Ising model and Grassmann algebra: What can possibly be new on the Ising model · 2012. 8. 17. ·...

Outline

Ising model and Grassmann algebra:What can possibly be new on the Ising model ?

Maxime Clusel1 Jean-Yves Fortin2

1Institut Laue-Langevin

2Universite Louis Pasteur/CNRS, StrasbourgLaboratoire de Physique Theorique

andLaboratoire Poncelet, Independent University of Moscow

23/06/2006

M. Clusel and J.-Y. Fortin Ising model and Grassmann Algebra

Outline

Outline

1 Ising modelIntroductionKnown resultsSome usual methods

2 Grassmann MethodGrassmann AlgebraOutline of the method (V.N. Plechko, 1985)2D Ising model in zero field

3 Ising model with a general boundary fieldProblemGeneral solutionHomogeneous boundary fieldApplication: wetting transition


IntroductionGrassmann Method

Ising model with a general boundary field

IntroductionKnown resultsSome usual methods





sOutline








Short history

E. Ising

(1900-1998)

Some dates1924: Ising’s thesis→ 1D casesolved1944: Onsager’s work→ exactsolution for the 2D case,magnetization1967-68: McCoy and Wu→ solutionwith homogeneous boundary field inthe 2D case1980’s: D. B. Abraham’s work onboundary field effects1990’s: conformal field theoryapplied to boundary perturbations





Solving the 2D Ising model

The questionGiven the Hamiltonian, we would like to know how to obtain thesurfacial free energy for a given configuration of boundary field

In practice :

H = −J∑〈i,j〉

σiσj

+∑

i∈border

hiσi︸︷︷︸surface effects impurities...

→ Z =∑{σ}

exp−βH ?





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Known results

1D caseWell known case.

2D casePF, Free energy and magnetisation in zero field (Onsager)PF, Free energy and magnetisation with homogeneousboundary field (McCoy Wu)Results with 2 opposite surface fields (D.B. Abraham)Few results with a bulk magnetic field (Zamolodchikov)

3D caseAlmost nothing...





Known results

1D caseWell known case.

2D casePF, Free energy and magnetisation in zero field (Onsager)PF, Free energy and magnetisation with homogeneousboundary field (McCoy Wu)Results with 2 opposite surface fields (D.B. Abraham)Few results with a bulk magnetic field (Zamolodchikov)

3D caseAlmost nothing...integrability ?





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Some usual methods (1)

Dimer statisticsPrinciple:

1 from Ising to dimer network2 Combinatorics on the dimers: Z = PfaffA3 Eigenvalues of A→ PF

Difficulties1 Mapping the Ising problem to a dimer problem2 Showing that Z is a Paffian3 Computing the Pfaffian





Some usual methods (2)

Transfer matrixPrinciple:

1 Generalisation of the 1D transfer matrix method2 Transfer matrix in terms of Pauli matrices3 Jordan-Wigner transformation: Fermionization4 Determinant calculus→ Z

Difficulties1 Manipulating quantum operators




Grassmann AlgebraOutline of the method (V.N. Plechko, 1985)2D Ising model in zero field

Grassmann Algebra ?

In the previous solutions, hidden Grassmann variables

Where:Dimer approach: PfaffianTransfer matrix: integral representation of fermions

Grassmann variables“Natural” representation of the Ising model

Question:Can we introduce directly the Grassmann algebra ?





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Grassmann algebra

DefinitionA Grassmann algebra over R or C is an associative algebraconstructed from an unit 1 and a set of generators {ai} withanti-commuting products:

∀i , j aiaj = −ajai

Consequences

a2i = 0

All the functions are finite degree polynomials !





Integration on Grassmann algebra

Definition

Derivation: If A = A1 + aiA2 then ∂A∂ai

= A2.Integration (Berezin):

∀A ∈ A,∫

daiA =∂A∂ai

.

Gaussian integrals∫ [ n∏i=1

daida∗i

]exp

(tA∗MA)

= det M.

∫ [ 2n∏i=1

dai

]exp

(tAMA)

= Pfaff M.





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Partition function (without field)

t ≡ tanh(βJ), Z ∝∑{σ}

L∏m,n=1

(1 + tσmnσmn+1)(1 + tσmnσm+1n)

Grassmann representation: “Fermionization”

1 + tσσ′ =∫

da∗da (1 + aσ)(1 + ta∗σ′)︸︷︷︸uncoupled spins

eaa∗

Strategy

∑{σ}

fermionization−−−−−−−−→∑{σ,a,a∗}

Trace on spins−−−−−−−−−−−−−→Grassmann calculus

∑{a,a∗}

Integral−−−−→ PF





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2D Ising model in zero field: Fermionization

Definition

Amn = 1 + amnσmn, A∗m+1n = 1 + ta∗mnσm+1n,

Bmn = 1 + bmnσmn, B∗mn+1 = 1 + tb∗mnσmn+1,

Mixed representation of the PF

Z ∝∑{σ}

L∏m,n=1






2D Ising model in zero field: Fermionization

Definition

Amn = 1 + amnσmn, A∗m+1n = 1 + ta∗mnσm+1n,

Bmn = 1 + bmnσmn, B∗mn+1 = 1 + tb∗mnσmn+1,

Mixed representation of the PF

Z ∝∑{σ}

L∏m,n=1


⇒ Z ∝ Tr{σ,a,b}

−−→L∏

m=1

−→L∏

n=1

[AmnA∗m+1nBmnB∗mn+1

]M. Clusel and J.-Y. Fortin Ising model and Grassmann Algebra




2D Ising model in zero field: Grassmann calculus

Fundamental operations1 Associativity:

(O0O∗1)(O1O∗2)(O2O∗3) = O0(O∗1O1)(O∗2O2)O∗3

2 Mirror ordering:

(O1O∗1)(O2O∗2)(O3O∗3) = O1O2O3O∗3O∗2O∗1

Final result (boundary terms are discarded)

Z ∼ Tr{σ,a,b}

−→L∏

n=1

−−→L∏m=1

(A∗mnB∗mnAmn)

←−−L∏

m=1

Bmn

.M. Clusel and J.-Y. Fortin Ising model and Grassmann Algebra




2D Ising model in zero field: Trace over the spins

Z ∼ Tr{σ,a,b}

−→L∏

n=1

−−→L−1∏m=1

(A∗mnB∗mnAmn) (A∗LnB∗LnALnBLn)︸︷︷︸Same spin σLn!

←−−L−1∏m=1

Bmn

.∑

σmn=±1

A∗mnB∗mnAmnBmn = exp Qmn,

Qmn = amnbmn + t2a∗m−1nb∗mn−1 + t(a∗m−1n + b∗mn−1)(amn + bmn).

Good news:Qmn is quadratic so it commutes with all other terms





2D Ising model in zero field: Final results

Gaussian action:

S =L∑

m,n=1

amna∗mn + bmnb∗mn + amnbmn + t2a∗m−1nb∗mn−1

+t(a∗m−1n + b∗mn−1)(amn + bmn).

Partition function (boundary terms are discarded):

Z2 ∼L∏

p,q=1

[(1 + t2)2 − 2t(1− t2)

(cos

2πpL

+ cos2πq

L

)].




ProblemGeneral solutionHomogeneous boundary fieldApplication: wetting transition

Outline








Problem

Notations

Ising model with amagnetic field hn on theline m = 1Periodic boundarycondition along n:σmL+1 = σm1

Free boundary conditionalong m:σL+1n = σ0n = 0





Hamiltonian and partition function

Hamiltonian

H = −JL∑

m,n=1

(σmnσm+1n + σmnσmn+1)−L∑

n=1

hnσ1n.

Partition function

un ≡ tanh(βJ)

Z ∝ Trσmn

L∏m,n=1

(1 + tσmnσm+1n)(1 + tσmnσmn+1)L∏

n=1

(1 + unσmn)





Outline








Fermionization

Mixed representation

Z ∝ Tr

−→L∏

n=1

B∗1nA1n(1 + unσ1n)

−−→L∏m=2

A∗mnB∗mnAmn ·

←−−L∏

m=2

Bmn

︸︷︷︸

Same as in zero field

B1n





Integration: 1D action

Strategy1 Trace over spins σmn, m 6= 12 Fermionization of the magnetic field→ (Hn,H∗n)

3 Trace over the boundary spins4 Action: S = Sbulk + Sint + Sfield5 Integration over the bulk Grassmann variables

amn,a∗mn,bmn,b∗mn

1D action

Z[h] ∼ Z0

∫dH∗dH exp (S1D)

S1D Gaussian action





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Homogeneous field: Thermodynamic limit

Field free energy: McCoy and Wu

βσfield =−14π

∫ π

−πdθ ln

(1 +

4u2t(1 + cos θ)(1 + t2)(1− 2t cos θ − t2) +

√R(θ)

)

R(θ) =[(1 + t2)2 + 2t(1− t2)(1− cos θ)

]×[

(1 + t2)2 − 2t(1− t2)(1 + cos θ)]

Boundary magnetisation: McCoy and Wu

m ∝ (t − tc)1/2 (u = 0), m ∝ −u ln u (t = tc)





Specific heat

C(T ) for L = 20





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Wetting transition at zero temperature (1)





Wetting transition at zero temperature (2)

Criterion for boundary spin flip (b)

ζ ≡ Lx

Ly≥ 1

4

(1 +

4Ly

)= ζs, and h ≥ J

(1 +

4Ly

)

Criterion for interface in the bulk (c)

Lx

Ly≤ 1

4

(1 +

4Ly

)= ζs, and h ≥ 4J

Lx

Ly= hs

Question:Can we compute the free energy of this system

and describe this transition ?





Exact results

Partition function

Z(h; k) ∝ Z0 Tr[eS1D

(1− 2u2∑k

m=1∑Ly

n=k+1 HmHn

)],

∝ Z(h; Ly )(

1− 2u2∑km=1

∑Lyn=k+1 〈HmHn〉S1D

).

Free interfacial energy

lnZ(h; k = Ly/2)⇒ −βσint

Exact expression for any T ,h,Lx ,Ly .Expression





Specific heat for finite system

Specific heat

Specific heat of the interface at ζ = 0.2 < ζsfor Lx = 40 and Ly = 200





Asymptotic analysis

Dirac like sum

S[F ] ≡ 2Ly

Ly/2−1∑q=0

(−1)q cot(θq+ 12/2)F (cos(θq+ 1

2))

Property of the sum

= F (1)− Cy exp(−AyLy ) + . . . , with θq =2πqLy

Cy and Ay depend on t and u.For Lx � 1 F (1) = 1− Cx exp(−AxLx), Ax > 0.





Interface stability

Asymptotic form of the interface free energy

−βσint ' ln(Cx exp(−AxLx) + Cy exp(−AyLy )

)Transition line: first order

If AxLx > AyLy , σint ∝ Ly : interface on the boundaryIf AxLx < AyLy , σint ∝ Lx : interface in the bulkIf Axζ = Ay ⇒ wetting transition T = Tw(h).





Transition line equation

Line equation

Transition line = quadratic polynomial in u2:

2t(1+v(4ζ))u4+(1+ t2)(1−2tv(4ζ)− t2)u2+2(v(4ζ)−1)t3 = 0

with

v(4ζ) = cosh[4ζ ln

(1− t

t(1 + t)

)]

Criterion of ζReal solutions for ζ 6 ζs = 1/4





Phase diagram

Phase diagram for the system at ζ = 0.2 < ζs = 1/4

First order transition ended by a critical point in zero fieldSimilar to the liquid/gas transition (βb = 1/2).





Crossover and correlation functions

Boundary correlations: 〈σ10σ1r 〉

As function of r/Ly and ζ, at T = 2, h = 0.1 and Ly = 100.Crossover 2D 1D behaviour at ζ ' 1/4





Conclusion:

Grassmann algebra deeply related to Ising modelAlternative method to solve Ising modelExtension to interface problems with inhomogeneousmagnetic fieldLimitations: operator ordering not always possible (bulkmagnetic field)

PerspectiveExact study of wetting problems induced by otherconfigurations ?Extension with 2 lines of magnetic field ?Random boundary magnetic field: link with randommatrices ?


For Further Reading

B. McCoy and T.T. Wu.The 2D Ising model.Harvard University Press, 1973.

V.N. PlechkoJ.Phys.Studies, 3(3):312-330, 1999.

Ming-Chya Wu, Chin-Kun HuJ. Phys. A: Math. Gen., 35:5189-5206, 2002.

M. Clusel and J.-Y. FortinJ. Phys.A: Math.Gen , 38, 2849, 2005.

M. Clusel and J.-Y. FortinJ. Phys.A: Math.Gen , 39, 995, 2006.


Expression of σint

−βσint = ln

1− 2Ly

Ly/2−1∑q=0

(−1)q cot(θq+ 12/2)F (θq+ 1

2)

θq =

2πqLy

, and F (x) = 4tu2G(x)/

(14[1−(1+t2)(t2+2tx−1)G(x)]2+2tu2(1+x)G(x)+4t4(1−x2)G(x)2)

G(x) =1Lx

Lx−1∑p=0

1(1 + t2)2 − 2t(1− t2)[cos(2πp

Lx) + x ]

Back


Property of the sum

S[1] =2Ly

Ly/2−1∑q=0

(−1)q cot[π

Ly(q +

12)

]= 1

∀ Ly even

Back


Ising model and Grassmann algebra: What can possibly be new on the Ising model · 2012. 8. 17. ·...

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