Tensor networks for frustrated Ising models:

partial lifting of a macroscopic ground-state degeneracy

JC, B. Vanhecke et al., arXiv:2206.11788 (2022)

B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)

Jeanne Colbois | Laboratoire de Physique Théorique, Université de Toulouse CNRS/UPS | France

IISER Pune  - Condensed Matter, Statistical, AMO and Nonlinear Physics Events -  Online - 29.09.2022

Andrew Smerald

KIT | Germany

Frank Verstraete

Ghent University | Belgium

Laurens Vanderstraeten

Ghent University | Belgium

Bram Vanhecke

University of Vienna | Austria

Frédéric Mila

EPFL | Switzerland

Frustration and motivation: artificial spin systems

A model with several macroscopically degenerate phases

Tensor networks for frustrated Ising models

Proving the ground-state energy?

> Motivation

> Introduction to tensor networks for classical spin systems

> A problem of contraction

Scope

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

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J. Colbois

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)

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J. Colbois

Frustration

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)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

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J. Colbois

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

Frustration

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)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

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J. Colbois

Frustration

H = J \sum_{\langle i,j\rangle} \vec{e}_i \cdot \vec{e}_j \sigma_i \sigma_j + D \sum_{(i,j)}\left(\frac{\vec{e_i} \cdot \vec{e_j}}{|\vec{r}_{ij}|^3} - \frac{3(\vec{e}_i \cdot \vec{r}_{ij})(\vec{e}_j \cdot \vec{r}_{ij})}{|\vec{r}_{ij}|^5}\right)\sigma_i \sigma_j

Anghinolfi et al.,

Nat. Commun. 6, (2015)

In-plane

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Artificial spin systems

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)

TbCo

Out-of-plane

Anghinolfi et al.,

Nat. Commun. 6, (2015)

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In-plane

H = J \sum_{\langle i,j\rangle} \vec{e}_i \cdot \vec{e}_j \sigma_i \sigma_j + D \sum_{(i,j)}\left(\frac{\vec{e_i} \cdot \vec{e_j}}{|\vec{r}_{ij}|^3} - \frac{3(\vec{e}_i \cdot \vec{r}_{ij})(\vec{e}_j \cdot \vec{r}_{ij})}{|\vec{r}_{ij}|^5}\right)\sigma_i \sigma_j

Artificial spin systems

 

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)

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J. Colbois

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Chioar et al., PRB 90, (2014)

Dipolar kagome Ising antiferromagnet (DKIAFM)

 

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)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

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Chioar et al., PRB 90, (2014)

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

Dipolar kagome Ising antiferromagnet (DKIAFM)

 

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)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

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Chioar et al., PRB 90, (2014)

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

Dipolar kagome Ising antiferromagnet (DKIAFM)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

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Chioar et al., PRB 90, (2014)

Dipolar kagome Ising antiferromagnet (DKIAFM)

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Chioar et al., PRB 90, (2014)

How does the degeneracy get lifted?

Dipolar kagome Ising antiferromagnet (DKIAFM)

Chioar et al., PRB 90, (2014)

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

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Short-range model and questions

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)

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

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Short-range model and questions

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

(How) Does the degeneracy get lifted?

Chioar et al., PRB 90, (2014)

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

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Short-range model and questions

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

(How) Does the degeneracy get lifted?

Proving the ground-state energy?

Evaluating the residual entropy precisely?

(How) Does the degeneracy get lifted?

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Expectations (1)

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

 

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

 

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (1)

J_2 < 0

(How) Does the degeneracy get lifted?

6

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

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

JC, K. Hofhuis, et al., PRB 104, (2022)

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J. Colbois

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Expectations (1) vs Reality

J_2 < 0
J_2 > 0

(How) Does the degeneracy get lifted?

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

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

JC, K. Hofhuis, et al., PRB 104, (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

7

Expectations (1) vs Reality

J_2 < 0
J_2 > 0

(How) Does the degeneracy get lifted?

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

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

JC, K. Hofhuis, et al., PRB 104, (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

7

Expectations (1) vs Reality

J_2 < 0
J_2 > 0

(How) Does the degeneracy get lifted?

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

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

JC, K. Hofhuis, et al., PRB 104, (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

7

Expectations (1) vs Reality

J_2 < 0
J_2 > 0

(How) Does the degeneracy get lifted?

The macroscopic g.s. degeneracy gets completely lifted by second-neighbor interactions.

 

T. Takagi and M. Mekata,  JSPS 62, (1993)

A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)

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

JC, K. Hofhuis, et al., PRB 104, (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

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Expectations (1) vs Reality

J_2 < 0
J_2 > 0

(How) Does the degeneracy get lifted?

The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

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Expectations (2)

(How) Does the degeneracy get lifted?

The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (2)

(How) Does the degeneracy get lifted?

J_2 = J_3

8

The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.

J_2 = J_3

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (2)

(How) Does the degeneracy get lifted?

This talk: a series of macroscopically degenerate phases

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Ising model  = easy to solve?

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (3)

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Ising model  = easy to solve?

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (3)

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  • The long-range model is challenging (~300 spins) (glassy behavior)

 

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)

Ising model  = easy to solve?

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Expectations (3)

9

  • The long-range model is challenging (~300 spins) (glassy behavior)

 

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

  • Short-range model :
    • MC falls out-of-equilibrium
    • Negative results from simple Pauling estimates

Proving the ground-state energy

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

Frustration and motivation: artificial spin systems

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Kanamori's method

10

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Constrain the possible values of the correlations:

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Kanamori's method

10

c_k := \frac{1}{2N} \sum_{\langle i,j \rangle_k} \sigma_i \sigma_j.
c_k := \frac{1}{2N} \sum_{\langle i,j \rangle_k} \sigma_i \sigma_j.

Constrain the possible values of the correlations:

\mod(n, 2) \leq \left(\sum_{i=1}^n s_i \sigma_i\right)^2 \leq n^2

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Kanamori's method

10

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Translations + rotations

to cover the lattice

\text{Cluster } A\, :\; c_1 \geq - \frac{1}{3},\\ \text{Cluster } B\, :\; c_2 \pm 2 c_1 \geq - 1,\\ \text{Cluster } C\, :\; c_2 \geq - \frac{1}{3}.

Constrain the possible values of the correlations:

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

Kanamori's method

10

c_k := \frac{1}{2N} \sum_{\langle i,j \rangle_k} \sigma_i \sigma_j.
\mod(n, 2) \leq \left(\sum_{i=1}^n s_i \sigma_i\right)^2 \leq n^2

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

J. Colbois

Kanamori's method : applied to our problem

11

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

c_1 = -1/3
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

J. Colbois

Kanamori's method : applied to our problem

11

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Kanamori's method : applied to our problem

11

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Kanamori's method : applied to our problem

11

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

Not all triangles can respect the UUD/DDU rule at the same time

H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Kanamori's method : applied to our problem

11

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

Corners directly correspond to G.S. phases:

\vec{J}
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Obtaining ground-state energy lower bounds

12

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

\vec{J}
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Corners directly correspond to G.S. phases:

J. Colbois

12

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

Obtaining ground-state energy lower bounds

\vec{J}
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Corners directly correspond to G.S. phases:

J. Colbois

12

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Obtaining ground-state energy lower bounds

Corners directly correspond to G.S. phases:

\vec{J}
\vec{J}
H = J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j+J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j +J_{3} \left(\sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j+\sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j\right)\\
2N c_1
2N c_2
2N c_3

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Edges correspond to g.s. phase boundaries

J. Colbois

12

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Obtaining ground-state energy lower bounds

All the corners must be verified

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Proving the ground-state energy

13

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All the corners must be verified

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Proving the ground-state energy

13

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-\frac{2}{3}J_1 + 2 J_2- J_3

All the corners must be verified

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Proving the ground-state energy

13

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-\frac{2}{3}J_1 + 2 J_2- J_3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3

All the corners must be verified

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Proving the ground-state energy

13

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-\frac{2}{3}J_1 + 2 J_2- J_3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3
-\frac{2}{3}J_1 - \frac{1}{3}J_3
-\frac{2}{3}J_1 + \frac{2}{3}J_2- J_3

All the corners must be verified

 Methods: J. Kanamori, Prog. Theor. Phys. 35, (1966)

F. Ducastelle, Cohesion and Structure No 3, (1990)

G. Rakala, K. Damle, PRE 96, (2017)

 

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

J. Colbois

Proving the ground-state energy

13

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Tensor networks for frustrated Ising models

Proving the ground-state energy

Frustration and motivation: artificial spin systems

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> Motivation

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15

Motivation (1) : why not Monte Carlo?

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior

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J. Colbois

15

Motivation (1) : why not Monte Carlo?

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior
N= 9L^2

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J. Colbois

15

Motivation (1) : why not Monte Carlo?

G. Rakala, K. Damle, PRE 96, (2017)

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior
N= 9L^2

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J. Colbois

15

Motivation (1) : why not Monte Carlo?

G. Rakala, K. Damle, PRE 96, (2017)

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior
N= 9L^2

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J. Colbois

15

Motivation (1) : why not Monte Carlo?

G. Rakala, K. Damle, PRE 96, (2017)

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior
N= 9L^2

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

15

Motivation (1) : why not Monte Carlo?

G. Rakala, K. Damle, PRE 96, (2017)

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Monte Carlo challenges:

  • "Good" update
  • Thermodynamic integration
  • Good control over the finite-size scaling behavior
N= 9L^2

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

15

Motivation (1) : why not Monte Carlo?

G. Rakala, K. Damle, PRE 96, (2017)

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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16

Motivation (2) : why tensor networks?

TNs are very powerful for 2D classical problems with short-range interactions

They give a direct access to the partition function "per site".

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

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Motivation (3) : idea

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

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17

Motivation (3) : idea

\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta E_{\rm{GS}}} {\color{teal}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\ = e^{-\beta E_{\rm{GS}}} {\color{teal}\tilde{\mathcal{Z}}_N}
\lim_{\beta \rightarrow \infty} {\color{teal}\tilde{\mathcal{Z}}_N} = {\color{teal}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{teal}\Lambda_0}^N

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Motivation (3) : idea

\lim_{\beta \rightarrow \infty} {\color{teal}\tilde{\mathcal{Z}}_N} = {\color{teal}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
\cong {\color{teal}\Lambda_0}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta E_{\rm{GS}}} {\color{teal}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\ = e^{-\beta E_{\rm{GS}}} {\color{teal}\tilde{\mathcal{Z}}_N}

We "just" have to compute the (normalized) partition function for one site...

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J. Colbois

17

Motivation (3) : idea

\lim_{\beta \rightarrow \infty} {\color{teal}\tilde{\mathcal{Z}}_N} = {\color{teal}W_N}
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)

We "just" have to compute the (normalized) partition function for one site...

 

... which is the result that naturally comes out with a large precision from tensor networks.

\cong {\color{teal}\Lambda_0}^N
\mathcal{Z}_N = \sum_{\{\sigma\}} e^{-\beta \mathcal{H}(\{\sigma\})} = e^{-\beta E_{\rm{GS}}} {\color{teal}\sum_{\{\sigma\}} e^{-\beta \left(\mathcal{H}(\{\sigma\})- E_{\rm{GS}}\right)}}\\ = e^{-\beta E_{\rm{GS}}} {\color{teal}\tilde{\mathcal{Z}}_N}

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J. Colbois

17

Motivation (3) : idea

Tensor networks for frustrated Ising models

Proving the ground-state energy

Frustration and motivation: artificial spin systems

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

> Motivation

> Introduction to tensor networks for classical spin systems

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18

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}}

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

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J. Colbois

18

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)

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J. Colbois

18

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
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)

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18

Tensor networks : inspired by transfer matrices

\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
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)

Tensor network language:

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

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18

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}}

"Tensor"

"Legs"

"Contraction"

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T = \begin{pmatrix} e^{\beta J} &e^{-\beta J}\\ e^{-\beta J} &e^{\beta J}\\ \end{pmatrix}

Tensor network language:

"vector"

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

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J. Colbois

18

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}}

"Legs"

"Contraction"

Open legs = number of indices

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T = \begin{pmatrix} e^{\beta J} &e^{-\beta J}\\ e^{-\beta J} &e^{\beta J}\\ \end{pmatrix}

Tensor network language:

"vector"

"Tensor"

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J. Colbois

18

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}}

"Tensor"

"Legs"

"Contraction"

Open legs = number of indices

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T = \begin{pmatrix} e^{\beta J} &e^{-\beta J}\\ e^{-\beta J} &e^{\beta J}\\ \end{pmatrix}

"vector"

\mathcal{Z}_L =

Tensor network language:

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19

Partition function of a 2D problem as a TN

\mathcal{Z}_N =

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

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J. Colbois

19

Partition function of a 2D problem as a TN

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

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

19

Partition function of a 2D problem as a TN

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

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

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

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J. Colbois

19

Partition function of a 2D problem as a TN

\mathcal{Z}_N =

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

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

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20

Contraction and leading eigenvector

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
\mathcal{Z}_L \rightarrow \lambda_0^L
T^L = \lambda_{+}^L \sum_{i = +, -} \left( \frac{\lambda_i}{\lambda_+} \right)^L \left|r_i\rangle \langle l_i\right| \xrightarrow[L \to \infty]{} \lambda_+^L \left|r_+\rangle \langle l_+\right|

"Exact contraction"

"Approximate contraction"

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

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Contraction and leading eigenvector

\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
\mathcal{Z}_L \rightarrow \lambda_0^L
T^L = \lambda_{+}^L \sum_{i = +, -} \left( \frac{\lambda_i}{\lambda_+} \right)^L \left|r_i\rangle \langle l_i\right| \xrightarrow[L \to \infty]{} \lambda_+^L \left|r_+\rangle \langle l_+\right|

"Exact contraction"

"Approximate contraction"

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21

|\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

Contraction and leading eigenvector (2D)

\mathcal{Z}_{N = M \times L} = \mathcal{T}_M^L
\mathcal{Z}_N \rightarrow \Lambda_M^L

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

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

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|\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

The Matrix Product State Ansatz

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

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21

|\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

The Matrix Product State Ansatz

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

High number of parameters

(2^L)

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21

|\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

The Matrix Product State Ansatz

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

High number of parameters

(2^L)

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21

|\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

The Matrix Product State Ansatz

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

High number of parameters

(2^L)
\chi

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|\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

The Matrix Product State Ansatz

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

High number of parameters

Much smaller number of parameters

\chi
(\chi \times 2 \times \chi)
(2^L)

(1 + 1)D

iTEBD / VUMPS

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(1+1)D versus 2D

2D

(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 

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22

(1+1)D versus 2D

2D

(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 

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22

(1+1)D versus 2D

2D

(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 

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22

(1+1)D versus 2D

2D

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

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

(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 

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22

(1+1)D versus 2D

2D

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

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

(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 

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22

(1+1)D versus 2D

2D

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

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

(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 

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J. Colbois

22

(1+1)D versus 2D

2D

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

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

(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 

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J. Colbois

22

(1+1)D versus 2D

2D

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

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

Environment

Iteratively solving fixed-point equations

T tensor can be replaced by observables

(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 

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

22

(1+1)D versus 2D

2D

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

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

Environment

Iteratively solving fixed-point equations

T tensor can be replaced by observables

(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 

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

22

(1+1)D versus 2D

2D

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

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

Frustration and motivation: artificial spin systems

Tensor networks for frustrated Ising models

Proving the ground-state energy?

> Motivation

> Introduction to tensor networks for classical spin systems

> A problem of contraction

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J. Colbois

The issue

23

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, arXiv:2104.13264, (2021)

Fails in the presence of frustration and macroscopic g.s. degeneracy

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J. Colbois

The issue

23

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, arXiv:2104.13264, (2021)

Fails in the presence of frustration and macroscopic g.s. degeneracy

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

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

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

 

Related to the inexact cancellation of very large

and very small factors.

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J. Colbois

The issue

23

\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}

Non-frustrated models:

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24

What is going on?

\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} e^{-2\beta |J|} & 1\\ 1 & e^{-2\beta |J|}\end{pmatrix}

Non-frustrated models:

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J. Colbois

What is going on?

24

\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} e^{-2\beta |J|} & 1\\ 1 & e^{-2\beta |J|}\end{pmatrix}

Non-frustrated models:

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

Triangular/ kagome lattices:

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J. Colbois

What is going on?

24

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Why is it intriguing?

J. Colbois

25

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

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Why is it intriguing?

J. Colbois

25

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

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Why is it intriguing?

J. Colbois

T_{d_1, d_2,d_3} =
= \begin{cases} 1 \quad & \text{if } \prod_i d_i = -1\\ 0 & \text{otherwise} \end{cases}

25

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

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Why is it intriguing?

J. Colbois

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

25

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, arXiv:2104.13264, (2021)

 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)

J. Colbois

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26

Systematically finding the local rule

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

1. Split the Hamiltonian in a different way

 

 

 

 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)

J. Colbois

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

1. Split the Hamiltonian in a different way

 

 

 

2. Lower bound on the ground-state energy:

 

 

 

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

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J. Colbois

Systematically finding the local rule

26

 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)

J. Colbois

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

1. Split the Hamiltonian in a different way

 

 

 

2. Lower bound on the ground-state energy:

 

 

 

3. This lower bound can be maximized with respect to the weights (max-min approach):

 

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

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J. Colbois

Systematically finding the local rule

26

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

J. Colbois

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

1. Split the Hamiltonian in a different way

 

 

 

2. Lower bound on the ground-state energy:

 

 

 

3. This lower bound can be maximized with respect to the weights (max-min approach):

 

 

 

4. When the lower bound is saturated:

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

All the ground states are made of configurations that minimize the local Hamiltonian.

\rightarrow
\rightarrow

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J. Colbois

Systematically finding the local rule

26

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

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27

Does it work?

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

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J. Colbois

Does it work?

27

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

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J. Colbois

Does it work?

27

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

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J. Colbois

Does it work?

27

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

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28

Spurious tiles

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

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Spurious tiles

28

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

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Spurious tiles

28

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

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Spurious tiles

28

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

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J. Colbois

Spurious tiles

28

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

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

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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TNs for frustrated Ising models

J. Colbois

29

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, arXiv:2104.13264, (2021)

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

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

A model with several macroscopically degenerate phases

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JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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Phase diagram

J. Colbois

30

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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Phase diagram

J. Colbois

30

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Residual entropies

J. Colbois

31

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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Residual entropies: chevrons phase

J. Colbois

31

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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

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31

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Residual entropies: chevrons phase

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

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JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Residual entropies: chevrons phase

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

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33

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Residual entropies: pinwheels phase

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

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33

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Residual entropies: pinwheels phase

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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33

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

Residual entropies: pinwheels phase

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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J. Colbois

34

Residual entropies: pinwheels phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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J. Colbois

34

Residual entropies: pinwheels phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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J. Colbois

34

Residual entropies: pinwheels phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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J. Colbois

35

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S = 0.01920 \pm 3 \cdot 10^{-5}

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J. Colbois

35

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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}

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J. Colbois

35

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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}

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J. Colbois

36

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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}

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J. Colbois

36

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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}

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J. Colbois

36

Residual entropies: strings phase

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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

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37

Discussion

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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

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37

Discussion

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

37

Discussion

JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

38

Open questions...

Tensor network "problem": does this solution work for TNR?

Contraction of PEPS wavefunctions?

Use of the ground-state local rule: existence of the cluster?

Nature of the phases at higher temperatures?

A different approach for spin glasses: extend to approximate contraction?

Ronceray and Le Floch, PRE 100, (2019)

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

 

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

38

Open questions...

Ongoing work with Afonso Rufino , Samuel Nyckees and Frédéric Mila

 

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

39

Take home...

JC, B. Vanhecke et al., arXiv:2206.11788 (2022)

B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)

If the ground-state rule is implemented at the level of the tensor, approximate contraction aglorithms converge

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

39

Take home...

JC, B. Vanhecke et al., arXiv:2206.11788 (2022)

B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)

If the ground-state rule is implemented at the level of the tensor, approximate contraction aglorithms converge

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

39

Take home...

Tensor networks are a powerful tool to study 2D classical spin systems!

JC, B. Vanhecke et al., arXiv:2206.11788 (2022)

B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)

If the ground-state rule is implemented at the level of the tensor, approximate contraction aglorithms converge

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

IISER PUNE | 09.2022 | TNS FOR FRUSTRATED ISING MODELS : PARTIAL LIFTING OF G. S.  DEG.

J. Colbois

39

Take home...

Tensor networks are a powerful tool to study 2D classical spin systems!

Thank you!

JC, B. Vanhecke et al., arXiv:2206.11788 (2022)

B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)

Tensor networks for frustrated Ising models and partial lifting of a macroscopic gs degeneracy

By Jeanne Colbois

Tensor networks for frustrated Ising models and partial lifting of a macroscopic gs degeneracy

Online seminar at IISER Pune, invited by Prof. Dr. G. J. Sreejith

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