Smectic-$A$ and -$C$ phases, and the transition between them, in uniaxial disordered environments

We present a theory of the elasticity and fluctuations of the smectic-$A$ and -$C$ phases in uniaxial, anisotropic disordered environments: e.g., stretched aerogel. We find that, bizarrely, the low-temperature, lower-symmetry smectic-$C$ phase is less translationally ordered than the high-temperature, higher-symmetry smectic-$A$ phase, with short-range “$m=1$ Bragg glass” and algebraic “$XY$ Bragg glass” order, respectively. The smectic-$A$–smectic-$C$ $\left(AC\right)$ phase transition belongs to a new universality class, whose fixed points and exponents we find in a $d=5-ϵ$ expansion. We give very detailed predictions for the very rich light-scattering behavior of both phases and the critical point.

Figure 1

The $q_z$ dependence of the x-ray-scattering intensity for $q_{\perp }=0$ in the smectic-$A$ phase. In the $C$ phase, the sharp, power-law peaks disappear, leaving only the broad peak.

Figure 2

The $q_{\perp }$ dependence of the x-ray-scattering intensity for $q_z=q_0$ in the smectic-$A$ phase. Again, the sharp peak vanishes in the $C$ phase.

Figure 3

Illustration of the three distinct regions in wave-vector space with different wave-vector dependences of ${\Delta }_{t,c}$ and $K$, which is given in Eq. (1.38).

Figure 4

Illustration of the eight regions listed in (1.44), each of which exhibits different scaling of the light-scattering intensity in the $A$ phase as $T\to T_{AC}^{+}$. The light-scattering intensity is dominated by random field fluctuations in regions 1 and 2, by random tilt fluctuations in regions 3, 4, 6, and 7, and by random compression fluctuations in regions 5 and 8. Above locus $OFH$ the common denominator in formula (1.42) is dominated by $B{q}_z^2$, while below $OFH$ it is dominated by $K{q}_{\perp }^4$ in region 6 and by $D{q}_{\perp }^2$ in regions 1 and 3, respectively.

Figure 5

Three-dimensional picture in the transformed reciprocal space ($\vec{q}$ space) illustrating the five distinct regions with different wave-vector dependences of $C_{ij}\left(\vec{q}\right)$ in the $C$ phase, which is given in (1.58).

Figure 6

Illustration of the ten distinct regions in the $q_z-q_x$ plane with different wave-vector dependences of $C_{xx}\left(\vec{q}\right)$, which is given in (1.69) for the $C$ phase. Line $OA$ is define by $q_{x^{\prime }}=0$.

Figure 7

The graphical correction to $B$ coming from the combination of the two cubic vertices.

Figure 8

Graphical corrections to the bend, tilt, and random tilt terms.

Figure 9

A graphical correction to the tilt term.

Figure 10

The graphical correction to the linear term ${\sum }_{n=1}^{\infty }{\partial }_{z}u$.

Figure 11

A graphical correction to the cubic term.

Figure 12

Another graphical correction to the cubic term.

Figure 13

One more graphical correction to the cubic term.

Figure 14

A graphical correction to the quartic term.

Figure 15

Another graphical correction to the quartic term.

Figure 16

Another graphical correction to the quartic term.

Figure 17

Yet another graphical correction to the quartic term.

Figure 18

One more graphical correction to the quartic term.

Figure 19

The final graphical correction to the quartic term.

Figure 20

Graphical RG flow of the dimensionless coupling $g_3$ and $g_4$. Point $a$ is the stable fixed point (0, $32ϵ∕15$).

Figure 21

The graphical correction to ${\Delta }_{x^{\prime }}^{\prime }$.