Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Jeanne Colbois - Institut Néel CNRS & UGA - Grenoble - France
eXTREME VALUE STATISTICS of local observables in Disordered chains
New generation in strongly correlated electron systems | Aarhus | September 2026
Jeanne Colbois - Institut Néel CNRS & UGA - Grenoble - France
eXTREME VALUE STATISTICS of local observables in Disordered chains
New generation in strongly correlated electron systems | Aarhus | September 2026
Natalia Chepiga
(Univerisity of Oxford)
Shaffique Adam
(Washington University in St Louis)
Gabriel Lemarié (INPHYNI, CNRS, France)
Nicolas Laflorencie
(LPT, CNRS, France)
Jeanne Colbois - Institut Néel CNRS & UGA - Grenoble - France
New generation in strongly correlated electron systems | Aarhus | September 2026
eXTREME VALUE STATISTICS of local observables in Disordered chains
1
Tossing a COIN
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
\(L = 176 \)
HTTHTHH
THTHHT
2
Tossing a COIN
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
TTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTTHTTHTHTHTHHHTHTHTHTTTHHTHTHTHTHHTTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (83 T / 93 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
Tossing a COIN
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
TTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTTHTTHTHTHTHHHTHTHTHTTTHHTHTHTHTHHTTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (83 T / 93 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
2
Tossing a COIN
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
TTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTTHTTHTHTHTHHHTHTHTHTTTHHTHTHTHTHHTTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (83 T / 93 H )
\mathcal{P}(\ell) \sim 2^{-\ell}
\mathcal{P}(\ell_{\max}) \sim \frac{1}{L}
\Rightarrow \ell_{\max} \sim \ln L / \ln 2 \sim 7.45
2
2
Extreme value statistics: Rare events
Fisher and Young, Phys. Rev. B 58, (1998)
R. Juhász, Y,C. Lin, and F, Iglói, Phys. Rev. B 73, (2006)
N. Pancotti, M. Knap, D. A. Huse, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 97, (2018)
I. A. Kovács, T.Pető, and F.Iglói, Phys. Rev. Res. 3, (2021)
W.-H. Kao and N, B. Perkins, Phys. Rev. B 106, (2022)
and many more
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
NOT GAUSSIAN
Market risks
Athletic records
Extreme floods
Large wildfires
2
Fisher and Young, Phys. Rev. B 58, (1998)
R. Juhász, Y,C. Lin, and F, Iglói, Phys. Rev. B 73, (2006)
N. Pancotti, M. Knap, D. A. Huse, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 97, (2018)
I. A. Kovács, T.Pető, and F.Iglói, Phys. Rev. Res. 3, (2021)
W.-H. Kao and N, B. Perkins, Phys. Rev. B 106, (2022)
and many more
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Market risks
Athletic records
Extreme floods
Large wildfires
NOT GAUSSIAN
Disordered chains
Condensed matter
Extreme value statistics: Rare events
SCOPE
Extreme statistics of local observables
And localization transitions
in disordered chains
SCOPE
Extreme statistics of local observables
And localization transitions
in disordered chains
1. What and why?
2. Warm up: extreme value statistics in an Anderson localized spin chain at high energy
3. Capturing the strong disorder Berezinskii-Kosterlitz-Thouless transition in the XXZ chain
Colbois, Laflorencie, PRB (2023), arXiv:2305.10574
Colbois et al., NJP (2026), arXiv:2512.09029
1. What & Why
Anderson localization in 1 Dimension
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
3
Jump
On-site energy
Anderson localization in 1 Dimension
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
3
Jump
On-site energy
\(|\psi(x)|^2\)
|\phi_i^{m}|^2 \lesssim \exp \left( \frac{|i-i_0^m|}{\xi_m}\right)
\(\forall W , \, \forall E \) : localization !!
(1D, NN)
\(|\psi(x)|^2\)
\(\xi(W, E)\)
Anderson localization in 1 Dimension
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics 10, 107 (1961)
\(W\)
\mathcal{P}(h_i)
\(-W\)
\(W\)
\(|\psi(x)|^2\)
3
Jump
On-site energy
Localization length
E
|\Psi \rangle = | \left\{\phi_m, m \in {\color{#56B4E9}\mathrm{occ}} \right\}\rangle
Slater determinant of Anderson localized orbitals
4
Many non-interacting fermions
\epsilon_m
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
Conserves \(N_f\)
\(\xi(W, E)\)
\epsilon_m
E
|\Psi \rangle = | \left\{\phi_m, m \in {\color{#56B4E9}\mathrm{occ}} \right\}\rangle
Slater determinant of Anderson localized orbitals
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
Conserves \(N_f\)
Many non-interacting fermions
\(\xi(W, E)\)
4
\epsilon_m
E
|\Psi \rangle = | \left\{\phi_m, m \in {\color{#56B4E9}\mathrm{occ}} \right\}\rangle
Slater determinant of Anderson localized orbitals
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
Conserves \(N_f\)
Many non-interacting fermions
\(\xi(W, E)\)
4
High energy eigenstates are like ground states!
5
\(\xi(\epsilon, {\color{orange}W}/J)\)
\(\epsilon\)
Kramer & MacKinnon (1993), Müller & Delande (2011), Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
"Many-body" Localization length (high energy)
5
Kramer & MacKinnon (1993), Müller & Delande (2011), Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
"Many-body" Localization length (high energy)
\(W/J\)
\xi = \frac{1}{\ln\left(1+\left(\frac{{W}}{W_0}\right)^2\right)}
\(\xi(\epsilon, {\color{orange}W}/J)\)
\(\epsilon\)
6
A wave interference phenomenon
Billy, J., Josse, V., Zuo, Z. et al. Nature 453 (2008)
"An example of a real physical system with an infinite number of degrees of freedom [...] in which the approach to equilibrium is simply impossible."
Anderson, 1958
\(n_i = S_i^z + 1/2\)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
Jordan-Wigner
Magnetic field
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
non-local transformation
fermions in random potential = spins in random field
7
\mathcal{P}(h_i)
\(-W\)
\(W\)
\(n_i = S_i^z + 1/2\)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
Jordan-Wigner
Magnetic field
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
non-local transformation
fermions in random potential = spins in random field
7
\mathcal{P}(h_i)
\(-W\)
\(W\)
Charge is conserved
Magnetization is conserved
\(n_i = S_i^z + 1/2\)
Jordan-Wigner
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
Charge is conserved
Magnetization is conserved
non-local transformation
fermions in random potential = spins in random field
\mathcal{P}(h_i)
\(-W\)
\(W\)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
Magnetic field
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{lightgreen}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}
Attraction / repulsion
7
What happens when you add interactions?
\(\epsilon= 1\)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
8
\(\epsilon= 0\)
What happens when you add interactions?
\(\epsilon= 1\)
High energy / finite temperature:
Fleischman, Anderson, (1980); Altshuler, et al (1997); Gornyi et al (2005); Basko, Aleiner, Altshuler (2006)
...and a whole field!... Most recent review: Sierant et al., Rep. Progr. Phys (2025)
?
?
?
?
High energy
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
\(\epsilon= 0\)
8
What happens when you add interactions?
\(\epsilon= 1\)
?
?
?
?
High energy
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
\(\epsilon= 0\)
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
8
High energy / finite temperature:
Fleischman, Anderson, (1980); Altshuler, et al (1997); Gornyi et al (2005); Basko, Aleiner, Altshuler (2006)
...and a whole field!...
Most recent review: Sierant et al., Rep. Progr. Phys (2025)
What happens when you add interactions?
\(\epsilon= 1\)
?
?
?
?
High energy
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
Absence of thermalization?
- Role of rare events: rare resonances.
- Chain breakings from local density?
\(\epsilon= 0\)
8
High energy / finite temperature:
Fleischman, Anderson, (1980); Altshuler, et al (1997); Gornyi et al (2005); Basko, Aleiner, Altshuler (2006)
...and a whole field!...
Most recent review: Sierant et al., Rep. Progr. Phys (2025)
What happens when you add interactions?
\(\epsilon= 1\)
High energy / finite temperature:
Fleischman, Anderson, (1980); Altshuler, et al (1997); Gornyi et al (2005); Basko, Aleiner, Altshuler (2006)
...and a whole field!...
Most recent review: Sierant et al., Rep. Progr. Phys (2025)
?
?
?
?
High energy
XX chain
Ground state
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);
Altman et al (2004, 2008); Refael & Altman (2013), Pielawa & Altman (2013); Pollet, Prokof'ev, Svistunov (2013, 2014), Doggen et al (2017), Pfeffer et al (2019)...
(and many more)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
8
What happens when you add interactions?
\(\epsilon= 1\)
?
?
?
?
High energy
XX chain
Ground state
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988);
Altman et al (2004, 2008); Refael & Altman (2013), Pielawa & Altman (2013); Pollet, Prokof'ev, Svistunov (2013, 2014), Doggen et al (2017), Pfeffer et al (2019)...
(and many more)
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z} + (H_{BC})
ISING INTERACTION
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
8
High energy / finite temperature:
Fleischman, Anderson, (1980); Altshuler, et al (1997); Gornyi et al (2005); Basko, Aleiner, Altshuler (2006)
...and a whole field!...
Most recent review: Sierant et al., Rep. Progr. Phys (2025)
9
Distribution of the local observable
Access to local density of charges or local magnetization
Can we tell that the system is localized?
9
Distribution of the local observable
J. C., N. Laflorencie, PRB (2023)
Can we tell that the system is localized?
Access to local density of charges or local magnetization
\(|\langle S_i^{z}\rangle| < 1/2\)
9
Distribution of the local observable
No localization transition
J. C., N. Laflorencie, PRB (2023)
Localization transition at \(W\) = 0.38
J. C., et al, NJP (2026)
Laflorencie et al. PRB (2020),
see also Khemani et al. ; Pollmann et al.
Can we tell that the system is localized?
High energy; transition at \(W \sim 3.5\) for these sizes
Access to local density of charges or local magnetization
9
Distribution of the local observable
No localization transition
J. C., N. Laflorencie, PRB (2023)
Localization transition at \(W\) = 0.38
J. C., et al, NJP (2026)
Laflorencie et al. PRB (2020),
see also Khemani et al. ; Pollmann et al.
Can we tell that the system is localized?
High energy; transition at \(W \sim 3.5\) for these sizes
Access to local density of charges or local magnetization
9
Distribution of the local observable
No localization transition
J. C., N. Laflorencie, PRB (2023)
Localization transition at \(W\) = 0.38
J. C., et al, NJP (2026)
Laflorencie et al. PRB (2020),
see also Khemani et al. ; Pollmann et al.
Can we tell that the system is localized?
High energy; transition at \(W \sim 3.5\) for these sizes
Access to local density of charges or local magnetization
9
Distribution of the local observable
1. Localization can have a signature in the tails
2. Extreme value theory relates tails of the distribution to extreme value statistics.
No localization transition
Localization transition at \(W\) = 0.38
High energy; transition at \(W \sim 3.5\) for these sizes
Laflorencie et al. PRB (2020),
see also Khemani et al. ; Pollmann et al.
J. C., N. Laflorencie, PRB (2023)
J. C., et al, NJP (2026)
Can we tell that the system is localized?
Access to local density of charges or local magnetization
Extreme value statistics
in the anderson chain at high energy
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{orange} h_i S_i^z}
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Spin-flip
Magnetic field
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}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}
(= no interactions)
Extreme values capture the localization length
Extreme magnetization
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
Extreme magnetization
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
Extreme magnetization
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
Extreme magnetization
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
Extreme magnetization
exponential
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
Extreme magnetization
exponential
SPIN FREEZING!
10
\(|\langle S_i^{z}\rangle| < 1/2\)
J. C., N. Laflorencie, PRB (2023)
Some eigenstate
\delta_i = 1/2 -| \langle S_i^z \rangle|
SPIN FREEZING!
Extreme magnetization
exponential
10
\(|\langle S_i^{z}\rangle| < 1/2\)
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
11
A toy model of spin freezing
\(\xi_1\) \(\xi_2\)
\(\xi_3\) \(\xi_4\) ...
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
A toy model of spin freezing
11
\(\xi\) \(\xi\)
\(\xi\) \(\xi\)...
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
r
\ell_{\mathrm{cluster}}
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\ell_{\mathrm{cluster}}
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\ell_{\mathrm{cluster}}
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min}= e^{\overline{\ln\delta_{\min}}} \propto e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} + ...
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\ell_{\mathrm{cluster}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min}= e^{\overline{\ln\delta_{\min}}} \propto e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} + ...
\delta^{\mathrm{typ}}_{\min} \propto L^{-\frac{1}{2\xi \ln2}}
A toy model of spin freezing
11
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}} |\phi_m(i)|^2
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\ell_{\mathrm{cluster}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \propto L^{-\frac{1}{2\xi \ln2}}
\Rightarrow \quad \delta_{\min} \propto e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}} + ...
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min}= e^{\overline{\ln\delta_{\min}}} \propto e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} + ...
Power-law decay
controlled by the localization length
A toy model of spin freezing
11
12
THe extreme magnetization captures the localization length
PBC, \(10^4\) samples, XX chain, high energy
JC, N. Laflorencie, PRB (2023)
\delta^{\mathrm{typ}}_{\min} \propto L^{-\gamma_{\mathrm{typ}}} = L^{-\frac{1}{2\xi \ln2}}
power-law decay
controlled by the localization length
12
THe extreme magnetization captures the localization length
PBC, \(10^4\) samples, XX chain, high energy
"Many-body" Anderson localization length
JC, N. Laflorencie, PRB (2023)
\delta^{\mathrm{typ}}_{\min} \propto L^{-\gamma_{\mathrm{typ}}} = L^{-\frac{1}{2\xi \ln2}}
power-law decay
controlled by the localization length
THe extreme magnetization captures the localization length
PBC, \(10^4\) samples, XX chain, high energy
\delta^{\mathrm{typ}}_{\min} \propto L^{-\gamma_{\mathrm{typ}}} = L^{-\frac{1}{2\xi \ln2}}
power-law decay
controlled by the localization length
ED
1/\gamma_{\mathrm{typ}}
"Many-body" Anderson localization length
JC, N. Laflorencie, PRB (2023)
12
THe extreme magnetization captures the localization length
PBC, \(10^4\) samples, XX chain, high energy
ED
1/\gamma_{\mathrm{typ}}
"Many-body" Anderson localization length
JC, N. Laflorencie, PRB (2023)
12
\delta^{\mathrm{typ}}_{\min} \propto L^{-\gamma_{\mathrm{typ}}} = L^{-\frac{1}{2\xi \ln2}}
power-law decay
controlled by the localization length
Difference at weak disorder: largest cluster
THe extreme magnetization captures the localization length
PBC, \(10^4\) samples, XX chain, high energy
ED
1/\gamma_{\mathrm{typ}}
"Many-body" Anderson localization length
Difference at weak disorder: largest cluster
JC, N. Laflorencie, PRB (2023)
12
\delta^{\mathrm{typ}}_{\min} \propto L^{-\gamma_{\mathrm{typ}}} = L^{-\frac{1}{2\xi \ln2}}
power-law decay
controlled by the localization length
2. Adding interactions:
Capturing a BKT scaling in the random-FIeld XXZ chain
ATTRACTION/REPULSION
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{lightgreen}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}
2. Adding interactions:
Capturing a BKT scaling in the random-FIeld XXZ chain
ATTRACTION/REPULSION
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightblue}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{lightgreen}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{orange}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 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
13
Ground state phase diagram: Two limits
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
Localization
Inhomogeneous
Gapless
Compressible
Exponentially decaying correlations
13
Ground state phase diagram: Two limits
Luttinger liquid, \(K\)
Quasi-long-range ordered
Finite superfluid stiffness
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
13
Ground state phase diagram: Two limits
Luttinger liquid, \(K\)
Quasi-long-range ordered
Finite superfluid stiffness
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
13
\begin{align*}
\langle S^{z}(x,0)\,S^{z}(0,0)\rangle &= C_1\,\frac{1}{x^{2}} + {\color{orange}C_2\,(-1)^{x}\left(\frac{1}{x}\right)^{2K}} \\[6pt]
\langle S^{+}(x,0)\,S^{-}(0,0)\rangle &= {\color{lightblue}C_3\left(\frac{1}{x}\right)^{2K+\frac{1}{2K}} + C_4\,(-1)^{x}\left(\frac{1}{x}\right)^{\frac{1}{2K}}}
\end{align*}
Ground state phase diagram: Two limits
Luttinger liquid, \(K\)
Quasi-long-range ordered
Finite superfluid stiffness
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
13
\begin{align*}
\langle S^{z}(x,0)\,S^{z}(0,0)\rangle &= C_1\,\frac{1}{x^{2}} + {\color{orange}C_2\,(-1)^{x}\left(\frac{1}{x}\right)^{2K}} \\[6pt]
\langle S^{+}(x,0)\,S^{-}(0,0)\rangle &= {\color{lightblue}C_3\left(\frac{1}{x}\right)^{2K+\frac{1}{2K}} + C_4\,(-1)^{x}\left(\frac{1}{x}\right)^{\frac{1}{2K}}}
\end{align*}
\(\leftarrow K\rightarrow\infty\)
"Almost" a superfluid
Ground state phase diagram: Two limits
Luttinger liquid, \(K\)
Quasi-long-range ordered
Finite superfluid stiffness
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightblue}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{lightgreen} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{orange} h_i S_i^z}
13
\(\leftarrow K\rightarrow\infty\)
\(K\rightarrow 1/2 \rightarrow\)
\begin{align*}
\langle S^{z}(x,0)\,S^{z}(0,0)\rangle &= C_1\,\frac{1}{x^{2}} + {\color{orange}C_2\,(-1)^{x}\left(\frac{1}{x}\right)^{2K}} \\[6pt]
\langle S^{+}(x,0)\,S^{-}(0,0)\rangle &= {\color{lightblue}C_3\left(\frac{1}{x}\right)^{2K+\frac{1}{2K}} + C_4\,(-1)^{x}\left(\frac{1}{x}\right)^{\frac{1}{2K}}}
\end{align*}
"Almost" a superfluid
\(\Delta \rightarrow 1\)
"Almost" a density wave
Ground state phase diagram: Two limits
General refs: Lee and Ramakrishan RMP (85)
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988)
\(K < 3/2\)
Ground state phase diagram
14
Doggen et al (2017)
General refs: Lee and Ramakrishan RMP (85)
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988)
\(K < 3/2\)
Ground state phase diagram
14
Doggen et al (2017)
General refs: Lee and Ramakrishan RMP (85)
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988)
\(K < 3/2\)
Ground state phase diagram
14
Doggen et al (2017)
General refs: Lee and Ramakrishan RMP (85)
Giamarchi & Schulz EPL 3 (1987); PRB 37, (1988)
\(K < 3/2\)
\(K\) decreases to \(K_c = 3/2\)
Ground state phase diagram
14
Doggen et al (2017)
Superfluid gets worse everywhere at once until it breaks
\(K_c > 3/2\), non-universal
Ground state phase diagram
14
Doggen et al (2017)
Superfluid breaks locally
*Take into account a renormalization of the weak-links! (Kane-Fisher mechanism)
General refs: Lee and Ramakrishan RMP (85)
Altman et al (2004, 2008); Refael & Altman (2013), Pielawa & Altman (2013); Pollet, Prokof'ev, Svistunov (2013, 2014), Pfeffer et al (2019)... (and many more)
extreme statistics at the weak-link transition
JC, et al, NJP (2026)
15
extreme statistics at the weak-link transition
JC, et al, NJP (2026)
15
10k samples for \(L \leq 64\),
5k samples for \(64 \leq L \leq 128\)
OBC - eliminate the edges
Fixed U(1) sector
\( W = 0.7\)
Weak-link transition: Two regimes
16
JC, et al, NJP (2026)
Weak-link transition: Two regimes
JC, et al, NJP (2026)
16
(log-log)
Power-law
Weak-link transition: Two regimes
JC, et al, NJP (2026)
16
(log-log)
(log-log)
Power-law
Weak-link transition: Two regimes
JC, et al, NJP (2026)
16
(log-log)
(log-log)
Power-law + const.
Power-law
Weak-link transition: Two regimes
JC, et al, NJP (2026)
16
(log-log)
(log-log)
Power-law + const.
Power-law
Find the transition:
1. where does the constant go to zero?
2. where does an effective power-law exponent start to decrease?
Weak-link transition: Two regimes
JC, et al, NJP (2026)
16
(log-log)
(log-log)
Power-law + const.
Power-law
Find the transition:
1. where does the constant go to zero?
2. where does an effective power-law exponent start to decrease?
Locating the transition
17
JC, et al, NJP (2026)
Locating the transition
17
JC, et al, NJP (2026)
Dashed: \(L \in [14,28]\)
Full: \(L\in [48,100]\)
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\rm{eff}} \ln(L)
Locating the transition
17
JC, et al, NJP (2026)
Dashed: \(L \in [14,28]\)
Full: \(L\in [48,100]\)
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\rm{eff}} \ln(L)
Locating the transition
17
JC, et al, NJP (2026)
Dashed: \(L \in [14,28]\)
Full: \(L\in [48,100]\)
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\rm{eff}} \ln(L)
Capturing the Scaling?
18
Capturing the Scaling?
18
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{-(\gamma_c - \gamma(W))\ln L}
JC, et al, NJP (2026) (see also Garcia Mata et al. (2022) for scaling laws)
Capturing the Scaling?
18
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{-(\gamma_c - \gamma(W))\ln L}
\ln(\delta^{\rm typ}_{\rm min}) - \ln(\delta^{\rm typ}_{\rm min}(W_c)) = f\left( \frac{\ln L}{\lambda}\right)
Neglecting log. corrections to scaling related to \(A(W) \neq A(W_c)\):
JC, et al, NJP (2026) (see also Garcia Mata et al. (2022) for scaling laws)
Capturing the Scaling?
18
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{-(\gamma_c - \gamma(W))\ln L}
\ln(\delta^{\rm typ}_{\rm min}) - \ln(\delta^{\rm typ}_{\rm min}(W_c)) = f\left( \frac{\ln L}{\lambda}\right)
Neglecting log. corrections to scaling related to \(A(W) \neq A(W_c)\):
\lambda \sim (W-W_c)^{-\nu}
JC, et al, NJP (2026) (see also Garcia Mata et al. (2022) for scaling laws)
Capturing the Scaling?
18
JC, et al, NJP (2026) (see also Garcia Mata et al. (2022) for scaling laws)
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{-(\gamma_c - \gamma(W))\ln L}
\ln(\delta^{\rm typ}_{\rm min}) - \ln(\delta^{\rm typ}_{\rm min}(W_c)) = f\left( \frac{\ln L}{\lambda}\right)
Neglecting log. corrections to scaling related to \(A(W) \neq A(W_c)\):
\lambda \sim (W-W_c)^{-\nu}
Capturing the Scaling?
18
JC, et al, NJP (2026) (see also Garcia Mata et al. for scaling laws)
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{-(\gamma_c - \gamma(W))\ln L}
\ln(\delta^{\rm typ}_{\rm min}) - \ln(\delta^{\rm typ}_{\rm min}(W_c)) = f\left( \frac{\ln L}{\lambda}\right)
Neglecting log. corrections to scaling related to \(A(W) \neq A(W_c)\):
\lambda \sim (W-W_c)^{-\nu}
\(\nu \sim 0.67\), non-trivial vanishing of \(\gamma -\gamma_c\)
19
JC, et al, NJP (2026)
Overall: extreme value statistics of local observables
19
In the random field XX chain:
JC, et al, NJP (2026)
Localization length
Overall: extreme value statistics of local observables
19
JC, et al, NJP (2026)
In the random-field spin chain (XXZ)
In the random field XX chain:
Captures a weak-link transition
Localization length
Overall: extreme value statistics of local observables
Overall: extreme value statistics of local observables
19
Transparent to the Giamarchi-Schulz transition
\(\rightarrow\) open questions:
strong finite size effects vs. different regime?
JC, et al, NJP (2026)
What I didn't cover :
Captures the ergodic to many-body localized crossover at high energy
In the random-field spin chain (XXZ)
In the random field XX chain:
Captures a weak-link transition
!
Localization length
19
JC, et al, NJP (2026)
What next?
Aubry-André : quasiperiodic
More realistic conditions & models (Rydberg, NMR?), other observables?
Captures the ergodic to many-body localized crossover at high energy
Dynamics
What I didn't cover :
In the random-field spin chain (XXZ)
In the random field XX chain:
Captures a weak-link transition
Localization length
!
Overall: extreme value statistics of local observables
Transparent to the Giamarchi-Schulz transition
\(\rightarrow\) open questions:
strong finite size effects vs. different regime?
Take-Home message
20
Tail of the distribution
Extreme events
Extreme value theory
Text
Take-Home message
20
- chain breaking vs long-range resonances in localization
Tail of the distribution
Extreme events
Extreme value theory
Text
When rare events / impurities dominate the physics
JC, Laflorencie, PRB (2023)
JC, et al, NJP (2026)
Take-Home message
20
Tail of the distribution
Extreme events
Extreme value theory
Text
When rare events / impurities dominate the physics
JC, Laflorencie, PRB (2023)
JC, et al, NJP (2026)
- chain breaking vs long-range resonances in localization
- domain walls / string defects/...?
- Nagaoka / counter-Nagaoka effect ?
....
Take-Home message
20
Tail of the distribution
Extreme events
Thank you for your attention!
Extreme value theory
Text
JC, Laflorencie, PRB (2023)
JC, et al, NJP (2026)
- chain breaking vs long-range resonances in localization
- domain walls / string defects/...?
- Nagaoka / counter-Nagaoka effect ?
....
When rare events / impurities dominate the physics
Supplemental
Extreme magnetizations capture entanglement bottlenecks
Deviation
Entanglement
Site
Dupont, Macé, Laflorencie (PRB 2019)
Extreme magnetizations capture entanglement bottlenecks
Deviation
Entanglement
Site
Dupont, Macé, Laflorencie (PRB 2019)
Distributions and tails
JC, et al (NJP 2026)
W = 1.6
High energy
Ground state
Distributions and tails
JC, et al (NJP 2026)
Locating the transition, V2
JC, et al (NJP 2026)
Extreme statistics at high energy
Eigenstate thermalization hypothesis: variance of local observables goes to zero
\(\min_i \delta_i\) \(\rightarrow\) 1/2
Laflorencie et al (2020);
see Aramthottil et al (2025) for important remarks about the energy window selection
Power-law decay:
\(\min_i \delta_i \rightarrow 0 \) (?)
Typical minimal deviation at high energy
XX chain (Anderson)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
Typical minimal deviation at high energy
XX chain (Anderson)
disorder
increases
JC, N. Laflorencie, PRB (2023)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
disorder
increases
Heisenberg
Typical minimal deviation at high energy
JC, N. Laflorencie, PRB (2023)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
disorder
increases
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
Heisenberg
Follows Anderson at strong disorder
(for these sizes)
Text
Typical minimal deviation at high energy
JC, N. Laflorencie, PRB (2023)
disorder
increases
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
Heisenberg
Follows Anderson at strong disorder
(for these sizes)
Text
Typical minimal deviation at high energy
Deviates from power-law at weak disorder
JC, N. Laflorencie, PRB (2023)
Effective exponent
Heisenberg
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\mathrm{typ}}(W) \ln(L)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
Effective exponent
Heisenberg
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\mathrm{typ}}(W) \ln(L)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
Effective exponent
Heisenberg
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\mathrm{typ}}(W) \ln(L)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
Effective exponent
Heisenberg
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\mathrm{typ}}(W) \ln(L)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\(W\)
Effective exponent
Heisenberg
\ln(\delta^{\mathrm{typ}}_{\min}) \propto -\gamma_{\mathrm{typ}}(W) \ln(L)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, PRB (2023)
\(W\)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
Scaling
If the decay is power-law at the critical point:
\frac{\delta^{\rm{typ}}_{\rm min}(W)}{\delta^{\rm{typ}}_{\rm min}(W_c)} = \frac{A(W)}{A(W_c)}e^{(-\gamma(W_c) - \gamma(W))\ln L}
\ln(\delta^{\rm typ}_{\rm min}) - \ln(\delta^{\rm typ}_{\rm min}(W_c)) = f\left( \frac{\ln L}{\lambda}\right)
Neglecting the log. corrections to scaling related to \(A(W) \neq A(W_c)\):
Exponent comparison
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
- \frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi} -c
The power-law exponent \(\gamma\) is not purely controlled by the localization length.
In the ground state
Correlations across the most polarized site
Localization from extreme value statistics in random spin chains
By Jeanne Colbois
Localization from extreme value statistics in random spin chains
Invited talk at NGSCES 2026
