Physical combinatorics and TBA: paths, ( m, n ) systems and finitized - PowerPoint PPT Presentation

Physical combinatorics and TBA: paths, ( m, n ) systems and finitized characters Giovanni Feverati Laboratoire d’Annecy-le-Vieux de physique theorique Paul A. Pearce Department of Mathematics and Statistics University of Melbourne Nucl. Phys. B 663, 409-442 (2003), hep-th/0211185, hep-th/0211186 &. . . in preparation

Physical Combinatorics C ∩ IM ∩ CFT = Physical Combinatorics C IM Phys. Comb. CFT

Integrable RSOS Models with boundaries Double Row Transfer Matrix: σ ′ σ ′ σ ′ r r 1 N − 1 1 2 λ u λ u λ u − − − � u D ( N, u, ξ ) σ,σ ′ = 2 τ 1 τ 2 τ N − 1 τ N ξ τ 1 ,...,τ N u u u σ r σ 1 σ N − 1 r 1 ✒ ✻ boundary fixed b.c. interaction Critical A L RSOS Face Weights: (ABF 1984) ℓ 4 ℓ 3 � = sin( λ − u ) δ ℓ 1 ,ℓ 3 + sin u S ℓ 1 S ℓ 3 δ ℓ 2 ,ℓ 4 u sin λ sin λ S ℓ 2 S ℓ 4 ℓ 1 ℓ 2 π λ = L + 1 ; S ℓ = sin ℓλ ℓ j = 1 , . . . , L ”height” variables: 1 2 3 L | ℓ i − ℓ j | = 1 nearest neighbor sites: Boundary Weights: (Behrend, Pearce 2001) r � sin( r ± 1) λ sin( ξ ± u ) sin( rλ + ξ ∓ u ) u r ± 1 = sin 2 λ sin rλ ξ r

D q +1 ∼ D q D 1 + D q − 1 , d q ∼ D q +1 D q − 1 ˜ fusion: YBE ⇒ [ D q ( u ) , D q ′ ( v )] = 0 Integrability: basis of eigenstates independent of u each entry of D 1 ( u ) is entire function (zeros); Analyticity: each entry of D q ( u ) ( q > 1 ) is the ratio of an entire function by some known function u + π ≡ u Periodicity: q = 1 , . . . , L − 2 Functional equations d q ( u − λ d q ( u + λ ˜ 2)˜ 1 + ˜ 1 + ˜ d q − 1 ( u ) d q +1 ( u ) � � � � 2) = d 0 ( u ) = ˜ ˜ d L − 1 ( u ) = 0 • true for the eigenvalues of ˜ d q ( u ) • solution given by the analytic properties in u ∈ C : ** L − 2 analyticity strips ** zeros of eigenvalues D q ( u ) , zeros of the numerical factors ** (with boundaries) additional numerical factors with zeros/poles

A 4 : zeros of the eigenvalues of D ( u ) periodicity u + π ≡ u ; λ = π/ 5 Two analyticity strips: − λ 2 < Re ( u ) < 3 λ (1) 2 , (2) 2 λ < Re ( u ) < 4 λ m i = { number of 1 -strings in strip i = 1 , 2 } n i = { number of 2 -strings in strip i = 1 , 2 } � m 1 + n 1 = N + m 2 2 ( m , n ) − system: = ⇒ m 1 , m 2 , ∈ 2 N m 2 + n 2 = m 1 2 • • • m 1 = 4 • m 2 =2 • • • • n 1 = 2 n 2 = 0 • • (1) (2) − λ − 4 λ 2 Relative order

Paths paths on A L : only diagonal paths are permitted; initial and final point at height=1 paths on T L : diagonal paths are permitted; horizontal paths permitted at heigth=1; initial and final point at height=1; fixed shape rectangle L = ⌊ N 2 ⌋ L = 5 �❅ � ❅ 4 �❅ �❅ � ❅ � ❅� ❅ � ❅ 3 � ❅ �❅ � ❅ � ❅� ❅� ❅ 2 � ❅ �❅ � ❅ � ❅ 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22= N  Local energy density � ❅ ❅ ❅ ❅ � 0 σ j +1 = σ j − 1 if � �   (Baxter80, ABF84)     � ❅   � ❅  1 σ j +1 − σ j − 1 = ± 2 if  s s  � ❅  h ( σ j − 1 , σ j , σ j +1 ) =  1 ( σ j − 1 , σ j , σ j +1 ) = (1 , 1 , 1) 1 if        1 ❅ �  ❅ �  1 ( σ j − 1 ,σ j ,σ j +1 )=(2 , 1 , 1) or (1 , 1 , 2)  if  2 N E ( σ ) = 1 � j h ( σ j − 1 , σ j , σ j +1 ) 1-dim configurational sums 2 j =1

Quasi-particles (Warnaar 1995) . . L . . n 4 n 3 n 2 n 1 n 0 . . 5 � ✉ ✉ �❅ � ❅ 4 � ✉ ✉ ✉ ✉ � ❅ �❅ � ❅ � ❅ 3 � ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ � ❅ � ❅ �❅ � ❅ � ❅ � ❅ 2 � ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ � ❅ � ❅ � ❅ �❅ � ❅� ❅� ❅� ❅ 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 N n -family • L − 1 types of pure particles (pyramids): � � � � · · · • tower particle ( T L case only) = { # of particles of type a = 0 , 1 , 2 , . . ., L − 1 } n a m -family: in sites where there is a straight line segment or in the middle of horizontal lines • L − 2 types of dual particles (strings): • • • · · · = { # of dual particles of type a = 0 , 1 , 2 , . . . , L − 2 } m a

• Geometric packing constraints = n 0 + 2( L − 1) n L − 1 + . . . + 6 n 3 + 4 n 2 + 2 n 1 N = n 0 + 2( L − 2) n L − 1 + . . . + 4 n 3 + 2 n 2 m 1 = n 0 + 2( L − 3) n L − 1 + . . . + 2 n 3 m 2 . . . = n 0 + 2 n L − 1 m L − 2 m 0 = n 0 A L case: m 0 = n 0 = 0 • ( m, n ) system L − 2 � m a + n a = 1 2 Nδ ( a, 1) + 1 a = 1 , 2 , . . . , L − 2 A a,b : A L − 2 adjacency matrix A a,b m b 2 b =1 L − 1 � m a + n a = 1 2 Nδ ( a, 1) + 1 A a,b : T ′ a = 0 , 1 , 2 , . . ., L − 2 A a,b m b L − 1 adjacency matrix 2 b =1 • Interaction: n -family particles can be sliced and diced and turned upside-down (geometric packing constraints are respected).

General paths: interactions Path = { non-interacting particles } + { complexes of overlapping particles } (c.f. Warnaar 1995) 4 � � �❅ �❅ �❅ � ❅ � ❅ � ❅ 3 t t t t t t � ❅ �❅ � ❅ � ❅ � ❅� ❅ � ❅� ❅ 2 t t t � t t t � t t � � ❅ � ❅ �❅ � ❅� ❅� ❅ 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 baseline and maximum peak

Decomposition Algorithm 7 6 5 4 3 2 4 5 7 1 2 3 6 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1. Identify any tower particles and decorate them with dual-particles. These tower particles automatically separate the path into non-overlapping complexes with respect to the initial baseline at height 1 . 2. For each current baseline, separate the pure particles from the complexes and decorate the pure particles. A complex with respect to the current baseline is any path that is not a pyramid. 3. For each current complex, identify the left-most and right-most global maxima and connect these with a new baseline. The left-most and right-most global maxima may coincide and in this case no new baseline is drawn. 4. From each left (right) maxima, outline the profile of the complex moving continuously down and to the left (right) drawing new baselines as needed. Decorate the sliced particles correponding to these maxima. 5. Stand on your head, identify all the current baselines and go to 2.

Energy-preserving bijection: RSOS paths ↔ strings There is a natural energy-preserving bijection between RSOS paths and string patterns: a n -particle ( m -particle) of type a at position j corresponds to a 2-string (1-string) in strip a ; the relative order within a strip is preserved. The RSOS particle decomposition matches the pattern of the zeros of the transfer matrix eigenvalues. 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 In the example, m 1 = 10 , m 2 = 4 , m 3 = 2 and E = 50 .

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PeriodicStatesMaster.nb 1 chiralPaths @ 1, 1, 5, 8 D In[98]:= 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 0, 0, 0 < n = 8 5, 0, 0, 0 < E = 0 1: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < E = 2 2: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < 3: E = 3 _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < E = 4 4: _____________________________________________

PeriodicStatesMaster.nb 2 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < E = 4 5: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < E = 5 6: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 2, 0 < n = 8 2, 0, 1, 0 < 7: E = 6 _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 2, 0, 0 < n = 8 3, 1, 0, 0 < E = 6 8: _____________________________________________

PeriodicStatesMaster.nb 3 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 2, 0 < n = 8 2, 0, 1, 0 < E = 7 9: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 2, 0 < n = 8 2, 0, 1, 0 < E = 8 10: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 0, 0 < n = 8 1, 2, 0, 0 < 11: E = 8 _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 2, 0 < n = 8 2, 0, 1, 0 < E = 9 12: _____________________________________________

PeriodicStatesMaster.nb 4 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 4, 2, 0 < n = 8 2, 0, 1, 0 < E = 10 13: _____________________________________________ 5 4 3 2 1 2 3 4 5 6 7 8 9 10 m = 8 6, 4, 2 < n = 8 1, 0, 0, 1 < E = 12 14: _____________________________________________