Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
Majulab day | NTU Singapore | 2024/07/24
Jeanne Colbois
NUS & Majulab
Instabilities in random spin chainS:
Extreme magnetizations and correlations
Majulab day | NTU Singapore | 2024/07/24
Nicolas Laflorencie
Laboratoire de Physique Théorique
Toulouse, France
Fabien Alet
Jeanne Colbois
NUS & Majulab
Instabilities in random spin chainS:
Extreme magnetizations and correlations
1
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(|\psi(x)|^2\)
1
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Jump
Random on-site
energy
\(|\psi(x)|^2\)
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics, 10 (1961)
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
\(|\psi(x)|^2\)
\(\xi(h, E)\)
\(h\)
\(h\)
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\mathcal{P}(h_i)
\(-h\)
\(h\)
1
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Jump
Random on-site
energy
\(|\psi(x)|^2\)
Anderson, Phys. Rev. 109, 1492 (1958)
Mott & Twose, Advances in Physics, 10 (1961)
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
\(|\psi(x)|^2\)
\(\xi(h, E)\)
\(h\)
\(h\)
Anderson localization
\(\forall h , \, \forall E \) : localization !!
(1D, NN)
\mathcal{P}(h_i)
\(-h\)
\(h\)
Magnetization / correlations
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
High energy eigenstates of an isolated many-body quantum spin chain in random field
... through simple observables
2
Many-body effects...
Scope
Why care about many-body effects?
3
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Can we still say that high-energy many-body eigenstates are localized?
Why care about many-body effects?
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Can we still say that high-energy many-body eigenstates are localized?
If yes: no transport.
Simple example of an isolated quantum system that cannot thermalize
See Anderson, 1958
3
Program
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COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
1. From single fermion to many-body Anderson localization
2. Extreme magnetization
3. The role of interactions : instabilities
1. From single- to many-body Anderson localization
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
From fermions to spins
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
Anderson, Phys. Rev. 109, 1492 (1958)
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Charge is conserved
1 fermion
5
\mathcal{P}(h_i)
\(-h\)
\(h\)
From fermions to spins
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
Anderson, Phys. Rev. 109, 1492 (1958)
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Spin-flip
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
Charge is conserved
Magnetization is conserved
1 fermion
1 spin up
5
\(|\psi(x)|^2\)
\(\xi(h, E)\)
\mathcal{P}(h_i)
\(-h\)
\(h\)
Jordan-Wigner
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
mANY-BODY aNDERSON LOCALIZATION
6
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Spin-flip
Charge is conserved
Magnetization is conserved
\mathcal{P}(h_i)
\(-h\)
\(h\)
Jordan-Wigner
Anderson, Phys. Rev. 109, 1492 (1958)
mANY-BODY aNDERSON LOCALIZATION
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(L/2\) fermions
Jordan-Wigner
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
Magnetic field
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Spin-flip
Magnetization is conserved
\mathcal{P}(h_i)
\(-h\)
\(h\)
Charge is conserved
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
Anderson, Phys. Rev. 109, 1492 (1958)
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COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
7
Many-body lOCALIZATION LENGTH?
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
7
\(\epsilon\)
\(\xi(\epsilon, W)\)
Many-body lOCALIZATION LENGTH?
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(\epsilon\)
\(\xi(\epsilon, W)\)
7
Many-body lOCALIZATION LENGTH?
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(\epsilon\)
\(\xi(\epsilon, W)\)
\xi \ll 1
h \gg 2
\xi = \frac{1}{\ln\left(1+\left(\frac{h}{h_0}\right)^2 \right)}
7
Many-body lOCALIZATION LENGTH?
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(\epsilon\)
\(\xi(\epsilon, W)\)
\xi \ll 1
h \gg 2
\xi = \frac{1}{\ln\left(1+\left(\frac{h}{h_0}\right)^2 \right)}
7
Many-body lOCALIZATION LENGTH?
Related to correlation length!
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
2. A simple many-body effect
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
J. C., N. Laflorencie, PRB (2023)
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
8
Extreme magnetization
\(|\langle S_i^{z}\rangle| < 1/2\)
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
J. C., N. Laflorencie, PRB (2023)
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
8
\(|\langle S_i^{z}\rangle| < 1/2\)
Extreme magnetization
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
J. C., N. Laflorencie, PRB (2023)
8
\(|\langle S_i^{z}\rangle| < 1/2\)
Extreme magnetization
\delta_i = 1/2 -| \langle S_i^z \rangle|
Some eigenstate
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
J. C., N. Laflorencie, PRB (2023)
8
\(|\langle S_i^{z}\rangle| < 1/2\)
Extreme magnetization
how to predict it?
8
A little game...
Tossing a coin
M. F. Schilling, The College Mathematics Journal 21(3), 196-207 (1990)
P. Révész, Proc. 1978 Int'l Cong. of Mathematicians, 749-754 (1980)
\(L = 176 \)
HTTHTHH
THTHHT
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
9
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)
Tossing a coin
10
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
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)
Tossing a coin
10
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
HTTHTHTTHTHHHTHTTHHTTHHTHTTTHHHTTHTHTHTHHTHTHTHTHTHHTHTHHTHTHTHHTHTHTHHTHTHTTHTHTHTHHHTHTHTHTTHHTHTHTHTHHTHHHTTTHHTHTHTHTHTHTHHTHTHTHHTHTHHTTHTHHTHTHTHHTHTHHTHTHTHTHTHHTHTHHTHT
176 (81 T / 95 H )
HTTTHTTTHTHTHHTHHHHHHTTTTHHHHHHHTHTHHHTTHTHHTHHTTTHHHTHHHTTHHHHTHHTHHHTTTHTHTTHTHTTHHTHTTHTHTTTTTTTHHTHTHHHTHHTTHHTTTTTHHHTTHTHTHHTHTTHTTHHHHTHTHHHTTTTTHTHTTHHTHTTHHTHHHHTHHTHT
176 (83 T / 93 H )
\ell_{\max} \sim \ln L / \ln 2 \sim 7.45
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
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
10
11
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Toy model
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
|\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, Laflorencie, PRB 108, 144206 (2023)
11
Toy model
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
|\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, Laflorencie, PRB 108, 144206 (2023)
11
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}}
Toy model
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
r
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
11
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}}
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
Toy model
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}}
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
r
\ell_{\mathrm{cluster}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
11
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
Toy model
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
|\phi_m(i)|^2 \propto \exp\left(-\frac{|i - i_0^m|}{{\xi}}\right)
\Rightarrow \langle n_i \rangle = \langle S_i^z \rangle + 1/2 = \sum_{m \in {\color{#56B4E9}\mathrm{occ}}}
\delta_i = \langle n_i \rangle\approx e^{-\frac{r}{\xi} } + \dots
:
r
\ell_{\mathrm{cluster}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
11
\Rightarrow \quad \delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
\ell_{\mathrm{cluster}}
Toy model
12
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
Toy model validity
\ell_{\mathrm{cluster}}
12
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
Toy model validity
\ell_{\mathrm{cluster}}
Chain breaking
13
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
Chain breaking
13
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
\gamma
Chain breaking
13
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
SPIN FREEZING !
\gamma
CHAIN BREAKING !
Chain breaking : not just a toy model
14
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
14
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
\quad h
Chain breaking : not just a toy model
14
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
h
Chain breaking : not just a toy model
Extreme magnetization outlook
15
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Extreme value transition
Novel numerical methods?
\(h\)
Distributions predicted by extreme value
3. Interactions
rOLE OF INTERACTIONS
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
16
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}\right) -\sum_i{\color{gray}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} \right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
rOLE OF INTERACTIONS
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{salmon}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{gray}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{salmon} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Attraction /Repulsion
Ising
16
rOLE OF INTERACTIONS
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{salmon}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{gray}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{salmon} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Polynomial \(\rightarrow\) Exponential
Simulations of MBL lattice models | Fabien Alet | Cargese
Attraction /Repulsion
Ising
16
rOLE OF INTERACTIONS
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{salmon}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{gray}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{salmon} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Polynomial \(\rightarrow\) Exponential
See for instance Pietracaprina et al., SciPost Phys. 5, 045
Simulations of MBL lattice models | Fabien Alet | Cargese
Attraction /Repulsion
Ising
16
rOLE OF INTERACTIONS
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\mathcal{H}_f = \sum_{i} \frac{J}{2}\left({\color{lightgreen}c_i^{\dagger} c_{i+1}^{\vphantom{\dagger}} + c_{i+1}^{\dagger}c_i^{\vphantom{\dagger}}}+{\color{salmon}2 \Delta n_i n_{i+1}} \right) -\sum_i{\color{gray}h_i n_i}
\mathcal{H} = \sum_{i} \frac{J}{2}\left({\color{lightgreen}S_i^{+} S_{i+1}^{-} + S_i^{-} S_{i+1}^{+}} + {\color{salmon} 2\Delta S_i^z S_{i+1}^z}\right) - \sum_{i} {\color{gray} h_i S_i^z} {\color{black}+ E_0 + \mathcal{H}_{\mathrm{BC}}}
Polynomial \(\rightarrow\) Exponential
See for instance Pietracaprina et al., SciPost Phys. 5, 045
Simulations of MBL lattice models | Fabien Alet | Cargese
Attraction /Repulsion
Ising
16
The question
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
17
Anderson localized
(insulator)
disorder \(h \)
Interaction \(\Delta\)
Recent review : Sierant et al., arXiv:2403.07111
The question
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
Interaction \(\Delta\)
Many-body localized
Recent review : Sierant et al., arXiv:2403.07111
17
The question
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
Many-body localized
Interaction \(\Delta\)
Recent review : Sierant et al., arXiv:2403.07111
Debate :
- Stability of MBL
- Location of the transition
- Intermediate phase
17
The question
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
?
Interaction \(\Delta\)
Many-body localized
Many-body localized
Debate :
- Stability of MBL
- Location of the transition
- Intermediate phase
Recent review : Sierant et al., arXiv:2403.07111
17
Ergodic instability
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
18
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Anderson localized
(insulator)
disorder \(h \)
?
Interaction \(\Delta\)
Ergodic instability
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Anderson localized
(insulator)
disorder \(h \)
?
Interaction \(\Delta\)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
18
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
Ergodic instability
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
18
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
Ergodic instability
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
In the XXZ spin-chain
Avalanche theory + numerical results:
weak interactions \(\rightarrow\) ergodic!
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
18
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
19
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
How to evaluate \(\xi\) in the presence of interactions?
Instabilities in the spin-spin correlations
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
How to evaluate \(\xi\) in the presence of interactions?
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
Instabilities in the spin-spin correlations
19
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
How to evaluate \(\xi\) in the presence of interactions?
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
Instabilities in the spin-spin correlations
19
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
Avalanche theory:
if \(\xi > \xi_{\mathrm{av.}}\) then the system becomes ergodic
How to evaluate \(\xi\) in the presence of interactions?
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
Instabilities in the spin-spin correlations
19
JC, F. Alet, N. Laflorencie, arXiv:2403.09608
tAKE-HOME MESSAGEs
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Anderson localization
in 1D: \(\forall E\)
20
tAKE-HOME MESSAGEs
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Anderson localization
in 1D: \(\forall E\)
SPIN FREEZING!
CHAIN BREAKING!
20
tAKE-HOME MESSAGEs
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\(|\psi(x)|^2\)
\(\xi(h, E)\)
Anderson localization
in 1D: \(\forall E\)
SPIN FREEZING!
CHAIN BREAKING!
disorder \(h \)
Interaction \(\Delta\)
@ High energy!
20
Four Questions for you!
COLBOIS | INSTABILITIES IN RANDOM SPIN CHAINS | MAJULAB | 07.2024
\ell_{\max} \sim \ln L / \ln 2 \sim 7.45
1. How to show
2. True or false: Anderson localization is a purely quantum phenomenon
3. Is many-body localization different from "many-body Anderson" localization?
4. How can you use tensor networks to count
the number of ground states of a generalized Ising model?
21
22
Weitao CHEN
Sen MU
Nyayabanta SWAIN
Noam IZEM
Logarithmically slow expansion of a
quantum logarithmic-multifractal
wave packet over time
Localization, Multifractality and Chaos in complex quantum systems;
Random Matrices
Anderson localization/transition in cold atom systems
Multifractality
Numerical simulations
Example of a
multifractal eigenstate for one electron in a disordered potential:
neither localized nor metallic
The logarithm of the wave density
of a localized wave packet in 2D
exhibits similar scaling behaviors as a
growing interface
in the KPZ class
Quantum information and topological phases of matter
Kardar-Parisi-Zhang
universality class
Gabriel
LEMARIE
Quantum Monte Carlo
Eigenstate to Hamiltonian construction
Kardar-Parisi-Zhang universality class
Using EHC, we map the ground state of a 2D Hamiltonian in random disorder to an exotic excited state of a target Hamiltonian
Bonus slide unlocked
\(L/2\) 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
\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 \(m\)
What about high temperatures / high energy eigenstates?
Do isolated quantum systems thermalize?
Thermal average
?
ETH
Time average
Anderson
No growth
of entanglement
J. H. Bardarson, F. Pollmann, and J. E. Moore, PRL 109, 017202 (2012)
M. Znidaric, T. Prosen, and P. Prelovsek PRB 77, 064426 (2008)
Log growth
of entanglement
Initial \(S^z\) basis random product state
+
TEBD
W = 5
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)?
W
Ergodic
MBL phase/ regimes?
Prethermal
regime?
2023
\(L\) increases
Anderson chain / XX chain
Heisenberg chain
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
\mathcal{P}(\ln\delta_{\min}) \rightarrow AL\delta_{\min}^{\alpha} \exp\left({-\frac{AL}{\alpha+1}\delta_{\min}^{\alpha+1}}\right)
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
\Delta_u= (\alpha+1)(\ln\delta_{\min}- \ln\delta_{\min}^{\mathrm{typ}})
Fréchet
\(10^5 \) samples
\mathcal{P}(\delta) \overset{\delta \rightarrow 0}{\sim} A\delta^{\alpha}
\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}})
\delta_{\mathrm{\min}}^{\mathrm{typ}}(L) \approx \left(\frac{A}{1+\alpha} L \right)^{-\frac{1}{1+\alpha}}
Instabilities in Random Spin Chains
By Jeanne Colbois
Instabilities in Random Spin Chains
Talk at the Majulab day
