# Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect

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## Abstract

**:**

## 1. Introduction

## 2. Ising Model for Bi-Polar Transition

#### 2.1. First Order Expansion

#### 2.2. Second Order Expansion

- $\langle {S}_{1}\rangle =-\langle {S}_{N}\rangle $As a result of the broken symmetries, a bi-polar state appears with opposite states at the ends of the chain.
- $\langle {S}_{N}\rangle $ has the sign of ${A}_{2,1}$ if $\rho >0$, opposite if $\rho <0$.
- ${S}_{k}^{+}$ is a strictly monotonic function of k, strictly increasing (resp. decreasing) if ${A}_{2,1}>0$ (resp. ${A}_{2,1}<0$).
- Since $\langle {S}_{k}^{+}\rangle $ has opposite values at the boundary, it must be zero in one point internal to the interval $(1,N)$. It is readily seen that if N is odd, $\langle {S}_{N/2}\rangle =0$.
- The point of coordinate $(k,{S}_{k}^{+})$ with $k=(N-1)/2$ is an inflection point.
- $\langle {S}_{k}^{-}\rangle $ is an oscillating function of k, bounded by the two sequences ${S}_{k}^{+}$ with opposite values of ${A}_{2,1}$ (also seen in Monte Carlo simulations).
- The sequence $\left\{{S}_{k}^{+}\right\}$ converges to 0 as $k\to \infty $ $\forall \rho :\left|\rho \right|<1$ and to 1 if $\rho =1$ (convergent sequence bounded theorem [4]). The existence of these limits makes a sound point for the consistency of the model in the thermodynamic limit (cf. Section 4).

#### 2.3. Higher Order Expansion

## 3. Molecular Symmetry and Symmetry Breaking in the Hamiltonian

#### 3.1. Point Group ${C}_{n}$, $n>2$

#### 3.2. Point Group ${C}_{2v}$

#### 3.3. Oscillating Behavior: The ${C}_{\mathrm{s}}$ Point Group

## 4. Macroscopic Systems

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**General behavior of the average order parameter vs. distance on the chain. For clarity, the curve corresponding to ${A}_{1,1}>0$ and ${A}_{2,1}<0$ has been omitted as it is simply the oscillating curve plotted with reversed sign. The values $\rho =\pm 0.9$ have been arbitrarily chosen.

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**MDPI and ACS Style**

Cannavacciuolo, L.; Hulliger, J.
Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect. *Symmetry* **2016**, *8*, 10.
https://doi.org/10.3390/sym8030010

**AMA Style**

Cannavacciuolo L, Hulliger J.
Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect. *Symmetry*. 2016; 8(3):10.
https://doi.org/10.3390/sym8030010

**Chicago/Turabian Style**

Cannavacciuolo, Luigi, and Jürg Hulliger.
2016. "Polarity Formation in Molecular Crystals as a Symmetry Breaking Effect" *Symmetry* 8, no. 3: 10.
https://doi.org/10.3390/sym8030010