Ondas de densidade de carga em 1D:

Hubbard vs. Luttinger?

Thereza Paiva (UC-Davis)

e

Raimundo R dos Santos (UFRJ)

Work supported by Brazilian agencies and

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 Tc superconductors

Striped phase?

Stripes in CuO2 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 superconductivitysets 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 couplingleads to a modulation of thecharge distribution:

Dynamics of collective modes (x,t)

t

ejCDW

e.g., TTF-TCNQ, NbSe3,...

Here: interested only in effects of e--e- interactions on CDW’s

[from Gruner (88)]

Luttinger Liquid (LL) description

• Excitations: Fermi Liquid theoryFermi gas Fermi liquid (interactions on)

quasi-particlesare fermions

n

FF

nT=0

• OK in 3D• ? in 2D• Breaks down in 1D (Peierls instability) need new framework

• The Luttinger model [Voit (‘94)]

q

kF

kF

g2

kF

kF

qg4

Linear dispersion

Gapless excitations

Forward scattering (i.e. momentum transfer q << 2kF):

Effect of dimensionality and spin-charge seperation:• Let us inject an e- in 2nd plane-wave state, |2, above Fermi surface• g4 only connects |2 to |1, the 1st plane wave state above Fermi surface

Effective Hamiltonian in this subspace:

L

v

L

gL

g

L

v

HF

d

dF

eff

42

24

4

4

,4

Thus, g4 irrelevant (RG: L ) for d=3, but marginal for d=1

Diagonalizing H4,eff yields

u= vF + g4/2 velocity of charge excitations

u= vF g4/2 velocity of spin excitations

u u spin-charge separation

Solution of the Luttinger modelNote low-T specific heat for fermions: C ~ Tc.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

KF

KF

x

x)kA

xx

xkA

x

Kxnn 422/31 12

4cos(

ln

)2cos(

)()()0(

K is a non-universal (interaction-dependent) exponent

2kF n, where n is electron density

2kF dominates if 1K 4K K 1/3

• Other measurable quantities– Specific heat: C = T

where 2 = 0 vF [u-1

+ u -1], with 0 = 2 kB

2 /3vF

– 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 g2 & g4

• 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:

2/12

)2/ln(8

1

R

v

eg

F

dielectric constanttube 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

i

iiiii

ii nnUcccctH ][ ,,1,

,1,

• Bethe ansatz solution [Lieb & Wu (‘68)] Ground state but not correlation functions

• Connection with LL [Schulz(90)]:

K

u

n

nE

L 2

)(12

02

system size

Calculated from Bethe ansatz solution

K (n,U)

K 1/2 2kF charge mode dominates over 4kF

c.f. early Renormalization Group predictions [Sólyom(‘79)]

• Quantum Monte Carlo (world-line) simulations [Hirsch & Scalapino (83,84)]:

first suggestions of 4kF charge mode dominating over 2kF as U increases

attributed to finite-temperature effects; should not prevail at lower temperatures

Is it really so?

x

= M

Ns

M

The space–imaginary-time lattice for QMC simulations

)(det)(det OOTrZ

/

/

s

sAA

The “minus-sign problem”:

Sign of det ·det

T 0: Quantum Monte Carlo (determinantal) simulations

Charge susceptibility:

As U increases, 4kF susceptibility still grows as T 0, while 2kF

seems to stabilize. (Ns 36 sites)

Neither finite-size nor finite-temperature effects: simulations with Ns 96

N(4kF) ln

n 1/6

[Paiva & dS (00a)]

0,0 0,2 0,4 0,6 0,8 1,0

-0,10

-0,05

0,00

0,05

0,10

0,15

0,20

U=0 U=2 U=4 U=6 U=9 U=12 U=20

n =1/6 NS=24

C(q

)

q/

T 0: Lanczos diagonalizations on finite-sized lattices

…and is notis not a finite-sizeeffect: cusps get sharperas Ns increases

As U increases the cuspmoves towards 4kF ...

n 1/6

0,0 0,2 0,4 0,6 0,8 1,0

0,00

0,05

0,10

0,15

0,20

U=3 NS=12

U=12 NS=12

U=3 NS=24

U=12 NS=24

n =1/6 NS=12 and NS=24

C(q

)

q/

The same happens for other occupations

n 1/3

n 1/2

0,0 0,2 0,4 0,6 0,8 1,0-0,2

-0,1

0,0

0,1

0,2

0,3

U=0 U=4 U=6 U=8 U=9 U=12 U=20

Ns=18 n=1/3

C(q

)

q/

0,0 0,2 0,4 0,6 0,8 1,0

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

U=0 U=3 U=6 U=9 U=12

Ns=16 n=1/2

C(q

)

q/

Thus, 4kF charge mode indeed dominates over 2kF, at least for sufficiently large values of U.

Agreement with LL description: 2kF amplitude A1(n,U) 0 for U U (n)

Schematically:

n1

0

U

2kF

4kF

U (n)

Hubbard superlattices

• Model for layered systems [Paiva & dS (96)]: e.g., (thin) magnetic metallic multilayers

U 0 U 0

L0 LU

• 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?

0,2 0,4 0,6 0,8 1,00,0

0,1

0,2

0,3

0,4 (c) L0=3

q/

0,0

0,1

0,2

0,3

0,4

0,5

(a) L0=1

n=1/6 Ns=24LU=1 U=12

C(q

)

(b) L0=2

0,2 0,4 0,6 0,8 1,0

(d) L0=5

q/

Important parameter is # of electrons per cell:

neff n (L0 LU)

Define 2kF

* neff

cusp is located at

4kF*

Conclusions

• For sufficiently large values of U, 4kF charge mode dominates over 2kF

• The LL description can only be made consistent if the amplitude of the 2kF mode vanishes

• For Hubbard superlattices the same results apply, with redefined neff and kF

* talk downloadable from

http://www.if.ufrj.br/~rrds/rrds.html

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)