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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
Why study quantum phase transitions ?
T
Quantum-critical
gc
• Theory for a quantum system with strong correlations:
describe phases on either side of gc by expanding in
deviation from the quantum critical point.
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.
Important property of ground state at g=gc :
temporal and spatial scale invariance;
characteristic energy scale at other values of g:  ~ g  g c
z
Outline
I.
Quantum Ising Chain
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.
IV. Superconductor-metal transition in nanowires
Universal conductance and sensitivity to leads
I. Quantum Ising Chain
I. Quantum Ising Chain
Degrees of freedom: j  1
N qubits, N "large"
 , 
or
 j

1
  
j
2
j

j
j
,  j

1
 
j
2

Hamiltonian of decoupled qubits:
H 0   Jg   xj
j
j

j
2Jg

j
Coupling between qubits:
H1   J   zj  zj 1
j

j

j

 

j 1
j 1
Prefers neighboring qubits
are either 
Full Hamiltonian
j

j 1
or 
j

j 1
(not entangled)

H  H 0  H1   J  g xj   zj  zj 1

j
leads to entangled states at g of order unity


Experimental realization
LiHoF4
Weakly-coupled qubits
g
1
Ground state:
G 

1

2g
   

Lowest excited states:
j

  j  

Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p
p  e
  p
ipx j
j
j
 pa 
1
Excitation energy   p     4 J sin 2 
O g
2 
 
 
Excitation gap   2 gJ  2 J  O g 1



a
p
Entire spectrum can be constructed out of multi-quasiparticle states

a
Dynamic Structure Factor S ( p, ) :
Weakly-coupled qubits  g
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  
Z      p  
Quasiparticle pole
Three quasiparticle
continuum
~3
Structure holds to all orders in 1/g

At T  0, collisions between quasiparticles broaden pole to
a Lorentzian of width 1   where the phase coherence time  
is given by
1


2k B T

e kBT
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
1
Ground states:
G 
Strongly-coupled qubits  g
1

g

2
  

Ferromagnetic moment
N0  G  z G  0
Second state G  obtained by   
G  and G  mix only at order g N
Lowest excited states: domain walls
dj 
   j    

Coupling between qubits creates new “domainwall” quasiparticle states at momentum p
p  e
ipx j
  p

dj
j
 pa 
2
Excitation energy   p     4 Jg sin 2 
O g
2 
 
 
Excitation gap   2 J  2 gJ  O g 2


a
p

a
Dynamic Structure Factor S ( p, ) :
Strongly-coupled qubits  g
1
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  
N 02  2       p 
2
Two domain-wall
continuum

~2
Structure holds to all orders in g
At T  0, motion of domain walls leads to a finite phase coherence time   ,
and broadens coherent peak to a width 1   where
1


S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)
2k B T

e kBT
Entangled states at g of order unity
Z ~  g  gc 
1/ 4
“Flipped-spin”
Quasiparticle
weight Z
A.V. Chubukov, S. Sachdev, and J.Ye,
Phys. Rev. B 49, 11919 (1994)
gc
g
N0 ~  gc  g 
1/ 8
Ferromagnetic
moment N0
P. Pfeuty Annals of Physics, 57, 79 (1970)
gc
g
Excitation
energy gap 
 ~ g  gc
gc
g
Dynamic Structure Factor S ( p, ) :
Critical coupling
 g  gc 
Cross-section to flip a  to a  (or vice versa)
while transferring energy  and momentum p
S  p,  

~  c p
c p
2
2
2

7 / 8

No quasiparticles --- dissipative critical continuum

H I   J  g ix   iz iz1

i
Quasiclassical
dynamics
Quasiclassical
dynamics
 ( ) 
i

  dt
k

 zj  t  ,  kz  0   ei t
0
A
T
7/4
1  i /  R  ...
 k T

 R   2 tan  B
16 

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).
S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).
 
z
j
z
k
~
1
jk
P. Pfeuty Annals of
Physics, 57, 79 (1970)
1/ 4
Outline
I.
Quantum Ising Chain
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.
IV. Superconductor-metal transition in nanowires
Universal 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 transition
Boson Hubbard model at integer filling.
IV. Superconductor-metal transition in nanowires
Universal conductance and sensitivity to leads
III. Superfluid-insulator transition
Boson Hubbard model at integer filling
Bosons at density f  1
Weak interactions:
superfluidity
Strong interactions:
Mott insulator which
preserves all lattice
symmetries
LGW theory: continuous quantum transitions between these states
M. 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, b†j , hopping between the
sites, j , of a lattice, with short-range repulsive interactions.
U
†
H  t  bi b j -  n j   n j ( n j  1) 
2 j
ij
j
nj  b b
†
j j
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
nj  3
nj  7 / 2
t

U
Ground state has “density wave” order, which
spontaneously breaks lattice symmetries
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Excitations of the insulator: infinitely long-lived, finite energy
quasiparticles and quasiholes
p2
Energy of quasi-particles/holes:  p ,h  p    p ,h 
2m*p ,h
Boson Green's function G ( p, ) :
Insulating ground state
Cross-section to add a boson
while transferring energy  and momentum p
G  p, 
Z      p  
Quasiparticle pole
~3
Continuum of
two quasiparticles +
one quasihole

Similar result for quasi-hole excitations obtained by removing a boson
Entangled states at g  t / U of order unity
Z ~  gc  g 

Quasiparticle
weight Z
A.V. Chubukov, S. Sachdev, and J.Ye,
Phys. Rev. B 49, 11919 (1994)
gc
g
 p ,h ~  gc  g  for g  gc

Excitation
energy gap 
 p ,h =0 for g  gc
g
rs ~  g  gc 
Superfluid
density rs
ggcc
g
( d  z  2)
Crossovers at nonzero temperature
Quasiclassical
dynamics
Quasiclassical
dynamics
Relaxational dynamics ("Bose molasses") with
phase coherence/relaxation time   given by
1

  Universal number 
k BT
Q2   
Conductivity (in d=2) =


h  kBT 
1 K  20.9kHz 
S. Sachdev and J. Ye,
Phys. Rev. Lett. 69, 2411 (1992).
K. Damle and S. Sachdev
Phys. Rev. B 56, 8714 (1997).
  universal function
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 transition
Boson Hubbard model at integer filling.
IV. Superconductor-metal transition in nanowires
Universal conductance and sensitivity to leads
IV. Superconductor-metal transition in
nanowires
T=0 Superconductor-metal transition
Repulsive BCS
interaction
Attractive BCS
interaction
R
Superconductor
Metal
Rc
R
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
yi
R
Continuum theory for quantum critical point
Consequences of hyperscaling
T
No transition for
T>0 in d=1
Normal
Superconductor
Rc
Sharp transition at T=0
R
Consequences of hyperscaling
Quantum Critical
Region
T
No transition for
T>0 in d=1
Normal
Superconductor
Rc
Sharp transition at T=0
R
Consequences of hyperscaling
Quantum Critical
Region
Consequences of hyperscaling
Quantum Critical
Region
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.