How to break tetrahedral symmetry

Transcription

1 How to brea tetrahedra syetry Etha Lae (Dated: Noveber 22, 2015) The goa of these otes s to carefuy ay dow the achery I be usg ater to costruct phases that brea tetrahedra syetry. The otvato coes fro y spedg the ast two wees gettg stuc o a cacuato because of a subte cosstecy the covetos for the duaty aps that I was usg. Here, I hope to ay dow a good set of covetos that ca be referred to ater. We w osty wor the fow of te pcture, ad w avod puttg arrows o graph edges. Ths aes soe of the coputatos a bt cearer, ad eas that we do t have to assue rotatoa varace. If we wat rotatoa varace (as L ad Lev s paper), we ust have to put arrows bac o the edges. Noe of our resuts chage as og as we are cosstet about oy appyg operators to graphs that have arrow drectos agreeg wth the fow of te deftos gve here. Ths pot shoud becoe ore cear ater o. The fuso vector space V ad ts dua, the spttg vector space, are preseted by the orthoora set of bass vectors (assug utpcty-free fuso rues throughout), ; = ( ) 1/4 d d d ( ) 1/4 d,, ; = d d (1) The orazato factors are portat, ad ths turs out (after a bt of experetato) to be the ost coveet coveto. Wth ths choce, we ca off bubbes by = δ d d d (2) We w draw the F -ove as F ; (3) Note that our coveto for orderg the row ad cou dex (.e., the ad ) s dfferet tha the oe used soe of We s wor. It s aways possbe (ad I w aways do so) to wor a utary gauge where the F -sybos are utary: (F ) = (F ) = (F ) 1. (4) Ths s pretty quc to see fro the gauge trasforato of F, but s aso a physca requreet. We w defe the quatu desos the theory as d = d = 1 F ;00, (5) athough they ay pc up otrva phases ater o whe we ru gauge trasforatos o the oste syetres. Now we coe to the trcy ssue of sotopy varace. I w adopt a dfferet coveto tha L ad Lev, sce t geerazes ore aturay to the represetato pcture. I w wrte rgdty as = d F ;00 = α (6)

2 where α s the Frobeus-Schur dcator of. We ca aways wor a gauge where α = α = ±1 f = ad α = 1 otherwse, but we w avod settg the gauge expcty for ow. To dea wth vacuu es ad tae to accout the FS dcators, I put red ad bue dots to abe the dfferet duaty aps, wth the foowg covetos: 2 = = α (7) ad = = (α ) (8) Thus, we ca reove the dots ad straghte a par of wgges whe we have a bue dot / red dot par: = = (9) If at ay te I forget to expcty coor a duaty ap, assue t to be coored bue. Cosder the vector space V = ho(1, ( )). Defe the cyce operator C by C : V = ho(1, ( )) V = ho(1, ( )). (10) I caed ths R the ast ote, but a swtchg to C to avod cofuso wth the R-atrx. The T -tesors defed the ast ote are equvaet to the oodroy of the cyce operators. As derved earer, the cyce operators are gve by C = α F 0; (11) Whe = =, the trace of C s essetay a sort of 3rd FS dcator for the abe. More o ths w foow the ext ote. I w be usg the F -sybo wth vacuu at ts ower vertex so ofte that I gve t ts ow sybo: F F 0;. (12) We w eed two usefu dettes vovg the F tesors. Oe s that we ca reverse F-oves by = 1 F (13) Aso, we ca act o fuso vertces wth F as foows: = F (14) Techcay, there w be prefactors frot of the F for a geera gauge choce of the put fuso data. However, the utary gauge I worg over, these prefactors w aways be oe. I aso woud e to defe aps that tae fuso spaces to spttg spaces ad vce versa. To ths ed, t s hepfu to troduce what I ca the H-ove (sce t oos e the etter H): = H ; (15)

3 Usg our resut for the spttg vertex / fuso vertex stacg rue, we have that the H-ove s reated to the F -ove through 3 H ; = d d d d (F ) (16) whch s derved by cosderg the foowg dagra ad the utarty of the F -ove: 1 H ; (F ) 1 q d d d (1) q d d d (17) We ow troduce the foowg eft ad rght owerg operators that w aow us to trasfor spttg vertces to fuso vertces ad vce versa. The eft owerg operator s preseted graphcay by = L L (18) ad gve agebracay as L L = α H0 ; = α d d d (F ) 0 (19) whch foows straghtforwardy fro appyg the H-ove drecty to the eft had sde of (18). We sary defe the rght owerg operator R L that acts by = R L (20) Agebracay, t s gve by R L = d d d F ;0 (21)

4 4 1 whch we ca derve through the foowg dagra: F ;0 R L q d d d d (22) (1) Sary, we ca defe eft ad rght rasg operators that tae fuso vertces to spttg vertces. They are defed graphcay by = R R (23) = L R (24) The expressos for the rasg operators ters of the F -sybos are derved usg the sae dea as for the owerg operators. I suary, our arsea of operators are as foows: Fro whch we ote that L L = (α ) d d d L R = α d d d (F ) 0, R L F ;0, R R = = d d d d d d F ;0 (F ) 0 (25) ( ) ( ) L R = L L, R R = R L (26) whch tur aows us to derve the usefu dettes ( ) ( ) L L L L = α, ( R L ) ( ) R L = α. (27) Iterestgy, L ad Lev assued that rght rasg ad owerg operators are trva for every vertex (equato 11 ther paper, after trasatg betwee ther otato ad e). They ever had to use owerg operators to copute the oste syetry, so t dd t atter for the. Whe cosderg tetrahedra syetry however, the owerg operators are certay portat. Wth a ths achery pace, we ca ow go through ad (carefuy!) derve the reatos for tetrahedra syetry breag. We start wth the order 3 rotato. Graphcay, the F -sybo trasfors as F ; F ; (28)

5 The goa s to trasfor the eft tetrahedro to the rght oe usg oy the operators preseted so far. After dog t a few ways, I th t s easest to wor the fow of te pcture. To accopsh ths, t becoes hepfu to draw the tetrahedro as the fow-of-te pcture: 5 F ; = (29) Ths way of thg aes coputg the trasforato uder the tetrahedra acto easer (athough deterg the age of the F sybo uder the trasforato s best doe usg the arrows-cuded pcture). The coputato

6 s doe as foows: 6 1 F ; = = F F! = F! ( ) /! R L /! R R! F! = F ; Fro whch we ca see that Ateratvey, we ca aso wrte Now for the order 2 rotato. Graphcay, we have ( ) F ; = α F F F F ( R L )(R R ) F ; (31) F ; = F F ( R R )( L L )( L L )( R R ) (32) (30) F ; F ; (33)

7 7 The F -sybo trasfors as ; = F F F F F F F ; (34) F whch s derved usg the steps show the ast ote. We dd t ae use of the dots the ast ote, but rather surprsgy, they tur out ot to be eeded for the dervato to wor. Fay, we ca vert the F -sybo by fppg the assocated dagra about the horzota axs. Ths s a tte trcy, sce we have vertces whose edges trasfor e, ;, ;, whch we ca t hade usg oy the toos deveoped so far. What saves us s the utary gauge codto we ve chose, whch eas that (soewhat scheatcay) refecto about the horzota e s the sae as cougato: = (35) Wth ths, we see that fppg the dagra about the horzota axs s equvaet to cougatg the correspodg F -sybo. After the fp the ceter curvy e the dgra sats dowwards, whereas the defto of the F -sybo t sats upwards. We the sert a bue duaty / red duaty par the curvy e ad use the rasg ad owerg operators to chage ts sat. Thus, we have ( ) F ; = ( L L )( R R )F ; (36) Whch s the fa part of the stadard tetrahedra syetry. Before ovg o, we shoud brefy dscuss the roe of party varace. Our costructo aows party varace to be expcty broe. Ideed, we ve see that a party fp s ocay equvaet to cougato. Addtoay, t chages red duaty aps to bue duaty aps ad vce versa, whch aso cotrbutes to the overa party trasforato. As og as we assue that (35) hods for arbtrary graph cofguratos, the trasforato of a graph uder party ca be wored out exacty usg the oca operators preseted above. I do t see ay rea reaso why ths shoud t be the case, at east the groud state. What about party fps about the te axs? These are accessbe wth curret techoogy. We ow how to do, ;, ;, but ot, ;,, ;. Oe ght guess that dog a fp about the horzota axs ad the repacg every abe wth ts dua woud be a syetry of the F -sybos, whch woud aow us to copute, ;,, ;. Ths s t the case geera though (the twsted quatu doube of Z 3 provdes a quc chec), ad so party fps about the te axs rea out of reach for us.