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

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)

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)

2

Extreme value theory

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

Condensed matter

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

Disordered spin chains

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)

3

Summary : Spin-1/2 chain in random field

4

Spin-1/2            $W = h =$ disorder strength for random fields along $S^z$

4

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

Spin-1/2            $W = h =$ disorder strength for random fields along $S^z$

Eigenstates in the middle of the many-body spectrum

Heisenberg chain

XX chain ("many-body Anderson")

Summary : Spin-1/2 chain in random field

Distribution over disorder realizations and high-energy eigenstates

5

Anderson chain / XX chain

Summary : Spin-1/2 chain in random field

Distribution over disorder realizations and high-energy eigenstates

5

Anderson chain / XX chain

Summary : Spin-1/2 chain in random field

Distribution over disorder realizations and high-energy eigenstates

5

Anderson chain / XX chain

Heisenberg chain

Summary : Spin-1/2 chain in random field

Distribution over disorder realizations and high-energy eigenstates

5

Anderson chain / XX chain

Heisenberg chain

Summary : Spin-1/2 chain in random field

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

W

Ergodic

? 

MBL regime(s)

Spin-1/2    $W = h =$ disorder strength for random fields

6

W

Ergodic

MBL regime(s)

? 

Fate of isolated

quantum systems?

QUestions

Spin-1/2    $W = h =$ disorder strength for random fields

• Toy model?

6

W

Ergodic

? 

MBL regime(s)

QUestions

Spin-1/2    $W = h =$ disorder strength for random fields

• Quantitative description?

• Toy model?

6

W

Ergodic

? 

MBL regime(s)

QUestions

Spin-1/2    $W = h =$ disorder strength for random fields

• 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

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

Spin-1/2 chain in a random field 

Jordan-Wigner

Spinless fermions

(hardcore bosons)

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

7

Spin-1/2 chain in a random field 

Jump

Magnetic field

Spin-flip

On-site energy

$-W$

$W$

Attraction/ repulsion

Attraction/ repulsion

Ising interaction

P. Jordan and E. Wigner,  Z. Physik 47, 631–651 (1928)

7

Anderson Localization

8

1 particle

Magnetic field

Spin-flip

Ising interaction

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

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

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

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)

9

Localization length

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)

Localization length

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)

10

Localization length

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)

10

Localization length

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

$S_z = 0$

12

$L/2$ fermions

$S_z = 0$