7/20/2010
Multiple Transitions and Order
Parameters: Higher Order Phase
Transitions
Bohdan Andraka, Pradeep Kumar
Department of Physics
University of Florida
Gainesville, FL 32611-8440
and
Avadh Saxena
T-4, MS B. 262
Los Alamos National Laboratory
Los Alamos, NM 87545
• Ehrenfest classification of
phase transitions (order
1,2):
– Derivatives of order less than
2
Tc ∆ χ
2 are continuous.
 dT 
 dH  = ∆ C


– Ehrenfest equation
– Derivatives of order less than
p are continuous.
F (T ) =
– Equation for pth order
 dT 
 dH 
•
F = aϕ 2 + c | ∇ϕ | 2 +bϕ 2 p /( p −1)
a = a o  T − 1
 Tc 
The free energy
b
C ≈ 1 − T 
Tc 

p −1
[
]
P =2
P=3
P=4
p −2
Superfluid Density ρs = (1-T/Tc) (p- 1) = Hc1 = λ-2
ρs
•
•
•
p=2
PrOs4 Sb12 (POS) is a Pr based
4000
heavy fermion
∆C/Tc = 103 mJ/K2 - mol
Superconductivity in PrOs4 Sb12 3500
there are two transitions.
• The lower transition (1.72K) is3000
sharp and robust.
• The upper transition (1.85K) is
2500
broad and sensitive to
perturbations. Tc varies widely.
C (mJ/K mol)
p=3
p=4
Tc
T
Tc
T
Large crystal #1
C/T (mJ/K2mol)
p
p
 p−2 χ

Tc ∆ ∂
∂ H p − 2 

=
p−2
∂
C
∂T p−2
b, c ≥ 0
mc 2
p −1
a ≤ 0 ρ s = 2 ϕo ≈ a
h
( p − 1) p −1 | a | p
F =−
T p−µ p
)
Tc
Specific heat C = (1-T/Tc) (p-2) for p = 2,3,4
The expression is a fixed point, arrived at, for a
description near Tc, regardless of the microscopic details.
( p −1) / 2
p
− f (1 −
C
This is no longer a polynomial expansion but φ might not
be an isolated solitary field. The power is the result after
the secondary fields have been averaged. A polynomial
only allows p =2.
 | a | p − 1
ϕ o (T ) = 

 b p 
=0
a>0
T 2 −α
)
Tc
• Order p
Interacting phase boundaries for superconducting transitions in PrOs4Sb12.
What to expect?
The resolution
F (T ) = − f (1 −
6000
The upper transition is proposed to
be of order three.
– The onset temperature is robust 5000
– There is curvature in the lower
critical field.
– Compressibility and thermal
expansion are similarly without
a clear discontinuity at Tc.
(Oeschler ‘04)
1.5
1.6
1.7
1.8
1.9
T (K)
1
7/20/2010
7000
C (mJ/K mol)
• Two phase boundaries:
Non Crossing boundary
• T1>T2
• T1 transition is a third
order phase transition.
T2 is a second order
transition
PrOs4Sb12
"stoichiometric" crystal
6000
Different samples, both C and C/T
5000
7000
C (mJ/K mol)
"small crystal"
3500
3000
1.5
1.6
1.7
1.8
1.9
2.0
6000
F 1 = a 1 ϕ 12 + b 1 ϕ 13
F 2 = a 2 ϕ 22 + b 2 ϕ 24
Fint = s1ϕ1 .ϕ 2 + s 2 | ϕ1 .ϕ 2 | 2 + s3ϕ1ϕ 22 + s 4 ∇ϕ1 .∇ϕ 2
5000
4000
T (K)
C/T (mJ/K2mol)
C/T (mJ/K2mol)
4000
F = F 1 + F 2 + F int
3500
3000
1.5
1.6
1.7
1.8
1.9
2.0
1 (Josephson coupling) Requires symmetry overlap, 2 and 4 are
quite general with 4 as the interaction between super-currents, 3
is special to the case of mixed order coupling. It is 2 with a
transformation φ1 = ψ2.
T (K)
Fint = s1ϕ1 .ϕ 2 + s 2 | ϕ1 .ϕ 2 | 2 + s3ϕ1ϕ 22 + s 4 ∇ϕ1 .∇ϕ 2
• Only s1 ≠ 0:
– There is only one transition at T1.
– The transition at T1 is of order 3. No specific heat
discontinuity, nor in any other free energy
derivative. Specific heat rises linearly below T1.
– Super fluid density ρs(T) ~a2 ~Hc1 (T).
• Only s2 ≠ 0
– Two transitions with renormalized T2. Upper
transition is third order the lower is second order.
– Hc1 near T2 is linear in a2. Near T1, it is quadratic in
a1.
Fint = s1ϕ1 .ϕ 2 + s 2 | ϕ1 .ϕ 2 | 2 + s3ϕ1ϕ 22 + s 4 ∇ϕ1 .∇ϕ 2
• Only s3 ≠ 0
– Arises under special situation, there are two
transitions at T1 and a renormalized T2 .
– Specific heat rises linearly below T1 followed by a
discontinuity at T2.
– Superfluid Density (or the penetration depth or the
lower critical field Hc1) is quadratic in a1 at T1 but
linear in a2 at T2.
• Only s4 ≠ 0
– Arises from interaction between supercurrents.
– Allows for magnetic field related changes.
Conclusions
Does POS have a third order phase transition to
superconductivity?
•Specific heat looks right but more information
needed for superfluid density. So far support only
for s2 coupling.
What should one expect to see in a higher order
phase transition near a lower order transition?
• Look for specific heat C, compressibility κ and
thermal expansion β. Also superfluid density ρs(T)
with a temperature dependence ap-1.
2