Zooming in on the Quantum Hall Effect
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1 Zooming in on the Quantum Hall Effect Cristiane MORAIS SMITH Institute for Theoretical Physics, Utrecht University, The Netherlands Capri Spring School p.1/31
2 Experimental Motivation Historical Summary: Hall Resistance Integral QHE ν = n I B électrons D electrons D R L R H longitudinal resistance I 1980 : Discovery of the IQHE (v. Klitzing) Magnetic Field Capri Spring School p./31
3 Experimental Motivation Hall Resistance Integral QHE ν = n Fractional QHE ν = p/(ps1) /3 3/5 1 4/7 1/ I 3/7 /5 4/9 1/3 B électrons D electrons D R L R H longitudinal resistance I Historical Summary: 1980 : Discovery of the IQHE (v. Klitzing) 1983 : Discovery of the FQHE (Tsui, Störmer); Laughlin : incompressible quantum Magnetic Field liquids Capri Spring School p./31
4 Experimental Motivation Hall Resistance Integral QHE ν = n Fractional QHE ν = p/(ps1) /3 3/5 1 4/7 1/ I 3/7 /5 4/9 1/3 B électrons D electrons D R L R H longitudinal resistance I Historical Summary: 1980 : Discovery of the IQHE (v. Klitzing) 1983 : Discovery of the FQHE (Tsui, Störmer); Laughlin : incompressible quantum Magnetic Field liquids 1989 : composite fermions (Jain, Read, Lopez/Fradkin,...) Capri Spring School p./31
5 Experimental Motivation Hall Resistance R xx (Ohms) R xy (h/e ) 0.35 a) h/3e / 31/5 34/5 0.5 h/4e b) Reentrant IQHE / Magnetic Field (Tesla) 3/5 1 4/7 1/ 3/7 /5 4/9 1/3 longitudinal resistance Historical Summary: 1980 : Discovery of the IQHE (v. Klitzing) 1983 : Discovery of the FQHE (Tsui, Störmer); Laughlin : incompressible quantum Magnetic field liquids 00 : discovery of Reentrant IQHE (Eisenstein et al.) Capri Spring School p./31
6 Experimental Motivation Hall Resistance R xx (Ohms) R xy (h/e ) 0.35 a) h/3e / 31/5 34/5 0.5 h/4e b) Reentrant IQHE / Magnetic Field (Tesla) 3/5 1 4/7 1/ 3/7 /5 4/9 1/3 5/13 4/11 3/8 Self similarity of the Hall curve longitudinal resistance Historical Summary: 1980 : Discovery of the IQHE (v. Klitzing) 1983 : Discovery of the FQHE (Tsui, Störmer); Laughlin : incompressible quantum Magnetic field liquids 00 : discovery of Reentrant IQHE (Eisenstein et al.) 003 : discovery of the FQHE (Pan et al.) Capri Spring School p./31
7 Theoretical Model D electrons in a perpendicular magnetic field no spin! 1 particle Hamiltonian Coulomb interactions : energy quantization (Landau levels) : impurity potential (pinning) : FQHE, electron-solid phases Capri Spring School p.3/31
8 IQHE: single particle picture one electron in : 4 degenerate Landau levels (LLs) Density of states per LL: Landau Levels 3 1 n = 0 heb/m m filling factor : Capri Spring School p.4/31
9 IQHE: single-particle localisation n ε n R xx R xy h/e n class. ν =n B electrons in full LLs: one quantum of conductance e /h per LL Capri Spring School p.5/31
10 IQHE: single-particle localisation n ε n th (n1) LL R xx R xy h/e n class. ν =n B electrons in full LLs: one quantum of conductance e /h per LL Capri Spring School p.5/31
11 IQHE: single-particle localisation n electrons in partially filled LL trapped by impurities ( ) ε n ε n th (n1) LL R xx R xy R xx R xy h/e n class. ν =n B B electrons in full LLs: "inert" background (c.f. noble gases, full shells) Capri Spring School p.5/31
12 IQHE: single-particle localisation n electrons in partially filled LL trapped by impurities ( ) ε n ε n ε n th (n1) LL R xx R R xy R xx R R xy xx xy h/e n ν =n class. B h/e n h/e (n1) electrons in full LLs: "inert" background (c.f. noble gases, full shells) B B Capri Spring School p.5/31
13 When Coulomb becomes essential ν ( = n ν < 1 ν < 1 ) ν = n Capri Spring School p.6/31
14 Capri Spring School p.6/31 When Coulomb becomes essential ) ( = n ν = n ν < 1 ν < 1 ν Hamiltonian in the th LL projected density:
15 When Coulomb becomes essential ν ( = n ν < 1 ν < 1 ) ν = n Hamiltonian in the th LL Capri Spring School p.6/31
16 Effective interaction potential M.Goerbig and C.M.S., Europhys. Lett. 63, 736 (003) 0.8 v (r) n 1.5 v ~ (r) universal function ~R c R = l (n1) c B 1/ r/10l B r/10r c length scales: and (interparticle separation) Capri Spring School p.7/31
17 Wigner Crystal and Bubbles A. Wigner Crystal (WC): Quasi-classical limit R c d Capri Spring School p.8/31
18 Wigner Crystal and Bubbles A. Wigner Crystal (WC): Quasi-classical limit R c d B. FQHE: Capri Spring School p.8/31
19 Wigner Crystal and Bubbles A. Wigner Crystal (WC): C. Bubbles (super-wc): Quasi-classical limit Not in LLL, R c v n(r) 1) d d ~ energy = u d d Rc d u r ) d d d >R c energy = ~ u B. FQHE: Capri Spring School p.8/31
20 Energy of competing ground states M=7 Wigner crystal Bubble crystal Stripe phase Wigner crystal and Bubble: Hartree-Fock impurities Excitations of the quantum liquid: Hamiltonian Theory Murthy and Shankar, Rev. Mod. Phys. 75, 1101 (003) Capri Spring School p.9/31
21 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 7/ 31/5 1/?? 3e Capri Spring School p.10/31
22 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 1/?? /3e 7/ 31/5 Capri Spring School p.10/31
23 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 1/?? /3e 7/ 31/5 Capri Spring School p.10/31
24 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 1/?? /3e 7/ 31/5 Capri Spring School p.10/31
25 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 1/?? /3e 7/ 31/5 Capri Spring School p.10/31
26 Results for Goerbig, Lederer, CMS, PRB 68, 4130(R) (003) R long R Hall I I Energy /9 1/7 1/5 1/3 n=1 M= (electrons per site) impurities M=1 quantum liquids Partial filling of the last level [h/e ] Hall resistance 1/3 1/3.5 filling of the last level 1/5 1/3 crystal liquid crystal liquid crystal 1/?? /3e 7/ 31/5 Capri Spring School p.10/31
27 Results for Energy M. O. Goerbig, P. Lederer, C. M. S., PRB 69, (004) /91/7 1/5 1/3 M=1 M= impurities quantum liquid stripes Partial filling factor No FQHE at, but indications at Capri Spring School p.11/31
28 Phase Transitions Goerbig, Lederer, and CMS, PRB 69, (004) energy (a.u.) n = M= M= mixed phase partial filling factor Mixed phase Wigner crystal/ Bubble Capri Spring School p.1/31
29 Phase Transitions Goerbig, Lederer, and CMS, PRB 69, (004) energy (a.u.) n = M= M= mixed phase partial filling factor Mixed phase Wigner crystal/ Bubble Pinning mode at Capri Spring School p.1/31
30 Phase Transitions Goerbig, Lederer, and CMS, PRB 69, (004) energy (a.u.) n = M= partial filling factor M= mixed phase Mixed phase Wigner crystal/ Bubble Pinning mode at Lewis et al., PRL 93, (04) Re [σ xx ] (µs) ν=4.16 ν=4.18 ν=4.1 ν=4.1 ν=4.6 Data fit peak 1 peak f(ghz) Capri Spring School p.1/31
31 Phase Transitions Goerbig, Lederer, and CMS, PRB 69, (004) energy (a.u.) n = M= partial filling factor M= mixed phase f pk (GHz) Mixed phase Wigner crystal/ Bubble f p1 f p Pinning mode at ν Capri Spring School p.1/31
32 Discovery of a new FQHE at CF theory : classified all the known FQHE plateaus... FQHE ν = p/(ps1) 1/3 Hall Resistance IQHE ν = n /3 3/5 1 4/7 1/ /5 3/7 4/ longitudinal resistance magnetic field Capri Spring School p.13/31
33 Discovery of a new FQHE at CF theory : classified all the known FQHE plateaus until 003 : new class of states (Pan et al.) FQHE ν = p/(ps1) 1/3 Hall Resistance IQHE ν = n /3 3/5 1 4/7 5/13 1/ /5 3/7 4/9 11/9 8/1 7/19 10/7 4/11 3/8 Self-similarity of the Hall curve longitudinal resistance magnetic field Capri Spring School p.13/31
34 Composite Fermions Idea: interpret strongly correlated electrons in terms of quasi-particles (CF) with negligible interactions ν = 1/3 electronic filling 1/3 CF theory 1 filled CF level electron "free" flux quantum pseudo vortex (with flux quanta) composite fermion (CF) Capri Spring School p.14/31
35 Composite Fermions Idea: interpret strongly correlated electrons in terms of quasi-particles (CF) with negligible interactions ν = 1/3 electronic filling 1/3 CF theory 1 filled CF level electron "free" flux quantum pseudo vortex (with flux quanta) composite fermion (CF) ν = /5 filled CF levels At FQHE of electrons, IQHE of CFs Capri Spring School p.14/31
36 # $ * Hamiltonian Theory of the FQHE Treat pseudo-vortex as new particle (charge "! ): electron (charge 1) (' &% ) Murthy/Shankar, Pasquier/Haldane, Read pseudo vortex (charge c ) Capri Spring School p.15/31
37 # $ * * Hamiltonian Theory of the FQHE Treat pseudo-vortex as new particle (charge "! ): electron (charge 1) (' &% ) Murthy/Shankar, Pasquier/Haldane, Read Constraint : (' &% pseudo vortex (charge c ) Capri Spring School p.15/31
38 # * * Hamiltonian Theory of the FQHE ): Treat pseudo-vortex as new particle (charge electron (charge 1) $ "! ) (' &% pseudo vortex (charge c ) Murthy/Shankar, Pasquier/Haldane, Read (' &% Constraint : Preferred combination (CF density): (' &% (' (' Capri Spring School p.15/31
39 # * * Hamiltonian Theory of the FQHE ): Treat pseudo-vortex as new particle (charge electron (charge 1) $ "! ) (' &% pseudo vortex (charge c ) Murthy/Shankar, Pasquier/Haldane, Read (' &% Constraint : Preferred combination (CF density): (' &% (' (' (non degenerate state): At completely filled CF levels Capri Spring School p.15/31
40 Second Generation of CFs At : CF levels are degenerate CF interactions ν = 11/3 filling 1/3 of first excited CF level CF of first generation = (with flux quanta) Capri Spring School p.16/31
41 Second Generation of CFs At : CF levels are degenerate CF interactions ν = 11/3 C F filling 1/3 of first excited CF level CF of first generation = (with flux quanta) theory 1filled CF level CF of second generation (C F) 1 filled C F level = 1 CF CF vortex (in first excited CF level) (with additional flux quanta) Capri Spring School p.16/31
42 Second Generation of CFs At : CF levels are degenerate CF interactions ν = 11/3 C F filling 1/3 of first excited CF level CF of first generation = (with flux quanta) theory 1filled CF level CF of second generation (C F) 1 filled C F level = 1 CF CF vortex (in first excited CF level) (with additional flux quanta) Explanation of IQHE of C Fs state (??): Capri Spring School p.16/31
43 Second Generation of CFs At : CF levels are degenerate CF interactions ν = 11/3 C F filling 1/3 of first excited CF level CF of first generation = (with flux quanta) theory 1filled CF level CF of second generation (C F) 1 filled C F level = 1 CF CF vortex (in first excited CF level) (with additional flux quanta) Explanation of IQHE of C Fs state (??): New hierarchy scheme of states Capri Spring School p.16/31
44 Interacting CFs at ν = 11/3 1.st generation CF Low energy excitations of CFs : intra-level Capri Spring School p.17/31
45 Interacting CFs at ν = 11/3 1.st generation CF Low energy excitations of CFs : intra-level - Wave functions : numerical calculations (finite size) Problem : ambiguous results (no thermodynamical limit) [Mandal and Jain, PRB 66, (00); Chang and Jain, PRL 9, (004)] Capri Spring School p.17/31
46 Interacting CFs at ν = 11/3 1.st generation CF Low energy excitations of CFs : intra-level - Wave functions : numerical calculations (finite size) Problem : ambiguous results (no thermodynamical limit) [Mandal and Jain, PRB 66, (00); Chang and Jain, PRL 9, (004)] - Hamiltonian theory : simple analytical frame Capri Spring School p.17/31
47 Model for interacting CFs at Goerbig, Lederer, C.M.S., Europhys. Lett. 68, 7 (004) (' ' Density restricted to level, : Capri Spring School p.18/31
48 Model for interacting CFs at Goerbig, Lederer, C.M.S., Europhys. Lett. 68, 7 (004) (' ' :, Density restricted to level Interaction potential Effective Hamiltonian: ' ' Capri Spring School p.18/31
49 Model for interacting CFs at Goerbig, Lederer, C.M.S., Europhys. Lett. 68, 7 (004) (' ' :, Density restricted to level Interaction potential Effective Hamiltonian: ' ' Self-similarity of FQHE Similarity with original model Capri Spring School p.18/31
50 Activation Gaps of C F States Goerbig, Lederer, C.M.S., PRB 69, (004) Inter-level excitations Finite width : screened interaction (RPA) Activation gaps (a) s=1 s=1, ~ p=1 ~ (ν= 4/11) p=1 s=1, ~ p= ~ (ν=6/17) s=, ~ p=1 ~ (ν=7/19) s=, ~ p= ~ (ν=11/31) Activation gaps s= p=1 s=1, ~ p=1 ~ s=1, ~ p= ~ s=, ~ p=1 ~ s=, ~ p= ~ (b) (ν=4/19) (ν=6/9) (ν=7/33) (ν=11/53) width in units of l B width in units of l B one order of magnitude smaller than for CF states! Capri Spring School p.19/31
51 Reentrant FQHE Goerbig, Lederer, C.M.S., PRL 93, 1680 (004) Self-similarity same approach as for electrons Energy Electronic filling factor /3 6/17 4/11 3/ (CF Wigner crystal) M= M= (CF bubbles) quant. liquids (FC ) -0.0 s=1 CF stripes p=1 1/5 1/ partial CF filling factor Capri Spring School p.0/31
52 Reentrant FQHE Goerbig, Lederer, C.M.S., PRL 93, 1680 (004) Self-similarity same approach as for electrons Energy Electronic filling factor /3 6/17 4/11 3/ (CF Wigner crystal) M= M= (CF bubbles) quant. liquids (FC ) -0.0 s=1 CF stripes p=1 1/5 1/3 FC state stable at partial CF filling factor Capri Spring School p.0/31
53 Reentrant FQHE Goerbig, Lederer, C.M.S., PRL 93, 1680 (004) Self-similarity same approach as for electrons Energy Electronic filling factor /3 6/17 4/11 3/ (CF Wigner crystal) M= M= (CF bubbles) quant. liquids (FC ) -0.0 s=1 CF stripes p=1 1/5 1/ partial CF filling factor FC state stable at Reentrance in the FQHE Capri Spring School p.0/31
54 Conclusions I: Self-similarity of QHE Hall Resistance Integral QHE ν Fractional QHE 1/3 = IQHE of CFs 4/11 ν = n /7 /5 4/9 1/ 4/7 3/8 3/5 /3 5/13 = p/(ps1) 1 Self Similarity of the Hall Curve Magnetic Field state due to residual CF interactions Capri Spring School p.1/31
55 Conclusions I: Self-similarity of QHE Hall Resistance Integral QHE ν Fractional QHE 1/3 = IQHE of CFs 4/11 ν = n /7 /5 4/9 1/ 4/7 3/8 3/5 /3 5/13 = p/(ps1) 1 Self Similarity of the Hall Curve Magnetic Field state due to residual CF interactions Interacting CF model derived in the Hamiltonian theory of the FQHE Capri Spring School p.1/31
56 Conclusions I: Self-similarity of QHE Hall Resistance Integral QHE ν = n Fractional QHE 1/3 = IQHE of CFs 4/11 ν = p/(ps1) /3 3/5 1 4/7 1/ 3/7 /5 4/9 Magnetic Field 5/13 3/8 Self Similarity of the Hall Curve Model reveals self-similarity of the FQHE state due to residual CF interactions Interacting CF model derived in the Hamiltonian theory of the FQHE new hierarchy scheme (higher CF generations) Capri Spring School p.1/31
57 Conclusions II: Phase Diagram electron crystals M n=0 n=1 n= n=3 ν Localisation LL partial filling of the last LL electronic filling 1/3 /5 1 CF filling C F C F 4/11 5/13 CF stripes CFs CF level p=1 insulating Zoom in CF phases quantum liquids stripes Capri Spring School p./31
58 Perspectives What about SPIN? Each Landau Level splits into two levels (Zeeman energy) Quantum Hall Ferromagnet at n=0 g m=0 m=1 m= m=n 1 φ : Capri Spring School p.3/31
59 Perspectives What about SPIN? Each Landau Level splits into two levels (Zeeman energy) Magneto-excitons bosons n=0 g m=0 m=1 m= m=n 1 φ Capri Spring School p.3/31
60 Bosonization theory: DES at Doretto, Caldeira, Girvin, PRB 71, (005) Capri Spring School p.4/31
61 Bosonization theory: DES at Doretto, Caldeira, Girvin, PRB 71, (005) Interacting DEG at Capri Spring School p.4/31
62 ) ) Bosonization theory: DES at Doretto, Caldeira, Girvin, PRB 71, (005) Interacting DEG at ) Capri Spring School p.4/31
63 Bosonization theory: DES at Capri Spring School p.4/31 Doretto, Caldeira, Girvin, PRB 71, (005) Interacting DEG at ) ' ) $
64 Bosonization theory: DES at Doretto, Caldeira, Girvin, PRB 71, (005) Interacting DEG at ) $ ' ) non-interacting bosons RPA interaction term Skyrmion/anti-Skyrmion pair Capri Spring School p.4/31
65 Bosonization theory: DES at Doretto et al. PRB 7, (005)? - include SPIN in Hamiltonian theory - use Bosonization theory for Spin-excitations of the QH FM of composite fermions Capri Spring School p.5/31
66 Bosonization theory: DES at Doretto et al. PRB 7, (005)? - include SPIN in Hamiltonian theory - use Bosonization theory for Spin-excitations of the QH FM of composite fermions What about PSEUDOSPIN? Bosonization theory for QH bilayers: next talk Capri Spring School p.5/31
67 Conclusion M : impureté : électron n n=0 n=1 n= n=3 ν aggrandissement : auto similarité M=3 îlots à électrons rubans (EHQF) localisation à une particule phases de FC 5/, 7/ EHQF 1/3 /5 1 remplissage él. remplissage FC 4/11 5/13 p=1 isolants FCs rubans de FC niveau FC FC Capri Spring School p.6/31
68 Construction of Low-Energy Model at ν = 11/3 Restriction to a single CF level Capri Spring School p.7/31
69 Construction of Low-Energy Model at ν = 11/3 Restriction to a single CF level Restriction of CF density: (' Capri Spring School p.7/31
70 Construction of Low-Energy Model at ν = 11/3 Restriction to a single CF level Restriction of CF density: (' (' Capri Spring School p.7/31
71 Construction of Low-Energy Model at ν = 11/3 Restriction to a single CF level Restriction of CF density: (' (' : Because of factorisation Capri Spring School p.7/31
72 ) Construction of Low-Energy Model at ν = 11/3 Restriction to a single CF level Restriction of CF density: (' (' : Because of factorisation (' ) ('! independent of Capri Spring School p.7/31
73 Hierarchical States, Haldane/Halperin (1983) Motivation: FQHE e.g. at (non-laughlin) QP form a Laughlin state due to QP interactions? Capri Spring School p.8/31
74 * Hierarchical States, Haldane/Halperin (1983) Motivation: FQHE e.g. at (non-laughlin) QP form a Laughlin state due to QP interactions? Continued fraction: $ : positive integer : odd integer Capri Spring School p.8/31
75 * Hierarchical States, Haldane/Halperin (1983) Motivation: FQHE e.g. at (non-laughlin) QP form a Laughlin state due to QP interactions? Continued fraction: Generation: (I) p =1 i 1/3 (q=3) (II) /5 /7 $ : positive integer : odd integer (III) (IV) 3/7 5/13 5/17 3/11 4/9 8/19 1/31 8/1 8/7 1/41 8/9 4/15 encircled states are stable Stable states at Capri Spring School p.8/31
76 * Self-Similarity Hierarchy of States Recursion formula for possible states of ( : electrons): -th CF generation : C F level filling factor; : number of attached flux pairs in C F : number of filled CF levels Capri Spring School p.9/31
77 * Self-Similarity Hierarchy of States Recursion formula for possible states of ( : electrons): -th CF generation : C F level filling factor; : number of attached flux pairs in C F : number of filled CF levels modular group Capri Spring School p.9/31
78 * $ Self-Similarity Hierarchy of States Recursion formula for possible states of ( : electrons): -th CF generation : C F level filling factor; : number of attached flux pairs in C F : number of filled CF levels modular group Continued fraction: Capri Spring School p.9/31
79 * $ Self-Similarity Hierarchy of States Recursion formula for possible states of ( : electrons): -th CF generation : C F level filling factor; : number of attached flux pairs in C F : number of filled CF levels modular group Continued fraction: Fixed point ( series, ): Capri Spring School p.9/31
80 $ $ CF Interaction Potential Haldane s pseudopotential expansion CF m pseudopotentials V V m p p=1 s=1 p= p= m Capri Spring School p.30/31
81 CF Phases in cohesive energy / M=1 s=1 p= electronic filling factor 11/7 7/17 5/ /5 quantum liquid (C F) 1/3 M= partial CF filling factor CF stripe Capri Spring School p.31/31
82 CF Phases in cohesive energy / M=1 s=1 p= electronic filling factor 11/7 7/17 5/ /5 quantum liquid (C F) 1/3 M= partial CF filling factor CF stripe Quantum liquid (C F) ceases to be ground state at Capri Spring School p.31/31
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