Outline
Ising model and Grassmann algebra:
What can possibly be new on the Ising model ?
Maxime Clusel1
1 Institut
Jean-Yves Fortin2
Laue-Langevin
2 Université
Louis Pasteur/CNRS, Strasbourg
Laboratoire de Physique Théorique
and
Laboratoire Poncelet, Independent University of Moscow
23/06/2006
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Outline
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Outline
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Outline
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
M. Clusel and J.-Y. Fortin
Introduction
Known results
Some usual methods
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
s
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Short history
E. Ising
Some dates
1924: Ising’s thesis → 1D case
solved
1944: Onsager’s work → exact
solution for the 2D case,
magnetization
1967-68: McCoy and Wu → solution
with homogeneous boundary field in
the 2D case
1980’s: D. B. Abraham’s work on
boundary field effects
(1900-1998)
1990’s: conformal field theory
applied to boundary perturbations
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Solving the 2D Ising model
The question
Given the Hamiltonian, we would like to know how to obtain the
surfacial free energy for a given configuration of boundary field
In practice :




H = −J
σi σj +

hi,ji


X
X
hi σi
i∈border
|
{z
}


X

exp −βH ?
→ Z=

{σ}

surface effects impurities...
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Known results
1D case
Well known case.
2D case
PF, Free energy and magnetisation in zero field (Onsager)
PF, Free energy and magnetisation with homogeneous
boundary field (McCoy Wu)
Results with 2 opposite surface fields (D.B. Abraham)
Few results with a bulk magnetic field (Zamolodchikov)
3D case
Almost nothing...
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Known results
1D case
Well known case.
2D case
PF, Free energy and magnetisation in zero field (Onsager)
PF, Free energy and magnetisation with homogeneous
boundary field (McCoy Wu)
Results with 2 opposite surface fields (D.B. Abraham)
Few results with a bulk magnetic field (Zamolodchikov)
3D case
Almost nothing...
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Known results
1D case
Well known case.
2D case
PF, Free energy and magnetisation in zero field (Onsager)
PF, Free energy and magnetisation with homogeneous
boundary field (McCoy Wu)
Results with 2 opposite surface fields (D.B. Abraham)
Few results with a bulk magnetic field (Zamolodchikov)
3D case
Almost nothing...
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Known results
1D case
Well known case.
2D case
PF, Free energy and magnetisation in zero field (Onsager)
PF, Free energy and magnetisation with homogeneous
boundary field (McCoy Wu)
Results with 2 opposite surface fields (D.B. Abraham)
Few results with a bulk magnetic field (Zamolodchikov)
3D case
Almost nothing...integrability ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Some usual methods (1)
Dimer statistics
Principle:
1
from Ising to dimer network
2
Combinatorics on the dimers: Z = PfaffA
3
Eigenvalues of A → PF
Difficulties
1
Mapping the Ising problem to a dimer problem
2
Showing that Z is a Paffian
3
Computing the Pfaffian
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Some usual methods (1)
Dimer statistics
Principle:
1
from Ising to dimer network
2
Combinatorics on the dimers: Z = PfaffA
3
Eigenvalues of A → PF
Difficulties
1
Mapping the Ising problem to a dimer problem
2
Showing that Z is a Paffian
3
Computing the Pfaffian
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Some usual methods (2)
Transfer matrix
Principle:
1
Generalisation of the 1D transfer matrix method
2
Transfer matrix in terms of Pauli matrices
3
Jordan-Wigner transformation: Fermionization
4
Determinant calculus → Z
Difficulties
1
Manipulating quantum operators
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Introduction
Known results
Some usual methods
Some usual methods (2)
Transfer matrix
Principle:
1
Generalisation of the 1D transfer matrix method
2
Transfer matrix in terms of Pauli matrices
3
Jordan-Wigner transformation: Fermionization
4
Determinant calculus → Z
Difficulties
1
Manipulating quantum operators
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline 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: Pfaffian
Transfer matrix: integral representation of fermions
Grassmann variables
“Natural” representation of the Ising model
Question:
Can we introduce directly the Grassmann algebra ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline 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: Pfaffian
Transfer matrix: integral representation of fermions
Grassmann variables
“Natural” representation of the Ising model
Question:
Can we introduce directly the Grassmann algebra ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline 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: Pfaffian
Transfer matrix: integral representation of fermions
Grassmann variables
“Natural” representation of the Ising model
Question:
Can we introduce directly the Grassmann algebra ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Grassmann algebra
Definition
A Grassmann algebra over R or C is an associative algebra
constructed from an unit 1 and a set of generators {ai } with
anti-commuting products:
∀i, j ai aj = −aj ai
Consequences
ai2 = 0
All the functions are finite degree polynomials !
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Integration on Grassmann algebra
Definition
∂A
= A2 .
Derivation: If A = A1 + ai A2 then ∂a
i
Integration (Berezin):
Z
∂A
∀A ∈ A,
dai A =
.
∂ai
Gaussian integrals
Z "Y
n
#
dai dai∗
exp t A∗ MA = det M.
i=1
Z "Y
2n
#
dai exp t AMA = Pfaff M.
i=1
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Integration on Grassmann algebra
Definition
∂A
= A2 .
Derivation: If A = A1 + ai A2 then ∂a
i
Integration (Berezin):
Z
∂A
∀A ∈ A,
dai A =
.
∂ai
Gaussian integrals
Z "Y
n
#
dai dai∗
exp t A∗ MA = det M.
i=1
Z "Y
2n
#
dai exp t AMA = Pfaff M.
i=1
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Partition function (without field)
t ≡ tanh(βJ), Z ∝
L
X Y
(1 + tσmn σmn+1 )(1 + tσmn σm+1n )
{σ} m,n=1
Grassmann representation: “Fermionization”
Z
∗
1 + tσσ 0 = da∗ da (1 + aσ)(1 + ta∗ σ 0 ) eaa
{z
}
|
uncoupled spins
Strategy
X
{σ}
fermionization
−−−−−−−−→
X
{σ,a,a∗ }
Trace on spins
−−−−−−−−−−−−−→
Grassmann calculus
M. Clusel and J.-Y. Fortin
X
Integral
−−−−→ PF
{a,a∗ }
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Partition function (without field)
t ≡ tanh(βJ), Z ∝
L
X Y
(1 + tσmn σmn+1 )(1 + tσmn σm+1n )
{σ} m,n=1
Grassmann representation: “Fermionization”
Z
∗
1 + tσσ 0 = da∗ da (1 + aσ)(1 + ta∗ σ 0 ) eaa
{z
}
|
uncoupled spins
Strategy
X
{σ}
fermionization
−−−−−−−−→
X
{σ,a,a∗ }
Trace on spins
−−−−−−−−−−−−−→
Grassmann calculus
M. Clusel and J.-Y. Fortin
X
Integral
−−−−→ PF
{a,a∗ }
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Partition function (without field)
t ≡ tanh(βJ), Z ∝
L
X Y
(1 + tσmn σmn+1 )(1 + tσmn σm+1n )
{σ} m,n=1
Grassmann representation: “Fermionization”
Z
∗
1 + tσσ 0 = da∗ da (1 + aσ)(1 + ta∗ σ 0 ) eaa
{z
}
|
uncoupled spins
Strategy
X
{σ}
fermionization
−−−−−−−−→
X
{σ,a,a∗ }
Trace on spins
−−−−−−−−−−−−−→
Grassmann calculus
M. Clusel and J.-Y. Fortin
X
Integral
−−−−→ PF
{a,a∗ }
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
2D Ising model in zero field: Fermionization
Definition
∗
Amn = 1 + amn σmn , A∗m+1n = 1 + tamn
σm+1n ,
∗
∗
Bmn = 1 + bmn σmn , Bmn+1
= 1 + tbmn
σmn+1 ,
Mixed representation of the PF
Z∝
L
X Y
(1 + tσmn σmn+1 )(1 + tσmn σm+1n )
{σ} m,n=1
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
2D Ising model in zero field: Fermionization
Definition
∗
Amn = 1 + amn σmn , A∗m+1n = 1 + tamn
σm+1n ,
∗
∗
Bmn = 1 + bmn σmn , Bmn+1
= 1 + tbmn
σmn+1 ,
Mixed representation of the PF
Z∝
L
X Y
(1 + tσmn σmn+1 )(1 + tσmn σm+1n )
{σ} m,n=1
−−→−→
L Y
L
Y
∗
⇒ Z ∝ Tr
Amn A∗m+1n Bmn Bmn+1
{σ,a,b}
m=1n=1
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
2D Ising model in zero field: Grassmann calculus
Fundamental operations
1
Associativity:
(O0 O1∗ )(O1 O2∗ )(O2 O3∗ ) = O0 (O1∗ O1 )(O2∗ O2 )O3∗
2
Mirror ordering:
(O1 O1∗ )(O2 O2∗ )(O3 O3∗ ) = O1 O2 O3 O3∗ O2∗ O1∗
Final result (boundary terms are discarded)

−→ −−→
←−−
L
L
L
Y
Y
Y
∗

Z ∼ Tr
(A∗mn Bmn
Amn )
Bmn  .
{σ,a,b}
n=1
m=1
M. Clusel and J.-Y. Fortin
m=1
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
2D Ising model in zero field: Trace over the spins


−→ −−→
←−−
L−1
L
L−1
Y Y
Y

∗
∗
∗
∗

.
(A
B
A
)
(A
B
A
B
)
Z ∼ Tr
B
mn
mn
Ln
Ln
mn
mn
Ln
Ln


|
{z
}
{σ,a,b}
n=1 m=1
m=1
Same spin σLn !
X
∗
A∗mn Bmn
Amn Bmn = exp Qmn ,
σmn =±1
∗
∗
∗
∗
Qmn = amn bmn + t 2 am−1n
bmn−1
+ t(am−1n
+ bmn−1
)(amn + bmn ).
Good news:
Qmn is quadratic so it commutes with all other terms
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
2D Ising model in zero field: Final results
Gaussian action:
S=
L
X
∗
∗
∗
∗
amn amn
+ bmn bmn
+ amn bmn + t 2 am−1n
bmn−1
m,n=1
∗
∗
+t(am−1n
+ bmn−1
)(amn + bmn ).
Partition function (boundary terms are discarded):
2
Z ∼
L Y
p,q=1
2πp
2πq
(1 + t ) − 2t(1 − t ) cos
+ cos
.
L
L
2 2
M. Clusel and J.-Y. Fortin
2
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Problem
Notations
Ising model with a
magnetic field hn on the
line m = 1
Periodic boundary
condition along n:
σmL+1 = σm1
Free boundary condition
along m:
σL+1n = σ0n = 0
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Hamiltonian and partition function
Hamiltonian
L
X
H = −J
(σmn σm+1n + σmn σmn+1 ) −
m,n=1
L
X
hn σ1n .
n=1
Partition function
un ≡ tanh(βJ)

Z ∝ Tr 
σmn
L
Y
(1 + tσmn σm+1n )(1 + tσmn σmn+1 )
m,n=1
L
Y

(1 + un σmn )
n=1
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Introduction
Grassmann Method
Ising model with a general boundary field
Fermionization
Mixed representation
Z ∝ Tr
−→
L
Y
−−→

←−−
L
L
Y
Y
∗
∗
B1n
A1n (1 + un σ1n ) 
A∗mn Bmn
Amn ·
Bmn  B1n
n=1
m=2
|
M. Clusel and J.-Y. Fortin
m=2
{z
Same as in zero field
Ising model and Grassmann Algebra
}
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Integration: 1D action
Strategy
1
Trace over spins σmn , m 6= 1
2
Fermionization of the magnetic field → (Hn , Hn∗ )
3
Trace over the boundary spins
4
Action: S = Sbulk + Sint + Sfield
5
Integration over the bulk Grassmann variables
∗ , b , b∗
amn , amn
mn mn
1D action
Z
Z[h] ∼ Z0
dH ∗ dH exp (S1D )
S1D Gaussian action
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Homogeneous field: Thermodynamic limit
Field free energy: McCoy and Wu
βσfield
−1
=
4π
Z
π
dθ ln 1 +
−π
!
4u 2 t(1 + cos θ)
(1 + t 2 )(1 − 2t cos θ − t 2 ) +
h
i
R(θ) = (1 + t 2 )2 + 2t(1 − t 2 )(1 − cos θ) ×
h
i
(1 + t 2 )2 − 2t(1 − t 2 )(1 + cos θ)
Boundary magnetisation: McCoy and Wu
m ∝ (t − tc )1/2 (u = 0), m ∝ −u ln u (t = tc )
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
p
R(θ)
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Specific heat
C(T ) for L = 20
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Outline
1
Ising model
Introduction
Known results
Some usual methods
2
Grassmann Method
Grassmann Algebra
Outline of the method (V.N. Plechko, 1985)
2D Ising model in zero field
3
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Wetting transition at zero temperature (1)
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Wetting transition at zero temperature (2)
Criterion for boundary spin flip (b)
Lx
1
4
4
ζ≡
≥
1+
= ζs , and h ≥ J 1 +
Ly
4
Ly
Ly
Criterion for interface in the bulk (c)
Lx
1
4
Lx
≤
1+
= ζs , and h ≥ 4J
= hs
Ly
4
Ly
Ly
Question:
Can we compute the free energy of this system
and describe this transition ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Wetting transition at zero temperature (2)
Criterion for boundary spin flip (b)
Lx
1
4
4
ζ≡
≥
1+
= ζs , and h ≥ J 1 +
Ly
4
Ly
Ly
Criterion for interface in the bulk (c)
Lx
1
4
Lx
≤
1+
= ζs , and h ≥ 4J
= hs
Ly
4
Ly
Ly
Question:
Can we compute the free energy of this system
and describe this transition ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Wetting transition at zero temperature (2)
Criterion for boundary spin flip (b)
Lx
1
4
4
ζ≡
≥
1+
= ζs , and h ≥ J 1 +
Ly
4
Ly
Ly
Criterion for interface in the bulk (c)
Lx
1
4
Lx
≤
1+
= ζs , and h ≥ 4J
= hs
Ly
4
Ly
Ly
Question:
Can we compute the free energy of this system
and describe this transition ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Exact results
Partition function
i
h
PLy
P
Z(h; k) ∝ Z0 Tr eS1D 1 − 2u 2 km=1 n=k+1
Hm Hn ,
P
PLy
∝ Z(h; Ly ) 1 − 2u 2 km=1 n=k
hH
H
i
m n S1D .
+1
Free interfacial energy
ln Z(h; k = Ly /2) ⇒ −βσint
Exact expression for any T , h, Lx , Ly .
Expression
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Exact results
Partition function
i
h
PLy
P
Z(h; k) ∝ Z0 Tr eS1D 1 − 2u 2 km=1 n=k+1
Hm Hn ,
P
PLy
∝ Z(h; Ly ) 1 − 2u 2 km=1 n=k
hH
H
i
m n S1D .
+1
Free interfacial energy
ln Z(h; k = Ly /2) ⇒ −βσint
Exact expression for any T , h, Lx , Ly .
Expression
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Specific heat for finite system
Specific heat
Specific heat of the interface at ζ = 0.2 < ζs
for Lx = 40 and Ly = 200
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Asymptotic analysis
Dirac like sum
S[F ] ≡
Ly /2−1
2 X
(−1)q cot(θq+ 1 /2)F (cos(θq+ 1 ))
2
2
Ly
q=0
Property of the sum
= F (1) − Cy exp(−Ay Ly ) + . . . , with θq =
Cy and Ay depend on t and u.
For Lx 1 F (1) = 1 − Cx exp(−Ax Lx ), Ax > 0.
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
2πq
Ly
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Asymptotic analysis
Dirac like sum
S[F ] ≡
Ly /2−1
2 X
(−1)q cot(θq+ 1 /2)F (cos(θq+ 1 ))
2
2
Ly
q=0
Property of the sum
= F (1) − Cy exp(−Ay Ly ) + . . . , with θq =
Cy and Ay depend on t and u.
For Lx 1 F (1) = 1 − Cx exp(−Ax Lx ), Ax > 0.
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
2πq
Ly
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Interface stability
Asymptotic form of the interface free energy
−βσint ' ln Cx exp(−Ax Lx ) + Cy exp(−Ay Ly )
Transition line: first order
If Ax Lx > Ay Ly , σint ∝ Ly : interface on the boundary
If Ax Lx < Ay Ly , σint ∝ Lx : interface in the bulk
If Ax ζ = Ay ⇒ wetting transition T = Tw (h).
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Interface stability
Asymptotic form of the interface free energy
−βσint ' ln Cx exp(−Ax Lx ) + Cy exp(−Ay Ly )
Transition line: first order
If Ax Lx > Ay Ly , σint ∝ Ly : interface on the boundary
If Ax Lx < Ay Ly , σint ∝ Lx : interface in the bulk
If Ax ζ = Ay ⇒ wetting transition T = Tw (h).
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Transition line equation
Line equation
Transition line = quadratic polynomial in u 2 :
2t(1+v (4ζ))u 4 +(1+t 2 )(1−2tv (4ζ)−t 2 )u 2 +2(v (4ζ)−1)t 3 = 0
with
v (4ζ) = cosh 4ζ ln
1−t
t(1 + t)
Criterion of ζ
Real solutions for ζ 6 ζs = 1/4
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Phase diagram
Phase diagram for the system at ζ = 0.2 < ζs = 1/4
First order transition ended by a critical point in zero field
Similar to the liquid/gas transition (βb = 1/2).
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Crossover and correlation functions
Boundary correlations: hσ10 σ1r i
As function of r /Ly and ζ, at T = 2, h = 0.1 and Ly = 100.
Crossover 2D
1D behaviour at ζ ' 1/4
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Conclusion:
Grassmann algebra deeply related to Ising model
Alternative method to solve Ising model
Extension to interface problems with inhomogeneous
magnetic field
Limitations: operator ordering not always possible (bulk
magnetic field)
Perspective
Exact study of wetting problems induced by other
configurations ?
Extension with 2 lines of magnetic field ?
Random boundary magnetic field: link with random
matrices ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Introduction
Grassmann Method
Ising model with a general boundary field
Problem
General solution
Homogeneous boundary field
Application: wetting transition
Conclusion:
Grassmann algebra deeply related to Ising model
Alternative method to solve Ising model
Extension to interface problems with inhomogeneous
magnetic field
Limitations: operator ordering not always possible (bulk
magnetic field)
Perspective
Exact study of wetting problems induced by other
configurations ?
Extension with 2 lines of magnetic field ?
Random boundary magnetic field: link with random
matrices ?
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
For Further Reading
B. McCoy and T.T. Wu.
The 2D Ising model.
Harvard University Press, 1973.
V.N. Plechko
J.Phys.Studies, 3(3):312-330, 1999.
Ming-Chya Wu, Chin-Kun Hu
J. Phys. A: Math. Gen., 35:5189-5206, 2002.
M. Clusel and J.-Y. Fortin
J. Phys.A: Math.Gen , 38, 2849, 2005.
M. Clusel and J.-Y. Fortin
J. Phys.A: Math.Gen , 39, 995, 2006.
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Expression of σint

−βσint = ln 1 −
θq =
2
Ly

Ly /2−1
X
q=0
(−1)q cot(θq+ 1 /2)F (θq+ 1 )
2
2
2πq
, and F (x) = 4tu 2 G(x)/
Ly
1
( [1−(1+t 2 )(t 2 +2tx−1)G(x)]2 +2tu 2 (1+x)G(x)+4t 4 (1−x 2 )G(x)2 )
4
Lx −1
1 X
1
G(x) =
2πp
2
2
2
Lx
p=0 (1 + t ) − 2t(1 − t )[cos( Lx ) + x]
Back
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra
Property of the sum
Ly /2−1
2 X
π
1
q
S[1] =
(−1) cot
(q + ) = 1
Ly
Ly
2
q=0
∀ Ly even
Back
M. Clusel and J.-Y. Fortin
Ising model and Grassmann Algebra