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Nonequilibrium Quantum Criticality in Open Electronic Systems
Nonequilibrium Quantum Criticality in
Open Electronic Systems
Yong Baek Kim
University of Toronto
Hong Kong, December 18, 2006
In collaboration with
A. Mitra (Toronto), S. Takei (Toronto), A. J. Millis (Columbia) Outline
nonequilibrium
quantum phase transitions

Equilibrium magnetic quantum phase transitions
in itinerant magnets
; Hertz-Millis-Moriya theory
Open system;
steady state nonequilibrium state
Development of
renormalization group
scheme
Study of
universality classes
PAST:
PRESENT:
FUTURE:
Future directions
General consideration: Past Equilibrium quantum phase transitions
Phase transitions at T=0 tuned by control parameters
(e.g. pressure, magnetic elds, chemical doping)
Change in broken symmetries at a quantum critical point
Motivation
Hertz pioneered in the study of ferromagnetic quantum
phase transition in thermal equilibrium. Millis later revisited
the problem noting that temperature is a relevant
perturbation.
Open question:
How does a system,
tuned at a quantum critical point,
behave as an applied voltage drives it
into a non-equilibrium steady state?
Forecast:
One can expect non-linear external perturbation to
have a similar role to temperature in the equilibrium case.
Decoupling of statics and dynamics should result because non-equilibrium
perturbations typically destroy phase coherence. Strong departures from equilibrium
should force the system to regain its effective dimensionality of d. Equilibrium quantum phase transitions
Quantum criticality: presence of
singular fluctuations of the OP in
space and time.
Temperature is not a tuning
parameter, but a finite size effect:
singularity in the phase diagram.
Need for new theories - a new
conceptual and computational
infrastructure for understanding
highly correlated materials.
Experiment a vital component in
elucidating the key physics of the
QPT - need for new materials, new
precision measurements.
36
Quantum criticality: meaning and challenge.
Observations in Heavy Fermion Systems.
Break down of the standard model
Local and deconfined quantum criticality.
3
3
Spatial and Temporal uctuations
are coupled in
quantum phase transitions
Effective dimensionality
d
e
= d + z > d

z
t
x
y Fermi Liquid Theory and Quasiparticle Picture
Landau, JETP 3, 920 (1957)
Landau:
interactions can be turned on
adiabatically, preserving the excitation
spectrum.
Interactions
adiabatically
-
-
Quasiparticle
5
f(E)
E
A
6
Landau, JETP 3, 920 (1957)
Landau:
interactions can be turned on
adiabatically, preserving the excitation
spectrum.
Interactions
adiabatically
-
-
Quasiparticle
5
Breakdown of Landau Fermi Liquid Theory

Quantum Critical Point
Singular Fermi
Liquid
T
What happens when the interaction becomes too
large ?
0
Ordered State.
Fermi Liquid
X
c
X = P, H . . . QCP
Hertz, 1976
9
Singular Fermi
Liquid
T
What happens when the interaction becomes too
large ?
0
Ordered State.
Fermi Liquid
X
c
X = P, H . . . QCP
Hertz, 1976
9
Singular Fermi
Liquid
T
What happens when the interaction becomes too
large ?
0
Ordered State.
Fermi Liquid
X
c
X = P, H . . . QCP
Hertz, 1976
9 P
c
P
AFM
metal
Heavy Fermion
Materials
H. Von Lohneyson (1996)
Quantum Criticality:
divergent specific
heat capacity
Quantum Critical
Point
= AT
1+
23 Theory of Quantum Criticality in Equilibrium
Itinerant Magnetic Phase Transitions
Effective eld theory of the order parameter in
d+z
dimensions (effective space-time dimensions)
q
z
(q) constant
Clean/Dirty Ferromagnet (z=3,4)
Open System (z=2)

z = 1 UN(0)
Stoner parameter
S
e
=
q, 1
(q, )|m(q, )|
2
+ u
d
d
x dt
[m(x, t)]
4
F
=
d
d
x
[ m(x)]
2
+ [m(x)]
2
+ u[m(x)]
4
(c.f.
Landau free energy) 1
(q, ) = i
(q) + q
2
+
(q) q, q
2 d
e
> 4
d
e
< 4
Renormalization Group Analysis;
quartic term is irrelevant ( ); relevant ( )

T is a relevant perturbation
Thermal uctuations decouple the dynamics and
statics
- classical transitions in d-dimensions
dT
(b)
d
ln b = zT (b)
T =

c
u
z/(d+z2)
T
= (
c
)
z/2
ordered phase
2 < d < z + 2
quantum
gaussian
classical
quantum
critical = (T ) T T
c Present The elds which drive the system out of equilibrium
typically increase its energy and destroy phase coherence;
this may be
analogous to temperature
?
similarity
between non-equilibrium transitions and
thermal transitions
?

General Consideration
Motivation
Hertz pioneered in the study of ferromagnetic quantum
phase transition in thermal equilibrium. Millis later revisited
the problem noting that temperature is a relevant
perturbation.
Open question:
How does a system,
tuned at a quantum critical point,
behave as an applied voltage drives it
into a non-equilibrium steady state?
Forecast:
One can expect non-linear external perturbation to
have a similar role to temperature in the equilibrium case.
Decoupling of statics and dynamics should result because non-equilibrium
perturbations typically destroy phase coherence. Strong departures from equilibrium
should force the system to regain its effective dimensionality of d.
T General Consideration
Motivation
Hertz pioneered in the study of ferromagnetic quantum
phase transition in thermal equilibrium. Millis later revisited
the problem noting that temperature is a relevant
perturbation.
Open question:
How does a system,
tuned at a quantum critical point,
behave as an applied voltage drives it
into a non-equilibrium steady state?
Forecast:
One can expect non-linear external perturbation to
have a similar role to temperature in the equilibrium case.
Decoupling of statics and dynamics should result because non-equilibrium
perturbations typically destroy phase coherence. Strong departures from equilibrium
should force the system to regain its effective dimensionality of d.
J
µ
L
µ
R
V = µ
L
µ
R
V
??
The elds which drive the system out of equilibrium
typically increase its energy and destroy phase coherence;
this may be
analogous to temperature
?
similarity
between non-equilibrium transitions and
thermal transitions
?
Departures from equilibrium may also
break basic
symmetries
(e.g. time reversal invariance, spatial
symmetries such as rotation and inversion)
General Consideration
These new effects may change the critical behavior. What problems do we want to solve ?
Theory of
nonequilibrium quantum criticality in
itinerant electron systems
Open systems coupled to reservoirs
-
non-conserved order parameter;
nonequilibrium by differences between reservoirs -
time-independent drive and steady state
J
left lead
right lead
µ
L
µ
R
V = µ
L
µ
R 2D interacting
electron system
2
V
Left Reservoir
Right Reservoir
z
x
y
Interacting electron system
FIG. 2: Schematic view of systems studied: central region
with interacting electron physics leading to critical behavior,
coupled to two leads.
may be periodic in time (quasiperiodic or chaotic solu-
tions seem possible in principle, but we have not encoun-
tered these so far).
One may calculate expectation values by use of a gen-
erating functional Z of source elds, , dened as a path
integral on the Keldysh two-time contour
3,17,18
. We for-
mally integrate out all of the degrees of freedom except
those connected with long wavelength order parameter
uctuations, introducing a wavevector cuto (as with
equilibrium critical phenomena the precise manner in
which the cuto is imposed is not important) and de-
note the modes we retain by m(x, t). We then obtain
3,18
Z() =
D[m
i
, m
f
]
SS,
(m
i
, m
f
)
D[m
+
(t), m (t)]
e
S
K
[{m
+
,m ,}]
(1)
Here m
±
are the uctuating order parameter elds of in-
terest and
D[{m
+
(t), m (t)}] denotes an integral over
all paths in function space beginning at m
i
on the +
contour at t = 0 and ending at m
f
at t = 0 on the
contour. The contributions of paths involving dif-
ferent endpoints weighted by the steady-state reduced
density matrix [{m
i
(x), m
f
(x)}], whose diagonal ele-
ments describe the probability that at an instant of time
the long wavelength components of the order parame-
ter eld take the conguration m(x). One may classify
phases by examining lim
0
. In the paramagnetic
phase,
0
= (m), whereas in a broken symmetry
phase
0
= (m ¯
m(h)) with ordered moment ¯
m de-
pendent on the direction of the symmetry breaking eld
h.
SS,
obeys a kinetic equation which may be deter-
mined from the requirement that correlation functions
calculated from Eq 1 are ca