Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Tensor networks
for classical frustrated spin systems
Jeanne Colbois | Institut Néel
Journée Théorie CPTGA 2025 - 08/10/2025
Tensor networks
for classical frustrated spin systems
Frustrated magnetism
Classical statistical mechanics
Jeanne Colbois | Institut Néel
Journée Théorie CPTGA 2025 - 08/10/2025
Tensor networks
for classical frustrated spin systems
Frustrated magnetism
Classical statistical mechanics
Tensor networks
Jeanne Colbois | Institut Néel
Journée Théorie CPTGA 2025 - 08/10/2025
AcknowledgEments
1
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Andrew Smerald
KIT | Germany
Frédéric Mila
EPFL | Switzerland
Frank Verstraete
Ghent University | Belgium
Laurens Vanderstraeten
Ghent University | Belgium
Samuel Nyckees
EPFL | Switzerland
Afonso Rufino
EPFL | Switzerland
Bram Vanhecke
University of Vienna | Austria
SCOPE
1. Classical frustrated magnetism?
2. Tensor networks for classical statistical mechanics
3. Frustrated magnetism : Ising models as weighted counting problems
4. Some applications & perspectives
2
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Frustrated magnetism
What is it?
Why study it?
Why is it difficult?
ising models of frustrated magnetism
3
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
ising models of frustrated magnetism
3
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
ising models of frustrated magnetism
3
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Magnetic order
Ideal paramagnet
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
ising models of frustrated magnetism
3
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Magnetic order
Ideal paramagnet
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
ising models of frustrated magnetism
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
Competing interactions
\(2100\) sites : \(2^{700} \) ground states!
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
W_{G.S.} \gtrsim 2^{N/3}
Magnetic order
Ideal paramagnet
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
3
ising models of frustrated magnetism
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
Competing interactions
\(2100\) sites : \(2^{700} \) ground states!
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
W_{G.S.} \gtrsim 2^{N/3}
Magnetic order
Ideal paramagnet
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
3
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
= 0.501833...
ising models of frustrated magnetism
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
Competing interactions
\(2100\) sites : \(2^{700} \) ground states!
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
W_{G.S.} \gtrsim 2^{N/3}
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
= 0.501833...
\xi = 1.2506...
A. Sütö, Z. Phys. B 44, (1981)
W. Apel, H.-U. Everts, J. Stat. Mech, (2011)
Magnetic order
Ideal paramagnet
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
Spin liquid
\(q_x\)
\(q_y\)
3
Emergent phenomena from the local constraints
4
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\(\xi = \infty\)
Fennell et al. (2009)
Neutron scattering on Ho2Ti2O7
Emergent phenomena from the local constraints
Ramirez et al (1999) : Dy2Ti2O7
4
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
Constraint
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\(\xi = \infty\)
Fennell et al. (2009)
Neutron scattering on Ho2Ti2O7
Emergent phenomena from the local constraints
Ramirez et al (1999) : Dy2Ti2O7
4
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
Constraint
Classical / quantum electrodynamics
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\(\xi = \infty\)
Fennell et al. (2009)
Neutron scattering on Ho2Ti2O7
Effective Coulomb interaction \(|F| \propto q^2/r^2\)
Henley (2005, 2010), Castelnovo etal(2008)
Emergent phenomena from the local constraints
5
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Magnetic order
Spin liquids
Emergent phenomena from the local constraints
5
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Magnetic order
Spin liquids
Magnetic moment fragmentation
Brooks-Bartlett et al (2014), Canals et al (2016), Petit et al (2016)
Each spin participates to both phases!
Solving ising models is hard
1D Ising
Solved exactly 1925
Planar Ising models
Solved exactly 1944 / 1960
Non-planar Ising models :
no closed forms
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
6
Ising
Onsager
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Solving ising models is hard
1D Ising
Solved exactly 1925
Planar Ising models
Solved exactly 1944 / 1960
Non-planar Ising models :
no closed forms
A version of the N-Body probleM
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
6
Ising
Onsager
Counting the number of ground states in spin glasses
#P-complete
Barahona (1982)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Solving ising models is hard
1D Ising
Solved exactly 1925
Planar Ising models
Solved exactly 1944 / 1960
Non-planar Ising models :
no closed forms
A version of the N-Body probleM
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
6
Ising
Onsager
Counting the number of ground states in spin glasses
#P-complete
Methods:
Monte Carlo, series expansion, RG & CFT
Barahona (1982)
Ergodicity issues
Limited to some regimes
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Tensor networks for statistical mechanics
What is a tensor network ?
Tensor networks as compression schemes
Ising models as tensor networks
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Extremely efficient classical computing methods
Based on "clever" compression of data
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Extremely efficient classical computing methods
Based on "clever" compression of data
1D quantum
\(S=1\) Heisenberg chain
Symmetry-protected topological phases
...
White (1992), ...
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Extremely efficient classical computing methods
Based on "clever" compression of data
1D quantum
2D quantum and more
\(S=1\) Heisenberg chain
Symmetry-protected topological phases
...
Topological order
Two-dimensional t-J model
...
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Extremely efficient classical computing methods
Based on "clever" compression of data
1D quantum
2D quantum and more
\(S=1\) Heisenberg chain
Symmetry-protected topological phases
...
Topological order
Two-dimensional t-J model
...
Simulation of quantum circuits Challenging quantum supremacy
...
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
Vidal (2003),
Zhou, Stoudenmire & Waintal (2020), ...
Quantum computing
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Extremely efficient classical computing methods
Based on "clever" compression of data
1D quantum
2D quantum and more
\(S=1\) Heisenberg chain
Symmetry-protected topological phases
...
Topological order
Two-dimensional t-J model
...
Simulation of quantum circuits Challenging quantum supremacy
...
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
Vidal (2003),
Zhou, Stoudenmire & Waintal (2020), ...
Chemistry, machine learning, mathematics...
Quantum computing
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
7
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
8
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Type
Notation
Visualization
Tensor networks notation
Scalar
Vector
Matrix
Rank-3 tensor
"Legs"
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
8
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Type
Notation
Visualization
Tensor networks notation
Scalar
Vector
Matrix
Rank-3 tensor
"Legs"
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
8
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Type
Notation
Visualization
Tensor networks notation
Scalar
Vector
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
"Legs"
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
8
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Type
Notation
Visualization
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Connecting legs = make the product
"CONTRACTION"
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Scalar product
Connecting legs = make the product
"CONTRACTION"
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Scalar product
Matrix-vector product
Connecting legs = make the product
"CONTRACTION"
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Scalar product
Matrix-vector product
Connecting legs = make the product
"CONTRACTION"
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
You can group indices:
\(\chi \times\chi \times \chi \times \chi\)
tensor
=
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
Tensor networks notation
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Scalar product
Matrix-vector product
Connecting legs = make the product
"CONTRACTION"
"Legs"
Size of the index = bond dimension = \(\chi\) or \(D\)
You can group indices:
\(\chi \times\chi \times \chi \times \chi\)
tensor
\(\chi^2 \times\chi^2\)
matrix
=
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
8
compressing many-body Wavefunctions
9
Many-body wavefunction
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
9
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
9
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
We want to "factorize" or compress it:
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
why (When) do We expect the bond dimension to be limited?
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
We want to "factorize" or compress it:
9
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
ENTANGLEMENT (area law)
We want to "factorize" or compress it:
9
why (When) do We expect the bond dimension to be limited?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
Many-body Hilbert space
We want to "factorize" or compress it:
9
why (When) do We expect the bond dimension to be limited?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
Many-body Hilbert space
\propto L
We want to "factorize" or compress it:
9
why (When) do We expect the bond dimension to be limited?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
Many-body Hilbert space
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
\propto L
We want to "factorize" or compress it:
why (When) do We expect the bond dimension to be limited?
9
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
Many-body Hilbert space
Ground states of gapped, local Hamiltonians
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
\propto L
We want to "factorize" or compress it:
9
why (When) do We expect the bond dimension to be limited?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
Many-body Hilbert space
Ground states of gapped, local Hamiltonians
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
\propto \mathrm{const}
\propto L
We want to "factorize" or compress it:
9
why (When) do We expect the bond dimension to be limited?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
Many-body Hilbert space
Ground states of gapped, local Hamiltonians
ENTANGLEMENT (area law)
|\phi\rangle = \sum_{\{s_j\}} \phi_{\dots, s_{j-1}, s_j, s_{j+1}, \dots} |\dots, s_{j-1}, s_{j}, s_{j+1}, \dots\rangle
High number of parameters
(2^L)
Much smaller number
(\chi \times 2 \times \chi)
\propto \mathrm{const}
\propto L
9
why (When) do We expect the bond dimension to be limited?
We want to "factorize" or compress it:
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\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}
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
The partition function is just the exponentiation of a 2x2 matrix!
\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 = (T^L)
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
The partition function is just the exponentiation of a 2x2 matrix!
\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}
1. Diagonalize
\mathcal{Z}_L = (T^L)
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P
The partition function is just the exponentiation of a 2x2 matrix!
\Lambda^{L}
= \lambda_{+}^{L} \begin{pmatrix}
1& 0 \\
0 & \left(\frac{\lambda_{-}}{\lambda_{+}}\right)^L
\end{pmatrix}
\xrightarrow[L \to \infty]{}
\lambda_{+}^{L} \begin{pmatrix}
1& 0 \\
0 & 0
\end{pmatrix}
\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 = (T^L)
1. Diagonalize
2. Compute
10
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Leading eigenvalue!
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
11
The 2D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
\mathcal{Z} =
11
The 2D ising model AS a tensor network
Decomposition & reshaping:
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
2^L
11
The 2D ising model AS a tensor network
Decomposition & reshaping:
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
2^L
Evaluate the partition function ?
Goldenfeld & Kadanoff, Science, 284 (1999)
12
Three main schemes
\mathcal{Z} =
2^L
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
12
Three main schemes
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
\chi
\mathcal{Z} =
2^L
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
12
Three main schemes
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
\chi
\mathcal{Z} =
2^L
2D classical is "like" 1D quantum
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
12
Three main schemes
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
R. J. Baxter, 1968; T. Nishino, K. Okunishi, 1996;
Corboz et al (2014), ...
\chi
\chi
\mathcal{Z} =
2^L
Building block for quantum problems : algorithms are already optimized
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
12
Three main schemes
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
R. J. Baxter, 1968; T. Nishino, K. Okunishi, 1996;
Corboz et al (2014), ...
Levin & Nave, 2007; Evenbly & Vidal (2014); Ebel, Kennedy, Rychkov (2025)....
\chi
\chi
\mathcal{Z} =
2^L
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
12
Three main schemes
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
R. J. Baxter, 1968; T. Nishino, K. Okunishi, 1996;
Corboz et al (2014), ...
Levin & Nave, 2007; Evenbly & Vidal (2014); Ebel, Kennedy, Rychkov (2025)....
\rightarrow
\chi
\chi
\chi
\mathcal{Z} =
2^L
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
13
Observables
\(\langle O \rangle= \)
Local observable:
\(\mathcal{Z}\)
Correlations:
\xi = - \ln \frac{\lambda_2}{\lambda_1}
Correlation length:
\(\langle O_i O_j \rangle =\)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Applications : Discrete spins
2D square lattice Ising model
14
Orús, Vidal, PRB 78, 2008
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Applications : Discrete spins
2D square lattice Ising model
14
Orús, Vidal, PRB 78, 2008
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
H = - \sum_{\vec{r}} \cos \left(\frac{2\pi}{3}(n_{\vec{r}+\vec{x}}-n_{\vec{r}}+\Delta)\right)\\ - \sum_{\vec{r}}\cos \left(\frac{2\pi}{3}(n_{\vec{r}+\vec{y}}-n_{\vec{r}})\right)
T
\Delta
\nu_x = 2/3\quad \nu_y = 1 \\
z = 3/2 \quad \bar{\beta} = 2/3
Nyckees, JC, Mila, NPB (2021)
2D chiral Potts model
Frustrated magnets
A simple case
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
15
A simple case
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
15
A simple case
Vanhecke, JC et al (2021)
Fails in the presence of frustration and macroscopic g.s. degeneracy
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
15
A simple case
Vanhecke, JC et al (2021)
Fails in the presence of frustration and macroscopic g.s. degeneracy
B. Vanhecke, JC, et al. PRR 3, (2021)
\(\rightarrow\) in spin glasses
\(\rightarrow \) in translation-invariant frustrated Ising models
\(\rightarrow\) in lattice gas models
\(\rightarrow\) in frustrated XY models
S. A. Akimenko, PRE 107, (2023)
F.F. Song, T.-Y. Lin, G. M. Zhang, PRB (2023)
C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)
Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)
J. G. Liu, L. Wang, P. Zhang, PRL 126, (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
15
The problem
16
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
The problem
16
Numerical problem
Cancellation of small and large factors
\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}
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?
\mathcal{Z}_{\triangle} = \tilde{t}^3
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
The problem
16
Numerical problem
Cancellation of small and large factors
\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}
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?
\mathcal{Z}_{\triangle} = \tilde{t}^3
Bad gauge
The transfer matrix is badly conditioned (e.g. not hermitian, ...)
W. Tang et al, (2024, 2025)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
The problem
16
Numerical problem
Ground-state rule
Cancellation of small and large factors
\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, PRB (2023)
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?
\mathcal{Z}_{\triangle} = \tilde{t}^3
Bad gauge
The transfer matrix is badly conditioned (e.g. not hermitian, ...)
W. Tang et al, (2024, 2025)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
The problem
16
Numerical problem
Ground-state rule
Cancellation of small and large factors
\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, PRB (2023)
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?
\mathcal{Z}_{\triangle} = \tilde{t}^3
Bad gauge
The transfer matrix is badly conditioned (e.g. not hermitian, ...)
W. Tang et al, (2024, 2025)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
17
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
18
17
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
17
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
17
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
Vanhecke, JC et al (2021)
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
17
Ground state:
Vanhecke, JC et al (2021)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
17
Finite temperature
\(2 \times 2 \times 2\)
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
18
\(2 \times 2 \times 2\)
\(=e^{-\beta J}\)
Same structure and size
Different entries
Finite temperature
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
18
\(2 \times 2 \times 2\)
\(=e^{-\beta J}\)
Same structure and size
Different entries
Finite temperature
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
18
More complex Ising models?
19
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
More complex Ising models?
H =
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
19
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
More complex Ising models?
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
19
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
More complex Ising models?
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
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
19
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
More complex Ising models?
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
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
Can we still find the constraint?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
19
20
Ising models as weigthed counting problems
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
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)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
20
Ising models as weigthed counting problems
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
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)
LINEAR PROGRAM:
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
20
Ising models as weigthed counting problems
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
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)
LINEAR PROGRAM:
1. Split with clusters that overlap
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
20
Ising models as weigthed counting problems
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
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)
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)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
20
Ising models as weigthed counting problems
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
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)
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)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
20
Ising models as weigthed counting problems
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
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)
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!
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Applications and perspectives
Three unexpected spin liquid phases
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
21
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3}} \sigma_i \sigma_j
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
21
Three unexpected spin liquid phases
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
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
21
Three unexpected spin liquid phases
JC, B. Vanhecke et. al., PRB 106 (2022)
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
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
21
JC, B. Vanhecke et. al., PRB 106 (2022)
Three unexpected spin liquid phases
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
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
22
A cascade of topological phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
An infinite series of plateaus
in the ratios of densities of 2 types of system-spanning strings
Perspectives
23
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Tensor network methods
"Classical" and quantum frustrated magnetism
Perspectives
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Consequences for studying 2D quantum many-body problems ?
Dealing with non-local constraints ?
Combining with Monte Carlo methods?
Wei Tang et al (2024, 2025)
Châtelain & Gendiar (2020)
Frias-Perez et al (2023)
Tensor network methods
"Classical" and quantum frustrated magnetism
23
Perspectives
Consequences for studying 2D quantum many-body problems ?
Dealing with non-local constraints ?
Range of frustration: hard versus "weak" frustration ?
Combining with Monte Carlo methods?
Wei Tang et al (2024, 2025)
Châtelain & Gendiar (2020)
Frias-Perez et al (2023)
Ronceray & Le Floch (2020)
Interpretation of "classical" entanglement ?
Carignano et al (2024)
Tensor network methods
"Classical" and quantum frustrated magnetism
A route to quantum-classical correspondences?
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Allegra et al (2016)
23
Take-home message
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
Constrained models in statistical mechanics shed light
on tensor network methods (and challenges)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
Thank you for your attention!
Constrained models in statistical mechanics shed light
on tensor network methods (and challenges)
COLBOIS|TNS FOR CLASSICAL FRUSTRATED MAGNETISM | 10.2025
Bonus slides
Wikipedia, CC BY license
Image compression
Julia Yeomans
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Wikipedia, CC BY license
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Wikipedia, CC BY license
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Wikipedia, CC BY license
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Wikipedia, CC BY license
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Wikipedia, CC BY license
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Wikipedia, CC BY license
\(\chi = 4\)
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Wikipedia, CC BY license
\(\chi = 4\)
\(\chi = 20\)
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Wikipedia, CC BY license
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
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Wikipedia, CC BY license
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
Image compression
Image compression
Always ask : why do I expect the bond dimension to be limited?
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Wikipedia, CC BY license
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
Tensor networks for frustrated classical Ising models
By Jeanne Colbois
Tensor networks for frustrated classical Ising models
Invited talk at CPTGA workshop
