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 physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Tensor networks for physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Tensor networks
Tensor networks for physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Ising model(s)
Tensor networks
Frustrated magnetism
Tensor networks for physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Ising model(s)
Tensor networks
Frustrated magnetism
Tensor networks for physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Ising model(s)
Tensor networks
Frustrated magnetism
Tensor networks for physics from counting:
Ising models, Ice, and tiling Dominoes
Jeanne Colbois | TMC
Ising model(s)
Tensor networks
Frustrated magnetism
1
Understanding collective behavior...
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
1
Understanding collective behavior...
... from interactions at the microscopic scale
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
1
Understanding collective behavior...
... from interactions at the microscopic scale
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
1
Understanding collective behavior...
... from interactions at the microscopic scale
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
1
Understanding collective behavior...
... from interactions at the microscopic scale
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Toy models,
effective hamiltonians
1
Understanding collective behavior...
... from interactions at the microscopic scale
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
Toy models,
effective hamiltonians
partition function
Disorder & competition in magnetic systems
2
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Disorder & competition in magnetic systems
2
Lausanne
Frustration
in artificial spin systems
PhD : Emergent disorder
2017-2022
Frédéric Mila
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Disorder & competition in magnetic systems
2
Lausanne
Frustration
in artificial spin systems
PhD : Emergent disorder
Tensor networks + Monte Carlo
Tensor networks to demonstrate magnetic disorder at zero temperature
2017-2022
Frédéric Mila
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Disorder & competition in magnetic systems
2
Lausanne
Frustration
in artificial spin systems
PhD : Emergent disorder
Postdocs : quenched disorder in spin chains
Tensor networks + Monte Carlo
Tensor networks to demonstrate magnetic disorder at zero temperature
2022-2024
2017-2022
Frédéric Mila
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Disorder & competition in magnetic systems
2
Toulouse
Lausanne
Anderson &
Many-body localization
Frustration
in artificial spin systems
PhD : Emergent disorder
Postdocs : quenched disorder in spin chains
Tensor networks + Monte Carlo
"Shift-invert" exact diagonalization
Tensor networks to demonstrate magnetic disorder at zero temperature
2022-2024
2017-2022
Frédéric Mila
Nicolas Laflorencie
Fabien Alet
Instability of Anderson
localization to weak interactions
vs Many-body localization
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Disorder & competition in magnetic systems
2
Toulouse
Lausanne
Anderson &
Many-body localization
Frustration
in artificial spin systems
PhD : Emergent disorder
Postdocs : quenched disorder in spin chains
Tensor networks + Monte Carlo
"Shift-invert" exact diagonalization
Localization,
Glassy physics
Singapore
DMRG
Tensor networks to demonstrate magnetic disorder at zero temperature
Localization transitions from extreme statistics
2022-2024
2017-2022
Frédéric Mila
Nicolas Laflorencie
Fabien Alet
Gabriel Lemarié
Shaffique Adam
Instability of Anderson
localization to weak interactions
vs Many-body localization
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
At Institut Néel
3
Tensor network investigation of fragmented magnetism
Each spin participates to both phases!
Magnetic order
Spin
liquid
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
At Institut Néel
3
Tensor network investigation of fragmented magnetism
Each spin participates to both phases!
Magnetic order
Spin
liquid
Benjamin Canals
Matthieu Deschamps
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
At Institut Néel
3
Tensor network investigation of fragmented magnetism
Each spin participates to both phases!
Magnetic order
Spin
liquid
Benjamin Canals
Matthieu Deschamps
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Arnaud Ralko
Remy Dangoisse
... and more frustrated systems !
Laurent Del Rey
Philippe David
Valérie Guisset
Johann Coraux
Nicolas Rougemaille
SCOPE
1. Why study Ising models?
2. Tensor networks and the many-body problem
3. Tensor networks should work for Ising models
4. Frustrated models as domino games
5. Some applications & perspectives
4
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The ising model
The ising model
Why it matters
Why we want to solve it
Why it is hard
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
Real space
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
Paramagnet
~ Gas
Real space
Diffraction
Reciprocal space
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\(q_x\)
\(q_y\)
\(\xi = 0\)
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
Paramagnet
~ Gas
Real space
Diffraction
Reciprocal space
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\(q_x\)
\(q_y\)
\(\xi = 0\)
Scale invariance!
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
Paramagnet
~ Gas
Real space
Diffraction
Reciprocal space
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\(q_x\)
\(q_y\)
\(\xi = 0\)
Scale invariance!
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
5
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\sigma_i\) is the orientation of a spin
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Lenz (1920), Ising (1925)
A simple model of magnetism...
Paramagnet
~ Gas
Real space
Diffraction
Reciprocal space
\(T>T_c\)
\(T= T_c\)
\(T< T_c\)
\(q_x\)
\(q_y\)
\(\xi = 0\)
Magnetic order
~ solid
\(q_x\)
\(\xi = \infty\)
\(q_y\)
Scale invariance!
\mathcal{Z}= \sum_{\mathrm{conf.}} e^{-\frac{1}{k_BT} H(\mathrm{conf})}
6
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
... with a claim to universality
6
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Wikimedia comons
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Brass!
\(\sigma_i\) : Cu / Zn
... with a claim to universality
Madsen et al PRB 2016
Same behavior at the transition!
(Ordered to disordered sublattice occupation)
6
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Hopfield networks
Voter models
\(\sigma_i\) : opinion
\(\sigma_i\) : off / on neuron
Wikimedia comons
H = - J \sum_{\langle i,j \rangle}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
\(\sigma_i\) : Cu / Zn
... with a claim to universality
Madsen et al PRB 2016
H = - \sum_{(i,j)}J_{i,j}\sigma_i \sigma_j \quad
\sigma_i = \pm 1
Same behavior at the transition!
Brass!
(Ordered to disordered sublattice occupation)
(c.f Nobel 2024)
7
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The example of ice
7
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Solid : only a few configurations in the ground state.
The example of ice
s_0 = \frac{1}{N} k_B \ln(\Omega)
7
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Solid : only a few configurations in the ground state.
The example of ice
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
Giauque and Ashley, (1933)
Missing entropy ?
7
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
The example of ice
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
Solid : only a few configurations in the ground state.
Missing entropy ?
7
Lattice of oxygens
Hydrogens form the molecules...
The example of ice
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
7
Lattice of oxygens
Hydrogens form the molecules...
Solid : only a few configurations in the ground state.
Missing entropy ?
The example of ice
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
Pauling (1935)
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Omega = 2^{N_{\mathrm{bonds}}}\left(\frac{6}{16}\right)^{N_{\mathrm{bonds}}/2}
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
All hydrogen configurations
Pauling (1935)
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Omega = 2^{N_{\mathrm{bonds}}}\left(\frac{6}{16}\right)^{N_{\mathrm{bonds}}/2}
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
\Omega = 2^{N_{\mathrm{bonds}}}\left(\frac{6}{16}\right)^{N_{\mathrm{bonds}}/2}
All hydrogen configurations
Valid configurations in tetrahedrons
Pauling (1935)
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
\Omega = 2^{N_{\mathrm{bonds}}}\left(\frac{6}{16}\right)^{N_{\mathrm{bonds}}/2}
All hydrogen configurations
Valid configurations in tetrahedrons
S_0/N \approx 0.2027 k_B > 0 (!!)
Pauling (1935)
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
The example of ice
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Giauque and Ashley, (1933)
Bernal and Fowler, (1933)
... under constraints:
1 hydrogen / bond
7
Solid : only a few configurations in the ground state.
Missing entropy ?
Lattice of oxygens
Hydrogens form the molecules...
\Omega = 2^{N_{\mathrm{bonds}}}\left(\frac{6}{16}\right)^{N_{\mathrm{bonds}}/2}
All hydrogen configurations
Valid configurations in tetrahedrons
S_0/N \approx 0.2027 k_B > 0 (!!)
Pauling (1935)
s_0 = \frac{1}{N} k_B \ln(\Omega)
\Delta S
\(\sim 2^{3\cdot 10^{23}}\) configurations in your ice cube
vs \(\sim 2^{265}\) atoms in the universe
8
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
8
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
\(\xi = \infty\)
Reciprocal space
Fennell et al. (2009) Ho2Ti2O7
8
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
\(\xi = \infty\)
Reciprocal space
Fennell et al. (2009) Ho2Ti2O7
8
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
\(\xi = \infty\)
Reciprocal space
Fennell et al. (2009) Ho2Ti2O7
Spin flip : 2 charges
Effective Coulomb interaction
\(|F| \propto q^2/r^2\)
8
Henley (2005, 2010), Castelnovo etal(2008)...
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
\(\xi = \infty\)
Reciprocal space
Fennell et al. (2009) Ho2Ti2O7
Spin flip : 2 charges
Effective Coulomb interaction
\(|F| \propto q^2/r^2\)
8
Henley (2005, 2010), Castelnovo etal(2008)...
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
ising models of frustrated magnetism
Ramirez et al (1999) : Dy2Ti2O7
\(\xi = \infty\)
Reciprocal space
Fennell et al. (2009) Ho2Ti2O7
Spin flip : 2 charges
Effective Coulomb interaction
\(|F| \propto q^2/r^2\)
8
Henley (2005, 2010), Castelnovo etal(2008)...
Gauss law \(\nabla \cdot \mathbf{B} = 0\)
Classical / quantum electrodynamics
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
9
Ising model as limits of quantum models
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
9
Ising model as limits of quantum models
H = J\sum_{\langle i,j \rangle } \sigma_i \sigma_j
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
Spin up
Spin down
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ising model as limits of quantum models
9
Wannier, Houttapel (1950)
With field: Blöte et al (1993), Qian et al (2004), Nyckees et al (JC) (2023)
H = J\sum_{\langle i,j \rangle } \sigma_i \sigma_j
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
Spin up
Spin down
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
9
H = J\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)
Wannier, Houttapel (1950)
With field: Blöte et al (1993), Qian et al (2004), Nyckees et al (JC) (2023)
Ising model as limits of quantum models
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
9
Wannier, Houttapel (1950)
With field: Blöte et al (1993), Qian et al (2004), Nyckees et al (JC) (2023)
Ising model as limits of quantum models
H = J\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)
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
9
M. Zhu et al, PRL (2024), NPJ quantum materials (2025)
Wannier, Houttapel (1950)
With field: Blöte et al (1993), Qian et al (2004), Nyckees et al (JC) (2023)
Ising model as limits of quantum models
\(\Omega > 2^{N/3}\)
H = J\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)
Solving ising models is hard
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
Solving ising models is hard
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
Solving ising models is hard
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
1D Ising
Solved exactly 1925
\(2\)x\(2\)
Ising
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
3D Ising
No closed form
2D Ising with a field
No closed form
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
3D Ising
No closed form
2D Ising with a field
No closed form
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
\mathcal{Z}= \sum_{\eta} e^{-\frac{1}{k_BT} E(\eta)}
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
3D Ising
No closed form
2D Ising with a field
No closed form
A version of the N-Body probleM
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
3D Ising
No closed form
2D Ising with a field
No closed form
A version of the N-Body probleM
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
Spin glasses
Counting the number of ground states:
#P-complete
(#P : asking how-many solutions in an NP problem.)
Barahona (1982)
Solving ising models is hard
1D Ising
Solved exactly 1925
2D Ising
Solved exactly 1944
3D Ising
No closed form
2D Ising with a field
No closed form
Spin glasses
A version of the N-Body probleM
\(2\)x\(2\)
\(2^{N}\)x\(2^{N}\)
Counting the number of ground states:
#P-complete
(#P : asking how-many solutions in an NP problem.)
Approximate methods: Monte Carlo, series expansion, field theory at the critical point, RG & conformal bootstrap, and...
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
10
Ising
Onsager
Barahona (1982)
Tensor networks
Tensor networks
What is a tensor network ?
Why do tensor networks matter?
What are key concepts ?
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
- Based on "clever" compression of data
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
- Based on "clever" compression of data
1D quantum many-body
\(S=1\) Heisenberg chain
Frustrated spin ladders
Disordered chains
...
White (1992), ...
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
- Based on "clever" compression of data
1D quantum many-body
2D and more quantum many-body
\(S=1\) Heisenberg chain
Frustrated spin ladders
Disordered chains
...
Topological order
Two-dimensional t-J model
Magnetization plateaus in Shastry-Sutherland
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
- Based on "clever" compression of data
1D quantum many-body
2D and more quantum many-body
Quantum computing
\(S=1\) Heisenberg chain
Frustrated spin ladders
Disordered chains
...
Topological order
Two-dimensional t-J model
Magnetization plateaus in Shastry-Sutherland
Classical simulation of quantum circuits
Challenging quantum supremacy claims
....
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
Vidal (2003), Zhou, Stoudenmire and Waintal (2020), ...
11
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
What are tensor networKs & why are they important
- Extremely efficient classical computing methods
- Based on "clever" compression of data
1D quantum many-body
2D and more quantum many-body
\(S=1\) Heisenberg chain
Frustrated spin ladders
Disordered chains
...
...and many more...
Topological order
Two-dimensional t-J model
Magnetization plateaus in Shastry-Sutherland
Classical simulation of quantum circuits
Challenging quantum supremacy claims
....
White (1992), ...
Verstraete et al (2004), Corboz (2014), ....
Vidal (2003), Zhou, Stoudenmire and Waintal (2020), ...
Chemistry, machine learning, mathematics...
Quantum computing
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Type
Notation
Visualization
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank-3 tensor
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank-3 tensor
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Connecting legs = make the product
"CONTRACTION"
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
Scalar
Vector
Type
Notation
Visualization
Matrix
Rank = # indices = # legs =#dimensions
Rank-3 tensor
Scalar product
Connecting legs = make the product
"CONTRACTION"
"Legs"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
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"
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
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"
Only 2 legs can meet!
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
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"
Only 2 legs can meet!
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Size of the index = bond dimension = \(\chi\) or \(D\)
You can group indices:
\(\chi \times\chi \times \chi \times \chi\)
tensor
=
Tensor networks notation
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, (1971)
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"
Only 2 legs can meet!
12
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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 networks Structure
13
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
A complicated tensor network product giving a matrix
Tensor networks Structure
13
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
A complicated tensor network product giving a matrix
Goldenfeld & Kadanoff, Science, 284 (1999)
Tensor networks Structure
13
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
A complicated tensor network product giving a matrix
Goldenfeld & Kadanoff, Science, 284 (1999)
Can we keep only the "main" information ?
Wikipedia, CC BY license
14
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
Julia Yeomans
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
14
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
14
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
14
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
14
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Image compression
14
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\chi = 4\)
Image compression
14
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\chi = 4\)
\(\chi = 20\)
Image compression
14
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
Image compression
14
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
Image compression
14
Image compression
Always ask : why do I expect the bond dimension to be limited?
\(M\)
\(U\)
\(S\)
\(V\)
\(=\)
Wikipedia, CC BY license
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\chi = 4\)
\(\chi = 20\)
\(\chi = 100\)
14
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
15
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| ISING, ICE AND DOMINOES | 09.2025
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)
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
|\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)
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I 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)
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I 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)
ENTANGLEMENT (area law)
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
compressing many-body Wavefunctions
Many-body wavefunction =huge tensor:
We want to compress it:
why (When) do I expect the bond dimension to be limited?
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
15
Why tensor networks
should be able to solve Ising models
Why tensor networks
should be able to solve Ising models
1. The 1D Ising model partition function is a TN
2. The solution of the 2D Ising model is TN-related
3. Successes / failures
16
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 1D ising model solution is a simple tensor network
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 1D ising model solution is a simple tensor network
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
16
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 1D ising model solution is a simple tensor network
\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}
16
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 1D ising model solution is a simple tensor network
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
16
\mathcal{Z}_L = (T^L)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 1D ising model solution is a simple tensor network
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
16
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 2d ising model : transfer matrix & tensor network
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
Generalized kronecker \(\delta\) tensor
17
R. J. Baxter, J. Math. Phys. 9, 1968
R. Orús, G. Vidal, PRB 78, 2008
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
The 2d ising model : transfer matrix & tensor network
Ferromagnet / antiferromagnet: Onsager
In a field : no closed form solution
\(L\)
\(M\)
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
(But the amount of information stored: \(2\times 2\times 2\times 2\))
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
17
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(2\times 2\times 2\)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
18
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(4\times 2\times 4\)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
18
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(8\times 2\times 8\)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
18
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(2^{r}\times 2\times 2^{r}\)
\(A\)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
18
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(2^{r}\times 2\times 2^{r}\)
\(A\)
Question : can we approximate \(A\) by a \(\chi\times 2\times \chi\) tensor ?
Answer: yes, if the area-law is respected!
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
18
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
"Compressing" the tensor network : Power method
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \(2^{r}\times 2\times 2^{r}\)
\(A\)
Question : can we approximate \(A\) by a \(\chi\times 2\times \chi\) tensor ?
Answer: yes, if the area-law is respected!
18
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \( \chi\times 2\times \chi\)
\(A\)
Question : can we approximate \(A\) by a \(\chi\times 2\times \chi\) tensor ?
Answer: yes, if the area-law is respected!
"Compressing" the tensor network : Power method
18
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \( \chi\times 2\times \chi\)
\(A\)
Question : can we approximate \(A\) by a \(\chi\times 2\times \chi\) tensor ?
Answer: yes, if the area-law is respected!
"Compressing" the tensor network : Power method
18
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\mathcal{T}\) : \(2^{L} \times 2^{L}\)
\(\mathcal{Z} = \mathcal{T}^{M}\)
Size : \( \chi\times 2\times \chi\)
\(A\)
Question : can we approximate \(A\) by a \(\chi\times 2\times \chi\) tensor ?
Answer: yes, if the area-law is respected!
"Compressing" the tensor network : Power method
18
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\mathcal{Z} = \)
"Compressing" the tensor network : Power method
\(A\)
18
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(\mathcal{Z} = \)
Back to 1d
19
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Back to 1d
\(\mathcal{Z} = \)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
19
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Back to 1d
\(\mathcal{Z} = \)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
19
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Back to 1d
\(\mathcal{Z} = \)
\(\overline{m} = \)
Local observable
\(\mathcal{Z}\)
R. J. Baxter, J. Math. Phys. 9, 1968; Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018; M. Fishman et. al PRB 98, 2018
19
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Successes and Failures
2D square lattice Ising model
... and many more, e.g. Huse-Fisher universality class
20
Orús, Vidal, PRB 78, 2008
Nyckees, JC, Mila (2021)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Successes and Failures
2D square lattice Ising model
20
Orús, Vidal, PRB 78, 2008
Vanhecke, JC et al (2021)
... and many more, e.g. Huse-Fisher universality class
Nyckees, JC, Mila (2021)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Successes and Failures
2D square lattice Ising model
20
Orús, Vidal, PRB 78, 2008
Vanhecke, JC et al (2021)
... and many more, e.g. Huse-Fisher universality class
Nyckees, JC, Mila (2021)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Successes and Failures
2D square lattice Ising model
20
Orús, Vidal, PRB 78, 2008
Vanhecke, JC et al (2021)
... and many more, e.g. Huse-Fisher universality class
Nyckees, JC, Mila (2021)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Successes and Failures
2D square lattice Ising model
20
Orús, Vidal, PRB 78, 2008
Vanhecke, JC et al (2021)
... and many more, e.g. Huse-Fisher universality class
Nyckees, JC, Mila (2021)
Solving domino games for frustrated Ising models
21
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
21
Another origin of tensor networks: Domino tilings
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
21
Another origin of tensor networks: Domino tilings
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
22
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
22
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
= 1
22
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
= 1
= 1
22
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
= 1
= 0
= 1
22
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
= 1
= 0
= 0
= 1
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
22
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Couting domino tilings
# of ways to place dominoes such that
- the lattice is fully covered
-no overlaps
\(\Omega \approx 1,3385^ N\)
\Omega =
\(2 \times 2 \times 2 \times 2\)
= 1
= 0
= 0
= 1
Baxter 1968 : First "tensor network" equations to solve this problem.
Kasteleyn || Temperley & Fisher (1961), R. J. Baxter, (1968)
22
23
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
23
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
23
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
Dimer coverings!
23
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
\(2 \times 2 \times 2\)
\(=0\)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
Dimer coverings!
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
\(2 \times 2 \times 2\)
\(=0\)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
23
Dimer coverings!
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Ground state:
Vanhecke, JC et al (2021)
\(2 \times 2 \times 2\)
\(=0\)
Simple case : triangular lattice Ising antiferromagnet
Kasteleyn, (1961), Fisher (1966)
23
Dimer coverings!
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Finite temperature
\(2 \times 2 \times 2\)
Vanhecke, JC et al (2021)
24
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(2 \times 2 \times 2\)
Vanhecke, JC et al (2021)
Finite temperature
24
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(2 \times 2 \times 2\)
Vanhecke, JC et al (2021)
\(=e^{-\beta J}\)
Same structure and size
Different entries
Finite temperature
24
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
\(2 \times 2 \times 2\)
Vanhecke, JC et al (2021)
\(=e^{-\beta J}\)
Same structure and size
Different entries
24
Finite temperature
1. Convergence depends on the formulation!
2. Weighted counting problem!
To retain:
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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)
25
25
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
25
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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
25
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
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 map to "domino" tilings ?
25
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
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)
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
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)
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
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)
LINEAR PROGRAM:
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
26
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Can we still map to "domino" tilings ? YES!
Split the Hamiltonian differently: ground states are tilings of local g.s. configurations!
LINEAR PROGRAM:
C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, (1969); M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , (1975);
B. Sriram Shastry and B. Sutherland, Physica 108 B+C, (1981); W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, (2016);
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
1. Split the lattice into clusters that overlap
2. Find the optimal energy lower-bound
3. Contract / extend to finite T
Applications and perspectives
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Exotic physics in kagome antiferromagnets
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
27
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Exotic physics in kagome antiferromagnets
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)
1. Three unexpected spin liquid phases
27
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Exotic physics in kagome antiferromagnets
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3}} \sigma_i \sigma_j
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
JC, B. Vanhecke et. al., PRB 106 (2022)
1. Three unexpected spin liquid phases
27
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Exotic physics in kagome antiferromagnets
1. Three unexpected spin liquid phases
2. Cascade of "topological" phase transitions
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3}} \sigma_i \sigma_j
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
27
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Exotic physics in kagome antiferromagnets
2. Cascade of "topological" phase transitions
Classical XY models for kagome superconductors, Understanding topological order, Studying quantum frustrated magnets, ...
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3}} \sigma_i \sigma_j
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
JC, B. Vanhecke et. al., PRB 106 (2022)
A. Rufino, S. Nyckees, JC, F. Mila, arXiv:2505.05889 (2025)
1. Three unexpected spin liquid phases
27
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Perspectives
28
Tensor network methods
"Classical" and quantum frustrated magnetism
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Perspectives
28
Tensor network methods
"Classical" and quantum frustrated magnetism
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)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Perspectives
28
Tensor network methods
"Classical" and quantum frustrated magnetism
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?
Generalized RK wavefunctions (variational ?) and topology
Wei Tang et al (2024, 2025)
Châtelain & Gendiar (2020)
Frias-Perez et al (2023)
Giudice et al (2022)
Ronceray & Le Floch (2020)
Interpretation of entanglement ?
Carignano et al (2024)
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Take-home message
29
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Take-home message
29
Nature can produce complex structures even in simple situations,
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Take-home message
29
Nature can produce complex structures even in simple situations,
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Take-home message
Nature can produce complex structures even in simple situations,
and can obey simple laws even in complex situations.
29
COLBOIS| ISING, ICE AND DOMINOES | 09.2025
Take-home message
Tensor networks:
a way to capture those complex structures
Boiling down to simple laws?
Thank you for your attention!
29
Nature can produce complex structures even in simple situations,
and can obey simple laws even in complex situations.
Ising, ice and dominos: an introduction to tensor networks
By Jeanne Colbois
Ising, ice and dominos: an introduction to tensor networks
Seminar at Institut Néel
