R Bonneville

Canonical Reduction of Tensors and the Physical Properties of Condensed Matter: Application to Optics

A general framework for the description of the physical properties of matter by a canonical reduction procedure of tensors is presented; besides geometrical symmetries, this paper emphasizes the role of intrinsic symmetries which are due either to the indiscernability of some of the physical quantities involved or to thermodynamical arguments. The intrinsic symmetries are expressed through the behaviour of the tensors describing the investigated property under the effect of some index permutation. The scheme of reduction of any tensor into parts that are irreducible not only with respect to rotations and inversion but also with respect to index permutations is shown and examples are given in the area of light-matter interaction.

INTRODUCTION

The interest in using an irreducible tensor formalism for describing the physical properties of condensed matter is well known. With such a formalism, for instance, there is easy writing of the selection rules imposed by the geometry of the system as well as the simplicity of performing such geometrical operations as the transformation from a microscopic frame to a macroscopic one. The reduction of a cartesian tensor into parts that are irreducible with respect to the orthogonal group or to a particular point-group, either in the cartesian or in the spherical formalism, has been systematically reviewed and many examples of the applications of that approach have been given (for examples see Reference 1 and bibliography therein).

Nevertheless, in most cases the decomposition of a rank-r cartesian tensor ... ijk T is not unique; alternative reduction schemes are possible whenever one can find in the decomposition results several linearly independent parts belonging to the same irreducible representation of the orthogonal group. For instance a rank-3 tensor ijk T has three mutually orthogonal vector parts; given such a triplet, any other tensor deduced from the former through a unitary transformation is also convenient. One thus may ask what is the best reduction scheme, i.e. the most suitable for a given physical problem; the aim of this paper is to clarify this point.

Any tensor may undergo two types of transformations; firstly, it may change the value (i.e. x or y or z) of the cartesian indices independently of their position (geometrical transformation); secondly, it may change the position of the cartesian indices independently of their value (intrinsic symmetry). Most often, physical effects have intrinsic symmetries which proceed either from the indiscernability of some among the quantities involved or from thermodynamical arguments, and the manifestation of those intrinsic symmetries is to affect the behaviour of the tensors associated with certain physical quantities under a class of index permutations. For instance, the second order electromagnetic susceptibility tensor (2) ( 2 , , ) ijk      describing second-harmonic generation is symmetric in the j-k permutation because of the indiscernability of the two fundamental fields; the third rank tensor ijk  which accounts for optical activity is antisymmetric in the permutation of its extreme indices i and k from a general property of the kinetic coefficients.

This gives the key of the problem: before decomposing a tensor into parts irreducible under geometrical operations, one first has to decompose it into parts irreducible under index permutations and to extract those which are compatible with the intrinsic symmetries of the studied effect 2 ; each of these latter parts can then be reduced under the three-dimension orthogonal group and ultimately under the particular point group of the system under investigation 3 .

Let T be any rank-r tensor; T will split under the permutation group of r objects   In the two following parts of this paper, we will present in more detail the two steps of this approach. The two last parts treat specific examples in the area of light-matter interaction: linear susceptibility, optical activity and magnetooptical effects, and nonlinear optical processes.

REDUCTION PROCESS, PERMUTATIONS

Whereas handling the irreducible representations of the three-dimension orthogonal group as well as those of the point groups is in current use in condensed matter physics, the properties of the irreducible representations of the permutation groups are far less well known and most often are only referred to in studies of the exchange of identical particles. The main results, which will be useful for the present study, are listed hereafter; for more details, the reader is referred, for instance, to References 4 and 5.

Considering the (non-commutative) permutation group of r objects  

          2 2 4
4 , 31 , 2 , 21 , 1 where the number of occurrences of a given integer has been put as an exponent. The irreducible representation associated with the partition   One can now return to the original problem of splitting a tensor into irreducible parts.

Given any group G of transformations within a N-dimension vector space E , the rank-r tensors subtend a tensor space

N E E E ... E E      
, which may be viewed as a N r -dimension reducible representation of G. A general way of reducing any tensor T of this type would consist of first reducing T with respect to a supergroup of G. Here, G will be the orthogonal group O(N) and as a supergroup we will take the full linear group GL(N). It can be shown that the only tensor operations which commute with GL(N) are the index permutations. The reducible N r -dimension space subtended by the rank-r tensors splits into irreducible subspaces whose dimensions are easy to find, as will be shown below. Firstly, the dimension of the tensor subspace possessing the permutation symmetry of the fully symmetric representation   r is the number of ways of giving a value to the r indices, with an allowed repetition, i.e.   From the relations between these numbers, it is easy to calculate, step by step, the explicit number of independent components for all the tensors irreducible under index permutations.

REDUCTION PROCESS, ORTHOGONAL GROUP

Henceforth N= 3 and each index i, j, k, ... will be given any of the three values x, y, z. The number of independent components of a rank-2, 3, 4 tensor irreducible under index permutations is shown in Figure 4. With the help of one can obtain the reduction spectrum of any cartesian tensor. This recursive reduction process is summarized in Figure 5, where a spectroscopic notation has been adopted for the irreducible tensor parts: an irreducible tensor part of weight J = 0, 1, 2, 3, 4, …, transforming like the base vector {} q  under index permutations, will be labelled by {} q  followed by a capital letter such as S, P, D, F,G,... . for a true tensor (parity =(1) J ) and by an overlined capital letter S, P, D, F, G,... for a pseudo-tensor (parity =(1) J+1 ). It has been assumed that the original rank-r cartesian tensor T is a true tensor (parity (1) r ); if T is a pseudo-tensor (parity (1) r+1 ), the representations S and S , P and P , D and D , F and F , ... respectively must be exchanged.

In order to obtain the explicit expressions of the irreducible components of a rank-r tensor T in terms of the original cartesian ones ijk T … one must first reduce

T under  
      ˆ1,
      ˆ1, 2 1, 2 / 2 A I P  .
More generally, the symmetrizer of any function of q objects (1,2, ...,q) is given by (1, 2,...,q)

(1, 2,...,q) / q ˆˆ! P SP   where the sum runs over the q! different permutations (1, 2, ) ˆ..., q P of (1,2, ...,q), and the anti-symmetrizer by ˆ(1, 2,...,q) ( ) (1, 2,...,q ˆ) / ! ˆq

Examples

for rank-2 and rank-3 cartesian tensors will be given below.

The reduction spectrum of a cartesian rank-2 true tensor is

     2 2 S + D , 1 P . The projector upon   2 is the symmetrizer     ˆ1, 2 / 2 IP  , hence   {2} /2 ij ij ji t T T    {2} {2}S / 3 /3 ij pq pq ij pq xx yy zz ij t T T T                {2} {2} {2}D {2}S /3 3 / 6 ij ij ij ip jq ij pq pq pq ip jq ij pq pq qp pq t t TT                The projector upon   2 1 is the anti-symmetrizer     ˆ1, 2 / 2 IP  , hence   2 {1 } 2 / 2 {1 }Pij ij ij ji ijk kpq pq k pq t T T T      
The reduction spectrum of a rank-3 cartesian true tensor is

         3 3 P F , 21 2 P D , 1 S 
The projector upon {3} is

                    1/ 6 , , ˆˆˆˆˆˆ, , , , , , , / 6
S i j k The projector upon  

I P i j P j k P k i P i k P k j P i j P j k      hence   {3} /6 ijk ijk jik ikj kji kij jki t T T T T T T      
3 1 is                     1/ 6 , , ˆˆˆˆˆˆ, , , , , , , / 6
A i j k

I P i j P j k P k i P i k P k j P i j P j k       Hence   3 {1 } 3 / 6
                  ˆˆˆˆˆ1 / 3 , , 1/ 3 , , , , A i k S i j I P i k P i j P i k P i j     for   21 ' and                   ˆˆˆˆˆ1 / 3 , , 1/ 3 , , , , A i j S i k I P i j P i k P i j P i k    
for  

"

Hence the corresponding base vectors are The explicit calculation of this scattering amplitude can be achieved through the cartesian formalism in a straightforward but rather tedious way and it appears somehow simpler for high rank tensors to use a spherical formalism. The methods given in Reference 1, which require the Clebsh-Gordan coefficients algebra, are cumbersome and a simpler technique of reduction of any rank-r cartesian tensor T will be presented now. The reduction spectrum of T has been previously determined (Figure 5); a given irreducible part of

  {21}' /3 ijk ijk kji jik jki t T T T T       {21}" /3 ijk ijk jik kji kij t T T T T     One derives   {21}'
T labelled as { } , qJ t  
has a weight J and a parity and transforms like the q th base vector of the irreducible representation {}  of the permutation group   σr.

One can first consider the case where

       
be the coordinates of a unit vector in the direction of polar angles  and  . From the explicit expression of the J M C (see Table 1), and using 1

x x y y z z u u u u u u  
 , it is a straightforward matter to build an homogeneous rank-r polynomial in , , ,,

x y z z y y z x x z z x y y x u u u u u u u u u u u u u u u      
; the resulting rank-r homogeneous polynomial behaves like a pseudo-tensor of weight J and henceforth will be noted as

J M
C . Finally, one applies to the latter polynomial the projector ˆ} /{ q P  as in the true tensor case.

An example of that process will now be given for the 0 th irreducible components of a cartesian rank-2 and a rank-3 true tensor. The resulting expressions are given at a normalization factor which will be commented upon later. The reduction spectrum of a cartesian rank-2 true tensor is   The reduction spectrum of a rank -3 cartesian true tensor is

         3 3 P F , 21 2 P D , 1 S  . From   1 0 z z x x y y z z C u u u u u u u u     and   3 0 2 3 3 / 2 z z z z x x z y y C u u u u u u u u u   
and by application of the projection operator ˆ} / {3 P , one can find    

3 P 0 3 xzx zxx xxz yzy zyy yyz zzz T T T T T T T            3 F 0 2 xzx zxx xxz yzy zyy yyz zzz T T T T T T T        
As for the representation {1 3 }, the comparison with 0 0

1 x x y y z z C u u u u u u    
which can be changed into the naturally fully anti-symmetrized expression 0 0 ()

x y z x z y y z x y x z z x y z y x C u u u u u u u u u u u u u u u u u u       yields 3 {1 }S xyz xzy yzx yxz zxy zyx T T T T T T       From   1 0 z z x x y y z z C u u u u u u u u    
T T T T T T       From 2 0 2 z z x x y y C u u u u u u    , which can be changed into 2 0 () 2 2 / 2 z x y z y x x y z x z y y z x y x x C u u u u u u u u u u u u u u u u u u      
     2 S 1/ 3 xx yy zz T T T            2 D 0 1/ 6 2 zz xx yy T T T          2 1 P 0 / {} 2 xy yx i T T         3 F 0 1/ 10 2 xzx zxx xxz yzy zyy yyz zzz T T T T T T T                3 P 0 1/ 15 3 xzx zxx xxz xyzy zyy yyz zzz T T T T T T T                21 'P 0 1/ 2 3 2 2 zxx xxz xzx zyy yyz yzy T T T T T T             {21 "P 0 1/ 2 3 2 2 } zxx xzx xxz zyy yzy yyz T T T T T T               21 'D 0 / 2 zxy zyx xzy yzx i T T T T          () 21 "D 0 / 2 zxy zyx yxz xyz i T T T T         3 { } 6 S / 1 xyz xzy yzx yxz zxy zyx i T T T T T T       
To conclude this part, it must be recalled that the norm

EXAMPLES

We will now briefly examine four well-known optical effects, namely linear susceptibility, optical activity, magneto-optical effects, and nonlinear susceptibilities; we will assume that a dielectric medium undergoes an optical field described by the vector potential

* . ( ) ( ) . aexp i t a exp i t         K A e K r e K r
(  : frequency, K : wave-vector, e : polarization).

Linear Susceptibility

In the electric dipole approximation, the linear polarizability tensor

  0 / 2 R n n n D resonant term E E i         h   0 / 2 NR n n n D antiresonant term E E i         h
In Table 2 the irreducible spherical components of  are expressed as combinations of the original cartesian components.

Optical Activity

Optical activity arises from the dependence of the dielectric polarization  is anti-symmetric in the exchange of its extreme indices i and k, i.e. j ijk ki   .

A quantum mechanical calculation of  on a multipolar expansion of the field-matter interaction hamiltonian in the radiation gauge, i.e.

. . : ...

H       d E M B Q E  ( d : electric dipole,
M : magnetic dipole, Q : electric quadrupole, E : electric field, B : magnetic field) shows both a magnetic dipole contribution In an isotropic or cubic medium where the pseudo quadrupole and the vector vanish, optical activity only arises from the pseudo-scalar term, which is entirely due to the magnetic dipole contribution; to our knowledge, the vector -type term, which is presumably very weak since it is the antisymmetric part of g , has not been observed yet.

In Table 3 the irreducible spherical components of  are expressed as combinations of the original cartesian components; the symmetry condition

Magneto-Optical Effects

In the presence of a strong static magnetic field, the dielectric polarization P at the frequency w can be expanded as a series of the applied magnetic field % is a rank-4 true tensor, symmetric in the permutation of its two extreme coefficients (a property of the kinetic coefficients) or its two middle indices (due to the indiscernability of the two magnetic fields), and accounts for quadratic magnetic birefringence (Cotton-Mouton effect); its reduction spectrum is

   2 { 4 S D G , 2 S }( D)   
.

NONLINEAR OPTICS

The (r-1) th order dielectric polarization at the frequency 1

 can be expanded as a series of the (r-1) incident electric fields at the respective frequencies 23 , ,..., r  is invariant in any simultaneous permutation of the indices and also of the frequencies.

             ( ) ( ) 1 1 2 3
Moreover, if the dispersion is weak the susceptibility tensor must approximately be symmetric in any index permutation. Thus, in the transparency range of the medium, the dominant parts of the reduction spectrum are those which appear in the spectrum of a fully symmetric tensor of the same rank while the other ones are, presumably, small. As a consequence, in the general case, the fully symmetric parts include resonant and non-resonant terms whereas the other ones only contain resonant terms.

Three-Wave Mixing

The first hyperpolarizability tensor   For instance, the fully antisymmetric pseudoscalar

  3 (2) (2) (2) (2) (2) (2)
{1 }S 6

CONCLUSION

We have emphasized the importance of index permutations in the reduction of cartesian tensors with respect to the orthogonal group and its subgroups. First the intrinsic symmetries of the physical effect under investigation must be considered; these symmetries are associated in the representative tensors with some index permutation symmetries which cause some of the t ensor parts, those that are irreducible under the permutation group, to vanish. The remaining parts can be reduced under the orthogonal group. Taking into account the effects of point-group invariances upon tensors constitutes the ultimate step of the process.



  transformation properties under   r  as the q th base vector {} q  of the representation {}  of   r  ; then each of the {} q t  s will in turn split into a set of irreducible tensors and belonging to the irreducible (2J + 1)-dimensional representation Jπ D of weight J and parity  of the three-dimension orthogonal group O(3). Finally the following equation constitutes the reduction spectrum of

r 2 . 4  4 

 244 , any of the r! elements of   r  can be expressed as a product of pair permutations; all the properties of   r  can thus be derived from those of   It can be shown that there is a one-to-one correspondence between the irreducible representations {}  of   r  and the so-called partitions of r: a partition of r is a set of integers   + n + ... + n = r . For instance, given the permutation group of four objects  hence has five irreducible representations which are respectively noted as

2  2 1 , both of dimension 1 ( 3  3 Figure 2 , 1 ,

 2213321 figured by filling the r squares of the array with the r objects in a proper arrangement; this arrangement can be derived through the following recursive process. One starts with two objects:   2 

  The tensors which belong to a given irreducible representation of the permutation group   r  belong to the same irreducible representation of GL(N). The tensor space N E  is therefore reducible with respect to GL(N) into tensor subspaces which are irreducible under   r  ; however, those subspaces are not irreducible under the orthogonal group O(N).

  of the tensor subspace having the permutation symmetry of the fully antisymmetric representation   r 1 is the number of ways of giving a value to the r indices with no repetition allowed, i.e.   N r if r <N, 0 otherwise. In Figure 3 the recursive generating process of the base vectors of the irreducible representations of   r  has been reprinted from Figure 1 but in addition under each base vector of   r  the number of independent components of a tensor transforming in the same way as the base vector under   r  has been indicated, either explicitly for the tensor subspaces of respective symmetries   r and   r 1 , or formally.

Figure 4

 4 Figure 4 and using the composition rule of the irreducible representations of O(3), i.e.

  upon the base vectors {} q  of the relevant irreducible representations {} q  of   r  and so build a set of linearly will be performed by applying the projection operators whose explicit expressions can be obtained by remembering that the eigenvector represented by a given Young array is symmetric in the permutation of any pair of indices on the same line and antisymmetric in the exchange of any pair of indices in the same column. I being the identity operator and   ˆ1, 2 P the permutation operator of two objects, 1 and 2, the symmetrizing operator of any function of the pair (1, 2) is

  (1)) if P writes as the product of an even (odd) number of pair permutations. Let one now consider the Young array associated with a given base vector {} q  of a d-dimension irreducible representation of   r  and the symmetrizers (1, 2,. ) ˆ..,q l S of the lines (l) of the array and the anti-symmetrizers ˆ(1, 2,...,q ) c A of the columns (c);indices which fill the squares of the array. The operator { } ( ) q ! (1, 2,...,q ) q ! /d (1, 2,...,q ) run over all the lines and all the columns of the array, is the projector upon the selected base vector. The d projectors associated with the dYoung arrays featuring the d base vectors of {}  thus provide a complete set of base vectors for that representation. Note that another, different but equivalent, set of base vectors could be obtained by the application of the projecting operators ( ) q ! (1, 2,...,q ) q ! (1, check that the sum of the projectors upon all the base vectors of all the irreducible representations of   r  is just the identity. Every tensor {} q t  generated as above will then be reduced under O(3) by projecting it upon the irreducible representations Jπ D of O(3); after noticing that a given representation Jπ D will appear at most only once in the projection of a given {} q t  , that reduction can actually be done by purely cartesian techniques by contracting {} q t  either with the symmetric scalar unit tensor ij  or with the antisymmetric pseudo-scalar unit tensor ijk  .

  physical quantity such as a scattering amplitude U writes as a scalar product between two tensors of the same rank, one of them,  , being a susceptibility of the medium and the other one, This scalar product can be expressed as a sum of scalar products of irreducible tensors:

.C

  By applying to that polynomial the projector ˆ} /{ q P  one generates a homogeneous polynomial having the required transformation properties under both permutations and rotations. The expression of the latter polynomial provides the expression of the irreducible tensor component having the same properties.Next, one can consider the case where { under any rotation; then one can make, for a single given index position (arbitrarily chosen but identical for all the terms of the polynomial), the following substitution:

.

  The above expressions of the irreducible spherical tensor components in terms of linear combinations of the cartesian components are not normalized, while the transformation from cartesian to spherical coordinates is a unitary one. Let  be any set of indices which allow one to discriminate among the various irreducible parts with the same weight J; taking normalization into account leads to The results are as follows (N.B. the phases of the coupling coefficients ... JM ijk   have been chosen to be   rJ i  so as to be consistent with Reference I):

  an important notion since it allows one to compare, for some property, the importance of a term of given symmetry in two different compounds or to compare the relative importance of the terms of various symmetries in the same compound (examples in the field of nonlinear optics are given in References 7 selection rules in spherical coordinates for the most frequently encountered point groups are given for example in Reference 7.

e

  permutation of their indices from a general property of the kinetic coefficients; the light-matter scattering amplitude U is proportional to are the polarization vectors of the incident and scattered fields respectively. Reversing the direction of time exchanges the roles of s e and i e but does not change U, hence ij ji   ; as a consequence there is no vector-type contribution to the linear susceptibility. Since ij  may be complex, ij ji   means that  is not hermitic; however the fields are real, The real and imaginary parts of  , which account for dispersion and absorption in the medium respectively, are connected by the well-known Kramers-Kronig relations 9 . If the losses in the medium are negligible,  is invariant in the simultaneous permutation of the indices and of the frequencies, i.e. the latter equality with the former one, one finds * ij ji   that is  is hermitic. The losses are thus taken into account by the antihermitic part of  .A quantum-mechanical calculation provides the explicit expression of  : electric dipole) with obvious notations and

  by a rank-3 cartesian tensor ijk  ; the light-matter scattering amplitude U is then proportional to  



  to the effect. The resulting expression for  is: contribution is a true rank-3 tensor anti-symmetric in the exchange of its extreme indices; since M is a pseudo-vector ( P type), () M  has three irreducible parts, a pseudo-quadrupole   21 D , a true vector   21 P , and a pseudo-scalar  3 1 S .The electric quadrupole contribution is also a true rank-3 cartesian tensor antisymmetric in the exchange of its extreme indices; since Q is a symmetric and traceless rank-2 tensor ( D type), () Q  has only two irreducible parts, a pseudoquadrupole   21 D and a true vector   21 P . When comparing the reduction that  can be written as D P S  ; that is the reduction spectrum of a cartesian rank-2 pseudo-tensor; optical activity can thus be described by a pseudo-tensor

3 ) ; conversely if p e is circularly polarized and if i e and s e

 3s mixing of three different frequencies in an optically active isotropic medium such as a solution of chiral molecules.The electro-optical effect is described by the tensor permutation of its extreme indices from a time-reversal argument (unlike the magnetic field in the magneto-optic effect, the electric field is unchanged); the reduction spectrum of the permutation of its two last indices from a time -reversal argument and shows a similar reduction spectrum. The second-harmonic generation effect is described by the tensor permutation of its two last indices because of the indiscernability of the two fundamental fields and again shows the same reduction spectrum, i.e. the original cartesian components; the symmetry condition ijk ikj implies that the irreducible parts   21 P ' and   21 P" are identical along with {21}D' and {21}D". be the polarization vectors of the pump beam, of the incident signal beam, and of the analysed signal beam respectively. The scattered amplitude is proportional to Since only the isotropic parts (J = 0) of(3) do not vanish, one deduces the following selection rules: if all the e are parallel to each other, one can only observe the fully symmetric part of (are linearly cross-polarized, one will only detect the non-fully-symmetric term.

2 S; with Young's methods, one can find the normalized expressions of those two scalars:

ACKNOWLEDGEMENT

The author feels indebted to J. Zyss (CNET) for fruitful discussions on the manuscript.

Four-Wave Mixing

The second hyperpolarizability tensor   For instance the coherent Raman effect tensor

, , ,

is symmetric in the permutation of its two extrema and of its two middle indices; its reduction spectrum thus reduces to      2 4 S+D+G , 2 (S+D) , and ultimately, in the degenerate configuration p   , to the fully symmetric term   4 S + D ( + G) . In the non-degenerate configuration the case of an isotropic medium will be considered; the reduction spectrum then reduces to two scalars, one totally symmetric   4S and one non- totally-symmetric  