Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Jeanne Colbois | CNRS - LPT Toulouse | France
Séminaire MCBT | Grenoble, 06.11.2023
Introduction to
tensor networks
for classical frustrated spin systems
SLIDES
and
PAPERS
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
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
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
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
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
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
5
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
S = 0.3230659...
G.H. Wannier, PR 79, (1950, 1973)
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
d_{i,j} = \sigma_i \sigma_j
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
S = 0.3230659...
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}
d_{i,j} = \sigma_i \sigma_j
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...
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
S = 0.3230659...
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}
d_{i,j} = \sigma_i \sigma_j
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...
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}
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}
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}
One question among others
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}
One question among others
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)
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)
\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}
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}
\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}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{orange}\lambda_+}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})}
= e^{-\beta E_{\rm{GS}}} {\color{orange}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\
= e^{-\beta E_{\rm{GS}}} {\color{orange}\tilde{\mathcal{Z}}_N}
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}}
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}}
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}}
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}
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}}
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}
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
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}}
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}
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}
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}}
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}
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^{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}}
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}
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^{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"
J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023
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
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
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}
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}
\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}
\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
J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023
2D partition function contraction
18
\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
EXPONENTIAL # of PARAMETERS
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
EXPONENTIAL # of PARAMETERS
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
\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
EXPONENTIAL # of PARAMETERS
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
\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
EXPONENTIAL # of PARAMETERS
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
\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
The frustration problem
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
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
J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023
T_{\sigma_1, \sigma_2, \sigma_3} =
= \begin{cases}
0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\
1 & \text{otherwise}
\end{cases}
T_{d_1, d_2,d_3} =
= \begin{cases}
1 \quad & \text{if } \prod_i d_i = -1\\
0 & \text{otherwise}
\end{cases}
A SURPRISE?
A SURPRISE?
23
J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023
T_{\sigma_1, \sigma_2, \sigma_3} =
= \begin{cases}
0 \quad & \text{if } \sigma_1 = \sigma_2 = \sigma_3\\
1 & \text{otherwise}
\end{cases}
T_{d_1, d_2,d_3} =
= \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}
23
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?
\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
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}
23
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}
Failure to minimize simultaneously all local Hamiltonians.
23
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}
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
23
J. COLBOIS | INTRO TO TNS. FOR CLASSICAL SPIN SYSTEMS | GRENOBLE | 06.11.2023
Bram Vanhecke
University of Vienna | Austria
Relaxing THE frustration
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
24
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)
1. Split
2. Lower bound on GS energy
\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
\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)
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)
\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
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)
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)
\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
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)
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)
\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
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)
24
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
24
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
24
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
25
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
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}}
Stephenson, J. Math. Phys. 11, 1970
25
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
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}}
Stephenson, J. Math. Phys. 11, 1970
25
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
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}}
Stephenson, J. Math. Phys. 11, 1970
25
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)}
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}}
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
25
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)}
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}}
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
26
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
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
26
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
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
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
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
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
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
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
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}
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}
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 =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
Tensor network construction
29
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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 = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}
S = 0.01920 \pm 3 \cdot 10^{-5}
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
H =
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
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}
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}
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}
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}
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}
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 frustrated spin systems
By Jeanne Colbois
Introduction to tensor networks for frustrated spin systems
Seminar at Institut Néel
