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 URu 2 Si 2  #9 Modern experimental methods in correlated electron systems  #10 Quantum phase transitions Present 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 (T C ) where t = (T – T C )/T C and quantum phase (all at T = 0) 2 nd order phase transitions QPT: Δ ~ J|r –r c | zν, ξ -1 ~ Λ|r – r c | ν, Δ ~ ξ -z, ћω >> k B T, T = 0 and r & r c 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  r c. 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 ħ ω typ << k B T 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 > k B T, long equilibration times: τ eq >> ħ/k B T classical dynamics If Δ < k B T, short equilibration times: τ eq ~ ħ/k B T 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 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 g C. Usually 1 st order phase transition –.…..NOT OF INTEREST HERE……

Experimental examples of tuning of QCP: LiHoF 4 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(Y 1-x Ho x )F 4 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 g C ) 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  CoNb 2 O 6 -- quantum Ising in H  with short range Heisenberg exchange, not long-range magnetic dipoles of LiHoF 4.  TiCuCl 3 -- Heisenberg dimers (single valence bond) due to crystal structure, under pressure forms an ordered Neél anitferromagnet via a QPT.  CeCu 6-x Au x -- heavy fermion antiferromagnet tuned into QPT via pressure, magnetic field and Au x - doping.  YbRh 2 Si mK antiferromagnetic to Fermi liquid with tiny fields. Sr 3 Ru 2 O 7 and URu 2 Si 2 “novel phases”, field-induced, masking QCP. Non-tuned QCP at ambients “serendipity” CeNi 2 Ge 2 YbAlB???. Let’s look in more detail at the first (1994)one CeCu 6-x Au x.

Ce Cu 6-x Au x 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 T  0 QCP

Susceptibility (M/H) vs T at x=0.1 in 0.1T(NFL -> QCP): χ = χ o ( 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 = ρ o + bT {NFL} but in fields: ρ = ρ o + AT 2 {FL}. Field restores HFL behavior. von Löhneysen et al. PRL(1994)

T – x phase diagram for CeCu 6-x Au x 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 T K AFM via SDW (c)local moments exist, only vanish at QCP Kondo breakdown W is magnetic coupling between conduction electrons and f- electrons, T N =0 at W c : 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 b 2i (coefficients) are diagramatically calculated. After intergrating out the the fermion quasiparticles: where |ω|/k z-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 [T N/C Néel/Curie & T I/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

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