Reduced dimensionality. T. Giamarchi
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1 Reduced dimensionality T. Giamarchi
2 References TG, arxiv/ (Salerno lectures) TG arxiv/ (Les Houches-Singapore) M.A. Cazalilla et al. arxiv/ (RMP) TG, Quantum physics in one dimension, Oxford (2004)
3 Plan of the lectures Why one dimensional systems 1D: Luttinger liquids Multicomponent systems Mott transition in one dimension Beyond Luttinger liquids
4
5 How to describe solids? Understood: free electrons Real systems : Coulomb interaction E» K! Properties of realistic systems? Free electron theory works quite well : Landau Fermi liquid
6 Transistor 1956 Supraconductivité (1913),1972, 1973,1987,2003 Magnétorésistance géante 2007
7 Densities Organics, fullerenes, novel compounds FM-I FM-M FM-insulator AF-insulator AF-M CMR-manganites AF-insulator SC metal High-T c cuprates insulator / semiconductor / metal SC metal Doped SrTiO 3 semiconductor (Wigner, FQHE) Silicon, GaAs n 2D ( e / cm 2 ) largest polarization reached by the field effect in oxides
8 Need to understand interactions!
9 23 10 particules + Interactions: Crucial fundamental problem ``non Fermi liquids tomorrow s materials Pb Ferroelectrics, High Tc, Manganites, Oxydes, Organics,, nanomaterials... 1 μm
10 Future electronic Need to worry about reduced dimensionality
11 One dimension is specially interesting No individual excitation can exist (only collective ones) Strong quantum fluctuations Difficult to order
12 One dimensional systems
13 Organic conductors b c D. Jaccard et al., J. Phys. C, 13 L89 (2001)
14 CARBON NANOTUBES Cees Dekker
15 Quantum wires O.M Ausslander et al., Science (2001)
16 Spin chains and ladders Spin chains Ladder systems
17 Spin dimer systems B. C. Watson et al., PRL (2001) M. Klanjsek et al., PRL (2008) B. Thielemann et al., arxiv: (2008)
18 Cold atoms
19 Cold atoms superfluid insulator T. Stoferle et al. PRL (2004) M. Greiner et al. Nature (2002) 1D physics (Luttinger Liquids)
20 Cold atoms: Interferences s 0 L h à y (r) Ã(0) i S. Hofferberth et al. Nat. Phys (2008) but K large (42) ½» ½ 0 hydrodynamic?
21 Methods Analytic methods: Exact methods (Bethe ansatz) Field theory (Bosonization) Conformal invariance Numerical methods: DMRG (real time dynamics of quantum systems)
22 Finite temperature DMRG Specific heat of spin chain
23 How to make a theory Tomonaga Luttinger Good excitations: collective ones 60 : A. Larkin, I. Dzialoshinskii, L. Gorkov, Bichkov E. Lieb, D. Mathis 70 : A. Luther, V.J. Emery 80 : F.D.M. Haldane
24 Luttinger liquid
25 S(q,!) J.S. Caux et al PRA (2006)
26 Luttinger parameters
27 Luttinger parameters
28
29 Attractive Hubbard model TG + B. S. Shastry PRB (1995)
30 Spin dimer systems B. C. Watson et al., PRL (2001) M. Klanjsek et al., PRL (2008) B. Thielemann et al., arxiv: (2008)
31 Luttinger parameters M. Klanjsek et al., PRL (2008) Red : Ladder (DMRG) Green: Strong coupling (J r 1 ) (BA)
32 Correlation functions M. Klanjsek et al., PRL (2008) R. Chitra, TG PRB (97); TG, AM Tsvelik PRB (99) NMR relaxation rate: Tc to ordered phase: 1/J = Â 1D (Tc)
33 NMR M. Klanjsek et al., PRL (2008)
34 LL: Cold atoms
35
36
37
38
39 Fractionalization
40 Neutron scattering < S r,t S 0,0 >
41
42 Conformal theory Finite temperature b c
43 Fermions Right (+k F ) and left (-k F ) particles
44 Systems with «spins»
45 Spin-Charge Separation Spin Charge Spinon Holon
46 Spin-Charge Separation higher D? Spin Charge Energy increases with spin-charge separation Confinement of spin-charge: «quasiparticle»
47 O.M Ausslander et al., Science (2001) Y. Tserkovnyak et al., PRL (2002) Y. Tserkovnyak et al., PRB (2003)
48 Proposal 87 Rb F=2,m F =-1i and F=2,m F =1i good potential candidates to observe spin-charge separation No problem of temperature ( fermions) Phase separation
49 A. Kleine, C. Kollath et al. PRA (2008); NJP (2008)
50 Effect of a lattice: Mott transition
51 How to treat? Incommensurate: Q 2 0 Commensutate: Q = 2 0
52 Competition dxd S = [ 1 ( ( x, )) 2 u( ( x, )) 2 ] x 2 K u Beresinskii- Kosterlitz-Thouless transition K=2
53 Mott insulator: is locked TG, Physica B (97) Gap in the excitation spectrum Density is fixed T. Kuhner et al. PRB (2000) n(k)
54
55 Beyond Luttinger liquids Luttinger liquids: allows to treat additional effects 1D additional perturbation: Lattice (Mott transition), disorder (Bose glass) etc. Multicomponents, mixtures,.. TG, cond-mat/ (Salerno lectures) + book
56
57 Effect of disorder
58 Disorder: Bose Glass TG + H. J. Schulz EPL (1987); PRB (1988) ``Two fourier components of disorder
59 Backward scattering (q» 2 0 ) Responsible for localization (Bose Glass) Pinning of a CDW of bosons Tonks limit: Anderson localization of free fermions
60 Bose glass phase 1D : TG + H. J. Schulz PRB (1988) Superfluid Localized (Bose glass) transition for K < 3/2 BKT like transition Higher dimensions: M.P.A. Fisher et al. PRB (1989) Bose glass also exists (scaling theory) continuous transition
61 Phase diagram D b Bose Glass Superfluid 1 3/2 K
62 Numerics S. Rapsch, U. Schollwoeck, W. Zwerger EPL (1999)
63 Experiments with interactions! L. Fallani et al. PRL 98, (2007)
64 Beyond Luttinger liquids Luttinger liquids: allows to treat additional effects 1D additional perturbation: Lattice (Mott transition), disorder (Bose glass) etc. Multicomponents, mixtures,.. TG, cond-mat/ (Salerno lectures) + book Major deviations from Luttinger liquids
65 Luttinger liquid + bath or noise
66 Quantum physics with a bath Reasonable (?) understanding of isolated systems Many systems are subjected to an external bath How is the physics modified?
67 Superconducting wire + gate
68 Trapped ions Noise on the electrodes: 1/f noise Disspative bath: laser cooling
69 Polar molecules Noise: EM waves Bath? : [immersion in a condensate]
70 Physics? Phase transition? E. Dalla Torre, E. Demler,TG,,E. Altman, Arxiv/
71 Coupled 1D chains: deconfinememt
72 Mott vs. Josephson R R Josephson coupling: delocalizes atoms Mott potential: localizes atoms AF Ho, M.A. Cazalilla, TG, PRL (2004); NJP (2006).
73 J/ Phase diagram transverse hopping Anisotropic 3D SF (BEC) repulsion D MI K 2D MI ( = ) ( = ) ( = ) ( = ) Array of atomic quantum dots
74
75 Activation energy, T* (K) Deconfinement c 600 TMTTF 2 PF a Spin-Peierls C SDW IC SDW T* b c Pressure (kbar) TG Chemical Review (2004) P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004) A. Pashkin, M. Dressel, M. Hanfland, C. A. Kuntscher, arxiv/ D. Jaccard et al., J. Phys. C, 13 L89 (2001)
76 Back to the (self-consistent) bath S. Biermann, A. Georges, A. Lichtenstein, TG, PRL (2001) C. Berthod et al. PRL 97, (2006) Bath must be self consistently determined
77 Non Luttinger liquids
78 Interacting 1D system (Luttinger liquid): e.g. bosons -- ``spin up Add one particle of ``spin down, or flip a spin G? (x; t) = h* js + (x; t)s (0; 0)j *i Example: two component bose-hubbard model
79 Ferromagnetic liquid M. B. Zvonarev, V. V. Cheianov, TG, PRL (2007)
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