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MSGCorep.wl
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(* ::Package:: *)
(* A package for irreducible corepresentations of magnetic space group using the conventions in the book of
C.J. Bradley and A.P. Cracknell, "The Mathematical Theory of Symmetry in Solids",
called "the BC book" afterwards. *)
(* Package Name: MSGCorep *)
(* Author: Gui-Bin Liu *)
(* Package verseion: see MSGCorep`Private`Version *)
(* Dependency: SpaceGroupIrep, for version see MSGCorep`Private`SpaceGroupIrepDependency *)
(* Mathematica version: >=11.2 *)
(* License: GPLv3 http://www.gnu.org/licenses/gpl-3.0.txt *)
With[{p=DirectoryName[$InputFileName]}, If[!MemberQ[$Path,p],AppendTo[$Path, p]]];
MSGCorep`Private`Version={1,0,0}; (*Specify version here.*)
MSGCorep`Private`SpaceGroupIrepDependency={1,2,3}; (*Specify dependency here.*)
Check[BeginPackage["MSGCorep`", {"SpaceGroupIrep`","LittleGroupIrepData`"}],
Print["MSGCorep package (version ",StringRiffle[ToString/@MSGCorep`Private`Version,"."],") needs the SpaceGroupIrep (version>=",
StringRiffle[ToString/@MSGCorep`Private`SpaceGroupIrepDependency,"."],") package, please download and install it first.\n",
"URL: https://github.com/goodluck1982/SpaceGroupIrep"]; Abort[],
{Needs::nocont, Get::noopen}]
(*---------- check dependency ---------*)
Module[{v1,v2,sv1,sv2,sv,out=False},
v1=MSGCorep`Private`SpaceGroupIrepDependency; v2=SpaceGroupIrep`Private`Version;
If[ValueQ[SpaceGroupIrep`Private`Version], If[Last@Sort[{v1,v2}]!=v2,out=True;sv2=StringRiffle[v2,"."]], out=True;sv2="<=1.2.2" ];
If[out, sv1=StringRiffle[v1,"."]; sv=StringRiffle[MSGCorep`Private`Version,"."];
Print[Style["Warning:",Red]," The current version of SpaceGroupIrep "<>sv2<>" does not meet the dependency >="<>sv1<>" required "<>
"by MSGCorep v"<>sv<>"."," Please update SpaceGroupIrep to the latest version and reload it manually by <<SpaceGroupIrep`."] ];
];
Remove[v1,v2,sv1,sv2,sv,out];
(*-------------------------------------*)
Unprotect@@Names["MSGCorep`*"];
ClearAll@@Names["MSGCorep`*"];
Get["Usage.wl"];
Get["MSGData.wl"];
Begin["`Private`"]
cont="MSGCorep`Private`";
(* ::Section:: *)
(*Magnetic Space Group (MSG) Symbols*)
(* ::Subsection:: *)
(*MSGSymStd (MSGSymBNS)*)
(* ::Subsubsection::Closed:: *)
(*MSGSymStd *)
Options[MSGSymStd]={"TeX"->False, "BC"->False};
MSGSymStd[symtext_String, OptionsPattern[]]:=
Module[{tmp,tex,bar,sub,p1,p2,p3,chs,prime,subp},
tex=(OptionValue["TeX"]===True);
sub[s1_,s2_]:=If[tex, s1<>"_{"<>s2<>"}", Subscript[s1,s2]];
bar[s1_]:=If[tex, "\\bar{"<>s1<>"}", OverBar[s1]];
prime[s1_]:=If[tex, s1<>"'", Superscript[s1,"\[Prime]"]];
subp[s1_,s2_]:=If[tex, s1<>"_"<>s2<>"'", Subsuperscript[s1,s2,"\[Prime]"]];
chs=Characters[symtext];
AppendTo[chs,"END"];
p1=Position[chs,"-"];
If[p1!={}, p1=p1[[1,1]];
chs=Join[chs[[1;;p1-1]], {bar[chs[[p1+1]]]}, chs[[p1+2;;]]];
];
p2=Position[chs,"_"];
While[p2!={}, p2=p2[[1,1]];
If[p2==2&&chs[[3]]==="2",
chs=Prepend[chs[[5;;]], sub[chs[[1]], "2"<>chs[[4]]]],
If[chs[[p2+2]]==="'",
chs=Join[chs[[1;;p2-2]], {subp[chs[[p2-1]],chs[[p2+1]]]}, chs[[p2+3;;]]],
chs=Join[chs[[1;;p2-2]], {sub[chs[[p2-1]],chs[[p2+1]]]}, chs[[p2+2;;]]]
];
];
p2=Position[chs,"_"];
];
p3=Position[chs,"'"];
While[p3!={}, p3=p3[[1,1]];
chs=Join[chs[[1;;p3-2]], {prime[chs[[p3-1]]]}, chs[[p3+1;;]]];
p3=Position[chs,"'"];
];
If[tex, StringJoin["$",chs[[;;-2]],"$"], Row[chs[[;;-2]]]]
]
(*We take the BNS notation (N. V. Belov, N. N. Neronova and T. S. Smirnova) of MSGs as standard*)
MSGSymStd[{sgno_, num_}, OptionsPattern[]]:=Module[{nums, pos, text,bc},
{nums,pos}=checkMSGinput[{sgno,num},"MSGSymStd"];
bc=If[OptionValue["BC"]===True, "BC",""];
text=ToExpression["type"<>ToString[pos[[1,1]]]<>"MSGSymText"<>bc][{sgno,num}];
MSGSymStd[text,"TeX"->OptionValue["TeX"]]
]
MSGSymStd[sgno_Integer, OptionsPattern[]]/;1<=sgno<=230:=Module[{nums,nums2,nums3,syms,bc,ogsyms},
nums={type1num[sgno],type2num[sgno],type3nums[sgno],type4nums[sgno]}//Flatten;
bc="BC"->OptionValue["BC"];
syms={"BNS:", ToString[sgno]<>"."<>ToString[#], MSGSymStd[{sgno,#},bc]}&/@nums;
ogsyms={"OG:", StringRiffle[ToString/@#,"."], MSGSymOG[#]}&/@(BNStoOG[{sgno,#}]&/@nums);
Flatten/@({getMSGType[{sgno,#}]&/@nums, syms, ogsyms}//Transpose)/.{1->"I", 2->"II", 3->"III", 4->"IV"}
]
(* ::Subsubsection::Closed:: *)
(*MSGSymBNS*)
MSGSymBNS=MSGSymStd; (* define an alias *)
(* ::Subsection::Closed:: *)
(*MSGSymBC*)
Options[MSGSymBC]={"TeX"->False};
MSGSymBC[{sgno_Integer,num_Integer}, OptionsPattern[]]:=
MSGSymStd[{sgno,num},"TeX"->OptionValue["TeX"], "BC"->True]
MSGSymBC[sgno_Integer, OptionsPattern[]]:=MSGSymStd[sgno, "BC"->True]
(* ::Subsection::Closed:: *)
(*MSGSymOG*)
Options[MSGSymOG]={"TeX"->False};
MSGSymOG[sgno_Integer, OptionsPattern[]]/;1<=sgno<=230:=Module[{nums,types,tmp,bnsnums},
nums=Select[OGDataList[[All,1]],#[[1]]==sgno&];
bnsnums=OGtoBNS/@nums; types=getMSGType/@bnsnums;
tmp={types, {"OG:", StringRiffle[ToString/@#,"."], MSGSymOG[#]}&/@nums,
{"BNS:", StringRiffle[ToString/@#,"."], MSGSymStd[#]}&/@bnsnums}//Transpose;
Flatten/@tmp/.{1->"I", 2->"II", 3->"III", 4->"IV"}
]
MSGSymOG[{p_Integer,q_Integer,r_Integer}, OptionsPattern[]]/;1<=p<=230&&1<=r<=1651:=Module[{nums},
nums=Select[OGDataList[[All,1]],#[[1]]==p&];
If[Select[nums,#=={p,q,r}&]=={}, Print["MSGSymOG: Error, {p,q,r} should be in ",nums]; Abort[]];
MSGSymStd[MSGSymTextOG[{p,q,r}], "TeX"->OptionValue["TeX"]]
]
(* ::Subsection::Closed:: *)
(*showMSGSym*)
Options[showMSGSym]={"family"->"BNS", "BCstyle"->1, "color"->True};
showMSGSym[OptionsPattern[]]:=showMSGSym[All, (#->OptionValue[#])&/@{"family","BCstyle","color"}]
showMSGSym[listOrSpan_, OptionsPattern[]]:=Module[{sglist,tab,head,tmp1,tmp2,pos,outBC,
lt=0.96,cls,bgs},
sglist=If[IntegerQ[listOrSpan], {listOrSpan}, Range[230][[listOrSpan]]];
head={Column[{"BNS","Number"}], Column[{"BNS","Symbol"}], Column[{"BNS(BC)","Symbol"}],
Column[{"OG","Number"}],Column[{"OG","Symbol"}], "Type"};
If[!MemberQ[{1,2,3,4},OptionValue["BCstyle"]],
Print["showMSGSym: The option \"BCstyle\" should be in {1,2,3,4} which controls how to ",
"show the BNS symbol in BC orientation if it is the same with the standard BNS symbol:\n",
"1, show \[LeftArrow] to mean the same with the left symbol\n2, does not show\n3, show in gray ",
"color\n4, show in black color"]; Abort[],
outBC[bc_]:=Switch[OptionValue["BCstyle"], 1, "\[LeftArrow]", 2, "", 3, Style[bc, Gray], 4, bc]
];
If[OptionValue["family"]=!="OG", (* for BNS *)
tmp1=MSGSymStd/@sglist;
pos={FoldList[Plus,1,#][[;;-2]],FoldList[Plus,#]}&[Length/@tmp1]+1;
tmp1=(Join@@tmp1)[[All,{3,4,6,7,1}]]\[Transpose];
tmp2=MSGSymBC[ToExpression/@StringSplit[#,"."]]&/@tmp1[[1]];
tmp2=If[#1===#2, outBC[#2], Style[#2,Red]]&@@@Transpose[{tmp1[[2]],tmp2}];
tab=Prepend[tmp1,tmp2][[{2,3,1,4,5,6}]]\[Transpose];
, (*---------------else: OG ---------------*)
tmp1=MSGSymOG/@sglist;
pos={FoldList[Plus,1,#][[;;-2]],FoldList[Plus,#]}&[Length/@tmp1]+1;
tmp1=(Join@@tmp1)[[All,{3,4,6,7,1}]]\[Transpose];
tmp2=MSGSymBC[ToExpression/@StringSplit[#,"."]]&/@tmp1[[3]];
tmp2=If[#1===#2, outBC[#2], Style[#2,Red]]&@@@Transpose[{tmp1[[4]],tmp2}];
tab=Append[tmp1,tmp2][[{1,2,3,4,6,5}]]\[Transpose];
head=head[[{4,5,1,2,3,6}]];
];
tab=Prepend[tab,head];
cls=Which[1<=#<=2, Lighter[Red,lt], 3<=#<=15, Lighter[Blue,lt], 16<=#<=74, Lighter[Orange,lt],
75<=#<=142, Lighter[Cyan,lt], 143<=#<=167, Lighter[Yellow,lt],
168<=#<=194, Lighter[Green,lt], 195<=#<=230, Lighter[Purple,lt]]&;
bgs=Table[{pos[[All,i]],{1,-1}}->cls[sglist[[i]]],{i,Length[sglist]}];
bgs=Prepend[bgs,{{1,1},{1,-1}}->Lighter[Gray,lt]];
Grid[tab, Alignment->Left, ItemSize->Full, Spacings->{2,0.3}, Frame->True,
Dividers->{{},Prepend[#->Directive[Thin,Gray]&/@pos[[1,2;;]],2->True]},
Background->If[OptionValue["color"]===True,{None,None,bgs},{}]
]
]
(* ::Section:: *)
(*Elements of MSG and MLG (Magnetic little group)*)
(* ::Subsection::Closed:: *)
(*Elements for type-2 MSG*)
Options[getType2MSGElem]={"double"->False};
getType2MSGElem[sgno_Integer,OptionsPattern[]]:=
getType2MSGElem[{sgno,type2num[sgno]},"double"->OptionValue["double"]]
getType2MSGElem[{sgno_Integer,num_Integer},OptionsPattern[]]:=Module[{tmp,MSG,G},
tmp=type2num[sgno];
If[num!=tmp, Print["getType2MSGElem: for type-2 MSG ",sgno,".num, num should be ",tmp]; Abort[]];
G=getLGElem[sgno,"\[CapitalGamma]"];
MSG=Join[Append[#,0]&/@G, Append[#,1]&/@G];
If[OptionValue["double"]===True, Join[MSG,{"bar"<>#1,#2,#3}&@@@MSG], MSG]
]
(* ::Subsection::Closed:: *)
(*Elements for type-3 MSG*)
RotNameIndex=SpaceGroupIrep`Private`RotNameIndex;
(* Get the type-III MSG elements from the "colored" generators in BC-Tab. 7.2 *)
Options[getType3MSGElem]={"double"->False};
getType3MSGElem[sgno_Integer,cgens_List,OptionsPattern[]]/;AllTrue[cgens,StringQ]:=Module[
{gens0,gens,brav,times,MSG0,SG,MSG,tmp},
gens0=SGGenElem[sgno][[1,All,1]];
If[!SubsetQ[gens0,cgens],
Print["getType3MSGElem: cgens=",cgens," is not a subset of the generators ",
"(rotation parts) ",gens0," in the BC-Tab. 3.7"];
Abort[];
];
gens=Join[{#,0}&/@Complement[gens0,cgens], {#,1}&/@cgens];
brav=getSGLatt[sgno];
times[{R1_,a1_},{R2_,a2_}]:={getRotName[brav,getRotMat[brav,R1].getRotMat[brav,R2]],Mod[a1+a2,2]};
MSG0=SortBy[generateGroup[gens,{"E",0},times],RotNameIndex[#[[1]]]&];
tmp=Association[Rule@@@MSG0];
SG=getLGElem[sgno,"\[CapitalGamma]"];
MSG={#[[1]],#[[2]],tmp[#[[1]]]}&/@SG;
If[OptionValue["double"]===True, Join[MSG,{"bar"<>#1,#2,#3}&@@@MSG], MSG]
]
getType3MSGElem[{sgno_Integer,num_Integer},OptionsPattern[]]:=Module[{tmp},
tmp=BCTab7d2[sgno]//Keys;
If[tmp=={}, Print["getType3MSGElem: There is no type-3 MSG begin with ",sgno]; Abort[],
If[!MemberQ[tmp,num],
Print["getType3MSGElem: for type-3 MSG ",sgno,".num, num should be in ",
Row[{#,"(", MSGSymStd[{sgno,#}],")"}]&/@tmp];
Abort[]]
];
getType3MSGElem[sgno,BCTab7d2[sgno,num],"double"->OptionValue["double"]]
]
(* ::Subsection::Closed:: *)
(*Elements for type-4 MSG*)
Options[getType4MSGElem]={"double"->False};
getType4MSGElem[{sgno_Integer,num_Integer},OptionsPattern[]]:=Module[{tmp,brav,BWlatt,A,MSG,G},
tmp=type4nums[sgno];
If[tmp=={}, Print["getType4MSGElem: There is no type-4 MSG begin with ",sgno]; Abort[],
If[!MemberQ[tmp,num],
Print["getType4MSGElem: for type-4 MSG ",sgno,".num, num should be in ",
Row[{#,"(",MSGSymStd[{sgno,#}],")"}]&/@tmp];
Abort[]]
];
brav=getSGLatt[sgno];
(*For orthorhombic system, we have to use the MSG symbol conforming to the BC orientation*)
If[16<=sgno<=74 (*orth system*),
BWlatt=StringTake[type4MSGSymTextBC[{sgno,num}],3],
BWlatt=StringTake[type4MSGSymText[{sgno,num}],3]
];
A=type4AUTrans[brav,BWlatt];
G=Append[#,0]&/@getLGElem[sgno,"\[CapitalGamma]"];
MSG=Join[G, MapAt[modone,MSGSeitzTimes[brav][A,#]&/@G,{All,2}]];
If[OptionValue["double"]===True, Join[MSG,{"bar"<>#1,#2,#3}&@@@MSG], MSG]
]
(* ::Subsection::Closed:: *)
(*getMSGElem: get elements of MSG*)
Options[getMSGElem]={"double"->False};
getMSGElem[{sgno_, num_},OptionsPattern[]]:=Module[{tmp,MSG,G,nums,pos,fun},
{nums,pos}=checkMSGinput[{sgno,num},"getMSGElem"];
pos=pos[[1,1]];
fun=ToExpression[cont<>"getType"<>ToString[pos]<>"MSGElem"];
If[pos==1,
Append[#,0]&/@getLGElem[sgno,"\[CapitalGamma]","DSG"->OptionValue["double"]],
fun[{sgno,num},"double"->OptionValue["double"]]
]
]
(* ::Subsection::Closed:: *)
(*getMLGElem: get elements of MLG*)
(* Get the magnetic little group. MSG is the element list of the magnetic space group.*)
getMLGElem[brav_String, MSG_List, k_]/;VectorQ[k]:=Module[{rotk,sele,LG,mLGu,mLGau},
rotk={getRotMatOfK[brav,StringReplace[#[[1]],"bar"->""]].k, #}&/@MSG;
sele=Select[rotk,If[#[[2,3]]==0,keqmod[k,#[[1]]],keqmod[-k,#[[1]]]]&];
mLG=sele[[All,2]]; mLGu=Select[mLG,#[[3]]==0&]; mLGau=Select[mLG,#[[3]]==1&];
Join[mLGu,mLGau] (*put all unitary elements in front*)
]
Options[getMLGElem]={"double"->False};
getMLGElem[{sgno_, num_},k_,OptionsPattern[]]:=
Module[{ks,brav,tmp,kco,BZs,MSG},
MSG=getMSGElem[{sgno,num},"double"->OptionValue["double"]];
brav=getSGLatt[sgno];
kco=If[StringQ[k], kBCcoord[sgno,k][[1,1]], k];
getMLGElem[brav,MSG,kco]
]
(* ::Subsection::Closed:: *)
(*showMSGSeitz*)
Options[showMSGSeitz]={"format"->"std", "fullbar"->True, "antiUcolor"->Red};
showMSGSeitz[{R_,v_,au_},OptionsPattern[]]:=Module[{Rv,color,c},
Rv=showSeitz[{R,v}, "format"->OptionValue["format"], "fullbar"->OptionValue["fullbar"]];
color=OptionValue["antiUcolor"];
If[!ColorQ[color], Print["showMSGSeitz: option \"antiUcolor\" should be a color."]; Abort[]];
Switch[OptionValue["format"],
"std"|"simple", If[au==0, Rv, Style[Superscript[Rv,"\[Prime]"],color]],
"TeX", c="\\color[rgb]"<>ToString@(List@@color);
If[au==0, Rv, "${"<>c<>StringTake[Rv,{2,-2}]<>"'}$"]
]
];
(* ::Section:: *)
(*Common utilities*)
(* ::Subsection::Closed:: *)
(*checkMSGinput*)
checkMSGinput[{sgno_,num_}, fun_String]:=Module[{nums,pos,spc,tab,sty1},
If[!IntegerQ[sgno]||sgno<1||sgno>230,
Print[fun<>": {sgno,num} should be any of the 1651 integer pairs listed as follow:\n",
Partition[MSGSymText//Keys//Sort,UpTo[10]]//Grid[#,Alignment->Left,Spacings->{0.5,0.2}]&];
Abort[];
];
nums={type1num[sgno],type2num[sgno],type3nums[sgno],type4nums[sgno]};
pos=Position[nums,num];
If[pos=={},
spc=StringReplace[fun,_->" "]<>" ";
sty1={{}, #->GrayLevel[0.7]&/@{2,3,Length[nums[[3]]]+3}};
tab=Grid[#,Alignment->Left,Frame->True,FrameStyle->Thin, Dividers->sty1,Spacings->2]&;
Print[fun<>": for MSG ",sgno,".num, num should be in ",nums,".\n",spc<>"Refer to:\n", spc,
MSGSymStd[sgno][[All,{3,4,1}]]//tab];
Abort[]
];
{nums,pos}
]
(* ::Subsection::Closed:: *)
(*Multiplication for (double) magnetic space group*)
MSGSeitzTimes[brav_][{Rname1_,v1_,antiU1_}, {Rname2_,v2_,antiU2_}]/;SubsetQ[{1,0},{antiU1,antiU2}]:=
Append[SeitzTimes[brav][{Rname1,v1},{Rname2,v2}], Mod[antiU1+antiU2,2]]
MSGSeitzTimes[brav_][Rvau1_,Rvau2_,more__]:=Fold[MSGSeitzTimes[brav],Rvau1,{Rvau2,more}]
MSGinvSeitz[brav_][{Rname1_,v1_,antiU1_}]/;MemberQ[{1,0},antiU1]:=
Append[invSeitz[brav][{Rname1,v1}],antiU1]
MSGpowerSeitz[brav_][{Rname1_,v1_,antiU1_},n_Integer]/;MemberQ[{1,0},antiU1]&&n>=0:=
Append[powerSeitz[brav][{Rname1,v1},n],Mod[n*antiU1,2]]
(*NOTE: For double MSG, theta^2\[Equal]barE, where theta is the time reversal *)
DMSGSeitzTimes[brav_][{Rname1_,v1_,antiU1_}, {Rname2_,v2_,antiU2_}]/;SubsetQ[{1,0},{antiU1,antiU2}]:=
Module[{au=antiU1+antiU2, Rv},
Rv=DSGSeitzTimes[brav][{Rname1,v1},{Rname2,v2}];
If[au<2, Append[Rv,au], Append[DSGSeitzTimes[brav][Rv,{"barE",{0,0,0}}],0]]
]
DMSGSeitzTimes[brav_][Rvau1_,Rvau2_,more__]:=Fold[DMSGSeitzTimes[brav],Rvau1,{Rvau2,more}]
DMSGinvSeitz[brav_][{Rname1_,v1_,antiU1_}]/;MemberQ[{1,0},antiU1]:=
With[{iRv=DSGinvSeitz[brav][{Rname1,v1}]},
If[antiU1==0, Append[iRv,0], Append[DSGSeitzTimes[brav][iRv,{"barE",{0,0,0}}],1]]]
DMSGpowerSeitz[brav_][{Rname1_,v1_,antiU1_},n_Integer]/;MemberQ[{1,0},antiU1]&&n>=0:=
Module[{Rvn},
Rvn=DSGpowerSeitz[brav][{Rname1,v1},n];
Switch[Mod[n*antiU1,4],
0, Append[Rvn,0], 1, Append[Rvn,1],
2, Append[DSGSeitzTimes[brav][Rvn,{"barE",{0,0,0}}],0],
3, Append[DSGSeitzTimes[brav][Rvn,{"barE",{0,0,0}}],1]
]
]
(* ::Subsection::Closed:: *)
(*Multiplication for (double) magnetic point group*)
Options[checkMagRotInput]={"double"->False};
checkMagRotInput[R_, OptionsPattern[]]:=Module[{rots,rots2},
rots=Keys[getSpinRotOp];
If[OptionValue["double"]=!=True, rots=rots[[;;Length[rots]/2]]];
rots2=Join[rots, #<>"'"&/@rots];
If[StringQ[R]&&MemberQ[rots2,R]||ListQ[R]&&Length[R]==2&&MemberQ[rots,R[[1]]]&&MemberQ[{0,1},R[[2]]],
Return@If[StringQ[R], If[StringTake[R,-1]=="'", {StringTake[R,{1,-2}],1},{R,0}], R],
Print["The input ",R," should be one of the two forms:\n",
"(1) The string of rotation name for an element in the magnetic point group, i.e. \"Rname\" or \"Rname'\".\n",
"(2) {\"Rname\",0} for unitary rotation (equivalent to \"Rname\") or {\"Rname\",1} for antiunitary rotation ",
"(equivalent to \"Rname'\").\nAnd Rname should be in the list\n",rots,".\n"
]; Abort[];
]
]
MagRotTimes[R1_,R2_]:=Module[{rot1,rot2},
rot1=checkMagRotInput[R1];
rot2=checkMagRotInput[R2];
{RotTimes[rot1[[1]],rot2[[1]]], Mod[rot1[[2]]+rot2[[2]],2]}
]
MagRotTimes[R1_,R2_,more__]:=Fold[MagRotTimes,R1,{R2,more}]
invMagRot[R_]:=With[{rot=checkMagRotInput[R]}, MapAt[invRot,rot,1]]
powerMagRot[R_,n_Integer]:=With[{rot=checkMagRotInput[R]}, {powerRot[rot[[1]],n],Mod[rot[[2]]*n,2]}]
DMagRotTimes[R1_,R2_]:=Module[{rot1,rot2,rot,au},
rot1=checkMagRotInput[R1,"double"->True];
rot2=checkMagRotInput[R2,"double"->True];
rot=DRotTimes[rot1[[1]],rot2[[1]]]; au=rot1[[2]]+rot2[[2]];
If[au==2, rot=DRotTimes[rot,"barE"]; au=0];
{rot,au}
]
DMagRotTimes[R1_,R2_,more__]:=Fold[DMagRotTimes,R1,{R2,more}]
invDMagRot[R_]:=Module[{rot=checkMagRotInput[R,"double"->True],r,au},
r=invDRot[rot[[1]]]; au=rot[[2]];
If[au==1, r=DRotTimes[r,"barE"]];
{r,au}
]
powerDMagRot[R_,n_Integer]:=Module[{rot=checkMagRotInput[R,"double"->True],r},
r=powerDRot[rot[[1]],n];
Switch[Mod[n*rot[[2]],4],
0, {r,0}, 1, {r,1}, 2, {DRotTimes[r,"barE"],0}, 3, {DRotTimes[r,"barE"],1}
]
]
(* ::Subsection::Closed:: *)
(*getMagKStar*)
Options[getMagKStar]={"cosets"->False};
getMagKStar[{sgno_, mno_}, kin_, OptionsPattern[]]:=Module[{k,MSG,brav,kall,star0,star,cosets},
MSG=getMSGElem[{sgno,mno}]; brav=getSGLatt[sgno];
k=If[!StringQ[kin], kin, kBCcoord[sgno,kin][[1,1]]];
kall={If[#[[3]]==0,1,-1]*getRotMatOfK[brav,#[[1]]].k, #}&/@MSG;
star0=Gather[kall,keqmod[#1[[1]],#2[[1]]]&];
star=star0[[All,1,1]]; cosets=star0[[All,All,2]];
If[OptionValue["cosets"]===True, {star,cosets}, star]
]
(* ::Subsection::Closed:: *)
(*getMSGType*)
getMSGType[{sgno_, mno_}]:=checkMSGinput[{sgno,mno},"getMSGType"][[2,1,1]]
(* ::Section:: *)
(*Magnetic point group (MPG)*)
(* ::Subsection::Closed:: *)
(*common*)
checkMPGinput[mpg_, fun_String]:=Module[{mpgno,mpgs,err=False,adjust},
If[IntegerQ[mpg]&&1<=mpg<=32,
Print[fun<>": Use any item as input listed as follow:\n",
TableForm[MapAt[InputForm,Select[MPGinfo[[All,2;;5]],#[[1,1]]==mpg&],{All,2;;4}],TableDepth->2]];
Abort[];
];
mpgno=Position[MPGinfo[[All,2;;5]],mpg,{2}];
If[mpgno!={}, mpgno=mpgno[[1,1]], err=True];
If[err,
adjust=MPGinfo[[All,2;;5]];
adjust=MapAt[InputForm,adjust,{All,2;;4}];
adjust=MapAt[Grid[{{#}},ItemSize->3.8,Alignment->Left]&, adjust, {All,1}];
adjust=MapAt[Grid[{{#}},ItemSize->3,Alignment->Left]&, adjust, {All,2}];
adjust=MapAt[Grid[{{#}},ItemSize->5.5,Alignment->Left]&, adjust, {All,3}];
adjust=MapAt[Grid[{{#}},ItemSize->5.5,Alignment->Left]&, adjust, {All,4}];
mpgs=TableForm@Partition[Row[#,"|"]&/@adjust,UpTo[3]];
Print[fun<>": To specify a mangnetic point group, use any one as input listed as follow:\n",mpgs];
Abort[]
];
mpgno
]
(* ::Subsection::Closed:: *)
(*getMPGElem and showMagRot*)
Options[getMPGElem]={"double"->False};
getMPGElem[mpg_, OptionsPattern[]]:=Module[{n,msg,elms},
n=checkMPGinput[mpg,"getMPGElem"]; msg=MPGinfo[[n,8]];
elms=getMLGElem[msg,"\[CapitalGamma]"][[All,{1,3}]];
If[OptionValue["double"]===True, elms=Join[elms, MapAt["bar"<>#&, elms, {All,1}]]];
elms
]
Options[showMagRot]={"format"->"std", "fullbar"->False, "antiUcolor"->Red};
showMagRot[R_,OptionsPattern[]]:=Module[{R0,hasbar,fmt,rot,fullbar,au,color},
fmt=OptionValue["format"]; fullbar=OptionValue["fullbar"];
If[!MemberQ[{"text","simple","std","TeX"},fmt],
Print["showMagRot: option \"format\" should be in ",InputForm/@{"text","simple","std","TeX"}]; Abort[];
];
rot=checkMagRotInput[R,"double"->True]; au=rot[[2]];
If[fmt=="text", Return@If[au==0, rot[[1]], rot[[1]]<>"'"]];
R0=showRot[rot[[1]],"format"->fmt,"fullbar"->fullbar];
If[au==0, Return[R0]];
R0=Switch[fmt, "simple"|"std", Superscript[R0,"\[Prime]"], "TeX", "{"<>R0<>"}'"];
color=OptionValue["antiUcolor"]; If[color===None, Return[R0]];
Switch[fmt, "simple"|"std", Style[R0,color],
"TeX", "{\\color[rgb]"<>ToString@(List@@color)<>R0<>"}" ]
]
(* ::Subsection::Closed:: *)
(*showMPGinfo*)
Options[showMPGinfo]={"long"->True, "color"->True, "elem"->False};
showMPGinfo[OptionsPattern[]]:=showMPGinfo[All,(#->OptionValue[#])&/@{"long","color","elem"}]
showMPGinfo[range_, OptionsPattern[]]:=Module[{stab,lt=0.90,bgc,bgs,cls,ncol,ltab,note,idx,nout,ifelm},
bgc[n1_,n2_,color_]:=#->color&/@Range[n1,n2];
idx=range;
If[IntegerQ[idx], idx=If[idx==0,All,{idx}]];
If[!(VectorQ[idx,IntegerQ]||Head[idx]===Span||idx===All),
Print["showMPGinfo: range can be an integer, a list of integers, a span, or just All ",
"to specify the magnetic point groups to show"]; Abort[]];
Check[idx=Range[Length[MPGinfo]][[DeleteCases[idx,0]]],
Print["showMPGinfo: out of range [0,",Length[MPGinfo],"]!"]; Abort[],
{Part::partw,Part::take}];
stab={Row[{"(",#[[1]],")"}],#[[3]],MSGSymStd[#[[5]]]}&/@MPGinfo[[idx]];
stab=Grid[{#}, ItemSize->{{3,3,Full}}, Alignment->Left, Spacings->0]&/@stab;
cls=Join[bgc[1,5,Lighter[Red,lt]], bgc[6,16,Lighter[Blue,lt]], bgc[17,28,Lighter[Orange,lt]],
bgc[29,59,Lighter[Cyan,lt]], bgc[60,75,Lighter[Yellow,lt]],
bgc[76,106,Lighter[Green,lt]],bgc[107,122,Lighter[Purple,lt]]]//Association;
ncol=5; nout=Length[idx];
bgs=Thread[Flatten[Table[{i,j},{i,Ceiling[nout/ncol]},{j,ncol}],1][[;;nout]] -> cls/@idx];
stab=Partition[stab, UpTo[ncol]]//Grid[#,Alignment->Left, Spacings->{2,0.3},
Background->If[OptionValue["color"],{None,None,bgs},{}]]&;
If[OptionValue["long"]==False, Return[stab]];
ifelm=If[OptionValue["elem"]===True,#,Nothing]&;
ltab={#[[1]], #[[3]], #[[4]], MSGSymStd[#[[5]]],
Row[{MSGSymStd[PGinfo[[#[[6]],3]]]," (",MSGSymStd[PGinfo[[#[[6]],2]]],")"}],
#[[7]], MSGSymStd[#[[8]]], ifelm[showMagRot/@getMPGElem[#[[2]]]]}&/@MPGinfo[[idx]];
ltab=Prepend[ltab, {Column[{"Seq.","No."}], Column[{"Number","(short)"}], Column[{"Number","(long)"}],
Column[{"Symbol"}], Column[{"Unitary","subgroup H"}],
Column[{"Type"}], "First MSG",ifelm["Elements of the MPG"]}];
cls=Prepend[Thread[Range[2,nout+1] -> Values[cls][[idx]]], 1->Lighter[Gray,lt]];
ltab=Grid[ltab, Alignment->Left, Spacings->{2,0.3}, Frame->True, Dividers->{{},{2->True}},
Background->If[OptionValue["color"],{{},cls},{}]];
note="The symbol of a magnetic point group (MPG) here is determined by the BNS symbol of\n"<>
"the first magnetic space group (MSG) with this MPG. And the order of MPGs also follows\n"<>
"the order of BNS MSG numbers. The symbols and long numbers here are consistent with\n"<>
"those on the BCS website. Note: the symbols and order of the type-3 MPGs here are not\n"<>
"always the same with those in BC-Tab. 7.1. For example, the mmm' in BC-Tab. 7.1 is \n"<>
"m'mm here, and 2/m' precedes 2'/m in BC-Tab. 7.1 while the order is reversed here.";
Column[{ltab,note}]
]
(* ::Subsection::Closed:: *)
(*getMPGCorep*)
Options[getMPGCorep]={"double"->False,"trace"->False};
getMPGCorep[mpg_,OptionsPattern[]]:=Module[{n,msgno,smap,dmap,MPG,subno,H,Hstd,A,nsir,subir,dbl,
matN,cordict,type,ir,nir,cor,lb,ncor,clb,ch,AH,elmidx,times,Hidx,AH2idx,inv,iAuA,i,j,subidx,
paired,aiA,aA,iAa,tmp,totrace},
n=checkMPGinput[mpg,"getMPGCorep"];
(*map the elements of the unitary subgroup to those of getPGElem*)
Switch[n,
22, smap=dmap={"\[Sigma]y"->"\[Sigma]z","bar\[Sigma]y"->"bar\[Sigma]z"},
26, smap=dmap={"C2x"->"C2z","\[Sigma]y"->"\[Sigma]x","\[Sigma]z"->"\[Sigma]y","barC2x"->"barC2z","bar\[Sigma]y"->"bar\[Sigma]x","bar\[Sigma]z"->"bar\[Sigma]y"},
46|50, smap=dmap={"\[Sigma]da"->"\[Sigma]y","\[Sigma]db"->"\[Sigma]x","bar\[Sigma]da"->"bar\[Sigma]y","bar\[Sigma]db"->"bar\[Sigma]x"},
56, smap=dmap={"C2a"->"C2y","C2b"->"C2x","\[Sigma]da"->"\[Sigma]y","\[Sigma]db"->"\[Sigma]x","barC2a"->"barC2y",
"barC2b"->"barC2x","bar\[Sigma]da"->"bar\[Sigma]y","bar\[Sigma]db"->"bar\[Sigma]x"},
73, smap={"\[Sigma]d1"->"\[Sigma]v2","\[Sigma]d2"->"\[Sigma]v3","\[Sigma]d3"->"\[Sigma]v1"}; dmap={"\[Sigma]d1"->"bar\[Sigma]v2","\[Sigma]d2"->"bar\[Sigma]v3",
"\[Sigma]d3"->"\[Sigma]v1","bar\[Sigma]d1"->"\[Sigma]v2","bar\[Sigma]d2"->"\[Sigma]v3","bar\[Sigma]d3"->"bar\[Sigma]v1"},
89, smap={"C21pp"->"C23p","C22pp"->"C21p","C23pp"->"C22p"}; dmap={"C21pp"->"C23p","C22pp"->"barC21p",
"C23pp"->"barC22p","barC21pp"->"barC23p","barC22pp"->"C21p","barC23pp"->"C22p"},
104,smap={"C21pp"->"C23p","C22pp"->"C21p","C23pp"->"C22p","\[Sigma]v1"->"\[Sigma]d3","\[Sigma]v2"->"\[Sigma]d1","\[Sigma]v3"->"\[Sigma]d2"};
dmap={"C21pp"->"C23p","C22pp"->"barC21p","C23pp"->"barC22p","\[Sigma]v1"->"\[Sigma]d3","\[Sigma]v2"->"bar\[Sigma]d1","\[Sigma]v3"->"bar\[Sigma]d2",
"barC21pp"->"barC23p","barC22pp"->"C21p","barC23pp"->"C22p","bar\[Sigma]v1"->"bar\[Sigma]d3","bar\[Sigma]v2"->"\[Sigma]d1","bar\[Sigma]v3"->"\[Sigma]d2"},
_, smap=dmap={}
];
msgno=MPGinfo[[n,2]]; subno=MPGinfo[[n,6]];
dbl=OptionValue["double"]===True;
MPG=getMPGElem[msgno,"double"->dbl];
subir=getPGIrepTab[subno,"double"->dbl,"trace"->False];
lb=subir["label"]; ir=subir["irep"]; nir=Length[ir]; Hstd=subir["elem"];
cordict=<||>;
cordict["number"]=MPGinfo[[n,{2,3}]]; cordict["symbol"]=MPGinfo[[n,5]];
cordict["USubG"]=PGinfo[[subno,1;;3]];
nsir=PGinfo[[subno,5]];
totrace=Map[If[MatrixQ[#],Simplify@Tr[#],#]&,#,{2}]&;
If[msgno[[2]]==1, (*------for type-1 MPG-------*)
cordict["elem"]={#,0}&/@Hstd;
cordict["A"]=None; cordict["N"]=None;
cordict["label"]=lb; cordict["type"]=Table["x",nir];
cordict["subidx"]=Range[nir];
cordict["sindex"]=Range[nsir];
cordict["dindex"]=Complement[Range[nir],Range[nsir]];
cordict["corep"]=If[OptionValue["trace"]===True, totrace[ir], ir];
Return[cordict]
];
ch=totrace[ir];
H=Select[MPG,#[[2]]==0&]; AH=Select[MPG,#[[2]]==1&]; A=AH[[1]];
elmidx=Association@Table[Hstd[[i]]->i,{i,Length[Hstd]}];
elmidx=elmidx/@(H[[All,1]]/.If[dbl,dmap,smap]);
ch=ch[[All,elmidx]]; ir=ir[[All,elmidx]];
times=If[dbl,DMagRotTimes,MagRotTimes];
Hidx=Association@Thread[H->Range[Length[H]]];
AH2idx=Hidx[times[#,#]]&/@AH;
type=Simplify[Total[#[[AH2idx]]]/Length[H]&/@ch]/.{1->"a",-1->"b",0->"c"};
inv=If[dbl,invDMagRot,invMagRot];
iAuA=Hidx@times[times[inv[A],#],A]&/@H;
matN=Table[None,nir];
For[i=1,i<=nir,i++, If[type[[i]]=="c", Continue[]];
matN[[i]]=findMatrixN[ir[[i]],ir[[i,iAuA]]\[Conjugate]]
];
subidx={}; paired=Table[False,nir];
For[i=1,i<=nir,i++, If[paired[[i]], Continue[]];
If[type[[i]]!="c", subidx=Append[subidx,i],
j=First@Select[Range[i+1,nir],Simplify[Conjugate@ch[[i,iAuA]]==ch[[#]]]&];
subidx=Append[subidx,{i,j}]; paired[[j]]=True;
]
];
ncor=Length[subidx]; clb=cor=Table[{},ncor];
aiA=Hidx@times[#,inv[A]]&/@AH;
aA=Hidx@times[#,A]&/@AH;
iAa=Hidx@times[inv[A],#]&/@AH;
For[i=1,i<=ncor,i++, j=subidx[[i]];
If[IntegerQ[j],
tmp=If[MatrixQ[ir[[j,1]]], #.matN[[j]]&/@ir[[j,aiA]], ir[[j,aiA]]];
If[type[[j]]=="a",
cor[[i]]=Join[ir[[j]], tmp]; clb[[i]]=lb[[j]], (*---next line else: b---*)
cor[[i]]=Join[ArrayFlatten[{{#,0},{0,#}}]&/@ir[[j]],ArrayFlatten[{{0,-#},{#,0}}]&/@tmp];
clb[[i]]=#<>#&/@lb[[j]]
], (*---------else: for type c----------*)
tmp=Mulliken2str@lb[[j[[1]],1]];
If[StringTake[tmp,1]=="1",
tmp=str2Mulliken@StringTake[tmp,{2,-1}],
If[StringLength[tmp]>4&&StringTake[tmp,4]=="bar1",
tmp=str2Mulliken["bar"<>StringTake[tmp,{5,-1}]], (*else*) tmp=StringJoin[lb[[j,1]]]
];
];
clb[[i]]={tmp,StringJoin[lb[[j,2]]]};
j=j[[1]];
cor[[i]]=Join[ArrayFlatten[{{#1,0},{0,#2}}]&@@@Transpose[{ir[[j]],ir[[j,iAuA]]\[Conjugate]}],
ArrayFlatten[{{0,#1},{#2,0}}]&@@@Transpose[{ir[[j,aA]],ir[[j,iAa]]\[Conjugate]}] ];
];
];
tmp=If[IntegerQ[#],#,#[[1]]]&/@subidx;
cordict["elem"]=Join[H,AH];
cordict["A"]=A; cordict["N"]=matN[[tmp]];
cordict["label"]=clb; cordict["type"]=type[[tmp]];
cordict["subidx"]=subidx;
tmp=Position[subidx,nsir][[1,1]];
cordict["sindex"]=Range[tmp];
cordict["dindex"]=Complement[Range[ncor],Range[tmp]];
cordict["corep"]=If[OptionValue["trace"]=!=True,cor,totrace[cor]];
cordict
]
(* ::Subsection::Closed:: *)
(*showMPGCorep*)
Options[showMPGCorep]={"double"->True,"rotmat"->True,"elem"->All,"corep"->All,"trace"->False,
"spin"->"downup","cartesian"->False,"linewidth"->2};
showMPGCorep[mpg_, OptionsPattern[]]:=Module[{n,mpgno,mpgcor,label,cor,elmopt,tmp,elmidx,dbl,type,
elems,nelm,elmerr,nscor,ndcor,ncor,sidx,didx,txtirl,iropt,irerr,iridx,row1,rots1,rots2,brav,tab,
nstart,grid,sty1,sty2,bg0,bg1,bg1a,bg2,bg3,bg4,bg5,spl,box},
(*-------check option "double"---------*)
dbl=OptionValue["double"];
If[!MemberQ[{True,False,Full},dbl],
Print["showMPGCorep: \"double\" should be one of True (default), False, or Full:\n",
"True: For coreps of double magnetic point groups. Only half of all elements, i.e. the ones without bar, are shown.\n",
"Full: For coreps of double magnetic point groups. All elements are shown.\n",
"False: For coreps of single magnetic point groups.\n"];
Abort[]
];
n=checkMPGinput[mpg,"showMPGCorep"]; mpgno=MPGinfo[[n,2]];
mpgcor=getMPGCorep[mpgno,"double"->OptionValue["double"]=!=False, "trace"->OptionValue["trace"]];
elems=mpgcor["elem"]; cor=mpgcor["corep"]; label=mpgcor["label"];
sidx=mpgcor["sindex"]; didx=mpgcor["dindex"]; type=mpgcor["type"];
nscor=Length[sidx]; ndcor=Length[didx]; ncor=nscor+ndcor;
nelm=Length[elems];
If[dbl=!=False,
tmp=Select[elems,StringTake[#[[1]]<>"xxx",3]!="bar"&];
tmp=Join[tmp, Select[elems,StringTake[#[[1]]<>"xxx",3]=="bar"&]];
tmp=Association@Table[tmp[[i]]->i,{i,nelm}]/@elems;
elems=elems[[tmp]];
cor=cor[[All,tmp]]
];
(*-------check option "elem"---------*)
elmopt=OptionValue["elem"];
elmerr:=Print["showMPGCorep: \"elem\" aims to reorder the columns and it can be:\n",
"A list of rotation names (elements). OR\n",
"A list of integers (or a span) for their sequence numbers. Refer to\n",
Grid[Partition[Grid[{#},ItemSize->{{1.5,Full}},Alignment->Left]&/@
Transpose@{Range[nelm],InputForm[showMagRot[#,"format"->"text"]]&/@elems},UpTo[10]],
ItemSize->Full, Frame->All, Alignment->Left, FrameStyle->Gray] ];
If[elmopt===0, elmopt=All];
If[StringQ[elmopt]||IntegerQ[elmopt], elmopt={elmopt}];
elmidx=elmopt;
If[VectorQ[elmidx,StringQ],
tmp=showMagRot[#,"format"->"text"]&/@elems;
If[SubsetQ[tmp,elmidx], elmidx=Position[tmp,#][[1,1]]&/@elmidx, elmerr; Abort[]], (*next line else*)
If[!(VectorQ[elmidx,IntegerQ]||Head[elmidx]===Span||elmidx===All), elmerr; Abort[]]
];
Check[elmidx=Range[nelm][[DeleteCases[elmidx,0]]],
Print["showMPGCorep: out of range! There are ",nelm," elements in total."]; Abort[],
{Part::partw,Part::take}];
(*-------check option "corep"---------*)
iropt=OptionValue["corep"];
irerr:=Print["showMPGCorep: \"corep\" aims to reorder the coreps (rows) and it can be:\n",
"An integer, a list of integers, a span, or just All (default) for the corep sequence numbers. ",
"The range is 1 to ", ncor, "."];
If[iropt===0, iropt=All];
iridx=If[StringQ[iropt]||IntegerQ[iropt], {iropt}, iropt];
If[!(VectorQ[iridx,IntegerQ]||Head[iridx]===Span||iridx===All), irerr; Abort[]]
Check[iridx=Range[ncor][[DeleteCases[iridx,0]]],
Print["showMPGCorep: out of range! There are ",ncor," coreps in total."]; Abort[],
{Part::partw,Part::take}];
If[dbl===True&&elmopt===All, nelm=nelm/2; elmidx=elmidx[[;;nelm]]];
cor=cor[[All,elmidx]]; elems=elems[[elmidx]];
tab=Map[formatRepMat,cor,{2}];
tab=Map[If[MatrixQ[#],MatrixForm[#],#]&, tab, {2}];
tab=Table[{i,Sequence@@label[[i]],type[[i]],Sequence@@tab[[i]]},{i,iridx}];
spl:=Sequence@@Table[SpanFromLeft,3];
tmp=Column[{Row[{"MPG: ", mpgcor["number"][[2]],", ",MSGSymStd@mpgcor["symbol"]}],
Row[{"USubG: ", MSGSymStd@mpgcor["USubG"][[3]], " (",str2Mulliken@mpgcor["USubG"][[2]],")"}]
}];
(*\:6ce8\:610f\:ff1a\:5982\:679c\:4e0d\:5bf9\:7fa4\:5143\:52a0\:4e2abox\:5c01\:88c5\:4e00\:4e0b\:ff0c\:5e26bar\:7684\:7fa4\:5143\:4f1a\:5bfc\:81f4\:4e0a\:4e00\:884c\:7684\:7f16\:53f7\:884c\:7684\:884c\:9ad8\:5f02\:5e38\:53d8\:5927*)
box=Grid[{{#}},Spacings->{0,0},Alignment->{Center,Center}]&;
row1={{tmp,spl,Sequence@@box/@elmidx},
{SpanFromAbove,Sequence@@Table[SpanFromBoth,3],Sequence@@(box@showMagRot[#]&/@elems)}};
If[OptionValue["rotmat"]=!=False,
brav=If[16<=mpgno[[1]]<=27, "HexaPrim", "CubiPrim"];
If[OptionValue["cartesian"]===True,
rots1=RotMatCart[StringReplace[#,"bar"->""]]&/@elems[[All,1]]; tmp="(cart.)",
rots1=getRotMat[brav,StringReplace[#,"bar"->""]]&/@elems[[All,1]]; tmp=Nothing
];
rots1={Column[{"Rotation","matrix",tmp}], spl, Sequence@@MatrixForm/@rots1};
rots2=ComplexExpand@First@getSpinRotOp[#]&/@elems[[All,1]];
tmp="(\[DownArrow]\[UpArrow])";
If[OptionValue["spin"]==="updown",
tmp="(\[UpArrow]\[DownArrow])"; rots2={{0,1},{1,0}}.#.{{0,1},{1,0}}&/@rots2
];
rots2={Column[{"Spin"<>tmp,"rotation","matrix"}], spl, Sequence@@MatrixForm/@rots2};
tab=Prepend[tab,rots1]; nstart=4;
If[dbl=!=False, tab=Insert[tab,rots2,2]; nstart=5];
, (*----else----*)
nstart=3
];
tab=Join[row1,tab];
sty1=Directive[Black,Thickness[OptionValue["linewidth"]]];
sty2=Directive[Thin,GrayLevel[0.8]];
bg0={1,1}->Lighter[Red,0.9];
bg1a={{2,nstart-1},{1,2}}->Lighter[Yellow,0.85];
bg1={{1,nstart-1},{2,-1}}->Lighter[Yellow,0.9];
sidx=Position[iridx,#][[1,1]]&/@Intersection[sidx,iridx];
didx=Complement[Range@Length[iridx],sidx];
bg2={{#,#}+nstart-1,{1,-1}}->Lighter[Green,0.95]&/@sidx;
bg3={{#,#}+nstart-1,{1,-1}}->Lighter[Blue,0.95]&/@didx;
bg4={{#,#}+nstart-1,{1,4}}->Lighter[Green,0.90]&/@sidx;
bg5={{#,#}+nstart-1,{1,4}}->Lighter[Blue,0.90]&/@didx;
grid=Grid[tab, Frame->All, Alignment->{Center,Center}, ItemSize->{{{},{2->3.1,3->2.9}},{}},
Dividers->{{{{sty2}},Join[#->sty1&/@{1,5,-1},#->sty2&/@{2}]},
{{{sty2}},#->sty1&/@{1,3,-1}}},
Background->{None,None,{bg0,bg1a,bg1,Sequence@@Join[bg2,bg3,bg4,bg5]}}
]
]
(* ::Subsection::Closed:: *)
(*MPGCorepDirectProduct*)
Options[MPGCorepDirectProduct]={"output"->1};
MPGCorepDirectProduct[mpg_, coreps1_, OptionsPattern[]]/;!VectorQ[Flatten[{coreps1}],MemberQ[{Rule,RuleDelayed},Head[#]]&]:=
MPGCorepDirectProduct[mpg, coreps1, coreps1, "output"->OptionValue["output"]]
MPGCorepDirectProduct[mpg_, OptionsPattern[]]:=MPGCorepDirectProduct[mpg, All, All, "output"->OptionValue["output"]]
MPGCorepDirectProduct[mpg_,coreps1_,coreps2_,OptionsPattern[]]/;
!Or@@(VectorQ[Flatten[{#}],MemberQ[{Rule,RuleDelayed},Head[#]]&]&/@{coreps1,coreps2}):=Module[{n,
mpgno,mpgcor,d,cijk,sub,type,subidx,subidx1,err,cr1idx,cr2idx,label,ncr,outopt,out1,out2,out3,dim},
n=checkMPGinput[mpg,"MPGCorepDirectProduct"]; mpgno=MPGinfo[[n,2]];
mpgcor=getMPGCorep[mpgno,"double"->True,"trace"->True];
sub=mpgcor["USubG"]; type=mpgcor["type"]; subidx=mpgcor["subidx"];
label=mpgcor["label"]; ncr=Length[label]; dim=mpgcor["corep"][[All,1]];
(*-------check input coreps1 and coreps2--------*)
err:=Print["MPGCorepDirectProduct: Inputs coreps1 and coreps2 can be:\n",
"An integer OR a list of integers for the corep sequence numbers. OR\n",
"A span such as 1;;5. OR All. Refer to\n",
Grid[Prepend[Transpose@label,Range[ncr]], Frame->All,FrameStyle->Gray] ];
If[coreps1===0||coreps2===0, err; Abort[]];
cr1idx=If[IntegerQ[coreps1], {coreps1}, coreps1];
If[!(VectorQ[cr1idx,IntegerQ]||Head[cr1idx]===Span||cr1idx===All), err; Abort[]]
Check[cr1idx=Range[ncr][[DeleteCases[cr1idx,0]]],
Print["MPGCorepDirectProduct: out of range! There are ",ncr," coreps in total."]; Abort[], {Part::partw,Part::take}];
cr2idx=If[IntegerQ[coreps2], {coreps2}, coreps2];
If[!(VectorQ[cr2idx,IntegerQ]||Head[cr2idx]===Span||cr2idx===All), err; Abort[]]
Check[cr2idx=Range[ncr][[DeleteCases[cr2idx,0]]],
Print["MPGCorepDirectProduct: out of range! There are ",ncr," coreps in total."]; Abort[], {Part::partw,Part::take}];
(*--------check option "output"---------------*)
outopt=OptionValue["output"];
If[!MemberQ[{1,2,3,4},outopt],
Print["MPGCorepDirectProduct: option \"output\" can be one of {1,2,3,4}:\n",
"1. Output the Mulliken label.\n2. Output the Gamma label.\n",
"3. Output the occurrence numbers of all coreps.\n4. Output all the above 1-3 results."
]; Abort[]
];
(*--------------------------------------------*)
subidx1=If[IntegerQ[#],#,#[[1]]]&/@subidx;
cijk=PGIrepDirectProduct[sub[[1]],"output"->3];
(* BC-Tab 7.8 *)
d[cri_Integer,crj_Integer]:=Module[{typei,typej,typek,facij,fack,si,sj,sij,re},
typei=type[[cri]]; typej=type[[crj]];
facij=1; If[typei=="b",facij*=2]; If[typej=="b",facij*=2];
si=subidx[[cri]]; If[typei!="c", si={si}];
sj=subidx[[crj]]; If[typej!="c", sj={sj}];
sij=Flatten[Table[{i,j},{i,si},{j,sj}],1];
fack=If[#!="b",1,1/2]&/@type;
Total[cijk/@sij][[subidx1]]*facij*fack
];
out3=Flatten[Table[{i,j}->d[i,j],{i,cr1idx},{j,cr2idx}],1]//Association;
out1=label[[#,1]]->(If[#[[1]]==1,#[[2]],#]&/@Select[Transpose@{out3[#],label[[All,1]]},#[[1]]!=0&])&/@Keys[out3]//Association;
out2=label[[#,2]]->(If[#[[1]]==1,#[[2]],#]&/@Select[Transpose@{out3[#],label[[All,2]]},#[[1]]!=0&])&/@Keys[out3]//Association;
(* (*Check dimensions*)
If[!And@@(Times@@dim[[#]]\[Equal]Total[out3[#]*dim])&/@Keys[out3], Print["MPGCorepDirectProduct: Corep dimensions error!"]];
*)
Switch[outopt, 1, out1, 2, out2, 3, out3, 4, {out1,out2,out3}]
]
(* ::Subsection::Closed:: *)
(*showMPGCorepDirectProduct*)
Options[showMPGCorepDirectProduct]={"label"->1, "double"->True, "linewidth"->2, "emph"->None};
showMPGCorepDirectProduct[mpg_, coreps1_, coreps2_, OptionsPattern[]]/;
!Or@@(VectorQ[Flatten[{#}],MemberQ[{Rule,RuleDelayed},Head[#]]&]&/@{coreps1,coreps2}):=Module[{n,
mpgno,mpgcor,ncor,dp,type,cor1,cor2,tmp,label,lopt,tab,scor1,dcor1,scor2,dcor2,sty1,sty2,bg0,bg1,
bg2,bg3,bg4,bg5,s1pos,d1pos,s2pos,d2pos,emopt,emidx,emerr,head,rowh,box},
lopt=OptionValue["label"];
If[!MemberQ[{1,2},lopt],
Print["showMPGCorepDirectProduct: option \"label\" can be 1 or 2:\n",
"1. Output the Mulliken label.\n2. Output the Gamma label."]; Abort[]
];
n=checkMPGinput[mpg,"showMPGCorepDirectProduct"]; mpgno=MPGinfo[[n,2]];
mpgcor=getMPGCorep[mpg, "double"->True];
label=mpgcor["label"]; ncor=Length[label]; type=mpgcor["type"];
(*-------check option "emph"--------*)
emopt=OptionValue["emph"]; If[emopt===None,emopt={}];
emerr:=Print["showMPGCorepDirectProduct: option \"emph\" specifies the corep(s) to be emphasized and can be:\n",
"An integer OR a list of integers for the corep sequence numbers. OR\n",
"A span such as 1;;5. OR All. Refer to\n",
Grid[Prepend[Transpose@label,Range[ncor]], Frame->All,FrameStyle->Gray] ];
If[emopt===0, emerr; Abort[]];
emidx=If[StringQ[emopt]||IntegerQ[emopt], {emopt}, emopt];
If[!(VectorQ[emidx,IntegerQ]||Head[emidx]===Span||emidx===All), emerr; Abort[]]
Check[emidx=Range[ncor][[DeleteCases[emidx,0]]],
Print["showMPGCorepDirectProduct: out of range! There are ",ncor," coreps in total."]; Abort[], {Part::partw,Part::take}];
dp=MPGCorepDirectProduct[mpg,coreps1,coreps2,"output"->4];
cor1=Keys[dp[[3]]][[All,1]]//DeleteDuplicates;
cor2=Keys[dp[[3]]][[All,2]]//DeleteDuplicates;
scor1=Intersection[cor1,mpgcor["sindex"]];
scor2=Intersection[cor2,mpgcor["sindex"]];
dcor1=Complement[cor1,scor1];
dcor2=Complement[cor2,scor2];
If[OptionValue["double"]===False, cor1=scor1; cor2=scor2];
tmp=Join@@(Position[dp,#]&/@label[[emidx,lopt]]);
If[emidx!={}, dp=MapAt[Style[#,{Red,Bold}]&,dp,tmp]];
dp=Row[#,"+"]&/@Map[If[ListQ[#],Row[#],#]&, dp[[lopt]], {2}];
If[emidx!={}, dp=Style[#,Gray]&/@dp];
tab=Table[dp[{label[[i,lopt]],label[[j,lopt]]}], {i,cor1}, {j,cor2}];
tab=Transpose@Join[{cor1}, Transpose@label[[cor1]], {type[[cor1]]}, Transpose@tab];
head=Column[{Row[{"MPG: ", mpgcor["number"][[2]],", ",MSGSymStd@mpgcor["symbol"]}],
Row[{"USubG: ", MSGSymStd@mpgcor["USubG"][[3]], " (",str2Mulliken@mpgcor["USubG"][[2]],")"}]
}];
box=Grid[{{#}},Spacings->{0,0},Alignment->{Center,Center}]&;
rowh={{head,SpanFromLeft,SpanFromLeft,SpanFromLeft,Sequence@@cor2},
{SpanFromAbove,SpanFromBoth,SpanFromBoth,SpanFromBoth,Sequence@@box/@label[[cor2,lopt]]}};
tab=Join[rowh,tab];
tmp=Association@Table[cor1[[i]]->i+2,{i,Length[cor1]}];
s1pos=tmp/@scor1; d1pos=tmp/@dcor1;
tmp=Association@Table[cor2[[i]]->i+4,{i,Length[cor2]}];
s2pos=tmp/@scor2; d2pos=tmp/@dcor2;
sty1=Directive[Black,Thickness[OptionValue["linewidth"]]];
sty2=Directive[Thin,GrayLevel[0.8]];
bg0={1,1}->Lighter[Red,0.9];
bg1=With[{c=Lighter[Green,0.85]}, Flatten@Join[Table[{#,i}->c,{i,4}]&/@s1pos,Table[{i,#}->c,{i,2}]&/@s2pos]];
bg2=With[{c=Lighter[Blue,0.85]}, Flatten@Join[Table[{#,i}->c,{i,4}]&/@d1pos,Table[{i,#}->c,{i,2}]&/@d2pos]];
bg3=Table[{i,j}->Lighter[Green,0.9],{i,s1pos},{j,s2pos}]//Flatten[#,1]&;
bg4=Table[{i,j}->Lighter[Yellow,0.9],{i,d1pos},{j,d2pos}]//Flatten[#,1]&;
bg5=Join[Table[{i,j}->Lighter[Blue,0.9],{i,s1pos},{j,d2pos}],
Table[{i,j}->Lighter[Blue,0.9],{i,d1pos},{j,s2pos}]]//Flatten[#,1]&;
Grid[tab, Frame->All, Alignment->{Center,Center}, ItemSize->{{{},{2->3.1,3->2.9}},{}},
Dividers->{{{{sty2}},Join[#->sty1&/@{1,5,-1},#->sty2&/@{2}]},
{{{sty2}},#->sty1&/@{1,3,-1}}},
Background->{None,None,{bg0,Sequence@@Join[bg1,bg2,bg3,bg4,bg5]}}
]
]
showMPGCorepDirectProduct[mpg_, coreps1_, OptionsPattern[]]/;!VectorQ[Flatten[{coreps1}],MemberQ[{Rule,RuleDelayed},Head[#]]&]:=
showMPGCorepDirectProduct[mpg, coreps1, coreps1, (#->OptionValue[#])&/@{"label","double","linewidth","emph"}]
showMPGCorepDirectProduct[mpg_, OptionsPattern[]]:=
showMPGCorepDirectProduct[mpg, All, All, (#->OptionValue[#])&/@{"label","double","linewidth","emph"}]
(* ::Section:: *)
(*MLG Corep (small corep)*)
(* ::Subsection::Closed:: *)
(*findMatrixN*)
(* find a unitary matrix N which satisfies \[CapitalDelta](R)=N*bar\[CapitalDelta](R)*N^-1, where bar\[CapitalDelta](R)=\[CapitalDelta](A^-1RA)^* *)
findMatrixN[Delta_, barDelta_]:=Module[{tryN,len,d,i,tmp,q,U1q},
(*----------method 1, only for 1D rep---------------*)
If[!MatrixQ[Delta[[1]]], Return[1]];
(*----------method 2, maybe not work in some cases--------------*)
len=Length[Delta]; d=Dimensions[Delta[[1]]]//First;
tryN=Total@Table[Delta[[i]].barDelta[[i]]\[ConjugateTranspose],{i,len}]/len//Simplify[#,u\[Element]Reals]&;
If[Total@Flatten@Abs[tryN.tryN\[ConjugateTranspose]-IdentityMatrix[d]]<1*^-8, Return[tryN]];
(*----------method 3, work in any case--------------*)
For[i=1,i<=d,i++,
tmp=Total[Delta[[All,1,1]]\[Conjugate]*barDelta[[All,i,i]]]*d/len//Simplify[#,u\[Element]Reals]&//Chop;
If[Abs[tmp]>1*^-8, q=i; U1q=Sqrt[tmp]; Break[]]
];
tryN=Table[Total[Delta[[All,1,i]]\[Conjugate]*barDelta[[All,q,j]]],{i,d},{j,d}]*d/(len*U1q)//Simplify[#,u\[Element]Reals]&//Chop;
If[Total@Flatten@Abs[tryN.tryN\[ConjugateTranspose]-IdentityMatrix[d]]>1*^-8, Print["findMatrixN: N is not unitary"]];
tryN
]
(* ::Subsection:: *)
(*For type-3 MSG*)
(* ::Subsubsection::Closed:: *)
(*SeitzConvert*)
(*Convert the SG Seitz symbol from cell1 (with lattice brav1) to cell2 (with lattice brav2).
Suppose that cell1=(t1,t2,t3) where t1,t2,t3 are column vectors of the basic vectors of cell1,
then cell2=R.cell1.T and os is the components of the origin shift O1-O2 under the bases of cell2.
T is the transformation matrix and R is the rotation matrix.
The returned value contains two Seitz symbols: the first for single SG, and the second for double SG.
NOTE: If Rspin is not give, it will be calculated in SeitzConvert, however this will slow down
this function. In practice, Rspin had better be calculated out of the function. *)
SeitzConvert[brav1_String, brav2_String, T_, os_, R_]:=SeitzConvert[brav1, brav2, T, os, R, None]
SeitzConvert[brav1_String, brav2_String, T_, os_, R_, Rspin_]:=Function[{Rv},
Module[{rot,rot2,v2,RSpin,Rang,Raxis,sx,srot,srot2,o3det,Rn2,sRn2,Rname,v,iT},
sx={{0,1},{1,0}}; Rname=Rv[[1]]; v=Rv[[2]];
rot=getRotMat[brav1,StringReplace[Rname,"bar"->""]];
(* Making the matrics numeric will speed up the calculations! *)
iT=Inverse[T//N]; rot2=iT.rot.T; v2= iT.v-rot2.os+os;
If[Rspin=!=None, RSpin=Rspin,
{Raxis,Rang}=rotAxisAngle[R//N][[{2,3}]];
RSpin=MatrixExp[-I*Rang*Raxis.(PauliMatrix/@{1,2,3})/2]//Simplify
];
{srot,o3det}=getSpinRotOp[Rname];
srot2=sx.RSpin.sx.srot.sx.RSpin\[ConjugateTranspose].sx//Simplify;
Rn2=getRotName[brav2,rot2];
sRn2=getSpinRotName[brav2,{srot2,o3det}];
If[StringReplace[sRn2,"bar"->""]!=Rn2, Print["SeitzConvert: error, ",Rn2,"!=",sRn2]];