Jeanne Colbois - Institut Néel - Grenoble

Quantum 2025 | Grenoble | 21 May 2025

Nicolas Laflorencie

Fabien Alet

LPT Toulouse - France

 Instabilities and Many-body localization

in the Random-Field XXZ Chain

(?)

Anderson insulator

disorder \(h \)

?

?

Ergodic 

Many-body localized

Interactions \(\Delta\)

PRL 133, 116502 (2024) - arXiv:2403.09608

PRB 110, 214210 (2024) - arXiv:2410.10325

arXiv:2504.10566

Jeanne Colbois - Institut Néel - Grenoble

Quantum 2025 | Grenoble | 21 May 2025

Nicolas Laflorencie

Fabien Alet

LPT Toulouse - France

 Instabilities and Many-body localization

in the Random-Field XXZ Chain

(?)

Anderson insulator

disorder \(h \)

?

?

Ergodic 

Many-body localized

Interactions \(\Delta\)

PRL 133, 116502 (2024) - arXiv:2403.09608

PRB 110, 214210 (2024) - arXiv:2410.10325

arXiv:2504.10566

Jeanne Colbois - Institut Néel - Grenoble

Quantum 2025 | Grenoble | 21 May 2025

Nicolas Laflorencie

Fabien Alet

LPT Toulouse - France

 Instabilities and Many-body localization

in the Random-Field XXZ Chain

(?)

Anderson insulator

disorder \(h \)

?

?

Ergodic 

Many-body localized

Interactions \(\Delta\)

PRL 133, 116502 (2024) - arXiv:2403.09608

PRB 110, 214210 (2024) - arXiv:2410.10325

arXiv:2504.10566

COLBOIS | INSTABILITIES AND MBL |  05.2025

1

Anderson localization in 1D

Anderson, Phys. Rev. 109, 1492 (1958)

Mott & Twose, Advances in Physics 10, 107 (1961)

COLBOIS | INSTABILITIES AND MBL |  05.2025

1

\(|\psi(x)|^2\)

\(\xi(h, E)\)

\(h\)

 \(\forall h , \, \forall E \) : localization !!

 (1D, NN)

\mathcal{P}(h_i)

\(-h\)

\(h\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Anderson localization in 1D

Anderson, Phys. Rev. 109, 1492 (1958)

Mott & Twose, Advances in Physics 10, 107 (1961)

COLBOIS | INSTABILITIES AND MBL |  05.2025

1

\(|\psi(x)|^2\)

\(\xi(h, E)\)

\(h\)

 \(\forall h , \, \forall E \) : localization !!

 (1D, NN)

\mathcal{P}(h_i)

\(-h\)

\(h\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Anderson localization in 1D

\(|\psi(x)|^2\)

several non-interacting fermions :

Anderson, Phys. Rev. 109, 1492 (1958)

Mott & Twose, Advances in Physics 10, 107 (1961)

COLBOIS | INSTABILITIES AND MBL |  05.2025

1

\(|\psi(x)|^2\)

\(\xi(h, E)\)

\(h\)

 \(\forall h , \, \forall E \) : localization !!

 (1D, NN)

\mathcal{P}(h_i)

\(-h\)

\(h\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

Anderson localization in 1D

What happens in the interacting case ?

\(|\psi(x)|^2\)

several non-interacting fermions :

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{orange}h_i n_i}

2

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

2

Is 1D localization preserved under weak interactions ?

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Jordan-Wigner

2

Is 1D localization preserved under weak interactions ?

\(n_i = S_i^z + 1/2\)

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Jordan-Wigner

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + \dots \right)
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}

Spin-flip

2

Is 1D localization preserved under weak interactions ?

\(n_i = S_i^z + 1/2\)

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Jordan-Wigner

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \dots

ISING INTERACTION

S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}

Spin-flip

2

Is 1D localization preserved under weak interactions ?

\(n_i = S_i^z + 1/2\)

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Jordan-Wigner

2

Is 1D localization preserved under weak interactions ?

\(n_i = S_i^z + 1/2\)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}

Spin-flip

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{P}(h_i)

\(-h\)

\(h\)

Attraction / repulsion

\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{cyan}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}

Jordan-Wigner

\(L/2\) fermions

Charge is conserved

\(\sum_i S_i^{z} = 0\)

Magnetization is conserved

2

Is 1D localization preserved under weak interactions ?

\(n_i = S_i^z + 1/2\)

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}

Spin-flip

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Is 1D AL Preserved under interactions (in spin chains) ?

3

Spin-flip

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Is 1D AL Preserved under interactions (in spin chains) ?

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Is 1D AL Preserved under interactions (in spin chains) ?

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Is 1D AL Preserved under interactions (in spin chains) ?

Counter example for thermalization of isolated quantum systems

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Is 1D AL Preserved under interactions (in spin chains) ?

Counter example for thermalization of isolated quantum systems

Polynomial \(\rightarrow\) Exponential

Absence of translation invariance

No typicality methods

No ground-state methods

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Is 1D AL Preserved under interactions (in spin chains) ?

Counter example for thermalization of isolated quantum systems

Expectation :

Interactions tend to delocalize

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Is 1D AL Preserved under interactions (in spin chains) ?

Counter example for thermalization of isolated quantum systems

High energy, \( \epsilon = 0.5\)

Ising interaction \(\Delta\)

Delocalized

Delocalized

disorder \(h \)

  MBL

  MBL

.... and a whole field! ....

Fleischman, Anderson, (1980);  Altschuler, et al  (1997); Gornyi et al (2005); Basko et al  (2006); Zidnarick et al (2008);  Aleiner et al (2010);  Pal and Huse (2010); Luitz et al (2015) [....]

Localization can survive!

COLBOIS | INSTABILITIES AND MBL |  05.2025

\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{cyan} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}

ISING INTERACTION

Spin-flip

4

\(\epsilon = 1\)

\(\epsilon = 0\)

Ground state, \(\epsilon = 0\)

Ising interaction \(\Delta\)

Localized

Delocalized

disorder \(h \)

Is 1D AL Preserved under interactions (in spin chains) ?

Counter example for thermalization of isolated quantum systems

High energy, \( \epsilon = 0.5\)

Ising interaction \(\Delta\)

Delocalized

Delocalized

disorder \(h \)

  MBL

  MBL

.... and a whole field! ....

Fleischman, Anderson, (1980);  Altschuler, et al  (1997); Gornyi et al (2005); Basko et al  (2006); Zidnarick et al (2008);  Aleiner et al (2010);  Pal and Huse (2010); Luitz et al (2015) [....]

?

Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);  

Ristivojevic, et al PRL 109, (2012);

Doggen et al, PRB 96,  (2017);

Lin et al, Scipost Phys 4 (2019)

Localization can survive!

Plan for the rest of the talk

(?)

Anderson insulator

disorder \(h \)

?

?

Ergodic 

Many-body localized

Interactions \(\Delta\)

Plan for the rest of the talk

(?)

Anderson insulator

disorder \(h \)

?

?

Ergodic 

Many-body localized

Interactions \(\Delta\)

MBL in one slide

 

Plan for the rest of the talk

MBL in one slide

A simple argument : ergodic instability

 

(?)

Anderson insulator

disorder \(h \)

Ergodic 

Many-body localized

Interactions \(\Delta\)

!

Plan for the rest of the talk

MBL in one slide

A simple argument : ergodic instability

Numerical results

(?)

Anderson insulator

disorder \(h \)

Ergodic 

Many-body localized

Interactions \(\Delta\)

!

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

2025

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Ergodic delocalized

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Ergodic delocalized

Many-body localized

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

Ergodic delocalized

Many-body localized

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

\(r\)

\(P(r)\)

Ergodic delocalized

Many-body localized

Ergodic thermalization hypothesis

Random matrix theory

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

\(r\)

\(P(r)\)

\(P(r)\)

\(r\)

Ergodic delocalized

Many-body localized

Ergodic thermalization hypothesis

Random matrix theory

Emergent integrability

"Localization length" \(\zeta\)

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

Entanglement entropy

...

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

A

\(r\)

\(P(r)\)

\(P(r)\)

\(r\)

Ergodic delocalized

Many-body localized

Ergodic thermalization hypothesis

Random matrix theory

Emergent integrability

"Localization length" \(\zeta\)

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

Entanglement entropy

...

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

A

\(r\)

\(P(r)\)

\(P(r)\)

\(r\)

Ergodic delocalized

Many-body localized

\(L/2\)

\(S/L\)

Ergodic thermalization hypothesis

Random matrix theory

Emergent integrability

"Localization length" \(\zeta\)

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

many-body localization in one slide

3

3

5

Spectral statistics

Entanglement entropy

...

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

A

\(r\)

\(P(r)\)

\(P(r)\)

\(r\)

\(L/2\)

\(S/L\)

\(L/2\)

\(S/L\)

Ergodic delocalized

Many-body localized

Ergodic thermalization hypothesis

Random matrix theory

Emergent integrability

"Localization length" \(\zeta\)

Avalanche theory

Runaway instability induced by rare regions of weak disorder

COLBOIS | INSTABILITIES AND MBL |  05.2025

De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),

Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)

3

3

6

Avalanche theory

Runaway instability induced by rare regions of weak disorder

COLBOIS | INSTABILITIES AND MBL |  05.2025

De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),

Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)

3

3

6

Adapted from Szoldra et at (2024)

Avalanche theory

 \(n_0\)

Runaway instability induced by rare regions of weak disorder

COLBOIS | INSTABILITIES AND MBL |  05.2025

De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),

Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)

Adapted from Szoldra et at (2024)

3

3

6

thermal "bubble" with level spacing \(\delta \sim 2^{-n_0}\)

Avalanche theory

 \(n_0\)

Runaway instability induced by rare regions of weak disorder

\(\Gamma \sim e^{-r/\zeta}\)

thermal "bubble" with level spacing \(\delta \sim 2^{-n_0}\)

COLBOIS | INSTABILITIES AND MBL |  05.2025

De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),

Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)

Adapted from Szoldra et at (2024)

3

3

6

Avalanche theory

 \(n_0\)

Runaway instability induced by rare regions of weak disorder

\(\Gamma \sim e^{-r/\zeta}\)

thermal "bubble" with level spacing \(\delta \sim 2^{-n_0}\)

Spin relaxes  if the interaction does not resolve the spectral gap of the grain

\(\zeta > \zeta_c\)

COLBOIS | INSTABILITIES AND MBL |  05.2025

De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),

Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)

Adapted from Szoldra et at (2024)

3

3

6

Avalanche theory At weak interactions

3

3

7

COLBOIS | INSTABILITIES AND MBL |  05.2025

Avalanche theory At weak interactions

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

7

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions
  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

Avalanche theory At weak interactions

COLBOIS | INSTABILITIES AND MBL |  05.2025

\(\epsilon\)

\(\epsilon_{\rm sp}\)

\xi_{\rm sp}(\epsilon_{\mathrm{sp}}, {\color{orange}h})

Colbois and Laflorencie (2023), Crowley and Chandran (2020)

3

3

7

Avalanche theory At weak interactions

COLBOIS | INSTABILITIES AND MBL |  05.2025

\(\epsilon\)

\(\epsilon_{\rm sp}\)

\xi_{\rm sp}(\epsilon_{\mathrm{sp}}, {\color{orange}h})

Colbois and Laflorencie (2023), Crowley and Chandran (2020)

\xi_{\rm MBA}

\(h/J\)

\(h/J \gg 1 \)

 

\(\xi_{\mathrm{MBA}} \ll L \)

\(\Rightarrow\)

3

3

7

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

Avalanche theory At weak interactions

\xi_{\rm MBA}

\(h\)

COLBOIS | INSTABILITIES AND MBL |  05.2025

 Crowley and Chandran (2020), Colbois and Laflorencie (2023)

3

3

8

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

2. The localization length tends to increase with the presence of interactions

Avalanche theory At weak interactions

\xi_{\rm MBA}

\(h\)

\xi_{\rm MBA} > \zeta_{\rm av.}

COLBOIS | INSTABILITIES AND MBL |  05.2025

 Crowley and Chandran (2020), Colbois and Laflorencie (2023)

3

3

8

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

2. The localization length tends to increase with the presence of interactions

Avalanche theory At weak interactions

\xi_{\rm MBA}

\(h\)

\(h^{\star}\)

\xi_{\rm MBA} > \zeta_{\rm av.}

COLBOIS | INSTABILITIES AND MBL |  05.2025

 Crowley and Chandran (2020), Colbois and Laflorencie (2023)

3

3

8

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

2. The localization length tends to increase with the presence of interactions

Avalanche theory At weak interactions

\xi_{\rm MBA}

\(h\)

High energy, \( \epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

!

\(h^{\star}\)

\(h^{\star}\)

\xi_{\rm MBA} > \zeta_{\rm av.}

COLBOIS | INSTABILITIES AND MBL |  05.2025

 Crowley and Chandran (2020), Colbois and Laflorencie (2023)

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

8

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

2. The localization length tends to increase with the presence of interactions

Avalanche theory At weak interactions

\xi_{\rm MBA}

\(h\)

High energy, \( \epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

!

\(h^{\star}\)

\(h^{\star}\)

\xi_{\rm MBA} > \zeta_{\rm av.}

COLBOIS | INSTABILITIES AND MBL |  05.2025

Can we probe this weak-interaction instability ?

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

 Crowley and Chandran (2020), Colbois and Laflorencie (2023)

3

3

8

  1. Any reasonable definition of \(\zeta\) should become the Anderson \(\xi_{\rm MBA}\) at weak interactions

2. The localization length tends to increase with the presence of interactions

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Analysis at fixed disorder

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

3

3

9

Entanglement entropy

COLBOIS | INSTABILITIES AND MBL |  05.2025

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

Analysis at fixed disorder

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

9

A

Entanglement entropy

Spectral statistics

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

Analysis at fixed disorder

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

9

\(P(r)\)

\(r\)

A

Entanglement entropy

Spectral statistics

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

0.1895 \quad \mathrm{Ergodic}
0 \quad \mathrm{Localized}

Kullback-Leibler divergence :

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

Analysis at fixed disorder

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

9

\(P(r)\)

\(r\)

A

Entanglement entropy

Spectral statistics

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

Analysis at fixed disorder

Delocalized at strong enough interactions

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

9

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

0.1895 \quad \mathrm{Ergodic}
0 \quad \mathrm{Localized}

Kullback-Leibler divergence :

\(P(r)\)

\(r\)

A

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

Analysis at fixed disorder

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

9

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

0.1895 \quad \mathrm{Ergodic}
0 \quad \mathrm{Localized}

Kullback-Leibler divergence :

\(P(r)\)

\(r\)

A

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

Critical interactions scaling

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

10

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

Critical interactions scaling

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

10

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

Critical interactions scaling

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

10

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

Critical interactions scaling

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

10

Delocalized at strong enough interactions

Entanglement entropy

Spectral statistics

Critical interactions scaling

KL divergence

COLBOIS | INSTABILITIES AND MBL |  05.2025

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

10

(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5;  Crowley and Chandran 2020 )

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

\(\Delta/J\)

\(h/J \)

Ergodic instability of Anderson localization

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

11

\(\Delta/J\)

\(h/J \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Ergodic instability of Anderson localization

(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5;  Crowley and Chandran 2020 )

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

(\xi_{\rm MBA} \approx 0.5)

3

3

11

\(\Delta/J\)

\(h/J \)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Ergodic instability of Anderson localization

(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5;  Crowley and Chandran 2020 )

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

11

\(\Delta/J\)

\(h/J \)

Ergodic

MBL

COLBOIS | INSTABILITIES AND MBL |  05.2025

Ergodic instability of Anderson localization

(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5;  Crowley and Chandran 2020 )

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

3

3

11

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

12

1) Ergodic instability

Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)

  • /!\ We cannot say that our results validate avalanche theory (there could be other mechanisms)
  • Finite-size effects work "in our favor"!

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

Conclusion

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

12

1) Ergodic instability

Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

2) Extrapolated transition

Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)

Conclusion

Conclusion

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

12

1) Ergodic instability

Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

2) Extrapolated transition

Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)

JC, F. Alet, N. Laflorencie, PRB 110, (2024)

N. Laflorencie, JC, F. Alet, arXiv (2025)

3) Advertisement :

Further crucial insight from the point of view of two-point correlation functions

Conclusion

COLBOIS | INSTABILITIES AND MBL |  05.2025

3

3

12

1) Ergodic instability

Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

2) Extrapolated transition

Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)

3) Advertisement :

Further crucial insight from the point of view of two-point correlation functions

Thank you !

JC, F. Alet, N. Laflorencie, PRB 110, (2024)

N. Laflorencie, JC, F. Alet, arXiv (2025)

Revisiting an old question

Interactions give rise to MBL
(unbounded Hamiltonians)

\(\Delta\)

\(\Delta_c\)

D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019)

Lin et al, Scipost (2018)

  • Fix \(\Delta, \epsilon \), vary the filling

 

  • Fix the filling and vary \(\epsilon\), \(\Delta\)

 

  • Fix the filling, \(\epsilon\) and W, vary \(\Delta\)

LeBlond et al. (2021)

Hopjan, Orso, Heidrich-Meisner (2021)

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

&

(for standard observables)

COLBOIS | INSTABILITIES AND MBL |  05.2025

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

GOE = Ergodic

Poisson = localized

Probes:

  1. spectral statistics

Probes : Spectral statistics

r_i = \min\left(\frac{s_i}{s_{i-1}}, \frac{s_{i-1}}{s_{i}}\right)
s_i
s_{i-1}

Gap ratio:

P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) [... a lot of works ...]

O. Giraud, N. Macé, E. Vernier, F. Alet, PRX 12, 011006 (2022)

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

Area-law at strong disorder

Volume-law at weak disorder

COLBOIS | INSTABILITIES AND MBL |  05.2025

 Probes : Entanglement entropy

Probes:

  1. spectral statistics
  2. entanglement transition

Khemani et al, PRX 7 (2017)

Disorder \(h\)

\(S_T =  (L-\log_2(e))/2\)

\(S = -\mathrm{Tr} \rho_A \ln \rho_A\)

A

spatial correlations results

SimplE

Experimentally

accessible

Somewhat

overlooked

\alpha = z
\alpha = x, y

Localized

Delocalized

It depends

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

 \(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle  - \langle n_i \rangle \langle n_{j} \rangle\)

From spin to bosons : \(n_i = S_i^{z} + 1/2\)

Density-density correlations

Main theoretical works*:

Aubry-André model

Pal & Huse, PRB (2010)

Lim, Sheng, PRB (2016)

Localization lengths are short in MBL

Varma et al., PRB (2019)

Character of short-range distributions

Colmenarez et al, SciPost (2019)

 \(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle  - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

e.g. : Herviou et al (2019), Hemery et al (2022),Weiner et al (2019), Morningstar et al (2022)

*Other works focus on QMI / on weaker disorders / on time evolution

Lukin et al, Science (2019)

Correlations as a probe of the transition...

Spin-Spin Correlations

SimplE

Experimentally

accessible

Somewhat

overlooked

\alpha = z
\alpha = x, y

Localized

Delocalized

It depends

\(|C^{\alpha,\alpha}_{r} |= A e^{-r /\xi_{\alpha}}\)

 \(C_{ij}^{zz} \rightarrow \langle n_i n_{j} \rangle  - \langle n_i \rangle \langle n_{j} \rangle\)

From spin to bosons : \(n_i = S_i^{z} + 1/2\)

Density-density correlations

Main theoretical works*:

Aubry-André model

Pal & Huse, PRB (2010)

Lim, Sheng, PRB (2016)

Localization lengths are short in MBL

Varma et al., PRB (2019)

Character of short-range distributions

Colmenarez et al, SciPost (2019)

 \(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle  - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

e.g. : Herviou et al (2019), Hemery et al (2022),Weiner et al (2019), Morningstar et al (2022)

*Other works focus on QMI / on weaker disorders / on time evolution

Lukin et al, Science (2019)

Correlations as a probe of the transition...

Spin-Spin Correlations

COLBOIS | INSTABILITIES AND MBL |  05.2025

Ergodic typical eigenstate = random vector

Anderson localized \(\Delta = 0\)

\(\xi_x \approx 2 \xi_z\) : XX correlations dominate

Random vector

  • bulk decay matches mid-chain decay
  • good proxy for the localization length

Two simple limits

Total spin conservation

Random state : XX

|C^{zz}_{L/2}| \sim 1/4(L-1)

\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)

|C^{xx}_{L/2}| \propto 2^{-L/2}
\xi_z \rightarrow \infty

finite \(\xi_x\)

Power-law

ZZ correlations dominate

\(\bra{R} S_i^{+} S_j^{-} \ket{R}\)

\(= \sum_{s} a_s a_{\mathrm{flip}(s)}\) \(\propto \frac{ \sqrt{\mathcal{N}}}{\mathcal{N}}\)

JC, F. Alet, N. Laflorencie, PRL 133 and  PRB 110, (2024)

 \(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle  - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)

COLBOIS | INSTABILITIES AND MBL |  05.2025

COLBOIS | INSTABILITIES AND MBL |  05.2025

Vertical cut

\(\Delta/J\)

\(h/J \)

Ergodic

MBL

\(\Delta/J\)

\(h/J \)

Ergodic

MBL

  1. \(\xi_{z,x}\) very short
  2. Directly connected to AL values
  3.  Flat  \(\xi_x\), fast  increase of \(\xi_z\)
  4. Instability

Ergodic

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Rare events I

JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024) & PRB 110, (2024)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Rare events II

N. Laflorencie, JC, F. Alet, arXiv (2025)

COLBOIS | INSTABILITIES AND MBL |  05.2025

Serbyn, Papic, Abanin (2013) ; Bauer, Nayak (2014) ; Huse, Nandkishore, Oganesyan (2014)

Reviews : Imbrie, Ros, Scardicchio (2017) Rademarker, Ortuno, Somoza (2017)

- out of equilibrium dynamics

M. Schreiber et al. Science (2015)

- log-growth of EE

J. H. Bardarson et al, PRL 109, 017202 (2012)

M. Znidaric et al PRB 77, 064426 (2008)

- analytical arguments / proof(s)

 

Nature / behavior of the transition ?

- Basko, Aleiner, Altschuler (2006), Ros, Müller (2017), Crowley, Chandran (2022),

- Imbrie (2016), ...

\mathcal{H} = \sum_{i} \tilde{h}_i \tau_i^{z} + \sum_{i>j} J_{ij} \tau_i^{z} \tau_j^{z} + \sum_{i>j>k} J_{ijk}\tau_i^{z}\tau_j^{z}\tau_k^{z} + ...

 

  • \(L\) commuting operators, commuting with \(\mathcal{H}\)
  • quasi-local

When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)

AL : Anderson orbitals

|\phi_i^{m}|^2 \lesssim \exp \left( \frac{i-i_0^m}{\xi_m}\right)
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
b_m = \sum_{i} \phi_i^{m} c_i

\(L\) conserved quantities

Phenomenological model : Emergent integrability

Interacting model

Strong finite-size effects

Ultraslow dynamics

Theory of instabilities

2025

Suntajs  et al, PRE (2020)

Suntajs  et al, PRB (2020)

Panda et al EPL (2019)

Abanin et al (2021)

Sels, Polkovnikov (2021)

LeBlond et al (2021)

Sierant & Zakrewski PRB (2022)

Morningstar et al (2022)

Evers, Modak, Bera (2023)

Long et al (2023)

Ha et al (2023)

Weisse, Gerstner, Sierker (2024)

...

2. MBL "Crisis"

COLBOIS | INSTABILITIES AND MBL |  05.2025

Some examples of Strong finite-size effects

Gap Ratio

Sierant, Lewenstein, Zakrewski PRL (2020)

Challenging finite-size scaling

Disorder \(h\)

Disorder \(h\)

\(h/L\)

\(h\)

Entanglement entropy

& other probes

JC, F. Alet, N. Laflorencie, PRL (2024)

Suntajs et al (2020)

(arXiv v1-v2)

COLBOIS | INSTABILITIES AND MBL |  05.2025

High energy, \(\epsilon = 0.5\)

Ising interaction \(\Delta\)

  MBL

Delocalized

disorder \(h \)

Hints in several works but extremely challenging to characterize

Jacobi method

Demixing

QMI

Fictitious evolution

|
|
\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}
\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}

\(r\)

|
|
\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}
\rangle_{\tau^z \, \mathrm{or}\, \sigma^z}

\(r\)

\ket{E', \pm} =

Phenomenological models

Relation to avalanches

Kjäll (2018)

 Colmenarez  et al (2019)

Villalonga and Clark (2020)

\(\pm\)

Crowley and Chandran (2020)

Garatt et al (2021)

Crowley and Chandran (2022)

Long et al (2023)

Ha, Morningstar and Huse (2023)

Morningstar et al (2022)

Perturb away from very strong disorder

Gopalakrishnan et al (2015)

Another possible mechanism for instabilities:

resonances between more localized many-body states

Gopalakrishnan et al (2015)

Villalonga and Clark (2020)

Garratt et al (2021)

Crowley and Chandran (2022)

Morningstar et al (2022)

Instabilities : Many-body resonances

COLBOIS | INSTABILITIES AND MBL |  05.2025

\(L \rightarrow \infty\)

Kiefer-Emmanouilidis et al (2020), Suntajs et al (2020), Sels, Polkovnikov (2021), Wiesse et al (2024),...

Evers et al (2023),Long (2023)...

JC, Alet, Laflorencie (2024), Laflorencie et al (2025)

Sierant et al(2020), Morningstar et al, (2022) ,  Crowley, Chandran (2022), Szoldra et al (2024), Nieda et al (2024), Scoquart et al (2025)....

Weiner et al (2019), Sierant, Zakrewski (2022),

Biroli et al (2024)

Morningstar et al, PRB 105 (2022)

(Recent) Phase diagram(s) and instabilities

avalanche instability

many-body resonances

here - end to end QMI

gap ratio

and minimum gap

Absence of MBL phase

Single transition from ergodic to MBL (potentially very large \(h_c\))

Intermediate phase (nature differs depending on the paper)

h

h

h

Dynamics

Dynamics

Anderson

No growth

of entanglement

J. H. Bardarson, F. Pollmann, and J. E. Moore, PRL 109, 017202 (2012)

M. Znidaric, T. Prosen, and P. Prelovsek PRB 77, 064426 (2008)

Log growth

of entanglement

Initial \(S^z\) basis random product state

+

TEBD

 

W = 5

D. Luitz, N. Laflorencie, F. Alet (2016)

Sierant and Zakrewski (2022)

Other probes

Some eigenstate

J. C., N. Laflorencie, PRB (2023)

\(|\langle S_i^{z}\rangle| < 1/2\)

\delta_i = 1/2 -| \langle S_i^z \rangle|
\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|

A simple many-body effect : Maximal magnetization?

Anderson chain / XX chain

Chain breaking

Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)

Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)

JC, Laflorencie, PRB 108, 144206 (2023)

Toy model:

\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta_{\rm min}^{\rm typ} = \exp(\overline{\ln \delta_{\min}})
\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|

SPIN FREEZING !

CHAIN BREAKING !

Participation entropy

Macé et al (2019)

Colbois, Alet, Laflorencie (2024)

Participation entropy

Landmarks

Landmarks

Other weak interactions results

COLBOIS | INSTABILITIES AND MBL |  05.2025

Model : t-V with usual XXZ units, except V = 2t

COLBOIS | INSTABILITIES AND MBL |  05.2025