VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE II SC. BKT
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Transcript VI. INTRODUCTION to THERMAL FLUCTUATIONS in TYPE II SC. BKT
VI. INTRODUCTION to THERMAL
FLUCTUATIONS in TYPE II SC. BKT
TRANSITION in 2D
A. Two scales of thermal fluctuations
1. The microscopic thermal fluctuations’ place in
the GL description
On the microscopic level temperature modifies
properties of the electron gas and the pairing
interaction responsible for the creation of Cooper pairs.
When we “integrated out” the microscopic (electronic)
degrees of freedom to obtain the effective mesoscopic
GL theory in terms of the distributions of the order
parameter x with ultraviolet (UV) cutoff a
1
One loosely describes the effective mesoscopic order
parameter
X
eiKX ck cK k
k /a
thermal quantum
as a classical field of Cooper pairs with center of mass
at x. More mathematically, the remaining mesoscopic
part of the statistical sum obeys the “scale matching”:
k
K
a
x
X
2
Z T
k /a
Dck Dck* Dck Dck* D X D* X
K / a
1/ T
iKX
*
*
X e cL cK L exp L ck , ck , cK , cK
L
0
1
*
D X D X exp F X , * X , T , a
T
Quantum effects on the mesoscopic level are usually
small (only when temperature is very close to T=0 they
might be of importance) and will be neglected. In this
case the mesoscopic classical field x is independent of
time. Dynamical generalization of the GL approach will
be introduced later.
3
The coefficients of the GL energy
F
2
2m *
2
D (T T )
L
c
2
2
4
are dependent on temperature expressing these
“microscopic” thermal fluctuations. The dependence can,
in principle, be derived from a microscopic theory
(example: Gor’kov’s derivation from the BCS theory of
“conventional superconductors). The coefficients also
depend on UV cutoff L or a, but we will see that this
dependence can be “renormalized away”.
In practice the “constants” m , , are also weakly
(typically logarithmically) temperature dependent
*
4
dH c 2 (T )
H 'c 2
| Tc
dT T Tc
Hc2
Normal
Mixed
state
H c1
leading for example to “curving
down” of the Hc2(T) line:
Tc
Meissner
5
2. Two kinds of mesoscopic thermal
fluctuations: “perturbative” and “topological”
The mesoscopic fluctuations
qualitatively are of two sorts:
“perturbative” small ones and
“topologically nontrivial” or
vortex ones.
Since under magnetic field the
order parameter takes a form of
vortices, the mesoscopic
fluctuations can be qualitatively
viewed as distortions of a system
of vortices or “thermal motion”:
vibrations, rotations, waves.
6
Broadening of the
resistivity
Resistance
Major thermal fluctuations effects include broadening
of the resistivity drop (paraconductivity) and
diamagnetism in the normal phase and melting of the
vortex solid into a homogeneous vortex liquid state
Nonfluctuating SC
Normal Metal
Fluctuating SC
TC
T
Magnetization or conductivity are greatly enhanced in
the “normal” region due to thermally generated “virtual”
Cooper pairs.
7
Influence of the fluctuations on the vortex
matter phase diagram
Due to the thermally induced vibrations lattice can melt
into “vortex liquid” and the vortex matter phase diagram
becomes more complicated.
8
Symmetries broken at the two transitons
Since symmetry of the normal and liquid phases are
same, the normal – liquid line becomes just a crossover.
H
Two different
E2
symmetries are at
breaking
Hc2
Just
play: the
crossover
geometrical E(2)
FLL
Normal
(including
U(1)
Vortex
translations and
breaking
liquid
H c1
rotations) and the
Meissner
Tc
electric charge U(1).
T
9
B. Nontopological excitations. The
Ginzburg criterion.
1. Gaussian fluctuations around the Meissner state
We ignore the thermal fluctuations of the magnetic
field which turn out to be very small even for high Tc
SC in magnetic field. The effect of thermal fluctuations
on the mesoscopic scale is determined by
1
Z D x D x exp F x , * x
T
*
2
2
2
4
2
2
L
F *
z (T Tc )
*
2mab
2mc
2
x
10
Ginzburg number
It is convenient to use units of coherence length (which
might be different in different directions), in units of
0 and energy scale in units of critical temperature
r r / ;z z / c z / ; 2 2 / 2 02
In the new units the Boltzmann factor becomes:
2
*
2
2
mab
2
z
2
*
2
*
4 mab 0
mc
F 4 D 04
D
f
d
x
T T
T
T
1
2
4
c
1
2
2
2 0 Tc
2
1
1 t 2 1 4
D 1
L
d x
t
T
/
T
c
2
2
2
11
with the anisotropy parameter mc / mab
Here an important dimensionless parameter
characterizing strength of thermal fluctuations is
T
T
2
D
2
Gi
t
4
/ 0 / 4 coh.volg0
8 2 e 2
9 2
T
8.10
AT K ;
2 2
c
Where the temperature independent Ginzburg number
characterizing he material was introduced:
1 8e Tc
Gi
2 c 2 2
2
2
2
12
Perturbation theory
To calculate the thermal effects via a somewhat bulky
functional integral, the simplest method is the saddle
point evaluation assuming that is small.
One minimized the free energy around the “classical” or
nonfluctuating solution (which we found in the preceding
parts also for the SC phase)
f
0
min
and then
min fl
expands in
2
2
f
“small”
f f min
| fl ...
*
fluctuations”
2
min
13
Now we consider the normal phase t>1 in which the
saddle point value of the order parameter is trivial:
min 0 0 fl f f fl
The thermal average of the order parameter is still zero
due to the U(1) symmetry:
x Z
1
D D
*
x e
f , *
0
However the superfluid density
x
2
x
*
x 0
due to fluctuations. First we calculate the fluctuation
correction to free energy and thereby to specific heat
which is impacted the most.
14
The fluctuation contribution to free energy
e
f
Z D fl D *fl e
f fl , *fl
From now on fl is dropped.
f ( x)G
*
x, y
1
x y ( y )
2
x
( x) f 2 f 4 ;
4
2
t 1
1
G x
( x)
2
2
The expansion of the partition function results in gaussian
integrals:
* f 2 f 4
Z D D e
e
f2
2
2
1 f 4 2 f 4 ...
15
which is presented graphically as “diagrams”:
Z Z 0 1 2
Z 0 e f2
G x x G x x
It is more convenient to perform the calculation in
momentum space
f L G 1 k k
k
D
k
2
l k l m f 2 f 4 ;
2 k ,l , m
integer,rangefrom
L
2
2
k
t
1
k
G 1 k
m2
2
2
2
*
k
*
m
a
to
a
16
since the basic gaussian integral in momentum space
becomes a product
The leading order
Z0
k /a
*
d
d
k ke
G1 k k
k
2
e
log G1 k
k
Therefore the fluctuations contribution to the free
energy density is:
T
2
2
k
m
1
3
F0 T Log G k
d k Log
3 3
2
2
k
T
3
/ a
k 0
k 2 m2
k 2 dk
T 3 m 2 m3
Log
2
2 3 3
2
a
2
2 2 a
3/ 2
3
T
T
T
t
Tc
c
c
2 3
3
2 a
a
2
17
The fact that free energy depends on the UV cutoff a
means that it is not a directly measurable mesoscopic
quantity: energy differences or derivatives are.
Entropy and specific heat
The mesoscopic Cooper pairs contribution to the
entropy density is less dependent on the division of
degrees of freedom into micro and meso: just a constant
2
2
df
m
d
m
dF
2
3
S
3
d k 2
1
3 3
2
2
dT
d m
dt
k m
2
4 2 3
1/ 2
t
a
1
d m2
dt
18
The second derivative, the fluctuation contribution to
specific heat, is already finite
dS
2
1
3
C T
d k
1 2 3
3 3
2
2
2
dT
8 m
2
k m
1/ 2
2 3 t
8
19
2. Interactions of the excitations and “critical”
fluctuations. Ginzburg criterion.
Closer than that to Tc the perturbation theory cannot
be used due to IR divergencies. Physics in critical
region is therefore dominated by fluctuations. A
nonperturbative method like RG required.
The fluctuation contribution outside the critical region
is called gaussian since fluctuations were considered
to
be non-interacting. Corrections already do depend on
interactions. They are smaller by a factor at least. It
is more instructive to see this on example of the
correlator.
20
Correlator and its divergence at criticality
A measure of coherence in the Meissner phase (density
of Cooper pairs) is
0 x 2
*
2
0
*
min
1 t
0 min x 2
2
2
0
A measure of the SC correlation (or “virtul density of
Cooper pairs”) in the normal phase is Fourier
transform ofikxcorrelator and small wave
2 2 t
2 vector:
2
G k e
x
* 0 x k* k
The leading correction is:
k 0
m
2
t 1
2
2
2
G k G l G k
/ a m
k
0
2
l
m
21
This expression is a small perturbation only when
2
2
2
2
2
2 m
a
m a
m
and therefore cannot be applied close to Tc. Perturbation
theory seems to be useless due to UV divergencies.
However it is natural to assume that quantities
measurable on the mesoscopiclevel should not depend on
cutoff a.
Renormalized perturbation theory
It is reasonable to assume that when the mesoscopic
fluctuations are “switched on” the superconducting
correlations are destroyed at TcR below TcL which takes
into account only the microscopic ones. How to quantify
it?
22
At TcR correlator decays slower:
One can try to improve on it by resumming some
diagrams
G k
...
G k G k Bub G k Bub2 ...
G k
1
1
2
2
2
2
1 G k Bub k / 2 m 2Bub k / 2 mR
2
2
3
mR2 m 2 2Bub m 2 2 L m
23
To first order in fluctuations criticality occurs at
mR2 0 m2 2L 1 TcR / TcL
TcR TcL 2LTcL
Since usually TcL is not known theoretically and is not a
quantity of interest, one expresses it via measured
critical temperature TcR used as an “input parameter”.
TcL TcR 2LTcR
Ginzburg criterion
After renormalization we return to the perturbation
theory applicability test:
2
2
2 m
m
2
2 m
m
1 t
2 Gi
24
This is known as Ginzburg criterion. Plugging correct
constants one obtains:
T Tc
T
8 2 Gi
This might be compared to the jump between Meissner
and normal phases before mesoscopic fluctuations are
taken into account:
2
1 1 t
1
f M g0 / units
cM
4
The condition that fluctuations do not become
“stronger” than the mean field effect is
1
c fl
m
1
cM
2
m
25
3. Fluctuations in the Meissner phase. Goldstone
modes.
The negative coefficient of the quadratic term
f [ ]
1
2
D
*
d x
a
2
1 t
a
2
leads to a nontrivial
minimum
a v
2
1
4
2
F
Re
Im
26
Feynman rules with shift and the Goldstone mode.
Since one of many possible “shifts” was chosen it is
convenient to present the real and imaginary parts of the
fluctuation separately:
( x) min ( x) fl ( x) v O( x) iA( x)
The energy in terms of two real fields O and A becomes:
1
1 4
F [ av v ] [O ( 2 eO )O A( 2 eA ) A]
2
-v (
2
+
) +1/2 (
+2
Where the “masses” of excitations are
+
)
27
eO a 3v 2v 0
2
2
eA a v 2 0
eA ( k ) k
2
Regular massive mode,
“optic”
Goldstone mode,
“acoustic”
Dispersion relation of the A mode is that of acoustic
phonons:
e (k ) k
A
A
And in the confuguration space the correlator is:
1 i kx
A( x ) A(0) d k 2 e
k
1
0
x
x
in 3D
D
L
log
0
x L
x
in 2D
28
The field A itself is not the order parameter since it
does not transforms linearly under the symmetry
transformation. The order parameter is ei ( x )
Where the phase of is related to the fields by:
( x ) arctg ( A / O ) A / v
Its correlator is:
ei ( x ) ei (0) e ( x ) (0) e
A ( x ) A (0)
1
x
e 1
log
e
L
x
in 3D
1
x
0
in 2D
29
The 2D correlator in the “ordered” phase decays albeit
slowly (as a power rather than exponential in the
disordered phase
x
i ( x ) i (0)
e
e
e
Fluctuations due to Goldstone bosons in 2D “destroy”
perfect order. Such a phase is called quasi – long range
order phase (or Berezinski-Kosterlitz-Thouless phase).
We started from the assumption of nonzero VEV. It
seems that fluctuations destroy this assumption!
30
4. Destructions caused by IR divergencies
in Db2 and the MWC theorem.
The energy to the one loop level is:
1
1 4 1
F [av v ] [Tr log( 2 eO ) Tr log( 2 eA )]
2
2
2
The corrected value of v is found by minimizing it
perturbatively in “loops”:
2
2
2
v a v
Noting that
1
2
Tr log( e) 2
2
v
k e
k
31
one obtains a logarithmically divergent correction to
VEV:
v 2 3
1
d k k 2 log L
2
To higher orders the logs can be resummed:
1
v a (1 log L log L ...) a e
0
L order
The VEV decays – do not diverges, indicating that
2
2
2
log L
is “slowly” restored. This is Hohenberg-Mermin-Wagner
theorem: in 2D continuous symmetry is not broken. More
importantly this does not mean the perturbation theory is
useless.
32
O(2) invariant quantities
For such quantities the “collective coordinates” method
simplifies into perturbation theory around “broken”
vacuum. All the IR divergencies cancel. Let us see this
for the energy to two loops order. Jevicki, (1987)
2
]
F2
[ 3 2 3 ] a[6
2
There also is correction due to change in v:
Fcorr
4
(3
)2
33
The leading log 2 L IR divergencies are easy to evaluate:
3
a 2
1
2
2
log L log L log 2 L 0
2
2 a
4
Subleading log L divergencies also cancel although it
is much less obvious. Cancellations occur to all orders
F.David (1990) in loop expansion.
What is the mechanism behind this cancellation of
“spurious divergencies?
It is hard to say generally, but at least in extreme case of
1D the answer is clear.
34
Physics below the lower critical dimension
For D=1 the model is equivalent to QM of particle on a
plane with Mexican hat potential
Pert.
ground
state
QM
ground
state
O
A
Ground state is O(2) invariant but is very far from the
origin (0,0): pert. Ground state is bad, but theory
“corrects” it using IR divergent matrix elements
Kao,B.R.,Lee PRB61, 12652 (2000)
35
5. Heuristic argument about destruction of order
by Goldstone bosons.
b * 2
F d x[ a ( ) ]
2
D
*
*
For the XY model (same universality class as GL) a
Typical excitation is a wave of phase
L
L
0
36
Its energy in various dimensions is
D
EL
L
2
1
for D 1
L
1
D2
L
D3
no order
order
D=2 is a border case in which there exists “almost
long range order”
37
C. Topological excitations. The dual picture
1. An extreme point of view: topological
fluctuations dominate thermodynamics.
Vortices are the most important degrees of freedom in an
extremely type II SC (even in the absence of magnetic
field ):
/
It is therefore advantageous in such a case to
reformulate the theory in terms of vortex degrees of
freedom only:
( x) xi ( )
38
The Feynman-Onsager excitation and the
“spaghetti” vacuum.
Minimal excitation (Cooper pair) = Smallest vortex ring
2(T 0) Lmin
The normal phase is reinterpreted as a proliferation od
loops:
Normal
SC
39
This point of view is not commonly accepted or used
with one notable exception: the “BKT” transition in thin
films. Reasons:
1. In 3D “non ligh Tc” materials k is not very large.
2. Vortex loop in 3D is a complicated object: infinity of
degrees of freedom (unlike in 2D)
The dual picture was nevertheless advanced after the
discovery of high Tc. Its extentions to include external
magnetic field were unsuccessful so far.
40
2. KT transition in thin films or layered SC
22
eff
d
Pearl (1964)
eff may be very large .
Interaction between 2D fluxons in logr up to scale eff
SC - normal phase transition in thin films is of a
novel type: the Berezinskii –Kosterlitz-Thouless
continuous type (71).
41
The basic picture is just 2D “slice” of the fluxon
proliferation 3D picture : instead of vortex loop dipoles.
Dipole unbinding triggers
the proliferation
Free “dual charges”:
normal
-Kosterlitz, Thouless
Dual dipoles:
SC
(72)
42
The magnetic flux symmetry
Noeter theorem ensures that (if the symmetry is not
spontaneously broken) any continuous symmetry has a
corresponding conservation law. Examples: the electric
charge global U(1) symmetry ei
Leads to charge conservation
d
i Ji
dt
Other examples: rotations – angular momentum…
43
Similarly one can interpret the Maxwell equation
d
B i E i ;E i ij E j
dt
As a conservation law of the magnetic flux, yet
another global U(1) symmetry, sometimes calle
“inverted” or “dual” U(1) .
The symmetry is unbroken in the SC, while
spontaneously breaks down (photon is a Golstone boson)
in the normal phase
-Kovner,
Rosenstein, PRL
(92)
The order parameter field was constructed and the GL
theory in terms of it was established
44
The dual picture
The analogy of the charge U(1) and the magnetic flux
U(1) is as follows:
duality
Q
B
duality
J i
E i ij E j
duality
charge
vortex
-Nelson,
Halperin, PRB (81)
45
3. A brief history of phase transitions with
continuous symmetry in 2D.
a. They do not exist
The Hohenberg-Mermin-Wagner theorem
demonstrates that fluctuations destroy long range order.
According to the dual picture a continuous magnetic
flux symmetry should be spontaneouly broken. This
seems to be impossible in 2D, hence according to
Landau’s “postulate” –no phase transition.
The correlator is a power decay at low temperatures
1
( 0) ( x )
*
x
(T )
-Berezinskii(71)
46
b. The high temperature expansion: something
happens in between.
High temp. expansion gives an exponential
(0) ( x) e
*
x
l
Some qualitative change should happen in between!
This “something” is unbinding of vortices.
47
c. The energy – entropy heuristic argument
r
E q log
a
2
Energy of a pair of size r:
-Kosterlitz, Thouless
-
(72)
+
2
Number of states: R S 2 log R
a
a
48
c. The energy – entropy argument
The free energy of a pair therefore is:
2
R
R
2
F q Log 2T Log 2
a
a
Where a is the core- size. This becomes negative for
2
TKT
q
4
which means that the pair proliferate and
superconductivity is lost
49
Kosterlitz (74) and Young (79) developed a heuristic
renormalization group (RG) approach to account for
differences between pair’s sizes. We will follow this
approach.
d. Systematic expansions and exact solution.
The XY model or the Coulomb gas maps onto the sineGordon field theory for which perturbation theory
exists
Wiegman (78)
Starting with Zamolodehikov’s (80) exact results
Nowadays that KT theory is one of the most solid
were obtained that confirm approximate ones.
in Recent
theoretical
physics. on BSSCO and other layered
experiments
high Tc superconductors found new area of
50
4. The RG theory of the BKT transition
The energy of the KT pair
neglecting interactions with
other pairs is
r
U 0 r 2 q Log
a
2
0
Let us assume that
screening can be
represented by the
dielectric function
e(r).
2
q0
q r
r
2
51
It takes into account the polarization of the pairs due to
smaller ones.
2
q
r
q
F r
r r
r
2
0
r
The energy therefore gets reduced:
r'
U r 2 q r ' Log
r 'a
a
r
2
52
The dielectric function itself is created by the
polarization due to dependence of the Boltzmann
weight on the orientation of the thermally created KT
dipoles:
r 1 4 r
To calculate the polarization due
consider constant electric field
we first write the polarizability
of a single dipole:
1
p r
2
d
q0 dE rcos
q0 r d 1
n r dE 2
T
E
q
r
|
E 0
U r Eq0 r cos
|
cos a exp
T
E 0
4
53
q0 r
1
p r
n r a 4 2
q0 r cos
U r
cos T exp T
U r q02 r 2
q02 r 2
exp
4
2Tn r a
T 2T
since the number density of such pairs is
U r
n r a exp
T
4
Therefore one gets
r
r 1 4 2 r ' n r ' p r '
r 'a
4 q
1
Ta
2 2
0
4
r
r 'a
r '3 exp U r ' / T
54
Thereby we have derived an integral form of the RG
eqs. for U and e. Differentiating it with respect to
the pair size one obtains a differential form of the RG
eqs.
d
q02
r
U r
Log
dr
r
a
d
4 2 q02 3
r
r exp U r / T
4
dr
Ta
with initial conditions
r a 1
U r a 2
55
The equations can be turned into a set of autonomous
ones by going to a log scale
r
l Log ;r ael
a
d
d
dr
q02
q02
U l U r
r
dl
dr
dl l r
l
d
d
dr 4 2 q02 4
l r
r exp U l / T
4
dl
dr
dl
Ta
and rescaling the density variable
N l r / a exp[U l / T ]
4
4
56
r4
d
r 4 q02 U l / T
q02
N l 4 4 4
4
e
N l
dl
T l
a a T l
d
4 2 q02
l
N l
dl
T
Exercise 5: Plot the “vector field” of this autonomous
system. Solve this system of differential equations
either numerically or approximately analytically using
the separatrix method.
It is clear that the character of solution changes at
4
q02
T l
0 T TKT
q02
4 l
57
Exact solution near the KT temperature
Let us slightly redefine again new variables to make the
solution evident
4T l
x l
1
2
q0
y l 2 N l
With x small near the transition
d
x l 4 y 2 l
dl
d
2
y l 2
y l 2 x l y l
dl
1 x l
58
It is clear that parabolas are the flow lines
C x 2 y 2 x 2 l 0 y 2 l 0 x02 y02
The first equation now can be integrated
d
x l 2 x 2 l C
dl
With the result
x l C coth A
forC 0 T TKT
y l C csch A
A 2 Cl cosh 1 x0 / y0
59
2 x0
x l l x 2
forC 0 T TKT
0
y l x l
x l C tan A
forC 0
y l C sec A
A 2 Cl sin 1 x0 / y0
The critical value of dielectric constant is finite
l , T Tc 1 x0 Tc
1
60
Singularities at transition
Tc
1
1
exp Log
T KT
Tc T
With criticality of the very weak KT type
Density of bound pairs is
n lc n l
lc
0
With the result
U r
61
It is clear that parabolas are the flow lines
C x 2 y 2 x 2 l 0 y 2 l 0
The first equation now can be integrated
d
x l 2 x 2 l C
dl
With the result
x l C coth A
C 0
y l C csch A
A 2 Cl cosh 1 x0 / y0
62
5. Phenomenology of the BKT transition
The energy of the KT pair
neglecting interactions with
other pairs is
r
U 0 r 2 q Log
a
2
0
Let us assume that
screening can be
represented by the
dielectric function
e(r).
2
q0
q r
r
2
63