Statistical Mechanics

Transcription

1 Kno Theory and Saisical Mechanics

2 KNOTS: A 3-dimensional loop projeced ono a 2-dimensional surface Unkno Trefoil Figure 8

3 LINK: The enanglemen of 2 or more loops

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10 Unkno Unkno Trefoil Hopf link

11 Unkno Unkno EQUIVALENT KNOTS: Two knos are equivalen if hey are he projecions of he same 3-dimensional kno Trefoil Hopf link

12 A n An unkno

13 An unkno An unkno KNOT INVARIANTS: Algebraic funcions (polynomials or numbers) consruced from he kno projecions A ha are he same for equivalen knos n

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15 Reviews of Modern Physics, (1992)

16 Unkno Unkno EQUIVALENT KNOTS: Two knos are equivalen if hey are he projecions of he same 3-dimensional kno Trefoil Hopf link

17 Equivalen knos can be ransformed ino each oher by Reidemeiser moves of lines There are 3 ypes of moves: I, II, and III IA IB IIA IIB

18 u v IIIA w

19 IIIB

20 ENTANGLED CONFIGURATIONS

21 Sufficien moves for all equivalen knos Any one of he IIIA moves Yang-Baxer equaion

22 Example of deducing a IIIB move using IIB and IIIA moves

23 Tradiionally, here are wo approaches o deduce kno invarians Algebraic approach: Conver knos ino braids and use group-heoreic properies of he braid group o deduce invarians Geomeric approach: Use kno graphs o deduce invarians

24 Algebraic approach Example of convering a kno ino a braid Braid group: σ i i i + 1 σ iσ i + 1 σ i = σ i + 1σ iσ i + 1 σ σ = σ σ, i j 2 i j j i

25 σ ii σ i + 1 σ i σ i i + 1 σ i σ i+ 1 σ i σ i+ i+1 1 i i + 1 σ σ i σ i+1 σ σ i+ 1 i i+1

26 Jones considers represenaions of he braid group Von Neumann algebra he Jones polynomial Using he Temperley-Lieb algebra akin o he Pos model leads o a one-variable polynomial invarian Hecke algebra Homfly polynomial Using he Hecke algebra leads o a wo-variable Using he Hecke algebra leads o a wo variable polynomial invarian

27 Temperley, 1998

28 Temperley, 1998 (Harold Neville Vazeille Temperley, )

29 Lieb (2003)

30 Geomerical approach (direced knos) Consider hree knos which are idenical excep a crossing L + L L0 Associae each kno wih a polynomial P(x, y, z) such ha The hree polynomials are relaed by The skein relaion xp Saring from L ( x, y, z ) + yp ( x, y, z ) = zp ( x, y, z ) + L L 0 P unkno ( x, y, z) = 1

31 Skein relaion Skein relaion ),,, ( z 1 y 1 2 x y x P x l = + z y x P + = 2l z xz y xy x P z z y x y z y x xp Hopf Hopf 2 2 1, ),, ( + = = + + xz z xz y xy x z y z y x xp Trefoil ),, ( = + xz ),, ( y xy z z y x P Trefoil = This gives 2 ) ( x y Trefoil

32 Alexander-Conway polynomial : z y x 1 1,, 1 = = = Δ() ) ( 1 ) ( ) ( 0 L L L Δ = Δ Δ + Skein relaion: Jones polynomial : ( ) V z y x 1,, 1 = = = Skein relaion: ) ( 1 ) ( ) ( 1 V 0 V V L L L = + Homfly polynomial : ), ( z P y x = =, 1 Skein relaion: ), ( ), ( ), ( 1 0 z P z z P z P L L L = +

33 Saisical Mechanical Approach

34 1. Consruc a laice from a given kno. 2. Define a saisical mechanical model on he laice. 3. Assign model parameers such ha he pariion funcion of he saisical mechanical model is invarian under Reidemeiser moves of lines. 4. The pariion funcion is by definiion a kno invarian. 5. Differen invarians are obained by using differen models.

35 There are hree main differen ypes of laice models: 1. Verex models 2. Ineracion-round-face (IRF) models 3. Edge-ineracion spin models

36 Verex models d b a c The saisical model is a verex model wih verex weighs. Each laice edge can be in q differen saes. Specify he saes of he 4 edges a a verex by variables a, b, c, d = 1, 2,, q and denoe he verex weigh by Z = ω ± ( a, b c, d) The pariion funcion of he laice is given by ω ± ( a, b c, d ) saes verices

37 For un-oriened knos, he Reidemeiser moves are

38 The Reidemeiser moves require he following condiions on he weighs:

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40 For oriened knos, i is convenien o inroduce a parameer ino he model. Then he Reidemeiser moves require λ 1,2,3 = ± (IIIA) The equaion (IIIA) is he Yang-Baxer equaion. 6 I is a se over-deermined d equaion wih q equaions 4 and 3q unknown verex weighs.

41 The Yang-Baxer equaion q equaions, 3q unknowns ns 6 4

42 The Yang-Baxer equaion q equaions, 3q unknowns ns 6 4 A soluion of he Y-B equaion exiss for each exacly solved models

43 Condiion I is a uniariy condiion which is usually saisfied. Condiion IIA can usually be saisfied by choosing model parameers. The Yang-Baxer equaion IIIA is saisfied for exacly solved models. This implies ha from each exacly solved model, one can consruc a kno invarian!

44 Ineracion-around-face (IRF) models Pariion funcion = W ( a, b, c, d) Verex weigh = W ( a, b, c, d )

45 Yang-Baxer equaion for IRF models W ' W W '' W '' W W ' c W ( a, b, c, a'') W '( a'', c, b', c') W ''( c, b, b'', b') = W ''( a'', a, c, a') W '( a, b, b'', c) W ( c, b'', b', a') c

46 Simple spin models wih pair ineracions Shade alernae faces and pu spins in shaded faces There are 2 kinds of pair ineracions along doed lines. a Z b a b 1 W ± ) = W± ( a, b) N / q ( 2

47 Type I Reidemeiser i moves 1 a m W± (, b) = α q b W± ( a, a ) = α m 1

48 Type II Reidemeiser moves 1 q W ( a, b ) W+ ( b. a ) = δ a b (, c IIA W ( a, b ) W ( a, b ) = 1 + b IIB

49 Type III Reidemeiser moves ), ( ), ( ), ( ), ( ), ( ), ( 1 a c W c b b W a W d c W d b W d a W q d ± ± ± ± = m m

50 Example: The Pos model a soluble (unphysical) case Take K±δ a, b W ( a, b) A e ± = ± wih K+ K = e = e A ± = q = + α = ±1/ / 4 1 Then his gives he Jones polynomial where n ± 1 ( ) 1/ 2 ( 3/ 4 n n+ V = q ) Z ( W = he number of ± ± ) crossings in he kno

51 Kno Jones polynomial

52 The Jones polynomial can also be derived from he The Jones polynomial can also be derived from he bracke polynomial of Kauffman (1987)

53 The Jones polynomial can also be derived from he bracke polynomial of Kauffman (1987) The bracke polynomial is idenical o a q-sae noninersecing sring model of Perk and Wu (1986)

54 Kauffman approach: Un-oriened kno Reidemeiser moves:

55 Theorem: Pariion funcion Z(w) is a regular Isoopy of kno invarians (Kauffman, 1987) The non-inersecing sring model of Perk and Wu (1986)

56 = polynomial in q, A, B

57 3 q 3 B 2 2 3q AB 3qA 2 B q 2 A Z = q B+ 3q AB+ 3qAB+ q A

58 Reidemeiser moves II: 1 B = A Reidemeiser moves I: q = ( A + A 2 2 ) α = A 3 A = 1/ 4 3/ 4 nn V ( ) = ( ) n + Z ( ) Is he Jones polynomial

59 Alexander-Conway polynomial Kno Alexander (1928) Conway (1970) z = 1

60 Kno invarians from he exac soluion of a 19-verex model Kno invarians from he exac soluion of a 19-verex model (Pan and Wu, 1995)

61 Jones polynomial V. F. R. Jones, Bull. Am. Mah. Soc. 12 (1985) Bracke polynomial L. H. Kauffman, Topology 26 (1987) Noninersecing sring model J. H. H. Perk and F. Y. Wu, J. Sa. Phys.42 (1986) Kno invarians and saisical mechanics V. F. R. Jones, Pacific J. Mah. 17 (1989) F. Y. Wu, Rev. Mod. Phys. 64 (1992) Kno invarians from he chiral Pos model F. Y. Wu, P. Pan, C. King, Phys. Rev. Le. 72 (1994)

62 The End