universal conductance of nanowires near the superconductor...

Universal conductance of nanowires near the superconductor-metal quantum transition

Subir Sachdev (Harvard)Philipp Werner (ETH)Matthias Troyer (ETH)

Physical Review Letters 92, 237003 (2004)

Talk online at http://sachdev.physics.harvard.edu

T Quantum-critical

Why study quantum phase transitions ?

ggc• Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point. • Critical point is a novel state of matter without quasiparticle excitations

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

~ zcg g ν∆ −

Important property of ground state at g=gc : temporal and spatial scale invariance;

characteristic energy scale at other values of g:

OutlineI. Quantum Ising Chain

II. Landau-Ginzburg-Wilson theory Mean field theory and the evolution of the excitation spectrum.

III. Superfluid-insulator transitionBoson Hubbard model at integer filling.

IV. Superconductor-metal transition in nanowiresUniversal conductance and sensitivity to leads

I. Quantum Ising Chain

I. Quantum Ising Chain

( ) ( )

Degrees of freedom: 1 qubits, "large"

,

1 1 or , 2 2

j j

j jj j j j

j N N=

=→

↓

↓

↑

↑ ↑+ =← − ↓

…

0

Hamiltonian of decoupled qubits: x

jj

H Jg σ= − ∑ 2Jg

j→

j←

1 1

Coupling between qubits: z z

j jj

H J σ σ += − ∑

( )( )1 1j j j j+ ++ +← ←→ ←→ →← →

1 1

Prefers neighboring qubits

are

(not entangle

d)j j j j

either or+ +

↓ ↓↑ ↑

( )0 1 1x z zj j j

jJ gH H H σ σ σ += + = − +∑

Full Hamiltonian

leads to entangled states at g of order unity

LiHoF4

Experimental realization

Weakly-coupled qubitsGround state:

1 2

G

g

→→→→→→→→→→

→→→→ →→→

=

−←←− →

Lowest excited states:

jj →→→→ +→←= →→→→

Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p

Entire spectrum can be constructed out of multi-quasiparticle states

jipxj

jp e= ∑

( ) ( )

( )

2 1

1

Excitation energy 4 sin2

Excitation gap 2 2

pap J O g

gJ J O g

ε −

−

⎛ ⎞= ∆ + +⎜ ⎟⎝ ⎠

∆ = − + p

∆

aπ

aπ

−

( )pε

( )1g

Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy

and momentum

( , )

p

S p

ω

ω→ ←

ω

( ),S p ω( )( )Z pδ ω ε−

Three quasiparticlecontinuum

Quasiparticle pole

~3∆

Weakly-coupled qubits ( )1g

Structure holds to all orders in 1/g

At 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the

21is given by

Bk TB

T

k Te

ϕ ϕ

ϕ

τ τ

τ π−∆

>

=

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Ground states:

2

G

g

=

−

↑ ↑↑↑↑↑↑↑↑↑↑

↑↑↑↑ −↑↑↓↑↑↑

Lowest excited states: domain walls

jjd ↓↓↑ ↓↑↑ ↓ +↑↑= ↓Coupling between qubits creates new “domain-

wall” quasiparticle states at momentum pjipx

jj

p e d= ∑( ) ( )

( )

2 2

2

Excitation energy 4 sin2

Excitation gap 2 2

pap Jg O g

J gJ O g

ε ⎛ ⎞= ∆ + +⎜ ⎟⎝ ⎠

∆ = − +

p

∆

aπ

aπ

−

( )pε

Second state obtained by

and mix only at order N

G

G G g

↓ ↓

↑↓

⇔↑ 0

Ferromagnetic moment0zN G Gσ= ≠

Strongly-coupled qubits ( )1g


and momentum

( , )

p

S p

ω

ω→ ←

Strongly-coupled qubits ( )1g

ω

( ),S p ω( ) ( ) ( )22

0 2N pπ δ ω δ

Two domain-wall continuum

Structure holds to all orders in g~2∆

At 0, motion of domain walls leads to a finite ,

21and broadens coherent peak to a width 1 where Bk TB

T

k Te

ϕ

ϕϕ

τ

ττ π

−∆

>

=

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Entangled states at g of order unity

ggc

“Flipped-spin” Quasiparticle

weight Z

( )1/ 4~ cZ g g−

A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)

ggc

Ferromagnetic moment N0

( )1/80 ~ cN g g−

P. Pfeuty Annals of Physics, 57, 79 (1970)

ggc

Excitation energy gap ∆ ~ cg g∆ −


and momentum

( , )

p

S p

ω

ω→ ←

Critical coupling ( )cg g=

ω

( ),S p ω

c p

( ) 7 /82 2 2~ c pω−

−

No quasiparticles --- dissipative critical continuum

Quasiclassicaldynamics


1/ 41~z z

j kj k

σ σ−

P. Pfeuty Annals of Physics, 57, 79 (1970)

( )∑ ++−=i

zi

zi

xiI gJH 1σσσ

( ) ( )

( )

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

i dt t e

AT i

k T

ωχ ω σ σ

ω

π

∞

⎡ ⎤= ⎣ ⎦

=− Γ +

⎛ ⎞Γ = ⎜ ⎟⎝ ⎠

∑∫

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).

II. Landau-Ginzburg-Wilson theory

Mean field theory and the evolution of the excitation spectrum

III. Superfluid-insulator transition

Boson Hubbard model at integer filling

Bosons at density f = 1Weak interactions:

superfluidity

Strong interactions: Mott insulator which preserves all lattice

symmetries

LGW theory: continuous quantum transitions between these statesM. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

I. The Superfluid-Insulator transition

Boson Hubbard model†

†

Degrees of freedom: Bosons, , hopping between the

sites, , of a lattice, with short-range repulsive interactions.

- ( 1)2

j

i j jj j

ji j

j

b

b b n n

jU nH t µ= − + − +∑ ∑ ∑

† j j jn b b≡M.PA. Fisher, P.B. Weichmann,

G. Grinstein, and D.S. Fisher Phys. Rev. B 40, 546 (1989).

For small U/t, ground state is a superfluid BEC with

superfluid density density of bosons≈

What is the ground state for large U/t ?

Typically, the ground state remains a superfluid, but with

superfluid density density of bosons

The superfluid density evolves smoothly from large values at small U/t, to small values at large U/t, and there is no quantum phase transition at any intermediate value of U/t.(In systems with Galilean invariance and at zero temperature, superfluid density=density of bosons always, independent of the strength of the interactions)

What is the ground state for large U/t ?

Incompressible, insulating ground states, with zero superfluid density, appear at special commensurate densities

tU

−3jn =

7 / 2jn = Ground state has “density wave” order, which spontaneously breaks lattice symmetries

Excitations of the insulator: infinitely long-lived, finite energy quasiparticles and quasiholes

( )2

, , *,

Energy of quasi-particles/holes: 2p h p h

p h

ppm

ε = ∆ +

Boson Green's function : Cross-section to add a boson while transferring energy and momentum

( , )

p

G p

ω

ω Insulating ground state

Continuum of two quasiparticles +

one quasihole

ω

( ),G p ω( )( )Z pδ ω ε−

Quasiparticle pole

~3∆

Similar result for quasi-hole excitations obtained by removing a boson

Entangled states at of order unity/g t U≡

ggc

Quasiparticleweight Z

( )~ cZ g g ην−

ggc

Superfluiddensity ρs

( )( 2)~ d zs cg g νρ + −−

gc

g

Excitation energy gap ∆

( ),

,

~ for

=0 for p h c c

p h c

g g g g

g g

ν∆ − <

∆ >

A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)



( ) ( )

Relaxational dynamics ("Bose molasses") with phase coherence/relaxation time given by

1 Universal number 1 K 20.9kHzBk Tϕ

ϕ

τ

µτ

= =

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).K. Damle and S. SachdevPhys. Rev. B 56, 8714 (1997).

Crossovers at nonzero temperature

2

Conductivity (in d=2) = universal functionB

Qh k T

ω⎛ ⎞∑ ∑ →⎜ ⎟

⎝ ⎠M.P.A. Fisher, G. Girvin, and G. Grinstein, Phys. Rev. Lett. 64, 587 (1990).K. Damle and S. Sachdev Phys. Rev. B 56, 8714 (1997).

IV. Superconductor-metal transition in nanowires

T=0 Superconductor-metal transition

Repulsive BCS interaction

Attractive BCS interaction

R

Rc

Metal SuperconductorR

M.V. Feigel'man and A.I. Larkin, Chem. Phys. 235, 107 (1998)B. Spivak, A. Zyuzin, and M. Hruska, Phys. Rev. B 64, 132502 (2001).

T=0 Superconductor-metal transition

R

ψi

Continuum theory for quantum critical point

Consequences of hyperscaling

R

SuperconductorNormal

Rc

No transition for T>0 in d=1

T

Sharp transition at T=0


R

SuperconductorNormal

Rc

Quantum CriticalRegion

No transition for T>0 in d=1

T

Sharp transition at T=0


Quantum CriticalRegion

Effect of the leads

Large n computation of conductance

Quantum Monte Carlo and large n computation of d.c. conductance

Conclusions• Universal transport in wires near the superconductor-metal

transition

• Theory includes contributions from thermal and quantum phase slips ---- reduces to the classical LAMH theory at high temperatures

• Sensitivity to leads should be a generic feature of the ``coherent’’ transport regime of quantum critical points.

Conclusions• Universal transport in wires near the superconductor-metal

transition

• Theory includes contributions from thermal and quantum phase slips ---- reduces to the classical LAMH theory at high temperatures

• Sensitivity to leads should be a generic feature of the ``coherent’’ transport regime of quantum critical points.

universal conductance of nanowires near the superconductor...

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