Circuitry with a Luttinger liquid K.-V. Pham Laboratoire de Physique des Solides Pascal’s Festschrifft Symposium

● Some Background: New playgrounds (< 10 yrs) for LL at the Meso/Nano scale: e.g. quantum wires, carbon nanotubes, cold atoms Finite-size: ergo New Physics due to the boundaries IMO Two quite relevant things: nature of the BOUNDARY CONDITION: Periodic (finite-size corrections, numerics…) Open (e.g. broken spin chains…), twisted Boundary conformal field theory (e.g. single impurity as a boundary problem, cf Kondo…) interaction with PROBES (are invasive) (e.g. transport)

● Towards Nanoelectronics / nanospintronics ● But before some more basic questions: What happens to a LL plugged into a (meso) electrical circuit? i.e. LL as an electrical component Impact of finite-size? Coupling to other electrical components?

How would an electrical engineer view a LL?

Condensed Matter theorist: Low-energy effective Field Theory (harmonic solid) Density: Current: LL phase fields

● Electrical engineer: How would an electrical engineer view a LL?

● Electrical engineer:

How would an electrical engineer view a LL? ● Electrical engineer: Capacitive energy !

How would an electrical engineer view a LL? ● Electrical engineer: Capacitive energy !

How would an electrical engineer view a LL? ● Electrical engineer: Capacitive energy ! Inductive energy !

● Electrical engineer: How would an electrical engineer view a LL? The LL is just a (lossless) Quantum Transmission line

● Electrical engineer: How would an electrical engineer view a LL? The LL is just a (lossless) Quantum Transmission line

● Electrical engineer: How would an electrical engineer view a LL? The LL is just a (lossless) Quantum Transmission line Further Ref:- Bockrath PhD Thesis ‘99, Burke IEEE ’02 - circuit theory (Nazarov, Blanter…) - K-V P., Eur Phys Journ B 2003

Zero modes (charged but dispersionless) Excitations (from bosonization): Density oscillations i.e. Plasmons (neutral)

Excitations (from bosonization): Density oscillations i.e. Plasmons (neutral) Zero modes (charged but dispersionless)

Excitations (from bosonization): Density oscillations i.e. Plasmons (neutral) Zero modes (charged but dispersionless) Electrical Engineer? Transmission line: telegrapher equation

Excitations (from bosonization): Density oscillations i.e. Plasmons (neutral) Zero modes (charged but dispersionless) Electrical Engineer? Transmission line: telegrapher equation Wave velocity excitations are also plasma waves

Excitations (from bosonization): Density oscillations i.e. Plasmons (neutral) Zero modes (charged but dispersionless) Electrical Engineer? Transmission line: telegrapher equation excitations are also plasma waves Wave velocity

DC Conductance of infinite LL:

A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible!

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave)

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) half-infiniteTransmission line resistance =

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) InfiniteTransmission line = 2 half-infinite TL half-infiniteTransmission line resistance =

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) InfiniteTransmission line = 2 half-infinite TL half-infiniteTransmission line resistance = => conductance: G=1/2Z 0

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) InfiniteTransmission line = 2 half-infinite TL half-infiniteTransmission line resistance = => conductance: G=1/2Z 0 Since:

DC Conductance of infinite LL: A little mystery: LL Conductivity is actually infinite ! Dissipation should be impossible! E.E. answer: resistance is non-zero because it’s not really a resistance but the characteristic impedance of the transmission line ! (quantifies the energy transported by a traveling wave) InfiniteTransmission line = 2 half-infinite TL half-infiniteTransmission line resistance = => conductance: G=1/2Z 0 Since: One recovers:

A simple Series circuit ● Ref: Lederer, Piéchon, Imura + K-V P., PRB 03

Rationale: Phenomenological Model for mesoscopic electrodes The 2 Resistors modelize contact resistances. Implementation: Are described in term of dissipative boundary conditions. Quantization not trivial (NO normal eigenmodes) but bosonization still holds (Ref: K- V P, Progr Th Ph 07)

Some Straightforward Properties (at least for an E.E.) (ref: K-V P, EPJB 03): DC resistance: AC conductance: is a 3 terminal measurement Conductance is a 3x3 matrix.

Resonances for G ij (i,j=1,2): Interpretation: Infinite Transmission Line (TL): Traveling waves

Resonances for G ij (i,j=1,2): Interpretation: Infinite Transmission Line (TL): Traveling waves Open TL: Standing waves (nodes: perfect reflections of plasma wave at boundaries)

Resonances for G ij (i,j=1,2): Interpretation: Infinite Transmission Line (TL): Traveling waves Open TL: Standing waves (nodes: perfect reflections of plasma wave at boundaries) TL+resistors: Standing waves are leaking (imperfect reflections => finite life-time)

Reflection coefficients for a TL (classical and quantum i.e. LL):

Reflections in a TL due to impedance mismatch (cf: Safi & Schulz, inhomogeneous LL, Fabry-Perot) Resonances:

Impedance matching of a TL and implications. Impedance mismatch leads to reflections => novel physics for Luttinger (E.E. :not so new, standing waves of a TL) Match impedances to Z 0 => kills reflections !

Impedance matching of a TL and implications. Impedance mismatch leads to reflections => novel physics for Luttinger (E.E. :not so new, standing waves of a TL) Match impedances to Z 0 => kills reflections ! => finite TL now behaves like infinite TL Property still true for quantum TL (i.e. Luttinger) ! (cf K-V P., Prog. Th. Ph. 07)

Impedance matching of a Luttinger Liquid: Remedy to invasiveness of probes The finite LL exhibits the same properties as the usual infinite LL: allows measurements of intrinsic properties of a LL in (and despite) a meso setup.

Impedance matching of a Luttinger Liquid: Remedy to invasiveness of probes The finite LL exhibits the same properties as the usual infinite LL: allows measurements of intrinsic properties of a LL in (and despite) a meso setup. Experimental realization: Rheostat??? Depends on type of measurement (DC or AC)

Tuning of (contact) resistances at the mesoscopic level in quantum wires (Yacoby): Electron density in the wire Ref: Yacoby et al, Nature Physics 07 Two-terminal conductance of a quantum wire

(unpublished; courtesy A. Yacoby) In this setup, contact resistances (barriers at electrodes) are equal: So that: Impedance matching if: (crossing of curves G=G(n L ) and Ke 2 /h=f(n L ) ) The two curves cross: impedance matching realized !

Applications of impedance matching: Shot noise (detection of fractional excitations in the LL) Issue: shot noise for infinite LL in various setups should exhibit anomalous charges (Kane, Fisher PRL 94; T. Martin et al 03) These charges are irrational in general and can be shown to correspond to exact eigenstates of the LL Description of LL spectrum in terms of fractional eigenstates (holons, spinons, 1D Laughlin qp, …) : K-V P, Gabay & Lederer PRB ’00 But probes are invasive so that it is predicted that fractional charges can not be extracted from shot noise (Ponomarenko ’99, Trauzettel+Safi ’04)

Interferences by probes circumvented by impedance matching: A promising setup (A. Yacoby expts): Two parallel quantum wires Spin-charge separation observed in this setup (Auslaender et al, Science ‘05) Current asymetry incompatible with free electrons observed (predicted by Safi Ann Phys ’97); can be ascribed to fractional excitations (K-V P, Gabay, Lederer PRB ’00). (Consistent with fractional excitations but not definite proof: more expts needed)

Other interesting things but no time for discussion… Gate conductance G 33, DC & AC shot noise, bulk tunneling, charge relaxation resistance

(Setup idea: Burke ’02)

Conclusion: Main message 1) The LL is a Quantum Transmission Line 2) The Physics of classical Transmission lines can bring many interesting insights into the LL physics at the meso scale

Conclusion: Main message 1) The LL is a Quantum Transmission Line 2) The Physics of classical Transmission lines can bring many interesting insights into the LL physics at the meso scale Thank You Thank you, Pascal, for many fruitful years of Physics !!!

Addenda: Gate conductance: Here R C is the contact resistance: R q is the charge relaxation resistance: NB: Recover earlier results of Blanter et al as special limit: