Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Topological Staircase in a
constrained kagome Ising antiferromagnet
Jeanne Colbois | Théorie de la Matière Condensée | Institut Néel
An application of tensor networks to classical frustrated magnets
Jeanne Colbois | Théorie de la Matière Condensée | Institut Néel
Samuel Nyckees
Afonso Rufino
Frédéric Mila
Topological Staircase in a
constrained kagome Ising antiferromagnet
An application of tensor networks to classical frustrated magnets
Andrew Smerald
KIT | Germany
Frédéric Mila
EPFL | Switzerland
Frank Verstraete
Ghent University | Belgium
Laurens Vanderstraeten
Ghent University | Belgium
Bram Vanhecke
University of Vienna | Austria
COLBOIS | KASTELEYN MECHANISM | 06.2025
COLBOIS | KASTELEYN MECHANISM | 06.2025
Unusual phase transitions Induced by constraints
2
COLBOIS | KASTELEYN MECHANISM | 06.2025
Unusual phase transitions Induced by constraints
2
\(T\)
\(C_V\)
\(T\)
\(m\) (order parameter)
Usual : 2nd order
COLBOIS | KASTELEYN MECHANISM | 06.2025
Unusual phase transitions Induced by constraints
2
Constraints / competition
\(T\)
\(C_V\)
\(T\)
\(m\) (order parameter)
\(T\)
"disorder" parameter
\(T\)
\(C_V\)
Usual : 2nd order
Kasteleyn
COLBOIS | KASTELEYN MECHANISM | 06.2025
Unusual phase transitions Induced by constraints
2
Constraints / competition
\(T\)
\(C_V\)
\(T\)
\(m\) (order parameter)
\(T\)
"disorder" parameter
\(T\)
\(C_V\)
\(C_V\)
Usual : 2nd order
\(T\)
wavevector
\(T\)
Kasteleyn
Staircases
COLBOIS | KASTELEYN MECHANISM | 06.2025
Unusual phase transitions Induced by constraints
2
Constraints / competition
\(T\)
\(C_V\)
\(T\)
\(m\) (order parameter)
\(T\)
"disorder" parameter
\(T\)
\(C_V\)
\(C_V\)
Usual : 2nd order
\(T\)
wavevector
\(T\)
Kasteleyn
Staircases
Today : a (new?) possibility induced by frustration
COLBOIS | KASTELEYN MECHANISM | 06.2025
sCOPE
3
1. Frustration, constraints and Kasteleyn
Goldenfeld & Kadanoff, Science, 284 (1999)
COLBOIS | KASTELEYN MECHANISM | 06.2025
sCOPE
3
1. Frustration, constraints and Kasteleyn
2. Tensor networks for classical spin systems
Goldenfeld & Kadanoff, Science, 284 (1999)
COLBOIS | KASTELEYN MECHANISM | 06.2025
sCOPE
3
1. Frustration, constraints and Kasteleyn
2. Tensor networks for classical spin systems
3. A Kasteleyn-driven staircase
Goldenfeld & Kadanoff, Science, 284 (1999)
COLBOIS | KASTELEYN MECHANISM | 06.2025
sCOPE
3
1. Frustration, constraints and Kasteleyn
2. Tensor networks for classical spin systems
3. A Kasteleyn-driven staircase
Goldenfeld & Kadanoff, Science, 284 (1999)
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
2-up 1-down (UUD),
2-down 1-up (DDU)
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
2-up 1-down (UUD),
2-down 1-up (DDU)
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
2-up 1-down (UUD),
2-down 1-up (DDU)
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
2-up 1-down (UUD),
2-down 1-up (DDU)
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
2-up 1-down (UUD),
2-down 1-up (DDU)
Entropy per site
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
S = 0.3230659...
G.H. Wannier, PR 79, (1950, 1973)
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
2-up 1-down (UUD),
2-down 1-up (DDU)
Entropy per site
COLBOIS | KASTELEYN MECHANISM | 06.2025
archetypes of frustration
4
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
S = 0.3230659...
S = 0.501833...
G.H. Wannier, PR 79, (1950, 1973)
K. Kano and S. Naya, Prog. Theor. Phys. 10, (1953)
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
2-up 1-down (UUD),
2-down 1-up (DDU)
\xi = 1.2506...
A. Sütö, Z. Phys. B 44, (1981)
W. Apel, H.-U. Everts, J. Stat. Mech, (2011)
Entropy per site
COLBOIS | KASTELEYN MECHANISM | 06.2025
Frustrated magnetism as constrained problems
5
C. Castlenovo, R. Moessner, S. L. Sondhi, Nature 451 (2008)
Spin ice in magnetic oxides
\(J \gg T\)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Frustrated magnetism as constrained problems
5
C. Castlenovo, R. Moessner, S. L. Sondhi, Nature 451 (2008) (and many others)
Spin ice in magnetic oxides
Divergence-free constraint
Pinched points
Emergent electrodynamics (when violated)
\(J \gg T\)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Frustrated magnetism as constrained problems
5
Divergence-free constraint
Pinched points
Emergent electrodynamics (when violated)
Spin ice in magnetic oxides
Frustrated Ising magnets
\(J \gg T\)
C. Castlenovo, R. Moessner, S. L. Sondhi, Nature 451 (2008) (and many others)
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Frustrated magnetism as constrained problems
5
Divergence-free constraint
Pinched points
Emergent electrodynamics (when violated)
C. Castlenovo, R. Moessner, S. L. Sondhi, Nature 451 (2008)
Spin ice in magnetic oxides
Frustrated Ising magnets
Hardcore dimers
Zero-temperature critical point
\(J \gg T\)
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
6
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
6
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Directed
non-crossing
non-terminating
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
6
ground state: Strings representation
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Directed
non-crossing
non-terminating
Consequences:
- power-law decay of correlations
- height field description
....
see e.g. Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986), Smeral et al (2016, 2019)
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
COLBOIS | KASTELEYN MECHANISM | 06.2025
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
COLBOIS | KASTELEYN MECHANISM | 06.2025
Now the only ground state
COLBOIS | KASTELEYN MECHANISM | 06.2025
Double domain wall : NOW EXCITATIONS
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
Constrained limit \(J \gg T, \delta \)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Double domain wall : NOW EXCITATIONS
Constrained limit \(J \gg T, \delta \)
Directed, non-crossing, non terminating
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
COLBOIS | KASTELEYN MECHANISM | 06.2025
Double domain wall : NOW EXCITATIONS
Directed, non-crossing, non terminating
Linear energy cost
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
Constrained limit \(J \gg T, \delta \)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Double domain wall : NOW EXCITATIONS
Directed, non-crossing, non terminating
Linear energy cost
Entropy gain
Perturbing away from the fine-tuned point
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
7
Constrained limit \(J \gg T, \delta \)
8
COLBOIS | KASTELEYN MECHANISM | 06.2025
Double domain wall : NOW EXCITATIONS
Constrained limit \(J \gg T\)
Directed, non-crossing, non terminating
Linear energy cost
Entropy gain
Kasteleyn transition
Nearest-neighbor anisotropic
Smerald & Mila, Scipost (2019)
COLBOIS | KASTELEYN MECHANISM | 06.2025
Kasteleyn mechanism
9
Ground state of some 2D classical
constrained model
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
\(E \propto L \)
\(F = E - TS\)
\(S \propto L \)
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
\(F = E - TS\)
\(S \propto L \)
\(T\)
No defects
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
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
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
\(F = E - TS\)
\(S \propto L \)
\(T\)
No defects
Strings
condense
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
\(F = E - TS\)
\(S \propto L \)
\(T\)
No defects
Strings
condense
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
\(E \propto L \)
\(F = E - TS\)
\(S \propto L \)
\(T\)
No defects
Strings
condense
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
EXCITATIONS:
- Directed, non-crossing, non-terminating
Energy cost : linear in system size
Entropy gain!
Kasteleyn mechanism
9
COLBOIS | KASTELEYN MECHANISM | 06.2025
10
Directed, non-crossing, non-terminating
Remarks
COLBOIS | KASTELEYN MECHANISM | 06.2025
10
Remarks
\(T\)
"disorder" parameter
\((T-T_K)^{1/2}\)
Directed, non-crossing, non-terminating
AKA the Pokrovsky-Talapov transition
n_{\mathrm{strings}} \propto \sqrt{(T-T_K)}
COLBOIS | KASTELEYN MECHANISM | 06.2025
10
\(T\)
"disorder" parameter
\((T-T_K)^{1/2}\)
Mapping to free fermions Hamiltonian
\(T \leftrightarrow\) chemical potential
\(n_{\mathrm{strings}} \leftrightarrow\) fermions density
Directed, non-crossing, non-terminating
AKA the Pokrovsky-Talapov transition
Remarks
n_{\mathrm{strings}} \propto \sqrt{(T-T_K)}
questions for frustrated magnets
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
questions for frustrated magnets
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
What constraints emerge ?
Can there be macroscopic degeneracy / classical spin liquids beyond fine-tuning ?
Can there be new examples of transitions ?
questions for frustrated magnets
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
What constraints emerge ?
Can there be macroscopic degeneracy / classical spin liquids beyond fine-tuning ?
Can there be new examples of transitions ?
1. Residual entropy
questions for frustrated magnets
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
What constraints emerge ?
Can there be macroscopic degeneracy / classical spin liquids beyond fine-tuning ?
Can there be new examples of transitions ?
1. Residual entropy
2. Correlations / structure factors
questions for frustrated magnets
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
What constraints emerge ?
Can there be macroscopic degeneracy / classical spin liquids beyond fine-tuning ?
Can there be new examples of transitions ?
1. Residual entropy
2. Correlations / structure factors
3. Controlled scaling (finite - size / finite - entanglement)
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
12
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
12
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
Farther neighbor interactions
12
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
Farther neighbor interactions
Exponential (extensive) number of ground states
12
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
Farther neighbor interactions
Exponential (extensive) number of ground states
1. Analytical (exact) methods
Planar Ising models
Fine-tuned points
The rest...
12
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
Farther neighbor interactions
Exponential (extensive) number of ground states
2. Monte Carlo methods
no sign problem (classical)
ergodicity
Planar Ising models
Fine-tuned points
The rest...
12
1. Analytical (exact) methods
COLBOIS | KASTELEYN MECHANISM | 06.2025
The challenge
Constraints / frustration
Farther neighbor interactions
Exponential (extensive) number of ground states
2. Monte Carlo methods
3. TODAY
Tensor networks
no sign problem (classical)
ergodicity
Planar Ising models
Fine-tuned points
The rest...
12
1. Analytical (exact) methods
COLBOIS | KASTELEYN MECHANISM | 06.2025
2. Tensor networks
for classical (frustrated) spin systems
COLBOIS | KASTELEYN MECHANISM | 06.2025
2. Tensor networks
for classical (frustrated) spin systems
Why ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
2. Tensor networks
for classical (frustrated) spin systems
Why ?
How ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
2. Tensor networks
for classical (frustrated) spin systems
Why ?
How ?
What is the catch ?
Why ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
- Tensor networks extremely useful in 1D
Why ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
- Tensor networks extremely useful in 1D
2. 2D classical is "like" 1D quantum
Why ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
- Tensor networks extremely useful in 1D
2. 2D classical is "like" 1D quantum
3. Building block for quantum problems : algorithms are already optimized
Why ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
- Tensor networks extremely useful in 1D
2. 2D classical is "like" 1D quantum
3. Building block for quantum problems : algorithms are already optimized
Infinite size for translation-invariant problems
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
\Lambda = \begin{pmatrix}
\lambda_{+} & 0 \\
0 & \lambda_{-}
\end{pmatrix} = \lambda_{+} \begin{pmatrix}
1& 0 \\
0 & \frac{\lambda_{-}}{\lambda_{+}}
\end{pmatrix}
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
"Exact contraction"
\Lambda = \begin{pmatrix}
\lambda_{+} & 0 \\
0 & \lambda_{-}
\end{pmatrix} = \lambda_{+} \begin{pmatrix}
1& 0 \\
0 & \frac{\lambda_{-}}{\lambda_{+}}
\end{pmatrix}
\Lambda^{L}
= \lambda_{+}^{L} \begin{pmatrix}
1& 0 \\
0 & \left(\frac{\lambda_{-}}{\lambda_{+}}\right)^L
\end{pmatrix}
How ? Transfer matrices
COLBOIS | KASTELEYN MECHANISM | 06.2025
11
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
"Exact contraction"
\Lambda = \begin{pmatrix}
\lambda_{+} & 0 \\
0 & \lambda_{-}
\end{pmatrix} = \lambda_{+} \begin{pmatrix}
1& 0 \\
0 & \frac{\lambda_{-}}{\lambda_{+}}
\end{pmatrix}
\Lambda^{L}
= \lambda_{+}^{L} \begin{pmatrix}
1& 0 \\
0 & \left(\frac{\lambda_{-}}{\lambda_{+}}\right)^L
\end{pmatrix}
\xrightarrow[L \to \infty]{}
\lambda_{+}^{L} \begin{pmatrix}
1& 0 \\
0 & 0
\end{pmatrix}
"Approximate contraction"
How ? Tensor network language
COLBOIS | KASTELEYN MECHANISM | 06.2025
12
"Contraction"
Matrix / tensor
Vector
Open legs = number of indices = "rank"
How ? Tensor network language
COLBOIS | KASTELEYN MECHANISM | 06.2025
12
"Contraction"
Matrix / tensor
Vector
Open legs = number of indices = "rank"
How ? Tensor network language
COLBOIS | KASTELEYN MECHANISM | 06.2025
12
"Contraction"
Matrix / tensor
Vector
Open legs = number of indices = "rank"
How ? Tensor network language
COLBOIS | KASTELEYN MECHANISM | 06.2025
12
"Contraction"
Matrix / tensor
Vector
Open legs = number of indices = "rank"
How ? MOving to 2D
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
R. J. Baxter, J. Math. Phys. 9, 1968
R. Orús, G. Vidal, PRB 78, 2008
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
How ? MOving to 2D
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
R. J. Baxter, J. Math. Phys. 9, 1968
R. Orús, G. Vidal, PRB 78, 2008
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases}
1 & \text{ all equal}\\
0 & \text{ otherwise}
\end{cases}
How ? MOving to 2D
COLBOIS | KASTELEYN MECHANISM | 06.2025
13
R. J. Baxter, J. Math. Phys. 9, 1968
R. Orús, G. Vidal, PRB 78, 2008
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases}
1 & \text{ all equal}\\
0 & \text{ otherwise}
\end{cases}
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
14
\mathcal{Z}_N =
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
2^L
14
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
2^L
14
Goldenfeld & Kadanoff, Science, 284 (1999)
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
2^L
14
Goldenfeld & Kadanoff, Science, 284 (1999)
Can we keep only the "main" information ?
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
2^L
14
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
\chi
2^L
14
How ? Evaluating the partition function
COLBOIS | KASTELEYN MECHANISM | 06.2025
\mathcal{Z}_N =
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
\chi
\chi
14
2^L
In both cases:
COLBOIS | KASTELEYN MECHANISM | 06.2025
15
EXPONENTIAL # of PARAMETERS
2^L
CONSTANT # of
PARAMETERS (poly. in \(\chi\))
(\chi \times 2 \times \chi)
\(\chi\) is the control parameter
In both cases:
COLBOIS | KASTELEYN MECHANISM | 06.2025
15
EXPONENTIAL # of PARAMETERS
2^L
CONSTANT # of
PARAMETERS (poly. in \(\chi\))
(\chi \times 2 \times \chi)
\(\chi\) is the control parameter
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
\(\langle m \rangle\) =
In both cases:
COLBOIS | KASTELEYN MECHANISM | 06.2025
15
EXPONENTIAL # of PARAMETERS
2^L
CONSTANT # of
PARAMETERS (poly. in \(\chi\))
(\chi \times 2 \times \chi)
\(\chi\) is the control parameter
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
\(\langle m \rangle\) =
Ueda, et al. JSPS 74, 111-124 (2005)
T. Viejira, et al, PRB 104, 235141 (2021)
Frustrated magnets ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
16
Vanderstraeten et al (2018)
Colbois et al, (2021)
Frustrated magnets ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
16
T_{\sigma_1, \sigma_2, \sigma_3} =
Vanderstraeten et al (2018)
Colbois et al, (2021)
Boltzmann
Weight
Kagome
Exact residual entropy with bond dimension 10
Correlation length to \(10^{-4}\)
Frustrated magnets ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
16
T_{\sigma_1, \sigma_2, \sigma_3} =
Vanderstraeten et al (2018)
Colbois et al, (2021)
Boltzmann
Weight
Kagome
Exact residual entropy with bond dimension 10
Correlation length to \(10^{-4}\)
3D Ice residual entropy and correlations...
Where is the catch ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
17
Vanhecke, JC et al (2021)
Where is the catch ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
17
Vanhecke, JC et al (2021)
Fails in the presence of
frustration and macroscopic g.s. degeneracy
Where is the catch ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
17
Vanhecke, JC et al (2021)
Fails in the presence of
frustration and macroscopic g.s. degeneracy
B. Vanhecke, JC, et al. PRR 3, (2021)
\(\rightarrow\) in spin glasses
\(\rightarrow \) in translation-invariant frustrated Ising models
\(\rightarrow\) in lattice gas models
\(\rightarrow\) in frustrated XY models
S. A. Akimenko, PRE 107, (2023)
F.F. Song, T.-Y. Lin, G. M. Zhang, PRB (2023)
C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)
Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)
J. G. Liu, L. Wang, P. Zhang, PRL 126, (2021)
Where is the catch ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
18
Vanhecke, JC et al (2021)
Implement the constraint locally.
Where is the catch ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
Vanhecke, JC et al (2021)
18
Implement the constraint locally.
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
H = \sum_{\langle i,j \rangle } \sigma_i \sigma_j = \frac{J}{2} \sum_{\triangle i,j,k \triangledown i,j,k } (\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
H = \sum_{\langle i,j \rangle } \sigma_i \sigma_j = \frac{J}{2} \sum_{\triangle i,j,k \triangledown i,j,k } (\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
Ground states = tiling of configurations that minimize the local Hamiltonian
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
H = \sum_{\langle i,j \rangle } \sigma_i \sigma_j = \frac{J}{2} \sum_{\triangle i,j,k \triangledown i,j,k } (\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
Ground states = tiling of configurations that minimize the local Hamiltonian
1. Split the Hamiltonian into clusters that overlap
2. Find the optimal energy lower-bound
1. Split the Hamiltonian into clusters that overlap
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
H = \sum_{\langle i,j \rangle } \sigma_i \sigma_j = \frac{J}{2} \sum_{\triangle i,j,k \triangledown i,j,k } (\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
Ground states = tiling of configurations that minimize the local Hamiltonian
1. Split the Hamiltonian into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract + extend to finite temperature
Good news: you can (HOPE TO) find the Constraint!
COLBOIS | KASTELEYN MECHANISM | 06.2025
19
Essential idea : Anderson bounds
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969)
M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975)
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981)
W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
H = \sum_{\langle i,j \rangle } \sigma_i \sigma_j = \frac{J}{2} \sum_{\triangle i,j,k \triangledown i,j,k } (\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
Ground states = tiling of configurations that minimize the local Hamiltonian
1. Split the Hamiltonian into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract + extend to finite temperature
Questions so far ?
3. A kasteleyn-driven staircase
COLBOIS | KASTELEYN MECHANISM | 06.2025
COLBOIS | KASTELEYN MECHANISM | 06.2025
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)
20
Frustrated models on the kagome lattice
COLBOIS | KASTELEYN MECHANISM | 06.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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
20
Frustrated models on the kagome lattice
COLBOIS | KASTELEYN MECHANISM | 06.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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
20
Frustrated models on the kagome lattice
COLBOIS | KASTELEYN MECHANISM | 06.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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
20
Frustrated models on the kagome lattice
COLBOIS | KASTELEYN MECHANISM | 06.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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
3 triangular sublattices
20
Frustrated models on the kagome lattice
COLBOIS | KASTELEYN MECHANISM | 06.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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
3 triangular sublattices
Highly frustrated
20
Macroscopically degenerate ground state phases
COLBOIS | KASTELEYN MECHANISM | 06.2025
21
JC, B. Vanhecke et. al., PRB 106 (2022)
J_1 \gg J_2, J_3
Macroscopically degenerate ground state phases
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
J_1 \gg J_2, J_3
All antiferromagnetic couplings :
3 phases due to the competition (exact g.s. energy)
19
21
Macroscopically degenerate ground state phases
COLBOIS | KASTELEYN MECHANISM | 06.2025
S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}
S = 0.01920 \pm 3 \cdot 10^{-5}
JC, B. Vanhecke et. al., PRB 106 (2022)
J_1 \gg J_2, J_3
All antiferromagnetic couplings :
3 phases due to the competition (exact g.s. energy)
21
Macroscopically degenerate ground state phases
COLBOIS | KASTELEYN MECHANISM | 06.2025
S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}
JC, B. Vanhecke et. al., PRB 106 (2022)
J_1 \gg J_2, J_3
All antiferromagnetic couplings :
3 phases due to the competition (exact g.s. energy)
S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}
200 tiles
21
"String phase" ground state
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
22
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
"String phase" ground state
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: frustrated Ising model on the triangular lattice
EXPONENTIAL NUMBER
22
"String phase" ground state
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: frustrated Ising model on the triangular lattice
EXPONENTIAL NUMBER
"Strings" representation
22
Zero-energy double domain walls (=DDW)
COLBOIS | KASTELEYN MECHANISM | 06.2025
23
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Zero-energy double domain walls (=DDW)
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
DDW Break the dense rows AF order
(Introduce vertical dimers)
23
Zero-energy double domain walls (=DDW)
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
DDW Break the dense rows AF order
(Introduce vertical dimers)
Entropic suppression
("partial order by disorder")
23
Zero-energy double domain walls (=DDW)
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
DDW Break the dense rows AF order
(Introduce vertical dimers)
Entropic suppression
("partial order by disorder")
What about finite-temperature?
(Our expectations: two second order phase transitions,
a single first-order or a Kasteleyn transition)
23
double domain wall Excitations
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
24
double domain wall Excitations
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Freedom inside the domain wall
Energy cost
Entropic gain
24
double domain wall Excitations
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
24
double domain wall Excitations
COLBOIS | KASTELEYN MECHANISM | 06.2025
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
24
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
25
\(T\)
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
\(T\)
25
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
\(T\)
25
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
\(T\)
\(T_c^{(1)}\)
\(n_c/n_A = 1\)
\(n_c =0\)
25
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
\(T\)
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(n_c/n_A = 1\)
\(n_c =0\)
\(n_c/n_A = 2\)
25
Kasteleyn mechanism for Plateaus of topological origin
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Freedom inside the domain wall
Energy cost
Entropic gain
Replacing a green string by a DDW
Entropic cost
Repulsion between domain walls
Energy cost for condensation
\(T\)
\(T_c^{(1)}\)
\(n_c/n_A = 1\)
\(n_c =0\)
\(T_c^{(2)}\)
\(n_c/n_A = 2\)
\(T_c^{(3)}\)
\(n_c/n_A = 3\)
25
Staircase in the density of strings
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
26
Staircase in the density of strings
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Series of 1st order transitions
Ratio of string densities takes all integer values
26
Staircase in the density of strings
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Ratio of string densities takes all integer values
Series of 1st order transitions
Not a "usual" Devil's staircase:
26
Density of strings is not exactly constant
(not commensurate)
Staircase in the density of strings
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Density of strings is not exactly constant
(not commensurate)
Series of 1st order transitions
Not a "usual" Devil's staircase:
Infinite number of locked portions
Ratio of string densities takes all integer values
26
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
27
Upcoming
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
27
Upcoming
Finite-\(J\) consequences ?
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
27
Upcoming
Finite-\(J\) consequences ?
Other models ?
Yes!
1D quantum / 2D Strings model
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
COLBOIS | KASTELEYN MECHANISM | 06.2025
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
28
Outlook
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Tensor network frustration problem
Starting point
Is there always a cell relaxing the frustration? (Hard vs weak frustration)
Can the problem be fixed at the level of the MPO?
Consequences for iPEPS?
Experimentally realizable ?
Ground for quantum models ?
COLBOIS | KASTELEYN MECHANISM | 06.2025
29
Take-home messages
- Directed, non-crossing, non-terminating
- energy cost versus entropic gain
Kasteleyn mechanism
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS | KASTELEYN MECHANISM | 06.2025
29
Take-home messages
- Directed, non-crossing, non-terminating
- energy cost versus entropic gain
Kasteleyn mechanism
Topological "devil's step"
1. Two kinds of system-spanning strings
2. Internal freedom within strings.
3. Effective repulsion between strings of the same kind
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS | KASTELEYN MECHANISM | 06.2025
29
Take-home messages
- Directed, non-crossing, non-terminating
- energy cost versus entropic gain
Kasteleyn mechanism
Topological "devil's step"
1. Two kinds of system-spanning strings
2. Internal freedom within strings.
3. Effective repulsion between strings of the same kind
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS | KASTELEYN MECHANISM | 06.2025
29
Take-home messages
- Directed, non-crossing, non-terminating
- energy cost versus entropic gain
Kasteleyn mechanism
Topological "devil's step"
1. Two kinds of system-spanning strings
2. Internal freedom within strings.
3. Effective repulsion between strings of the same kind
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
Thank you!
COLBOIS | KASTELEYN MECHANISM | 06.2025
20
Take-home messages
- Directed, non-crossing, non-terminating
- energy cost versus entropic gain
Kasteleyn mechanism
Topological "devil's step"
1. Two kinds of system-spanning strings
2. Internal freedom within strings.
3. Effective repulsion between strings of the same kind
Thank you!
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
2D kasteleyn transition in Spin ice
COLBOIS | KASTELEYN MECHANISM | 06.2025
2D kasteleyn transition in Spin ice
COLBOIS | KASTELEYN MECHANISM | 06.2025
COLBOIS | KASTELEYN MECHANISM | 06.2025
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)
Staircases
COLBOIS | KASTELEYN MECHANISM | 06.2025
\kappa = -J_2/J_1
von Boehm & Bak, PRB (1980)
Fisher and Selke, PRL (1980)
COLBOIS | KASTELEYN MECHANISM | 06.2025
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
\varepsilon_2
1
2
3
\varepsilon_3
\varepsilon_1
<
=
COLBOIS | KASTELEYN MECHANISM | 06.2025
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
\varepsilon_2
1
2
3
\varepsilon_3
\varepsilon_1
<
=
COLBOIS | KASTELEYN MECHANISM | 06.2025
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
\(\varepsilon_b\) : cost of putting a dimer on \(b = 1,2,3\)
z_b = e^{-\frac{\varepsilon_b}{k_B T}}
\mathcal{Z}(\mathbf{z}) = \sum_{\mathrm{coverings}} \prod_b z_b^{N_b}
\varepsilon_2
\varepsilon_1 = 0
1
2
3
\varepsilon_3
\varepsilon_1
<
=
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
- Directed, non-crossing, non-terminating
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
- Directed, non-crossing, non-terminating
- Linear energy cost
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
- Directed, non-crossing, non-terminating
- Linear energy cost
- Entropic gain
COLBOIS | KASTELEYN MECHANISM | 06.2025
Ground state v.s. Excitations
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
\varepsilon_2
\varepsilon_3
\varepsilon_1
<
=
Constraint : hardcore
- Directed, non-crossing, non-terminating
- Linear energy cost
- Entropic gain
10
COLBOIS | KASTELEYN MECHANISM | 06.2025
Contracting the TN of a frustrated model
Numerical problem
Ground-state rule
Cancellation of small and large factors
C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)
Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)
\(\rightarrow\) precision?
J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)
\(\rightarrow\) log?
(For TN experts)
MPO
The MPO is badly conditioned (e.g. not hermitian, ...). Fix it?
\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\
e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}
Failure to minimize simultaneously all local Hamiltonians.
B. Vanhecke, JC, et al. PRR 3, (2021)
F.F. Song, T.-Y. Lin, G. M. Zhang, arXiv:2309.05321
Entropy
COLBOIS | KASTELEYN MECHANISM | 06.2025
\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{orange}\lambda_+}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})}
= e^{-\beta E_{\rm{GS}}} {\color{orange}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\
= e^{-\beta E_{\rm{GS}}} {\color{orange}\tilde{\mathcal{Z}}_N}
COLBOIS | KASTELEYN MECHANISM | 06.2025
\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{orange}\lambda_+}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})}
= e^{-\beta E_{\rm{GS}}} {\color{orange}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\
= e^{-\beta E_{\rm{GS}}} {\color{orange}\tilde{\mathcal{Z}}_N}
Partition function for one site:
Most precise result
Direct access to zero temperature
Entropy
Topological staircase in a constrained kagome Ising antiferromagnet
By Jeanne Colbois
Topological staircase in a constrained kagome Ising antiferromagnet
Seminar at CPHT - Host : Karyn Le Hur
