Hierarchy of orientational phases and axial anisotropies in the gauge theoretical description of generalized nematic liquid crystals

The paradigm of spontaneous symmetry breaking encompasses the breaking of the rotational symmetries $\text{O}\left(3\right)$ of isotropic space to a discrete subgroup, i.e., a three-dimensional point group. The subgroups form a rich hierarchy and allow for many different phases of matter with orientational order. Such spontaneous symmetry breaking occurs in nematic liquid crystals, and a highlight of such anisotropic liquids is the uniaxial and biaxial nematics. Generalizing the familiar uniaxial and biaxial nematics to phases characterized by an arbitrary point-group symmetry, referred to as generalized nematics, leads to a large hierarchy of phases and possible orientational phase transitions. We discuss how a particular class of nematic phase transitions related to axial point groups can be efficiently captured within a recently proposed gauge theoretical formulation of generalized nematics [K. Liu, J. Nissinen, R.-J. Slager, K. Wu, and J. Zaanen, Phys. Rev. X 6, 041025 (2016)]. These transitions can be introduced in the model by considering anisotropic couplings that do not break any additional symmetries. By and large this generalizes the well-known uniaxial-biaxial nematic phase transition to any arbitrary axial point group in three dimensions. We find in particular that the generalized axial transitions are distinguished by two types of phase diagrams with intermediate vestigial orientational phases and that the window of the vestigial phase is intimately related to the amount of symmetry of the defining point group due to inherently growing fluctuations of the order parameter. This might explain the stability of the observed uniaxial-biaxial phases as compared to the yet to be observed other possible forms of generalized nematic order with higher point-group symmetries.

Figure 1

Point-group symmetric orientational degrees of freedom $R_i,R_j^{\prime }=U_{ij}{R}_j$ on a lattice with local identifications $R_{i,j}\simeq {\mathrm{\Lambda }}_i{R}_{i,j}$, for ${\mathrm{\Lambda }}_i\in G$, the associated gauge fields $U_{ij}\simeq {\mathrm{\Lambda }}_i{U}_{ij}{\mathrm{\Lambda }}_j^T$ on links $\langle ij\rangle$, and the nearest-neighbor Hamiltonian $\mathrm{Tr}\left[R_i^T\mathbb{J}{U}_{ij}{R}_j\right]$ between triads ${\mathbf{n}}_i^{\alpha }=\{{\mathbf{l}}_i,{\mathbf{m}}_i,{\mathbf{n}}_i\}$, ${\mathbf{n}}_j^{\prime \alpha }=\{{\mathbf{l}}_j^{\prime },{\mathbf{m}}_j^{\prime },{\mathbf{n}}_j^{\prime }\}$ with parameters $\mathbb{J}=\mathrm{diag}(J_1,J_2,J_3)$. For clarity we show the couplings of the triads on several different nearest neighbors sites $i,j$. For more details on lattice model and the gauge theoretical description of nematics, see Sec. 3.

Figure 2

The schematic temperature-anisotropy phase diagram of axial nematics with conventional twofold biaxial symmetries. Small and large $\frac{J_1}{J_3}$ correspond to weak and strong in-plane order, respectively. ${\left(\frac{J_1}{J_3}\right)}_c^U$ and ${\left(\frac{J_1}{J_3}\right)}_c^B$ are the critical anisotropies where the generalized biaxial-uniaxial transitions in Eq. (8) and the biaxial-biaxial${}^{*}$ transitions in Eq. (11) terminate, respectively. Solid lines in the phase diagram are present for all axial symmetries $\{C_n,C_{nv},S_{2n},C_{nh},D_n,D_{nh},D_{nd}\}$ with finite $n$, while the dashed line transition is present only for the symmetries $\{C_n,C_{nv},S_{2n},D_n,D_{nd}\}$.

Figure 3

The schematic $(\beta J_3,\beta J_3)$ phase diagrams of axial nematics. (a) The phase diagram in Fig. 2 in terms of $(\beta J_1,\beta J_3)$. As in Fig. 2, for low-order groups with two- and threefold symmetries the effective couplings stabolizing the biaxial and uniaxial order are of the same order and lead to a transition directly to the biaxial phase. (b) For higher in-plane symmetries, the biaxial phase is suppressed in comparison to the uniaxial phase. When allowed by the symmetries, axial terms with a vector or second rank uniaxial order parameter appear always in the Hamiltonian even at $J_3=0$ due to the $\text{O}\left(3\right)$ constraints. These always stabilize the uniaxial order while the higher order biaxial order is still fluctuating. Solid lines in the phase diagram are present for all axial symmetries $\{C_n,C_{nv},S_{2n},C_{nh},D_n,D_{nh},D_{nd}\}$ with finite $n$, while the dashed biaxial${}^{*}$-transition is present only for the symmetries $\{C_n,C_{nv},S_{2n},D_n,D_{nd}\}$ and the dotted uniaxial${}^{*}$-transition for $\{C_n,C_{nv}\}$.

Figure 4

The temperature-anisotropy phase diagram of (a) $D_{2h}$ and (b) $D_2$ biaxial nematics. At small $\frac{J_1}{J_3}$, there is a sequence of biaixal-uniaxial-liquid transition with a vestigial $D_{\infty }$-uniaxial phase. The uniaxial phase terminates at a triple point (the red star), after which the transition sequence truns to a direct biaxial-liquid transition. This directly reproduces the well-known phase transitions for $D_{2h}$ and $D_2$ nematics from the gauge theoretical setup (12). In addition, for large $\frac{J_1}{J_3}$ in the $D_2$ case, there is a vestigial $D_{2h}$-biaxial phase right to another triple point (the blue star), realizing the biaxial-biaxial${}^{*}$ transition in Eq. (11).

Figure 5

The $J_1\text{−}{J}_3$ phase diagram of (a) $D_{2h}$, (b) $D_{3h}$ and (c) $D_{4h}$ nematics. The red stars in (a) $D_{2h}$ and (b) $D_{3h}$ highlight a triple point, in which the three transition lines meet. Similar to the temperature-anisotropy phase diagram in Fig. 4, there is a vestigial uniaxial phase appearing from the region with small $J_1$ and large $J_3$ (small $\frac{J_1}{J_3}$), realizing the generalized biaxial-uniaxial transition in Eq. (8). As the symmetry increases, this vestigial uniaxial phase becomes more prominent and the fully ordered biaxial phase is remarkably squeezed. When the symmetry is sufficiently high, the vestigial uniaxial phase appears adjacent to the isotropic liquid due to the symmetry allowed axial terms. Moreover, our simulations indicate that depending on strength of the in-plane coupling, the biaxial-uniaxial transition may be either first order or second order. Therefore a tricritical point may exist in the biaxial-uniaxial transition line.

Figure 6

The $J_1\text{−}{J}_3$ phase diagram of (a) $D_2$, (b) $D_3$, and (c) $D_4$ nematics. This is similar to Fig. 5, but there is in addition a vestigial biaxial phase at small $J_3$ region, realizing the generalized biaxial-biaxial${}^{*}$ transition in Eq. (11). Both this vestigial biaxial phase and the fully ordered biaxial phase are squeezed considerably as the symmetry increases. The associated triple points at where transition lines meet are highlighted by large stars. As in Fig. 5, there may be a tricritical point in the biaxial-uniaxial transition line.