Tunneling Into a Luttinger Liquid...
Transcript of Tunneling Into a Luttinger Liquid...
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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
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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
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Single-channel quantum wires (aka Nanowires)
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Single-channel quantum wires (aka Nanowires)
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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
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Scaling of the tunneling conductance
Bockrath et al , Nature,1999 6
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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:
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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
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αe ≈ g
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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
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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
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Friedel oscillations in 1D 11
Hartree potential oscillates with the periodtat
Exchange contribution — similar15
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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
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Single impurity
Tunneling contact
Key difference: absent
Phase of the Friedel osc's is important
Weak impurity:
Point tunnel contact: 17
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Renormalization of backscattering
Weak impurity:
Point tunnel contact:
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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
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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
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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
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Impurity in the isolated wire
RG equations
Matveev, Yue, Glazman '93
beyond
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RG equations for the conductances
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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