Jeanne Colbois PRO
Physicist @ CNRS. Here you find slides for *some* of my presentations, as well as visual abstracts for recent publications.
IPS Meeting 2024 | NTU Singapore | 2024/09/24
Jeanne Colbois
NUS & Majulab
Interaction-Driven Instabilities
in the Random-Field XXZ Chain
Nicolas Laflorencie
Laboratoire de Physique Théorique
Toulouse, France
Fabien Alet
1
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(|\psi(x)|^2\)
Anderson localization
1
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)
\(-h\)
\(h\)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Anderson localization
2
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
"many-body" Anderson localization :
stable against interactions?
2
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
"many-body" Anderson localization :
stable against interactions?
See Anderson, 1958
"An example of a real physical system with an infinite number of degrees of freedom [...] in which the approach to equilibrium is simply impossible."
2
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
"many-body" Anderson localization :
stable against interactions?
See Anderson, 1958
Effect of interactions?
1. Many-body Anderson localization in \(U(1)\) symmetric spin chain
2. Ergodic instability at weak disorder
"An example of a real physical system with an infinite number of degrees of freedom [...] in which the approach to equilibrium is simply impossible."
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
1. From single- to many-body Anderson localization
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
From fermions to spins
\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)
3
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
1 fermion
Charge is conserved
From fermions to spins
\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)
S^{x,y,z} = \frac{1}{2} \sigma^{x,y,z}
3
Jordan-Wigner
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
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{white}+ (E_0 + \mathcal{H}_{\mathrm{BC}})}
Spin-flip
1 fermion
1 spin up
Magnetization is conserved
Charge is conserved
\mathcal{P}(h_i)
\(-h\)
\(h\)
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
mANY-BODY aNDERSON LOCALIZATION
4
\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}
Jordan-Wigner
Anderson, Phys. Rev. 109, 1492 (1958)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(L/2\) fermions
\(\sum_i S_i^{z} = 0\)
Magnetization is conserved
Charge is conserved
\mathcal{P}(h_i)
\(-h\)
\(h\)
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{white}+ (E_0 + \mathcal{H}_{\mathrm{BC}})}
Spin-flip
mANY-BODY aNDERSON LOCALIZATION
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{white}+ (E_0 + \mathcal{H}_{\mathrm{BC}})}
Spin-flip
Magnetization is conserved
Charge is conserved
P. Jordan and E. Wigner, Z. Physik 47, 631–651 (1928)
Anderson, Phys. Rev. 109, 1492 (1958)
4
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\mathcal{P}(h_i)
\(-h\)
\(h\)
5
Many-body lOCALIZATION LENGTH?
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
\(\epsilon\)
\(\xi(\epsilon, W)\)
Many-body lOCALIZATION LENGTH?
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
5
Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
\(\epsilon\)
\(\xi(\epsilon, W)\)
Many-body lOCALIZATION LENGTH?
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(h\)
5
Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
\(\epsilon\)
\(\xi(\epsilon, W)\)
\(h\)
\xi \ll 1
h \gg 2
\xi = \frac{1}{\ln\left(1+\left(\frac{h}{h_0}\right)^2 \right)}
Many-body lOCALIZATION LENGTH?
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
5
Crowley and Chandran, PRR (2020), J. C., N. Laflorencie, PRB (2023)
Averaging over the density of states
Correlation lengths
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
6
\(C_{ij}^{\alpha\alpha} = \langle S_i^{\alpha} S_{j}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{j}^{\alpha} \rangle\)
Correlation lengths
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(C_{ij}^{\alpha\alpha} = \langle S_i^{\alpha} S_{j}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{j}^{\alpha} \rangle\)
Typical value
\alpha = z
\alpha = x, y
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
Disorder
average
6
Correlation lengths
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(C_{ij}^{\alpha\alpha} = \langle S_i^{\alpha} S_{j}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{j}^{\alpha} \rangle\)
Typical value
\alpha = z
\alpha = x, y
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
Disorder
average
Flattening due to PBC
6
Correlation lengths
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(C_{ij}^{\alpha\alpha} = \langle S_i^{\alpha} S_{j}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{j}^{\alpha} \rangle\)
Typical value
Flattening due to PBC
Bulk & mid-chain : same correlation length
\alpha = z
\alpha = x, y
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
Disorder
average
6
Correlation lengths
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(C_{ij}^{\alpha\alpha} = \langle S_i^{\alpha} S_{j}^{\alpha} \rangle - \langle S_i^{\alpha} \rangle \langle S_{j}^{\alpha} \rangle\)
Typical value
Flattening due to PBC
Bulk & mid-chain : same correlation length
Similar trends
\(\xi_x \approx 2 \xi_z\)
\alpha = z
\alpha = x, y
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
Disorder
average
6
J. C., N. Laflorencie, PRB (2023)
7
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
A simple many-body effect : Maximal magnetization?
J. C., N. Laflorencie, PRB (2023)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Some eigenstate
\(|\langle S_i^{z}\rangle| < 1/2\)
7
A simple many-body effect : Maximal magnetization?
Some eigenstate
J. C., N. Laflorencie, PRB (2023)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
A simple many-body effect : Maximal magnetization?
\delta_i = 1/2 -| \langle S_i^z \rangle|
\(|\langle S_i^{z}\rangle| < 1/2\)
7
Some eigenstate
J. C., N. Laflorencie, PRB (2023)
\(|\langle S_i^{z}\rangle| < 1/2\)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\delta_i = 1/2 -| \langle S_i^z \rangle|
\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|
8
A simple many-body effect : Maximal magnetization?
Chain breaking
9
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Toy model:
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta_{\rm min}^{\rm typ} = \exp(\overline{\ln \delta_{\min}})
\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|
SPIN FREEZING !
CHAIN BREAKING !
Chain breaking
Dupont, Macé, Laflorencie, PRB 100, 134201, (2019)
Laflorencie, Lemarié, Macé, PRR 2, 042033(R), (2020)
JC, Laflorencie, PRB 108, 144206 (2023)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\delta_{\rm min}^{\rm typ} = \exp(\overline{\ln \delta_{\min}})
\delta_{\rm min} = 1/2 -\max_{i}| \langle S_i^z \rangle|
\quad h
\delta^{\mathrm{typ}}_{\min} \approx L^{-\gamma}
512 sites
9
Delocalized:
\delta^{\mathrm{typ}}_{\min} \rightarrow \mathrm{const}
2. Interactions
rOLE OF INTERACTIONS
10
\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}}}
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
rOLE OF INTERACTIONS
\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
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
10
rOLE OF INTERACTIONS
\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
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
See for instance Pietracaprina et al., SciPost Phys. 5, 045
10
rOLE OF INTERACTIONS
\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
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Polynomial \(\rightarrow\) Exponential
Simulations of MBL lattice models | Fabien Alet | Cargese
See for instance Pietracaprina et al., SciPost Phys. 5, 045
\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 \(m\)
Tendency to delocalize
10
Ground-state : well understood
11
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Anderson localized
(insulator)
disorder \(h \)
Interaction \(\Delta\)
The question
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
Recent review : Sierant et al., arXiv:2403.07111
High energy eigenstates!!
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Finite-size numerics show a transition e.g. in extreme magnetization
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
Interaction \(\Delta\)
Many-body localized
The question
Other probes: gap ratio, fidelity, participation entropy, entanglement entropy...
11
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
Recent review : Sierant et al., arXiv:2403.07111
High energy eigenstates!!
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
Finite-size numerics show a transition e.g. in extreme magnetization
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
Interaction \(\Delta\)
Many-body localized
The question
Other probes: gap ratio, fidelity, participation entropy, entanglement entropy...
11
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
Recent review : Sierant et al., arXiv:2403.07111
\(\delta_{\min}^{\rm typ} \rightarrow 1/2\)
\(\delta_{\min}^{\rm typ} \rightarrow 0\)
High energy eigenstates!!
Recent review : Sierant et al., arXiv:2403.07111
- Stability of MBL ?
- Location of the transition ? (strong drifts)
- Existence of an intermediate phase?
- .....
Largely focused on
\(\Delta = 1\)
"Standard model"
12
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
The Debate
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
High energy eigenstates!!
13
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
an old question
Anderson localized
(insulator)
disorder \(h \)
Ergodic
("metal")
Interaction \(\Delta\)
Many-body localized
?
What happens at weak interactions?
Recent review : Sierant et al., arXiv:2403.07111
C.f. Basko, Aleiner, Altshuler (2006) and many, many others
Crowley and Chandran 2020
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
High energy eigenstates!!
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
14
Ergodic instability of Anderson localization
After extrapolation
High energy eigenstates!!
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5; Crowley and Chandran 2020 )
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
14
Ergodic instability of Anderson localization
After extrapolation
High energy eigenstates!!
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024)
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
\(\Delta/J\)
\(h/J \)
Anderson localized
(insulator)
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
14
Ergodic instability of Anderson localization
After extrapolation
High energy eigenstates!!
Avalanche theory : An intuition
15
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
De Roeck & Huveneers 2017, Luitz, De Roeck & Huveneers 2017, Thiery et al 2018
Adapted from Szoldra et at (2024)
Rare regions of effective weak disorder will always happen in the limit of large systems.
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
De Roeck & Huveneers 2017, Luitz, De Roeck & Huveneers 2017, Thiery et al 2018
Adapted from Szoldra et at (2024)
Rare regions of effective weak disorder will always happen in the limit of large systems.
They are thermal.
Avalanche theory : An intuition
15
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
If LIOM characteristic length \(\zeta > \zeta_{\mathrm{av.}}\) :
runaway avalanche is triggered and the system thermalizes.
De Roeck & Huveneers 2017, Luitz, De Roeck & Huveneers 2017, Thiery et al 2018; Crowley and Chandran 2020
Adapted from Szoldra et at (2024)
Rare regions of effective weak disorder will always happen in the limit of large systems.
They are thermal.
Avalanche theory : An intuition
15
Ergodic instability : consequence of avalanche theory?
(See also LeBlond et al., PRB 104 L201117 (2021), @ h = 0.5)
16
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (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
\(\delta_{\min} \rightarrow 1/2\)
\(\delta_{\min} \sim L^{-\gamma}\rightarrow 0\)
17
Spin-Spin correlations
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
AL
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
ETH
The longitudinal correlations decay as a power-law
The transverse still decay exponentially
18
correlations At strong disorder
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
- Fast growth of \(\xi_z \rightarrow \) crossing
- Instability at strong interactions
1
0.75
0.5
0.25
0
0
1
2
3
4
5
6
7
8
9
10
\(\Delta/J\)
\(h/J \)
Instabilities in the spin-spin correlations
18
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
On the MBL side
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
Instabilities in the spin-spin correlations
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
On the MBL side
Does it stop?
JC, F. Alet, N. Laflorencie, PRL 133, 116502 (2024); JC, F. Alet. N. Laflorencie, in preparation.
18
tAKE-HOME MESSAGEs
CHAIN BREAKING!
disorder \(h \)
Interaction \(\Delta\)
@ High energy!
19
COLBOIS | INTERACTION DRIVEN INSTABILITIES | IPS MEETING | 09.2024
SPIN FREEZING!
Bonus slideS unlocked
Anderson vs MBL
\(L/2\) 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
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
Extreme value statistics
\(L\) 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}}
\overline{\ell_{\mathrm{cluster}}} \approx \frac{\ln L}{ \ln 2}
\delta^{\mathrm{typ}}_{\min} \approx L^{-\frac{1}{2\xi \ln2}}
\delta_{\min} \approx e^{-\frac{\ell_{\mathrm{cluster}}}{2\xi}}
h
MBL
Question:
Does the seed hybridize (absorb) the l-bits?
Answer:
it depends on
(1) \(V_{ij}\) the matrix element coupling the seed to the l-bit
(2) \(1/ \rho\) the level spacing.
Typically \(V_{ij} \gg 1/\rho\).
The challenge is to quantify this, see Crowley and Chandran.
Essentially:
\(J_{\alpha}\) should be large enough.
Interaction-driven instabilities in the Random-Field XXZ chain
By Jeanne Colbois
Interaction-driven instabilities in the Random-Field XXZ chain
Invited talk at the IPS meeting in Singapore
