vi. tÓpicos avanzados puntos críticos cuánticos ... · 4 0.01 0.1 1 10 100 1000 universe temp....
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VI. TÓPICOS AVANZADOSPuntos Críticos cuánticos. Transiciones de fase cuánticas. Fases de Griffiths.
Frustración: Spin Ice. Fases de Shastry Sutherlnad.. Dyslochinsky-Moriya.
2
Magn. Ord.
Paramagn.
which bed is better to hibernate?
‘ A QCP is defined when a 2nd order transition is driven to T = 0 by a non-thermal parameter ‘ T. Vojta, Annal. Phys. 2000
< 2K : Q vs Therm fluct.
free energy
Quantum critical point like a rainbow end-point
3
keep away from my buried treasure !
0 .0 0 .2 0 .4 0 .6 0 .80
1
2
3
4
5
6CePd
1-x R h
x
TC [
K]
x [conc.]
QCP
Sereni et al, Phys.Rev.B 2007
Magn. Ord.
Paramagn.
4
0.01
0.1
1
10
100
1000
Universe temp.
Adiabatic demagn.
50mK dilution He3/He4 frig.
450mK-Bariloche
Liquid He
Liquid H2
Liquid O2 & N
2
Room (bio) temp.
TC of Fe
Log
(T /
K)
Sereni J Low Temp Phys 2007
Thermal energy scale ~kBT
a) pre-critical region: change of TN,C(x) curvature at ~ 2K
Fingerprints for the proximity of a QCP in exemplary magnetic phase diagrams
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
10
0.6 0.7 0.8 0.9 1.0
0.1
1
0.6 0.7 0.8 0.9 1.0
0.0
0.5
1.0
1.5
2.0
TN
,C [
K ]
x / xcr
CeIn3 - xSn
x
CePd1 - xRh
x
CeCu2 (Ge
1-xSix )2
(a)
(c)
pre-critical
T* C
[K]
Cmax
/ Tχ'
ac
xcr
CePd1-xRh
x
x [Rh conc.]
(b)
TN [K
]
x / xcr
CeIn3-x Sn
x
CeCu2 (Ge
1-xSix )2
“The southernmost
cosmological singularity”
Prof. Dieter Wollhardt”
Bche. Feb.25,2011
5
0.0
0.4
0.8
1.2
1 10
10.0
0.5
1.0
1.5
2.0
0.60
0.500.55 0.47
0.37 0.300.41
0.25
x=0.15
CeIn3-x
Snx
T [K]
T [K]
x=0.50
0.65 0.600.75
0.70
CePd1-x
Rhx
Cm
/ T
[J/
mol
K2 ]
Sereni J Low Temp Phys 2007
0.1 10
1
2
3
Cm
/ T
[J/
mol
K2 ]
0.85
1.0
0.9
0.80.7
0.5 z=0.35
CeAu1-z
Cu5+z
T [K]
b) maximum value of (specific heat) Cmax / T => const.c) Cm / T ~ - log(T/T0)
0.35 0.5 1 2 4 80.0
0.5
1.0
1.5
2.0 x= 0 x=0.05 x=0.1 x=0.2 x=0.3 x=0.4 x=0.5
CeCu2(Si
1-xGe
x)2
T (K)
CP
/ T
(J/
mol
K2 )
6
Entropy
SS = ∫ Cm / T dT from Cm (T) = Q/∆T
A) 3rd. Law of Thermodynamics 3rd. Law of Thermodynamics ::
i) SS(T → 0) => SS0 , usually (but not necessarily) SS0 = 0 ii) no singularity at T = 0, then ∂SS / ∂T = Cm / T → const.
B) Cosequences Cosequences ::
i) Once in the Ground State → no further modifications are possible (singlet: SS0 = 0)ii) In metastable phases (e.g. quenched alloys) → SS0 ≠ 0 , but for t→ ∞: S0 → 0
~ µeVspectrometry
iii) Quantum fluctuationsQuantum fluctuations or frustration: different scenario since ≠ f f (T)
How to determine whether S = or ≠ 0?If SS0 ≠ 0 but unknown, S(T → ∞) is the alternative reference
since all states will be equally occupied
7
AlternativeAlternative:: If SS0 is unknown, S(T → ∞) is the alternative reference
00.0
0.5
1.0
0.2
Paramagn.
Magn. Ord.
∆ S
/ R
Ln2
T / ∆
∆ = splitting of Ce-magnetic levelsin cubic FCC Ce(In,Sn)3
Simplest case:a 2 level system
“Wie hoch ist der plateau ?”
“Wie tief ist die Elbe”
8
Experimental results from Ce(In1-xSnx)3
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2
4
6
8
10
?
QCP
xcr
Anti Ferro.
xcCeIn
3-xSn
x
TN [K
]
x [Sn conc.]
AFI
-4 0 4 80.0
0.2
0.4
0.6
0.8
1.0
0.5 Klimit
∆T = T-TN [K]
MO
0.80
0.4
CeIn3-x
Snx
Cm
/ T
[J /
mol
K2 ]
x 0.450.470.500.600.70Within linear (pre-critical) region
Cm / T curves coincide if plottedvs. ∆T =T-TN (T shifted by TN)
2 4 6 8
2 4 6 80.0
0.2
0.4
0.6
0.8
1.0
CeIn3-xSn
x
Sn conc [x]
x=0.25
0.30
0.47
0.60
0.50
0.37
0.55
0.450.41
T [K]
Cm
/ T
[J/m
olK
2 ]
linear TN (x) dependence: 0.4 < x < 0.65
Thermod. T = 0 shifts towards 0 as TN => 0
Magnetic Specific heat contribution as Cm / T
Pedrazzini, PhD Bariloche 2004
0.4
9
0.0
0.2
0.4
0.6
0.8
1.0-4 0 4 8 12
MO0.80
0.4
CeIn3-x
Snx
∆T = T-TN [K]
Cm
/ T
[J /
mol
K2 ]
x 0.450.470.500.550.600.70
0 4 8 12 16-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Sm [
R L
n2 ]
x=0.41
T [K]
0 4 8 12 16 200.0
0.2
0.4
0.6
0.8
1.0
SM
O
S
m [
R L
n2 ]
∆T + 4 K SS(T) of x = 0.4 as SSm + SSMOMO
Full RLn2 entropy computed from ∆T = - 4K
Computing the Entropy
60% RLn2 – paramag .20% RLn2 - MO20%RLn2 – missing entropy
SS(T) of x = 0.7 as SSm + SSMOMO
60% RLn2 – paramag .. 0% RLn2 - MO40%RLn2 – missing entropy
10
0 1 2 3 4 5 6-50
-40
-30
-20
-10
0
0 1 2 3 4
-4
-2
0
2
0.80
0.70CeIn
3-xSn
x
0.65
0.55
(VT -
V 6K
) /
V
T [K]
0.80
( Vx -
V0.
8 ) /
V
[10
-6 ]
T [K]
TN
0.55
0.650.70
0.0 0.2 0.4 0.6 0.80.0
0.2
0.4
0.6
0.8
1.0
5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
S0 = RLn2 - S
m
Sm
(20K)
Sm /
RLn
2
x [Sn Conc.]
0.38
x = 0.70
x = 0.15
CeIn3-x
Snx
Sm /
RLn
2
T [K]
Sms(x,T) + S0 (x) = RLn 2
Independent observation on thermal expansion:Independent observation on thermal expansion:Since Smeas(T) = ST→0 + S(x,T)from Maxwell relations- ∂S/∂P = ∂V/ ∂T = β (therm. expan.)
V(T) = ∫ β dT thenVmeas = V0 (x) + V(x,T)
V(T) normalizaed at 8K
Beyond x = xcr, Kondo effect takes over
Corresponding Vol. anomalyComputing the Entropy ........
β(T) after Kuechler PHD Darmstad 2003
11
0.1
1
100.0 0.2 0.4 0.6 0.8
0.0 0.2 0.4 0.6 0.8
Therm. vs Q.fluctuations
Pre-critical region
xcr
TI
TN
V 0(x
)
x [Sn conc.]T
N [K
]
S 0 (x)
x [Sn conc.]
Conclusions
S0(x) increases as x → xcr up to 0.38 RLn2
S=RLn2 RLn3/2new degree of freedom introduced by quantum fluctuations between MO & Paramag. statesas x → xcr & T → 0
There is a pre-critical region with Quantum Critical features at T ≤ 2K
Corresponding V0(x) anomalous dependence
100% - 60% => 40%
Alternative scenario:
12
EntropySS = ∫ Cm / T dT from Cm (T) = Q/∆T
A) 3rd. Law of Thermodynamics 3rd. Law of Thermodynamics ::
i) SS(T → 0) => SS0 , usually (but not necessarily) SS0 = 0 ii) no singularity at T = 0, then ∂SS / ∂T = Cm / T → const.
B) Cosequences Cosequences ::
i) Once in the Ground State → no further modifications are possible (singlet: SS0 = 0)ii) In metastable phases (e.g. quenched alloys) → SS0 ≠ 0 , but for t→ ∞: S0 → 0
thermal fluct.
~ µeVspectrometry
for t→ ∞
iii) Quantum fluctuations: different scenario since tuneling ≠ f f (T)
How to evaluate whether S = or ≠ 0?If SS0 ≠ 0 but unknown, S(T → ∞) is the alternative reference
since all states will be equally occupied
however
13
Divergencias (teória) vs.comportamientos observados
Régimen cuántico
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