V.Yu. KachorovskiiIoffe Physico-Technical Institute, St.Petersburg, Russia

Co-authors:

Alexander Dmitriev (Ioffe)

Igor Gornyi (KIT/Ioffe)

Dmitry Polyakov (KIT)

Dmitry Aristov (KIT/Petersburg Nucl. Phys. Inst.)

Peter Wölfle (KIT)

Tunneling Into a Luttinger Liquid Revisited

Phys. Rev. Lett. 105, 266404 (2010)

Petersburg

Nuclear

Physics

Institute

1

Outline• Introduction:

experimental motivation why 1D is special..experimental motivation, why 1D is special

• Simplest model of the tunneling junction: . p g j•.point contact, tunneling Hamiltonian

Infinitesimally weak tunneling coupling:• Infinitesimally weak tunneling coupling:suppression of tunneling density of .states (conventional zero bias anomaly)

• Coupling is not infinitesimally weak new physics: non trivial zero bias anomaly intermediatephysics:.non-trivial zero bias anomaly, intermediate-

..coupling ..fixed point suppression vs enhancement“phase transition”

• Generic tunneling junction2

Single-channel quantum wires (aka Nanowires)

3

Single-channel quantum wires (aka Nanowires)

4

Single-channel quantum wires (aka Nanowires)

backscattering disorder = random interedge tunnelingbackscattering disorder random interedge tunneling1D barrier in 2D : Kang et al., Nature '00; Yang et al., PRL '04L-shaped quantum wells : Grayson et al., APL '05, PRB '07

5

Scaling of the tunneling conductance

Bockrath et al , Nature,1999 6

Interacting electrons in 1D Luttinger liquid exact excitations: plasmons (+spinons)

Fermionic description:

μ = (+,-) right/left movers,

linear dispersion relation, short-range interaction V0 , spinlesslinear dispersion relation, short range interaction V0 , spinless

dimensionless interaction strength

Bosonic description:

,

-plasmon velocity,Electron operators:

Zero bias anomaly Dzyaloshinskii& Larkin (1973)

Luther & Peschel; Luther & Emery (1974)

Kane & Fisher (1992)

“bulk” tunneling DOS

Kane & Fisher (1992) 

αb ≈ g2/2

Scaling of the tunneling conductance usual assumption: tunneling matrix element i i fi it i ll

Scaling of the tunneling conductancefor

is infinitesimally small !!! for

tunneling into the end α ≈ gof a semi-infinite wire

8

αe ≈ g

Setup

tunnel electrode: noninteracting single channeltunnel electrode: noninteracting, single channel

ti ltime reversal symmetry

quantum wire:quantum wire: interacting, single channel, spinless, ballistic,

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Contact: 3x3 scattering matrix1

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Symmetry: 1) time-reversal 2) 2 3

for finite tunneling !!!

Most general parametrization

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Simplest model of the tunneling contact

point-like contact described by the tunneling Hamiltonian

1) Noninteracting case: hat happens if t iswhat happens if t0 is

not small ?

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Tunneling Hamiltonian: noninteracting case

Specific choice of the S-matrix

single parameter

tunneling transparency of the contact

tunnel conductance

parameter

of the contact

1tunnel contact decouples

2 3

1

from the wire

all three 1

2 3channels decouple

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Tunneling Hamiltonian: small γ and g

Lowest-order processes leading to the renormalization of the reflection amplitude

GF in the wireGF in the electrode

opposite signln (Λ/|ε|) ln (Λ/|ε|)

13Both diagrams are logarithmically divergent renormalization of γ

RG in the weak-tunneling limit

t li t- tunneling transparency of the contact

non-trivial geometry, junctionof 3 wires (Das, Rao, Sen ’02, ’04)

renormalization of the bulk tunneling density of states

stable point

unstable point

renormalization of the density of states dominates

g/2

enhancement of the tunneling

g/

Physics behind: scattering off Friedel oscillations vs. bulk ZBA14

Friedel oscillations in 1D 11

Hartree potential oscillates with the periodtat

Exchange contribution — similar15

Leading-log summation: sums up the most divergent terms, , of the perturbation theory

Split the important interval, , on smaller pieces,

so that but

… …

Single impurity: fermionic RG for the backscattering and transmission

two fixed pointsβ-function in the first order in g

p

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Matveev, Yue, Glazman '93

Single impurity

Tunneling contact

Key difference: absent

Phase of the Friedel osc's is important

Weak impurity:

Point tunnel contact: 17

Renormalization of backscattering

Weak impurity:

Point tunnel contact:

18

Tunneling Hamiltonian (point contact): arbitrary γ

t – amplitudet0 – amplitudeentering tunn.Ham.

1) unstable point: γ = γ c  2) stable points: γ =0, γ =1,

1 1γ =0 γ =11/2

Phase transition ??? 2 3 2 319

Generic tunnel contactfermionic S-matrix RG: exact in S-matrix and perturbative in g

thick lines: exact noninteractingGF (fully dressed by the tunneling vertices)

One-loop skeletons (a),(b) contribute to the exact-in-γβ−function, diagrams (c)-(e) do not

For tunneling problem the quadratic inFor tunneling problem the quadratic in g terms has to be included into β-function

S-matrix of the tunnel contact is renormalizable to order g2

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Four (two)-parameter scaling for the tunnel contact

Equations for tb and tin are closed two “leading equations"

Point tunnel contact: , four RG ti “ ll " t i lequations “collapse" onto a single one

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In fact, γ =0 is not a stable point, it is a saddle point no phase transition!!!

true stable point:γ = 1, ϕ = 0three wires are

fdisconnected from each other

dashed line:impurity in the isolated wire

saddle pointvertical line:point contact,tunneling hamiltonian

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Conductances

time reversal and 2 3 symmetries 2 independent conductances:

current in the wire ifno current flows in the tunnel electrode

tunneling conductance

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RG flows for

ideal contact

truefixed point

unstable point

saddle point

Limiting curves:

1) - point tunnel contact, 2) - impurity in the isolated wire

24Fixed points bosonic RGKane, Fisher’92

Scaling of the tunneling conductance

1) Point contact

ideal contact

bulk ZBAenhanced ZBAenhanced ZBA

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2) Generic contact

ideal contact iti lit t dcontact criticality retards

ZBA

I - enhancement of ZBAI - enhancement of ZBAII,III- interaction first make the tunneling contact more

generic tunnel contact point tunnel

t t transparentIV- conventional (bulk) ZBA

contact

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• Tunneling into a Luttinger liquid is described

Summary

• Tunneling into a Luttinger liquid is described by RG flow of 3x3 S-matrix

• Conventional fixed point has a finite basin..of attraction only in the point contact model

• Bulk ZBA generically unstableto the breakup of the liquid into two parts..to the breakup of the liquid into two parts

• Interaction-induced enhancement of tunnelingInteraction induced enhancement of tunneling

• Nonmonotonic behavior of the tunneling current ..with temperature or bias voltage

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Stability analysis: point-contact fixed point γ = 0,ϕ = 0 is a saddle point in (γ, ϕ) space

RG equations for small γ :

instability

Compare topoint contact(tunneling ham.)

Linear in gγ term

no exact cancelation at order (gγ) of the contributions to β-function from the Friedel osc's that screen the junction at x < 0 and x > 0

Impurity in the isolated wire

RG equations

Matveev, Yue, Glazman '93

beyond

RG equations for the conductances

Tunneling DOS at the site of a weak impurity

Linearizing the RG equations in Gt around the stable fixed point tunneling DOS ρ(ε) at ε 0 for an arbitrary bare strength of impurity U0