Jeanne Colbois | Institut Néel CNRS | Grenoble, France

Tensor networks

 

Statistical mechanics

Tensor networks

 

Frustrated magnetism

What can classical frustrated models

 teLL us about tensor networks ?

Fixnet | UZH | 14-16 January 2026, Zurich, Switzerland

What can classical frustrated models

 teLL us about tensor networks ?

Jeanne Colbois | Institut Néel CNRS | Grenoble, France

Contracting the TN

partition function

of a

frustrated model

Fixnet | UZH | 14-16 January 2026, Zurich, Switzerland

What can classical frustrated models

 teLL us about tensor networks ?

Jeanne Colbois | Institut Néel CNRS | Grenoble, France

Contracting the TN

partition function

of a

frustrated model

Numerical problem

MPO properties

Emergent d.o.fs

Fixnet | UZH | 14-16 January 2026, Zurich, Switzerland

COLBOIS | FRUSTRATED TNS |  01.2026

sCOPE

1

1. Motivation

 

2. The case of frustrated spin systems

 

3. Range of constraints

 

4. Conclusions

COLBOIS | FRUSTRATED TNS |  01.2026

Acknowledgments

2

Samuel Nyckees

Afonso Rufino

Andrew Smerald

Frédéric Mila

Frank Verstraete

Laurens Vanderstraeten

Bram Vanhecke

TN contraction problem instances:

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Vanderstraeten et al., PRE 98 (2018)

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721, (2019)

J. G. Liu, L. Wang, P. Zhang, PRL 126, (2021)

B. Vanhecke, JC, et al. PRR 3, (2021)

FF Song, GM Zhang, PRB 105, (2022)

G. Giudice, F. Surace, H. Pichler, G. Giudici, PRB 106, (2022)

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, (2023)

F.F. Song, H. Numoin, N. Kawashima,  PRB 111, (2025)

W. Tang, F. Verstraete, J. Haegeman, PRB 111, (2025)

Homma et al., PRB 111, (2025)

Our applications:

JC, B. Vanhecke, et al, PRB 106, (2022)

Nyckees et al, PRE 108, (2023)

A. Rufino, et al., arXiv:2505.05889 (to appear in PRL)

A. Rufino et al., in prep

Nathan Perruchoud

Motivation

Frustrated magnetism and constrained models: why?

How? (a priori?)

Frustrated Spin systems

3

COLBOIS | FRUSTRATED TNS |  01.2026

Frustrated Spin systems

3

Incompatible  constraints

COLBOIS | FRUSTRATED TNS |  01.2026

Cannot simultaneously minimize all terms

Frustrated Spin systems

3

Incompatible  constraints

Macroscopic ground-state degeneracy

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}

COLBOIS | FRUSTRATED TNS |  01.2026

Cannot simultaneously minimize all terms

Frustrated Spin systems

3

Incompatible  constraints

Macroscopic ground-state degeneracy

Spin liquid  with algebraic or exponential correlations; Partial order

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}

COLBOIS | FRUSTRATED TNS |  01.2026

Cannot simultaneously minimize all terms

Frustrated Spin systems

3

Incompatible  constraints

Macroscopic ground-state degeneracy

Spin liquid  with algebraic or exponential correlations; Partial order

Emergent degrees-of-freedom

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}

C. Castlenovo, R. Moessner, S. L. Sondhi, Nature 451 (2008)

COLBOIS | FRUSTRATED TNS |  01.2026

Cannot simultaneously minimize all terms

Classical constrained / Frustrated stat Mech models: why?

4

COLBOIS | FRUSTRATED TNS |  01.2026

Classical constrained / Frustrated stat Mech models: why?

4

COLBOIS | FRUSTRATED TNS |  01.2026

Exotic classical phenomena / directly relevant for classical degrees-of-freedom

Classical constrained / Frustrated stat Mech models: why?

4

COLBOIS | FRUSTRATED TNS |  01.2026

Exotic classical phenomena / directly relevant for classical degrees-of-freedom

"Simple" limits of quantum many-body systems

Fig. from Giudice etal.

PRB 106, (2022)

|\psi\rangle =\frac{1}{\sqrt{Z}} \sum_{c} \sqrt{\mathcal{W}(c)}|c\rangle

(Generalized-)RK type wavefunctions

Classical limit of the dof.

Effective models
...

Classical constrained / Frustrated stat Mech models: why?

4

COLBOIS | FRUSTRATED TNS |  01.2026

Partition functions as exact* tensor networks

- particularly hard to contract?

* when the dofs are discrete

Exotic classical phenomena / directly relevant for classical degrees-of-freedom

"Simple" limits of quantum many-body systems

Fig. from Giudice etal.

PRB 106, (2022)

|\psi\rangle =\frac{1}{\sqrt{Z}} \sum_{c} \sqrt{\mathcal{W}(c)}|c\rangle

(Generalized-)RK type wavefunctions

Classical limit of the dof.

Effective models
...

Classical constrained / Frustrated stat Mech models: why?

4

COLBOIS | FRUSTRATED TNS |  01.2026

* when the dofs are discrete

Partition functions as exact* tensor networks

- particularly hard to contract?

Exotic classical phenomena / directly relevant for classical degrees-of-freedom

"Simple" limits of quantum many-body systems

Fig. from Giudice etal.

PRB 106, (2022)

|\psi\rangle =\frac{1}{\sqrt{Z}} \sum_{c} \sqrt{\mathcal{W}(c)}|c\rangle

(Generalized-)RK type wavefunctions

Classical limit of the dof.

Effective models
...

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Ergodicity issues

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

Ergodicity issues

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

Entropy as the first outcome (free energy per site)

Ergodicity issues

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

A priori: 

Entropy as the first outcome (free energy per site)

Ergodicity issues

image/svg+xml
image/svg+xml

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

A priori: 

image/svg+xml
image/svg+xml

Entropy as the first outcome (free energy per site)

Ergodicity issues

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

1. Associate a Boltzmann weight with each interaction

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

A priori: 

image/svg+xml
image/svg+xml

Entropy as the first outcome (free energy per site)

Ergodicity issues

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

1. Associate a Boltzmann weight with each interaction

2. split

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

A priori: 

image/svg+xml
image/svg+xml

1. Associate a Boltzmann weight with each interaction

2. split

3. group and reshape

Entropy as the first outcome (free energy per site)

Ergodicity issues

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

Why TNS for (not exactly solvable) constrained models?

5

COLBOIS | FRUSTRATED TNS |  01.2026

No sign problem!

Monte Carlo:

Finite-size scaling

Tensor networks:

Finite-entanglement scaling / RG 

A priori: 

image/svg+xml
image/svg+xml

1. Associate a Boltzmann weight with each interaction

2. split

3. group and reshape

4. contract the 2D TN with your favourite tool

Entropy as the first outcome (free energy per site)

Ergodicity issues

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

The case of frustrated spin systems

Understanding the triangular lattice Ising antiferromagnet

How to go beyond

Current limitations

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

6

COLBOIS | FRUSTRATED TNS |  01.2026

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

6

COLBOIS | FRUSTRATED TNS |  01.2026

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
S = 0.3230659...

G.H. Wannier, PR 79, (1950, 1973)

6

COLBOIS | FRUSTRATED TNS |  01.2026

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
S = 0.3230659...

G.H. Wannier, PR 79, (1950, 1973)

6

COLBOIS | FRUSTRATED TNS |  01.2026

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
S = 0.3230659...

G.H. Wannier, PR 79, (1950, 1973)

Boltzmann Weight

t = \begin{pmatrix} e^{-\beta J} & e^{\beta J}\\ e^{\beta J} & e^{-\beta J}\end{pmatrix}

6

COLBOIS | FRUSTRATED TNS |  01.2026

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
S = 0.3230659...

G.H. Wannier, PR 79, (1950, 1973)

\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}

Boltzmann Weight

t = \begin{pmatrix} e^{-\beta J} & e^{\beta J}\\ e^{\beta J} & e^{-\beta J}\end{pmatrix}

6

COLBOIS | FRUSTRATED TNS |  01.2026

6

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Ground state:

S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
S = 0.3230659...

G.H. Wannier, PR 79, (1950, 1973)

\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}
{\color{orange}\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}

Boltzmann Weight

COLBOIS | FRUSTRATED TNS |  01.2026

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

1. Cancellation of big / small factors

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)

 

\(\rightarrow\) precision?

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

 

J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)

\(\rightarrow\) log / tropical algebra

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Rams et al, PRE 104, (2021)

Note: no approximate contraction scheme

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

Local tensor: 

{\color{orange}\tilde{t}} = \begin{pmatrix} 0.069 & 3.89\\ 3.89 & 0.069\end{pmatrix}

1. Cancellation of big / small factors

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)

 

\(\rightarrow\) precision?

 

J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)

\(\rightarrow\) log / tropical algebra

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Rams et al, PRE 104, (2021)

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

2. Low-temperature limit / emergent dofs

Vanderstraeten et al., PRE 98 (2018)

B. Vanhecke, JC, et al. PRR 3, (2021)

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, (2023)

1. Cancellation of big / small factors

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)

 

\(\rightarrow\) precision?

 

J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)

\(\rightarrow\) log / tropical algebra

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Rams et al, PRE 104, (2021)

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

3. MPO properties

W. Tang, F. Verstraete, J. Haegeman, PRB 111, (2025)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

Vanderstraeten et al., PRE 98 (2018)

B. Vanhecke, JC, et al. PRR 3, (2021)

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, (2023)

2. Low-temperature limit / emergent dofs

1. Cancellation of big / small factors

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)

 

\(\rightarrow\) precision?

 

J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)

\(\rightarrow\) log / tropical algebra

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Rams et al, PRE 104, (2021)

7

An exactly solvable example : the TIAFM

H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1

Vanhecke, JC et al (2021)

3. MPO properties

2. Low-temperature limit / emergent dofs

Vanderstraeten et al., PRE 98 (2018)

B. Vanhecke, JC, et al. PRR 3, (2021)

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, (2023)

W. Tang, F. Verstraete, J. Haegeman, PRB 111, (2025)

{\color{orange}\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}

COLBOIS | FRUSTRATED TNS |  01.2026

1. Cancellation of big / small factors

Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)

 

\(\rightarrow\) precision?

 

J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)

\(\rightarrow\) log / tropical algebra

C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)

Rams et al, PRE 104, (2021)

9

The ground-state local rule / emergent DOF

COLBOIS | FRUSTRATED TNS |  01.2026

{\color{orange}\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}
{\color{orange}\tilde{\omega}}

with all entries smaller or equal to 1? 

9

The ground-state local rule / emergent DOF

COLBOIS | FRUSTRATED TNS |  01.2026

{\color{orange}\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}
{\color{orange}\tilde{\omega}}

with all entries smaller or equal to 1? 

Dimer counting problem on a honeycomb lattice!

see e.g. Vanderstraeten et al, PRE 98 (2018)

9

The ground-state local rule / emergent DOF

COLBOIS | FRUSTRATED TNS |  01.2026

Dimer counting problem on a honeycomb lattice!

 Wannier,  PR  79 1950;

Kasteleyn, 1960;

Levin, Nave, PRL 99, 2007

{\color{orange}\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}
{\color{orange}\tilde{\omega}}

with all entries smaller or equal to 1? 

\tilde{\omega} (i,j,k) = \begin{cases} 1 & \text{if one index has value 2}\\ 0 & \text{otherwise} \end{cases}

see e.g. Vanderstraeten et al, PRE 98 (2018)

9

The ground-state local rule / emergent DOF

COLBOIS | FRUSTRATED TNS |  01.2026

Dimer counting problem on a honeycomb lattice!

\tilde{\omega} (i,j,k) = \begin{cases} 1 & \text{if one index has value 2}\\ \exp(-2 \beta J) & \text{if } i = j = k = 2\\ 0 & \text{otherwise} \end{cases}
{\color{orange}\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}
{\color{orange}\tilde{\omega}}

with all entries smaller or equal to 1? 

see e.g. Vanderstraeten et al, PRE 98 (2018)

 Wannier,  PR  79 1950;

Kasteleyn, 1960;

Levin, Nave, PRL 99, 2007

Xie et al, Phys. Rev. X 4 (2014) 

10

Hamiltonian splitting

COLBOIS | FRUSTRATED TNS |  01.2026

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Text

\(\omega\) on all triangles

Vanhecke, JC et al (2021)

\(E_{\mathrm{GS}}/N_{\triangle}\) = \(\min_{\vec{\sigma }}H_{\triangle}(\vec{\sigma})\)

Hamiltonian splitting

COLBOIS | FRUSTRATED TNS |  01.2026

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Text

Vanhecke, JC et al (2021)

10

\(\omega\) on all triangles

\(E_{\mathrm{GS}}/N_{\triangle}\) = \(\min_{\vec{\sigma }}H_{\triangle}(\vec{\sigma})\)

all entries smaller or equal to 1

Hamiltonian splitting

COLBOIS | FRUSTRATED TNS |  01.2026

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Text

Vanhecke, JC et al (2021)

10

\(\omega\) on all triangles

\(E_{\mathrm{GS}}/N_{\triangle}\) = \(\min_{\vec{\sigma }}H_{\triangle}(\vec{\sigma})\)

all entries smaller or equal to 1

Hamiltonian splitting

COLBOIS | FRUSTRATED TNS |  01.2026

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Text

Vanhecke, JC et al (2021)

10

\(\omega\) on all triangles

\(E_{\mathrm{GS}}/N_{\triangle}\) = \(\min_{\vec{\sigma }}H_{\triangle}(\vec{\sigma})\)

all entries smaller or equal to 1

Hamiltonian splitting

COLBOIS | FRUSTRATED TNS |  01.2026

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Text

Vanhecke, JC et al (2021)

JC, PhD Thesis

10

\(\omega\) on all triangles

\(E_{\mathrm{GS}}/N_{\triangle}\) = \(\min_{\vec{\sigma }}H_{\triangle}(\vec{\sigma})\)

all entries smaller or equal to 1

11

COLBOIS | FRUSTRATED TNS |  01.2026

Text

Vanhecke, JC et al (2021)

H_1 = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
H_2 = \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Does the way of splitting matter?

11

COLBOIS | FRUSTRATED TNS |  01.2026

Text

Same Hamiltonian in PBC

Vanhecke, JC et al (2021)

H_1 = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
H_2 = \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Does the way of splitting matter?

11

COLBOIS | FRUSTRATED TNS |  01.2026

Text

Boltzmann weight

(half for each bond)

Boltzmann weight

(half for each bond)

Kronecker Delta

Boltzmann weight

Does the way of splitting matter?

Vanhecke, JC et al (2021)

Same Hamiltonian in PBC

H_1 = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
H_2 = \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

COLBOIS | FRUSTRATED TNS |  01.2026

Text

11

Vanhecke, JC et al (2021)

Does the way of splitting matter?

Same Hamiltonian in PBC

H_1 = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)
H_2 = \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

A non-local Gauge transformation

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

A non-local Gauge transformation

12

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

See also, Nourhani et al, PRE 98 (2018)

Tang, Vestraete, Haegeman, PRB 111, (2025)

A non-local Gauge transformation

12

1. Transfer matrix \(\mathcal{T}_1\)

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

See also, Nourhani et al, PRE 98 (2018)

Tang, Vestraete, Haegeman, PRB 111, (2025)

A non-local Gauge transformation

12

1. Transfer matrix \(\mathcal{T}_1\)

- is normal : \(\mathcal{T}_1^{\dagger} \mathcal{T}_1  = \mathcal{T}_1 \mathcal{T}_1^{\dagger}\)

- conserves U(1) (conserves the sector)

\(U(1)\)

COLBOIS | FRUSTRATED TNS |  01.2026

See e.g. Kasteleyn 1960

1. Transfer matrix \(\mathcal{T}_1\)

- is normal : \(\mathcal{T}_1^{\dagger} \mathcal{T}_1  = \mathcal{T}_1 \mathcal{T}_1^{\dagger}\)

- conserves U(1) (conserves the sector)

See also, Nourhani et al, PRE 98 (2018)

1. Transfer matrix \(\mathcal{T}_2 = \mathcal{T}_1 + \Delta \mathcal{T}\)

- non-normal

- related to \(\mathcal{T}_1\) by a bond-dimension 2 MPO \(\mathcal{P}\) that allows to interpolate between them

- affects scaling of the entanglement entropy

Tang, Vestraete, Haegeman, PRB 111, (2025)

A non-local Gauge transformation

12

\(U(1)\)

13

What is needed to go beyond the tiafm

COLBOIS | FRUSTRATED TNS |  01.2026

What is needed to go beyond the tiafm

Numerical problem

Tropical algebra: approximate contraction?

COLBOIS | FRUSTRATED TNS |  01.2026

13

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Tropical algebra: approximate contraction?

COLBOIS | FRUSTRATED TNS |  01.2026

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

13

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Tropical algebra: approximate contraction?

Finding the ground-state local rule / emergent dof?

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

COLBOIS | FRUSTRATED TNS |  01.2026

13

Emergent d.o.fs: dual construction?

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Tropical algebra: approximate contraction?

Finding the ground-state local rule / emergent dof?

1. Beyond nearest-neighbor couplings

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

COLBOIS | FRUSTRATED TNS |  01.2026

image/svg+xml
image/svg+xml
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

13

Emergent d.o.fs: dual construction?

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Tropical algebra: approximate contraction?

Finding the ground-state local rule / emergent dof?

1. Beyond nearest-neighbor couplings

2. Beyond discrete spins: XY models

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

COLBOIS | FRUSTRATED TNS |  01.2026

image/svg+xml
image/svg+xml
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

13

Emergent d.o.fs: dual construction?

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Emergent d.o.fs: dual construction?

Tropical algebra: approximate contraction?

Finding the ground-state local rule / emergent dof?

1. Beyond nearest-neighbor couplings

2. Beyond discrete spins: XY models

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

COLBOIS | FRUSTRATED TNS |  01.2026

image/svg+xml
image/svg+xml
\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

13

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

What is needed to go beyond the tiafm

Numerical problem

MPO properties

Tropical algebra: approximate contraction?

Finding the ground-state local rule / emergent dof?

1. Beyond nearest-neighbor couplings

2. Beyond discrete spins: XY models

Given the (badly conditioned) MPO \(\rightarrow\) how to find the (non-local) gauge?

COLBOIS | FRUSTRATED TNS |  01.2026

image/svg+xml
image/svg+xml
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
\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

13

Emergent d.o.fs: dual construction?

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

14

LINEAR PROGRAM:

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

1. Split with clusters that overlap

H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

1. Split with clusters that overlap

2. Minimize : G.S. lower-bound 

\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

\max_{\alpha}\,\,\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)

3.  Maximize w.r.t the weights:

1. Split with clusters that overlap

2. Minimize : G.S. lower-bound 

\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

\max_{\alpha}\,\,\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)

3.  Maximize w.r.t the weights:

1. Split with clusters that overlap

2. Minimize : G.S. lower-bound 

\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

Obtain the ground states by tiling!

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

\max_{\alpha}\,\,\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)

3.  Maximize w.r.t the weights:

1. Split with clusters that overlap

2. Minimize : G.S. lower-bound 

\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

Obtain the ground states by tiling!

And more generally : 

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_{c'})

14

Farther-neighbor couplings:

Ising models as weighted counting problems

COLBOIS | FRUSTRATED TNS |  01.2026

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); Nagy et al; PRE 109 (2024)

Essential idea : Anderson bounds

LINEAR PROGRAM:

\max_{\alpha}\,\,\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)

3.  Maximize w.r.t the weights:

1. Split with clusters that overlap

2. Minimize : G.S. lower-bound 

\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)

Obtain the ground states by tiling!

And more generally : 

"Interactions round a face"

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_{c'})

14

R. J. Baxter, J. Stat. Phys. 19, (1978), T. Nishino, J.  Phys. Soc. Jpn. 64,  (1995), T. Nishino and K. Okunishi, J. Phys. Soc. Jpn. 66, (1997)

Xie et al, Phys. Rev. X 4 (2014) 

15

Remarks on The linear program and  symMetries

COLBOIS | FRUSTRATED TNS |  01.2026

The linear program is invariant under the cluster symmetry group \(G\)

 

15

Remarks on The linear program and  symMetries

COLBOIS | FRUSTRATED TNS |  01.2026

The linear program is invariant under the cluster symmetry group \(G\)

 

1. We can look for optimal solutions in the subspace of \(G\)-invariant solutions.

 

 

2. We can keep only one configuration per orbit when solving the linear program

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

16

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

16

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

What happens when your partition function cannot be exactly expressed as a TN?

16

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

What happens when your partition function cannot be exactly expressed as a TN?

1. Similar issue as TIAFM (on square XY)

16

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

What happens when your partition function cannot be exactly expressed as a TN?

1. Similar issue as TIAFM (on square XY)

2. A slightly different issue on the kagome: truncation leads to wrong results

16

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

 

(1) decomposition onto cylindrical harmonics

(2) split and group

Comparing two constructions for frustrated XY Models

\(t = \exp(-\beta J \cos(\theta_i - \theta_j)\)

17

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

 

(2) split and group

Comparing two constructions for frustrated XY Models

\(t = \exp(-\beta J \cos(\theta_i - \theta_j)\)

W_{\triangle} = \exp(-\beta \sum_{\langle i, j\rangle \in p} \cos(\theta_i - \theta_j))

(1) Fourier transform or decomposition onto Bessel functions

(2) split and group

17

(1) decomposition onto cylindrical harmonics

COLBOIS | FRUSTRATED TNS |  01.2026

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

see also FF. Song; GM Zhang, Ch. Phys. Letters (2025)

Comparing two constructions for frustrated XY Models

18

COLBOIS | FRUSTRATED TNS |  01.2026

First-order phase transition?

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

see also FF. Song; GM Zhang, Ch. Phys. Letters (2025)

Comparing two constructions for frustrated XY Models

18

COLBOIS | FRUSTRATED TNS |  01.2026

First-order phase transition?

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

see also FF. Song; GM Zhang, Ch. Phys. Letters (2025)

Comparing two constructions for frustrated XY Models

Converges

but not reliably

18

COLBOIS | FRUSTRATED TNS |  01.2026

First-order phase transition?

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

see also FF. Song; GM Zhang, Ch. Phys. Letters (2025)

Comparing two constructions for frustrated XY Models

Converges

but not reliably

18

COLBOIS | FRUSTRATED TNS |  01.2026

First-order phase transition?

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

see also FF. Song; GM Zhang, Ch. Phys. Letters (2025)

Comparing two constructions for frustrated XY Models

Converges

but not reliably

BKT

18

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

1. Finding the ground state rule / emergent dof: range of frustration?

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

Ronceray & Le Floch (2020)

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

1. Finding the ground state rule / emergent dof: range of frustration?

2. Making sure that no "spurious tile" spoils the convergence?

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

Ronceray & Le Floch (2020)

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

1. Finding the ground state rule / emergent dof: range of frustration?

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

Ronceray & Le Floch (2020)

3. More fundamentally: tiling problems 

2. Making sure that no "spurious tile" spoils the convergence?

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

1. Finding the ground state rule / emergent dof: range of frustration?

3. More fundamentally: tiling problems 

\(\rightarrow\) undecidability in general

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

Ronceray & Le Floch (2020)

2. Making sure that no "spurious tile" spoils the convergence?

19

Do we have a systematic solution?

COLBOIS | FRUSTRATED TNS |  01.2026

1. Finding the ground state rule / emergent dof: range of frustration?

3. More fundamentally: tiling problems 

\(\rightarrow\) practical cases?

\(\rightarrow\) undecidability in general

\mathcal{Z} = \sum_{s} \prod_{c} \omega^{\alpha}(s|_c) \prod_{\langle c, c' \rangle} \delta(s|_c, s|_c')

Ronceray & Le Floch (2020)

2. Making sure that no "spurious tile" spoils the convergence?

Range of the constraint : Beyond dimers

Nathan Perruchoud

Afonso Rufino

Multimer modelS

20

COLBOIS | FRUSTRATED TNS |  01.2026

Fig. from N. Perruchoud's Master thesis

Given a lattice : packing polymers of length \(k\)

Multimer modelS

20

COLBOIS | FRUSTRATED TNS |  01.2026

Fig. from N. Perruchoud's Master thesis

Given a lattice : packing polymers of length \(k\)

Multimer modelS

20

COLBOIS | FRUSTRATED TNS |  01.2026

Fig. from N. Perruchoud's Master thesis

RVB state properties?

Given a lattice : packing polymers of length \(k\)

See eg. Giudice et al, PRB 106 (2022),

Zhang et al, Commun. Mat. 6 (2025)

Multimer modelS

20

COLBOIS | FRUSTRATED TNS |  01.2026

Fig. from N. Perruchoud's Master thesis

RVB state properties?

Statistical mechanics ? 

Given a lattice : packing polymers of length \(k\)

See eg. Giudice et al, PRB 106 (2022),

Zhang et al, Commun. Mat. 6 (2025)

See next slide

Focusing on straight rods

21

COLBOIS | FRUSTRATED TNS |  01.2026

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Low-density disordered

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Low-density disordered

Intermediate density nematic

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Low-density disordered

Intermediate density nematic

High density disordered

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Low-density disordered

Intermediate density nematic

High density disordered

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

A. Ghosh and D. Dhar. Euro. Letters, 78 (2007).

  • strong evidence of nematic phase for \(k \geq 7\)

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Low-density disordered

Intermediate density nematic

High density disordered

  • strong evidence of nematic phase for \(k \geq 7\)
  • proof of the existence of the nematic phase for \(k \rightarrow \infty\)

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Disertori and Giuliani.,Comm. Math.Phys, 323, (2013).

A. Ghosh and D. Dhar. Euro. Letters, 78 (2007).

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Low-density disordered

Intermediate density nematic

High density disordered

  • strong evidence of nematic phase for \(k \geq 7\)
  • proof of the existence of the nematic phase for \(k \rightarrow \infty\)
  • first phase transition in the Ising universality class

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Disertori and Giuliani.,Comm. Math.Phys, 323, (2013).

A. Ghosh and D. Dhar. Euro. Letters, 78 (2007).

Matoz-Fernandez, et al, Europhys. Letters, 82 (2008).

21

Focusing on straight rods

COLBOIS | FRUSTRATED TNS |  01.2026

Low-density disordered

Intermediate density nematic

High density disordered

  • strong evidence of nematic phase for \(k \geq 7\)
  • proof of the existence of the nematic phase for \(k \rightarrow \infty\)
  • first phase transition in the Ising universality class
  • nature of the second phase transition?

Fig: Shah et al, PRE 105 (2022)

Nice review: Shah et al, PRE 105 (2022)

Disertori and Giuliani.,Comm. Math.Phys, 323, (2013).

A. Ghosh and D. Dhar. Euro. Letters, 78 (2007).

Matoz-Fernandez, et al, Europhys. Letters, 82 (2008).

21

Grand-canonical partition function

COLBOIS | FRUSTRATED TNS |  01.2026

\mathcal{Z} = \sum_{N_{-}} \Omega(N_{-}) z_{-}^{N_{-}} \propto \sum_{\mathcal{C}(-)} \prod_{e} \omega_{e}
\omega_e = e^{-\beta\mu/3}

22

Grand-canonical partition function

COLBOIS | FRUSTRATED TNS |  01.2026

22

\mathcal{Z} = \sum_{N_{-}} \Omega(N_{-}) z_{-}^{N_{-}} \propto \sum_{\mathcal{C}(-)} \prod_{e} \omega_{e}
\omega_e = e^{-\beta\mu/3}

Contraction with CTMRG :

  • issues (see e.g.  Chatelain & Gendiar, (2020))
  • group at least 2x2

N. Perruchoud Master's thesis

Giudice etal. PRB 106, (2022)

  • trimers: \(U(1) \times U(1)\) symmetry
  • translation-invariant representation requires a 3x3 block

Grand-canonical partition function

22

COLBOIS | FRUSTRATED TNS |  01.2026

Contraction with CTMRG :

  • issues (see e.g.  Chatelain & Gendiar, (2020))
  • group at least 2x2

N. Perruchoud Master's thesis

Giudice etal. PRB 106, (2022)

  • trimers: \(U(1) \times U(1)\) symmetry
  • translation-invariant representation requires a 3x3 block
\mathcal{Z} = \sum_{N_{-}} \Omega(N_{-}) z_{-}^{N_{-}} \propto \sum_{\mathcal{C}(-)} \prod_{e} \omega_{e}
\omega_e = e^{-\beta\mu/3}

CTMRG (extraplation): 0.158496(14)    

TM with extapolation: 0.158539(37);  0.158520(15)

Serra et al,  (2023)

Ghosh et al,  (2023)

23

COLBOIS | FRUSTRATED TNS |  01.2026

Remark on Emergent non-local dofs 

Afonso Rufino

image/svg+xml
image/svg+xml
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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

A. Rufino, et al., arXiv:2505.05889 (to appear in PRL)

 

COLBOIS | FRUSTRATED TNS |  01.2026

Remark on Emergent non-local dofs 

Afonso Rufino

image/svg+xml
image/svg+xml

\(J_1, J_3 \rightarrow \infty\)

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

some of the constraints

A. Rufino, et al., arXiv:2505.05889 (to appear in PRL)

 

23

COLBOIS | FRUSTRATED TNS |  01.2026

Remark on Emergent non-local dofs 

Afonso Rufino

image/svg+xml
image/svg+xml

\(J_1, J_3 \rightarrow \infty\)

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_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
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

some of the constraints

emergent string degrees of freedom

leading to a topological staircase

A. Rufino, et al., arXiv:2505.05889 (to appear in PRL)

 

23

Conclusions

Take-home messages

24

COLBOIS | FRUSTRATED TNS |  01.2026

Take-home messages

COLBOIS | FRUSTRATED TNS |  01.2026

1. Simple stat mech models written as exact TNs can lead to contraction failure

24

Take-home messages

COLBOIS | FRUSTRATED TNS |  01.2026

1. Simple stat mech models written as exact TNs can lead to contraction failure

2. Strategies: tropical algebra; gauging; finding the frustration-free cluster

Rules-of-thumb:

- normal MPO

- lattice symmetries

- emergent dofs : interactions-round-a-face

24

Take-home messages

COLBOIS | FRUSTRATED TNS |  01.2026

1. Simple stat mech models written as exact TNs can lead to contraction failure

3. Families of frustrated spin systems where we have a well-behaved formulation

Learn strategies to fix the gauge?

Topological devil's staircase: 

non-local emergent dofs.

2. Strategies: tropical algebra; gauging; finding the frustration-free cluster

Rules-of-thumb:

- normal MPO

- lattice symmetries

- emergent dofs : interactions-round-a-face

24

Take-home messages

COLBOIS | FRUSTRATED TNS |  01.2026

1. Simple stat mech models written as exact TNs can lead to contraction failure

3. Families of frustrated spin systems where we have a well-behaved formulation

Learn strategies to fix the gauge?

Thank you!

Topological devil's staircase: 

non-local emergent dofs.

2. Strategies: tropical algebra; gauging; finding the frustration-free cluster

Rules-of-thumb:

- normal MPO

- lattice symmetries

- emergent dofs : interactions-round-a-face

24

Bonus slides

Remark on U(1) symmetries and Back to dimers

COLBOIS | FRUSTRATED TNS |  01.2026

Giudice etal. PRB 106, (2022)

  • trimers: \(U(1) \times U(1)\) symmetry
  • translation-invariant representation requires a 3x3 block

Remark by Juraj Hasik: 

  • enforcing U(1) symmetry for dimers on the honeycomb lattice \(\rightarrow\) /!\ boundary conditions

 

  • could explain missing scaling of the entanglement entropy? 

F. Pollmann, et al. PRL 102 (2009)

L. Tagliacozzoet al. PRB 78 (2008)

S. Nyckees et al. PRE 108 (2023)

F. Pollmann, et al. PRL 102 (2009)

L. Tagliacozzoet al. PRB 78 (2008)

Multi-site CTMRG failure

COLBOIS | FRUSTRATED TNS |  01.2026

3x3

Afonso Rufino

3x3

2x2

Remark on non-local emergent dofs

COLBOIS | FRUSTRATED TNS |  01.2026

\(E \propto L \)

Nearest-neighbor anisotropic

Smerald & Mila, Scipost (2019)

Constrained limit \(J \gg T, \delta \)

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

Ground-states: U(1) + 3-states clock

Huse, Rosenfeld, PRB 45, (1992)

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

Ground-states: U(1) + 3-states clock

Residual entropy due to 3-states: \(S = 0.126...\)

Huse, Rosenfeld, PRB 45, (1992)

R. J. Baxter, J. Math. Phys. 11, (1970)

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

Beyond discrete spins: frustrated XY Models

COLBOIS | FRUSTRATED TNS |  01.2026

Ground-states: U(1) + 3-states clock

Residual entropy due to 3-states: \(S = 0.126...\)

Huse, Rosenfeld, PRB 45, (1992)

R. J. Baxter, J. Math. Phys. 11, (1970)

F.F. Song, G. M. Zhang, PRB 108, 014424 (2023)

 

F.F. Song, T.-Y. Lin, G. M. Zhang, PRB 108, 224404 (2023)

(see also FF Song, H Nuomin, N Kawashima PRB 111 (2025))

H = J\sum_{\langle i,j \rangle} \cos(\theta_i-\theta_j)

Question:

- BKT transition, or

- pre-empted by a first-order or 2nd order transition in the chiralities?

R. J. Baxter, J. Math. Phys. 11, (1970)

COLBOIS | FRUSTRATED TNS |  01.2026

\chi = 40

\(10^{-8}\) precision on the partition function per site

CTMRG (extraplation): 0.158496(14)    

TM with extapolation: 0.158539(37);  0.158520(15)

Serra et al,  (2023)

N. Perruchoud Master's thesis

Ghosh et al,  (2023)

High-density limit

Tropical algebra