Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect

Polar molecules located at lattice sites as found in molecular crystals represent a system to discuss the effects of symmetry breaking in the statistical calculation of a property such as electrical polarity. Given the point symmetry of the lattice and its finite size, the system will thermalize into a bi-polar state featuring zero overall polarity [1].

Starting from our previous work [2], we consider a generalized Ising model whose Hamiltonian is given as an infinite series of multipoles. The well known spin flip symmetry of the Ising Hamiltonian is found to be broken by the interaction of the first non-zero multipole with the next order multipole, when such an interaction is present. This is analogous to the effect of an external magnetic field acting on a paramagnetic system where the symmetry breaking in the Hamiltonian generates an induced magnetization in the material. As a result of the coupling of the symmetry breaking in the Hamiltonian with the translational symmetry breaking due to the free boundary conditions, a spatial ordering in the system is achieved.

The simplest model one can imagine to reproduce bi-polar state formation in media is an Ising-like one. It consists in a three-dimensional cubic lattice Γ and the binary variables $S_i=\pm 1$ defined on each of its site i. Only interaction between neighboring variables are included. The energy of a given configuration $\left\{S_i\right\}$ is given by the Ising Hamiltonian where the symbol $\langle i,j\rangle$ indicates that the sum is between the next neighboring $S_i$ only. Since the Hamiltonian (1) is invariant under the transformation $S_i\to -S_i$, $S_j\to -S_j$ (global spin flip symmetry) the thermal average $\langle S_i\rangle =0$, unless this symmetry is broken. Thus, Equation (1), in this simple form, cannot reproduce a net average polarity. In order to analyze the issue in more detail, consider two non-overlapping charge distributions centered around ${\mathbf{R}}_{\mathbf{A}}$ and ${\mathbf{R}}_{\mathbf{B}}$, respectively. The electrostatic energy in multipole expansion can be written as: where ${\mathbf{R}}_{\mathbf{AB}}={\mathbf{R}}_{\mathbf{B}}-{\mathbf{R}}_{\mathbf{A}}$, $k_C$ is the Coulomb constant whose value depends on the units, and where for each $l_A$, $l_B$ $\in (0,\infty )$ the $(2l_A+1)\times (2l_B+1)$ matrix $B^{l_A,l_B}$ is defined as where $I_l^m\left(r\right)$ are the irregular solid harmonics, $C_{l_A{m}_A,l_B{m}_B}^{l_A+l_B}$ are the Clebsch–Gordan coefficients. and ${\left(Q_A\right)}_l^m$ (resp. ${\left(Q_B\right)}_l^m)$ are the multipole moments of distribution A (resp. B) given by where ${\mathbf{r}}_{\mathbf{i}}$ is the position of the charge $q_i$ of the distribution considered, and $R_l^m\left(r_i\right)$ are the regular solid harmonics. Since the $Q_l^m$ have parity ${(-1)}^l$ $\forall m$, it turns out that each term in the sum of Equation (2) has a well definite parity. The interactions of multipoles such that $l_A+l_B$ is odd are, therefore, the origin of the symmetry breaking in Equation (1). This condition is, however, not sufficient for the onset of a bi-polar state because, as we will show in the following, the translational symmetry of the system should also be broken in one direction at least. This is not surprising as in the thermodynamic limit we need to have a "free" (of interaction from the outside) surface where the bi-polar state shows up.

According to the above discussion, if the system has a net charge, the lowest order terms of the series Equation (2) with broken (parity) symmetry are those with $l_A=0$, $l_B=1$ and $l_A=1$, $l_B=0$, which correspond to charge-dipole interactions. If the system is electrically neutral but polar, the symmetry is broken by the dipole-quadrupole interactions, at lowest order.

Let us consider a one-dimensional array of N identical molecules located at the sites i of the lattice and assume that each molecule i can be in one of the two possible states $S_i$ differing by an overall rotation of 180${}^{\circ }$ around the center of mass, i.e., an inversion of the dipolar direction. The multipole moments will, in general, be dependent on the state of the molecule according to some relation of the kind: where the $q_l^m$ are the multipole moments of molecule i in a reference state arbitrarily chosen for all molecules, $S_i^l$ is the usual effective “spin” variable associated to molecule i to the power of l. Due to the parity symmetry, only l-odded multipoles are affected by molecule flip.

The Ising Hamiltonian (1) can be re-written as where $A_{l_i,l_j}$ are given by

It is straightforward to see that the transformation $\{l_i,m_i\}⟷\{l_j,m_j\}$ gives $A_{l_i,l_j}={(-1)}^{l_i+l_j}{A}_{l_j,l_i}$, and, therefore, the matrix of interactions has parity $l_i+l_j$. The diagonal terms $A_{l_i,l_i}$ can be expressed in the more compact form where, for each $l\in (0,\infty )$,

By using the symmetry properties of the Clebsch–Gordan coefficients, it can easily be seen that $B^l$ is symmetric $\forall l\in (0,\infty )$. For one-dimensional lattices, the matrices $B^l$ are of a particular simple form. Let us assume that the lattice extends over the z-axis, from the symmetry of the spherical harmonics and of the Clebsch–Gordan coefficients, we then get which, for each l, is an anti-diagonal matrix.

To highlight the properties of the Hamiltonian let us separate the sums in Equation (6) over odd and even $l_i$ and $l_j$ where the superscript g (resp. u) indicates that the corresponding sum runs over the even (resp. odd) indices. Using the definition of $S_i$, the previous Equation simplifies to give and after swapping the index in the third sum and using the rule of transformation for $A_{l_i,l_j}$, we finally get

In respect to the $S_i$ variables, the Hamiltonian is therefore decomposed into the sum of three terms: one is a constant, one anti-symmetric and one symmetric where

Note that $H_g$ is the usual Ising Hamiltonian with coupling constants given by and, therefore, it is the same Hamiltonian as an Ising ferromagnetic or anti-ferromagnetic depending on the sign of the $J_{ij}$s.

In Section 2, we have seen that, as long as the molecules on the chain have a non-zero odd-ranked multipole l together with a non-zero $l+1$ multipole, a bi-polar state may appear as a result of the broken spin flip symmetry of the Hamiltonian (6). This occurrence is of course closely connected with the molecule symmetry. We stress that no other assumption has been made on the molecules forming the system. The bi-polar state is, in that respect, a universal feature. From a standpoint of group symmetry, we notice that molecules with a center of inversion, i.e., of the $C_i$ point group, cannot have an l-ranked quadrupole with l odd and, therefore, no symmetry breaking can occur. On the contrary, $C_n$, $C_{n\nu }$, and $C_s$ have a non-zero dipole and quadrupole, and, therefore, the symmetry breaking occurs and a bi-polar state may show up.

In the next subsections, we will examine three more significant point groups and calculate the expectation value of the order parameter. The method outlined can, however, be employed for all point groups using the corresponding character tables [5,6]. Attention should be paid considering the appropriate orientation of the molecules in the chain to match the symmetry of the molecule.

We have already noticed that, roughly speaking, symmetry breaking cannot occur if the building blocks do not have an appropriate symmetry, such as $C_i$. On the other side, symmetries can greatly simplify calculation and allow major generalization.

A d-dimensional Ising lattice model is defined on a finite subset ${\Gamma }_L$ of integers $Z^d$, the number of spin variable configurations ${\left\{S_i\right\}}_{i\in \Gamma }$ is therefore finite. In order to extrapolate results for macroscopic system, we consider the limit of ${\Gamma }_L$ with free boundary condition for $L\to \infty$. Note that averaged quantities depend in general on the nature of the boundary conditions. We demonstrate that in the canonical ensemble, the thermodynamic limit of the free energy per site exists and is finite for the model considered. The free energy is given by $F=-k_{B}TlnZ$. Using Equation (34), the free energy per site at the second order can be expressed in the limit of large systems as At every finite temperature, therefore, the limit exists and is finite. Analogously, the averaged energy of the system $\langle E\rangle$, which is the thermodynamical internal energy U, is given by

Note the explicit appearance of the extensivity in the energy due to the factor N. The energy fluctuations are given by

The specific heat is given by

We notice that the non-diagonals terms of the coupling parameters $A_{i,j}$, $i\ne j$, which are essential for the symmetry breaking in the Hamiltonian and, ultimately, to the onset of spontaneous polarization do not enter the fundamental extensive thermodynamic quantities.

We conclude this section by showing that the free energy per site is a concave function of the absolute temperature. This is easily accomplished by observing that, since $f\in C^2\left(\mathcal{R}\right)$

In consideration that the thermodynamic properties at equilibrium are determined by the free energy, this last statement ensures the thermodynamic stability of the model [7].

Starting from a generalized Ising model with the electrostatic interactions given in multipole expansion, we have shown that the onset of polarity in a molecular system can be described as the result of two symmetry breaking effects: the translational and the global spin flip invariance. The former is of geometrical origin and is reflected by the presence of free boundary conditions, the latter is explained as the effect of specific multipole interactions. The spontaneous polarization is assessed by the thermal average of the order parameter $\langle S\rangle$ analogous to the average magnetization of the standard Ising model. The analysis of $\langle S\rangle$ as a function of the length inside the system shows two different behaviors depending on the sign of the asymmetric interaction of same odd order multipoles (e.g., dipole-dipole): an S-shaped odd function characteristic of spin correlations fading out inside the system, and an oscillating function corresponding to an alternate ordering of plus/minus spins.

The thermodynamic limit is shown to exist under very general conditions, and the concavity of the free energy per lattice site demonstrates the thermodynamic stability of the model. At a second order treatment, the asymmetric interactions (e.g., the dipole-quadrupole one), which are crucial for the Hamiltonian symmetry breaking, do not enter any of the macroscopic thermodynamic quantities such as the free energy and specific heat. These latter quantities appear to be only determined by the symmetric (e.g., dipole-dipole) interactions. Asymmetric interactions seem, therefore, to only trigger the onset of polarity by a symmetry breaking process, leaving other thermodynamic properties, including the general behavior of $\langle S\rangle$, unaffected.

The extension of the previous analysis to higher spatial dimensions and to off-lattice models is a priority for us, and, to this goal, works are already on the way. Along with theoretical calculations, large scale Monte Carlo and molecular dynamics simulations with more complex force fields are running and will be the object of forthcoming publications.