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

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

Outline

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 ?

Outline

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

3D caseAlmost nothing...integrability ?

Outline

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

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

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

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 ?

Grassmann Algebra ?

Outline

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

(tA∗MA)

= det M.

∫ [ 2n∏i=1

(tAMA)

= Pfaff M.

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

(tA∗MA)

= det M.

∫ [ 2n∏i=1

(tAMA)

= Pfaff M.

Outline

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

L∏m,n=1

1 + tσσ′ =∫

eaa∗

Strategy

∑{σ}

∑{a,a∗}

L∏m,n=1

1 + tσσ′ =∫

eaa∗

Strategy

∑{σ}

∑{a,a∗}

Outline

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∏

−→L∏

[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∏

−−→L∏m=1

(A∗mnB∗mnAmn)

←−−L∏

.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∏

−−→L−1∏m=1

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

←−−L−1∏m=1

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∑

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∏

[(1 + t2)2 − 2t(1− t2)

+ cos2πq

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∑

(σmnσm+1n + σmnσmn+1)−L∑

hnσ1n.

Partition function

un ≡ tanh(βJ)

Z ∝ Trσmn

L∏m,n=1

(1 + tσmnσm+1n)(1 + tσmnσmn+1)L∏

(1 + unσmn)

Outline

Fermionization

Mixed representation

Z ∝ Tr

−→L∏

B∗1nA1n(1 + unσ1n)

−−→L∏m=2

A∗mnB∗mnAmn ·

←−−L∏

︸︷︷︸

Same as in zero field

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

1D action

Z[h] ∼ Z0

S1D Gaussian action

1D action

Z[h] ∼ Z0

S1D Gaussian action

1D action

Z[h] ∼ Z0

S1D Gaussian action

1D action

Z[h] ∼ Z0

S1D Gaussian action

1D action

Z[h] ∼ Z0

S1D Gaussian action

Outline

Homogeneous field: Thermodynamic limit

Field free energy: McCoy and Wu

βσfield =−14π

∫ π

−πdθ ln

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

Outline

Wetting transition at zero temperature (1)

Criterion for boundary spin flip (b)

ζ ≡ Lx

Ly≥ 1

)= ζs, and h ≥ J

Criterion for interface in the bulk (c)

Ly≤ 1

)= ζs, and h ≥ 4J

Ly= hs

Question:Can we compute the free energy of this system

and describe this transition ?

ζ ≡ Lx

Ly≥ 1

)= ζs, and h ≥ J

Ly≤ 1

Ly= hs

ζ ≡ Lx

Ly≥ 1

)= ζs, and h ≥ J

Ly≤ 1

Ly= hs

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

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

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.

Asymptotic analysis

Dirac like sum

S[F ] ≡ 2Ly

Ly/2−1∑q=0

(−1)q cot(θq+ 12/2)F (cos(θq+ 1

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).

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

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 ?

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

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 ]

Property of the sum

S[1] =2Ly

Ly/2−1∑q=0

(−1)q cot[π

Ly(q +

∀ Ly even

ising model and grassmann algebra: what can possibly be new on the ising model · 2012. 8. 17. ·...

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