Lecture schedule October 3 – 7, 2011

#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

Present basic experimental phenomena of the above topicsPresent basic experimental phenomena of the above 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.,2D Heisenberg AF

e.g.,3D 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 , 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.

cr r

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

1/

1 1/

( ) / unimportant for QC

crossover line

C / , B / ( ) defines Gr neisen parameter

B/C where C is the specific heat

and B is the thermal expansion with pres

νzC C C

νz

B typ c

ν zc

T T T T

k T ω r r

T S T S r P ü

r r T

h

sure tuning

At 0, 1/ [imag. time] / ( is the real time)

new time axis and new "space dimension"

[imag. time] diverges as 0, "black hole" analogy

" / scaling"at QC

BT k T iΘ Θ

D d z

T

ω T

h

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

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

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……

Experimental examples of tuning of QCP: LiHoF4 Ising

ferromagnet in transverse magnetic field (H)

Bitko et al. PRL(1996)

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.

Solution of quantum Ising model in transverse field where Jg = µH and J exchange coupling: F (here) or AF

Somewhere (at gC) between these two states there is non-analyticity, i.e., QPT/QCP

nonmagnetic

ferromagnetic

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

von Löhneysen et al. PRL(1994)

At x = 0.1 as T0 QCP

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

W is magnetic coupling between conduction electrons and f-electrons, TN=0 at Wc : QCP

Which materials obey scenario (b) or (c)??

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); 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(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

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