Plan

Computing the residual entropy of (classical) frustrated systems
- Motivation: classical frustrated spin systems
- "Standard" TN formulation of partition functions
- Issues with the low temperature limit
Solving the convergence problem of tensor networks on frustrated spin systems
- Finding the ground state local rule
- An example of ground state phase

MPQ Theory seminar - 27.01.2021

1

MPQ Theory seminar - 27.01.2021

Motivation

Frustrated models, residual entropies and methods to compute them

MPQ Theory seminar - 27.01.2021

Nearest neighbour Ising antiferromagnet on the triangular lattice

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

2

We cannot minimize the Hamiltonian on each bond simultaneously

MPQ Theory seminar - 27.01.2021

Nearest neighbour Ising antiferromagnet on the triangular lattice

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

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

S \geq \frac{1}{3} \ln(2) = 0.23105... \quad S = 0.32306...

S \geq \frac{1}{3} \ln(2) = 0.23105... \quad S = 0.32306...

Wannier, Antiferromagnetism. The triangular Ising net. PR 79 1950 (Erratum PRB 7, 1973)

2

MPQ Theory seminar - 27.01.2021

Computing/estimating residual entropies

Exactly solvable cases
- Mapping to free fermions, Kasteleyn's method, height mappings
- In further-neighbour models : integrable points, mapping to charge models

Baxter, Exactly solved models in statistical mechanics

3

MPQ Theory seminar - 27.01.2021

Computing/estimating residual entropies

Exactly solvable cases
- Mapping to free fermions, Kasteleyn's method, height mappings
- In further-neighbour models : integrable points, mapping to charge models
What about non-integrable models?
- Pauling estimates (entropy of ice)
- Monte Carlo simulations
- Tensor network contractions

Baxter, Exactly solved models in statistical mechanics

Pauling, J. Am. Chem. Soc. 57 , 1935

3

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

4

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

Difficulties

1. Critical slowing down, ergodicity

2. Thermodynamic integration

4

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

Difficulties

1. Critical slowing down, ergodicity

2. Thermodynamic integration

H = J_1\sum_{\langle i, j \rangle} \sigma_i \sigma_j + J_2 \sum_{\langle\langle i, j \rangle\rangle} \sigma_i \sigma_j + J_3 \sum_{\langle\langle\langle i, j \rangle\rangle\rangle} \sigma_i \sigma_j

H = J_1\sum_{\langle i, j \rangle} \sigma_i \sigma_j + J_2 \sum_{\langle\langle i, j \rangle\rangle} \sigma_i \sigma_j + J_3 \sum_{\langle\langle\langle i, j \rangle\rangle\rangle} \sigma_i \sigma_j

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

Mizoguchi, Jaubert, Udagawa, PRL 119, 2017

4

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

Mizoguchi, Jaubert, Udagawa, PRL 119, 2017

(2 digits)

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

5

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

Zukovic, Eur. Phys. J. B, 2013

6

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using Monte Carlo simulations

Zukovic, Eur. Phys. J. B, 2013

S = 0.3230659669...

S = 0.3230659669...

Wannier:

6

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using tensor networks and ground state local rules

Vanderstraeten, Vanhecke, Verstraete, PRE 98, 2018

7

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using tensor networks and ground state local rules

TNs : 10 digits, vs 5 digits for MC

Vanderstraeten, Vanhecke, Verstraete, PRE 98, 2018

7

MPQ Theory seminar - 27.01.2021

Computing residual entropies

Using tensor networks and ground state local rules

No need to understand the whole temperature range to understand the ground state

Vanderstraeten, Vanhecke, Verstraete, PRE 98, 2018

TNs : 10 digits, vs 5 digits for MC

7

TN

"Standard" tensor network construction for partition functions

MPQ Theory seminar - 27.01.2021

"Standard" construction

Partition functions of classical spin systems as a tensor network

\mathcal{Z} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} e^{\beta J\sigma_i \sigma_j}

\mathcal{Z} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} e^{\beta J\sigma_i \sigma_j}

Baxter, Exactly solved models in statistical mechanics;

Nishino, Okunishi, J. Phys. Soc. Jpn 65, 1996; Orús, Vidal, PRB 78, 2008;

Zhao, Xie, et. al. PRB 81, 2010; Xie, Chen et. al. PRB 86, 2012;

Zhu, Katzgraber, arXiv:1903.0772, 2019;

Nyckees, JC, Mila, arXiv:2008.08408, 2020

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H = - J \sum_{\vec{\sigma}} \sigma_i \sigma_j

H = - J \sum_{\vec{\sigma}} \sigma_i \sigma_j

MPQ Theory seminar - 27.01.2021

"Standard" construction

Partition functions of classical spin systems as a tensor network

\mathcal{Z} = \sum_{\vec{\sigma_1}, \vec{\sigma_2}, \vec{\sigma_3}, \vec{\sigma_4}} \prod_{i} \Delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} t_{\sigma_{i-x,3}, \sigma_{i,1}} t_{\sigma_{i,2}, \sigma_{i+y,4}}

\mathcal{Z} = \sum_{\vec{\sigma_1}, \vec{\sigma_2}, \vec{\sigma_3}, \vec{\sigma_4}} \prod_{i} \Delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}} t_{\sigma_{i-x,3}, \sigma_{i,1}} t_{\sigma_{i,2}, \sigma_{i+y,4}}

\Delta_{\sigma_{i,1},\sigma_{i,2},\sigma_{i,3},\sigma_{i,4}} =

\Delta_{\sigma_{i,1},\sigma_{i,2},\sigma_{i,3},\sigma_{i,4}} =

1

1

2

2

3

3

4

4

Baxter, Exactly solved models in statistical mechanics;

Nishino, Okunishi, J. Phys. Soc. Jpn 65, 1996; Orús, Vidal, PRB 78, 2008;

Zhao, Xie, et. al. PRB 81, 2010; Xie, Chen et. al. PRB 86, 2012;

Zhu, Katzgraber, arXiv:1903.0772, 2019;

Nyckees, JC, Mila, arXiv:2008.08408, 2020

= e^{\beta J \sigma_i \sigma_j}

= e^{\beta J \sigma_i \sigma_j}

t_{\sigma_i, \sigma_j} =

t_{\sigma_i, \sigma_j} =

8

MPQ Theory seminar - 27.01.2021

Contraction algorithms

1 + 1

2 D

CTMRG

TRG

iTEBD

VUMPS

Nishino, Okunishi, J. Phys. Soc. Jpn 65, 1996;

Levin, Nave, PRL 99, 2007

9

Orús, Vidal, PRB 78, 2008;

Zauner-Stauber et. al. PRB 97,2018; Fishman et. al PRB 98, 2018

MPQ Theory seminar - 27.01.2021

Contraction algorithms

1 + 1

2 D

CTMRG

TRG

iTEBD

VUMPS

Nishino, Okunishi, J. Phys. Soc. Jpn 65, 1996;

Levin, Nave, PRL 99, 2007

9

Orús, Vidal, PRB 78, 2008;

Zauner-Stauber et. al. PRB 97,2018; Fishman et. al PRB 98, 2018

MPQ Theory seminar - 27.01.2021

Contraction and leading eigenvalue

(1+1)D : Look for the leading eigenvector of the row-to-row transfer matrix

10

Haegeman, Verstraete, Annu. Rev. Condens. Matter Phys. 2017

MPQ Theory seminar - 27.01.2021

Contraction and leading eigenvalue

(1+1)D : Look for the leading eigenvector of the row-to-row transfer matrix

\frac{1}{N}\ln(\mathcal{Z}_N) \rightarrow \ln\left(\lambda_1\right)

\frac{1}{N}\ln(\mathcal{Z}_N) \rightarrow \ln\left(\lambda_1\right)

Just like in the 1D chain, because the partition function is the trace of the transfer matrix:

10

Haegeman, Verstraete, Annu. Rev. Condens. Matter Phys. 2017

Residual entropy from the leading eigenvalue

MPQ Theory seminar - 27.01.2021

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

11

Residual entropy from the leading eigenvalue

MPQ Theory seminar - 27.01.2021

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

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

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

Residual entropy from the leading eigenvalue

MPQ Theory seminar - 27.01.2021

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta N\epsilon_0} \sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})-N\epsilon_0\right)}\\ = e^{-\beta N\epsilon_0} \tilde{\mathcal{Z}}_N

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

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

\Rightarrow S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right) = \lim_{\beta\rightarrow\infty} \ln\left(\tilde{\lambda}_1\right)

\Rightarrow S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right) = \lim_{\beta\rightarrow\infty} \ln\left(\tilde{\lambda}_1\right)

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Issues with taking the low temperature limit

MPQ Theory seminar - 27.01.2021

Non-frustrated models

(e.g. ferromagnet)

MPQ Theory seminar - 27.01.2021

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} 1 & e^{-2\beta |J|}\\ e^{-2\beta |J|} & 1\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} 1 & e^{-2\beta |J|}\\ e^{-2\beta |J|} & 1\end{pmatrix}

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Non-frustrated models

(e.g. antiferromagnet, square lattice)

MPQ Theory seminar - 27.01.2021

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-2\beta |J|} & 1\\ 1 & e^{-2\beta |J|}\end{pmatrix}

\tilde{t} = e^{\beta \frac{E_{\text{G.S.}}}{N_{\text{bonds}}}} t = \begin{pmatrix} e^{-2\beta |J|} & 1\\ 1 & e^{-2\beta |J|}\end{pmatrix}

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Triangular lattice Ising antiferromagnet

E_{\text{G.S.}} = -NJ \,\,\,\, N_{\text{bonds}} = 3N

E_{\text{G.S.}} = -NJ \,\,\,\, N_{\text{bonds}} = 3N

Kagome lattice Ising antiferromagnet

E_{\text{G.S.}} = -\frac{2}{3}NJ \,\,\,\, N_{\text{bonds}} = 2N

E_{\text{G.S.}} = -\frac{2}{3}NJ \,\,\,\, N_{\text{bonds}} = 2N

MPQ Theory seminar - 27.01.2021

13

Triangular lattice Ising antiferromagnet

E_{\text{G.S.}} = -NJ \,\,\,\, N_{\text{bonds}} = 3N

E_{\text{G.S.}} = -NJ \,\,\,\, N_{\text{bonds}} = 3N

Kagome lattice Ising antiferromagnet

E_{\text{G.S.}} = -\frac{2}{3}NJ \,\,\,\, N_{\text{bonds}} = 2N

E_{\text{G.S.}} = -\frac{2}{3}NJ \,\,\,\, N_{\text{bonds}} = 2N

\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}

MPQ Theory seminar - 27.01.2021

13

Why was that not a problem?

MPQ Theory seminar - 27.01.2021

Why was that not a problem?

... in general ...

MPQ Theory seminar - 27.01.2021

It is a problem, e.g. in spin glasses

Wang, Qin, Zhou, PRB 90, 2014

Zhu, Katzgraber, arXiv:1903.07721, 2019

14

Why was that not a problem?

... in general ...

MPQ Theory seminar - 27.01.2021

It is a problem, e.g. in spin glasses

Wang, Qin, Zhou, PRB 90, 2014

Zhu, Katzgraber, arXiv:1903.07721, 2019

2. The problem can be avoided when one knows how to write the partition function in a different way

Kramers, Wannier, PR 60, 1941; Wannier, PR 79 1950;

Kano, Naya, Prog. Theor. Phys. 10 , 1953

Levin, Nave, PRL 99, 2007

14

Why was that not a problem?

... in general ...

MPQ Theory seminar - 27.01.2021

It is a problem, e.g. in spin glasses

Wang, Qin, Zhou, PRB 90, 2014

Zhu, Katzgraber, arXiv:1903.07721, 2019

2. The problem can be avoided when one knows how to write the partition function in a different way

Kramers, Wannier, PR 60, 1941; Wannier, PR 79 1950;

Kano, Naya, Prog. Theor. Phys. 10 , 1953

Levin, Nave, PRL 99, 2007

=> First build a tensor network which describes the ground states and can be contracted

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MPQ Theory seminar - 27.01.2021

Can we just split the Hamiltonian differently?

15

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;

Z. Wang, M. Navascues, Proc. Roy. Soc. A 474, 2018

Splitting the Hamiltonian

... on the triangular lattice

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

16

Splitting the Hamiltonian

... on the triangular lattice

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

G.S. configurations

Delta tensor with bond dimension 6

16

Splitting the Hamiltonian

... on the triangular lattice

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

Spins must match

G.S. configurations

16

Splitting the Hamiltonian

... on the triangular lattice

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

17

How come?

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

What happens to configurations where a down triangle is ferromagnetic?

18

How come?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

What happens to configurations where a down triangle is ferromagnetic?

The local rule is imposed locally

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

18

How come?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

What happens to configurations where a down triangle is ferromagnetic?

The local rule is imposed locally

The local rule has to be computed

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\vartriangle_{i,j,k}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j =\\ \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} \frac{J}{2}(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)

18

MPQ Theory seminar - 27.01.2021

Can we perform this splitting in a systematic way?

Huang, et al., Phys. Rev. B 94, 2016

Proving the ground state energies of generalized Ising models

MPQ Theory seminar - 27.01.2021

Side question: can we detect problematic cases without prior knowledge of the model?

Can we perform this splitting in a systematic way?

Huang, et al., Phys. Rev. B 94, 2016

Proving the ground state energies of generalized Ising models

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

19

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

Select a cluster, with which one can tile the lattice with overlaps

19

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

For each bond, a set of weight describes how to split it among the clusters

19

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

For each bond, a set of weight describes how to split it among the clusters

\alpha_{1,2}'=1-\alpha_{1,2}

\alpha_{1,2}'=1-\alpha_{1,2}

19

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}}\\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}}\\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

\alpha_{1,2}'=1-\alpha_{1,2}

\alpha_{1,2}'=1-\alpha_{1,2}

20

Splitting the Hamiltonian

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

H(\{\sigma\}) = \sum_n h_n(\{\sigma\}_n)

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

\alpha_{1,2}'=1-\alpha_{1,2}

\alpha_{1,2}'=1-\alpha_{1,2}

20

Ground state lower bounds

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

21

Ground state lower bounds

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

For a given cluster and weights, the

local ground state energy imposes a

lower bound

on the global g.s. energy (per cluster)

E^{\{\alpha\}}_0

E^{\{\alpha\}}_0

21

Ground state lower bounds

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

For a given cluster and weights, the

local ground state energy imposes a

lower bound

on the global g.s. energy (per cluster)

E^{\{\alpha\}}_0

E^{\{\alpha\}}_0

E_0 \geq E^u_0 = \max_{\{\alpha\}} E^{\{\alpha\}}_0\\

E_0 \geq E^u_0 = \max_{\{\alpha\}} E^{\{\alpha\}}_0\\

21

Restricting the weights

MPQ Theory seminar - 27.01.2021

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

H = \sum_{c\in\mathcal{T}_u} H_c^{\{\alpha\}} = \sum_{c \in \mathcal{T}_u} \sum_{n \in c} \alpha_n^c h_n \\ \text{Constraints: } \sum_{c\in \mathcal{T}_u|n \in c} \alpha_n^c = 1 \, \forall n

E_0 \geq E^u_0 = \max_{\{\alpha\}} E^{\{\alpha\}}_0\\

E_0 \geq E^u_0 = \max_{\{\alpha\}} E^{\{\alpha\}}_0\\

\alpha_1 = \alpha_2 = \alpha \in [0,1]

\alpha_1 = \alpha_2 = \alpha \in [0,1]

\alpha_{1,2}'=1-\alpha_{1,2}

\alpha_{1,2}'=1-\alpha_{1,2}

\rightarrow

\rightarrow

22

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

\rightarrow

\rightarrow

Getting the ground state entropy

We have a g.s. energy lower bound, what if it is saturated?

MPQ Theory seminar - 27.01.2021

23

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

Getting the ground state entropy

We have a g.s. energy lower bound, what if it is saturated?

MPQ Theory seminar - 27.01.2021

\{\alpha^u_n\}

\{\alpha^u_n\}

realize the g.s. energy

All the local cluster Hamiltonians can be minimised simultaneously

23

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

Getting the ground state entropy

We have a g.s. energy lower bound, what if it is saturated?

MPQ Theory seminar - 27.01.2021

The frustration is "relaxed"

\{\alpha^u_n\}

\{\alpha^u_n\}

realize the g.s. energy

All the local cluster Hamiltonians can be minimised simultaneously

23

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

Getting the ground state entropy

We have a g.s. energy lower bound, what if it is saturated?

ALL the ground states are described as tilings of the local g.s. configurations

MPQ Theory seminar - 27.01.2021

The frustration is "relaxed"

\{\alpha^u_n\}

\{\alpha^u_n\}

realize the g.s. energy

All the local cluster Hamiltonians can be minimised simultaneously

(Local rule; vertex model)

G^{\{\alpha\}}:=

G^{\{\alpha\}}:=

23

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

What happened in the triangular case?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

24

What happened in the triangular case?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

24

What happened in the triangular case?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

24

What happened in the triangular case?

MPQ Theory seminar - 27.01.2021

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

24

What happened in the triangular case?

MPQ Theory seminar - 27.01.2021

On the boundary : additional "spurious" tiles. Always select weights in the interior.

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

24

Selecting the weights

We are trying to maximize a function under linear constraints
Linear programming
With this we can find weights in the interior of the set of weights which maximise the g.s. energy lower bound.
Systematically try clusters

MPQ Theory seminar - 27.01.2021

(Credit goes fully to Bram Vanhecke)

25

Once you have the tiles...

Tensor network based on the g.s. local rule
bond tensors which impose overlapping spins to match

MPQ Theory seminar - 27.01.2021

G^{\{\alpha\}}:=

G^{\{\alpha\}}:=

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

26

Once you have the tiles...

Tensor network based on the g.s. local rule
bond tensors which impose overlapping spins to match

Bond dimension?

# local g.s.?
Actually, P matrices are rank-deficient : SVD and group, much lower bond dimension
Note that Ps are typically not symmetric

MPQ Theory seminar - 27.01.2021

G^{\{\alpha\}}:=

G^{\{\alpha\}}:=

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

26

Proving the g.s. energy

Rigorously:

1. Find an upper bound, or

2. Find a state with the lower bound energy, or

3. Show that the tiles in can tile the lattice

G^{\{\alpha\}}_u

G^{\{\alpha\}}_u

Z. Wang, M. Navascues, Proc. Roy. Soc. A 474, (2018)

W. Huang, D. A. Kitchaev, S. T. Dacek et. al. , Phys. Rev. B 94, (2016)

MPQ Theory seminar - 27.01.2021

27

An example

Further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

A further neighbour model on the kagome lattice

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

MPQ Theory seminar - 27.01.2021

28

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

28

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

132 G.S. tiles

Bond dimension 18

28

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

29

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

30

Type I

Type II

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Type I

Type II

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

30

Only type I tiles

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

31

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

32

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Type I

Type II

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

33

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

Type I

Type II

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

A further neighbour model on the kagome lattice

MPQ Theory seminar - 27.01.2021

34

Conclusion

Facilitates a quick exploration of ground state phases (on top of the ground state energy).

1. Allows extremely precise extraction of the residual entropy with very limited computational effort

2. Tiles and exact contractions allow for some understanding

MPQ Theory seminar - 27.01.2021

35

Conclusion

Facilitates a quick exploration of ground state phases (on top of the ground state energy).

1. Allows extremely precise extraction of the residual entropy with very limited computational effort

2. Tiles and exact contractions allow for some understanding

Maps frustrated problems onto tiling problems

1. Question of the existence of the scale at which frustration is relaxed

2. Still have to explore how this mapping could help through well-studied questions on tilings.

MPQ Theory seminar - 27.01.2021

35

Conclusion

Facilitates a quick exploration of ground state phases (on top of the ground state energy).

1. Allows extremely precise extraction of the residual entropy with very limited computational effort

2. Tiles and exact contractions allow for some understanding

Maps frustrated problems onto tiling problems

1. Question of the existence of the scale at which frustration is relaxed

2. Still have to explore how this mapping could help through well-studied questions on tilings.

MPQ Theory seminar - 27.01.2021

Alternative approaches?

1. Tropical tensor networks: Jin-Guo Liu, Lei Wang, and Pan Zhang, arXiv:2008.06888

2. A systematic way of making the MPO hermitian?

35

Acknowledgments

MPQ Theory seminar - 27.01.2021

Bram Vanhecke

Laurens Vanderstraeten, Frank Verstraete,

Frédéric Mila

Thank you for your attention!

Manuel Stathis, Patrick Emonts, Mithilesh Nayak, Jonathan D'Emidio

(And the Centro de ciencias de Benasque Pedro Pascual)

Discussions:

Collaborators:

Finite temperature

\mathcal{Z}_0 =

\mathcal{Z}_0 =

D^{C_1, C_2, C_3, C_4}(\beta,\{\alpha\}) = \delta^{C_1, C_2, C_3,C_4} B(C_1, \beta, \{\alpha\})\\ B(C_1, \beta, \{\alpha\}) = e^{-\beta (H^{\{\alpha\}}_u(C_1)-E_u)}

D^{C_1, C_2, C_3, C_4}(\beta,\{\alpha\}) = \delta^{C_1, C_2, C_3,C_4} B(C_1, \beta, \{\alpha\})\\ B(C_1, \beta, \{\alpha\}) = e^{-\beta (H^{\{\alpha\}}_u(C_1)-E_u)}

MPQ Theory seminar - 27.01.2021

Linear programming ideas

Linear program gives extreme points, i.e. get weights on the boundary
So instead find an interior simplex of the same dimension as A
Find a point in the interior of A

Number of constraints exponential in number of sites
Incoroporate only useful ones

MPQ Theory seminar - 27.01.2021

(Credit goes fully to Bram Vanhecke)

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

H = J_1\sum_{(i,j)_1} \sigma_i \sigma_j+J_2\sum_{(i,j)_2} \sigma_i \sigma_j +J_3\sum_{(i,j)_3} \sigma_i \sigma_j

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

J_1 < 0 \quad J_2 = J_3 = J > 0 \quad J_2 = J_3 \gg |J_1|

1. Type-II tiles can connect reflection-symmetry-broken sectors

2. Each type-II tile has to have exactly two neighbours of the same sub-type (can have of other sub-type)

3. No sharp angles (only straight or 120°)

4. No way to go back

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

More about the type-II tiles

MPQ Theory seminar - 27.01.2021

Unending strings separating symmetry-broken sectors

Idea used in the classical context:

Inequalities based on a systematic splitting of the Hamiltonian

M. Kaburagi, J. Kanamori, Prog. Theor. Phys. 54 , 1975

Finding ground states of generalised Ising models

W. Huang, D. A. Kitchaev, et. al. , Phys. Rev. B 94, 2016

Exact results for square lattice Ising models

Z. Wang, M. Navascues, Proc. Roy. Soc. A 474, 2018

(and many more)

Splitting the Hamiltonian to learn about ground states

MPQ Theory seminar - 27.01.2021

Idea used in the quantum context:

Exact ground state in the J1-J2, spin-1/2 Heisenberg chain (Majumdar-Ghosh point)

C. K. Majumdar and D. K. Ghosh, J. Math. Phys. 10, 1969

Eigenstate in the Shastry-Sutherland model (for small J'/J)

B. Sriram Shastry and B. Sutherland, Physica 108 B+C, 1981

(and many more)

Splitting the Hamiltonian to learn about ground states

MPQ Theory seminar - 27.01.2021

Proving the g.s. energy

- Numerical evidence -

Consider the "partition function" at zero temperature

If one cannot make a ground state, then the partition function is zero
Reciprocally, if the exact leading eigenvalue is 1 or larger, then the tiles can be combined into a state
Approximate contraction = numerical evidence of tiling

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

MPQ Theory seminar - 27.01.2021

Proving the g.s. energy

- Numerical evidence -

Cases

1. The approximate contraction converges consistently to a leading eigenvalue which is 1 or larger

2. The approximate contraction converges only for some bond dimensions

3. The approximate contraction does not converge

MPQ Theory seminar - 27.01.2021

Remark on tilings

Given a Hamiltonian and a cluster, we have a tiling rule (corresponding to the maximal lower bound).

1. Does it have a solution, i.e. does it admit a tiling?

2. Does it admit a periodic tiling, or only non-periodic ones?

The first problem is undecidable. It relates to the undecidability of the energy problem in 2D (Wang, Navascues 2017)

Fh, Fv associate an energy to the bonds.

They correspond to our bond P tensors here.

Z. Wang, M. Navscues, Proc. Roy. Soc. A 474, (2018)

MPQ Theory seminar - 27.01.2021

Remark on tilings

Given only the Hamiltonian

1. Does it admit a cluster which gives rise to a solvable tiling rule?

2. Does it admit a cluster whose tiling rule admits periodic tilings?

And for the approximate contractions:

How do the algorithms react if given a rule for aperiodic tilings? What if only some of the tilings corresponding to the rule are periodic?

MPQ Theory seminar - 27.01.2021

Experimental motivations

MPQ Theory seminar - 27.01.2021

Chioar,et. al. PRB 90, 2014

Luo et. al. , Science 363, 2019

JC et. al. , In preparation

MPQ Theory seminar - 27.01.2021

Why was that not a problem?

... on the kagome lattice

MPQ Theory seminar - 27.01.2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

Kano, Naya, Prog. Theor. Phys. 10 , 1953

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

Why was that not a problem?

... on the kagome lattice

MPQ Theory seminar - 27.01.2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

G.S. configurations

Spins must match

Why was that not a problem?

... on the kagome lattice

MPQ Theory seminar - 27.01.2021

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

H = \sum_{\langle i,j \rangle} J \sigma_i \sigma_j = \sum_{\substack{\vartriangle_{i,j,k}\\ \triangledown_{i,j,k}}} J(\sigma_i \sigma_j + \sigma_j \sigma_k + \sigma_k \sigma_i)\\ := \sum_{\vartriangle_{i,j,k}} H_{\vartriangle_{i,j,k}} + \sum_{\triangledown_{i,j,k}} H_{\triangledown_{i,j,k}}.

G.S. configurations

Spins must match

Vanhecke, JC, Vanderstraeten, Verstraete, Mila, PRR 3, 2021

MPQ Theory seminar - 27.01.2021

1D Ising model, transfer matrix and TN

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

Your statistical physics lecture, or Baxter's Exactly solved models in statistical mechanics

MPQ Theory seminar - 27.01.2021

1D Ising model, transfer matrix and TN

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\

Your statistical physics lecture, or Baxter's Exactly solved models in statistical mechanics

MPQ Theory seminar - 27.01.2021

1D Ising model, transfer matrix and TN

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\ = \sum_{\{\sigma\}} T_{\sigma_1, \sigma_2} T_{\sigma_2, \sigma_3} \cdots T_{\sigma_N, \sigma_1} = \sum_{\sigma} (T^N)_{\sigma, \sigma}

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\ = \sum_{\{\sigma\}} T_{\sigma_1, \sigma_2} T_{\sigma_2, \sigma_3} \cdots T_{\sigma_N, \sigma_1} = \sum_{\sigma} (T^N)_{\sigma, \sigma}

Your statistical physics lecture, or Baxter's Exactly solved models in statistical mechanics

1D Ising model, transfer matrix and TN

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{H} = - J \sum_{i} \sigma_i \sigma_{i+1}, \quad \sigma_i = \pm 1

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\ = \sum_{\{\sigma\}} T_{\sigma_1, \sigma_2} T_{\sigma_2, \sigma_3} \cdots T_{\sigma_N, \sigma_1} = \sum_{\sigma} (T^N)_{\sigma, \sigma}

\mathcal{Z} = \sum_{\{\sigma\}} \exp{\left(-\beta \mathcal{H}\right)} = \sum_{\{\sigma\}} \prod_{\langle i,j \rangle}\exp{\left( \beta J \sigma_i \sigma_{j}\right)}\\ = \sum_{\{\sigma\}} T_{\sigma_1, \sigma_2} T_{\sigma_2, \sigma_3} \cdots T_{\sigma_N, \sigma_1} = \sum_{\sigma} (T^N)_{\sigma, \sigma}

Your statistical physics lecture, or Baxter's Exactly solved models in statistical mechanics

\mathcal{Z} = \sum_{m}\lambda_m^{N} = \lambda_1^{N}\left(1 + \sum_{m \neq 1} \left[\frac{\lambda_m}{\lambda_1}\right]^N \right) \rightarrow \lambda_1^N

\mathcal{Z} = \sum_{m}\lambda_m^{N} = \lambda_1^{N}\left(1 + \sum_{m \neq 1} \left[\frac{\lambda_m}{\lambda_1}\right]^N \right) \rightarrow \lambda_1^N

MPQ Theory seminar - 27.01.2021

"Standard" construction

Partition functions of classical spin systems as a tensor network

\mathcal{H} = - J \sum_{\langle i, j \rangle } \sigma_i \sigma_{j}, \quad \sigma_i = \pm 1

\mathcal{H} = - J \sum_{\langle i, j \rangle } \sigma_i \sigma_{j}, \quad \sigma_i = \pm 1

MPQ Theory seminar - 27.01.2021

"Standard" construction

Partition functions of classical spin systems as a tensor network

\mathcal{Z} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} e^{\beta J\sigma_i \sigma_j} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} t_{\sigma_i, \sigma_j}

\mathcal{Z} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} e^{\beta J\sigma_i \sigma_j} = \sum_{\vec{\sigma}} \prod_{\left< i,j \right>} t_{\sigma_i, \sigma_j}