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
GDR Physique quantique mésoscopique - Aussois 2025 - 03/12/2025
Tensor networks
for classical (FRUSTRATED) spin systems
Jeanne Colbois | Institut Néel
everything commutes
Classical Models on a lattice?
GDR Physique quantique mésoscopique - Aussois 2025 - 03/12/2025
AcknowledgEments
2
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
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
SCOPE
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
3
SCOPE
1. Motivation: classical models
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
3
SCOPE
1. Motivation: classical models
2. Statistical mechanics and tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
3
SCOPE
1. Motivation: classical models
2. Statistical mechanics and tensor networks
3. Frustrated magnetism and tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
3
SCOPE
1. Motivation: classical models
2. Statistical mechanics and tensor networks
3. Frustrated magnetism and tensor networks
4. Broader perspective: open challenges
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
3
3
Motivation: Classical models
Why study them?
Why I study them: frustration, exponential ground state degeneracy
(Why) do we need new techniques?
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
4
Why study classical models
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Position of a molecule
I. A. Chioar, et al PRB 93, (2016)
Afonso-Moro et al. (2023)
Degree of freedom is captured by a classical variable
Nanomagnet magnetization
4
Why study classical models
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Simplified models of quantum (many-body) systems
Position of a molecule
I. A. Chioar, et al PRB 93, (2016)
Afonso-Moro et al. (2023)
Degree of freedom is captured by a classical variable
Nanomagnet magnetization
4
Why study classical models
Why study classical models
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Degree of freedom is captured by a classical variable
Simplified models of quantum (many-body) systems
Quantum models mapped to classical models -- see next talk!
Position of a molecule
I. A. Chioar, et al PRB 93, (2016)
Nanomagnet magnetization
Afonso-Moro et al. (2023)
4
Characterizing classical systems
5
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
partition function
Characterizing classical spin systems
6
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
6
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
T
\(C_v\)
Spin up
Spin down
Thermodynamic properties
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N} \\ \rightarrow 0
Characterizing classical spin systems
6
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Thermodynamic properties
Magnetic order
\(q_x\)
\(q_x\)
\(\xi = \infty\)
\(\xi = 0\)
\(q_y\)
\(q_y\)
Ideal paramagnet
T
\(C_v\)
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N} \\ \rightarrow 0
Spin up
Spin down
Correlation functions
Characterizing classical spin systems
classical models of Frustrated magnets
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
Spin up
Spin down
7
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
W_{G.S.} \gtrsim 2^{N/3}
Spin up
Spin down
7
\(2100\) sites : \(2^{700} \) ground states!
classical models of Frustrated magnets
(vs \(2^{265} \) atoms in the universe)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
W_{G.S.} \gtrsim 2^{N/3}
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N} \\ \rightarrow 0.501833...
Spin up
Spin down
Kano, Naya (1958)
7
Thermodynamic properties
\(2100\) sites : \(2^{700} \) ground states!
classical models of Frustrated magnets
(vs \(2^{265} \) atoms in the universe)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
\(2100\) sites : \(2^{700} \) ground states!
W_{G.S.} \gtrsim 2^{N/3}
\xi = 1.2506...
A. Sütö, Z. Phys. B 44, (1981)
W. Apel, H.-U. Everts, J. Stat. Mech, (2011)
\(q_x\)
\(q_y\)
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N} \\ \rightarrow 0.501833...
Spin up
Spin down
Kano, Naya (1958)
7
Correlation functions
Thermodynamic properties
classical models of Frustrated magnets
(vs \(2^{265} \) atoms in the universe)
8
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Exotic phases and phase transitions
(From local constraints)
8
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Exotic phases and phase transitions
(From local constraints)
"Mixed-order"
Kasteleyn phase transition
\(T\)
\(C_V\)
Kasteleyn (1960's), Nagle et al(1989), Jaubert et al (2008)
More complex Ising models?
9
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
More complex Ising models?
H =
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
9
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
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
9
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
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
9
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
Z. Luo et al. Science 363, (2019)
JC et al., PRB 104 (2021)
Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
10
High-temperature series expansion
Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
10
High-temperature series expansion
Monte Carlo methods
Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
10
High-temperature series expansion
Renormalization group
Monte Carlo methods
Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
10
High-temperature series expansion
Renormalization group
Transfer matrix
Monte Carlo methods
Solving Classical models can be hard
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
10
High-temperature series expansion
Renormalization group
Transfer matrix
Monte Carlo methods
Tensor networks And statistical mechanics
Writing partition functions as tensor networks?
Observables?
One-Slide: tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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One-Slide: tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
11
One-Slide: tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
11
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
One-Slide: tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
11
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
One-Slide: tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
11
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
One-Slide: tensor networks
11
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
One-Slide: tensor networks
11
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Scalar
Vector
Matrix
Rank-3 tensor
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Text
Scalar product
Matrix-vector product
CONTRACTION
Only 2 legs can meet!
Conceptual shift
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Given:
a complicated function
Conceptual shift
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Given:
a complicated function
Question:
approximate it as a tensor network?
Conceptual shift
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Given:
a complicated function
OR
Given:
a Hamiltonian
Question:
approximate it as a tensor network?
Conceptual shift
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
Given:
a complicated function
OR
Given:
a Hamiltonian
Question:
approximate its ground-state wavefunction as a tensor network?
Question:
approximate it as a tensor network?
Conceptual shift
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Given:
a complicated function
OR
Given:
a Hamiltonian
Question:
approximate its ground-state wavefunction as a tensor network?
Given:
a partition function
exact tensor network
Question:
approximate it as a tensor network?
Conceptual shift
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
12
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Given:
a complicated function
OR
Given:
a Hamiltonian
Question:
Can you evaluate it?
Given:
a partition function
exact tensor network
Question:
approximate its ground-state wavefunction as a tensor network?
Question:
approximate it as a tensor network?
Conceptual shift
Yuriel's talk
(mostly 1D quantum)
This talk
(mostly 2D classical)
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
13
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Ising (1924)
\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}
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
13
Ising (1924)
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)
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Ising (1924)
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)
The 1D ising model AS a tensor network
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
13
Ising (1924)
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
The 1D ising model AS a tensor network
Leading eigenvalue!
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ising (1924)
2 examples of 2d tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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2 examples of 2d tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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2D Ising model
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Generalized kronecker \(\delta\) tensor
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
14
Dimer counting
# of ways to put dimers on the edges of a square lattice?
Baxter 1968 : First "tensor network" equations to solve this problem.
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
14
Dimer counting
# of ways to put dimers on the edges of a square lattice?
Baxter 1968 : First "tensor network" equations to solve this problem.
\(\Omega \approx 1,3385^ N\)
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
14
Dimer counting
# of ways to put dimers on the edges of a square lattice?
\(2 \times 2 \times 2 \times 2\)
Baxter 1968 : First "tensor network" equations to solve this problem.
\(\Omega \approx 1,3385^ N\)
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
2 examples of 2d tensor networks
\mathcal{Z} =
2D Ising model
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
14
Dimer counting
\(\Omega \approx 1,3385^ N\)
# of ways to put dimers on the edges of a square lattice?
\(2 \times 2 \times 2 \times 2\)
=
= ... = 1
=
= ... = 0
Baxter 1968 : First "tensor network" equations to solve this problem.
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Given by the Hamiltonian
Evaluate the partition function? Three main schemes
15
1. Boundary-MPS methods
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
2. Corner transfer matrix renormalization group
3. Tensor network renormalization
2d classical is like 1d quantum
Approximate the infinite environment
Real-space renormalization
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Given by the Hamiltonian
R. J. Baxter, J. Math. Phys. 9, 1968;
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Levin & Nave, 2007; Gu & Wen (2009); Evenbly & Vidal (2014); Ebel, Kennedy, Rychkov (2025)....
Evaluate the partition function? Three main schemes
15
1. Boundary-MPS methods
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
2. Corner transfer matrix renormalization group
3. Tensor network renormalization
2d classical is like 1d quantum
Approximate the infinite environment
Real-space renormalization
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Given by the Hamiltonian
R. J. Baxter, J. Math. Phys. 9, 1968;
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Levin & Nave, 2007; Gu & Wen (2009); Evenbly & Vidal (2014); Ebel, Kennedy, Rychkov (2025)....
Evaluate the partition function? Three main schemes
15
BoundarY matrix product states methods
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
16
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
BoundarY matrix product states methods
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
16
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
BoundarY matrix product states methods
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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\chi
BoundarY matrix product states methods
Baxter, 1968; Orús, Vidal, 2008; Zauner-Stauber et. al. 2018; Fishman et. al 2018
\mathcal{Z} =
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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\chi
Free energy per site
2^L
Corner transfer matrix renormalization group
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
Corboz et al (2014)
\mathcal{Z} =
17
Corner transfer matrix renormalization group
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Approximate the infinite-size environment with a few tensors of finite bond dimension
17
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
Corboz et al (2014)
Corner transfer matrix renormalization group
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Approximate the infinite-size environment with a few tensors of finite bond dimension
17
\chi
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
Corboz et al (2014)
Corner transfer matrix renormalization group
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Approximate the infinite-size environment with a few tensors of finite bond dimension
17
\chi
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
Corboz et al (2014)
Corner transfer matrix renormalization group
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\mathcal{Z} =
Approximate the infinite-size environment with a few tensors of finite bond dimension
Building block for quantum problems : algorithms are already optimized
17
\chi
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
Fishman et al. PRB 98, 2018
Corboz et al (2014)
18
Observables
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Observables
\(\langle O \rangle= \frac{1}{\mathcal{Z}} \sum_{\vec{\sigma}} O(\mathrm{\vec{\sigma}}) e^{-\beta H}\) =
1. Local observable:
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Observables
\(\langle O \rangle= \frac{1}{\mathcal{Z}} \sum_{\vec{\sigma}} O(\mathrm{\vec{\sigma}}) e^{-\beta H}\) =
1. Local observable:
2. Correlation function
\(\langle O_i O_j \rangle =\)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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Observables
\(\langle O \rangle= \frac{1}{\mathcal{Z}} \sum_{\vec{\sigma}} O(\mathrm{\vec{\sigma}}) e^{-\beta H}\) =
1. Local observable:
2. Correlation function
\xi = - \ln \frac{\epsilon_2}{\epsilon_1}
Correlation length:
\(\langle O_i O_j \rangle =\)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Example: 2D Ising model
Orús, Vidal, PRB 78, 2008
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
\(T \in [2.2666, 2.2698]\)
Example: 2D Ising model
Orús, Vidal, PRB 78, 2008
Vanhecke et al. PRL (2019)
So far
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Partition functions of short-range models as exact, infinite-size tensor networks
Capture correlation functions, critical exponents, etc.
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 and tensor networks
Ground-state local rules
Example of results
A simple case
Vanhecke, JC et al (2021)
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
A simple case
Vanhecke, JC et al (2021)
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
A simple case
Vanhecke, JC et al (2021)
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COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
A simple case
Vanhecke, JC et al (2021)
Numerical problem
Cancellation of small and large factors
C. Wang, S.-M. Qin, H.-J. Zhou, PRB 90, (2014)
Z. Zhu, H. G. Katzgraber, arXiv:1903.07721 (2019)
J. G. Liu, L. Wang, P. Zhan, PRL 126, (2021)
20
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
A simple case
Vanhecke, JC et al (2021)
Numerical problem
Cancellation of small and large factors
Bad gauge
The transfer matrix is badly conditioned
(e.g. not hermitian, ...)
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. Zhan, PRL 126, (2021)
W. Tang et al, (2024, 2025)
20
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
A simple case
Vanhecke, JC et al (2021)
Numerical problem
Ground-state rule
Cancellation of small and large factors
Failure to minimize simultaneously all local Hamiltonians.
Bad gauge
The transfer matrix is badly conditioned
(e.g. not hermitian, ...)
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. Zhan, PRL 126, (2021)
W. Tang et al, (2024, 2025)
B. Vanhecke, JC, et al. PRR 3, (2021)
F.F. Song, T.-Y. Lin, G. M. Zhang, PRB (2023)
20
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
21
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
21
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
Kasteleyn, (1961), Fisher (1966)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
21
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
Kasteleyn, (1961), Fisher (1966)
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
21
Ground state:
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
Dimer coverings!
\(2 \times 2 \times 2\)
\(=0\)
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
21
Finite temperature
22
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Finite temperature
22
Vanhecke, JC et al (2021)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\(2 \times 2 \times 2\)
\(=e^{-\beta J}\)
Same structure and size
Different entries
Finite temperature
Vanhecke, JC et al (2021)
22
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\(2 \times 2 \times 2\)
Finite temperature
Vanhecke, JC et al (2021)
22
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
\(2 \times 2 \times 2\)
\(=e^{-\beta J}\)
Same structure and size
Different entries
More complex Ising models?
23
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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
More complex Ising models?
23
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
We can still write a well-behaved local tensor
Systematic linear program
Finite-range models
Exact ground state energy
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
Vanhecke, JC et al (2021)
JC et al (2022)
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)
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
24
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
1. ground state properties: three unexpected spin liquids
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)
24
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
1. ground state properties: three unexpected spin liquids
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)
24
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
JC et. al., PRB 106 (2022)
1. ground state properties: three unexpected spin liquids
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)
JC et. al., PRB 106 (2022)
1. ground state properties: three unexpected spin liquids
Exact ground-state energy
\(10^{-5}\) precision on the entropy
24
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
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)
JC et. al., PRB 106 (2022)
1. ground state properties: three unexpected spin liquids
Exact ground-state energy
\(10^{-5}\) precision on the entropy
24
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
25
2. A cascade of topological phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
25
2. A cascade of topological phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
What causes this cascade of phase transitions?
25
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
2. A cascade of "topological" phase transitions
25
2. A cascade of "topological" phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
25
2. A cascade of "topological" phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Sampling in real space
25
2. A cascade of "topological" phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Sampling in real space
Correlation functions
25
2. A cascade of "topological" phase transitions
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Sampling in real space
Correlation functions
An infinite series of phases not fixed by commensurability
Plateaus in the ratios of densities of 2 types of system-spanning strings
Broader perspective
2D systems & 3D systems
Promising directions
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
2D
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ising
XY
Heisenberg
NN
NN frustrated
Farther Neighbours
Farther-N. frustrated
Vanhecke et al.
Colbois et al.
Vanderstraeten et al.
Song et al.
Ueda et al.
Schmoll et al.
Ueda et al.
2D
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ising
XY
Heisenberg
NN
NN frustrated
Farther Neighbours
Farther-N. frustrated
Vanhecke et al.
Colbois et al.
Vanderstraeten et al.
Song et al.
Ueda et al.
Schmoll et al.
Ueda et al.
2D
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ising
XY
Heisenberg
NN
NN frustrated
Farther Neighbours
Farther-N. frustrated
?
?
Vanhecke et al.
Colbois et al.
Vanderstraeten et al.
Song et al.
Ueda et al.
Schmoll et al.
Ueda et al.
2D
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Ising
XY
Heisenberg
NN
NN frustrated
Farther Neighbours
Farther-N. frustrated
?
?
Vanhecke et al.
Colbois et al.
Vanderstraeten et al.
Song et al.
Ueda et al.
Schmoll et al.
Ueda et al.
2D
1. Large system limits for disordered systems
2. Long-range interactions
3. Dealing with non-local constraints
Wishlist
Liu et al. (2021)
1D: Nunez-Fernandez et al. (2025)
Status and Open challenges in 2D and 3D
26
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
SOTA : 3D Ising, 3D Ice : bond dimension 2
Ising
XY
Heisenberg
NN
NN frustrated
Farther Neighbours
1. Large system limits for disordered systems
2. Long-range interactions
Farther-N. frustrated
3. Dealing with non-local constraints
?
?
Vanhecke et al.
Colbois et al.
Vanderstraeten et al.
Song et al.
Ueda et al.
Schmoll et al.
Wishlist
Liu et al. (2021)
1D: Nunez-Fernandez et al. (2025)
Ueda et al.
2D
3D
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
directions under exploration in the field
27
Conformal field theory
Renormalization group
Contracting 2D and 3D tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Properties of the fixed-point tensor / RG
Grasping 3D CFTs using original regularization of the sphere
directions under exploration in the field
27
Läuchli et al (2025), Rey et al (2025)
Ueda et al. (2023), Kennedy & Rychkov (2024), Ebel et al (2024,2025)
Conformal field theory
Renormalization group
Contracting 2D and 3D tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Properties of the fixed-point tensor / RG
Grasping 3D CFTs using original regularization of the sphere
Taking advantage of lattice symmetries
directions under exploration in the field
27
Läuchli et al (2025), Rey et al (2025)
Lukin et al (2023), Nyckees et al (JC) (2023),
Yang & Corboz (2025), Naumann et al (2025)
Ueda et al. (2023), Kennedy & Rychkov (2024), Ebel et al (2024,2025)
Conformal field theory
Renormalization group
Contracting 2D and 3D tensor networks
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Properties of the fixed-point tensor / RG
Contraction in 3D:
1. Advanced CTMRG and tensor-RG schemes
2. taking advantage of the normality of the transfer operator
Grasping 3D CFTs using original regularization of the sphere
Taking advantage of lattice symmetries
directions under exploration in the field
27
Läuchli et al (2025), Rey et al (2025)
Lukin et al (2023), Nyckees et al (JC) (2023),
Yang & Corboz (2025), Naumann et al (2025)
Ueda et al. (2023), Kennedy & Rychkov (2024), Ebel et al (2024,2025)
Nishino et al (2000), [many more],
Vanderstraeten et al (2018),
Ebel (2025), Xu,Lin,Zhang(2025), Tang et al (2025)
Conformal field theory
Renormalization group
Contracting 2D and 3D tensor networks
directions under exploration in the field
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Properties of the fixed-point tensor / RG
Combining with classical Monte Carlo
Contraction in 3D:
1. Advanced CTMRG and tensor-RG schemes
2. taking advantage of the normality of the transfer operator
Frias-Perez et al (2023)
Nishino et al (2000), [many more],
Vanderstraeten et al (2018),
Ebel (2025), Xu,Lin,Zhang(2025), Tang et al (2025)
Grasping 3D CFTs using original regularization of the sphere
Läuchli et al (2025), Rey et al (2025)
Taking advantage of lattice symmetries
Lukin et al (2023), Nyckees et al (JC) (2023),
Yang & Corboz (2025), Naumann et al (2025)
27
Ueda et al. (2023), Kennedy & Rychkov (2024), Ebel et al (2024,2025)
Conformal field theory
Renormalization group
Contracting 2D and 3D tensor networks
Take-home message
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Constrained models shine light
on tensor network methods
Take-home message
Tensor networks:
a way to capture complex behavior in statistical mechanics
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Thank you for your attention!
P.S. I also work on quantum many-body disordered systems -
come talk if you are curious!
Constrained models shine light
on tensor network methods
Bonus slides
Kagome
Kagome
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)
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:
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)
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)
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)
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
Residual entropy and Normalized partition function
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)
\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{orange}\lambda_+}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})}
= e^{-\beta E_{\rm{GS}}} {\color{orange}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\
= e^{-\beta E_{\rm{GS}}} {\color{orange}\tilde{\mathcal{Z}}_N}
Wikipedia, CC BY license
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Julia Yeomans
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\(\chi = 4\)
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\(\chi = 4\)
\(\chi = 20\)
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\(\chi = 4\)
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Always ask : why do I expect the bond dimension to be limited?
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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
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
Examples
2D square lattice Ising model
Vanhecke et al. PRL (2019)
19
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
2D chiral Potts model (1d Rydberg arrays)
\(T \in [2.2666, 2.2698]\)
T
Anisotropy
3-states
Highly anisotropic, not conformal critical point
Nyckees, JC, Mila, NPB (2021) and refs. therein
Examples
2D square lattice Ising model
Vanhecke et al. PRL (2019)
19
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
2D chiral Potts model (1d Rydberg arrays)
\(T \in [2.2666, 2.2698]\)
T
Anisotropy
3-states
Highly anisotropic, not conformal critical point
Nyckees, JC, Mila, NPB (2021) and refs. therein
Examples
2D square lattice Ising model
Vanhecke et al. PRL (2019)
19
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
2D chiral Potts model (1d Rydberg arrays)
\(T \in [2.2666, 2.2698]\)
T
Anisotropy
3-states
Highly anisotropic, not conformal critical point
Nyckees, JC, Mila, NPB (2021) and refs. therein
Examples
2D square lattice Ising model
Vanhecke et al. PRL (2019)
19
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
2D chiral Potts model (1d Rydberg arrays)
\(T \in [2.2666, 2.2698]\)
T
Anisotropy
?
3-states
Highly anisotropic, not conformal critical point
Nyckees, JC, Mila, NPB (2021) and refs. therein
Examples
2D square lattice Ising model
Vanhecke et al. PRL (2019)
19
COLBOIS|TNS FOR CLASSICAL SPIN SYSTEMS | 11.2025
Critical point
\(\rightarrow\) finite-bond dimension scaling
2D chiral Potts model (1d Rydberg arrays)
Anisotropy
\(T \in [2.2666, 2.2698]\)
Nyckees, JC, Mila, NPB (2021) and refs. therein
Tensor networks for classical (frustrated) spin systems
By Jeanne Colbois
Tensor networks for classical (frustrated) spin systems
Invited talk at GDR quantum meso
