Download Lecture schedule October 3 – 7, 2011

Lecture schedule October 3 – 7, 2011
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Kondo effect
Spin glasses
Giant magnetoresistance
Magnetoelectrics and multiferroics
High temperature superconductivity
Applications of superconductivity
Heavy fermions
Hidden order in URu2Si2
Modern experimental methods in correlated electron systems
Quantum phase transitions
Present
Present basic
basic experimental
experimental phenomena
phenomena of
of the
the above
above topics
topics
# 10 Quantum Phase Transitions:
Theoretical driven 1975 … Experimentally first found
1994 … : T = 0 phase transition tuned by pressure,
doping or magnetic field [Also quantum well structures]
e.g.,3D AF
e.g.,2D Heisenberg AF
r is the tuning parameter: P, x; H
e.g., FL
Critical exponents – thermal (TC) where t = (T – TC)/TC
and quantum phase (all at T = 0) 2nd order phase
transitions
QPT: Δ ~ J|r –rc|zν , ξ-1 ~ Λ|r – rc|ν, Δ ~ ξ-z, ћω >> kBT,
T = 0 and r & rc finite
Parameters of QPT describing T = 0K singularity, yet they
strongly influence the experimental behavior at T > 0
Δ is spectral density fluctuation scale at T = 0 for r  rc , e.g., energy of
lowest excitation above the ground state or energy gap or qualitative change
in nature of frequency spectrum. Δ  0 as r  rc.
J is microscopic coupling energy.
z and ν are the critical exponents.
 is the diverging characteristic length scale.
Λ is an inverse length scale or momentum.
ω is frequency at which the long-distance degrees of freedom fluctuate.
For a purely classical description ħωtyp << kBT with classical critical
exponents. Usually interplay of classical (thermal) fluctuations and quantum
fluctuations driven by Heisenberg uncertainty principle.
Beyond the T = O phase transition: How about the
dynamics at T > 0?
eq
is thermal equilibration time, i.e., when local thermal
equilibrium is established. Two regimes:
If Δ > kBT, long equilibration times: τeq >> ħ/kBT classical dynamics
If Δ < kBT, short equilibration times: τeq ~ ħ/kBT quantum critical
Note dashed crossover lines
Crossover lines, divergences and imaginary time:
Some unique properties of QPT
(T  TC ) / TC  TC1/ νz unimportant for QC
k BT  h ωtyp  r  rc
νz
crossover line
C  T S / T , B  S / r ( P) defines Grüneisen parameter
B/C  r  rc
1
 T 1/ ν z where C is the specific heat
and B is the thermal expansion with pressure tuning
At T  0, 1/ k BT  [imag. time]  iΘ / h (Θ is the real time)
 new time axis and new "space dimension" D  d  z
[imag. time] diverges as T  0, "black hole" analogy
"ω / T scaling"at QC
Hypothesis: Black hole in space – time is the quantum
critical matter (droplet) at the QCP (T = 0). Material
event horizon – separates the electrons into their spin
and charge constituents through two new horizons.
Subtle ways of non-temperature tuning QCP: (i) Level
crossings/ repulsions and (ii) layer spacing variation in
2D quantum wells
Excited state becomes ground state: Varying green layer thickness changes ferricontinuously or gapping: light or
magnetic coupling (a) to quantum paramagnet
frequency tuning. Non-analyticity at (dimers) with S =1 triplet excitations.
gC. Usually 1st order phase transition –
.…..NOT OF INTEREST HERE……
Experimental examples of tuning of QCP: LiHoF4 Ising
ferromagnet in transverse magnetic field (H)
H┴ induces quantum tunneling between the two states: all ↑↑↑↑ or all ↓↓↓↓. Strong
tunneling of transverse spin fluctuations destroys long-range ferromagnetic order at QCP.
Note for dilute/disordered case of Li(Y1-xHox )F4 can create a putative quantum spin glass.
Bitko et al. PRL(1996)
Solution of quantum Ising model in transverse field where
Jg = µH and J exchange coupling: F (here) or AF
nonmagnetic
ferromagnetic
Somewhere (at gC) between these two states there is non-analyticity, i.e., QPT/QCP
Some experimental systems showing QCP at T = 0K
with magnetic field, pressue or doping (x) tuning
 CoNb2O6 -- quantum Ising in H with short range Heisenberg
exchange, not long-range magnetic dipoles of LiHoF4.
 TiCuCl3 -- Heisenberg dimers (single valence bond) due to crystal
structure, under pressure forms an ordered Neél anitferromagnet via a
QPT.
 CeCu6-xAux -- heavy fermion antiferromagnet tuned into QPT via
pressure, magnetic field and Aux- doping.
 YbRh2Si2 -- 70 mK antiferromagnetic to Fermi liquid with tiny fields.
Sr3Ru2O7 and URu2Si2 “novel phases”, field-induced, masking QCP.
Non-tuned QCP at ambients “serendipity” CeNi2Ge2 YbAlB???.
Let’s look in more detail at the first (1994)one CeCu6-xAux.
Ce Cu6-x Aux experiments: Low-T specific heat tuned with x
(i) x=0, C/T  const. Fermi liquid (ii) x=0.05;0.1  logT NFL
behavior and (iii) x= 0.15,0.2;0.3 onset of maxima AF order
At x = 0.1
as T0 QCP
von Löhneysen et al. PRL(1994)
Susceptibility (M/H) vs T at x=0.1 in 0.1T(NFL -> QCP):
χ = χo( 1 – a√T ) and in 3T(normal FL): χ = const.
Field restores HFL behavior. Pressure also.
(1 - a√T)
von Löheneysen et al.PRL(1994)
Resistivity vs T field at x=0.1 in 0-field: ρ = ρo + bT {NFL}
but in fields: ρ = ρo + AT2 {FL}. Field restores HFL behavior.
von Löhneysen et al. PRL(1994)
T – x phase diagram for CeCu6-xAux in zero field and at
ambient pressure. Green arrow is QCP at x=0.1
Pressure and magnetic field
Pressure dependence of C/T as fct.(x,P) where P is the
hydrostatic pressure. Note how AFM 0.2 and 0.3 are
shifted with P to NFL behavior and 0.1 at 6kbar is HFLiq.
Two “famous” scenarios for QCP (here at x=0.1)
(b)local moments are quenched at a finite TK AFM via SDW
(c)local moments exist, only vanish at QCP Kondo breakdown
Which materials obey
scenario (b) or (c)??
W is magnetic coupling between
conduction electrons and felectrons, TN=0 at Wc : QCP
Weak vs strong coupling models for QPT with NFL.
Top] From FL to magnetic instability (SDW)
Bot] Local magnetic moments (AFM) to Kondo lattice
Many disordered materials-NFL, yet unknown effects of disorder
So what is all this non-Fermi liquid (NFL) behavior?
See Steward, RMP (2001 and 2004)
Hertz-PRB(1976), Millis-PRB(1993); MoriyaBOOK(1985) theory of QPT for itinerant fermions.
Order parameter(OP) theory from microscopic model
for interacting electrons.
S is effective action for a field tuned QCP with  vector field OP
-1 (propagator) and b2i (coefficients) are diagramatically calculated.
After intergrating out the the fermion quasiparticles:
where |ω|/kz-2 is the damping term of the OP fluct. of el/hole paires at
the FS and d = 2 or 3 dims., and z the dynamical critical exponent.
Use renormalization-group techniques to study QPT in 2 or 2 dims. for
Q vectors that do not span FS. Results depend critically on d and z
Predictions of theories for measureable NFL quantities -over -
Predictions of different SF theories: FM & AFM in d & z
(a) Millis/Hertz [TN/C Néel/Curie & TI/II crossover T’s]
(b) Moriya et al.
{All NFL behaviors]
(c) Lonzarich
Millis/Hertz theory-based T – r (tuning) phase diagram
I) Disordered quantum regime-HFLiq., II) perturbed classical
regime, III) quantum critical-NFL, and V) magnetically ordered
Néel/Curie [SDW] phase transitions. Dashed lines are crossovers.
Summary: Quantum Phase Transitions
Apologies being too brief and superficial
The end of Lectures
STOP
 ħħħħ τeqττ ξξ
EXP LiHoF4
xxx