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

Lattice: in-states

• ut-states

Yang-Baxter equation: =

• Transfer matrix:

T(u) ∼

• 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:

[T(u), T(v)] = 0 u, v ∈ C

• Integrals of motions:

T(u) = I0 + u I1 + 1

2u2 I2 + . . .

[T(u), Ik] = 0

Dimer model TL Integrability Conclusion

Dimer model

Covering of the (M, N) = (6, 9) cylinder Corresponding spin configuration

• Classical counting problem

solved by Kasteleyn, Fisher, Temperley, Stephenson, Lieb, Ferdinand, Wu, ..., in the 60’s

• Partition function:

Z =

• σ

αh

• Transfer matrix approach

initiated in ’67 paper by Lieb

• Bijective map to spin

configurations on the cylinder: → →

Dimer model TL Integrability Conclusion

Transfer matrix approach

• T(α) ∈ (C2)⊗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:

T(α) = V3V1 V1 =

N

• j=1

σx

j

V3 =

N−1

• j=1

(I + α σ−

j σ− j+1)

Action of V1 and V3:

V1

− →

V3

− → + α + α

• Partition function for dimers:

Z = Tr

• TM(α)

Dimer model TL Integrability Conclusion

Properties of T(α)

• T(α) is symmetric → diagonalisable with real spectra
• No commutativity:

[T(α), T(α′)] = 0

• More convenient to study T2(α):

T2(α) = V3VT

3 = N−1

• j=1

(I + α σ−

j σ− j+1) N−1

• j=1

(I + α σ+

j σ+ j+1)

• One-dimensional subspaces invariant under T2(α):

and

• Invariant subspaces Ev

N of T2(α) labelled by eigenvalues v of the

variation index operator V: [T2(α), V] = 0 V = 1

2 N

• j=1

(−1)jσz

j

v ∈ {− N

2 , − N 2 + 1, . . . , N 2 }

Dimer model TL Integrability Conclusion

Exact solvability and integrability

• Jordan-Wigner

transformation → (lattice) free fermions → closed expressions for eigenvalues Λν,τ =

N−1

• k=1

k=N−1 mod 2

• α sin pk +
• 1 + α2 sin2 pk

2(1−νk−τk) νk, τk ∈ {0, 1} pk = π k 2(N + 1)

• Partial partition function

in each v-sector → Verma character in the scaling limit Zv(q) = q

v2 2

η(q), η(q) = q

1 24

∞

• j=1

(1 − qn), q = e−απM/N

• Exact solvability but no Yang-Baxter integrability ?!
• Unknown integrals of motions.

Dimer model TL Integrability Conclusion

Temperley-Lieb algebra TLn(β)

Generators A connectivity I =

...

1 2 3 n

ej =

... ...

1 n j j+1

c =

• Multiplication is by vertical concatenation

Example ej ej+1 ej =

... ...

1 n j j+1

=

... ...

1 n j j+1

= ej Algebraic definition TLn(β) =

• I, ej ; j = 1, . . . , n − 1
• I A = A I = A
• A ∈ TLn(β)
• e2

j = βej,

ejej±1ej = ej, eiej = ejei, |i − j| > 1

Dimer model TL Integrability Conclusion

Temperley-Lieb algebra TLn(β)

Generators A connectivity I =

...

1 2 3 n

ej =

... ...

1 n j j+1

c =

• Multiplication is by vertical concatenation

Example (ej)2 =

... ... ... ...

1 n j j+1

= β

... ...

1 n j j+1

= β ej Algebraic definition TLn(β) =

• I, ej ; j = 1, . . . , n − 1
• I A = A I = A
• A ∈ TLn(β)
• e2

j = βej,

ejej±1ej = ej, eiej = ejei, |i − j| > 1

Dimer model TL Integrability Conclusion

Dimer representation of TL(β = 0)

• Rewrite T2(α) by grouping terms of the products 2 by 2:

T2(α) =

• ⌊N−2

2 ⌋

• j=1

I+α (σ−

2j−1 σ− 2j+σ− 2j σ− 2j+1)

• ⌊N−1

2 ⌋

• j=1

I+α (σ+

2j−2 σ+ 2j−1+σ+ 2j−1 σ+ 2j)

• Representation of TLn(β = 0) on (C2)⊗n−1:

τ(I) = I τ(ej) = σ−

j−1σ− j + σ− j σ− j+1

j even σ+

j−1σ+ j + σ+ j σ+ j+1

j odd

• T2(α) can be written as the matrix representation of the tilted

transfer tangle of TLn(β = 0) for n = N + 1: T2(α) = (1 + α2)N/2 τ(D(v)) α = tan v D(v) =

v v . . . v v v . . . v

= cos v + sin v

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Spanning forests and loop models

Series of maps: dimer coverings →

• riented

spanning forests → critical dense polymer configurations → TL algebra

Dimer model TL Integrability Conclusion

Module structure of τ

Variation index decomposition: (C2)⊗n−1 =

• v

Ev

n−1

• Id

n ≡ irreducible rep. of TLn(β = 0)

• Structure for n odd:

Ev

n−1 =

• n−1−2|v|

4

• i=0

I2|v|+4i+1

n

Structure for n even, v > 0 : Ev

n−1 =

                     I2v+1

n

I2v+3

n

I2v+5

n

. . . In−2

n

In

n

ց ւ ց ւ ց

n−1−2v 2

• dd

I2v+1

n

I2v+3

n

I2v+5

n

. . . In−4

n

In−2

n

In

n

ց ւ ց ւ ց ւ

n−1−2v 2

even E−v

n−1 is the module contragredient to Ev n−1

Dimer model TL Integrability Conclusion

TL representations at β = 0

n odd: TLn(β = 0) is semi-simple. Direct sums of irreducibles only. n even: TLn(β = 0) is non semi-simple and admits reducible yet indecomposable representations.

• Standard modules:

Vd

n ≃

         Id

n

d = 0, n Id

n

Id+2

n

ց 0 < d < n

• XXZ spin chain at ∆ = 0:

(C2)⊗n ≃ I4

n

I2

n

I2

n

ւ ց ⊕     

n 2 −1

• r=2

r I2r−2

n

I2r+2

n

I2r

n

I2r

n

ց ւ ց ւ     ⊕ n

2

In−2

n

In

n

In

n

ց ւ ⊕ n+2

2

In

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:

D(u, ξ) = 1 sin 2u . . . . . . . . . . . .

u+ξ1 u−ξ1 u+ξ2 u−ξ2 u+ξn u−ξn

∈ TLn(β = 0) u = cos u + sin u

• ξ = (ξ1, ξ2, . . . , ξn) are the inhomogeneities
• Commuting transfer tangles:

[D(u, ξ), D(v, ξ)] = 0

• An example for n = 2:

D(u, 0) = sin3 u cos u

= I

+ sin2 u cos2 u

= e1

+ cos4 u

= 0

+ . . .

Dimer model TL Integrability Conclusion

Tilted transfer tangle

• Special choice of spectral parameter and inhomogeneities:

u = v

2,

ξ = ξv = ( v

2, − v 2, v 2, − v 2, . . . )

• Resulting transfer tangle:

D( v

2, ξv) =

1 sin v . . . . . . . . . . . .

v v v v

• Using the planar identity

v

= sin v , we find D( v

2, ξv) =

= D(v)

Dimer model TL Integrability Conclusion

Commuting transfer matrices

• Absence of commutativity for T2(α) ≃ τ
• D(v)
• :

ξv = ξv ′ → [D(v, ξv)

• D(v)

, D(v′, ξv ′)

• D(v ′)

] = 0 → [T2(α), T2(α′)] = 0

• Family of transfer tangles that commutes with D(v):

[D(u, ξv), D( v

2, ξv)

• =D(v)

] = 0, u ∈ C

• Family of matrices that commutes with T2(α):

[τ

• D(u, ξv)
• , T2(α)] = 0,

u ∈ C, α = tan v

• Different values of α belong to different integrable families!

Dimer model TL Integrability Conclusion

Integrals of motion

• Possible to write integrals of motion in terms of TL generators
• Inversion identity for D(u, ξv) → closed form for eigenvalues

D(u, ξv)D(u + π

2 , ξv) = I

• f(u, v)

2

• Asymptotic expansion of spectra:

− 1

2 log

• Eig
• D(u, ξv)
• ν,τ
• = n fbulk + fbdy +

∞

• p=1

a(ν,τ)

2p−1

n2p−1 .

• Coefficients decompose into two parts:

a(ν,τ)

2p−1 = λ2p−1(u, v) I(ν,τ) 2p−1

• Integrals of motion of the Virasoro algebra:

I1 = L0 − c 24, I3 = 2

∞

• n=1

L−nLn + L2

0 − c + 2

12 L0 + c(5c + 22) 2880 , I5 = . . .

• In each fixed v-sector, I(ν,τ)

2p−1 reproduces spectra of I2p−1 for c = −2

for the lowest-lying states

Dimer model TL Integrability Conclusion

Summary and outlook

Summary

• Dimer model → TL loop model for β = 0
• YB integrability and free fermion solvability reconciliated
• Indecomposables in the form of long chains of irreducibles
• Spectra of integrals of motion in each sector correspond to c = −2

Outlook

• Construct lattice approximate realisations of Virasoro modes
• Can the dimer representation be generalized to other β?

Dimer model TL Integrability Conclusion

Some references

Dimer transfer matrix

Lieb (1967); Rasmussen, Ruelle (2012); Brankov, Poghosyan, Priezzhev, Ruelle (2014)

Temperley-Lieb algebra

Temperley, Lieb (1971); Pasquier-Saleur (1989); Martin, Saleur (1993); Jones (1999); Ridout, Saint-Aubin (2012)

Tilted transfer matrices

Yung, Batchelor (1995); Dubail, Jacobsen, Saleur (2009)

Integrals of motion

Bazhanov, Lukyanov, Zamolodchikov (1996-1997); Nigro (2009)

Critical dense polymers

Pearce, Rasmussen (2007, . . . ); AMD, Pearce, Rasmussen (2013)

In preparation

• AMD, Rasmussen, Ruelle, Dimer representation of the TL algebra
• AMD, Rasmussen, Ruelle, Integrability and conformal data of the dimer model