Introduction to tensor networks for frustrated spin systems

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Tensor networks

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Tensor networks

Revolution in 1D (2D) quantum spin systems

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Revolution in 1D (2D) quantum spin systems

Tensor networks

A new language

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Tensor networks

Revolution in 1D (2D) quantum spin systems

A new language

Large potential for

2D (3D) classical spin systems

2

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Aim for today

What are tensor networks (TNs)? Transfer matrix!

2

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Aim for today

What are tensor networks (TNs)? Transfer matrix!

Why use them for classical systems?

2

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Aim for today

What are tensor networks (TNs)? Transfer matrix!

Why use them for classical systems?

What do we learn by using TNs for frustrated systems?

Numerical challenge
Physics!

3

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Hello!

PhD @ EPFL ( Lausanne)

with Frédéric Mila

Classical frustrated magnetism
Tensor networks
Artificial spin systems

3

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Hello!

PhD @ EPFL ( Lausanne)

with Frédéric Mila

Classical frustrated magnetism
Tensor networks
Artificial spin systems

Postdoc @ LPT (Toulouse)

with Nicolas Laflorencie

Quantum spin chains in disordered field (localization - delocalization)
Extreme value theory
More tensor networks!

3

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Hello!

Postdoc @ LPT (Toulouse)

with Nicolas Laflorencie

Quantum spin chains in disordered field (localization - delocalization)
Extreme value theory
More tensor networks!

Seminar on

Wednesday, 11:00 @ LPMMC

PhD @ EPFL ( Lausanne)

with Frédéric Mila

Classical frustrated magnetism
Tensor networks
Artificial spin systems

Acknowledgments

4

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Acknowledgments

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

4

Acknowledgments

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

4

Acknowledgments

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

4

Motivation : Classical frustration...

What are frustrated (Ising) models?

5

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

5

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

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

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

2-up 1-down (UUD),

2-down 1-up (DDU)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

5

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

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

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

2-up 1-down (UUD),

2-down 1-up (DDU)

2-up 1-down (UUD),

2-down 1-up (DDU)

5

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

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

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

2-up 1-down (UUD),

2-down 1-up (DDU)

2-up 1-down (UUD),

2-down 1-up (DDU)

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

5

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

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

S = 0.3230659...

S = 0.3230659...

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

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

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

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

2-up 1-down (UUD),

2-down 1-up (DDU)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

2-up 1-down (UUD),

2-down 1-up (DDU)

5

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

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

S = 0.3230659...

S = 0.3230659...

S = 0.501833...

S = 0.501833...

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

K. Kano and S. Naya, Prog. Theor. Phys. 10, (1953)

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

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

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

2-up 1-down (UUD),

2-down 1-up (DDU)

\xi = 1.2506...

\xi = 1.2506...

A. Sütö, Z. Phys. B 44, (1981)

W. Apel, H.-U. Everts, J. Stat. Mech, (2011)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

5

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

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

S = 0.3230659...

S = 0.3230659...

S = 0.501833...

S = 0.501833...

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

K. Kano and S. Naya, Prog. Theor. Phys. 10, (1953)

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}

d_{i,j} = \sigma_i \sigma_j

d_{i,j} = \sigma_i \sigma_j

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

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

2-up 1-down (UUD),

2-down 1-up (DDU)

\xi = 1.2506...

\xi = 1.2506...

A. Sütö, Z. Phys. B 44, (1981)

W. Apel, H.-U. Everts, J. Stat. Mech, (2011)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What are frustrated (Ising) models?

What about farther-neighbor interactions ?

T. Mizoguichi, L. Jaubert, M. Udagawa,

PRL 119, 077207 (2017)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Some examples

6

Fine-tuning : $J = J_2 = J_3$

Macroscopic degeneracy

D. Kiese, F. Ferrari, N. Astrakhantsev, N. Niggemann, P. Ghosh et al. , PRR 5, L012025 (2023)

T. Mizoguichi, L. Jaubert, M. Udagawa,

PRL 119, 077207 (2017)

Fine-tuning : $J = J_2 = J_3$

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Some examples

6

Macroscopic degeneracy

T. Lugan, L.D.C. Jaubert, M. Udagawa, A. Ralko,

PRB 106, L140404 (2022)

Some examples

7

Complex behavior

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Spin ice

Emergent electrodynamics

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Complex behavior

Some examples

7

In-plane artificial kagome ice

L. Anghinolfi et al.,

Nat. Commun. 6, (2015)

Long-range vs short-range interactions

$\rightarrow$ nature of the transition

Emergent electrodynamics

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Complex behavior

Some examples

7

Spin ice

In-plane artificial kagome ice

L. Anghinolfi et al.,

Nat. Commun. 6, (2015)

Long-range vs short-range interactions

$\rightarrow$ nature of the transition

local constraint

Emergent electrodynamics

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Complex behavior

Some examples

7

Spin ice

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

8

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

Artificial spin systems

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

8

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

Artificial spin systems

Luo et al. Science 363, (2019)

Colbois et al., PRB 104 (2021)

DMI : possibility of tuning the nearest-neighbor coupling

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

9

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

One question among others

Q_{\bigtriangleup} = \sum_i \sigma_i\\ Q_{\bigtriangledown} = - \sum_i \sigma_i

Q_{\bigtriangleup} = \sum_i \sigma_i\\ Q_{\bigtriangledown} = - \sum_i \sigma_i

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)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

9

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

One question among others

Q_{\bigtriangleup} = \sum_i \sigma_i\\ Q_{\bigtriangledown} = - \sum_i \sigma_i

Q_{\bigtriangleup} = \sum_i \sigma_i\\ Q_{\bigtriangledown} = - \sum_i \sigma_i

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)

How does the

degeneracy get lifted?

Can there be macroscopic degeneracy beyond fine-tuning?

"Cahier des charges"

10

1. Residual entropy

2. Correlations / structure factors

3. Controlled scaling (finite - size / finite - entanglement)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Tensor networks primer

The DMRG revolution...

11

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Steve White, PRL (1992), PRB (1993)

Fannes, Nachtergale and Werner (1992)

The DMRG revolution...

11

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Steve White, PRL (1992), PRB (1993)

The DMRG webpage (T. Nishino) : 5 papers in 1993 ... ~25 papers in October 2023

Fannes, Nachtergale and Werner (1992)

the curse of dimensionality - the power of tns

12

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144 (1992)

Pasquale Calabrese and John Cardy J. Stat. Mech. (2004) P06002

M B Hastings J. Stat. Mech. (2007) P08024

the curse of dimensionality - the power of tns

12

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$2^L$

Many-body Hilbert space

M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144 (1992)

Pasquale Calabrese and John Cardy J. Stat. Mech. (2004) P06002

M B Hastings J. Stat. Mech. (2007) P08024

the curse of dimensionality - the power of tns

12

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$2^L$

Many-body Hilbert space

Ground states of gapped, local Hamiltonians

M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144 (1992)

Pasquale Calabrese and John Cardy J. Stat. Mech. (2004) P06002

M B Hastings J. Stat. Mech. (2007) P08024

the curse of dimensionality - the power of tns

12

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$2^L$

Many-body Hilbert space

M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144 (1992)

Pasquale Calabrese and John Cardy J. Stat. Mech. (2004) P06002

M B Hastings J. Stat. Mech. (2007) P08024

Ground states of gapped, local Hamiltonians

$\Rightarrow$ TNs are very good at describing these states

No sign problem!

TNs For 2D classical systems - Why?

13

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

TNs For 2D classical systems - Why?

13

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D classical is "like" 1D quantum

TNs For 2D classical systems - Why?

13

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D classical is "like" 1D quantum

Building block for 2D quantum problems - algorithms already optimized

TNs For 2D classical systems - Why?

13

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D classical is "like" 1D quantum

Building block for 2D quantum problems - algorithms already optimized

Infinite size for translation-invariant problems

$\rightarrow$ T. Nishino & K. Okunishi, 1995, 1996

Why Should tn be useful for frustrated models?

14

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Why Should tn be useful for frustrated models?

14

Monte Carlo?

no sign problem (classical)

ergodicity

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Why Should tn be useful for frustrated models?

14

Monte Carlo?

no sign problem (classical)

ergodicity

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

Why Should tn be useful for frustrated models?

14

Monte Carlo?

no sign problem (classical)

ergodicity

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

\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}

\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}

Why Should tn be useful for frustrated models?

14

Monte Carlo?

no sign problem (classical)

ergodicity

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}

\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

\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}

\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}

\cong {\color{orange}\lambda_+}^N

\cong {\color{orange}\lambda_+}^N

Why Should tn be useful for frustrated models?

14

Monte Carlo?

no sign problem (classical)

ergodicity

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}

\lim_{\beta \rightarrow \infty} {\color{orange}\tilde{\mathcal{Z}}_N} = {\color{orange}W_N}

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

\cong {\color{orange}\lambda_+}^N

\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}

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta E_{\rm{GS}}} {\color{orange}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\ = e^{-\beta E_{\rm{GS}}} {\color{orange}\tilde{\mathcal{Z}}_N}

Partition function for one site...

... Most precise result out of the TN

Direct access to zero temperature

Tensor networks : inspired by transfer matrices

15

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

15

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

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15

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

15

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

15

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

"Exact contraction"

15

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

\Lambda^{L} = \lambda_{+}^{L} \begin{pmatrix} 1& 0 \\ 0 & \left(\frac{\lambda_{-}}{\lambda_{+}}\right)^L \end{pmatrix}

\Lambda^{L} = \lambda_{+}^{L} \begin{pmatrix} 1& 0 \\ 0 & \left(\frac{\lambda_{-}}{\lambda_{+}}\right)^L \end{pmatrix}

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}

\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}

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

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

T^L = \left(P^{-1} \Lambda P\right)^L = P^{-1} \Lambda^L P

"Exact contraction"

15

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

\Lambda = \begin{pmatrix} \lambda_{+} & 0 \\ 0 & \lambda_{-} \end{pmatrix} = \lambda_{+} \begin{pmatrix} 1& 0 \\ 0 & \frac{\lambda_{-}}{\lambda_{+}} \end{pmatrix}

\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}

\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}

"Approximate contraction"

TN language

16

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

"Contraction"

Matrix / tensor

Vector

Open legs = number of indices = "rank"

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TN language

16

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

"Contraction"

Matrix / tensor

Vector

Open legs = number of indices = "rank"

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

TN language

Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)

"Contraction"

Matrix / tensor

Vector

Open legs = number of indices = "rank"

16

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2D partition function

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D partition function

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

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D partition function

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

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

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

\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases} 1 & \text{ all equal}\\ 0 & \text{ otherwise} \end{cases}

\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases} 1 & \text{ all equal}\\ 0 & \text{ otherwise} \end{cases}

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

2D partition function

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

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

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

\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases} 1 & \text{ all equal}\\ 0 & \text{ otherwise} \end{cases}

\delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} = \begin{cases} 1 & \text{ all equal}\\ 0 & \text{ otherwise} \end{cases}

17

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2D partition function contraction

18

\mathcal{Z}_N =

\mathcal{Z}_N =

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Row to row transfer matrix -> "Matrix product operator"

2D partition function contraction

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

18

Row to row transfer matrix -> "Matrix product operator"

2D partition function contraction

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

18

Row to row transfer matrix -> "Matrix product operator"

2D partition function contraction

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

18

Row to row transfer matrix -> "Matrix product operator"

2D partition function contraction

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

18

19

Matrix - Product - State Ansatz

R. J. Baxter, J. Math. Phys. 9, 1968

S. R. White, PRL 69, 2863-2866 (1992) & PRB 49, 10345-10356 (1993)

G. Vidal, PRL 91 147902 (2003) & PRL 98, 070201 (2007)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

19

Matrix - Product - State Ansatz

R. J. Baxter, J. Math. Phys. 9, 1968

S. R. White, PRL 69, 2863-2866 (1992) & PRB 49, 10345-10356 (1993)

G. Vidal, PRL 91 147902 (2003) & PRL 98, 070201 (2007)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

|\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

|\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

EXPONENTIAL # of PARAMETERS

2^L

2^L

|\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

|\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

EXPONENTIAL # of PARAMETERS

2^L

2^L

R. J. Baxter, J. Math. Phys. 9, 1968

S. R. White, PRL 69, 2863-2866 (1992) & PRB 49, 10345-10356 (1993)

G. Vidal, PRL 91 147902 (2003) & PRL 98, 070201 (2007)

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Matrix - Product - State Ansatz

19

\chi

\chi

\chi

\chi

|\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

|\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

EXPONENTIAL # of PARAMETERS

2^L

2^L

R. J. Baxter, J. Math. Phys. 9, 1968

S. R. White, PRL 69, 2863-2866 (1992) & PRB 49, 10345-10356 (1993)

G. Vidal, PRL 91 147902 (2003) & PRL 98, 070201 (2007)

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Matrix - Product - State Ansatz

19

CONSTANT # of

PARAMETERS (poly. in $\chi$ )

(\chi \times 2 \times \chi)

(\chi \times 2 \times \chi)

\chi

\chi

\chi

\chi

|\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

|\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

EXPONENTIAL # of PARAMETERS

2^L

2^L

R. J. Baxter, J. Math. Phys. 9, 1968

S. R. White, PRL 69, 2863-2866 (1992) & PRB 49, 10345-10356 (1993)

G. Vidal, PRL 91 147902 (2003) & PRL 98, 070201 (2007)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Matrix - Product - State Ansatz

19

CONSTANT # of

PARAMETERS (poly. in $\chi$ )

(\chi \times 2 \times \chi)

(\chi \times 2 \times \chi)

\chi

\chi

\chi

\chi

Control parameter!

tWO MAIN SCHEMES

20

(1 + 1)D

iTEBD / VUMPS

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

tWO MAIN SCHEMES

(1 + 1)D

iTEBD / VUMPS

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

2D

R. J. Baxter, J. Math. Phys. 9, 1968

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

Fishman et al. PRB 98, 2018

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

20

tWO MAIN SCHEMES

(1 + 1)D

iTEBD / VUMPS

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

R. J. Baxter, J. Math. Phys. 9, 1968

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

Fishman et al. PRB 98, 2018

2D

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

20

tWO MAIN SCHEMES

(1 + 1)D

iTEBD / VUMPS

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

R. J. Baxter, J. Math. Phys. 9, 1968

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

Fishman et al. PRB 98, 2018

2D

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

20

tWO MAIN SCHEMES

(1 + 1)D

iTEBD / VUMPS

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

R. J. Baxter, J. Math. Phys. 9, 1968

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

Fishman et al. PRB 98, 2018

2D

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

20

LOcal observables

21

R. J. Baxter, J. Math. Phys. 9, 1968

Orús, Vidal, PRB 78, 2008;

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

V. Zauner-Stauber et. al. PRB 97,2018;

M. Fishman et. al PRB 98, 2018

$\langle m \rangle$ =

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$\propto\mathcal{Z}$

LOcal observables

18

R. J. Baxter, J. Math. Phys. 9, 1968

Orús, Vidal, PRB 78, 2008;

T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996

V. Zauner-Stauber et. al. PRB 97,2018;

M. Fishman et. al PRB 98, 2018

$\langle m \rangle$ =

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Conceptually simple and numerically efficient

way of representing and computing

- classical partition functions

- ground states of local Hamiltonians

Tensor networks?

The frustration problem

22

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The frustration problem

Fails in the presence of

frustration and macroscopic g.s. degeneracy

22

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

The frustration problem

Fails in the presence of

frustration and macroscopic g.s. degeneracy

22

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$\rightarrow$ in spin glasses

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

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

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

The frustration problem

Fails in the presence of

frustration and macroscopic g.s. degeneracy

B. Vanhecke, JC, et al. PRR 3, (2021)

$\rightarrow$ in spin glasses

$\rightarrow$ in translation-invariant frustrated Ising models

22

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

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

The frustration problem

Fails in the presence of

frustration and macroscopic g.s. degeneracy

B. Vanhecke, JC, et al. PRR 3, (2021)

$\rightarrow$ in spin glasses

$\rightarrow$ in translation-invariant frustrated Ising models

$\rightarrow$ in lattice gas models

S. A. Akimenko, PRE 107, (2023)

22

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

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

The frustration problem

Fails in the presence of

frustration and macroscopic g.s. degeneracy

B. Vanhecke, JC, et al. PRR 3, (2021)

$\rightarrow$ in spin glasses

$\rightarrow$ in translation-invariant frustrated Ising models

$\rightarrow$ in lattice gas models

$\rightarrow$ in frustrated XY models

S. A. Akimenko, PRE 107, (2023)

F.F. Song, T.-Y. Lin, G. M. Zhang, arXiv:2309.05321

22

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

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

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

23

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T_{\sigma_1, \sigma_2, \sigma_3} =

T_{\sigma_1, \sigma_2, \sigma_3} =

= \begin{cases} 0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\ 1 & \text{otherwise} \end{cases}

= \begin{cases} 0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\ 1 & \text{otherwise} \end{cases}

T_{d_1, d_2,d_3} =

T_{d_1, d_2,d_3} =

= \begin{cases} 1 \quad & \text{if } \prod_i d_i = -1\\ 0 & \text{otherwise} \end{cases}

= \begin{cases} 1 \quad & \text{if } \prod_i d_i = -1\\ 0 & \text{otherwise} \end{cases}

A SURPRISE?

23

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T_{\sigma_1, \sigma_2, \sigma_3} =

T_{\sigma_1, \sigma_2, \sigma_3} =

= \begin{cases} 0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\ 1 & \text{otherwise} \end{cases}

= \begin{cases} 0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\ 1 & \text{otherwise} \end{cases}

T_{d_1, d_2,d_3} =

T_{d_1, d_2,d_3} =

= \begin{cases} 1 \quad & \text{if } \prod_i d_i = -1\\ 0 & \text{otherwise} \end{cases}

= \begin{cases} 1 \quad & \text{if } \prod_i d_i = -1\\ 0 & \text{otherwise} \end{cases}

The convergence of the tensor network contraction depends on the

formulation of the partition function

Points of view

Contracting the TN of a frustrated model

Numerical problem

Cancellation of small and large factors

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

23

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Points of view

Contracting the TN of a frustrated model

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)

$\rightarrow$ precision?

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

$\rightarrow$ log?

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Points of view

Contracting the TN of a frustrated model

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)

$\rightarrow$ precision?

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

$\rightarrow$ log?

(For TN experts)

MPO

The MPO is badly conditioned (e.g. not hermitian, ...). Fix it?

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Points of view

Contracting the TN of a frustrated model

Numerical problem

Ground-state rule

Cancellation of small and large factors

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

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

$\rightarrow$ precision?

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

$\rightarrow$ log?

(For TN experts)

MPO

The MPO is badly conditioned (e.g. not hermitian, ...). Fix it?

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

Failure to minimize simultaneously all local Hamiltonians.

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Points of view

Contracting the TN of a frustrated model

Numerical problem

Ground-state rule

Cancellation of small and large factors

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

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

$\rightarrow$ precision?

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

$\rightarrow$ log?

(For TN experts)

MPO

The MPO is badly conditioned (e.g. not hermitian, ...). Fix it?

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-\frac{4}{3}\beta J} & e^{\frac{2}{3}\beta J}\\ e^{\frac{2}{3}\beta J} & e^{-\frac{4}{3}\beta J}\end{pmatrix}

Failure to minimize simultaneously all local Hamiltonians.

B. Vanhecke, JC, et al. PRR 3, (2021)

F.F. Song, T.-Y. Lin, G. M. Zhang, arXiv:2309.05321

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Bram Vanhecke

University of Vienna | Austria

Relaxing THE frustration

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

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

1. Split

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

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)

P. W. Anderson, PR 83, (1951).

Essential idea : Anderson bounds

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Relaxing THE frustration

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

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

1. Split

2. Lower bound on GS energy

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

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

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

\rightarrow

\rightarrow

P. W. Anderson, PR 83, (1951).

Essential idea : Anderson bounds

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

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)

Relaxing THE frustration

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

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

1. Split

2. Lower bound on GS energy

3. Maximize with respect to the weights:

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

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

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

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

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

\rightarrow

\rightarrow

P. W. Anderson, PR 83, (1951).

Essential idea : Anderson bounds

24

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

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)

Relaxing THE frustration

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

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

1. Split

2. Lower bound on GS energy

3. Maximize with respect to the weights:

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

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

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

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

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

\rightarrow

\rightarrow

\rightarrow

\rightarrow

P. W. Anderson, PR 83, (1951).

Essential idea : Anderson bounds

24

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Ground states

tiling of configurations that minimize the local Hamiltonian

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)

Relaxing THE frustration

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

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

1. Split

2. Lower bound on GS energy

3. Maximize with respect to the weights:

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

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

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

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

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

\rightarrow

\rightarrow

\rightarrow

\rightarrow

P. W. Anderson, PR 83, (1951).

Essential idea : Anderson bounds

24

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Ground states

tiling of configurations that minimize the local Hamiltonian

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)

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B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Contraction

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Contraction

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B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Contraction

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Jacobsen, Fogedby, Physica A 246, 1997

G.S. critical

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

Stephenson, J. Math. Phys. 11, 1970

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Jacobsen, Fogedby, Physica A 246, 1997

G.S. critical

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

Stephenson, J. Math. Phys. 11, 1970

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Jacobsen, Fogedby, Physica A 246, 1997

G.S. critical

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

Stephenson, J. Math. Phys. 11, 1970

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet

No transition at finite temperature

\langle \sigma_i \sigma_j \rangle \sim \frac{e^{-r/\xi}}{\sqrt{r}}\\ \xi = -\frac{1}{\ln\tanh(1/T)}

\langle \sigma_i \sigma_j \rangle \sim \frac{e^{-r/\xi}}{\sqrt{r}}\\ \xi = -\frac{1}{\ln\tanh(1/T)}

Jacobsen, Fogedby, Physica A 246, 1997

Wannier, PR 79 , 1950

Houtappel, Physica 16, 1950

G.S. critical

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

Stephenson, J. Math. Phys. 11, 1970

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Samuel Nyckees

EPFL | Switzerland

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

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J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet

No transition at finite temperature

\langle \sigma_i \sigma_j \rangle \sim \frac{e^{-r/\xi}}{\sqrt{r}}\\ \xi = -\frac{1}{\ln\tanh(1/T)}

\langle \sigma_i \sigma_j \rangle \sim \frac{e^{-r/\xi}}{\sqrt{r}}\\ \xi = -\frac{1}{\ln\tanh(1/T)}

Jacobsen, Fogedby, Physica A 246, 1997

Wannier, PR 79 , 1950

Houtappel, Physica 16, 1950

G.S. critical

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

S = 0.3230659...\\ \langle \sigma_i \sigma_j \rangle \sim \frac{1}{\sqrt{r}}

Stephenson, J. Math. Phys. 11, 1970

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Samuel Nyckees

EPFL | Switzerland

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j \quad J > 0 \quad \sigma_i = \pm 1

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B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet In a field

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Racz PRB 21, 1980;

Qian, Wegewijs, Blöte PRE 69, 2004;

Baxter, Exactly solved models

Alexander P.L.A 54 (1975)

Kinzel & Schick PRB 23 (1981)

Noh & Kim, Int. J. Phys. B 06 ( 1992)

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

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B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet In a field

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Racz PRB 21, 1980;

Qian, Wegewijs, Blöte PRE 69, 2004;

Baxter, Exactly solved models

Alexander P.L.A 54 (1975)

Kinzel & Schick PRB 23 (1981)

Noh & Kim, Int. J. Phys. B 06 ( 1992)

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet In a field

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Racz PRB 21, 1980;

Qian, Wegewijs, Blöte PRE 69, 2004;

Baxter, Exactly solved models

Alexander P.L.A 54 (1975)

Kinzel & Schick PRB 23 (1981)

Noh & Kim, Int. J. Phys. B 06 ( 1992)

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

26

$h = 3$

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet In a field

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Racz PRB 21, 1980;

Qian, Wegewijs, Blöte PRE 69, 2004;

Baxter, Exactly solved models

Alexander P.L.A 54 (1975)

Kinzel & Schick PRB 23 (1981)

Noh & Kim, Int. J. Phys. B 06 ( 1992)

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

26

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)

Triangular lattice ising antiferromagnet In a field

Nyckees, Rufino, Mila & JC, arXiv:2306.0904 (2023)

Racz PRB 21, 1980;

Qian, Wegewijs, Blöte PRE 69, 2004;

Baxter, Exactly solved models

Alexander P.L.A 54 (1975)

Kinzel & Schick PRB 23 (1981)

Noh & Kim, Int. J. Phys. B 06 ( 1992)

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

\nu = 5/6, \eta = 4/15, c = 4/5, \beta = 1/9

26

$h = 3$

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

H = J \sum_{\langle i, j \rangle} \sigma_i \sigma_j - h \sum_i \sigma_i \quad J, h > 0 \quad \sigma_i = \pm 1

What have we learned so far?

27

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

What have we learned so far?

Tensor network approaches can be understood from the point of view of transfer matrices

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

27

What have we learned so far?

The convergence of the tensor network contraction depends on the formulation of the partition function

Tensor network approaches can be understood from the point of view of transfer matrices

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

27

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

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

B. Vanhecke, JC, et al. PRR 3, (2021)

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

M. Friaz-Pérez, M, Mariën et al, SciPost 14(2023)

What have we learned so far?

B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete,

F. Mila, PRR 3, (2021)

JC, K. Hofhuis, et al, PRB 104 (2021)

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

F.F. Song, G. M. Zhang, PRB 105, (2022)

The convergence of the tensor network contraction depends on the formulation of the partition function

Tensor network approaches can be understood from the point of view of transfer matrices

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

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

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

B. Vanhecke, JC, et al. PRR 3, (2021)

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

M. Friaz-Pérez, M, Mariën et al, SciPost 14(2023)

If the ground-state rule is implemented at the level of the tensor, the algorithms converge

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

27

Farther-neighbors on kagome

28

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

Chioar et al., PRB 90, (2014)

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Artificial Out-of-plane kagome

28

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}

Chioar et al., PRB 90, (2014)

I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)

J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Artificial Out-of-plane kagome

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

Tensor network construction

29

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Ground state Phase diagram

30

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}

S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}

S = 0.01920 \pm 3 \cdot 10^{-5}

S = 0.01920 \pm 3 \cdot 10^{-5}

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

H =

H =

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j

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

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

+

+

+

+

+

+

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Emergent degrees-of-freedom

31

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Emergent degrees-of-freedom

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

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

MC algorithm : G. Rakala, K. Damle, PRE 96 (2017)

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

31

Emergent degrees-of-freedom

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

31

Emergent degrees-of-freedom

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

32

Emergent degrees-of-freedom

S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}

S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}

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

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$\mathbb{Z}_2 \times \mathbb{Z}_3$ symmetry breaking

Relation to the TIAFM

33

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Outlook

36

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Outlook

TNs

Is there always a cell relaxing the frustration? (Hard vs weak frustration)

Consequences for tensor networks for 2D quantum systems?

36

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Outlook

TNs

Beyond

classical, short-range

Ising

Effect of quantum fluctuations?

NN Frustrated XY models. Farther-neighbors? Heisenberg?

Other classical constrained models?

Long-range interactions? (TNMH)

Spin glasses ? (Tropical TNs)

36

Is there always a cell relaxing the frustration? (Hard vs weak frustration)

Consequences for tensor networks for 2D quantum systems?

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Take home message

37

Why use them for classical systems?

What are tensor networks (TNs)?

Transfer matrix!

Complement to Monte Carlo
Different scaling

What do we learn?

Macroscopic ground-state degeneracy beyond fine-tuning in kagome

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Take home message

37

Why use them for classical systems?

What are tensor networks (TNs)?

Transfer matrix!

Complement to Monte Carlo
Different scaling

What do we learn?

Macroscopic ground-state degeneracy beyond fine-tuning in kagome

Thank you for your attention!

SLIDES

and

PAPERS

Thank you for the question!

Area-Law of entanglement

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

$2^L$

Many-body Hilbert space

M. Fannes, B. Nachtergaele, R.F. Werner, Commun. Math. Phys. 144 (1992)

M B Hastings J. Stat. Mech. (2007) P08024

Ground states of gapped, local Hamiltonians

SAMPLING

Ueda, et al. JSPS 74, 111-124 (2005)

A. Rufino, S. Nyckees, J. Colbois and F. Mila, in preparation

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

SAMPLING

Ueda, et al. JSPS 74, 111-124 (2005)

A. Rufino, S. Nyckees, J. Colbois and F. Mila, in preparation

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

SAMPLING

Ueda, et al. JSPS 74, 111-124 (2005)

A. Rufino, S. Nyckees, J. Colbois and F. Mila, in preparation

SAMPLING

$\rightarrow$ patches of the infinite lattice

Ueda, et al. JSPS 74, 111-124 (2005)

A. Rufino, S. Nyckees, J. Colbois and F. Mila, in preparation

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

IPEPS

J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023

Jiménez, J.L., Crone, S.P.G., Fogh, E. et al. Nature 592, 370–375 (2021)

SrCu2(BO3)2 under pressure

finite-temperature iPEPS simulations

Dimer

Plaquette

Introduction to

tensor networks

for classical frustrated spin systems