Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter
Author
Matthias Geilhufe
Title
Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter
Description
GTPack - A Wolfram Language Group Theory Package
Category
Essays, Posts & Presentations
Keywords
Group Theory, paclets
URL
http://www.notebookarchive.org/2022-10-cfxwbcm/
DOI
https://notebookarchive.org/2022-10-cfxwbcm
Date Added
2022-10-27
Date Last Modified
2022-10-27
File Size
1.16 megabytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter
Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter
Matthias Geilhufe
Load GTPack
In[]:=
Needs["GroupTheory`"]
In[]:=
?GT*
Out[]=
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Symbols
Symbols
transformation symbols
In[]:=
C3z
Out[]=
C
3z
In[]:=
C4x
Out[]=
C
4x
In[]:=
C2α
Out[]=
C
2α
In[]:=
IC2a
Out[]=
IC
2a
representation matrices
In[]:=
GTGetMatrix[C3z]
Out[]=
-,,0,-,-,0,{0,0,1}
1
2
3
2
3
2
1
2
In[]:=
GTGetMatrix[C4x]
Out[]=
{{1,0,0},{0,0,1},{0,-1,0}}
In[]:=
GTGetMatrix[C2α]
Out[]=
-,,-,,-,-,-,-,-
1
3
2
3
2
3
2
3
1
3
2
3
2
3
2
3
1
3
In[]:=
eα
Out[]=
-,-,
1
3
1
3
1
3
SU(2) matrices?
In[]:=
GTGetSU2Matrix[C3z]
Out[]=
--,0,0,-+
1
2
3
2
1
2
3
2
In[]:=
GTGetSU2Matrix[DC3z]
Out[]=
+,0,0,-
1
2
3
2
1
2
3
2
Groups
Groups
GTPack focus is on crystallographic groups, i.e., point and space groups
Install a group from generators
In[]:=
GTGroupFromGenerators[{C3z}]
Out[]=
Ee,,
-1
C
3z
C
3z
In[]:=
GTGroupQ[%]
Out[]=
True
Install a pre-defined point group
In[]:=
GTInstallGroup[C3]
The standard representation has changed to O(3)
Out[]=
Ee,,
-1
C
3z
C
3z
Install a pre-defined space group
In[]:=
GTInstallGroup[19]
Out[]=
Ee,{0,0,0},,,0,,,0,,,,,,0
C
2z
1
2
1
2
C
2y
1
2
1
2
C
2x
1
2
1
2
Install a pre-defined double group
In[]:=
GTInstallGroup[C3,GORepresentation->"SU(2)"]
The standard representation has changed to SU(2)
Out[]=
Ee,,,,,
__
Ee
C
3z
-1
C
3z
-1
_
C
3z
_
C
3z
In[]:=
GTGetMatrix[C3z]
Out[]=
--,0,0,-+
1
2
3
2
1
2
3
2
In[]:=
GTGetRotationMatrix[C3z]
Out[]=
-,,0,-,-,0,{0,0,1}
1
2
3
2
3
2
1
2
Representation theory
Representation theory
Install a more interesting group
In[]:=
d4hgroup=GTInstallGroup[D4h]
The standard representation has changed to O(3)
Out[]=
Ee,,,,,,,,IEe,,,,,,,
C
2z
C
2y
C
2b
C
2a
C
2x
-1
C
4z
C
4z
IC
2x
IC
2a
IC
2b
IC
2y
IC
2z
IC
4z
-1
IC
4z
Calculate the character table
In[]:=
ctd4h={classesd4h,charsd4h,namesd4h}=GTCharacterTable[d4hgroup,GOIrepNotation->"Mulliken"]
Ee | 2 C 2y | 2 IC 2x | 2 C 2b | 2 IC 2a | 2 IC 4z | 2 -1 C 4z | IEe | C 2z | IC 2z | |
A 1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A 2g | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 |
B 2g | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 |
A 2u | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 |
B 2u | 1 | -1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 |
B 1u | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 |
A 1u | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 |
B 1g | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 |
E u | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 |
E g | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 |
C
1
C
2
C
2y
C
2x
C
3
IC
2x
IC
2y
C
4
C
2b
C
2a
C
5
IC
2a
IC
2b
C
6
IC
4z
-1
IC
4z
C
7
-1
C
4z
C
4z
C
8
C
9
C
2z
C
10
IC
2z
Out[]=
{Ee},,,,,,,,,,,,,{IEe},{},{},{{1,1,1,1,1,1,1,1,1,1},{1,-1,-1,-1,-1,1,1,1,1,1},{1,-1,-1,1,1,-1,-1,1,1,1},{1,-1,1,-1,1,-1,1,-1,1,-1},{1,-1,1,1,-1,1,-1,-1,1,-1},{1,1,-1,-1,1,1,-1,-1,1,-1},{1,1,-1,1,-1,-1,1,-1,1,-1},{1,1,1,-1,-1,-1,-1,1,1,1},{2,0,0,0,0,0,0,-2,-2,2},{2,0,0,0,0,0,0,2,-2,-2}},,,,,,,,,,
C
2y
C
2x
IC
2x
IC
2y
C
2b
C
2a
IC
2a
IC
2b
IC
4z
-1
IC
4z
-1
C
4z
C
4z
C
2z
IC
2z
A
1g
A
2g
B
2g
A
2u
B
2u
B
1u
A
1u
B
1g
E
u
E
g
representation matrices
In[]:=
ΓEg=GTGetIrep[d4hgroup,10,ctd4h];Map[MatrixForm,%]
Out[]=
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1 | 0 |
0 | 1 |
-1 | 0 |
0 | -1 |
-1 | 0 |
0 | 1 |
0 | -1 |
-1 | 0 |
0 | 1 |
1 | 0 |
1 | 0 |
0 | -1 |
0 | -1 |
1 | 0 |
0 | 1 |
-1 | 0 |
1 | 0 |
0 | 1 |
1 | 0 |
0 | -1 |
0 | 1 |
1 | 0 |
0 | -1 |
-1 | 0 |
-1 | 0 |
0 | 1 |
-1 | 0 |
0 | -1 |
0 | 1 |
-1 | 0 |
0 | -1 |
1 | 0 |
In[]:=
ΓB1g=GTGetIrep[d4hgroup,8,ctd4h];Map[MatrixForm,%]
Out[]=
{(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
),(
)}
1 |
1 |
1 |
-1 |
-1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
1 |
1 |
-1 |
-1 |
Play with representations - superconductivity
Play with representations - superconductivity
symmetry-adapted free energy expansion (available with GTPack 1.4)
In[]:=
GTLandauExpansion[{ΓB1g},{{Δ}},4,GOSuperconductor->{True}]
Out[]=
2
Abs[Δ]
4
Abs[Δ]
In[]:=
fl=GTLandauExpansion[{ΓEg},{{η[1],η[2]}},4,GOSuperconductor->{True}]
Out[]=
2
Abs[η[1]]
2
2
Abs[η[2]]
2
4
Abs[η[1]]
2
4
Abs[η[2]]
2
2
2
Abs[η[1]η[2]]
2
Conjugate[η[2]]
2
η[1]
2
2
Conjugate[η[1]]
2
η[2]
2
Calculate bilinears
Calculate bilinears
bilinear basis
In[]:=
basis=Flatten[KroneckerProduct[Conjugate[{η[1],η[2]}],{η[1],η[2]}]]
Out[]=
{Conjugate[η[1]]η[1],Conjugate[η[1]]η[2],Conjugate[η[2]]η[1],Conjugate[η[2]]η[2]}
characters of the direct product ⊗
*
E
g
E
g
In[]:=
charsd4h[[10]]
Out[]=
{2,0,0,0,0,0,0,2,-2,-2}
In[]:=
charsd4h[[10]]^2
Out[]=
{4,0,0,0,0,0,0,4,4,4}
decompose into Clebsch-Gordan sum
In[]:=
GTIrep[charsd4h[[10]]^2,ctd4h]
A
1g
A
2g
B
2g
B
1g
Out[]=
{1,1,1,0,0,0,0,1,0,0}
e.g. :compute Clebsch-Gordan coefficients
B
1g
In[]:=
cgB1g=GTClebschGordanCoefficients[ΓEg,ΓEg,ΓB1g]
Out[]=
-,0,0,
1
2
1
2
In[]:=
Flatten[cgB1g].basis//FullSimplify
Out[]=
-+
2
Abs[η[1]]
2
Abs[η[2]]
2
can emerge as a nematic vestigial order within the above free energy
[Fernandes, et al., Annu. Rev. Condens. Matter Phys., 10:133-154]
[Fernandes, et al., Annu. Rev. Condens. Matter Phys., 10:133-154]
Chemistry - crystal field theory
Chemistry - crystal field theory
Phenomenological theory
Phenomenological theory
Install octahedral group
In[]:=
ohgroup=GTInstallGroup[Oh];ctoh={classesoh,charsoh,namesoh}=GTCharacterTable[ohgroup,GOIrepNotation->"Mulliken"];
The standard representation has changed to O(3)
Ee | 3 C 2z | 3 IC 2x | 6 C 2f | 6 IC 4x | 6 IC 2a | 6 -1 C 4z | 8 IC 3δ | 8 C 3β | IEe | |
A 1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
A 2u | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 |
A 1u | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 |
A 2g | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 |
E u | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 1 | -1 | -2 |
E g | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 |
T 1g | 3 | -1 | -1 | -1 | 1 | -1 | 1 | 0 | 0 | 3 |
T 2g | 3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 | 0 | 3 |
T 1u | 3 | -1 | 1 | -1 | -1 | 1 | 1 | 0 | 0 | -3 |
T 2u | 3 | -1 | 1 | 1 | 1 | -1 | -1 | 0 | 0 | -3 |
C
1
C
2
C
2z
C
2y
C
2x
C
3
IC
2x
IC
2y
IC
2z
C
4
C
2f
C
2e
C
2b
C
2d
C
2c
C
2a
C
5
IC
4x
-1
IC
4x
IC
4z
-1
IC
4y
IC
4y
-1
IC
4z
C
6
IC
2a
IC
2c
IC
2d
IC
2b
IC
2e
IC
2f
C
7
-1
C
4z
C
4y
-1
C
4y
C
4z
-1
C
4x
C
4x
C
8
IC
3δ
IC
3α
-1
IC
3δ
-1
IC
3β
-1
IC
3α
-1
IC
3γ
IC
3γ
IC
3β
C
9
C
3β
C
3γ
-1
C
3γ
-1
C
3α
-1
C
3β
-1
C
3δ
C
3α
C
3δ
C
10
Characters of the angular momentum representation for d-electrons
In[]:=
GTAngularMomentumChars[classesoh,2]
Out[]=
{5,1,1,1,-1,1,-1,-1,-1,5}
Decomposition into irreducible representations of - splitting of d-level in cubic symmetry
O
h
In[]:=
GTIrep[%,ctoh]
E
g
T
2g
Out[]=
{0,0,0,0,0,1,0,1,0,0}
And now for - splitting of d-level in square symmetry
D
4h
In[]:=
GTAngularMomentumChars[classesd4h,2]GTIrep[%,ctd4h]
Out[]=
{5,1,1,1,1,-1,-1,5,1,1}
A
1g
B
2g
B
1g
E
g
Out[]=
{1,0,1,0,0,0,0,1,0,1}
What about the decomposition of representations into irreducible representations of
O
h
D
4h
In[]:=
GTCrystalFieldSplitting[ohgroup,d4hgroup,ctoh,ctd4h]
Out[]=
Repr. in group 1 | Ee | C 2y | IC 2x | C 2b | IC 2a | IC 4z | -1 C 4z | IEe | C 2z | IC 2z | Splitting in group 2 |
A 1g | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | A 1g |
A 2u | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | B 1u |
A 1u | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | A 1u |
A 2g | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | B 1g |
E u | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | B 1u A 1u |
E g | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | A 1g B 1g |
T 1g | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 3 | -1 | -1 | A 2g E g |
T 2g | 3 | -1 | -1 | 1 | 1 | -1 | -1 | 3 | -1 | -1 | B 2g E g |
T 1u | 3 | -1 | 1 | -1 | 1 | -1 | 1 | -3 | -1 | 1 | A 2u E u |
T 2u | 3 | -1 | 1 | 1 | -1 | 1 | -1 | -3 | -1 | 1 | B 2u E u |
quantitative crystal field theory
quantitative crystal field theory
Derive the crystal field expansion in cubic symmetry
In[]:=
cf=GTCrystalField[ohgroup,4]
Out[]=
A[0,0]Y[0,0]+A[4,-4]Y[4,-4]+A[4,-4]Y[4,0]+A[4,-4]Y[4,4]
4
r
14
5
Use e.g. Buckmaster-Smith-Thornley operator equivalents
In[]:=
GTBSTOperator[4,0,2]//MatrixForm
Out[]//MatrixForm=
3 2 | 0 | 0 | 0 | 0 |
0 | -6 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | -6 | 0 |
0 | 0 | 0 | 0 | 3 2 |
In[]:=
cf/.r->1/.A[0,0]->0/.A[4,-4]->0.5/.Y[l_,m_]:>GTBSTOperator[l,-m,2];Eigenvalues[%]
Out[]=
{7.52994,7.52994,-5.01996,-5.01996,-5.01996}
Alternatively, one can include values for r^l from databases
In[]:=
path=$UserBaseDirectory<>"/Applications/GroupTheory/Documentation/English/ReferencePages/Symbols/datasets/CF_Database";GTCFDatabaseInfo[path];
Reference | < 2 r | < 4 r | < 6 r | < -3 r | |
Ce_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 1.34 | 4.22 | 25.4 | 4.55 |
Dy_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.79 | 1.55 | 6.21 | 9.9 |
Er_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.724 | 1.33 | 5.15 | 11.5 |
Eu_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.925 | 2.06 | 9.11 | 7.69 |
Gd_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.872 | 1.84 | 7.79 | 8.4 |
Ho_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.756 | 1.43 | 5.3 | 10.7 |
Lu_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.65 | 1.11 | 4.2 | 14.1 |
Nd_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 1.14 | 3.04 | 15.8 | 5.73 |
Pm_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 1.06 | 2.64 | 13. | 6.35 |
Pr_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 1.23 | 3.55 | 19.8 | 5.13 |
Sm_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.986 | 2.32 | 10.8 | 7.01 |
Tb_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.829 | 1.68 | 6.91 | 9.14 |
Tm_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.695 | 1.24 | 4.72 | 12.4 |
Yb_SIC | J. Forstreuter, et al., Phys. Rev. B, 55, 15, 1997 | 0.669 | 1.16 | 4.39 | 13.3 |
In[]:=
GTCFDatabaseRetrieve[path,"Ho_SIC"]
Out[]=
0.756,1.43,5.3,10.7
2
r
4
r
6
r
1
3
r
In[]:=
?GTCrystalFieldParameter
Out[]=
molecular symmetry-adapted spin space (mSASS)
molecular symmetry-adapted spin space (mSASS)
R. M. Geilhufe, J. D. Rinehart, arXiv: arXiv:2209.03123
Compute the representation matrices of SU (2) , for J=2.
In[]:=
Orep=GTAngularMomentumRep[ohgroup,2,GOHarmonics->"Real"];
In[]:=
Map[MatrixForm,Orep]
Out[]=
Generate a generic 5x5 Hamiltonian matrix
In[]:=
dim=5;ham=Table[u[i,j],{i,1,dim},{j,1,dim}];
In[]:=
MatrixForm[ham]
Out[]//MatrixForm=
u[1,1] | u[1,2] | u[1,3] | u[1,4] | u[1,5] |
u[2,1] | u[2,2] | u[2,3] | u[2,4] | u[2,5] |
u[3,1] | u[3,2] | u[3,3] | u[3,4] | u[3,5] |
u[4,1] | u[4,2] | u[4,3] | u[4,4] | u[4,5] |
u[5,1] | u[5,2] | u[5,3] | u[5,4] | u[5,5] |
The mSASS Hamiltonian is projected using the representation matrices of the angular momentum representation,In the present case we choose the group G = O, and J=2. The right hand-side of the above equation is evaluated as follows.
H=(g)H
∑
g∈G
J
D
J
D
-1
(g)
In[]:=
projham=1/Length[Orep]Sum[mat.ham.Inverse[mat],{mat,Orep}]//Simplify;
To match the right hand side with the left-hand side, we solve for the symmetry invariant terms:
In[]:=
sol=Quiet@Solve[projhamham,Flatten[ham]][[1]];
Imposing H to be Hermitian, the final Hamiltonian is given by:
In[]:=
HamCombined=ham/.u[i_,j_]If[i>j,Conjugate[u[j,i]],u[i,j]]/.sol;
In[]:=
MatrixForm[HamCombined]
Out[]//MatrixForm=
u[1,1] | 0 | 0 | 0 | 0 |
0 | u[2,2] | 0 | 0 | 0 |
0 | 0 | u[1,1] | 0 | 0 |
0 | 0 | 0 | u[2,2] | 0 |
0 | 0 | 0 | 0 | u[2,2] |
Graphene - band structure calculations
Graphene - band structure calculations
Define the structure in the GTPack style
In[]:=
hcp={{"C","Honeycomb"},"hP1","","P6/mmm",191,{{Sqrt[3]a,0,0},{Sqrt[3]a/2,3a/2,0},{0,0,0}},{{{0,0,0},"C1"},{{0,-a,0},"C2"}},{a240}};
A
f
In[]:=
GTPlotStructure2D[hcp,5,GOLattice{a1}]
61 atoms
Atoms in cluster: {{C1,31},{C2,30}}
Out[]=
 |
|
Generate an atomic cluster and neighbor shells
In[]:=
cl=GTCluster[hcp,3,GOLattice{a->1}];bas0=hcp[[7]]/.{a1};shells={2,2};
25 atoms
Atoms in cluster: {{C1,13},{C2,12}}
In[]:=
lat=GTShells[cl,bas0,shells,GOVerboseTrue,GOTbLattice{{"C1,C1",{0}},{"C2,C2",{0}},{"C1,C2",{1}}},GOPosition"Relative"];
Basis | dist | Atoms | dist | Atoms |
C1 | 1 | {{C2,3}} | 3 | {{C1,6}} |
C2 | 1 | {{C1,3}} | 3 | {{C2,6}} |
Construct 2-center tight-binding Hamiltonian
In[]:=
bas={{"C1",1,{0,1}},{"C2",1,{0,1}}};hamc=GTTbHamiltonian[bas,lat];GTHamiltonianPlot[hamc,bas]
Out[]=
s | p y | p z | p x | s | p y | p z | p x | |
s | ||||||||
p y | ||||||||
p z | ||||||||
p x | ||||||||
s | ||||||||
p y | ||||||||
p z | ||||||||
p x |
Extract the -orbital contributions
p
z
In[]:=
hpp={{hamc[[3,3]],hamc[[3,7]]},{hamc[[7,3]],hamc[[7,7]]}};
In[]:=
hamc[[3,3]]
Out[]=
C1
(pp0)
In[]:=
hamrule=0,0,2.4;hparm=hpp/.hamrule;
"C1"
"(pp0)"
"C2"
"(pp0)"
"C1,C2"
"(ppπ)"
1
Plot the band structure
In[]:=
kp=GTBZPath["Honeycomb"]
Out[]=
-,0,0,{0,0,0},,-,0,,0,0,{K',Γ,M,K}
2
3
3
1
2
3
1
6
2
3
3
In[]:=
GTBandStructure[hparm,kp,25,2,JoinedTrue,FrameLabel{" ","Energy (arb. units)"},PlotLabel"π-electrons in Graphene"]
Maximum Abscissa = 0.910684
Out[]=
Cite this as: Matthias Geilhufe, "Group Theory Package (GTPack) and Symmetry Principles in Condensed Matter" from the Notebook Archive (2022), https://notebookarchive.org/2022-10-cfxwbcm
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