Persistent currents in two dimension: New regimes induced by the interplay between electronic correlations and disorder Zoltán Ádám Németh Jean-Louis Pichard CEA - Saclay, Service de Physique de l'Etat Condensé
Outline: • Overview of the strongly correlated 2D electron-gas problem; • Introduction of the numerical model: few interacting particles on a lattice • Persistent current maps with disorder References: Z.Á. Németh and J.-L. Pichard, Eur. Phys. J. B 45 , 111 (2005) J.-L. Pichard and Z.Á. Németh, J. Phys. IV France 131 , 155 (2005)
Quantum solid state physics • Fermi: weakly-interacting quantum particles • Wigner: strongly interacting particles TWO LENGTH SCALES: • average interparticle spacing a • Bohr-radius a B Dimensionless scaling parameter: r s = a/a B
Weak interaction limit • Electrons localized in k -space (Fermi liquid behavior). → 0 high density limit, r s kinetic energy ( ) [ ] a a = + + × 1 2 ln Fermi E O r NRy 0 2 s r r s s exchange energy a 1 = 2.0 a 2 = –1.6972 M. Gell-Mann and K. A. Brueckner Phys. Rev. 106 , 364 (1957)
Strong interaction limit • Electrons localized in real space (Wigner crystal) → ∞ low density limit, r s classical elctrostatic energy ( ) [ ] c c = + + − 2 × 1 3 / 2 Wigner E O r NRy 0 3 / 2 s r r s s quantum zero-point motion c 1 = –2.21 c 3/2 = 1.63 W. J. Carr Jr., Phys. Rev. 122 , 1437 (1961)
Quantum Monte-Carlo • Fixed node Monte-Carlo GS energy polarized method: liquid solid B. Tanatar and D. M. Ceperley PRB 39 , 5005 (1989) unpolarized liquid r s r s = 37±5
Semiconductor heterostructures Since the ’70s it is possible to fabricate 2D electron gas in semiconductor devices. Electron density and r s are varied through voltage gates. Example: quantum point-contact
Unexpected metallic behavior in 2D In ultra-clean heterostructures: resistivity r s can reach ≈ 40 Observed metallic behavior at low density intermediate r s • S.V. Kravchenko et al. , PRB 50 , 8039 (1994) • J. Yoon et al. , PRL 82 , 1744 (1999) intermediate density temperature
Hybrid phase in QMC Density-density correlation function r s Hybrid phase: nodal structure of Slater determinants in the crystal potential: mixed liquid-solid behavior H. Falakshahi, X. Waintal: Phys. Rev. Lett. 94 , 046801 (2004)
Theory for intermediate r s Still mainly speculations... • Andreev-Lifshitz « supersolid » state (relation with He -physics) • Inhomogeneous phases, stripes and bubbles ( B. Spivak ) Relation with the physics of high- T c cuprates and with Hubbard model (high lattice filling, contact interaction).
Lattice model N spinless fermions on L × L square lattice with periodic boundary conditions (lattice spacing s ): ˆ ˆ ⎛ ⎞ n n U ∑ + = ⋅ + ⋅ + ε ˆ ˆ ˆ H 4 ∑ ∑ ⎜ ⎟ i j t N- c c W n ⎝ i j ⎠ 2 j j d < > ≠ i,j i j j ij discrete Laplacian disorder the t and U parameters of THE r s AND r l PARAMETERS: this Hamiltonian: UL a r l = = → l r r s π 2 2 t 2 a N h e B = = U t 2 2 ms s
Persistent current Longitudinal and transverse currents + π Φ Φ = Ψ 2 Ψ •local: ( ) 2 Im long i L j c c e + 0 ( 1 , 0 ) 0 j j j 1 = ∑ •total: ∆ Φ = Φ = − Φ ~ ( ) ( 0 ) ( ) long I j E E E 0 0 0 long j L j Transverse current is analogous.
Persistent currents with disorder • Effects of an infinitesimal disorder: new lattice perturbative regime – Ballistic motion – Coulomb Guided Stripes – Localization if the Wigner crystal
Strong interaction, lattice regimes Can be relevant in real materials e.g. Cobalt-oxides (Na x CoO 2 ) • effective mass: m*/m = 175 relative dielectric constant: ε r = 20 • • lattice spacing s = 2.85 Å Lemmens et al. , cond-mat/0309186 • carrier density depends on Na + concentration
Lattice perturbation theory when t → 0 Example: the persistent current perturbation ⎛ ⎞ n n U + = ⋅ + ⋅ H 4 ∑ ∑ ⎜ ⎟ i j t N- c c -t ⎝ i j ⎠ 2 d < > ≠ i,j i j ij -t ( ) ( ) Φ = = − + 0 2 cos cos E t K K eff x y -t ⎛ ⎞ π ⎛ ⎞ Φ ⎛ ⎞ N Φ = = − ⎜ + + ⎟ ⎜ ⎟ 2 cos cos ⎜ ⎟ 0 E t ⎜ K K ⎟ 2 ⎝ ⎠ eff x y rigid hopping ⎝ ⎠ ⎝ ⎠ L π 2 9 Φ ⎛ ⎞ ( ) Φ = − Φ = ≈ ~ 0 ⎜ ⎟ 0 I E E t 2 ⎝ ⎠ 2 lattice eff L 3 6 1296 t L t eff = where teff eff describes the rigid hopping of the three particle π 2 2 49 « molecule » U
Ballsitic motion (BWM) Effective Hamiltonian: ∑ + = − − H 4 Nt E t C C ' eff Coul eff j j , ' j j N t − 3 3 ~ Persistent current map Density map N t L − 1 eff N U C + j C ' j N=3 L=9 W=0.01 t=1 U=300
Coulomb Guided Stripes (CGS) Disorder correction in the effective Hamiltonian: ∑ ∑ ˆ + = − − + ε H 4 ' Nt E t C C W N ' eff Coul eff j j j j , ' j j j 2 / 3 π Φ L cos( 2 ) t ∆ Φ ~ ( ) ~ eff I E 0 − 2 / 3 1 long L W N t − 3 3 ~ N t L − 1 eff N U current of a collective motion N=3 L=9 W=1 t=1 U=1000 = transverse current: I I trans long
Localized Wigner Molceule (LWM) Standard perturbation therory: − 3 3 π Φ cos( 2 ) L L t L ∆ Φ ~ ( ) ~ I E 0 − 1 long L U current of independent particles N=3 L=6 W=20 t=1 U=1000 = 0 transverse current: I trans
Numerical check of the different regimes L=6, t=1 L=6, W=1 ballistic stripe stripe localized localized longitudinal & transverse longitudinal
Phase diagram for weak disorder Critical lines: 2 W stripe ∝ t 6 L localized 2 t U 1 / 2 W loc ∝ t stripe 3 / 2 L liq. hyb. solid 1 / 2 t U ballistic continuum lattice
In case of long-range interaction, lattice models without disorder exhibit a latice-continuum transition.
Continuum perturbation theory when r s → ∞ Zero point motion in the vibrating mode of the molecule 2nd order expansion around equilibrium r r 2 ( ) h ˆ ≈ − ∇ + ∇ + ∇ + + ⇒ 2 2 2 H E X M X D 1 2 3 2 el m ∂ 2 2 ( ) ( ) 6 h ∑ ≈ − + + χ 2 + χ 2 + χ 2 + χ 2 H 10 4 E B B 3 4 5 6 ∂ χ 2 2 el m α = 1 α Longitudinal modes Transverse modes 2 π 6 e = where B 3 24 D ( ) 20 8 = + ω + ω B B h E E ω = ω = = 0 K el L T L T m m
Limit for the zero point motion electrostatic energy − = ( , , ) ( , , 0 ) E L U t E L U t E 0 : ground state energy = ( , , ) 0 0 F N L U t = ( , 0 , ) E L U t F N : scaling function 0 Lattice behavior: Example: Three spinless fermions on L × L lattice − = 4 E E Nt 0 el L = 18 L = 15 L = 12 Harmonic vibration of the L = 9 solid in the continuum: L = 6 3 = 0 . 2327 N = 3 F r l r l = UL/t
Persistent currents with disorder Do we see similar thing with disorder? • Effect of an intermediate disorder in the continuum limit: – Coulomb guided stripes – Level crossing and supersolid behavior
Coulomb guided stripe on a lattice N=3 L=9 W=1 t=1 U=1000 Parallel density and current Continuum version of the Coulomb guided stripe N=3 L=9 W=1 t=1 U=50
Crossover regime Disconnected current and density N=3 L=9 W=1 t=1 U=15 N=3 L=9 W=1 t=1 U=7 Legett’s rule: 1D motion diamagnetic means even number of particles Sign of supersolid Sign of supersolid
Crossover regime Disconnected current and density N=3 L=9 W=1 t=1 U=15 N=3 L=9 W=1 t=1 U=7 Legett’s rule: 1D motion diamagnetic means even number of particles Sign of supersolid Sign of supersolid
Level crossing & strong disorder Weak disorder The presence of a diamagentic level crossing is paramagnetic specific to N. Strong disorder Crossover: W glass ~ U
Phase diagram ( N=3 ) glass localized stripe liq. hyb. solid ballistic continuum lattice
Conclusion • In the presence of a weak disorder, we have identified for large U/t three lattice regimes, characterized by different power-law decays as a function of U , t and W , L , N . • The physics of the continuum is also affected by disorder (signatures of the supersolid).
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