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.
Top Related