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

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

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

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

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

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)]

... quantum wires, carbon nanotubes, etc.

Here: focus on charge distribution Charge-density waves

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)]

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

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 ):

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

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 

Charge-density correlation function    K F K F x x)x)k A xx xk A x K x nn 4 2 2/ cos( 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

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

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

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)]

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

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)]

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?

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

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)]

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

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

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)

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?

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 *

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

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