Symmetry & its Applications

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Symmetry & its Applications


Antisymmetry of Distortions, Brian K. VanLeeuwen, V. Gopalan, Nature Communications, 6, 8818 (2015).

Distortions are ubiquitous in nature. Under perturbations such as stresses, fields or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a distortion. Here we introduce an antisymmetry operation called distortion reversal that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal. Given its isomorphism to magnetic groups, distortion groups could have a commensurate impact in the study of distortions, as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

Here is a brief youtube video explaining the idea of distortion reversal symmetry.

Distortion symmetry: The above distortion of B3O6 rings in β-BaB2O4 (BBO), a well-known nonlinear optical crystal can be tagged by a distortion group, R-3*c.  It has all the information about the distortion, and determines how every crystal property change as a function of this distortion path.


Discovering minimum energy pathways via distortion symmetry groups, Jason M. Munro, Hirofumi Akamatsu,Haricharan Padmanabhan, Vincent S. Liu, Yin Shi, Long-Qing Chen, Brian K. VanLeeuwen, Ismaila Dabo, and Venkatraman Gopalan, Physical Rev. B, 98, 085107 (2018).

Physical systems evolve from one state to another along paths of least energy barrier. Without a priori knowledge of the energy landscape, multidimensional search methods aim to find such minimum energy pathways between the initial and final states of a kinetic process. However, in many cases, the user has to repeatedly provide initial guess paths, thus implying that the reliability of the final result is heavily user-dependent. Recently, the idea of “distortion symmetry groups” as a complete description of the symmetry of a path has been introduced. Through this, a new framework is enabled that provides a powerful means of classifying the infinite collection of possible pathways into a finite number of symmetry equivalent subsets, and then exploring each of these subsets systematically using rigorous group theoretical methods. The method, which we name the distortion symmetry method, is shown to lead to the discovery of previously hidden pathways for the case studies of bulk ferroelectric switching and domain wall motion in proper and improper ferroelectrics, as well as in multiferroic switching. These provide novel physical insights into the nucleation of switching pathways at experimentally observed domain walls in Ca3Ti2O7, as well as how polarization switching can proceed without reversing magnetization in BiFeO3. Furthermore, we demonstrate how symmetry-breaking from a highly symmetric pathway can be used to probe the non-Ising (Bloch and Néel) polarization components integral to transient states involved in switching in PbTiO3. The distortion symmetry method is applicable to a wide variety of physical phenomena ranging from structural, electronic and magnetic distortions, diffusion, and phase transitions in materials.

Pathways obtained for 180◦ ferroelectric domain wall motion in PbTiO3. (a) The structure of the supercells of the initial and final states used to construct the initial path for 180◦ domain-wall motion. Each box indicates a PbTiO3 unit cell, with the red and blue arrows indicating polarization direction. (b) The energy relative to the initial and final states as a function of reaction coordinate for the final paths obtained from NEB calculations. The colors indicate the distortion group of the final path shown in the tree in panel (c). (c) The tree of distortion symmetry groups resulting from path perturbations. (d) The Ising, Bloch, and Néel components of the Ti displacement for the first unit cell in the structures shown in panel (a) as a function of reaction coordinate. The energy of the complete structure is indicated by the color of the line, with the colored labels indicating which paths are present in the data.



Implementation of Distortion Symmetry for the Nudged Elastic Band Method with DiSPy . Jason M. Munro, Vincent S. Liu, Venkatraman Gopalan, and Ismaila Dabo,  Computational Materials.5, 52 (2019).

The nudged elastic band (NEB) method is a commonly used approach for the calculation of minimum energy pathways of kinetic processes. However, the final paths obtained rely heavily on the nature of the initially chosen path. This often necessitates running multiple calculations with differing starting points in order to obtain accurate results. Recently, it has been shown that the NEB algorithm can only conserve or raise the distortion symmetry exhibited by an initial pathway. Using this knowledge, symmetry- adapted perturbations can be generated and used as a tool to systematically lower the initial path symmetry, enabling the exploration of other low-energy pathways that may exist. Here, the group and representation theory details behind this process are presented and implemented in a standalone piece of software (DiSPy). The method is then demonstrated by applying it to the calculation of ferroelectric switching pathways in LiNbO3. Previously reported pathways are more easily obtained, with new paths also being found which involve a higher degree of atomic coordination.

Distortion Symmetry in Python (DiSPy) is a Python code written by Jason Munro and Vincent Liu to implement the idea of distortion symmetry introduced by Brian Van Leeuwen and V. Gopalan (2015) into the commercial Nudged Elastic Band Method codes to calculate minimum energy pathways.  It can be downloaded from Github at the link here.


Forty-One Types of Physical Quantities in Arbitrary Dimensions Venkatraman Gopalan, arXiv:1910.09286 [physics.gen-ph]

It is shown that there are 41 types of multivectors representing physical quantities in non-relativistic physics in arbitrary dimensions within the formalism of Clifford Algebra.  The classification is based on the action of three symmetry operations on a general multivector: spatial inversion, , time-reversal, , and a third that is introduced here, namely, wedge reversion. Examples of these multivector types from non-relativistic physics are presented.

A wedge product reversal operation (1-dagger) leaves scalars and vectors invariant,  reverses bivectors and trivectors, leaves quad- and pentavectors invariant, reverses hexa- and heptavectors and so on. ^ indicates wedge products between linearly independent vectors. 3D contains only up to trivectors. Quadvectors and higher order wedges exist in 4D and above.


Antisymmetry: Fundamentals and Applications (A Review article), Hari Padmanabhan, Jason Munro, Ismaila Dabo, Venkatraman Gopalan, Annual Reviews of Materials Research, in print (2020)

Symmetry is fundamental to understanding our physical world.  An antisymmetry operation switches between two different states of a trait, such as two time-states, position-states, charge-states, spin-states, chemical-species etc.   This review covers the fundamental concepts of antisymmetry, and focuses on four antisymmetries, namely spatial inversion in point groups, time reversal, distortion reversal and wedge reversion.  The distinction between classical and quantum mechanical descriptions of time reversal is presented.  Applications of these antisymmetries in crystallography, diffraction, determining the form of property tensors, classifying distortion pathways in transition state theory, finding minimum energy pathways, diffusion, magnetic structures and properties, ferroelectric and multiferroic switching, classifying all physical quantities in arbitrary dimensions, and antisymmetry-protected topological phenomena are presented.

Time Reversal and Magnetic Symmetry in Crystals: The first reported neutron diffraction study on MnO revealing magnetic symmetry by  Shull et. al. (a) Unit-cell doubling and four extra antiferromagnetic reflections are observed below the Neel temperature of 120 K, since neutrons are sensitive to the spin of the scattering electrons. (b) The unit cell of MnO, shown with only the Mn atoms and the magnetic moments at each site.


Double antisymmetry and rotation reversal space groups, K. VanLeeuwen, V. Gopalan, D. B. Litvin, Acta Crystallographica A, 70, 24-38 (2014). & Crystallographic data of double antisymmetry space groups, M. Huang, B. K. VanLeeuwen, D. B. Litvin, V. Gopalan, Acta Crystallographica A, A70, 373-381 (2014).

When two antisymmetry elements such as distortion reversal (which is a generalization of rotation reversal) and time- reversal symmetry are considered in conjunction with space-group symmetry, it is found that there are 17 803 types of symmetry which a crystal structure can exhibit. These symmetry groups have the potential to advance understanding of polyhedral rotations in crystals, the magnetic structure of crystals and the coupling thereof. The full listing of the double antisymmetry space groups can be found in the supplementary materials of the present work and at the following link.

Double antisymmetry space groups: (a) Multiplication table of 1’1*. To evaluate the product of two elements, we find the row associated with the first element and the column associated with the second element, e.g. for 1′-1*, go to the second row, third column to find 1’*. (b) Cayley graph generated by 10 , 1* and 1’*. To evaluate the product of two elements, we start from the circle representing the first element and follow the arrow representing the second, e.g. for 1′ -1*, we start on the red circle (1′ ) and take the blue path (1*) to the green circle (1’*).


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