1. Ondas de densidade de carga em 1D: Hubbard vs. Luttinger? Thereza Paiva (UC-Davis) e Raimundo R dos Santos (UFRJ) Work supported by Brazilian agenciesand

2. Outline Motivation Luttinger liquid description Hubbard model Hubbard superlattices Conclusions (References)

3. Motivation Strongly correlated electrons: interplay between charge and spin degrees of freedom determines magnetic and transport (including superconducting) properties

4. Quasi-2D example: high T c superconductors Striped phase?

5. Stripes in CuO 2 planes [from Kivelson et al., (‘99)] Direction of charge modulation

6. 1D examples: organic conductors,… [from Gruner (‘94)] Chain direction SeC F P Spin density waves disappear for P ~ 6.0 kbar and triplet superconductivity sets in [Lee et al. (00)]

7. ... quantum wires, carbon nanotubes, etc.

8. Here: focus on charge distribution Charge-density waves

9. Well known example of CDW: the Peierls instability Electron-phonon coupling leads to a modulation of the charge distribution: Dynamics of collective modes     (x,t)  e.g., TTF-TCNQ, NbSe 3,... Here: interested only in effects of e - -e - interactions on CDW’s [from Gruner (88)]

10. Luttinger Liquid (LL) description Excitations: Fermi Liquid theory Fermi gasFermi liquid (interactions on) quasi-particles are fermions n  FF  FF n T=0 OK in 3D ? in 2D Breaks down in 1D (Peierls instability)  need new framework

11. The Luttinger model [Voit (‘94)] q  kF kF kF kF g2g2 kFkF kF kF q g4g4  Linear dispersion  Gapless excitations  Forward scattering (i.e. momentum transfer q << 2k F ):

12. Effect of dimensionality and spin-charge seperation: Let us inject an e - in 2 nd plane-wave state, |  2 , above Fermi surface g 4 only connects |  2  to |  1 , the 1 st plane wave state above Fermi surface Effective Hamiltonian in this subspace: Thus, g 4 irrelevant (RG: L   ) for d=3, but marginal for d=1

13. Diagonalizing H 4,eff yields u  = v F + g 4 /2   velocity of charge excitations u  = v F  g 4 /2   velocity of spin excitations u   u   spin-charge separation Solution of the Luttinger model Note low-T specific heat for fermions: C ~ T c.f., low-T specific heat for d-dimensional bosons with   k s : C ~ T d/s  linear for d=s=1 Quasi-particles are bosons  soluble via bosonization 

14. Charge-density correlation function    K F K F x x)x)k A xx xk A x K x nn 4 2 2/3 1 1 2 4cos( ln )2cos( )( )()0(    K  is a non-universal (interaction-dependent) exponent  2k F   n, where n is electron density 2k F dominates if 1  K   4K   K   1/3

15. Other measurable quantities – Specific heat: C =  T where 2  =  0 v F [u  -1 + u  -1 ], with  0 = 2  k B 2 / 3 v F – Spin susceptibility :  = 2 K  /  u  – Compressibility:  = 2 K  /  u  – Drude weight (DC conductivity): D = 2 u  K  Parametrization of theory (u , K  ) and (u , K  ) depend on the coupling constants g 2 & g 4

16. The Luttinger Liquid conjecture The LL is believed to provide the (gapless) low-energy phenomenology for all 1D metals

17. LL theory of single-wall metallic nanotubes: dielectric constant tube length tube radius  g ~ 0.2; c.f. g = 1 for Fermi gas LL behaviour observed through tunnelling experiments [see Egger et al. (‘00)]

18. The Hubbard model Simplest lattice model to include correlations :  Tight binding with one orbital per site  Coulomb repulsion: on-site only  Nearest neighbour hoppings only Bethe ansatz solution [Lieb & Wu (‘68)]  Ground state but not correlation functions

19. Connection with LL [Schulz(90)]: system size Calculated from Bethe ansatz solution  K  (n,U) K   1/2  2k F charge mode dominates over 4k F c.f. early Renormalization Group predictions [Sólyom(‘79)]

20. Quantum Monte Carlo (world-line) simulations [Hirsch & Scalapino (83,84)]:  first suggestions of 4k F charge mode dominating over 2k F as U increases  attributed to finite-temperature effects; should not prevail at lower temperatures Is it really so?

21. x  = M   NsNs M The space–imaginary-time lattice for QMC simulations The “minus-sign problem”: Sign of det ·det 

22. T  0: Quantum Monte Carlo (determinantal) simulations Charge susceptibility: As U increases, 4k F susceptibility still grows as T  0, while 2k F seems to stabilize. (N s  36 sites) Neither finite-size nor finite- temperature effects: simulations with   N s  96  N(4k F )  ln  n  1/6 [Paiva & dS (00a)]

23. T  0: Lanczos diagonalizations on finite-sized lattices is not …and is not a finite-size effect: cusps get sharper as N s increases As U increases the cusp moves towards 4k F... n  1/6

24. The same happens for other occupations n  1/3 n  1/2

25. Thus, 4k F charge mode indeed dominates over 2k F, at least for sufficiently large values of U. Agreement with LL description: 2k F amplitude A 1 (n,U)  0 for U  U  (n) Schematically: n 1 0 U 2kF2kF 4kF4kF U  (n)

26. Hubbard superlattices Model for layered systems [Paiva & dS (96)]: e.g., (thin) magnetic metallic multilayers U  0U  0 L0L0 LULU Interesting magnetic behaviour and metal-insulator transitions [Paiva & dS (‘98,’00)]; see also LL superlattices [Silva-Valencia et al. (‘00)]. With attractive interactions leads to coexistence between superconductivity and magnetism [Paiva (‘99)] Which is the dominant CDW mode?

27. Important parameter is # of electrons per cell: n eff  n (L 0  L U ) Define 2k F *   n eff  cusp is located at 4k F *

28. Conclusions For sufficiently large values of U, 4k F charge mode dominates over 2k F The LL description can only be made consistent if the amplitude of the 2k F mode vanishes For Hubbard superlattices the same results apply, with redefined n eff and k F * talk downloadable from http://www.if.ufrj.br/~rrds/rrds.html

29. References R Egger at al., cond-mat/0008008 G Grüner, Rev.Mod.Phys. 60, 1129 (1988) G Grüner, Rev.Mod.Phys. 66, 1 (1994) J E Hirsch and D J Scalapino, Phys.Rev.B 27, 7169 (1983) J E Hirsch and D J Scalapino, Phys.Rev.B 29, 5554 (1984) S Kivelson et al., cond-mat/9907228 I J Lee et al., cond-mat/0001332 E H Lieb and F Y Wu, Phys.Rev.Lett. 20, 1445 (1968) T Paiva, PhD thesis, UFF (1999) T Paiva and R R dos Santos, Phys.Rev.Lett. 76, 1126 (1996) T Paiva and R R dos Santos, Phys.Rev.B 58, 9607 (1998) T Paiva and R R dos Santos, Phys.Rev.B 61, 13480 (2000) T Paiva and R R dos Santos, Phys.Rev.B 62, 7004 (2000) H J Schulz, Phys.Rev.Lett. 64, 2831 (1990) J Silva-Valencia, E Miranda, and R R dos Santos, preprint (2000) J Sólyom, Adv.Phys. 28, 209 (1979) J Voit, Rep.Prog.Phys. 57, 977 (1994)