Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Quantum information, quantum matter and quantum gravity | Kyoto | 27.09.2023
Extreme Value Theory
And
LocalIZation IN RANDOM spin chains
Jeanne Colbois
Nicolas Laflorencie
LPT | CNRS & Toulouse University | France
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Tossing a coin
1
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
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Tossing a coin
2
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 | QIMG 2023 , KYOTO | 27.09.2023
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)
\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
Extreme value theory
Athletic records
Market risks
Extreme floods
Large wildfires
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
3
Extreme value theory
Athletic records
Market risks
Extreme floods
Large wildfires
Condensed matter
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
3
Extreme value theory
Disordered spin chains
Athletic records
Market risks
Extreme floods
Large wildfires
Condensed matter
R. Juhász, Y,C. Lin, and F, Iglói, Phys. Rev. B 73, 224206 (2006)
N. Pancotti, M. Knap, D. A. Huse, J. I. Cirac, and M. C. Bañuls, Phys. Rev. B 97, 094206 (2018)
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)
J. C., N. Laflorencie, arXiv:2305.10574
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
3
Summary : Spin-1/2 chain in random field
4
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
4
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
Heisenberg chain
XX chain ("many-body Anderson")
Summary : Spin-1/2 chain in random field
4
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Spin-1/2 \(W = h = \) disorder strength for random fields along \(S^z\)
E_m
Eigenstates in the middle of the many-body spectrum
Heisenberg chain
XX chain ("many-body Anderson")
Summary : Spin-1/2 chain in random field
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distribution over disorder realizations and high-energy eigenstates
5
Anderson chain / XX chain
Heisenberg chain
Summary : Spin-1/2 chain in random field
QUestions
Spin-1/2 \(W = h = \) disorder strength for random fields
6
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
W
Ergodic
?
MBL regime(s)
Spin-1/2 \(W = h = \) disorder strength for random fields
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
6
W
Ergodic
MBL regime(s)
?
Fate of isolated
quantum systems?
QUestions
Spin-1/2 \(W = h = \) disorder strength for random fields
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
- Toy model?
6
W
Ergodic
?
MBL regime(s)
QUestions
Spin-1/2 \(W = h = \) disorder strength for random fields
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
- Quantitative description?
- Toy model?
6
W
Ergodic
?
MBL regime(s)
QUestions
Spin-1/2 \(W = h = \) disorder strength for random fields
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
- Toy model?
- Quantitative description?
- Consequences?
6
W
Ergodic
?
MBL regime(s)
QUestions
Scope
1. Spin chains in random field and localization : introduction
2. Exact diagonalization
4. Quantitative analysis: extreme value theory
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
3. Minimal deviations in the XX chain
5. Consequences
Spin chains in Random field and localization
Spin-1/2 chain in a random field
7
\mathcal{H} = \sum_{i} \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} h_i S_i^z
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Spin-1/2 chain in a random field
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
\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]
Jordan-Wigner
Spinless fermions
(hardcore bosons)
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
\mathcal{H} = \sum_{i} \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} h_i S_i^z
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
7
Spin-1/2 chain in a random field
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Jump
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{P}(h_i) =
Spin-flip
On-site energy
\(-W\)
\(W\)
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
Attraction/ repulsion
Attraction/ repulsion
Ising interaction
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
7
Anderson Localization
8
1 particle
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Spin-flip
Ising interaction
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
Anderson Localization
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{#76a5af}h_i n_i}\Bigr]
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
9
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
Anderson Localization
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{#76a5af}h_i n_i}\Bigr]
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
9
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
Anderson Localization
1 particle
{\xi}(E, {\color{#76a5af}W})
\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{#76a5af}h_i n_i}\Bigr]
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\epsilon_m
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{H}_f = \sum_{m} \epsilon_m b_m^{\dagger}b_m
9
Localization length
{\xi}(E, {\color{#76a5af}W})
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
10
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(\epsilon\)
\(\xi(\epsilon, W)\)
Localization length
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(\epsilon\)
\(\xi(\epsilon, W)\)
{\xi}(E, {\color{#76a5af}W})
10
Localization length
{\xi}(E, {\color{#76a5af}W})
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
\(\epsilon\)
\(\xi(\epsilon, W)\)
10
Localization length
{\xi}(E, {\color{#76a5af}W})
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\(\epsilon\)
\(\xi(\epsilon, W)\)
\xi \ll 1
W \gg 2
10
A. C. Potter, R. Vasseur, and S. A. Parameswaran, PRX 5, 031033 (2015)
P. W. Anderson, Phys. Rev. 109, 1492 (1958); N.F. Mott & W.D. Twose, Advances in Physics 10, 107-163, (1961)
B. A. Van Tiggelen, In: J. P. Fouque (eds), Diffuse Waves in Complex Media, NATO Science Series, 531, Springer, Dordrecht, (1999)
"Many-Body" Anderson Insulator (= XX chain)
\(L/2\) fermions
\(S_z = 0\)
\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{#76a5af}h_i n_i}\Bigr]
\epsilon_m
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
12
\(L/2\) fermions
\(S_z = 0\)
|\Psi \rangle = | \left\{\phi_m, m \in {\color{#56B4E9}\mathrm{occ}} \right\}\rangle
\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{#76a5af}h_i n_i}\Bigr]
\epsilon_m
E_m
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
"Many-Body" Anderson Insulator (= XX chain)
12
Heisenberg: Introducing interactions
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
13
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\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}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{H} = \sum_m \epsilon_m b_m^{\dagger} b_m + \sum_{j,k,l,m} {\color{#ff9900}V_{j,k,l,m} b_j^{\dagger} b_k^{\dagger} b_l b_m}
In the Anderson basis:
Anderson
orbitals \(m\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
14
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
Effect of interactions?
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\mathcal{H} = \sum_m \epsilon_m b_m^{\dagger} b_m + \sum_{j,k,l,m} {\color{#ff9900}V_{j,k,l,m} b_j^{\dagger} b_k^{\dagger} b_l b_m}
In the Anderson basis:
Anderson
orbitals \(m\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
14
P. W. Anderson, Phys. Rev. 109, 1492 (1958)
Effect of interactions?
Interactions favor delocalization. Do they fully destroy localization?
Effect of interactions? GRound state
15
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\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}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
Effect of interactions? GRound state
15
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\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}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
Effect of interactions?
16
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\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}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
What about high temperatures / high energy eigenstates?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
Effect of interactions?
16
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\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}}}+{\color{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
Do interactions destroy localization?
T. Giamarchi and H. J. Schulz, EPL 3 1287 (1987); PRB 37, 325 (1988)
Z. Ristivojevic, et al PRL 109, 026402 (2012);
Doggen et al, PRB 96, 180202(R) (2017)
What about high temperatures / high energy eigenstates?
J. M. Deutsch , PRA. 43, 2046–2049, (1991) ,
M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
Do isolated quantum systems thermalize?
Thermal average
?
ETH
Time average
Weak interactions and disorder
17
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Analytical, general picture:
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
Weak interactions and disorder
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Analytical, general picture:
Interactions \(\Rightarrow\) transition between weak and strong disorder
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
disorder
interactions
Anderson localized
Delocalized
Ergodic
Insulator
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
17
18
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
18
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
Gaussian orthogonal ensemble statistics = random matrix
level repulsion
\(\leftrightarrow\) ergodic
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
18
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
Gaussian orthogonal ensemble statistics = random matrix
level repulsion
\(\leftrightarrow\) ergodic
Poisson statistics
non-ergodic
A. Pal, D. Huse, PRB 82, 174411 2010
(See series of works by V. Oganesyan, A. Pal, D. Huse, 2007-2010)
Exact diagonalization
disorder
gap ratio
Probes: gap ratio
19
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
Probes: gap ratio | entanglement entropy
Initial \(S^z\) basis random product state
+
TEBD
W = 5
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)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
Probes: gap ratio | entanglement entropy
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)
Many body
Log growth
of entanglement
19
Initial \(S^z\) basis random product state
+
TEBD
W = 5
20
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
\(\frac{E -E_{\min}}{E_{\max}-E_{\min}}\)
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
A few among many...
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)
Probes: gap ratio | entanglement entropy | participation entropy
|\Psi\rangle = \sum_{\alpha = 1}^{\mathcal{N}} \psi_{\alpha} |\alpha \rangle
S_q = \frac{1}{1-q} \ln\left(\sum_{\alpha=1}^{\mathcal{N}} |\psi_{\alpha}|^{2q}\right)
Configuration space
S_q = a_q \ln(\mathcal{N})
20
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Heisenberg in random field : A paradigmatic example
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
A few among many...
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)
Probes: gap ratio | entanglement entropy | participation entropy | imbalance [...]
M. Schreiber et al. (I. Bloch) , Science 349, 842 (2015)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
W
Ergodic
MBL
2016
Finite L
debate
21
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
W
Ergodic
MBL
2016
Finite L
- Finite-size scaling? Location of the transition?
- Destabilization by ergodic bubbles even at strong disorder?
- Immediate onset of quantum chaos? Intermediate phase(s)?
debate
21
W
Ergodic
MBL phase/ regimes?
Prethermal
regime?
2023
J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, PRE 102, 062144 (2020); D.A. Abanin, et al, Annals of Physics 427, 168415, (2021);
D. Sels, A. Polkovnikov, JCCM January 2023_1 (2023); Tyler LeBlond, Dries Sels, Anatoli Polkovnikov, and Marcos Rigol, PRB 104, L201117 (2021); A. Morningstar et al, PRB 105, 174205 (2022); L. Colmenarez, D. Luitz, W. De Roeck, arXiv:2308.01350 (2023); P, Sierant and J. Zakrzewski, PRB 105, 224203 (2022)...
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
W
Ergodic
MBL
2016
Finite L
- Finite-size scaling? Location of the transition?
- Destabilization by ergodic bubbles even at strong disorder?
- Immediate onset of quantum chaos? Intermediate phase(s)?
debate
21
W
Ergodic
MBL phase/ regimes?
Prethermal
regime?
For today : Magnetization, ED data and comparison to the Anderson line
J. Šuntajs, J. Bonča, T. Prosen, and L. Vidmar, PRE 102, 062144 (2020); D.A. Abanin, et al, Annals of Physics 427, 168415, (2021);
D. Sels, A. Polkovnikov, JCCM January 2023_1 (2023); Tyler LeBlond, Dries Sels, Anatoli Polkovnikov, and Marcos Rigol, PRB 104, L201117 (2021); A. Morningstar et al, PRB 105, 174205 (2022); L. Colmenarez, D. Luitz, W. De Roeck, arXiv:2308.01350 (2023); P, Sierant and J. Zakrzewski, PRB 105, 224203 (2022)...
2023
Exact diagonalization
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
challenge
22
- Many-body
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
challenge
- Many-body
- Disorder \(\rightarrow\) translation invariance, high number of realisations
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
22
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
challenge
22
- Many-body
- Disorder \(\rightarrow\) translation invariance, high number of realisations
- High-energy eigenstates
- High density of eigenstates
- Potential absence of thermalization
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
Do not invert!
Solve \((H-\sigma) \vec{y} = \vec{x}\)
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
G = "\sum_k \alpha_k H^k"
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
23
spectral transformation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{N} = \frac{L!}{\left(\frac{L}{2}!\right)^2}
High energy eigenstates close to \(\epsilon = \sigma\), high dos
Idea : transform the spectrum!
F = (H - \sigma)^2
G = (H - \sigma)^{-1}
G = "\sum_k \alpha_k H^k"
\(L = 14, S^z_{\mathrm{tot}} = 0\), clean case
D. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
F. Pietracaprina, et al, .SciPost Phys. 5, 045 (2018)
P. Sierant et al, PRL 125, 156601 (2020)
Up to 22, 24 sites
\(\mathcal{N} > 2\cdot 10^6\)
# non-zero el. \(> 3 \cdot 10^7 \)
23
Magnetization distributions
24
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Anderson chain / XX chain
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
Magnetization distributions
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Anderson chain / XX chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
24
Magnetization distributions
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
Magnetization distributions
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
Magnetization distributions
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
24
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
Magnetization distributions
25
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
\delta_i = 1/2 - | \langle S_i^z \rangle |
V. Khemani, F. Pollmann, and S. L. Sondhi, PRL 116, 247204 (2016)
S. P. Lim and D. N. Sheng, PRB 94, 045111 (2016)
D. J. Luitz and Y. Bar Lev, PRL 117, 170404 (2016)
M. Dupont and N. Laflorencie, PRB 99, 020202(R) (2019)
M. Hopjan and F. Heidrich-Meisner, Phys. Rev. A 101, 063617 (2020)
N. Laflorencie, G. Lemarié, N. Macé, PRR 2, 042033(R) (2020)
J. C., N. Laflorencie, arXiv:2305.10574
Minimal deviations in the XX chain - TOy model
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
26
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
ED on one sample - XX (ANDERSON) CHAIN
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
Spin Freezing
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
SPIN FREEZING!
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
\delta_i = 1/2 -| \langle S_i^z \rangle|
SPIN FREEZING!
Some eigenstate
Spin Freezing
J. C., N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
26
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Toy model : analytical description
27
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Toy model : analytical description
27
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
Toy model : analytical description
\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, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
27
Minimal deviation?
28
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle
\ell_{\mathrm{cluster}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
r
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
\ell_{\mathrm{cluster}}
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
:
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\delta_i = 1/2 -|\langle n_i \rangle -1/2|
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
\Rightarrow \quad \delta^{\mathrm{typ}}_{\min}= e^{\overline{\ln\delta_{\min}}} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
28
r
\ell_{\mathrm{cluster}}
29
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
29
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
29
r
\ell_{\mathrm{cluster}}
Minimal deviation?
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
29
\gamma
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
30
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
30
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Exponent : toy model
JC, N. Laflorencie, arXiv:2305.10574
\xi = \frac{1}{\ln\left(1+\left(\frac{W}{W_0}\right)^2 \right)}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
30
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
31
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
31
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma_{\mathrm{typ}}(W)}
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
31
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma_{\mathrm{typ}}(W)}
Chain breaking
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
31
\delta^{\mathrm{typ}}_{\min} = e^{\overline{\ln\delta_{\min}}} \approx L^{-\gamma_{\mathrm{typ}}(W)}
Exponents: XX CHain
32
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
JC, N. Laflorencie, arXiv:2305.10574
ED
1/\gamma_{\mathrm{typ}}
Exponents: XX CHain
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
32
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
JC, N. Laflorencie, arXiv:2305.10574
Excellent agreement ED - Toy model !
ED
1/\gamma_{\mathrm{typ}}
Exponents: XX CHain
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
32
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
JC, N. Laflorencie, arXiv:2305.10574
Excellent agreement ED - Toy model !
ED
1/\gamma_{\mathrm{typ}}
Exponents: XX CHain
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
32
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
Scaling
33
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
Scaling
JC, N. Laflorencie, arXiv:2305.10574
\(L \gg \xi\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
33
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
Scaling
JC, N. Laflorencie, arXiv:2305.10574
33
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\(L \gg \xi\)
\quad \delta^{\mathrm{typ}}_{\min} \approx e^{-\frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi}} ?
- \frac{\overline{\ell_{\mathrm{cluster}}}}{2\xi} -c
Quantitative description : Extreme value statistics
Tails and extremes
34
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Tails
Tails and extremes
34
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
So far :
Extreme value
Tails
Tails and extremes
34
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
So far :
Extreme value
Tails
Extreme value theory
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Z = \mathrm{rescaled}(Y)
Extreme value theory
Extreme value
Tails
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Power-law tail
Z = \mathrm{rescaled}(Y)
Extreme value theory
Extreme value
Tails
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Power-law tail
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Power-law tail
Exponential tail
Gaussian tail
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
Tails and extremes
35
\{X_i\}_{i = 1, 2, \dots, L} \sim p(x)
Y = \max(X_i)
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Power-law tail
Exponential tail
Gaussian tail
z \sim e^{-z - e^{-z}}
Gumbel law
Z = \mathrm{rescaled}(Y)
z \sim \frac{\beta}{z^{1+\beta}} e^{-z^{-\beta}}
Fréchet law
Extreme value theory
Extreme value
Tails
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
36
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
36
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
37
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
37
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
37
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
E. J. Gumbel, Statistics of Extremes, Dover, (1958, 2004)
S. N. Majumdar, A. Pal, G. Schehr, Physics Reports, 840, 1 (2020)
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
Extreme value theory - XX Chain
JC, N. Laflorencie, arXiv:2305.10574
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
37
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
EXPONENT : POWER-LAW
JC, N. Laflorencie, arXiv:2305.10574
38
ED
1/\gamma_{\mathrm{typ}}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
JC, N. Laflorencie, arXiv:2305.10574
1+\alpha
ED
1/\gamma_{\mathrm{typ}}
EXPONENT : POWER-LAW
38
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma_{\mathrm{typ}}(W)}
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
Consequences : interacting system
39
Effect of interactions?
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\mathcal{H} = \mathcal{H}_{XX} + \Delta \sum_i S_i^z S_{i+1}^z
"Stability" of the cluster with respect to the interactions?
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
40
Effect of interactions?
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
disorder
increases
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
40
Effect of interactions?
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
disorder
increases
\(\delta_i\) \(\rightarrow\) 1/2 (\(\langle S^z_i\rangle \rightarrow 0\))
Empty circles:
Heisenberg
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
40
Effect of interactions?
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
disorder
increases
Empty circles:
Heisenberg
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\(\delta_i\) \(\rightarrow\) 1/2 (\(\langle S^z_i\rangle \rightarrow 0\))
41
Exponent
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
At strong disorder, \(\gamma \sim 1/\xi\)
\gamma
\(\Rightarrow \) Interpretation of the exponent as related to a many-body localization length
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
42
Exponent
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
At strong disorder, \(\gamma \sim 1/\xi\)
\gamma
\(\Rightarrow \) Interpretation of the exponent as related to a many-body localization length
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, N. Laflorencie, arXiv:2305.10574
\(\Lambda\) : disorder-dependent non-ergodicity volume
\(\lambda\) : interpreted as a localization length
43
Extreme value distributions
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
JC, N. Laflorencie, arXiv:2305.10574
\(W = 2\)
43
Extreme value distributions
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
JC, N. Laflorencie, arXiv:2305.10574
\(W = 2\)
43
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
\(\Delta = 0\)
\(\Delta = 1\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Extreme value distributions
JC, N. Laflorencie, arXiv:2305.10574
43
Conjecture : Gumbel (?) on the Ergodic side, Fréchet on the MBL side.
\(\Delta = 0\)
\(\Delta = 1\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Extreme value distributions
JC, N. Laflorencie, arXiv:2305.10574
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
44
Kullback-Leibler divergence
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
44
Kullback-Leibler divergence
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
44
Consequences : Heisenberg
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
- Transition in the extreme value distributions
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
44
Consequences : Heisenberg
- Transition in the extreme value distributions
- Coinciding with the MBL transition?
E. H. V. Doggen et al., PRB 98, 174202 (2018)
See e.g. D. Sels, PRB 106 , L020202 (2022)
S. Kullback and R. A. Leibler, The annals of mathematical statistics 22, 79 (1951)
JC, N. Laflorencie, arXiv:2305.10574
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
{\rm{KL}}(p|q)= \sum_i q_i \ln \frac{q_i}{p_i}
Kullback-Leibler divergence :
Take home message
SPIN FREEZING!
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
- Excellent fits & collapses with a Fréchet Law
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
W
No chain breaks
Chain breaks
- Excellent fits & collapses with a Fréchet Law
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
W
No chain breaks
Chain breaks
- Excellent fits & collapses with a Fréchet Law
Comparing \(\delta_{\min}\) deviation in Heisenberg vs Many-body Anderson:
Extreme value transition characterized by the
KL divergence.
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
Take home message
SPIN FREEZING!
XX chain :
- controlled by largest cluster of occupied orbitals
Heisenberg chain at strong disorder:
Chain breaks!
W
No chain breaks
Chain breaks
- Excellent fits & collapses with a Fréchet Law
Comparing \(\delta_{\min}\) deviation in Heisenberg vs Many-body Anderson:
Extreme value transition characterized by the
KL divergence.
Thank you for your attention!
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
45
good question!
Anderson Localization : example
1 particle
11
Billy J, et al. Direct observation of Anderson localization of matter waves in a controlled disorder. Nature. 2008
\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{#76a5af}h_i n_i}\Bigr]
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
AL vs MBL
No spreading of entanglement
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Weak interactions and disorder
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Analytical, general picture:
Interactions \(\Rightarrow\) transition between weak and strong disorder
L. Fleischman, P. W. Anderson, PRB 2, 2336 (1980) \(\rightarrow\) single-particle excitations and conditions for Anderson transition
B. Altschuler, Y. Gefen, A. Kamenev, L. S. Levitov, PRL 78, 2803, (1997) \(\rightarrow\) quasi particle lifetime & localization in Fock space
P. Jacquod, D. L. Shepelyansky, PRL 79, 1837 (1997) \(\rightarrow\) Gap ratio statistics, finite systems
I. V. Gornyi, A. D. Mirlin, D. G. Polyakov, PRL 95, 206603 (2005) \(\rightarrow\) zero conductivity at low temperature
*D. M. Basko, I. L. Aleiner, B. L. Altschuler, Annals of Physics 321, 1126 (2006) \(\rightarrow\) metal-insulator transition, localization in Fock space
I.L. Aleiner, B. L. Altshuler, G. V Shlyapnikov, Nature Physics 6, 900-904 (2010) \(\rightarrow\) weakly interacting bosons
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
disorder
interactions
Anderson localized
Delocalized
Ergodic
Insulator
ED : Why
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
High energy eigenstates, high dos
Potential absence of thermalization
Simulations of Many-Body Localizable (MBL) lattices models | Fabien Alet | Cargese
Cannot use stochastic methods
Usual condensed matter methods target the ground state.
\(\rightarrow\) DMRG-X, RSRG-X, time evolution with MPS, Unitary flow, ...
Ideally should work on both sides of the transition
Eigenstate thermalization hypothesis
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Fate of isolated many-body quantum systems?
\(\leftrightarrow\) A question from quantum chaos:
Thermal average
Time average
?
ETH
J. M. Deutsch , PRA. 43, 2046–2049, (1991) , M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
Eigenstate thermalization hypothesis
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Fate of isolated many-body quantum systems?
\(\leftrightarrow\) A question from quantum chaos:
Thermal average
Time average
?
ETH
J. M. Deutsch , PRA. 43, 2046–2049, (1991) , M. Srednicki, PRE 50, 888–901, (1994)
L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Adv. Phys. 65, 239 (2016)
(i) diagonal elements of local observables are smooth functions of the energy and take their microcanonical average value
(ii)off-diagonal elements vanish in the thermodynamic limit like \(e^{(-E_m -E_n)/2}\)
Statement about high energy.
Heisenberg: Many-Body Localization
Morningstar et al, PRB 105, 174205 (2022)
Fate of isolated quantum many-body systems ?
\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{#b4a7d6}2 \Delta n_i n_{i+1}} \right) -{\color{#76a5af}h_i n_i}\Bigr]
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{#b4a7d6} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{#76a5af} h_i S_i^z}
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Kane-fisher
LL + weakened link can flow to
an opened chain.
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Entanglement entropy
N. Laflorencie, in A. Bayat et al. (eds.), Entanglement in Spin Chains,
Quantum Science and Technology, https://doi.org/10.1007/978-3-031-03998-0_4
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distributions of minimal deviations
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Distributions of minimal deviations
JC, Laflorencie, In preparation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Cluster lengths
JC, Laflorencie, In preparation
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Power-law tails?
JC, N. Laflorencie, arXiv:2305.10574
\delta_{\min} \sim e^{-\frac{\ell}{2\xi}}
\( \Rightarrow \delta\) occurs if there is \(\ell \geq -2 \xi \ln(\delta)\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\ell
\mathcal{P}(\ell) \propto 2^{-\ell}
Very roughly:
\Rightarrow \mathcal{P}_L(\ln(\delta)) \propto \exp\left(2\xi\ln2 \times \ln(\delta)\right)
Power-law tails?
JC, N. Laflorencie, arXiv:2305.10574
\delta_{\min} \sim e^{-\frac{\ell}{2\xi}}
\( \Rightarrow \delta\) occurs if there is \(\ell \geq -2 \xi \ln(\delta)\)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\ell
\mathcal{P}(\ell) \propto 2^{-\ell}
Very roughly:
\mathcal{P}_L(\delta) \propto \delta^{\left(2\xi\ln2-1\right)}
\Rightarrow \mathcal{P}_L(\ln(\delta)) \propto \exp\left(2\xi\ln2 \times \ln(\delta)\right)
Experiments
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
PROBES
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
"Dynamical" probes
"Static" probes
Entanglement
entropy (EE)
Multifractality (participation entropy)
Entanglement spectrum (ES)
and these probes -->
Spectral repulsion
Magnetization & extremal magnetization
-> Lcluster, delta min
Evolution / autocorrelation of a prepared state
Out-of-Time-Order Correlator (OTOC)
KL divergence between eigenstates
Increasing number of involved states
Gap ratio
two-eigenstates correlation functions
Spectral form factor
Level compressibility
Imbalance
Dream: LIOMs
(minimal correlator)
Many-body resonances between eigenstates
Minimal gap
Distribution of matrix elements?
Heisenberg: Ergodic to MBL
Fate of isolated quantum many-body systems ?
\(\frac{E -E_{\min}}{E_{\max}-E_{\min}}\)
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
\(S_E\) : Entanglement entropy (red: visual estimate)
\(r\) : Gap ratio
\(\mathcal{F}\) : Bipartite fluctuations
\(\mathcal{F} = \langle (S_A^z)^2 \rangle - \langle S_A^z \rangle ^2 \)
\(S_1^{P} = a_1 \ln(\mathrm{dim} H) = - \sum_i p_i \ln(p_i)\) : partitipation entropy (multifractality)
\( f \) : dynamic fraction of an
initial spin polarization
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Hilbert-space localization
|\Psi\rangle = \sum_{\alpha = 1}^{\mathcal{N}} \psi_{\alpha} |\alpha \rangle
S_q = \frac{1}{1-q} \ln\left(\sum_{\alpha=1}^{\mathcal{N}} |\psi_{\alpha}|^{2q}\right)
\mathcal{N} = \text{ Hilbert space dim.}
Participation entropies
Basis-dependent
Macé, Alet, Laflorencie, PRL 123, 180601 (2019)
POLFED
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Ground state phase diagram
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
\Delta
h
\Delta = 0
\Delta = -1
Bose glass:
gapless, but exponentially decaying correlations
finite compressibility
insulating; localized
infinite superfluid susceptibility
\Delta = -1/2
BKT from SDRG:
K > 3/2
Weak link physics
Superfluid
Dynamic fraction
D. J. Luitz, N. Laflorencie, F. Alet, PRB 91, 081103(R) (2015)
J. COLBOIS | QIMG 2023 , KYOTO | 27.09.2023
Extreme Statistics in Random spin chains
By Jeanne Colbois
Extreme Statistics in Random spin chains
Talk at "Quantum information, quantum matter and quantum gravity", Kyoto 2023
