Jeanne Colbois | TMC

Jeanne Colbois | TMC

Acknowledgement for previous work : 

the kasteleyn Phase transition

the kasteleyn Phase transition

AKA the Pokrovsky-Talapov transition

the kasteleyn Phase transition

AKA the Pokrovsky-Talapov transition

Constrained models

Incommensurate systems

the kasteleyn Phase transition

2nd order phase transition: 

\(T\)

\(T\)

\(C_V\)

\(m\) (order parameter)

the kasteleyn Phase transition

2nd order phase transition: 

\(T\)

\(T\)

\(T\)

\(T\)

\(C_V\)

\(m\) (order parameter)

"disorder" parameter

\(C_V\)

Kasteleyn / Pokrovsky-Talapov:

the kasteleyn Phase transition

2nd order phase transition: 

Kasteleyn / Pokrovsky-Talapov:

\(T\)

\(T\)

\(T\)

\(T\)

\(C_V\)

\(m\) (order parameter)

"disorder" parameter

\(C_V\)

"Mixed order"

the kasteleyn Phase transition

2nd order phase transition: 

Kasteleyn / Pokrovsky-Talapov:

\(T\)

\(T\)

\(T\)

\(T\)

\(C_V\)

\(m\) (order parameter)

"disorder" parameter

\(C_V\)

"Mixed order"

Fluctuationless at LOw-T

the kasteleyn Phase transition

2nd order phase transition: 

Kasteleyn / Pokrovsky-Talapov:

\(T\)

\(T\)

\(T\)

\(T\)

\(C_V\)

\(m\) (order parameter)

"disorder" parameter

\(C_V\)

"Mixed order"

Fluctuationless at LOw-T

Square-root singularity

sCOPE

sCOPE

1. The Kasteleyn mechanism

sCOPE

1. The Kasteleyn mechanism

2. Examples in frustrated magnetism

sCOPE

1. The Kasteleyn mechanism

2. Examples in frustrated magnetism

3. Staircases in Ising models

sCOPE

1. The Kasteleyn mechanism

2. Examples in frustrated magnetism

3. Staircases in Ising models

4. A Kasteleyn-driven topological staircase

Kasteleyn mechanism : IDEA

Ground state of some 2D classical

constrained model

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Kasteleyn mechanism : IDEA

DEFECTS /EXCITATIONS:

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Kasteleyn mechanism : IDEA

DEFECTS /EXCITATIONS:

• Directed

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Kasteleyn mechanism : IDEA

DEFECTS /EXCITATIONS:

• Directed, non-crossing

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Kasteleyn mechanism : IDEA

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Kasteleyn mechanism : IDEA

\(E \propto L \)

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Kasteleyn mechanism : IDEA

\(E \propto L \)

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

\(E \propto L \)

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

\(E \propto L \)

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

\(E \propto L \)

\(F = E - TS\)

\(S \propto L \)

\(T\)

No defects

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

\(E \propto L \)

\(F = E - TS\)

\(S \propto L \)

\(T\)

No defects

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Kasteleyn mechanism : IDEA

\(E \propto L \)

\(F = E - TS\)

\(S \propto L \)

\(T\)

No defects

Strings

condense

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

\(E \propto L \)

\(F = E - TS\)

\(S \propto L \)

\(T\)

No defects

Strings

condense

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

Kasteleyn mechanism : IDEA

\(E \propto L \)

\(F = E - TS\)

\(S \propto L \)

\(T\)

No defects

Strings

condense

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

DEFECTS /EXCITATIONS:

• Directed, non-crossing, non-terminating

Energy cost : linear in system size

Entropy gain!

An example : the dimer model 

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Adsorption of diatomic molecules (dimers) on crystal surfaces

An example : the dimer model 

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Adsorption of diatomic molecules (dimers) on crystal surfaces

1

2

3

An example : the dimer model 

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Adsorption of diatomic molecules (dimers) on crystal surfaces

Hardcore (close-packed) dimers

1

2

3

An example : the dimer model 

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Adsorption of diatomic molecules (dimers) on crystal surfaces

Hardcore (close-packed) dimers

1

2

3

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

• Directed, non-crossing, non-terminating

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

• Directed, non-crossing, non-terminating

• Linear energy cost

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

• Directed, non-crossing, non-terminating

• Linear energy cost

• Entropic gain

Ground state v.s. Excitations

J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13  (1989)

Constraint : hardcore

• Directed, non-crossing, non-terminating

• Linear energy cost

• Entropic gain

Disorder parameter : density of strings

Directed, non-crossing, non-terminating

Disorder parameter : density of strings

\(T\)

"disorder" parameter

\((T-T_K)^{1/2}\)

Directed, non-crossing, non-terminating

Disorder parameter : density of strings

\(T\)

"disorder" parameter

\((T-T_K)^{1/2}\)

0

in the ground state

1

at most

2/3

in the large \(T\) limit

Directed, non-crossing, non-terminating

Disorder parameter : density of strings

\(T\)

"disorder" parameter

\((T-T_K)^{1/2}\)

0

in the ground state

1

at most

2/3

in the large \(T\) limit

Mapping to free fermions Hamiltonian 

\(T \leftrightarrow\) chemical potential

\(n_{\mathrm{strings}} \leftrightarrow\) fermions density

Directed, non-crossing, non-terminating

Questions so far ? 

Kasteleyn / Pokrovsky-Talapov transition:

entropy-driven condensation of

directed, non-crossing, non-terminating string excitations

in a constrained model.

Questions so far ? 

Kasteleyn / Pokrovsky-Talapov transition:

entropy-driven condensation of

Not discussed here: 

algebraic decay of correlations

fundamental anisotropy in the critical exponents

etc. 

directed, non-crossing, non-terminating string excitations

in a constrained model.

2. Kasteleyn transition in frustrated magnetism

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Ground state

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations : double domain wall

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations : double domain wall

NB: NN model vs anisotropic one

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations : double domain wall

NB: NN model vs anisotropic one

2D : triangular lattice Ising antiferromagnets

Constrained limit \(J \gg T\)

Excitations : double domain wall

NB: NN model vs anisotropic one

3D kasteleyn transition in Spin ice

L, Jaubert et. al. J.Phys.:Conf.Ser. 145, 012024 (2009)

L, Jaubert et. al. PRL 100 (2008)

Spin ice in a [100] field

2-in 2-out ground state

3D kasteleyn transition in Spin ice

L, Jaubert et. al. J.Phys.:Conf.Ser. 145, 012024 (2009)

L, Jaubert et. al. PRL 100 (2008)

Spin ice in a [100] field

2-in 2-out ground state

Magnetization in each (100) plane is the same

3D kasteleyn transition in Spin ice

L, Jaubert et. al. J.Phys.:Conf.Ser. 145, 012024 (2009)

L, Jaubert et. al. PRL 100 (2008)

Spin ice in a [100] field

2-in 2-out ground state

Magnetization in each (100) plane is the same

1. Start from fully polarized state

2 . Lower the field

3D kasteleyn transition in Spin ice

L, Jaubert et. al. J.Phys.:Conf.Ser. 145, 012024 (2009)

L, Jaubert et. al. PRL 100 (2008)

Spin ice in a [100] field

2-in 2-out ground state

Magnetization in each (100) plane is the same

1. Start from fully polarized state

2 . Lower the field

Defects are strings

directed

non-terminating

non-crossing

3. Staircases

ANNNI model devil's staircase

Ferro \(J_1\)

AF \(J_2\) in one direction

In 3D : 

Macroscopic degeneracy of arrangements for successive ferromagnetic layers

CeSb

von Boehm & Bak, PRB (1980)

commensurate wavevectors

von Boehm & Bak, PRB (1980)

Fisher and Selke, PRL (1980)

4. A kasteleyn-driven staircase

Frustrated models on the kagome lattice

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)

Z. Luo et al. Science 363, (2019)

JC et al., PRB 104 (2021)

Frustrated models on the kagome lattice

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)

Z. Luo et al. Science 363, (2019)

JC et al., PRB 104 (2021)

Frustrated models on the kagome lattice

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)

Z. Luo et al. Science 363, (2019)

JC et al., PRB 104 (2021)

Kagome lattice

Frustrated models on the kagome lattice

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)

Z. Luo et al. Science 363, (2019)

JC et al., PRB 104 (2021)

Kagome lattice

3 Kagome sublattices

Frustrated models on the kagome lattice