[PPT] - Correlation effects in transport through quantum wires: a PowerPoint Presentation

SLIDE 1

Correlation effects in transport through quantum wires: a functional renormalization group approach

Kurt Sch¨

nhammer

Institut f¨ ur Theoretische Physik Universit¨ at G¨

ttingen

SLIDE 2

Collaborators

from Stuttgart
Sabine Andergassen
Tilman Enss
Walter Metzner
from Aachen
Ulrich Schollw¨
ck
from G¨
ttingen
Volker Meden
Xavier Barnab´

e-Th´ eriault

Abdelouahab Sedeki

SLIDE 3

Outline

Experimental systems — carbon nanotubes and other quantum wires
Transport problems to be studied — correlations and impurities
The method — functional renormalization group
A single impurity — power-laws and one-parameter scaling
Resonant tunneling — “universal” and “non-univeral” behavior
Transport in more complex geometries — Y-junctions, X-junctions, . . .
Summary and outlook

SLIDE 4

Experimental systems

single-walled carbon nanotube

(theory ’97: Egger & Gogolin; Kane, Balents, & Fisher) (Dekkers group)

ther quantum wires
semiconductor heterostructures
chains of atoms on surfaces
. . .

SLIDE 5

Transport problems:

junction

lead impurity contact gate voltage t U

ingredients:

1d quantum wire
electron correlations — Tomonaga-Luttinger liquid
non-interacting leads
contacts
impurities, gate voltage
more complex geometries

SLIDE 6

Scattering on 1d lattices: noninteracting fermions

The Hamiltonian for a single particle on the lattice is given by H = Hleads + VLR + Hs, with Hleads = −

n=−∞

(|n − 1n| + H.c.) −

∞

n=N+1

(|nn + 1| + H.c.) VLR = tL (|01| + H.c) + tR (|NN + 1| + H.c.) , and Hs is an arbitrary Hamiltonian in the scattering segment from site 1 to N which is connected to ideal leads with nearest neighbor hopping equal t = −1, corresponding to the energy dispersion ǫk = −2 cos k and the bandwidth B = 4. The transmission probability |t(ǫk)|2 can be expressed in terms of the full resolvent as |t(ǫk)|2 = 4t2

Lt2 R sin2 k|N|G(ǫk + i0)|1|2

SLIDE 7

Landauer-B¨ uttiker-formula

For spinless fermions on the lattice the Landauer-B¨ uttiker formula for the stationary current reads ˆ j∞ = e h B/2

−B/2

[fL(ǫ) − fR(ǫ)] |t(ǫ)|2dǫ , where the fa(ǫ) =

eβa(ǫ−µa) + 1

−1 are the Fermi functions of the leads. For βR = βL the linear conductance G(T) is given by G(T) = e2 h B/2

−B/2

(−d f dǫ)|t(ǫ)|2dǫ

T =0

= e2 h |t(µ)|2

SLIDE 8

Example for Hs: quantum dot with ND sites

dot region: 1 < jl ≤ n ≤ jr < N, jr = jl + ND − 1 Hs = ˜ Hs + (tl|jl − 1jl| + tr|jrjr + 1| + H.c.) , where ˜ Hs has no matrix elements connecting the dot to the wire. The sites neighboring the dot are given the indices ˜ ja, i.e. ˜ jl = jl − 1, ˜ jr = jr + 1. With (Λa,∆a real functions) Γa(ǫ + i0) = t2

a˜

ja|(ǫ + i0 − Hta=0)−1|˜ ja = Λa(ǫ) − i∆a(ǫ) the transmission probability can then be expressed as |t(ǫ)|2 = 4∆l(ǫ)∆r(ǫ)|Gjr,jl(ǫ + i0)|2

SLIDE 9

Example continued: resonant tunneling

For well separated dots (|ta| ≪ 1) ND narrow resonances

ccur near the

eigenvalues ǫα of the isolated dot. For ǫ ≈ ǫα one obtains the generalized Breit-Wigner form |t(ε)|2 ≈ 4∆(α)

l

(ε)∆(α)

r (ε)

ε − ǫα − Λ(α)

l

(ε) − Λ(α)

r (ε)

2 +

∆(α)

l

(ε) + ∆(α)

r (ε)

2 , where ∆(α)

a

= |ja|ǫα|2∆a and correspondingly for the Λ(α)

a

. Questions: What replaces the LB-formula when the interaction is turned on? What happens to the resonances when the interaction is turned on?

SLIDE 10

Meir-Wingreen-formula

Using the Keldysh nonequilibrium Green function technique Meir and Wingreen (PRL 68,2512 (1992)) derived a formula for the stationary current for the transport through an interacting region connected to noninteracting leads. For

ur geometry the (nonsymmetrized) stationary current (on the link 0 → 1) is

given by ˆ j∞ = e h B/2

−B/2

2i∆L(ǫ)

fL(ǫ) (Gr

11(ǫ) − Ga 11(ǫ)) + G< 11(ǫ)

dǫ,

which can easily shown to reduce to the LB-formula in the noninteracting case. The simplification under the special assumption discussed by MW cannot be used for our geometry (in the interacting model). The correction to the LB-formula as presented vanishes if Σr − Σa = 0 and Σ< = 0 holds (approximately).

SLIDE 11

Luttinger liquids with impurities

much is known from bosonization and exactly solved models

density response of homogeneous system:

(Luther & Peschel ’74; Mattis ’74)

χ(q ≈ 2kF) ∝ |q − 2kF|2K−2

spinless fermions, nearest neighbor interaction, half-filling:

(Haldane ’80)

K−1 = 2 π arccos

−U

2

local sine-Gordon model :

(e.g. Apel & Rice ’82)

perturbative RG for K < 1:

(Kane & Fisher ’92)

VkF ,−kF relevant; hopping between open ends irrelevant

numerics: no intermediate fixed point

(Moon et al. ’93; Egger & Grabert ’95)

Bethe ansatz: no intermediate fixed point

(Saleur et al. ’95)

alternative fermionic approach for weak two body interaction:

leading-log resummation for transmission

(Glazman et al. ’93)

SLIDE 12

Functional renormalization group

(Polchinski ’84, Wetterich ’93, Morris ’94, Salmhofer ’98, . . . )

idea

generating functional Γ of grand can. pot., self-energy, m-particle interaction
cutoff Λ in free propagator G0,Λ, cuts out infrared divergencies (if necessary)
take ∂Λ, expand in sources ⇒ exact hierachy of differential equations

formalism

“interacting” part of action:

Sint

{ ¯

ψ}, {ψ}

=
k′,k

Vk′,k ¯ ψk′ψk + 1 4

k′

1,k′ 2,k1,k2

¯ uk′

1,k′ 2,k1,k2 ¯

ψk′

1 ¯

ψk′

2ψk2ψk1

generating functional of connected Green functions:

Wc,Λ ({¯ η}, {η}) = ln 1 ZΛ

D ¯

ψψ e

¯

ψ,[G0,Λ]

−1ψ

−Sint({ ¯

ψ},{ψ})−( ¯ ψ,η)−(¯ η,ψ)

Legendre transformation:

ΓΛ {¯ φ}, {φ}

= −Wc,Λ

{¯ ηΛ}, {ηΛ}

−

¯ φ, ηΛ −

¯

ηΛ, φ

+
¯

φ,

G0,Λ−1 φ

SLIDE 13

differentiate:

∂ΛΓΛ = Tr

∂Λ
G0,Λ−1G0,Λ

− Tr

GΛ∂Λ
G0,Λ−1V

δ2ΓΛ δφδφ, GΛ

expand in sources:

ΓΛ {¯ φ}, {φ}

=

∞

m=0

(−1)m (m!)2

k′

1,...,km

γΛ

m (k′ 1, . . . , k′ m; k1, . . . , km) ¯

φk′

1 . . . ¯

φk′

mφkm . . . φk1

exact hierachy of flow equations:
with GΛ =
G0,Λ−1 + γΛ

1

−1 , SΛ = GΛ∂Λ

G0,Λ−1GΛ
and G0,Λ0 = 0 , G0,Λ=0 = G0 , ΓΛ0

{¯ φ}, {φ}

= Sint
{¯

φ}, {φ}

approximation: γΛ

mc+1 = γΛ0 mc+1 = 0 for mc ≥ 2; γΛ=0 m

correct to order mc

SLIDE 14

Transport in a quantum wire – fRG approach (linear response)

t U

set up:

cutoff in Matsubara frequency G0,Λ(iω) = Θ (|ω| − Λ) G0(iω), for T = 0
approximations:
γΛ

3 = γΛ=∞ 3

= 0

no self-energy corrections in ∂ΛγΛ

2

γΛ

2 frequency independent ⇒ ΣΛ freq. indep. ⇒ no bulk power-laws

interaction remains nearest neighbor

equations: ∂ΛΣΛ

j,j = − 1

2πU Λ GΛ

j+1,j+1(iΛ) + GΛ j−1,j−1(iΛ) + (iΛ → −iΛ)

∂ΛΣΛ

j,j±1 = 1

2πU Λ GΛ

j,j±1(iΛ) + GΛ j,j±1(−iΛ)

GΛ(iω) =
G0(iω)

−1 − ΣΛ − Σleads(iω) −1 , U Λ = f(U, Λ) Σleads(z) =

t2

l |1 1| + t2 r |N N|

G0,leads

boundary(z)

initial condition: ΣΛ=∞

j,j(±1) = Vj,j(±1)

conductance:

G(N) = e2 h |t(0, N)|2 , |t(ε, N)|2 ∼ |G1,N(ε)|2

SLIDE 15

Two examples for the effective scattering potential

400 450 500 550 600 j 1 2 V

j,j

⇒

400 450 500 550 600 j 1 2 Σ Σ j,j+1

j,j

400 450 500 550 600 j 0.2 0.4 0.6 0.8 1 V j,j+1

⇒

400 450 500 550 600 j 0.3 0.32 0.34 Σ j,j+1

long range oscillatory potentials → power law behaviour in |t(ǫ)|2

SLIDE 16

Comparison with DMRG for “small” N

G, no impurity, abrupt contacts:

1 2 3 4 5

U

0.2 0.4 0.6 0.8 1

G/(e

2/h)

DMRG fRG

N=12

no impurity, smooth contacts: G = e2

h

Friedel oscillations, open boundaries:

50 100

j

0.4 0.5 0.6 0.7

<nj>

DMRG fRG

N=128, U=1

nj requires its own flow-equation

(composite operator)

SLIDE 17

Limiting cases

infinitesimal imp., smooth contacts:

e2 h − G ∝ N 2(1−KRG)

2
1

1 2 U

1
0.5

0.5 1 1-K

exact leading order fRG

weak link from jimp to jimp + 1,

smooth contacts: ρj≈jimp ∝ N −αRG

B

⇒ G ∝ N −2αRG

B

2
1

1 2 U

1
0.5

0.5 1 αB

exact leading order fRG

KRG and αRG

B

“almost” fulfill the Tomonaga-Luttinger relation αB = 1/K−1

we can distinguish between 1 − K and 1/K − 1

SLIDE 18

Single impurity, smooth contacts — one parameter scaling

G = e2

h ˜

GK(x) , x = [T/T0(U, V )]K−1 or x = [N0(U, V )/N]K−1

10

2

10

1

10 10

1

x=[T/T0(U,V)]

K-1

10

3

10

2

10

1

10 G/(e

2/h)

U=0.5,n=1/2

1-x

2

x

2/K

10

1

10 10

1

10

2

x=[N0(U,V)/N]

1/2

10

8

10

6

10

4

10

2

10

G/(e

2/h)

U=2.23, n=1/2

0.0 0.2 0.4 0.6 0.8 1.0 |R| 10

6

10 10

6

10

12

N0(U,V) U=0.5 U=1 U=2.23

(V.M. et al. ’03)

leading-log approach always gives

K = 1 scaling function: ˜ GK=1(x) = 1 1 + x2

SLIDE 19

Transport at finite T — smooth contacts

10

4

10

3

10

2

10

1

10 10

1

T 10

3

10

2

G/(e

2/h)

U=0.5 U=0

V=10, N=10

4

T

2αB

10

4

10

3

10

2

10

1

10 10

1

T 10

3

10

2

10

1

10 G/(e

2/h)

V=10 V=1 V=0.1

U=0.5, N=10

4

T

2αB

V. Meden, T. Enss, S. Andergassen, W. Metzner, and K.S., PRB 71, 041302(R)

(2005)

SLIDE 20

“Universality”, but . . . 10

2

10

1

10 10

1

x=[T/T0(U,n,V)]

K-1

10

3

10

2

10

1

10 G/(e

2/h)

K=0.85

length band

SLIDE 21

Resonant tunneling I — experiment versus theory

(Dekkers group ’01)

theory

no exact solution for model with two local sine-Gordon terms
Gp(T)meas not consistent with UST exponent αB − 1 (Furusaki & Nagaosa ’93)
new suggestion: CST with exponent 2αB − 1

(Thorwart et al. ’02)

not found in “leading-log” method (Nazarov & Glazman ’03; Polyakov & Gornyi ’03)
not found in exactly solvable model

(Komnik & Gogolin ’03)

QMC data interpreted to be consistent with CST

(H¨ ugle & Egger ’03)

SLIDE 22

Resonant tunneling – Conductance as function of the gate voltage

gate voltage smooth

2

2 Vg 0.00 0.01 0.02 0.03 G/(e

2/h)

T=0.3 T=0.1 T=0.03 T=0

ND=6, Vl/r=10, U=0.5, N=10

4

at T = 0, width ∼ N K−1

SLIDE 23

Simplest case: ND = 1

V t’ t’

With g = µ − V − 2t′2Gaux({Σ}, ǫ = µ + i0) the transmission probability is

|t(0)|2 = (Im g)2/|g|2

0.5 1 1.5

Re g

0.5 1 1.5

Im g U=0

0.5 1 1.5

Re g

0.5 1 1.5

Im g U=0

“perfect chain” FP: stable for U < 0
“decoupled chain” FP: stable for U > 0

(Kane & Fisher ’92, Enss et al. ’05)

SLIDE 24

Resonant tunneling — “universal” scaling of Gp(T)

1 2 3 4 5

p

ln G (T) lnT

regime 1: e2

h − Gp(T) ∼ T 2 for T < ∆W

regime 2: e2

h − Gp(T) ∼ T 2K for ∆W < T < T ∗ (small ND, small Vl/r)

regime 3: Gp(T) ∼ T αB−1 and w(T) ∝ T for T ∗ < T < ∆D
regime 4: Gp(T) ∼ T 2αB for ∆D < T < B (large ND, large Vl/r)
regime 5: Gp(T) ∼ T −1 for B < T
T ∗ ∝ (t2

bar/ND)1/(1−αB)

SLIDE 25

Resonant tunneling — details for the scaling of Gp(T)

10

4

10

3

10

2

10

1

10 10

1

T

1
0.5

exponent 10

3

10

2

10

1

10 Gp(T)/(e

2/h)

ND=2 ND=100

Vl/r=10, U=0.5, N=10

4

2αB αB-1 2αB-1

10

4

10

3

10

2

10

1

10 10

1

T

1
0.5

0.5 1 exponent 10

1

10 Gp(T)/(e

2/h)

ND=10 ND=50

Vl/r=0.8, U=1.5, N=10

4

2αB αB-1 2αB-1

no indications of 2αB − 1 (“correlated sequential tunneling”)

SLIDE 26

Resonant tunneling — off resonance scaling

Vl/r = 10, U = 0.5, ND = 100, N = 104:

10

4

10

3

10

2

10

1

10 10

1

T

1

1 2 exponent 10

5

10

4

10

3

10

2

G(T)/(e

2/h)

∆Vg=0.04 ∆Vg=0.001

two different temperature regimes with exponent 2αB

SLIDE 27

Resonant tunneling — asymmetric barriers

Vl = 5, Vr = 50, U = 0.5, ND = 100, N = 104:

10

3

10

2

10

1

10 10

1

T

1
0.5

0.5 exponent 10

4

10

3

10

2

Gp(T)/(e

2/h)

ND=2 ND=20

SLIDE 28

X,Y,. . . -junctions

dot junction of Nleg ≥ 2 legs, made of N sites, smoothly coupled to leads
perfect and open chain fixed points of wire → more fixed points
example: symmetric case is a fixed point with GFP = e2

h

2

Nleg

2

scaling of spectral weight: ρdot ∼ N αB(1−2/Nleg)
more general junction (e.g. “ring” with flux)

(Nayak et al. ’99, Chen et al. ’02, Lal et al. ’02, Egger et al. ’03, Chamon et al. ’03, . . . )

SLIDE 29

Y-junction with flux – set up

1 2 3

V t t∆

Y

φ

single-particle scattering theory:

g = [−V − t2

Y Gaux ({Σ})]/|t△|

G1,2 = e2 h 4 (Im g)2 e−iφ − g

2

|g3 − 3g + 2 cos φ|2 G2,1 = e2 h 4 (Im g)2 e+iφ − g

2

|g3 − 3g + 2 cos φ|2

φ = 0.4 π

0.2 0.4 0.6 0.8 1 Re g Im g

3
2
1

1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.4 0.6 0.8 1 Re g Im g

3
2
1

1 2 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

SLIDE 30

Y-junction with flux – flow for U > 0

−3 −2 −1 1 2 3 Re g 0.5 1 1.5 2 Im g −3 −2 −1 1 2 3 0.5 1 1.5 2

φ=0.4π

−3 −2 −1 1 2 3 Re g 0.5 1 1.5 2 Im g −3 −2 −1 1 2 3 0.5 1 1.5 2

φ=0.1π

G1,2 = G2,1 = 0 G1,2 = G2,1 = 4

9 e2 h

G1,2 = e2

h , G2,1 = 0

on light blue curve: flow to blue FPs – new scaling dimension
general: flow to red line of FPs, but

|G1,2−G2,1| G1,2+G2,1 → 0

⇒ restoring time reversal symmetry by interaction

(Barnab´ e-Th´ eriault et al. PRL 94, 136405 (2005))

SLIDE 31

Summary . . .

fRG is reliable and flexible tool to study microscopic models of inhomogeneous

1d correlated electrons

provides simple physical picture for e.g. transport
allows to describe the interplay of contacts, impurities, and correlations
the method covers all energy scales
example: resonant tunneling — “universal” and “non-universal” behavior

. . . and outlook

consider more realistic models for the contacts and leads
consider more realistic models for the quantum wire
in the long run: combine “ab initio” methods with fRG
extend to non-linear transport

SLIDE 32

The role of more realistic contacts

interaction turned on abruptly at contacts
additional reduced hopping in and out of quantum wire

10

2

10

4

10

6

N

10

9

10

6

10

3

10 G/(e

2/h)

no imp., tl/r=1 timp=0.5, tl/r=1 timp=0.5, tl/r=0.7

U=1.5 N

2αB

N

4αB
what happens as a function of T?

SLIDE 33

Transport at finite T — abrupt contacts

10

4

10

3

10

2

10

1

10 10

1

T 10

2

10

1

10 G/(e

2/h)

tl/r=0.3 tl/r=0.8

U=1, N=10

4

T

αB

10

4

10

3

10

2

10

1

10 10

1

T 10

5

10

4

10

3

10

2

10

1

G/(e

2/h)

jimp=N/2 jimp=N/4 jimp=N/2-20 timp=0.1, tl/r=0.8, U=1, N=10

4

T

αB

T

2αB

10

4

10

3

10

2

10

1

10 10

1

T 10

2

10

1

G/(e

2/h)

jimp=N/2 jimp=N/4 timp=0.5, tl/r=0.8, U=1, N=10

4

T

αB

T

2αB

10

4

10

3

10

2

10

1

10 10

1

T 10

5

10

4

10

3

10

2

10

1

G/(e

2/h)

timp=0.1, tl/r=0.3, U=1, N=10

4

T

αB

T

2αB

SLIDE 34

Resonant tunneling V — details for ND = 1 (U = 1)

10

4

10

3

10

2

10

1

10 10

1

T

10

3

10

2

10

1

10

Gp(T)/(e

2/h)

tl/r=0.05 tl/r=0.1 tl/r=0.2 tl/r=0.5 tl/r=0.7

10

4

10

3

10

2

10

1

10 10

1

T

1.0
0.5

0.0 exponent of Gp(T)/(e

2/h)

αB-1

Γl(ε, T ) = t2

l Im ˜

G0

jl−1,jl−1(ε, T )

Ωl(ε, T ) = t2

l Re ˜

G0

jl−1,jl−1(ε, T )

|t(ε, T )|2 =

10

3

10

2

10

1

10 10

1

T 1 2 exponent of 1-Gp(T)/(e

2/h)

2K

4Γl(ε, T )Γr(ε, T ) [ε − Vg − Ωl(ε, T ) − Ωr(ε, T )]2 + [Γl(ε, T ) + Γr(ε, T )]2

SLIDE 35

Resonant tunneling — |t(ε, T, N)|2

gate voltage smooth

2
1

1 2 ε 0.0 0.5 1.0 |t(ε,T)|

2

T=10 T=1 T=0.1 T=0

ND=10, Vl/r=10, U=0.5, N=10

4, Vg=0.2404...

important T-dependence
2

2 Vg 0.00 0.01 0.02 0.03 G/(e

2/h)

T=0.3 T=0.1 T=0.03 T=0

ND=6, Vl/r=10, U=0.5, N=10

4

at T = 0, w ∼ N K−1