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

Universal conductance of nanowires near the superconductor-metal quantum transition

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

Physical Review Letters 92, 237003 (2004)

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.

~z

cg g

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

characteristic energy scale at other values of g:

• 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.

Outline

I. 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:

xj

j

H Jg 2Jg

j

j

1 1

Coupling between qubits:

z zj j

j

H J

1 1

Prefers neighboring qubits

are

(not entangle

d)

j j j jeither or

1 1j j j j

0 1 1x z zj j j

j

J gH H H Full Hamiltonian

leads to entangled states at g of order unity

LiHoF4

Experimental realization

Weakly-coupled qubits Ground 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

j

p e

2 1

1

Excitation energy 4 sin2

Excitation gap 2 2

pap J O g

gJ J O g

p

a

a

p

1g

Dynamic Structure Factor :

Cross-section to flip a to a (or vice versa)

while transferring energy

and momentum

( , )

p

S p

,S p Z p

Three quasiparticlecontinuum

Quasiparticle pole

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

phase coherence time

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

~3

Weakly-coupled qubits 1g

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

p

a

a

p

Second state obtained by

and mix only at order N

G

G G g

0

Ferromagnetic moment

0zN G G

Strongly-coupled qubits 1g

Dynamic Structure Factor :

Cross-section to flip a to a (or vice versa)

while transferring energy

and momentum

( , )

p

S p

,S p 22

0 2N p

Two domain-wall continuum

Structure holds to all orders in g

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

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

T

k Te

phase coherence time

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

~2

Strongly-coupled qubits 1g

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

1/8

0 ~ cN g g

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

ggc

Excitation energy gap ~ cg g

Dynamic Structure Factor :

Cross-section to flip a to a (or vice versa)

while transferring energy

and momentum

( , )

p

S p

,S p

Critical coupling cg g

c p

7 /82 2 2~ c p

No quasiparticles --- dissipative critical continuum

Quasiclassical dynamics

Quasiclassical dynamics

1/ 4

1~z z

j kj k

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

i

zi

zi

xiI gJH 1

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

idt t e

A

T i

k T

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

Outline

I. 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

II. Landau-Ginzburg-Wilson theory

Mean field theory and the evolution of the excitation spectrum

Outline

I. 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

III. Superfluid-insulator transition

Boson Hubbard model at integer filling

Bosons at density f

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Weak interactions: superfluidity

Strong interactions: Mott insulator which preserves all lattice

symmetries

LGW theory: continuous quantum transitions between these states

I. The Superfluid-Insulator transition

†

†

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

j

UnH t

† j j jn b b

Boson Hubbard model

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

superfluid density density of bosons

M.PA. Fisher, P.B. Weichmann, G. Grinstein, and D.S. Fisher

Phys. Rev. B 40, 546 (1989).

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

3jn t

U

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

pp

m

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

2

, , *,

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

p h

pp

m

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

2

, , *,

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

p h

pp

m

Boson Green's function :

Cross-section to add a boson

while transferring energy and momentum

( , )

p

G p

Continuum of two quasiparticles +

one quasihole

,G p Z p

Quasiparticle pole

~3

Insulating ground state

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

Entangled states at of order unity

ggc

Quasiparticle weight Z

~ cZ g g

ggc

Superfluid density s

( 2)~

d z

s cg g

gc

g

Excitation energy gap

,

,

~ for

=0 for

p h c c

p h c

g g g g

g g

/g t U

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

Quasiclassical dynamics

Quasiclassical dynamics

Relaxational dynamics ("Bose molasses") with

phase coherence/relaxation time given by

1 Universal number 1 K 20.9kHzBk T

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

Q

h k T

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).

Outline

I. 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

IV. Superconductor-metal transition in nanowires

T=0 Superconductor-metal transition

Attractive BCS interaction

R

Repulsive BCS interaction

RRc

Metal Superconductor

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

i

Continuum theory for quantum critical point

Consequences of hyperscaling

R

SuperconductorNormal

Rc

Sharp transition at T=0

T

No transition for T>0 in d=1

Consequences of hyperscaling

R

SuperconductorNormal

Rc

Sharp transition at T=0

T

Quantum CriticalRegion

No transition for T>0 in d=1

Consequences of hyperscaling

Quantum CriticalRegion

Consequences of hyperscaling

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.