Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Jeanne Colbois, Nicolas Laflorencie | LPT Toulouse
NanoX-Fermi Days 2023 | 06.04.2023
Extreme statistics
IN
RANDOM spin chains
Tossing a coin
1
J. COLBOIS | NANOX-FERMI | 04.2023
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
Tossing a coin
2
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
J. COLBOIS | NANOX-FERMI | 04.2023
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
Tossing a coin
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
J. COLBOIS | NANOX-FERMI | 04.2023
2
Tossing a coin
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
J. COLBOIS | NANOX-FERMI | 04.2023
2
\mathcal{P}(\ell) \sim 2^{-\ell}
\mathcal{P}(\ell_{\max}) \sim \frac{1}{L}
Tossing a coin
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
J. COLBOIS | NANOX-FERMI | 04.2023
2
\mathcal{P}(\ell) \sim 2^{-\ell}
\mathcal{P}(\ell_{\max}) \sim \frac{1}{L}
\ell_{\max} \sim \ln L / \ln 2 \\ \sim 7.45
\Rightarrow
Extreme value theory
3
J. COLBOIS | NANOX-FERMI | 04.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Extreme value theory
3
J. COLBOIS | NANOX-FERMI | 04.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Extreme floods
Extreme value theory
3
J. COLBOIS | NANOX-FERMI | 04.2023
Market risks
Extreme floods
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Extreme value theory
3
J. COLBOIS | NANOX-FERMI | 04.2023
Athletic records
Market risks
Extreme floods
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Extreme value theory
3
J. COLBOIS | NANOX-FERMI | 04.2023
Athletic records
Market risks
Extreme floods
Large wildfires
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Extreme value theory
3
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | NANOX-FERMI | 04.2023
Spin chains
Athletic records
Market risks
Extreme floods
Large wildfires
Condensed matter
R. Juhász, Y,C. Lin, and F, Iglói, “S Phys. Rev. B 73, 224206 (2006)
I. A. Kovács, T.Pető, and F.Iglói, Phys. Rev. Res. 3, 033140 (2021)
W.-H. Kao and N, B. Perkins, Phys. Rev. B 106, L100402 (2022)
4
J. COLBOIS | NANOX-FERMI | 04.2023
h
Spin chain in A Random magnetic field
ED result - "typical" eigenstate
Spin chain in A Random magnetic field
4
J. COLBOIS | NANOX-FERMI | 04.2023
h
Maximal magnetization
4
J. COLBOIS | NANOX-FERMI | 04.2023
h
Maximal magnetization
Where?
How?
Consequences?
Spin chain in A Random magnetic field
Scope
5
1. Spin chain in random field and localization
2. Chain breaking
3. Extreme values & Chain breaking
J. COLBOIS | NANOX-FERMI | 04.2023
Spin chain in Random field and localization
Spin-1/2 chain in a random field
6
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left(S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L h_i S_i^z
Spin-1/2 chain in a random field
6
J. COLBOIS | NANOX-FERMI | 04.2023
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left(S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L h_i S_i^z
Spin-1/2 chain in a random field
J. COLBOIS | NANOX-FERMI | 04.2023
Spin-flip
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
6
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L h_i S_i^z
Spin-1/2 chain in a random field
J. COLBOIS | NANOX-FERMI | 04.2023
Spin-flip
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
6
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L h_i S_i^z
Spin-1/2 chain in a random field
J. COLBOIS | NANOX-FERMI | 04.2023
Spin-flip
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
6
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L h_i S_i^z
-1/2 < \langle S_i^z \rangle < 1/2
Spin-1/2 chain in a random field
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
J. COLBOIS | NANOX-FERMI | 04.2023
Magnetic field
7
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
h
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + 2\Delta S_i^z S_{i+1}^z\right) - \sum_{i=1}^L {\color{#20B2AA} h_i S_i^z}
Spin-flip
Spin-1/2 chain in a random field
Ising interaction
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H} = \sum_{i = 1}^{L-1} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#FF4500} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i=1}^L {\color{#20B2AA} h_i S_i^z}
Spin-flip
Magnetic field
7
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
From spins to fermions
8
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left(c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}+2 \Delta n_i n_{i+1} \right) -h_i n_i\Bigr]
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
8
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+2 \Delta n_i n_{i+1} \right) -h_i n_i\Bigr]
Jump
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
8
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+2 \Delta n_i n_{i+1} \right) -h_i n_i\Bigr]
Jump
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+2 \Delta n_i n_{i+1} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
On-site energy
8
Jump
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
On-site energy
8
Jump
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
Attraction/ repulsion
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
On-site energy
8
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
Attraction/ repulsion
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
On-site energy
8
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
Attraction/ repulsion
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
From spins to fermions
J. COLBOIS | NANOX-FERMI | 04.2023
On-site energy
8
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
Attraction/ repulsion
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
Anderson Localization
10
J. COLBOIS | NANOX-FERMI | 04.2023
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
10
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
1 particle
Anderson Localization
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(L \times L \) Tridiagonal matrix \(\rightarrow\) large systems
10
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
1 particle
Anderson Localization
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(L \times L \) Tridiagonal matrix \(\rightarrow\) large systems
\mathcal{H}_f = \sum_{m} \epsilon_m |\phi_m \rangle \langle \phi_m |
\epsilon_m
Anderson Localization
J. COLBOIS | NANOX-FERMI | 04.2023
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
11
h_i = h \,\, \forall i
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
|\phi_m \rangle
Anderson Localization
J. COLBOIS | NANOX-FERMI | 04.2023
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
11
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
h_i = h \,\, \forall i
\epsilon_m
|\phi_m \rangle
Anderson Localization
J. COLBOIS | NANOX-FERMI | 04.2023
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
h_i = h \,\, \forall i
\epsilon_m
- Wave interference
- Absence of diffusion
11
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
{\xi}
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
12
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
{\xi}(E)
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
12
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
{\xi}(E, {\color{#20B2AA}W})
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
12
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
JC, Laflorencie, in preparation
{\xi}({\color{#20B2AA}W})
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
12
\mathcal{P}(h_i) = \begin{cases}
\frac{1}{2W} & \text{ if } h_i \in [-W, W]\\
0 & \text{otherwise}
\end{cases}
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\xi = \frac{1}{\ln\left(1+\left(\frac{{\color{#20B2AA}W}}{W_0}\right)^2\right)}
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
{\xi}({\color{#20B2AA}W})
JC, Laflorencie, in preparation
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
12
J. COLBOIS | NANOX-FERMI | 04.2023
Localization length
1 particle
\xi \ll 1
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
{\xi}({\color{#20B2AA}W})
JC, Laflorencie, in preparation
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
\xi = \frac{1}{\ln\left(1+\left(\frac{{\color{#20B2AA}W}}{W_0}\right)^2\right)}
12
MANY Non-interacting fermions
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
13
\epsilon_m
\sum_i \langle n_i \rangle= N_f
MANY Non-interacting fermions
J. COLBOIS | NANOX-FERMI | 04.2023
\sum_i \langle n_i \rangle= N_f
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
\epsilon_m
13
\sum_i \langle n_i \rangle= N_f
Chain breaking, Spin freezing
14
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
ED on one sample
Some eigenstate
ED on one sample
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
14
Some eigenstate
ED on one sample
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
14
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
14
\delta_i = 1/2 -| \langle S_i^z \rangle|
ED on one sample
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
14
\delta_i = 1/2 -| \langle S_i^z \rangle|
ED on one sample
Some eigenstate
14
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
\delta_i = 1/2 -| \langle S_i^z \rangle|
ED on one sample
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
Chain Breaking
JC, Laflorencie, in preparation
\delta_i = 1/2 -| \langle S_i^z \rangle|
15
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
Chain Breaking
JC, Laflorencie, in preparation
\delta_i = 1/2 -| \langle S_i^z \rangle|
15
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
Chain Breaking
JC, Laflorencie, in preparation
\delta_i = 1/2 -| \langle S_i^z \rangle|
15
CHAIN BREAKING!
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
Chain Breaking
JC, Laflorencie, in preparation
\delta_i = 1/2 -| \langle S_i^z \rangle|
15
CHAIN BREAKING!
Some eigenstate
J. COLBOIS | NANOX-FERMI | 04.2023
Toy model
16
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
J. COLBOIS | NANOX-FERMI | 04.2023
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Toy model
16
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
J. COLBOIS | NANOX-FERMI | 04.2023
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Toy model
|\Psi \rangle = \otimes_{m \in {\color{#56B4E9}\mathrm{occ}}} | \phi_m\rangle \Rightarrow \langle n_i \rangle = \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, Laflorencie, in preparation
16
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
17
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
:
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
17
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
:
r
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
17
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
:
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
r
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\ell_{\max}
17
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
:
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
r
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\ell_{\max}
17
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
r
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
\ell_{\max}
17
expecTations
18
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\overline{\ell_{\max}} \approx \frac{\ln L}{ \ln 2}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
r
\ell_{\max}
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\overline{\ell_{\max}} \approx \frac{\ln L}{ \ln 2}
\Rightarrow
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
r
\ell_{\max}
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
18
expecTations
J. COLBOIS | NANOX-FERMI | 04.2023
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\overline{\ell_{\max}} \approx \frac{\ln L}{ \ln 2}
\Rightarrow
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
r
\ell_{\max}
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\max}}}{2\xi}}
18
EXPONENT
19
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
=2\xi \ln2
\xi = \frac{1}{\ln\left(1+\left(\frac{{\color{#20B2AA}W}}{W_0}\right)^2\right)}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
ED results
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
\(10^4\) samples, \(L = 512\)
19
ED results
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
\(10^4\) samples, \(L = 512\)
19
Spin Freezing
J. COLBOIS | NANOX-FERMI | 04.2023
20
JC, Laflorencie, in preparation
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma(W)}
Spin Freezing
J. COLBOIS | NANOX-FERMI | 04.2023
20
JC, Laflorencie, in preparation
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma(W)}
Spin Freezing
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
20
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma(W)}
Spin Freezing
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
20
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma(W)}
EXPONENTS
J. COLBOIS | NANOX-FERMI | 04.2023
21
JC, Laflorencie, in preparation
EXPONENTS
J. COLBOIS | NANOX-FERMI | 04.2023
ED
JC, Laflorencie, in preparation
21
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
1/\gamma_{\mathrm{typ}}
EXPONENTS
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
Excellent agreement ED - Toy model !
21
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
ED
1/\gamma_{\mathrm{typ}}
Extreme Values statistics & Chain breaking
J. COLBOIS | NANOX-FERMI | 04.2023
22
J. COLBOIS | NANOX-FERMI | 04.2023
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
Power-Law Tails
Power-Law Tails
J. COLBOIS | NANOX-FERMI | 04.2023
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
22
Extreme value theory
J. COLBOIS | NANOX-FERMI | 04.2023
\int_0^{\delta_{\min}(L)} \mathcal{P}(\delta)\, \mathrm{d} \delta \sim \frac{1}{L}
JC, Laflorencie, in preparation
23
23
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory
\int_0^{\delta_{\min}(L)} \mathcal{P}(\delta)\, \mathrm{d} \delta \sim \frac{1}{L} \Rightarrow
\delta_{\mathrm{\min}}(L) \sim L^{-\frac{1}{1+\alpha}}
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet \(\rightarrow \mathcal{P}(\ln\delta_{\min}) \)
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
\int_0^{\delta_{\min}(L)} \mathcal{P}(\delta)\, \mathrm{d} \delta \sim \frac{1}{L} \Rightarrow
\delta_{\mathrm{\min}}(L) \sim L^{-\frac{1}{1+\alpha}}
Extreme value theory
23
\(10^5 \) samples
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet \(\rightarrow \mathcal{P}(\ln\delta_{\min}) \)
\int_0^{\delta_{\min}(L)} \mathcal{P}(\delta)\, \mathrm{d} \delta \sim \frac{1}{L} \Rightarrow
\delta_{\mathrm{\min}}(L) \sim L^{-\frac{1}{1+\alpha}}
Extreme value theory
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
\delta_{\mathrm{\min}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
23
\(10^5 \) samples
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet \(\rightarrow \mathcal{P}(\ln\delta_{\min}) \)
\int_0^{\delta_{\min}(L)} \mathcal{P}(\delta)\, \mathrm{d} \delta \sim \frac{1}{L} \Rightarrow
\delta_{\mathrm{\min}}(L) \sim L^{-\frac{1}{1+\alpha}}
Extreme value theory
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
\delta_{\mathrm{\min}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
23
\(10^5 \) samples
Exponents
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
24
ED
1/\gamma_{\mathrm{typ}}
Exponents
J. COLBOIS | NANOX-FERMI | 04.2023
JC, Laflorencie, in preparation
24
1+\alpha
ED
1/\gamma_{\mathrm{typ}}
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
25
Maximal magnetization
Where
How
Consequences
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
25
Maximal magnetization
Where
How
Consequences
\ell_{\max}
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
25
Maximal magnetization
Where
How
Consequences
\ell_{\max}
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
25
Maximal magnetization
Where
How
Consequences
\ell_{\max}
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
25
Maximal magnetization
Where
How
Consequences
\ell_{\max}
+ Attraction/ repulsion
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}}\\
+ c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019)
F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, in preparation
Conclusion
J. COLBOIS | NANOX-FERMI | 04.2023
Anderson localization: Introduction and known results | Dominique Delande | SOAL 2020
From one extreme to another: the statistics of extreme events | Jon Keating | Oxford Mathematics Public Lectures
J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer (1999)
F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008)
D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019)
F. Alet, N. Laflorencie,
C. R. Phys. 19,498 (2018)
Extreme
Values
Anderson
MBL
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
25
Where
How
Consequences
\ell_{\max}
\mathcal{H}_f = \sum_{i} \Bigl[\frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}}\\
+ c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{#FF4500}2 \Delta n_i n_{i+1}} \right) -{\color{#20B2AA}h_i n_i}\Bigr]
Attraction/ repulsion
good question :D
Many interacting fermions...
J. COLBOIS | NANOX-FERMI | 04.2023
F. Alet, N. Laflorencie, C. R. Phys. 19,498 (2018)
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
Power Law Tails
J. COLBOIS | NANOX-FERMI | 04.2023
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} \delta^{\alpha}
\mathcal{P}(\ln \delta) \overset{\delta \rightarrow 0}{\sim} e^{(1+\alpha)\ln\delta}
JC, Laflorencie, in preparation
- XX Chain: all localized
- XXZ : Adding interactions.
In the Anderson basis
H = \sum_m \epsilon_m b_m^{\dagger} b_m + \sum_{j,k,l,m} V_{j,k,l,m} b_j^{\dagger} b_k^{\dagger} b_l b_m
N. Laflorencie, slides
J. COLBOIS | NANOX-FERMI | 04.2023
From XX to XXZ
Anderson vs Many-body localization
Anderson
Many body
No spreading of entanglement
Log spreading of entanglement
J. H. Bardarson, F. Pollmann, and J. E. Moore Phys. Rev. Lett. 109, 017202
J. COLBOIS | NANOX-FERMI | 04.2023
Distributions of minimal deviations
JC, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
Distributions of minimal deviations
JC, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
Cluster lengths
JC, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
Consequences for MBL?
JC, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
Magnetization
XX
XXZ
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
XX vs XXZ
J. C, Laflorencie, In preparation
J. COLBOIS | NANOX-FERMI | 04.2023
Extreme Statistics in Random spin chains
By Jeanne Colbois
Extreme Statistics in Random spin chains
Presentation for the NanoX-Fermi days 2023 in Toulouse
