Unresolved phase-diagram questions of the Nelson–Halperin model of a smectic layer with tilted molecules: order of symmetry-equivalent transitions, the strong-coupling fate of the unlocked tilted hexatic, and the stacking of coupled layers

The 1980 theory of Nelson and Halperin (Phys. Rev. B 21, 5312) applied dislocation-mediated two-dimensional melting to a freely suspended smectic layer whose molecules tilt relative to the layer normal, coupling a sixfold bond-orientational field to a full $2\pi$ -periodic tilt-angle field through stiffnesses $K_6$ , $K_1$ , a gradient cross-coupling $g$ , and a local locking field $h\cos[6(\theta-\phi)]$ . The theory predicted up to seven phases, including locked and unlocked tilted hexatics, but left its own phase diagram incomplete: the order (continuous versus first-order) of transitions between regions sharing the same symmetry, the fate of the unlocked tilted hexatic $C^*$ when the locking field is strong, the character of several phase boundaries left unanalyzed in the proposed renormalization-group flows, and the three-dimensional consequences of stacking weakly coupled layers were all left open. Later work answered parts of this program — Selinger and Nelson’s theory of hexatic-to-hexatic transitions, the experimental observation of a smectic-C–smectic-I critical point, and the stacked-hexatic experimental and phenomenological program — but the strong-coupling fate of the unlocked phase for the $p=6$ bond–tilt model remains undetermined, as do the coupled defect (disclination–vortex–locking) treatment of the symmetry-equivalent boundaries and the effect of weak interlayer coupling on the two-dimensional flows. The problem is to complete this phase-diagram determination, at the renormalization-group or an equivalent nonperturbative level, for the single layer and for stacked layers.

Background

Smectic liquid crystals are layered fluids: molecules organize into stacks of two-dimensional sheets with density modulation along the layer normal but fluid-like order within each layer. In a freely suspended film — a smectic film spanning a hole in a plate, with free surfaces and a controllable number of layers down to two — the physics of a single sheet can be probed almost in isolation, which makes such films model systems for two-dimensional phase transitions.

The framework on which the source paper builds is the theory of two-dimensional melting developed around 1979 (the Kosterlitz–Thouless–Halperin–Nelson–Young theory, KTHNY; see Drouin-Touchette et al. 2022 for a modern exposition and bibliography). In two dimensions, a solid can melt through two continuous topological transitions instead of one first-order event: bound pairs of dislocations unbind first, destroying translational quasi-long-range order, then bound disclinations (point defects carrying fractional bond-orientational charge) unbind, destroying bond-orientational order. Between the two transitions lies the hexatic phase: a fluid with short-range translational order but quasi-long-range (algebraically decaying) sixfold bond-orientational order, where bond-orientational correlations $C_6(r)=\langle e^{6i[\theta(r)-\theta(0)]}\rangle$ decay as $r^{-\eta_6}$ . The renormalization-group (RG) treatment works by mapping the angle field onto a gas of topological defects interacting logarithmically (a Coulomb gas) and integrating the flows of stiffnesses and defect fugacities.

A further ingredient specific to the source paper is molecular tilt: in the smectic- $C$ phase, molecules lean relative to the layer normal, so each layer carries a two-component tilt field whose local direction is described by an angle $\phi$ defined modulo $2\pi$ . Nelson and Halperin (1980) combined the sixfold bond-orientational field $\theta$ (defined modulo $2\pi/6$ ) with this tilt field in a single effective Hamiltonian for one layer,

\frac{H}{k_B T}=\frac{1}{2}\int d^2r\;\Big[K_6|\nabla\theta|^2+K_1|\nabla\phi|^2+2g\,(\nabla\theta)\cdot(\nabla\phi)\Big]-h\int d^2r\;\cos\big[6(\theta-\phi)\big],

where $K_6$ and $K_1$ are the bond-orientational and tilt stiffnesses, $g$ is a gradient cross-coupling, and $h$ is a local anisotropy that locks the tilt direction to the local hexatic bond directions. The abstract of the paper summarizes the outcome: as many as seven distinct phases are possible, including ones the authors identify with smectic- $A$ , - $B$ , - $C$ , and - $H$ liquid crystals, and some of these phases may survive when layers are stacked to form bulk smectics.

The phase inventory of the model includes: the isotropic fluid; the untilted hexatic (algebraic bond order, no tilt order); the locked tilted hexatic, in which $\phi$ is pinned to $\theta$ so the two orders fluctuate together (the analogs observed in experiments are the smectic- $I$ and smectic- $F$ phases, where the tilt points toward a nearest-neighbor bond or toward a bond midpoint respectively, and the oblique smectic- $L$ phase); the unlocked tilted hexatic $C^*$ , in which both bond and tilt correlations decay algebraically but the two angles fluctuate independently; and tilted or untilted solids. Necessary stability conditions for $C^*$ are expressed through renormalized stiffnesses as $\eta_\Delta>4$ , $K_1^R>2/\pi$ , $K_6^R>72/\pi$ , where $\eta_\Delta$ governs the algebraic decay of the relative-orientation correlation $C_\Delta(r)=\langle e^{6i[\theta(r)-\phi(r)]}e^{-6i[\theta(0)-\phi(0)]}\rangle$ . The RG analysis couples three defect species — disclinations in $\theta$ , vortices in $\phi$ , and $p$ -fold periodic (locking) excitations — into a multi-component Coulomb gas with cross-coupling, and the proposed phase diagrams in the plane of inverse stiffnesses contain transition lines of undetermined character, some drawn as double lines in the paper’s figures and not analyzed.

In the limit of infinite locking $h\to\infty$ , the bond and tilt fields are forced locally parallel, $\theta=\phi+2\pi s/6$ with integer $s$ , and the model collapses to a single-field lattice Hamiltonian containing both $\cos[6(\phi-\phi')]$ and $\cos[\phi-\phi']$ nearest-neighbor terms, for which the authors sketched a plausible reduced phase diagram (liquid, hexatic, locked tilted hexatic) in which the unlocked $C^*$ is absent — but they stated that they could not be certain the unlocked phase is absent at large finite $h$ without a more detailed analysis, e.g. by Migdal recursion (a real-space renormalization-group approximation in which couplings are recursively thinned by bond-moving and decimation) or other nonperturbative methods. The same authors noted that first-order transitions are possible between regions of the diagrams that share the same symmetries, such as the two subregions of the locked tilted hexatic distinguished by whether the tilt locks toward a bond or between bonds, without determining the conditions. Finally, the paper speculated about which layered phases survive stacking into bulk smectics, but the stability and critical properties of stacked weakly coupled hexatic or tilted layers were left to be worked out.

Follow-up literature gives the experimental context for these questions. Synchrotron x-ray scattering established hexatic bond-orientational order in freely suspended films and its two-dimensional character in films down to tens of layers (Brock et al. 1989), light scattering measured bond-orientational correlations in tilted hexatic films (Sprunt and Litster 1987), and optical-texture studies mapped the rich sequence of tilted hexatic phases in thin films (Dolganov et al. 2006; Chao et al. 2003).

Problem Statement

Complete the phase-diagram program that the 1980 Nelson–Halperin theory of a single smectic layer with tilted molecules left unfinished. The common objective is a full determination of the phase behavior of the coupled bond–tilt model — its phase topology, the order of its transitions, and the structure of its renormalization-group flows — together with its extension to stacked layers. Specifically, determine, by renormalization-group, nonperturbative, or controlled numerical analysis of the model with bond field $\theta$ (modulo $2\pi/6$ ), tilt field $\phi$ (modulo $2\pi$ ), stiffnesses $K_6$ and $K_1$ , gradient cross-coupling $g$ , and locking field $h$ :

(a) the detailed character — continuous or first-order, and the conditions selecting each — of the transitions between regions of the phase diagram that share the same symmetries. The general class of such same-symmetry boundaries is required; the specific instances that must be treated are the two locked tilted hexatic subregions in which the tilt locks toward a bond direction versus between bond directions (the smectic- $I$ -like and smectic- $F$ -like subregions) and the symmetry-equivalent variants of the uniaxial tilted solid phase (distinguished by the same tilt-locking directions). For the $I$ -like/ $F$ -like subregions, determine explicitly whether the oblique smectic- $L$ -like phase or the unlocked tilted hexatic intervenes between them (the strong-locking fate of the unlocked tilted hexatic is treated separately in (b)). This part also requires how these boundaries are modified when free disclinations and vortices are retained in the coupled defect (Coulomb-gas) treatment;

(b) the fate of the unlocked tilted hexatic phase $C^*$ in the strong-locking regime: whether it survives at large but finite $h$ , or whether the phase diagram collapses toward the three-fluid-phase structure sketched in the $h\to\infty$ limit, and what transitions (or crossovers) separate the locked, unlocked, and untilted hexatic regimes as $h$ grows — settling the question the authors explicitly left open as unconfirmed;

(c) the character of the phase boundaries and RG flows left unanalyzed in the proposed phase diagrams (the boundaries drawn as double lines in the 1980 figures), including the flow of the gradient cross-coupling $g$ , determining which boundaries are Kosterlitz–Thouless-type defect-unbinding lines, which are locking or Potts-like transitions of the relative angle (transitions in the universality class of the $q$ -state Potts model, the standard discrete-symmetry spin model with $q$ competing states), and which are first-order;

and (d) the three-dimensional consequences of stacking: the stability and critical properties of stacks of weakly coupled hexatic or tilted layers — in particular how weak interlayer coupling modifies the two-dimensional renormalization-group flows and the layer phase diagram, and which of the single-layer phases survive into the bulk smectics identified with smectic- $A$ , - $B$ , - $C$ , and - $H$ .

The four parts (a)–(d) are jointly posed, in the $(K_1^{-1},K_6^{-1})$ plane with $g$ and $h$ as parameters, and none of them resolves the task on its own.

References

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