Anderson Localization Looking Forward

Transcription

1 Anderson Localzaton Lookng Forward Bors Altshuler Physcs Department, Columba Unversty Collaboratons: Also Igor Alener Dens Basko, Gora Shlyapnkov, Vncent Mchal, Vladmr Kravtsov, Lecture3 September, 10, 015

2 Outlne 1. Introducton. Anderson Model; Anderson Metal and Anderson Insulator 3. Localzaton beyond the real space. Integrablty and chaos. 4. Spectral Statstcs and Localzaton 5. Many-Body Localzaton. 6. Many-Body Localzaton of the nteractng fermons. 7. Many-Body localzaton of weakly nteractng bosons. 8. Many-Body Localzaton and Ergodcty

3 extended f = 3.04 GHz f = 7.33 GHz Anderson Model W, I localzed Localzed j I j E Ĥ x E x I Fnte moton 0 DoS x I 1

4 N N N z z z x ˆ ˆ ˆ ˆ ˆ x ˆ j j 0 1 j 1 1 Hˆ B J I H I t t Irreversblty = delocalzaton x

5 Mchal, Alener, Shlyapnkov and BA, 015 1D bosons n the random potental couplng constant Two transtons

6 Dsordered nteractng bosons n two dmensons

7 Arnold dffuson I I Vˆ 0 Each pont n the space of the ntegrals of moton corresponds to a torus and vce versa I 1 Fnte moton? I 1 d All classcal trajectores correspond to a fnte moton

8 When a theorst s asked to evaluate the stablty of a table wth 4 legs he/she 1. Evaluates the stablty of a table wth 1 leg, then. Evaluates the stablty, of a table wth nfnte number of legs and after that 3. Spends the rest of the lfe n attempts to evaluate the stablty of the table wth an arbtrary number of legs.? When a mathematcan s asked to evaluate the stablty of a table wth 4 legs he/she 1. Evaluates the stablty of a table wth 1 leg, then. Evaluates the stablty, of a table wth nfnte number of legs and after that 3. Spends the rest of the lfe n attempts to evaluate the stablty of the table wth an arbtrary number of legs.

9 Arnold dffuson I I Vˆ 0 Each pont n the space of the ntegrals of moton corresponds to a torus and vce versa I 1 Fnte moton? I 1 d d All classcal trajectores correspond to a fnte moton Most of the trajectores correspond to a fnte moton However small fracton of the trajectores goes nfntely far

10 Arnold dffuson 1. Most of the tor survve KAM. Classcal trajectores do not cross each other space # of dmensons real space phase space d d energy shell d-1 tor d d d d 1 Each torus has nsde and outsde en. shell tor nsde d d d 1 en. shell tor A torus does not have nsde and outsde as a rng n > dmensons

11 Conductvty nsulator 0 Many body localzaton! metal d 1 0 nteracton strength localzaton spacng 0 Conductvty 0 Bad metal Drude metal Bad metal temperature T temperature T Q:? What happens n the classcal lmt 0 Speculatons: 1.No transton c.bad metal stll exsts Reason: Arnold dffuson T 0

12 Large number d of the degrees of freedom Conventonal Boltzmann- Gbbs Statstcal Physcs Equpartton Postulate Ergodcty: tme average = space (ensemble) average Chaos Hamltonan H p, q? H H0 V Integrable Systems d d degrees of freedom ntegrals of moton Ergodcty s volated Invarant tor dmenson Hamltonan H Energy shell, dmenson p q 0, d d 1 Classcal Dynamcs Ergodcty Equpartton Quantum Dynamcs??? Ferm, Pasta, Ulam system (connected nonlnear oscllators) Solar system... KAM regon Arnold dffuson Non-ergodc 0

13 One quantum partcle Anderson Localzaton: n a random potental Strong enough dsorder the egenstates are localzed Weak dsorder maybe the egenstates are extended Localzaton Delocalzaton n real space Not only quantum dynamcs any wave dynamcs Isolated quantum system, Many-Body Localzaton: many degrees of freedom Close to the ntegrablty the egenstates are localzed Far from the ntegrablty the egenstates are extended Localzed Extended: space of quantum numbers Genune quantum phenomenon

14 Classcal Dynamcal Systems: Are the dynamcs ergodc outsde the KAM regme? For some low-dmensonal systems one can prove the ergodcty: Sna bllard, Bunmovch bllard, etc. At least some systems wth hgh number of dmensons are known to be non-ergodc: Solar System Ferm-Pasta-Ulam system of connected non-lnear oscllators...

15 The results of the calculatons (performed on the old MANIAC machne) were nterestng and qute surprsng to Ferm. He expressed to me the opnon that they really consttuted a lttle dscovery n provdng lmtatons that the prevalent belefs n the unversalty of mxng and thermalzaton n non-lnear systems may not always be justfed. [S.Ulam]

16 Age: ~4.5 Bllon years Sun des n ~8 Bllon years Mass Solar masses Newton: Moton of a sngle planet around the Sun. However, there are 8 planets (Newton knew 6). Each one exerts forces on the others small and perodcally varyng,. Newton: the Planets move one and the same way n Orbs concentrc, some nconsderable Irregulartes excepted, whch may have arsen from the mutual Actons of Comets and Planets upon one another, and whch wll be apt to ncrease, tll ths System wants a Reformaton., God has to ntervene contnuously to stablze the world?! Lebnz sneered at Newton s concepton, as beng that God so ncompetent as to be reduced to mracles n order to rescue hs machnery from collapse.

17 Age: ~4.5 Bllon years Sun des n ~8 Bllon years Mass Solar masses Isaac Newton: Moton of a sngle planet around the Sun. However, there are 8 planets (Newton knew 6). Each one exerts small and perodcally varyng forces on the others The postons of the planets n >10 8 years are unpredctable: they are too senstve to ntal condton - chaos. In 8 bllon years (just before the Sun des) the orbts wll most lkely be smlar to ther present ones. The unpredctablty s mostly n the orbtal phases, collsons between planets are unlkely n spte of the chaos. Ensemble of solar systems wth slghtly dfferent parameters at the present tme (random shfts ~1mm): ~1% percent probablty that Mercury colldes wth Venus before the death of the Sun. The solar system s nether absolutely stable nor ergodc

18 Glassy States of Matter: Glass n Egyptan tombs no tendency for orderng/thermalzaton n ~3000 years

19 Ideal (no dsorder) 1D Josephson array S, q Superconductng sland # q Phase of the order parameter Electrc charge n unts of e Canoncally conjugated varables:, q j k jk S -, q S -1, q, 1 1 S q S +1 1, q 1 S +, q C G E J

20 Ideal (no dsorder) 1D Josephson array S -, q S -1 S, q 1, q 1 S +1 1, q 1 S +, q C G E J EJ Josephson energy 1 ˆ H E 1 cos E E c C J 1 c G Classcal Lmt Chargng energy E d 0 q 0 Non-ergodc classcal and quantum dynamcs Small entropy at nfnte temperature. M. G. Pno, L.B. Ioffe, BA q c E J dt

21 Ideal (no dsorder) 1D Josephson array ˆ H E 1 cos E J 1 c q Quantum (1+1) BKT - transton Ins-r Superconductor

22 charge current q t j E sn, 1 J 1 L Low temperatures elementary exctatons plasmons. q 0

23 charge current q t j E sn, 1 J 1 L Fnte current state q 0 j 0

24 charge current q t j E sn, 1 J 1 L Hgh temperatures elementary exctatons plasmons. q 0 It makes more sense to buld the descrpton n terms of the charges rather than n terms of the phases.

25 Quantum Transton: ˆ q EJ 1 cos 1 ˆ ˆ ˆ ˆ Ec H EJ E c b b 1 b b 1 q Matrx element of the q, q q 1, q E J transton s Energy dfference of the two states Ec q q 1 1 E q T q c T E c Rato T c T T c E E J c Therefore T T c T T c metal nsulator Localzed phase at hgh temperatures! Freezng wth coolng! Classcal lmt: E J E c Tc

26 Classcal lmt: equatons of moton: sn sn 1 1 t E E J c U T u E L 1 J U L Total energy Length = # of slands u 1 1 cos L 1 Dmensonless energy per sland. 6

27 charge current q t j E sn, 1 J 1 L l Averagng over the macroscopc subsystem of the length l L

28 Slow relaxaton n the classcal lmt u u 1 exp ln 0 du d U U F T u Red 1 ln n 0( / ) T l J 1 curve F TL T du u T dt! Averagng over the macroscopc subsystem of the length l L df dt 8

29 Effectve temperature ` Effectve temperature grows as u ncreases, but s dfferent from both thermodynamc and pseudothermodynamc temperature. For example at u=3.5: T T T FDT Th

30 Quantum Smulatons Fnte number (5) of the charged states per ste q 0, 1, T Entanglement entropy concdes wth the conventonal entropy Entropy around crtcal pont 30

31 Quantum Smulatons Fnte number (5) of the charged states per ste q 0, 1,