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Lecture schedule October 3 – 7, 2011 PowerPoint Presentation

Lecture schedule October 3 – 7, 2011

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Lecture schedule October 3 – 7, 2011

Present basic experimental phenomena of the above topics

Present basic experimental phenomena of the above topics

#1 Kondo effect

#2 Spin glasses

#3 Giant magnetoresistance

#4 Magnetoelectrics and multiferroics

#5 High temperature superconductivity

#6 Applications of superconductivity

#7 Heavy fermions

#8 Hidden order in URu2Si2

#9 Modern experimental methods in correlated electron systems

#10 Quantum phase transitions

# 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., FL

e.g.,2D Heisenberg AF

r is the tuning parameter: P, x; H

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 , 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.

Λis an inverse length scale or momentum.

ω is frequency at which the long-distance degrees of freedomfluctuate.

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

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: continuously or gapping: light or frequency tuning. Non-analyticity at gC. Usually 1st order phase transition –.…..NOT OF INTEREST HERE……

Varying green layer thickness changes ferri- magnetic coupling (a) to quantum paramagnet (dimers) with S =1 triplet excitations.

Experimental examples of tuning of QCP: LiHoF crossings/ repulsions and (ii) layer spacing variation in 2D quantum wells4 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 crossings/ repulsions and (ii) layer spacing variation in 2D quantum wells 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 Isingin 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 Cu magnetic field, pressue or doping (x) tuning6-xAux 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 T0 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 infields: ρ =ρo+ AT2 {FL}. Field restores HFL behavior.

von Löhneysen et al. PRL(1994)

T – x phase diagram for CeCu 6-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) hydrostatic pressure. Note how AFM 0.2 and 0.3 are shifted with P to NFL behavior and 0.1 at 6kbar is HFLiq.(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 f-electrons, TN=0 at Wc :QCP

Weak vs strong coupling models for QPT with NFL. hydrostatic pressure. Note how AFM 0.2 and 0.3 are shifted with P to NFL behavior and 0.1 at 6kbar is HFLiq.Top] From FL to magnetic instability (SDW)Bot] Local magnetic moments (AFM) to Kondo lattice

So what is all this non-Fermi liquid (NFL) behavior? disorderSee Steward, RMP (2001 and 2004)

Hertz-PRB(1976), Millis-PRB(1993); Moriya-BOOK(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 of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons.(a) Millis/Hertz [TN/C Néel/Curie & TI/II crossover T’s](b) Moriya et al.(c) Lonzarich

{All NFL behaviors]

Millis/Hertz theory-based T – r (tuning) phase diagram of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons.

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 of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons. Apologies being too brief and superficial

The end of Lectures

STOP of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons.

- ħħħħ of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons.τeqττξξ

EXP LiHoF4 of QPT for itinerant fermions. Order parameter(OP) theory from microscopic model for interacting electrons.

xxx

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