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Spatial Inversion Symmetry and Non-linear Response

Popularity:521 ℃/2024-07-26 16:28:41

We define a parity transformation as the inversion of all coordinates:

\[\mathcal{P}: \begin{pmatrix} x \\ y \\ z \end{pmatrix} \rightarrow \begin{pmatrix} -x \\ -y \\ -z \end{pmatrix} \]

If in a one-dimensional world, a metric transformation is like looking at the world through a "mirror"; in a three-dimensional world, it is a point of symmetry of the whole system with respect to a reference point.

Spatial inversion symmetry refers to the property of a "lattice" system in which the atomic positions, physical formulae, and other characteristics remain unchanged before and after an u-symmetric transformation, also known as theconservation of cosmic volume. Any even number of responses is forbidden under the condition of conservation of the Ucronyms. We can also understand that:

In crystals with spatial inversion symmetry, even order nonlinear effects are prohibited.

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Let's say I apply an electric field to a crystal.\(E\), then experimentally we can measure its electrode polarization rate\(P\). Now we fix the applied electric field in the x-axis direction, assuming that the response polarization vector is in the y-axis direction (as in the figure above) . We can set the\(P\)-\(E\) The response relationship is expressed as:

\[P=\chi^1 E + \chi^2 E^2 + ... \]

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Now we reverse the applied electric field (as shown above). All the physical laws should remain the same before and after the transformation, if the crystal satisfies the conservation of Utopia. Then the direction of reversal should lead:\(E\rightarrow -E\), \(P\rightarrow -P\). I.e.:

\[-P = \chi^1 (-E) + \chi^2 (-E)^2 + ... \]

Connecting the two equations above, we can find that only the\(\chi^2=0\) conditions for space inversion symmetry to hold. By a simple generalization we can see that all even-ordered responses should not exist if spatial inversion symmetry holds.

So conversely, if the crystal does not satisfy spatial inversion symmetry, would there still be such a restriction? The answer is no.

Spatial inversion symmetry breaking allows for nonlinear effects.

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We still apply an applied electric field, and at this point the electrodeposition response has uncertainty. Because crystals do not have spatial inversion symmetry (conservation of Usuymmetrics), if we flip the electric field, the direction of the polarization vector is not necessarily flipped and its magnitude is not necessarily unchanged. We can use the\(P^{'}\) to make a distinction. At this point, all nonlinear effects can exist (\(\chi^2\) (can be a finite value).

Above we have discussed all crystal structures, if we consider spin, another symmetry is involved:Time-reversal symmetry. A single time-reversal operation does not change the coordinates of the lattice, but flips all spin directions.