Jeanne Colbois | Institut NEEL | CNRS & Université Grenoble Alpes

Samuel Nyckees

Afonso Rufino

Frédéric Mila

Topological Staircase in a

constrained kagome Ising antiferromagnet

Phys. Rev. Letters 136, 086701 (2026)

arXiv:26??.????

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}
image/svg+xml
\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}
image/svg+xml
\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}
image/svg+xml
\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}
image/svg+xml
\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

  1. Two kinds of strings 
  2. Internal freedom
  3. 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

  1. Two kinds of strings 
  2. Internal freedom
  3. 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

  1. Two kinds of strings 
  2. Internal freedom
  3. 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

  1. Two kinds of strings 
  2. Internal freedom
  3. 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!