Integrability of the dimer model Alexi Morin-Duchesne Universit e - PowerPoint PPT Presentation

Dimer model TL Integrability Conclusion Integrability of the dimer model Alexi Morin-Duchesne Universit´ e Catholique de Louvain (UCL) Supported by the Natural Sciences and Engineering Research Council of Canada Integrability and Combinatorics, Stroganov memorial June 25, 2014 Joint work with Jørgen Rasmussen and Philippe Ruelle

Dimer model TL Integrability Conclusion Outline Dimer model • Statistical model • Transfer matrix formulation • Yang-Baxter Integrability? Temperley-Lieb algebra • Dimer representation of TL at β = 0 • Spanning forests and critical dense polymers Integrability • Commuting transfer matrices • Integrals of motion Outlook and references

Dimer model TL Integrability Conclusion Transfer matrices and YB integrability • Transfer matrix : Lattice: T ( u ) ∼ in-states • T ( u ) is an operator that acts on in-states and outputs the possible out-states with the correct Boltzmann weights • The spectral parameter u ∈ R measures the lattice anisotropy • Yang-Baxter integrability : out-states [ T ( u ) , T ( v )] = 0 u , v ∈ C Yang-Baxter equation: • Integrals of motions: 2 u 2 I 2 + . . . = T ( u ) = I 0 + u I 1 + 1 [ T ( u ) , I k ] = 0

Dimer model TL Integrability Conclusion Dimer model • Classical counting problem solved by Kasteleyn, Fisher, Temperley, Stephenson, Lieb, Ferdinand, Wu, ..., in the 60’s • Partition function: � α h Z = Covering of the ( M , N ) = ( 6 , 9 ) cylinder σ • Transfer matrix approach initiated in ’67 paper by Lieb • Bijective map to spin configurations on the cylinder: � → � → Corresponding spin configuration

Dimer model TL Integrability Conclusion Transfer matrix approach • T ( α ) ∈ ( C 2 ) ⊗ N acts on spin states of a row and constructs all possible spin states of the next row, with the correct weights: T ( α ) + α + α − → • Expression in terms of Pauli matrices: N N − 1 � � σ x ( I + α σ − j σ − T ( α ) = V 3 V 1 V 1 = V 3 = j + 1 ) j j = 1 j = 1 Action of V 1 and V 3 : V 1 − → V 3 + α + α − → • Partition function for dimers: � T M ( α ) � Z = Tr

Dimer model TL Integrability Conclusion Properties of T ( α ) • T ( α ) is symmetric → diagonalisable with real spectra [ T ( α ) , T ( α ′ )] � = 0 • No commutativity: • More convenient to study T 2 ( α ) : N − 1 N − 1 � � T 2 ( α ) = V 3 V T ( I + α σ − j σ − ( I + α σ + j σ + 3 = j + 1 ) j + 1 ) j = 1 j = 1 • One-dimensional subspaces invariant under T 2 ( α ) : and • Invariant subspaces E v N of T 2 ( α ) labelled by eigenvalues v of the variation index operator V : N � [ T 2 ( α ) , V ] = 0 V = 1 (− 1 ) j σ z v ∈ { − N 2 , − N 2 + 1 , . . . , N 2 } j 2 j = 1

Dimer model TL Integrability Conclusion Exact solvability and integrability Jordan-Wigner (lattice) closed expressions • → → transformation free fermions for eigenvalues ν k , τ k ∈ { 0 , 1 } N − 1 � � 2 ( 1 − ν k − τ k ) � � 1 + α 2 sin 2 p k Λ ν,τ = α sin p k + π k p k = k = 1 2 ( N + 1 ) k = N − 1 mod 2 Partial partition function Verma character • → in each v -sector in the scaling limit v 2 ∞ � Z v ( q ) = q 2 1 ( 1 − q n ) , q = e − απ M / N η ( q ) , η ( q ) = q 24 j = 1 • Exact solvability but no Yang-Baxter integrability ?! • Unknown integrals of motions.

Dimer model TL Integrability Conclusion Temperley-Lieb algebra TL n ( β ) Generators A connectivity ... ... ... I = e j = c = n j j + 1 n 1 2 3 1 • Multiplication is by vertical concatenation Example e j e j + 1 e j = ... ... = ... ... = e j j j + 1 n 1 j j + 1 n 1 Algebraic definition � � TL n ( β ) = I , e j ; j = 1 , . . . , n − 1 � � I A = A I = A A ∈ TL n ( β ) e 2 j = β e j , e j e j ± 1 e j = e j , e i e j = e j e i , | i − j | > 1

Dimer model TL Integrability Conclusion Temperley-Lieb algebra TL n ( β ) Generators A connectivity ... ... ... I = e j = c = n j j + 1 n 1 2 3 1 • Multiplication is by vertical concatenation Example ... ... ( e j ) 2 = = β ... ... = β e j ... ... j j + 1 n 1 j j + 1 n 1 Algebraic definition � � TL n ( β ) = I , e j ; j = 1 , . . . , n − 1 � � I A = A I = A A ∈ TL n ( β ) e 2 j = β e j , e j e j ± 1 e j = e j , e i e j = e j e i , | i − j | > 1

Dimer model TL Integrability Conclusion Dimer representation of TL ( β = 0 ) • Rewrite T 2 ( α ) by grouping terms of the products 2 by 2: ⌊ N − 2 ⌊ N − 1 2 ⌋ 2 ⌋ � � � �� � T 2 ( α ) = I + α ( σ − 2 j − 1 σ − 2 j + σ − 2 j σ − I + α ( σ + 2 j − 2 σ + 2 j − 1 + σ + 2 j − 1 σ + 2 j + 1 ) 2 j ) j = 1 j = 1 • Representation of TL n ( β = 0 ) on ( C 2 ) ⊗ n − 1 : � σ − j − 1 σ − j + σ − j σ − j even j + 1 τ ( I ) = I τ ( e j ) = σ + j − 1 σ + j + σ + j σ + j odd j + 1 • T 2 ( α ) can be written as the matrix representation of the tilted transfer tangle of TL n ( β = 0 ) for n = N + 1: T 2 ( α ) = ( 1 + α 2 ) N / 2 τ ( D ( v )) α = tan v . . . v v v D ( v ) = = cos v + sin v v . . . v v

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical TL dimer → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical TL dimer → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical TL dimer → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical TL dimer → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical dimer TL → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical dimer TL → spanning → dense polymer → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical dimer TL spanning dense polymer → → → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Spanning forests and loop models Series of maps: oriented critical dimer TL spanning dense polymer → → → coverings algebra forests configurations

Dimer model TL Integrability Conclusion Module structure of τ Variation index decomposition: Structure for n odd: ( C 2 ) ⊗ n − 1 = � E v � � n − 1 − 2 | v | n − 1 4 v � I 2 | v | + 4 i + 1 E v n − 1 = n � � I d n ≡ irreducible rep. of TL n ( β = 0 ) i = 0 Structure for n even, v > 0 :  I 2 v + 1 I 2 v + 5 I n − 2   n n n   n − 1 − 2 v  ց ւ ց ւ ց odd   2   . . . I n I 2 v + 3  n n E v n − 1 =  I n  I 2 v + 1 I 2 v + 5 I n − 4  n  n n n   n − 1 − 2 v  ց ւ ց ւ ց ւ even   2  . . . I 2 v + 3 I n − 2 n n E − v n − 1 is the module contragredient to E v n − 1

Dimer model TL Integrability Conclusion TL representations at β = 0 n odd: TL n ( β = 0 ) is semi-simple . Direct sums of irreducibles only. n even: TL n ( β = 0 ) is non semi-simple and admits reducible yet indecomposable representations. • Standard modules:  I d d = 0 , n  n    V d n ≃ I d n   0 < d < n  ց  I d + 2 n • XXZ spin chain at ∆ = 0: I 2 I 2 r I n   n n n ց n ւ ց ւ 2 − 1   ( C 2 ) ⊗ n ≃ I 4 � I 2 r + 2 r I 2 r − 2  ⊕ n I n − 2 ⊕ n + 2 I n ⊕   n n n n n 2 2   ւ ց ւ ց  r = 2 I n I 2 I 2 r n n n

Dimer model TL Integrability Conclusion Critical dense polymers • Critical dense polymers: Loop model on the lattice with non local degrees of freedom. Contractible loops are prohibited ( β = 0) • Double-row transfer tangle: . . . . . . u − ξ 1 u − ξ 2 u − ξ n 1 D ( u , ξ ) = ∈ TL n ( β = 0 ) sin 2 u . . . . . . u + ξ 1 u + ξ 2 u + ξ n = cos u + sin u u • ξ = ( ξ 1 , ξ 2 , . . . , ξ n ) are the inhomogeneities • Commuting transfer tangles: [ D ( u , ξ ) , D ( v , ξ )] = 0 • An example for n = 2: D ( u , 0 ) = sin 3 u cos u + sin 2 u cos 2 u + cos 4 u + . . . � �� � � �� � � �� � = I = e 1 = 0

Dimer model TL Integrability Conclusion Tilted transfer tangle • Special choice of spectral parameter and inhomogeneities: u = v ξ = ξ v = ( v 2 , − v 2 , v 2 , − v 2 , 2 , . . . ) • Resulting transfer tangle: . . . . . . v v 1 D ( v 2 , ξ v ) = sin v . . . . . . v v • Using the planar identity = sin v , we find v D ( v 2 , ξ v ) = = D ( v )