hat{\bf n}$, where $m=1/a$ is
the density of layers, $\hat{\bf n}$ is a unit vector normal to the layers and
$a_{0}$ a reference layer spacing. The vector field $m\hat{\bf n}$ satisfies a
general balance equation

$\partial_{t}(m\hat{\bf n})+\nabla J_{m}=-\hat{z}\times{\bf J}_{d}\quad.$
(A.1)

Here, the dislocation current is

${\bf J}_{d}=\sum_{\alpha}b_{\alpha}{\bf
v}_{\alpha}\delta(\mathbf{r}-\mathbf{r}_{\alpha})\quad,$ (A.2)

where ${\bf v}_{\alpha}$ is the velocity of disloctation $\alpha$ and
$b_{\alpha}$ its Burgers number. The layer rate $J_{m}$ can in the absence of
dislocations be identified with the rate of change of layer displacement:

$J_{m}a_{0}=-\partial_{t}u\ \ \ ,\ \ \ {\rm no}\,{\rm dislocations}\,.$ (A.3)

Eqns. (A.1) and (A.2) imply Eq. (II.9).

In a hydrodynamic theory, we write phenomenological constitutive equations for
$J_{m}$ and for the force balance, using terms allowed by symmetry at lowest
order in spatial derivatives. For an up-down and rotationally symmetric
smectic, these can be written as

$\displaystyle J_{m}$ $\displaystyle=$ $\displaystyle m{\bf v}\cdot\hat{\bf
n}-D_{m}\hat{\bf n}\cdot\nabla m+\lambda_{m}\nabla\cdot\hat{\bf n}$ (A.4)
$\displaystyle\nabla\cdot\bm{\sigma}$ $\displaystyle=$
$\displaystyle\bm{\mu}\cdot{\bf v}+\bm{\nu}\cdot\nabla
m+\lambda_{b}^{v}(\hat{\bf n}\cdot\nabla)\hat{\bf n}$ (A.5) $\displaystyle+$
$\displaystyle\lambda_{s}^{v}\hat{\bf n}(\nabla\cdot\hat{\bf n})$

where $\bf v$ is the material flow velocity and we have introduced
phenomenological coefficients $D_{m}$, $\lambda_{m}$, $\lambda_{b,s}^{v}$ as
well as mobility and kinetic tensors $\bm{\mu}$, $\bm{\nu}$ and we have
omitted the noise for simplicity.

The stress tensor $\bm{\sigma}$ also obeys a constitutive relation which is of
the form

$\sigma_{ij}=-P\delta_{ij}-B\bigg{(}\hat{n}_{i}\hat{n}_{j}-\frac{1}{2}\delta_{ij}\bigg{)}\frac{m-m_{0}}{m_{h}}\quad,$
(A.6)

where $P$ is pressure and we have omitted higher order and viscous terms which
are subdominant in the hydrodynamic limit. To study small deformations, we
write $m=1/a_{0}+\delta m$ and $\hat{\bf
n}=\sin(\phi)\hat{x}-\cos(\phi)\hat{y}$. To linear order in $\phi$ and $\delta
m$ we then have

$a_{0}m\hat{\bf n}\simeq\phi\hat{x}-(1+a_{0}\delta m)\hat{y}\quad,$ (A.7)

which is the same as Eq. (II.10). Furthermore, we have to lowest order
$w_{x}\simeq\phi$ and $w_{y}\simeq-a_{0}\delta m$. Finally, Eq. (A.4) becomes

$-J_{m}a_{0}=v_{y}+D_{x}\partial_{x}w_{x}+D_{y}\partial_{y}w_{y}$ (A.8)

where $D_{y}=D_{m}$ and $D_{x}=-\lambda_{m}a_{0}$. In steady state $J_{m}=0$
and we obtain Eq. (IX.2).

The velocity $v_{y}$ can be determined using Eqns. (A.5) and (A.6). Doing so,
we have to distinguish the Malthusian from the incompressible case. In the
Malthusian case, material is not conserved and $\nabla\cdot{\bf v}$ does not
vanish. In this case, $P=-\eta_{b}\nabla\cdot{\bf v}$, where $\eta_{b}$ is a
bulk viscosity and the contributions from viscosity and pressure are
irrelevant in the hydrodynamic limit. We then have

$\mu_{yy}v_{y}\simeq\left(\lambda_{s}^{v}-B\left(\frac{m-m_{0}}{m_{h}}\right)\right)\partial_{x}\phi-\left(\nu_{yy}+\frac{B}{2m_{h}}\right)\partial_{y}m\quad.$
(A.9)

Using this expression in (A.8) to eliminate $v_{y}$ we arrive in steady state
at Eq. (IV.1) of a Malthusian active smectic but with renormalized
coefficients

$\displaystyle D_{x}$ $\displaystyle=$
$\displaystyle-\lambda_{m}a_{0}+\frac{\lambda_{s}^{v}-B(m_{h}-m_{0})/m_{h}}{\mu_{yy}}$
(A.10) $\displaystyle D_{y}$ $\displaystyle=$ $\displaystyle
D_{m}+\frac{\nu_{yy}+B/(2m_{h})}{a_{0}\mu_{yy}}\quad.$ (A.11)

In the incompressible case the pressure acts as a Lagrange multiplier to
impose the incompressibility constraint $\nabla\cdot{\bf v}=0$. This modifies
the hydrodynamic modes, see section IX.

The velocity of dislocation $\alpha$ can also be written on symmetry grounds
to lowest order as

${\bf v}_{\alpha}=\psi{\bf
v}+B^{\prime}\left(\frac{m-m_{h}}{m_{h}}\right)b_{\alpha}\hat{\bf
n}\times\hat{z}$ (A.12)

where $m_{h}$ is the layer density in the homeostatic state and which is
equivalent to Eq. (V.1).

In the presence of a symmetry breaking field ${\bf H}$ rotation invariance is
broken. In this case additional symmetry breaking terms involving a tensor
$S_{ij}=g\hat{H}_{i}\hat{H}_{j}$ are permitted. These additional terms do not
affect Eq. (A.8) at linear order, see Eq. (IX.2). However, $S_{ij}$ allows for
an additional term in the dislocation velocity when rotation invariance is
broken

${\bf v}_{\alpha}=\psi{\bf
v}+B^{\prime}\left(\frac{m-m_{h}}{m_{h}}\right)b_{\alpha}\hat{\bf
n}\times\hat{z}+B^{\prime\prime}b_{\alpha}{\bf S}\cdot\hat{\bf
n}\times\hat{z}$ (A.13)

where the field $\bf H$ is aligned such that ${\bf
S}=gH^{2}\hat{x}\otimes\hat{x}$, which corresponds to Eq. (VIII.1). The
dislocation velocities thus obey Eq. (VIII.6) with
$\mu_{x}=B^{\prime}b_{\alpha}$ and $\mu_{y}=gH^{2}B^{\prime\prime}b_{\alpha}$
and we obtain Eq. (VIII.7).

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