Jeanne Colbois | CNRS researcher at Institut NEEL | Grenoble | France
jeanne.colbois@neel.cnrs.fr
Jeanne Colbois | CNRS researcher at Institut NEEL | Grenoble | France
jeanne.colbois@neel.cnrs.fr
Anderson localization
Jeanne Colbois | CNRS researcher at Institut NEEL | Grenoble | France
jeanne.colbois@neel.cnrs.fr
Thermalization in isolated quantum many-body systems
Anderson localization
Jeanne Colbois | CNRS researcher at Institut NEEL | Grenoble | France
jeanne.colbois@neel.cnrs.fr
Instabilities in the random field XXZ chain
Nicolas Laflorencie
Fabien Alet
LPT Toulouse - France
Thermalization in isolated quantum many-body systems
Anderson localization
COLBOIS | FROM AL TO MBL | 07.2025
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
1
Conserves \(N_f\)
\(|\psi(x)|^2\)
COLBOIS | FROM AL TO MBL | 07.2025
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
1
Conserves \(N_f\)
\(|\psi(x)|^2\)
\(h\)
\(-h\)
\(h\)
COLBOIS | FROM AL TO MBL | 07.2025
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
1
Conserves \(N_f\)
\(|\psi(x)|^2\)
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\(h\)
\(-h\)
\(h\)
\(|\psi(x)|^2\)
\(\xi(h, E)\)
COLBOIS | FROM AL TO MBL | 07.2025
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
Constructive interference phenomenon
\(\langle x^{2}(t)\rangle \) saturates : absence of diffusion
1
Conserves \(N_f\)
\(|\psi(x)|^2\)
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\(h\)
\(-h\)
\(h\)
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Constructive interference phenomenon
\(\langle x^{2}(t)\rangle \) saturates : absence of diffusion
1
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
\(|\psi(x)|^2\)
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\(h\)
\(-h\)
\(h\)
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
\(|\psi(x)|^2\)
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\(h\)
\(-h\)
\(h\)
\(|\psi(x)|^2\)
Constructive interference phenomenon
\(\langle x^{2}(t)\rangle \) saturates : absence of diffusion
\(\xi(h, E)\)
1
Absence of thermalization in isolated many-body quantum systems
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
Quenched disorder
Eigenstates in the middle of the spectrum
2
COLBOIS | FROM AL TO MBL | 07.2025
Quenched disorder
Eigenstates in the middle of the spectrum
Non-perturbative quantum many-body physics on a lattice
Entanglement
(Random matrix theory ?)
2
COLBOIS | FROM AL TO MBL | 07.2025
3
1. From "many-body" Anderson localization to the question
2. Thermalization of isolated quantum systems
COLBOIS | FROM AL TO MBL | 07.2025
3
1. From "many-body" Anderson localization to the question
2. Thermalization of isolated quantum systems
On a spin chain:
3. Many-body localization vs thermalization
4. The MBL "crisis"
COLBOIS | FROM AL TO MBL | 07.2025
3
1. From "many-body" Anderson localization to the question
2. Thermalization of isolated quantum systems
On a spin chain:
3. Many-body localization vs thermalization
4. The MBL "crisis"
5. Avalanche instability at weak interactions
6. Resonances through correlations
COLBOIS | FROM AL TO MBL | 07.2025
4
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
4
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
commute with
commute with each other
(quasi-)local
5
Slater determinant based on quasi-local wavefunctions
\(\xi(h, E)\)
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
Slater determinant based on quasi-local wavefunctions
\(\xi(h, E)\)
COLBOIS | FROM AL TO MBL | 07.2025
Conserves \(N_f\)
5
6
A
\(\rho_A = \mathrm{Tr}_B | \Psi \rangle \langle \Psi |\)
COLBOIS | FROM AL TO MBL | 07.2025
\(\{| n_1, n_2, \dots, n_B \rangle \}\)
6
A
\(\rho_A = \mathrm{Tr}_B | \Psi \rangle \langle \Psi |\)
\(\{| n_1, n_2, \dots, n_B \rangle \}\)
COLBOIS | FROM AL TO MBL | 07.2025
\(S = -\mathrm{Tr} \rho_A \ln \rho_A\)
Von-Neumann EE
6
\(S = -\mathrm{Tr} \rho_A \ln \rho_A\)
A
\(\rho_A = \mathrm{Tr}_B | \Psi \rangle \langle \Psi |\)
Von-Neumann EE
\(\{| n_1, n_2, \dots, n_B \rangle \}\)
Entanglement entropy controlled by the boundary
(exponentially small contribution from the bulk)
COLBOIS | FROM AL TO MBL | 07.2025
7
Gap ratio:
COLBOIS | FROM AL TO MBL | 07.2025
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)
7
Gap ratio:
COLBOIS | FROM AL TO MBL | 07.2025
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)
7
Gap ratio:
Single-body:
\(P(r)\)
\(r\)
1. Limit \(h \gg J\) : \(E_i\) random \(\rightarrow\) Poisson
COLBOIS | FROM AL TO MBL | 07.2025
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)
7
Gap ratio:
Single-body:
\(P(r)\)
\(r\)
1. Limit \(h \gg J\) : \(E_i\) random \(\rightarrow\) Poisson
COLBOIS | FROM AL TO MBL | 07.2025
2. Smaller disorder : \(\rightarrow\) still Poisson
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)
7
Gap ratio:
Single-body:
Many-body (high in the spectrum)
\(P(r)\)
\(r\)
1. Limit \(h \gg J\) : \(E_i\) random \(\rightarrow\) Poisson
2. Smaller disorder : \(\rightarrow\) still Poisson
3. Neighbouring \(\mathcal{E}_n\) correspond to very different \(\sum E_m\)
COLBOIS | FROM AL TO MBL | 07.2025
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)
7
Gap ratio:
Single-body:
Many-body (high in the spectrum)
\(P(r)\)
\(r\)
1. Limit \(h \gg J\) : \(E_i\) random \(\rightarrow\) Poisson
2. Smaller disorder : \(\rightarrow\) still Poisson
3. Neighbouring \(\mathcal{E}_n\) correspond to very different \(\sum E_m\)
COLBOIS | FROM AL TO MBL | 07.2025
8
Attraction / repulsion
\(-h\)
\(h\)
at high energy
COLBOIS | FROM AL TO MBL | 07.2025
8
In the Anderson basis:
Anderson orbitals \(m\)
Attraction / repulsion
\(-h\)
\(h\)
at high energy
COLBOIS | FROM AL TO MBL | 07.2025
8
In the Anderson basis:
Anderson orbitals \(m\)
Attraction / repulsion
\(-h\)
\(h\)
at high energy
COLBOIS | FROM AL TO MBL | 07.2025
Interactions favor delocalization. Do they fully destroy localization?
COLBOIS | FROM AL TO MBL | 07.2025
9
Isolated
many-body system
?
Description with statistical mechanics
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
9
Isolated
many-body system
?
Description with statistical mechanics
Long-time average of few-body observables
Microcanonical average
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
9
Isolated
many-body system
?
Description with statistical mechanics
Long-time average of few-body observables
Microcanonical average
Unitary time evolution
Long-time behavior completely controlled by eigenstates
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Smooth function
Equal to microcanonical average
Equilibrium
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Smooth function
Equal to microcanonical average
Entropy
Smooth
Random variable
Approach to equilibrium
Equilibrium
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Smooth function
Equal to microcanonical average
Entropy
Smooth
Random variable
Approach to equilibrium
Equilibrium
Narrow energy window: random matrix theory
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
COLBOIS | FROM AL TO MBL | 07.2025
10
A sufficient condition on eigenstates
Smooth function
Equal to microcanonical average
Entropy
Smooth
Random variable
Approach to equilibrium
Equilibrium
Narrow energy window: random matrix theory
Deutsch (1991), Srednicki (1994)
D'Alessio et al (2016) for a review
2015
2018
2019
2025
3
3
11
\(-h\)
\(h\)
COLBOIS | FROM AL TO MBL | 07.2025
\(-h\)
\(h\)
Attraction / repulsion
3
3
11
COLBOIS | FROM AL TO MBL | 07.2025
3
3
\(-h\)
\(h\)
Attraction / repulsion
Jordan-Wigner
non-local transformation
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
3
3
11
COLBOIS | FROM AL TO MBL | 07.2025
\(-h\)
\(h\)
Attraction / repulsion
Jordan-Wigner
\(n_i = S_i^z + 1/2\)
\(c_i = \exp(i \pi \sum_{j=1}^{i-1} n_j) S_i^{-}\)
non-local transformation
Spin-flip
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
3
3
11
COLBOIS | FROM AL TO MBL | 07.2025
\(-h\)
\(h\)
Attraction / repulsion
Jordan-Wigner
non-local transformation
ISING INTERACTION
Spin-flip
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\(n_i = S_i^z + 1/2\)
\(c_i = \exp(i \pi \sum_{j=1}^{i-1} n_j) S_i^{-}\)
3
3
11
COLBOIS | FROM AL TO MBL | 07.2025
\(-h\)
\(h\)
Attraction / repulsion
Jordan-Wigner
non-local transformation
ISING INTERACTION
Spin-flip
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\(n_i = S_i^z + 1/2\)
\(c_i = \exp(i \pi \sum_{j=1}^{i-1} n_j) S_i^{-}\)
3
3
11
COLBOIS | FROM AL TO MBL | 07.2025
\(-h\)
\(h\)
Attraction / repulsion
Jordan-Wigner
non-local transformation
ISING INTERACTION
Spin-flip
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\(n_i = S_i^z + 1/2\)
\(c_i = \exp(i \pi \sum_{j=1}^{i-1} n_j) S_i^{-}\)
3
3
11
\(L/2\) fermions
Charge is conserved
\(\sum_i S_i^{z} = 0\)
Magnetization is conserved
COLBOIS | FROM AL TO MBL | 07.2025
ISING INTERACTION
Spin-flip
3
3
12
COLBOIS | FROM AL TO MBL | 07.2025
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
3
3
12
COLBOIS | FROM AL TO MBL | 07.2025
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
3
3
12
COLBOIS | FROM AL TO MBL | 07.2025
Counter example for thermalization of isolated quantum systems
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
Counter example for thermalization of isolated quantum systems
Polynomial \(\rightarrow\) Exponential
Absence of translation invariance
No typicality methods
No ground-state methods
3
3
12
COLBOIS | FROM AL TO MBL | 07.2025
\(\epsilon = 1\)
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
Counter example for thermalization of isolated quantum systems
Polynomial \(\rightarrow\) Exponential
Absence of translation invariance
No typicality methods
No ground-state methods
3
3
13
COLBOIS | FROM AL TO MBL | 07.2025
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
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!
3
3
13
COLBOIS | FROM AL TO MBL | 07.2025
ISING INTERACTION
Spin-flip
\(\epsilon = 1\)
\(\epsilon = 0\)
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)
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!
?
3
3
13
COLBOIS | FROM AL TO MBL | 07.2025
Disorder
\(\Delta > 0\)
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Disorder
\(\Delta > 0\)
Ergodic delocalized
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Disorder
\(\Delta > 0\)
Ergodic delocalized
Many-body localized
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Many-body localized
Spectral statistics
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)
Disorder
\(\Delta > 0\)
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Many-body localized
Spectral statistics
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)
Disorder
\(\Delta > 0\)
\(r\)
\(P(r)\)
GOE = Ergodic
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Text
Random matrix theory
Many-body localized
Spectral statistics
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)
Disorder
\(\Delta > 0\)
GOE = Ergodic
\(r\)
\(P(r)\)
\(P(r)\)
Poisson = localized
3
3
14
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Random matrix theory
Many-body localized
Disorder
\(\Delta > 0\)
Entanglement entropy
A
\(r\)
\(P(r)\)
\(P(r)\)
Khemani et al, PRX 7 (2017)
Disorder \(h\)
3
3
15
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Many-body localized
Disorder
\(\Delta > 0\)
Entanglement entropy
A
\(r\)
\(P(r)\)
\(P(r)\)
Volume-law at weak disorder
Khemani et al, PRX 7 (2017)
Disorder \(h\)
\(L/2\)
\(S/L\)
\(S_T = (L-\log_2(e))/2\)
3
3
15
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Eigenstates are thermal
Many-body localized
Disorder
\(\Delta > 0\)
Entanglement entropy
A
\(r\)
\(P(r)\)
\(P(r)\)
Area-law at strong disorder
Volume-law at weak disorder
Khemani et al, PRX 7 (2017)
Disorder \(h\)
\(L/2\)
\(S/L\)
\(S_T = (L-\log_2(e))/2\)
3
3
15
\(L/2\)
\(S/L\)
COLBOIS | FROM AL TO MBL | 07.2025
Ergodic delocalized
Eigenstates are thermal
3
3
16
Initial \(S^z\) basis random product state
+
time evolution \(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
16
Initial \(S^z\) basis random product state
+
time evolution \(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
J. H. Bardarson, F. Pollmann, and J. E. Moore, PRL 109, 017202 (2012)
M. Znidaric, T. Prosen, and P. Prelovsek PRB 77, 064426 (2008)
3
3
16
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)
Initial \(S^z\) basis random product state
+
time evolution \(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
16
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)
MBL
Log growth
of entanglement
Initial \(S^z\) basis random product state
+
time evolution \(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
Serbyn, Papic, Abanin (2013) ; Bauer, Nayak (2014) ; Huse, Nandkishore, Oganesyan (2014)
Reviews : Imbrie, Ros, Scardicchio (2017) Rademarker, Ortuno, Somoza (2017)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
Serbyn, Papic, Abanin (2013) ; Bauer, Nayak (2014) ; Huse, Nandkishore, Oganesyan (2014)
Reviews : Imbrie, Ros, Scardicchio (2017) Rademarker, Ortuno, Somoza (2017)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
Serbyn, Papic, Abanin (2013) ; Bauer, Nayak (2014) ; Huse, Nandkishore, Oganesyan (2014)
Reviews : Imbrie, Ros, Scardicchio (2017) Rademarker, Ortuno, Somoza (2017)
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
Serbyn, Papic, Abanin (2013) ; Bauer, Nayak (2014) ; Huse, Nandkishore, Oganesyan (2014)
Reviews : Imbrie, Ros, Scardicchio (2017) Rademarker, Ortuno, Somoza (2017)
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
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)
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
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)
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
- log-growth of EE
J. H. Bardarson et al, PRL 109, 017202 (2012)
M. Znidaric et al PRB 77, 064426 (2008)
AL : L quasilocal conserved quantities
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)
- Basko, Aleiner, Altschuler (2006), Ros, Müller (2017), Crowley, Chandran (2022),
- Imbrie (2016), ...
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
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)
- Basko, Aleiner, Altschuler (2006), Ros, Müller (2017), Crowley, Chandran (2022),
- Imbrie (2016), ...
When MBL : \(J_{i,...,j} \propto e^{-\frac{-|i-j|}{\zeta}}\)
Interacting model
COLBOIS | FROM AL TO MBL | 07.2025
3
3
17
AL : L quasilocal conserved quantities
As of 2016
Strong finite-size effects
Ultraslow dynamics
Theory of instabilities
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)
...
Strong finite-size effects
Ultraslow dynamics
Theory of instabilities
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)
...
Strong finite-size effects
Ultraslow dynamics
Theory of instabilities
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)
...
Strong finite-size effects
Ultraslow dynamics
Theory of instabilities
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)
...
3
3
17
COLBOIS | FROM AL TO MBL | 07.2025
Gap Ratio
Challenging finite-size scaling
\(h/L\)
\(h\)
Suntajs et al (2020)
(arXiv v1-v2)
3
3
18
COLBOIS | FROM AL TO MBL | 07.2025
Gap Ratio
Sierant, Lewenstein, Zakrewski PRL (2020)
Suntajs et al (2020)
(arXiv v1-v2)
Challenging finite-size scaling
Disorder \(h\)
Disorder \(h\)
\(h/L\)
\(h\)
3
3
18
COLBOIS | FROM AL TO MBL | 07.2025
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)
3
3
18
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
3
3
19
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
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
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
3
3
19
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
avalanche instability
De Roeck & Huveneers (2017), Luitz, De Roeck & Huveneers (2017),
Thiery et al (2018); Crowley and Chandran (2020), Crowley and Chandran (2022)
Small ergodic region triggers runaway delocalization
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
3
3
19
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
many-body resonances
Gopalakrishnan et al (2015)
Villalonga and Clark (2020)
Garratt et al (2021)
Crowley and Chandran (2022)
Morningstar et al (2022)
MBL is destabilized by resonances between localized eigenstates finite-size crossover
avalanche instability
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
3
3
19
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
avalanche instability
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
3
3
19
many-body resonances
COLBOIS | FROM AL TO MBL | 07.2025
\(L \rightarrow \infty\)
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)
See Nicolas' talk for a discussion
Morningstar et al, PRB 105 (2022)
avalanche instability
3
3
8
many-body resonances
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
3
3
19
COLBOIS | FROM AL TO MBL | 07.2025
Avalanches
Resonances
Avalanches
Anderson localization has an ergodic instability at weak disorder
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
Resonances
Avalanches
Anderson localization has an ergodic instability at weak disorder
Resonances
We find long-range cat states deep in the MBL regime
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB (2024)
N. Laflorencie, JC, F. Alet, arXiv (2025)
?
Anderson insulator
disorder \(h \)
Ising interaction \(\Delta\)
Ergodic
Many-body localized
?
?
?
Anderson insulator
disorder \(h \)
Ising interaction \(\Delta\)
Ergodic
Many-body localized
?
\(n_0\)
Runaway instability induced by rare regions of weak disorder
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
20
COLBOIS | FROM AL TO MBL | 07.2025
\(n_0\)
Runaway instability induced by rare regions of weak disorder
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
20
COLBOIS | FROM AL TO MBL | 07.2025
thermal "bubble" with level spacing \(\delta \sim 2^{-n_0}\)
\(n_0\)
Runaway instability induced by rare regions of weak disorder
\(\Gamma \sim e^{-r/\zeta}\)
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
20
COLBOIS | FROM AL TO MBL | 07.2025
thermal "bubble" with level spacing \(\delta \sim 2^{-n_0}\)
\(n_0\)
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
\(\zeta > \zeta_c\)
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
20
COLBOIS | FROM AL TO MBL | 07.2025
3
3
21
COLBOIS | FROM AL TO MBL | 07.2025
3
3
21
COLBOIS | FROM AL TO MBL | 07.2025
\(\epsilon\)
\(\epsilon_{\rm sp}\)
Colbois and Laflorencie (2023), Crowley and Chandran (2020)
3
3
21
COLBOIS | FROM AL TO MBL | 07.2025
\(\epsilon\)
\(\epsilon_{\rm sp}\)
Colbois and Laflorencie (2023), Crowley and Chandran (2020)
\(h/J\)
\(h/J \gg 1 \)
\(\xi_{\mathrm{MBA}} \ll L \)
\(\Rightarrow\)
3
3
21
COLBOIS | FROM AL TO MBL | 07.2025
\(h\)
Crowley and Chandran (2020), Colbois and Laflorencie (2023)
3
3
22
COLBOIS | FROM AL TO MBL | 07.2025
\(h\)
Crowley and Chandran (2020), Colbois and Laflorencie (2023)
3
3
22
COLBOIS | FROM AL TO MBL | 07.2025
\(h\)
\(h^{\star}\)
Crowley and Chandran (2020), Colbois and Laflorencie (2023)
3
3
22
COLBOIS | FROM AL TO MBL | 07.2025
\(h\)
High energy, \( \epsilon = 0.5\)
Ising interaction \(\Delta\)
MBL
Delocalized
disorder \(h \)
!
\(h^{\star}\)
\(h^{\star}\)
Crowley and Chandran (2020), Colbois and Laflorencie (2023)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
3
3
22
COLBOIS | FROM AL TO MBL | 07.2025
(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)
\(\Delta/J\)
\(h/J \)
3
3
23
Delocalized at strong enough interactions
COLBOIS | FROM AL TO MBL | 07.2025
(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)
\(\Delta/J\)
\(h/J \)
Delocalized at strong enough interactions
Critical interaction drifts to zero!
3
3
23
COLBOIS | FROM AL TO MBL | 07.2025
\(\Delta/J\)
\(h/J \)
(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
23
COLBOIS | FROM AL TO MBL | 07.2025
\(\Delta/J\)
\(h/J \)
(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
23
COLBOIS | FROM AL TO MBL | 07.2025
\(\Delta/J\)
\(h/J \)
(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
23
COLBOIS | FROM AL TO MBL | 07.2025
High energy, \(\epsilon = 0.5\)
Ising interaction \(\Delta\)
MBL
Delocalized
disorder \(h \)
Gopalakrishnan et al (2015)
Villalonga and Clark (2020)
Garratt et al (2021)
Crowley and Chandran (2022)
Morningstar et al (2022)
3
3
24
COLBOIS | FROM AL TO MBL | 07.2025
Another possible mechanism for instabilities:
resonances between more localized many-body states
High energy, \(\epsilon = 0.5\)
Ising interaction \(\Delta\)
MBL
Delocalized
disorder \(h \)
Gopalakrishnan et al (2015)
Villalonga and Clark (2020)
Garratt et al (2021)
Crowley and Chandran (2022)
Morningstar et al (2022)
\(r\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
24
Another possible mechanism for instabilities:
resonances between more localized many-body states
High energy, \(\epsilon = 0.5\)
Ising interaction \(\Delta\)
MBL
Delocalized
disorder \(h \)
\(r\)
\(r\)
Gopalakrishnan et al (2015)
Villalonga and Clark (2020)
Garratt et al (2021)
Crowley and Chandran (2022)
Morningstar et al (2022)
\(\pm\)
Perturb away from very strong disorder
Another possible mechanism for instabilities:
resonances between more localized many-body states
COLBOIS | FROM AL TO MBL | 07.2025
3
3
24
High energy, \(\epsilon = 0.5\)
Ising interaction \(\Delta\)
MBL
Delocalized
disorder \(h \)
\(r\)
\(r\)
Gopalakrishnan et al (2015)
Villalonga and Clark (2020)
Garratt et al (2021)
Crowley and Chandran (2022)
Morningstar et al (2022)
\(\pm\)
Perturb away from very strong disorder
Another possible mechanism for instabilities:
resonances between more localized many-body states
COLBOIS | FROM AL TO MBL | 07.2025
3
3
24
Hints in several works but extremely challenging to characterize
Jacobi method
Demixing
QMI
Fictitious evolution
Phenomenological models
Relation to avalanches
Kjäll (2018)
Colmenarez et al (2019)
Villalonga and Clark (2020)
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)
Gopalakrishnan et al (2015)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Morningstar et al (2022)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Morningstar et al (2022)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Morningstar et al (2022)
see e.g. Varma et al., PRB (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Morningstar et al (2022)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Morningstar et al (2022)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Distance-dependent \(|C^{\alpha\alpha}_r|\) :
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L\)
Systemwide \(|C^{\alpha\alpha}_r|\) : \(r = L/2\)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Morningstar et al (2022)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
25
see e.g.
Pal & Huse (2010)
Varma et al. (2019)
Villalonga and Clark (2020)
Colmenarez et al (2019)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
Simple, experimentally accessible, somewhat overlooked
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
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 | FROM AL TO MBL | 07.2025
3
3
26
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
26
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi^z_{\mathrm{bulk}} = 0.448 \pm 0.005\)
\(\xi^z_{\mathrm{mid}} = 0.449 \pm 0.003\)
\(\xi^x_{\mathrm{bulk}} = 0.87 \pm 0.01\)
\(\xi^x_{\mathrm{mid}} = 0.89 \pm 0.01\)
\(h = 5\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
26
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
26
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
26
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
Total spin conservation
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
Total spin conservation
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
Power-law
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
Total spin conservation
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
Power-law
Random state : XX
\(\bra{R} S_i^{+} S_j^{-} \ket{R}\)
\(= \sum_{s} a_s a_{\mathrm{flip}(s)}\) \(\propto \frac{ \sqrt{\mathcal{N}}}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
Total spin conservation
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
finite \(\xi_x\)
Power-law
Random state : XX
\(\bra{R} S_i^{+} S_j^{-} \ket{R}\)
\(= \sum_{s} a_s a_{\mathrm{flip}(s)}\) \(\propto \frac{ \sqrt{\mathcal{N}}}{\mathcal{N}}\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
Random vector
Total spin conservation
Random state : XX
\(\ket{R} = \sum_{s= 1}^{\mathcal{N}} a_s \ket{s}\), \(\ket{s} = \ket{\uparrow, \downarrow, \dots}\),\( |a_s|^2\propto\frac{1}{\mathcal{N}}\)
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)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Ergodic typical eigenstate = random vector
Anderson localized \(\Delta = 0\)
\(\xi_x \approx 2 \xi_z\) : XX correlations dominate
ZZ correlations dominate
Random vector
XXZ, \(h = 1, \Delta = 1\)
finite \(\xi_x\)
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
\(C_{r}^{\alpha\alpha} = \langle S_i^{\alpha} S_{i+r}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{i+r}^{\alpha} \rangle\)
\(\xi_{\mathrm{MBA}}\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
27
Random vector
finite \(\xi_x\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
28
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
28
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
28
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
COLBOIS | FROM AL TO MBL | 07.2025
3
3
28
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
An inversion of the dominant correlations occurs deeper in the MBL regime
Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
COLBOIS | FROM AL TO MBL | 07.2025
3
3
28
JC, F. Alet, N. Laflorencie, PRL 133 and PRB 110, (2024)
Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
An inversion of the dominant correlations occurs deeper in the MBL regime
Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
?
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
Limited broadening with \(L\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
Limited broadening with \(L\)
Limited weight at large correlations
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
ZZ
Limited broadening with \(L\)
Limited weight at large correlations
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
ZZ
Limited broadening with \(L\)
Limited weight at large correlations
Broadening with \(L\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
29
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
XX
ZZ
Limited broadening with \(L\)
Limited weight at large correlations
Broadening with \(L\)
Non-zero probability of large correlations
COLBOIS | FROM AL TO MBL | 07.2025
3
3
30
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
ZZ
Broadening with \(L\)
Non-zero probability of large correlations
COLBOIS | FROM AL TO MBL | 07.2025
3
3
30
ZZ
Broadening with \(L\)
Non-zero probability of large correlations
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
COLBOIS | INSTABILITIES AND MBL | 05.2025
3
3
31
Typical vs average correlations:
Full analysis of correlations statistics: JC, F. Alet, N. Laflorencie, PRB 110, (2024)
\(\exp(\overline{\ln|C^{zz}_{L/2}|})\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
3
3
31
Full analysis of correlations statistics: JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Typical vs average correlations:
\(\exp(\overline{\ln|C^{zz}_{L/2}|})\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
3
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31
Average : better described by a power-law decay
Full analysis of correlations statistics: JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Typical vs average correlations:
\(\exp(\overline{\ln|C^{zz}_{L/2}|})\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
3
3
31
Typical : better described by an exponential decay
Average : better described by a power-law decay
Full analysis of correlations statistics: JC, F. Alet, N. Laflorencie, PRB 110, (2024)
Typical vs average correlations:
\(\exp(\overline{\ln|C^{zz}_{L/2}|})\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
32
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
Average of the maximum over all eigenstates
COLBOIS | FROM AL TO MBL | 07.2025
3
3
32
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
Average of the maximum over all eigenstates
Up to \(h \sim 20-25\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
33
\(L = 20, h = 12\)
COLBOIS | FROM AL TO MBL | 07.2025
3
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33
\(L = 20, h = 12\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
33
\(L = 20, h = 12\)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
33
\(L = 20, h = 12\)
Rigorous analysis of cat states through toy model: N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
COLBOIS | FROM AL TO MBL | 07.2025
3
3
34
Diagram of the correlations regimes:
COLBOIS | FROM AL TO MBL | 07.2025
3
3
34
Diagram of the correlations regimes:
Standard observables
COLBOIS | FROM AL TO MBL | 07.2025
3
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34
Diagram of the correlations regimes:
Average: power-law
Standard observables
COLBOIS | FROM AL TO MBL | 07.2025
3
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Diagram of the correlations regimes:
Average: power-law
Average: exponential
Standard observables
Fate of this region ?
Ergodic?
Non-ergodic delocalized?
MBL?
COLBOIS | FROM AL TO MBL | 07.2025
3
3
34
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv:2504.10566 (2025)
Pawlik et al (2025) in a Quantum Sun Model
(Others closer to the transition)
Diagram of the correlations regimes:
Rare events' importance in the MBL debate
Rare thermal regions
Rare paths in Hilbert space
Rare large correlations
Average: power-law
Average: exponential
Biroli, Hartmann, Tarzia (2024), ....
De Roeck, Huveneers (2017), Crowley, Chandran (2020),...
Rare, highly localized regions
De Roeck et al (2024), Dupont et al (2019), ...
COLBOIS | INSTABILITIES AND MBL | 05.2025
3
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"Beyond the Superconducting Super Collider story:"
Anderson localization
In Anderson localization (1D, half-filling) all states are "like" ground states
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
3
3
36
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
3
3
36
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
3
3
36
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
2) Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
3
3
36
3) Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
2) Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
3
3
36
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv (2025)
3) Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
2) Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
4) Fate of this intermediate MBL regime ?
Driven by rare cat states?
3
3
36
1) Ergodic instability
Below \(h^{\star} \sim 2-3\), the Anderson insulator immediately turns ergodic for \(\Delta > 0\)
2) Extrapolated transition
Standard estimates lead to an extrapolated transition line \(h_c(\Delta)\)
3) Deep MBL
MBL and AL are connected from the point of view of correlations \(\xi_z, \xi_x\) at weak interactions
COLBOIS | INSTABILITIES AND MBL | 05.2025
JC, F. Alet, N. Laflorencie, PRB 110, (2024)
N. Laflorencie, JC, F. Alet, arXiv (2025)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
4) Fate of this intermediate MBL regime ?
Driven by cat states?
3
3
36
Delocalized at strong enough interactions
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
\(\xi_x > \xi_z\)
Inversion
Extrapolated \(h_c\)
\(\xi_z \rightarrow \infty\)
\(h/J \)
\(h/J \)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
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)
Some eigenstate
J. C., N. Laflorencie, PRB (2023)
\(|\langle S_i^{z}\rangle| < 1/2\)
Anderson chain / XX chain
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
Toy model:
SPIN FREEZING !
CHAIN BREAKING !
Luitz et al (2015)
Macé et al (2019)
Colbois, Alet, Laflorencie (2024)
De Roeck & Huveneers 2017, Luitz, De Roeck & Huveneers 2017, Thiery et al 2018; Crowley and Chandran 2020
Condition for spin at \(r\) to relax thanks to the grain:
Avalanche criterion:
Question:
Does the seed hybridize (absorb) the l-bits?
Answer: it depends on
(1) \(V_{ij}\) the matrix element coupling the seed to the l-bit
(2) \(1/ \rho\) the level spacing.
Typically \(V_{ij} \gg 1/\rho\).
The challenge is to quantify this, see Crowley and Chandran.
DEEP MBL :
COLBOIS | INSTABILITIES AND MBL | 05.2025
Model : t-V with usual XXZ units, except V = 2t
COLBOIS | INSTABILITIES AND MBL | 05.2025