Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
J1−J2−J3
J_1 - J_2 - J_3
Partial lifting of degeneracy in the J1-J2-J3
Ising antiferromagnet on the kagome lattice
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
Andrew Smerald
KIT | Germany
Frank Verstraete
Ghent University | Belgium
Laurens Vanderstraeten
Ghent University | Belgium
Bram Vanhecke
University of Vienna | Austria
Frédéric Mila
EPFL | Switzerland
Magnetism and Magnetic Materials 2022 - Minneapolis and Online - Nov. 2022
Frustration and motivation: artificial spin systems
A model with several macroscopically degenerate phases
Tensor networks for frustrated Ising models
Proving the ground-state energy?
Scope
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
1
J. Colbois
H=J∑⟨i,j⟩σiσjσi=±1
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
di,j=σiσj
d_{i,j} = \sigma_i \sigma_j
2-up 1-down (UUD),
2-down 1-up (DDU)
2
J. Colbois
Frustration
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J∑⟨i,j⟩σiσjσi=±1
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
di,j=σiσj
d_{i,j} = \sigma_i \sigma_j
2-up 1-down (UUD),
2-down 1-up (DDU)
2
J. Colbois
WG.S.=# configurations ≳2N/3
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
Frustration
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J∑⟨i,j⟩σiσjσi=±1
H = J\sum_{\langle i,j \rangle} \sigma_i \sigma_j \qquad \sigma_i = \pm 1
S=0.3230659...
S = 0.3230659...
S=0.501833...
S = 0.501833...
G.H. Wannier, PR 79, (1950, 1973)
K. Kano and S. Naya, Prog. Theor. Phys. 10, (1953)
WG.S.=# configurations ≳2N/3
W_{G.S.} = \# \text{ configurations } \gtrsim 2^{N/3}
di,j=σiσj
d_{i,j} = \sigma_i \sigma_j
S:=limN→∞Nln(WG.S.)
S := \lim_{N \rightarrow \infty} \frac{\ln(W_{\text{G.S.}})}{N}
2-up 1-down (UUD),
2-down 1-up (DDU)
2
J. Colbois
Frustration
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Anghinolfi et al.,
Nat. Commun. 6, (2015)
J. Colbois
3
In-plane
H=J∑⟨i,j⟩ei⋅ejσiσj+D∑(i,j)(∣rij∣3ei⋅ej−∣rij∣53(ei⋅rij)(ej⋅rij))σiσj
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
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
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)
J. Colbois
3
In-plane
H=J∑⟨i,j⟩ei⋅ejσiσj+D∑(i,j)(∣rij∣3ei⋅ej−∣rij∣53(ei⋅rij)(ej⋅rij))σiσj
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
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
J. Colbois
4
Chioar et al., PRB 90, (2014)
Dipolar kagome Ising antiferromagnet (DKIAFM)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
4
Chioar et al., PRB 90, (2014)
Dipolar kagome Ising antiferromagnet (DKIAFM)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
J. Colbois
4
Chioar et al., PRB 90, (2014)
Dipolar kagome Ising antiferromagnet (DKIAFM)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
I. A. Chioar, N. Rougemaille, B. Canals, PRB 93, (2016)
J. Hamp, C. Castelnovo, R. Moessner, PRB 98, (2018)
L. Cugliandolo, L. Foini, M. Tarzia, PRB 101 (2020)
How does the degeneracy get lifted?
J. Colbois
5
Short-range model and questions
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=
H =
J2∑⟨i,j⟩2σiσj
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J3∑⟨i,j⟩3∣∣σiσj
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J3∑⟨i,j⟩3⋆σiσj
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
+
+
+
J1∑⟨i,j⟩1σiσj
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
J. Colbois
5
Short-range model and questions
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=
H =
J2∑⟨i,j⟩2σiσj
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J3∑⟨i,j⟩3∣∣σiσj
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J3∑⟨i,j⟩3⋆σiσj
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
+
+
+
J1∑⟨i,j⟩1σiσj
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
J. Colbois
5
Short-range model and questions
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Proving the ground-state energy?
Evaluating the residual entropy precisely?
H=
H =
J2∑⟨i,j⟩2σiσj
J_2 \sum_{\langle i,j \rangle_{2}} \sigma_i \sigma_j
J3∑⟨i,j⟩3∣∣σiσj
J_{3} \sum_{\langle i,j \rangle_{3||}} \sigma_i \sigma_j
J3∑⟨i,j⟩3⋆σiσj
J_{3} \sum_{\langle i,j \rangle_{3\star}} \sigma_i \sigma_j
+
+
+
+
+
+
J1∑⟨i,j⟩1σiσj
J_1 \sum_{\langle i,j \rangle_1} \sigma_i \sigma_j
J. Colbois
5
Short-range model and questions
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
- Short-range model :
- MC falls out-of-equilibrium
- Negative results from simple Pauling estimates assuming the UUD/DDU rule is respected on each triangle
Proving the ground-state energy?
Evaluating the residual entropy precisely?
(How) Does the degeneracy get lifted?
J. Colbois
6
Expectations (1)
T. Takagi and M. Mekata, JSPS 62, (1993)
A. Wills, R. Ballou, C. Lacroix, PRB 66, (2002)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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. Colbois
Expectations (1)
J2<0
J_2 < 0
(How) Does the degeneracy get lifted?
6
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
J. Colbois
7
Expectations (1) vs Reality
J2<0
J_2 < 0
J2>0
J_2 > 0
(How) Does the degeneracy get lifted?
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
J. Colbois
7
Expectations (1) vs Reality
J2<0
J_2 < 0
J2>0
J_2 > 0
(How) Does the degeneracy get lifted?
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
J. Colbois
7
Expectations (1) vs Reality
J2<0
J_2 < 0
J2>0
J_2 > 0
(How) Does the degeneracy get lifted?
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.
J. Colbois
Expectations (2)
(How) Does the degeneracy get lifted?
8
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.
J2=J3
J_2 = J_3
J. Colbois
Expectations (2)
(How) Does the degeneracy get lifted?
8
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
The macroscopic degeneracy can survive in the presence of third-neighbor interactions but only at very fine-tuned points.
J2=J3
J_2 = J_3
J. Colbois
Expectations (2) vs Reality
(How) Does the degeneracy get lifted?
This talk: a series of macroscopically degenerate phases
8
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Frustration and motivation: artificial spin systems
Ground-state energy lower bounds
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
2N c_3
J. Colbois
Kanamori's method : applied to our problem
9
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)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
c1=−1/3
c_1 = -1/3
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
2N c_3
J. Colbois
Kanamori's method : applied to our problem
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)
9
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
c1=−1/3
c_1 = -1/3
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
2N c_3
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
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
9
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
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
c1=−1/3
c_1 = -1/3
9
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J1∑⟨i,j⟩1σiσj+J2∑⟨i,j⟩2σiσj+J3(∑⟨i,j⟩3∣∣σiσj+∑⟨i,j⟩3⋆σiσj)
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)\\
2Nc1
2N c_1
2Nc2
2N c_2
2Nc3
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
c1=−1/3
c_1 = -1/3
Not all triangles can respect the UUD/DDU rule at the same time
9
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
10
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
10
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
Proving the ground-state energy
−32J1+2J2−J3
-\frac{2}{3}J_1 + 2 J_2- J_3
J. Colbois
10
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
Proving the ground-state energy
−32J1+2J2−J3
-\frac{2}{3}J_1 + 2 J_2- J_3
−32J1−32J2+J3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3
J. Colbois
10
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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)
Proving the ground-state energy
−32J1+2J2−J3
-\frac{2}{3}J_1 + 2 J_2- J_3
−32J1−32J2+J3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3
−32J1−32J2+J3
-\frac{2}{3}J_1 - \frac{2}{3} J_2 + J_3
−32J1−31J3
-\frac{2}{3}J_1 - \frac{1}{3}J_3
−32J1+32J2−J3
-\frac{2}{3}J_1 + \frac{2}{3}J_2- J_3
Frustration and motivation: artificial spin systems
Tensor networks for frustrated Ising models
Proving the ground-state energy?
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
S=limN→∞N1ln(WN)
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
J. Colbois
11
Idea: why tensor networks
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
S=limN→∞N1ln(WN)
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
ZN=∑{σ}e−βH({σ})=e−βEGS∑{σ}e−β(H({σ})−EGS)=e−βEGSZ~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}
J. Colbois
11
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Idea: why tensor networks
limβ→∞Z~N=WN
\lim_{\beta \rightarrow \infty} {\color{teal}\tilde{\mathcal{Z}}_N} = {\color{teal}W_N}
S=limN→∞N1ln(WN)
S = \lim_{N \rightarrow \infty} \frac{1}{N} \ln\left(W_N\right)
≅Λ0N
\cong {\color{teal}\Lambda_0}^N
ZN=∑{σ}e−βH({σ})=e−βEGS∑{σ}e−β(H({σ})−EGS)=e−βEGSZ~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}
J. Colbois
11
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Idea: why tensor networks
limβ→∞Z~N=WN
\lim_{\beta \rightarrow \infty} {\color{teal}\tilde{\mathcal{Z}}_N} = {\color{teal}W_N}
S=limN→∞N1ln(WN)
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.
≅Λ0N
\cong {\color{teal}\Lambda_0}^N
ZN=∑{σ}e−βH({σ})=e−βEGS∑{σ}e−β(H({σ})−EGS)=e−βEGSZ~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}
J. Colbois
11
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Idea: why tensor networks
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)
J. Colbois
12
Tensor networks : inspired by transfer matrices
ZL=∑σ1,σ2,…σL∏ieβJσiσi+1
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
ZL=∑σ0(TL)σ0,σ0
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T=(eβJe−βJe−βJeβJ)
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Tensor notation: R. Penrose, in Combinatorial Mathematics and its applications, Academic Press (1971)
J. Colbois
Tensor networks : inspired by transfer matrices
ZL=∑σ1,σ2,…σL∏ieβJσiσi+1
\mathcal{Z}_L = \sum_{\sigma_1,\sigma_2,\dots\sigma_L} \prod_{i}e^{\beta J \sigma_i \sigma_{i+1}}
ZL=∑σ0(TL)σ0,σ0
\mathcal{Z}_L = \sum_{\sigma_0} (T^L)_{\sigma_0, \sigma_0}
T=(eβJe−βJe−βJeβJ)
T = \begin{pmatrix}
e^{\beta J} &e^{-\beta J}\\
e^{-\beta J} &e^{\beta J}\\
\end{pmatrix}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
"Legs"
"Contraction"
Open legs = number of indices
Tensor network language:
"vector"
"Tensor"
12
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
Partition function of a 2D problem as a TN
ZN=
\mathcal{Z}_N =
T=(eβJe−βJe−βJeβJ)
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
13
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
Partition function of a 2D problem as a TN
ZN=
\mathcal{Z}_N =
T=(eβJe−βJe−βJeβJ)
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
Δσi,1,σi,2,σi,3,σi,4={10 all equal otherwise
\Delta_{\sigma_{i,1}, \sigma_{i,2}, \sigma_{i,3}, \sigma_{i,4}}\\ = \begin{cases}
1 & \text{ all equal}\\
0 & \text{ otherwise}
\end{cases}
13
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
Partition function of a 2D problem as a TN
ZN=
\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
13
(1 + 1)D
iTEBD / VUMPS
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
J. Colbois
(1+1)D versus 2D
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
14
(1 + 1)D
iTEBD / VUMPS
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
J. Colbois
(1+1)D versus 2D
2D
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
14
(1 + 1)D
iTEBD / VUMPS
R. J. Baxter, J. Math. Phys. 9, 1968
Orús, Vidal, PRB 78, 2008;
V. Zauner-Stauber et. al. PRB 97,2018;
M. Fishman et. al PRB 98, 2018
J. Colbois
(1+1)D versus 2D
2D
R. J. Baxter, J. Math. Phys. 9, 1968
T. Nishino, K. Okunishi, J. Phys. Soc. Jpn 65, 1996
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
Environment
Iteratively solving fixed-point equations
T tensor can be replaced by observables
14
J. Colbois
The issue
15
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
The issue
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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
15
J. Colbois
The issue
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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.
15
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
H=∑cHcα(σ∣c)
H = \sum_{c} H_c^{\alpha}(\vec{\sigma}|_c)
1. Split the Hamiltonian in a different way
J. Colbois
Systematically finding the local rule
16
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
H=∑cHcα(σ∣c)
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:
minσ∣cHcα(σ∣c)
\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
J. Colbois
Systematically finding the local rule
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
16
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
H=∑cHcα(σ∣c)
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):
minσ∣cHcα(σ∣c)
\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
maxαminσ∣cHcα(σ∣c)
\max_{\alpha}\,\,\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
→
\rightarrow
J. Colbois
Systematically finding the local rule
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
16
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
H=∑cHcα(σ∣c)
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:
minσ∣cHcα(σ∣c)
\min_{\vec{\sigma}|_c} H_c^{\alpha}(\vec{\sigma}|_c)
maxαminσ∣cHcα(σ∣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
J. Colbois
Systematically finding the local rule
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
16
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
Does it work?
17
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
J. Colbois
Does it work?
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
17
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
J. Colbois
Does it work?
17
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
B. Vanhecke, JC, L. Vanderstraeten, F. Verstraete, F. Mila, PRR 3, (2021)
J. Colbois
Does it work?
17
Frustration and motivation: artificial spin systems
A model with several macroscopically degenerate phases
Tensor networks for frustrated Ising models
Proving the ground-state energy?
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
Phase diagram
J. Colbois
18
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
S=0.01920±3⋅10−5
S = 0.01920 \pm 3 \cdot 10^{-5}
J. Colbois
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
Residual entropies
19
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
S=0.01920±3⋅10−5
S = 0.01920 \pm 3 \cdot 10^{-5}
J. Colbois
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
Residual entropies
19
S=0.026922±3⋅10−6=12S△
S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
S=0.01920±3⋅10−5
S = 0.01920 \pm 3 \cdot 10^{-5}
J. Colbois
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
Residual entropies
19
S=0.026922±3⋅10−6=12S△
S = 0.026922 \pm 3 \cdot 10^{-6} = \frac{S_\triangle}{12}
S=0.107689±2⋅10−6≅3S△
S = 0.107689 \pm 2 \cdot 10^{-6} \cong \frac{S_\triangle}{3}
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
J. Colbois
20
Discussion
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
H=J∑⟨i,j⟩σiσj+D∑(i,j)rij3σiσj
H = J \sum_{\langle i, j \rangle } \sigma_i \sigma_j + D \sum_{(i,j)} \frac{\sigma_i \sigma_j}{r_{ij}^3}
J. Colbois
Discussion
JC, B. Vanhecke, L. Vanderstraeten, et al., arXiv:2206.11788 (2022)
20
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
The convergence of the tensor network contraction depends on the formulation of the partition function
J. Colbois
21
Take home...
JC, B. Vanhecke et al., arXiv:2206.11788 (2022)
B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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
J. Colbois
Take home...
JC, B. Vanhecke et al., arXiv:2206.11788 (2022)
B. Vanhecke, JC, et al., Phys. Rev. Research 3 (2021)
21
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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
J. Colbois
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)
21
MMM2022 | 11.2022 | PARTIAL LIFTING OF G. S. DEG. IN THE J1-J2-J3 AF ISING MODEL ON KAGOME
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
J. Colbois
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)
Thank you!
See you in the chat!
21
J 1 − J 2 − J 3 J_1 - J_2 - J_3 Partial lifting of degeneracy in the J1-J2-J3 Ising antiferromagnet on the kagome lattice 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 Andrew Smerald KIT | Germany Frank Verstraete Ghent University | Belgium Laurens Vanderstraeten Ghent University | Belgium Bram Vanhecke University of Vienna | Austria Frédéric Mila EPFL | Switzerland Magnetism and Magnetic Materials 2022 - Minneapolis and Online - Nov. 2022
Partial lifting of degeneracy in the J1-J2-J3 kagome Ising antiferromagnet
By Jeanne Colbois
Partial lifting of degeneracy in the J1-J2-J3 kagome Ising antiferromagnet
Contributed talk at the MMM 2022, Online
- 129
