Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Jeanne Colbois | Institut NEEL | CNRS & Université Grenoble Alpes
Samuel Nyckees
Afonso Rufino
Frédéric Mila
Topological Staircase in a
constrained kagome Ising antiferromagnet
Frustrated Ising magnets as constrained problems
2
Energy minimization
Local constraint
Unusual phase transitions
Macroscopic degeneracy
absence of long-range order
\(H = \sum_{i,j} J_{i,j} \sigma_i \sigma_j\)
\(\sigma_i = \pm 1\)
Frustrated Ising magnets as constrained problems
3
2-in 2-out constraint
Spin ice
Pinch points in diffuse scattering
\(H = \sum_{i,j} J_{i,j} \sigma_i \sigma_j\)
\(\sigma_i = \pm 1\)
see e.g. Moessner and Sondhi (2003), Fennell et al. (2007,2009), Henley (2005, 2010), Castelnovo et al (2008), ...
Frustrated Ising magnets as constrained problems
3
Spin ice
Wannier (1950), Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986),
Smeral et al (2016, 2019)
Pinch points in diffuse scattering
down
up
see e.g. Moessner and Sondhi (2003), Fennell et al. (2007,2009), Henley (2005, 2010), Castelnovo et al (2008), ...
\(H = \sum_{i,j} J_{i,j} \sigma_i \sigma_j\)
\(\sigma_i = \pm 1\)
?
UUD /DDU
2-in 2-out constraint
Triangular lattice NN Ising AF
Frustrated Ising magnets as constrained problems
3
Spin ice
Wannier (1950), Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986),
Smeral et al (2016, 2019)
Pinch points in diffuse scattering
down
up
see e.g. Moessner and Sondhi (2003), Fennell et al. (2007,2009), Henley (2005, 2010), Castelnovo et al (2008), ...
\(H = \sum_{i,j} J_{i,j} \sigma_i \sigma_j\)
\(\sigma_i = \pm 1\)
UUD /DDU - dimer constraint
Critical correlations
2-in 2-out constraint
Triangular lattice NN Ising AF
Frustrated Ising magnets as constrained problems
3
Spin ice
Wannier (1950), Kasteleyn (1960), Stephenson (1963), Fisher (1966), Yokoi et al (1986),
Smeral et al (2016, 2019)
Pinch points in diffuse scattering
UUD /DDU - dimer constraint
Critical correlations
down
up
see e.g. Moessner and Sondhi (2003), Fennell et al. (2007,2009), Henley (2005, 2010), Castelnovo et al (2008), ...
\(H = \sum_{i,j} J_{i,j} \sigma_i \sigma_j\)
\(\sigma_i = \pm 1\)
Kagome lattice Ising AF
UUD/DDU
Exponentially decaying correls.
Kano & Naya (1953), Suto (1981)
2-in 2-out constraint
Triangular lattice NN Ising AF
Today
4
Constraints leading to a topological staircase
Today
4
Kasteleyn transition (spin ice, triangular)
\(T\)
\((T-T_K)^{1/2}\)
string density
Topological staircase
- Integer ratios of strings
- Not commensurate
Kagome model and tensor network method
\(T\)
commensurate wavevector
Devil's staircase
\(J_1 - J_2-J_3\)
Constraints leading to a topological staircase
Kasteleyn transition and devil's staircase
Unique ground state
Kasteleyn transition
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
5
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
Forbidden in this toy model
Kasteleyn transition
5
\(J+\delta\)
\(J+\delta\)
\(J\)
No ferromagnetic triangles
Kasteleyn (1963)
\(4J+2\delta\)
Constrained limit \(J \rightarrow \infty\)
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Constrained limit \(J \rightarrow \infty\)
No ferromagnetic triangles
Kasteleyn (1963)
\(E = 2 \delta L_y\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(E = 2 \delta L_y\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
\(T\)
Directed, non-crossing, non terminating
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
Directed, non-crossing, non terminating
\(T\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
Directed, non-crossing, non terminating
\(T\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
Directed, non-crossing, non terminating
\(T\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
Directed, non-crossing, non terminating
\(T\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
Excitations are system-spanning strings
Kasteleyn transition
\(J+\delta\)
\(J+\delta\)
\(J\)
Kasteleyn (1963)
\(E = 2 \delta L_y\)
No ferromagnetic triangles
Constrained limit \(J \rightarrow \infty\)
\(S_{\mathrm{DW}} = \ln(2) L_y\)
\(F = E - TS\)
Directed, non-crossing, non terminating
\(T\)
5
Toy model:
Nearest-neighbor anisotropic
ising antiferromagnet
5
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
6
Kasteleyn transition
n_{\mathrm{strings}} \propto \sqrt{(T-T_K)}
5
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
6
Kasteleyn transition
n_{\mathrm{strings}} \propto \sqrt{(T-T_K)}
Quantum 1D Hamiltonian :
\(T \leftrightarrow\) chemical potential
\(n_{\mathrm{strings}} \leftrightarrow\) fermions density
5
J. F. Nagle et al, Domb & Lebowitz Phase transitions and critical phenomena 13 (1989)
6
Kasteleyn transition
Quantum 1D Hamiltonian :
\(T \leftrightarrow\) chemical potential
\(n_{\mathrm{strings}} \leftrightarrow\) fermions density
n_{\mathrm{strings}} \propto \sqrt{(T-T_K)}
Pokrovsky-Talapov
commensurate-incommensurate transition
5
7
Devil's staircase
locked at rational values
Bak, Rep. Prog. Phys.(1982)
Sequence of phase transitions
phases with commensurate modulation
5
7
Devil's staircase
locked at rational values
Ferro \(J_1\)
Ferro \(J_1\)
Antiferro \(J_2\)
Sequence of phase transitions
phases with commensurate modulation
3D ANNNI model
Bak, Rep. Prog. Phys.(1982)
5
7
Devil's staircase
locked at rational values
Fisher and Selke, PRL (1980)
Ferro \(J_1\)
wavevector (\(\pi/2a)\)
Ferro \(J_1\)
Antiferro \(J_2\)
\((J_2/J_1 - 1/2)/ \mathrm{scale}(T/J_1)\)
Sequence of phase transitions
phases with commensurate modulation
3D ANNNI model
Bak, Rep. Prog. Phys.(1982)
2/3
4/5
Kagome LatTice J1-J2-J3 Ising antiferromagnet
Model, tensor networks and ground state
Kagome LatTice J1-J2-J3 Ising antiferromagnet
Model, tensor networks and ground state
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)
Frustrated models on the kagome lattice
8
Frustrated models on the kagome lattice
8
Kagome lattice
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Frustrated models on the kagome lattice
8
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Kagome lattice
3
Kagome sublattices
Frustrated models on the kagome lattice
8
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Kagome lattice
3
triangular sublattices
3
Kagome sublattices
Frustrated models on the kagome lattice
8
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Kagome lattice
3
Kagome sublattices
3
triangular sublattices
Takagi & Mekata (1996)
J. Hamp, C. Castelnovo, R. Moessner (2018)
Frustrated models on the kagome lattice
8
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Kagome lattice
3
triangular sublattices
?
All antiferromagnetic:
highly frustrated
Takagi & Mekata (1996)
J. Hamp, C. Castelnovo, R. Moessner (2018)
3
Kagome sublattices
\(\rightarrow\) Tensor networks
Tensor networks for 2D statistical mechanics:
transfer matrix revisited!
9
\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}
Ising model:
Tensor networks for 2D statistical mechanics:
transfer matrix revisited!
9
\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}
Ising model:
\mathcal{Z}_N =
\(2^L\)
Tensor networks for 2D statistical mechanics:
transfer matrix revisited!
9
\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}
Ising model:
\mathcal{Z}_N =
- leverage existing methods for 2D quantum
- directly inifinite-size limit
- direct access to free energy
- efficient correlation functions
Nishino, Okunishi (1996)
Levin, Nave (2007)
....
\(2^L\)
\(\chi\)
Tensor networks for 2D statistical mechanics:
transfer matrix revisited!
9
\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}
Ising model:
\mathcal{Z}_N =
/!\ In frustrated systems: implement ground-state constraint locally /!\
B. Vanhecke, JC et. al., PRR (2021)
19
10
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
?
Antiferromagnetic couplings?
3 Macroscopically degenerate ground state phases
19
10
(For the experts: this is obtained with VUMPS,\(\chi \sim 200\))
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
Antiferromagnetic couplings?
Three macroscopically degenerate phases
3 Macroscopically degenerate ground state phases
19
10
(For the experts: this is obtained with VUMPS,\(\chi \sim 200\))
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_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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
Antiferromagnetic couplings?
Three macroscopically degenerate phases
Strings phase
\(J_3 > J_2 > 0\)
\(S_{\mathrm{TIAFM}}/3 \pm 6\cdot 10^{-6}\)
3 Macroscopically degenerate ground state phases
Results : constrained limit \(J_1 \rightarrow \infty, J_3 \rightarrow \infty\)
1. Understanding the strings phase ground state
2. Directed strings leading to a "topological" staircase
11
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
down
up
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
An exponential family of ground states
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
down
up
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
An exponential family of ground states
11
\(\mathbb{Z}_2\) (translation) \(\times \mathbb{Z}_3\) (rotation) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: triangular lattice Ising antiferromagnet
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
down
up
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
An exponential family of ground states
11
\(\mathbb{Z}_2\) (translation) \(\times \mathbb{Z}_3\) (rotation) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: triangular lattice Ising antiferromagnet
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
Dimer mapping
down
up
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
An exponential family of ground states
11
\(\mathbb{Z}_2\) (translation) \(\times \mathbb{Z}_3\) (rotation) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: triangular lattice Ising antiferromagnet
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
Dimer mapping
down
up
An exponential family of ground states
Partially ordered family of states:
directed strings on the triangular lattice
11
\(\mathbb{Z}_2\) (translation) \(\times \mathbb{Z}_3\) (rotation) symmetry breaking
\Psi_{\mathbb{Z}_2} = \lim_{x \rightarrow \infty} | \sigma_0 \sigma_{x} |
Dense rows: AF order
Sparse rows: triangular lattice Ising antiferromagnet
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
JC, B. Vanhecke, L. Vanderstraeten, A. Smerald, F. Verstraete, F. Mila (2022)
\(J_1 \rightarrow \infty, J_3 \rightarrow \infty\) : UUD /DDU constraints on NN and 3rdNN triangles
\(J_2\) : partially lifts the degeneracy
Dimer mapping
down
up
Partially ordered family of states:
directed strings on the triangular lattice
An exponential family of ground states
Partial order?
11
\(\mathbb{Z}_2\) (translation) \(\times \mathbb{Z}_3\) (rotation) symmetry breaking
12
Full ground state description
Zero energy double domain walls (replacing B)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Full ground state description
Zero energy double domain walls (replacing B)
Break AF order
Restore rotation
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
12
12
Full ground state description
Zero energy double domain walls (replacing B)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Entropically suppressed in the thermodynamic limit
Break AF order
Restore rotation
12
Full ground state description
Zero energy double domain walls (replacing B)
ENTROPY-DRIVEN PARTIAL ORDER
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Entropically suppressed in the thermodynamic limit
Break AF order
Restore rotation
12
Full ground state description
Zero energy double domain walls (replacing B)
What about finite-temperature?
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
ENTROPY-DRIVEN PARTIAL ORDER
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Entropically suppressed in the thermodynamic limit
Break AF order
Restore rotation
12
Full ground state description
Zero energy double domain walls (replacing B)
What about finite-temperature?
Expectactions:
Two second order PTs / 1 first-order PT
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
ENTROPY-DRIVEN PARTIAL ORDER
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Entropically suppressed in the thermodynamic limit
Break AF order
Restore rotation
12
Full ground state description
Zero energy double domain walls (replacing B)
What about finite-temperature?
Expectactions:
Two second order PTs / 1 first-order PT
or a Kasteleyn transition driven by
C
strings
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
ENTROPY-DRIVEN PARTIAL ORDER
\(\mathbb{Z}_2 \times \mathbb{Z}_3\) symmetry breaking
Entropically suppressed in the thermodynamic limit
Break AF order
Restore rotation
13
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
A staircase
Series of jumps
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
A staircase
Series of jumps
Single jump in the AF order
13
A staircase
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Series of jumps
Single jump in the AF order
Rotation restorded through series of transitions
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
13
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Devil's / Harmless staircase?
\(\rightarrow\) constant commensurate wavevector within the plateau?
A staircase
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
13
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Devil's / Harmless staircase?
\(\rightarrow\) constant commensurate wavevector within the plateau?
A staircase
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
13
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Devil's / Harmless staircase?
\(\rightarrow\) constant commensurate wavevector within the plateau?
\(\rightarrow\) really plateaus in the rotation symmetry breaking order parameter?
A staircase
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
13
strings Densities
14
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Recall: \(\Psi_{\mathbb{Z}_3}\) directly related to \(n_C\)
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
strings Densities
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Recall: \(\Psi_{\mathbb{Z}_3}\) directly related to \(n_C\)
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
14
Not constant!
strings Densities
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Recall: \(\Psi_{\mathbb{Z}_3}\) directly related to \(n_C\)
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
14
Not constant!
strings Densities
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
14
Not constant!
Recall: \(\Psi_{\mathbb{Z}_3}\) directly related to \(n_C\)
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
\(n_A\)
strings Densities
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
14
Not constant!
Recall: \(\Psi_{\mathbb{Z}_3}\) directly related to \(n_C\)
\(\Psi_{\mathbb{Z}_3} = \frac{1}{2}- \frac{3n_C}{4}\)
Ratio of string densities is fixed to an integer
\(n_A\)
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Physical picture (1st plateau)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
Physical picture (1st plateau)
Physical picture (1st plateau)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
C-strings host defects
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
Physical picture (1st plateau)
Internal defects: change in arrow direction
Energy cost, entropy gain
C-strings host defects
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
Physical picture (1st plateau)
C-strings host defects
External "decorations":
entropy gain
Internal defects: change in arrow direction
Energy cost, entropy gain
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
Physical picture (1st plateau)
C-strings host defects
External "decorations":
entropy gain
Internal defects: change in arrow direction
Energy cost, entropy gain
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Competition:
entropy + energy cost, entropy gain
Effective repulsion between strings
15
Snapshot sampled from tensor network
environment, \(T_c^{(1)} < T <T_c^{(2)}\)
Physical picture (1st plateau)
Real space picture
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
16
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Real space picture
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
16
Real space picture
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
16
Real space picture
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
16
Number of C-strings between A-strings is fixed
Wavevector can change with T
Signature in magnetic structure factor
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
17
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Signature in magnetic structure factor
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
17
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Signature in magnetic structure factor
\(n_C/n_A= 1\)
\(n_C/n_A= 2\)
\(n_C/n_A= 3\)
\(T\)
17
\(T_c^{(1)}\)
\(T_c^{(2)}\)
\(T_c^{(3)}\)
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Equal time structure factor: wavevectors related to the average distance between A-strings
phase diagram
18
Constrained limit \(J_1, J_3 \rightarrow \infty\) : topological staircase
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
phase diagram
18
Constrained limit \(J_1, J_3 \rightarrow \infty\) : topological staircase
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Finite \(J_1, J_3\):
Staircase
Crossovers
Nematic
Partially ordered
Paramagnetic
phase diagram
18
Constrained limit \(J_1, J_3 \rightarrow \infty\) : topological staircase
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
A. Rufino, S. Nyckees, JC, F. Mila, PRL (2026)
Finite \(J_1, J_3\):
Staircase
Crossovers
Nematic
Partially ordered
Paramagnetic
Is the staircase infinite?
- cannot conclude from numerics
- \(n_C \rightarrow \infty\) to restore \(\mathbb{Z}_3\)
- long-range effective repulsion?
19
Final remarks
Possible experimental realizations?
Main challenge: Ising and larger third-neighbour
Artificial spin ice?
(but 3rd coupling)
Vesignieite (BaCu3V2O8(OH)) ? / spinels (GeFe2O4)?
(but Heisenberg / XY spins...)
19
Final remarks
Possible experimental realizations?
Artificial spin ice?
(but 3rd coupling)
Other models:
1D Hubbard model
String-Ising model
H = \sum_{i, \sigma} \left[-t\left(c_{i, \sigma}^{\dagger} +h.c.\right)+ Un_{i, \uparrow}n_{i, \downarrow} + \sum_{r, \sigma, \sigma'} V_{r}^{\sigma, \sigma'}n_{i, \sigma}n_{i+r, \sigma'} \right]
\(\rightarrow\) quantum 1D equivalent
\(\rightarrow\) analytical evidence
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Main challenge: Ising and larger third-neighbour
Vesignieite (BaCu3V2O8(OH)) ? / spinels (GeFe2O4)?
(but Heisenberg / XY spins...)
20
Take-home messages
Text
20
Take-home messages
Directed non-crossing strings
Energy cost vs entropic gain
Kasteleyn mechanism
Kasteleyn (1963), Jaubert & Holdsworth (2008), Fennell et al. (2009), Smerald & Mila (2018)
Text
20
Take-home messages
Directed non-crossing strings
Energy cost vs entropic gain
Kasteleyn mechanism
Topological (incommensurate) staircase
- Two kinds of strings
- Internal freedom
- Effective repulsion
Internal entropic gain vs. thickness entropic gain
A. Rufino, S. Nyckees, JC, F. Mila, Phys. Rev. Letters (2026)
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Kasteleyn (1963), Jaubert & Holdsworth (2008), Fennell et al. (2009), Smerald & Mila (2018)
Text
20
Take-home messages
Directed non-crossing strings
Energy cost vs entropic gain
Kasteleyn mechanism
- Two kinds of strings
- Internal freedom
- Effective repulsion
Internal entropic gain vs. thickness entropic gain
A. Rufino, S. Nyckees, JC, F. Mila, Phys. Rev. Letters (2026)
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Kasteleyn (1963), Jaubert & Holdsworth (2008), Fennell et al. (2009), Smerald & Mila (2018)
Text
Topological (incommensurate) staircase
20
Take-home messages
Directed non-crossing strings
Energy cost vs entropic gain
Kasteleyn mechanism
- Two kinds of strings
- Internal freedom
- Effective repulsion
Internal entropic gain vs. thickness entropic gain
A. Rufino, S. Nyckees, JC, F. Mila, Phys. Rev. Letters (2026)
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Kasteleyn (1963), Jaubert & Holdsworth (2008), Fennell et al. (2009), Smerald & Mila (2018)
Text
Tensor networks: also for "classical", 2D frustrated magnetism!
Vanhecke, JC et al. PRR (2021); JC et al. PRB (2022); Song et al PRB (2023); Nyckees et al (JC) PRE (2023), ...
Topological (incommensurate) staircase
20
Take-home messages
Directed non-crossing strings
Energy cost vs entropic gain
Kasteleyn mechanism
- Two kinds of strings
- Internal freedom
- Effective repulsion
Internal entropic gain vs. thickness entropic gain
A. Rufino, S. Nyckees, JC, F. Mila, Phys. Rev. Letters (2026)
A. Rufino, S. Nyckees, JC, F. Mila, in preparation
Kasteleyn (1963), Jaubert & Holdsworth (2008), Fennell et al. (2009), Smerald & Mila (2018)
Text
Vanhecke, JC et al. PRR (2021); JC et al. PRB (2022); Song et al PRB (2023); Nyckees et al (JC) PRE (2023), ...
Thank you!
Topological (incommensurate) staircase
Tensor networks: also for "classical", 2D frustrated magnetism!
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)
String Ising model
Entropy
\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}
\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
Tensor network contraction
\mathcal{Z}_N =
2^L
Goldenfeld & Kadanoff, Science, 284 (1999)
Can we keep only the "main" information ?
Tensor network contraction
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
\mathcal{Z}_N =
2^L
Tensor network contraction
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)
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
Good news: you can (HOPE TO) find the Constraint!
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!
Copy of Infinite topological staircase in a constrained kagome Ising antiferromagnet
By Jeanne Colbois
Copy of Infinite topological staircase in a constrained kagome Ising antiferromagnet
Invited talk at TEMM
