Time Reversal Symmetry and Space Groups

Chapter 3 Chapter 3 Group Theory and the Analysis of Periodic Electromagnetic

3.4 Space Groups

3.4.4 Time Reversal Symmetry and Space Groups

D(*k,a), and (j, i) label the row and column within each block.

An example may help clarify this notation. Consider an arbitrary k-point in the IBZ³¹. Assuming for simplicity that the point group contained within R is C_{2 ,} the *k then consists of { ±k }. In this case R

=

'lr0 { e, C2 } and

gk =

'lr0 { e}

=

1r

= {

e!Rh}.

Thus, the only possible IRREP of 9ok is the identity REP (A). Since the *k consists of ±k, the identity REPs come in pairs which have their spaces coupled through the only coset generator ( C2 ) 0):

(3.85)

(3.86)

Time reversal symmetry can best be described physically as a reversal of motion [72}: if one records the dynamical evolution of a field (particle, object, etc.), and then replays the recording backwards, the resulting motion can also described by the same set of dynamical equations. From a (scientific) observer's standpoint, both playbacks of the recording (in forward or reverse) would seem equally as physical.

Mathematically one is performing an inversion in the time dimension (t ~ -t).

In order to fully understand time reversal symmetry in the context of Maxwell's equations, we must first determine how the electric and magnetic fields transform.

The transformation properties of the electric and magnetic field are formally de- termined by the transformation properties of the electromagnetic 4-vector potential Aµ [73}³³. Based on physical intuition alone, however, we expect under time rever- sal symmetry charge density to remain unchanged and current density (movement of charge) to reverse direction. Considering the effects of time reversal symmetry on an electric and magnetic dipole, we then expect the electric field to be even under time reversal and the magnetic field to be odd [66]:

TE(x, t) = E(x, -t), TB(x, t)

=

-B(x, -t).

(3.87) (3.88) This is the representation of T on the classical real fields. The full spectrum of eigenfunctions permitted by Maxwell's equations includes complex field amplitudes from which the real fields are a special linear combination. The representation of T on the complex fields is a little more difficult to motivate, however, is essential as most spectra of the electromagnetic fields are computed using complex fields in order to take advantage of the fact that the harmonic functions are the IRREPs of

33To describe light relativistically, one must use the concept of an electromagnetic 4-vector poten- tial Aµ and an electromagnetic stress-energy tensor Fµv. Our analysis is done in "3+ l" -space so to speak. One may question whether or not we will miss some symmetries (within our single inertial frame) because of the narrower scope of our analysis. The answer is undoubtedly yes; however, for our purposes a non-relativistic picture will be adequate.

time translations, as well as to help bridge the gap from classical to quantum fields.

Consider the electric field for example³⁴:

(3.89)

where m(w) is an index into the degenerate space of modes for each frequency, and the positive and negative frequency components are combined such that E is real.

The Cm(w) are complex constants in the context of the classical fields; however, in quantum field theory these coefficients are upgraded to operators which act upon a Hilbert space representing the quantum states of the field. The complex spatial fields Em(r,w) are eigenfunctions of eE in eq. (3.47) with eigenvalue,\= (w/c)².

As discussed earlier ,\ must be real and positive. It follows that the frequency w must be real. However, the spectrum of 8E permits both positive and negative frequencies, w = ±c./X. Furthermore, since Maxwell's equations are real for real E(r, t), Em(r, w)* is also an eigenfunction with frequency ±c~³⁵. The degeneracy of Em and E~ is a consequence of the time reversal symmetry embedded in Maxwell's equations, and is present regardless of the space group of E(r).

The representation of the action of T on the complex spatial fields is that of complex conjugation (up to a multiplicative phase factor)³⁶:

(3.90)

With this definition the action of T on the real classical fields and the action of T on

34The magnetic field has a slightly different form, B(r, t)

=

J;'

g:

l:m(w) -i [dmBm(r)e-iwt -

d~B~(r)eiwt],

in order to satisfy the time reversal relation in eq. (3.88). This makes intuitive sense as the curl(B) is proportional to the derivative of E.

35To see this simply take the complex conjugate of eq. (3.47).

36Similarly for the complex spatial magnetic fields.

the complex spatial fields are consistent. The consequences are also the same, namely that Em and E:n are degenerate in frequency. Also, consider the operation of T on a plane wave:

Te-i(k·x) = e-i(-k·xl. (3.91)

We see that the direction of motion of the plane wave has indeed been reversed (k ~ -k), which is physically satisfying as we expect time reversal to correspond to a reversal of motion. However, the action of complex conjugation not only reverses the direction of each individual plane wave but also reverses the phase relationship between different plane waves³⁷. The operation of T is now nonlinear on the complex spatial fields. It is what is called anti-linear or anti-unitary [65, 72, 7 4, 75). As described below this complicates the group theoretical analysis of such a system³⁸.

In order to analyze a symmetry group which includes anti-linear operators such as complex conjugation requires the use of corepresentation theory [69, 72, 76]. It can be shown that the corepresentations are completely defined in terms of the normal space group representations. A simple criterion can then be used on the normal representation in order to determine any additional degeneracies that may appear in the photonic bandstructure [69):

(3.92)

d

0,

37The simple replacement of k with - k actually corresponds to the parity operation, or spatial inversion.

38Why do we need the added complication of complex conjugation? The answer lies in the un- derpinnings of quantum mechanics, and experimental verification of such a symmetry. It was E. P.

Wigner who first attributed the added degeneracy present in the experiments of H. A. Kramers to the time inversion symmetry present in the Schroedinger equation [72].

where { dJs} are all the elements of R/T for which dk

=

-k

+

Gg, ^9ak is the order of 9ok, and x(k,a) ( {dis

}2)

is the character of the IRREP being tested. If the sum equals g₀k, then there is not an additional degeneracy due to T, whereas if the sum is -g₀k

or 0, then there is. This test, in the case of representations of the electromagnetic field, is equivalent to a test for the reality of the normal IRREPs.

Regardless of the outcome of the above test, if there is time reversal symmetry present then the spectrum at k and -k are degenerate. The complication occurs if - k is already a member of the *k and this degeneracy has already been captured by the normal IRREPs. In this case it is possible that additional degeneracies are formed between the normal IRREPs by the presence of time reversal symmetry. It is interesting to note that for the same Bravais lattice that electrons and photons may have different degeneracies due to time reversal symmetry. This is because of the spin of the electron.