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Quantum Criticality in Polar Materials I : New Perspectives on Quantum Criticality from Polar Materials (pedagogical) P . Chandra (Rutgers) I. Title Unpacked II. Why ?? III. Historical Perspectives and Current Challenges PC, G.G. Lonzarich, S.E. Rowley and J.F. Scott, Reports on the Progress of Physics 80, 112502 (2017) � 1

What does quantum critical mean? Aren’t quantum fluctuations only important at T = 0 ? “The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.” Sidney Coleman � 2

Simple Harmonic Oscillator (1d) ⇢ 1 + 1 � h x 2 i = Ω Variance ( ~ = 1 , k B = 1) Ω 2 K T � 1 e h x 2 i Quantum Fluctuations 1 Ω 2 K Classical Fluctuations Ω T h x 2 i ⇠ T Thermal (classical) Fluctuations Ω < T K Thermal-Quantum Fluctuations 0 < T < Ω h x 2 i = Ω Pure Quantum Fluctuations T = 0 � 3 2 K

What does the behavior of a SHO have to do with phase transitions and criticality ? Order Parameter Fluctuations Variance of each of their Fourier Components, a mode of wavevector q whose behavior can be mapped onto a SHO � 4

h x 2 i ! 1 At a continuous phase transition K ! 0 ) What about the specific heat ? In diamond effects of quantum fluctuations present at room temperatures ! � 5

How to think about temperature near a quantum critical point? • Temperature is NOT a tuning parameter • Temperature is a boundary effect! Fluctuations are Purely Quantum up to this Time-scale � 6

Back to the SHO ...... ⇢ � n Ω + 1 χ = 1 h x 2 i = K (= Re χ ω =0 ) Ω χ 2 Im χ ω = π 2 ω χ δ ( ω − Ω ) ( ω > 0) Z ∞ ⇢ � h x 2 i = 2 n ω + 1 d ω Im χ ω 2 π 0 Nyquist Theorem � 7

Generalization to all modes in the Brillouin Zone Z ∞ ⇢ � h δφ 2 i = 2 n ω + 1 X d ω Im χ q ω π 2 0 q φ = ¯ Scalar Order Parameter φ + δφ Average h δφ i = 0 Im χ q ω = π 2 ω χ q δ ( ω − ω q ) ( ω > 0) (propagating limit, simplest case) � 8

Generalization to all modes in the Brillouin Zone Z ∞ ⇢ � h δφ 2 i = 2 n ω + 1 X d ω Im χ q ω π 2 0 q h δφ 2 T i Our Focus: Strongly Temperature-Dependent Contribution Dominant in Determining Temperature-Dependence of Observable Properties � 9

Dispersion and Important Wavevectors ω q z 1 q T ∝ T z ( ~ = 1 , k B = 1) T q BZ q T q Purely Classical Fluctuations q BZ < q T Quantum Fluctuations Present q BZ > q T � 10

Generalization to all modes in the Brillouin Zone Z ∞ ⇢ � h δφ 2 i = 2 n ω + 1 X d ω Im χ q ω π 2 0 q Focus: h δφ 2 T i X h δφ 2 T i ⇡ T ( T � ω q for q < q BZ ) χ q q<q BZ X h δφ 2 T i ⇡ T ( T ⌧ ω q for q < q T ) χ q q<q T ∝ κ 2 + q 2 ) ( χ − 1 ∝ κ 2 ) ( χ − 1 � 11 q

Ferroelectrics: The Simplest Polar Materials A FE is a material that has a spontaneous polarization that is switchable by an electric field of practical magnitude � 12

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SrTiO 3 - Almost a Ferroelectric (Muller and Burkhard 79) Ferroelectricity induced by Uniaxial Stress, Ca and O-18 Substitution � 14

SrTiO 3 - Almost a Ferroelectric (Rowley et al. 14) Ferroelectricity induced by Uniaxial Stress, Ca and O-18 Substitution � 15

What do FEs have to do with Quantum Criticality ? Insulators (link to novel metals and superconductivity?) !? Classical FE transitions usually 1st order !? � 16

Many, many (magnetic) settings to study quantum criticality.... why do we need more ?? � 17

Important Role in (Classical) 
 Critical Phenomena First Calculation of Logarithmic Corrections to Mean-Field Theory in d=d* !! � 18

Uniaxial Ferroelectric All dipoles in z direction TO phonon dispersion (1) Application of Simple Scaling (2) Simultaneous Satisfaction of (1) and (2) k=2 “counts” for effectively two dimensions � 19

Why Study FE Quantum Criticality ? Quest for Universality Simplicity Controlled Additional Degrees of Freedom (and maybe novel metals and exotic superconductors) ( Possible Applications) � 20

χ − 1 ∝ T 2 Simpler way to get this result ?? � 21

S. Rowley, L. Spalek, R. Smith, M. Dean, M. Itoh, J.F. Scott, G.G. Lonzarich and S. Saxena, Nature Physics 10, 367-72 (2014) � 22

Self-consistent Landau Approach Minimization Observed moment requires fluctuation-averaging (due to coarse-graining over q T ) � 23

We can Fourier transform in the limit to obtain Most probable vs. average values...coarse-graining over q T ! Z q T ✓ κ ( ◆ d − 2 ) q d − 1 T X ≈ T q d − 2 κ 2 ∝ κ 2 + q 2 ≈ T 1 − T q 2 q T κ q<q T Temptation ... ( d + z − 2) χ − 1 ∝ κ 2 ∝ T z � 24

When is this approach valid ? ✓ κ ✓ κ ( ◆ d − 2 ) ◆ 2 ( d + z − 4) ∝ T 1 − z q T q T ✓ κ ◆ lim → 0 if d eff ≡ d + z > 4 q T T → 0 Ferroelectrics d = 3, z = 1 ( d + z − 2) χ − 1 ∝ κ 2 ∝ T = T 2 (log terms) z Agrees with previous calculation by different methods � 25

Finite-Size Scaling in Space and T • Space (near CCP) ξ ∼ t − ν � L � L ⇥ ⇥ χ ∼ t − γ Φ ∼ t − γ Φ ξ t − ν For L << ξ χ = χ ( L ) � L ⇥ γ ⇥ p � L ν γ χ ∼ t − γ ∼ t − γ Φ ∼ L ν ξ t − ν � 26

• Time ξ ∼ g − ν − → ξ τ ∼ g − z ν � (near QCP) L τ = k B T � L τ � L τ ⇥ ⇥ χ ∼ g − γ Φ ∼ g − γ Φ ξ τ g − z ν For L τ << ξ τ χ = χ ( L τ ) � L τ ⇥ γ ⇥ p � L τ z ν γ ∼ T − γ χ ∼ g − γ ∼ g − γ Φ ∼ L z ν z ν τ ξ τ g − z ν χ − 1 ∝ T 2 (near FE-QCP) � 27 (here z = 1 , ν = 1 / 2 , γ = 1 → γ z ν = 2)

Zhu et al (13) Gruneisen Ratio Minimization + Fluctuation-Averaging Near Classical Phase Transition Γ CF E ( T → T c ) ∝ ( T − T c ) 0 (supported by experiment) � 28

Zhu et al (13) Gruneisen Ratio Minimization + Fluctuation-Averaging In the vicinity of a (d=3) FE-QCP � 29

Zhu et al (13) Scaling Approach to the Gruneisen Ratio ∂ S/ ∂ V Γ = α 1 = − c P V m T ∂ S/ ∂ T Dimensionally  1 � [ Γ ] = g Near a (FE)-QCP ✓ L τ ✓ L τ ◆ ◆ Γ = 1 = 1 1 = Γ 0 T − 1 = ˜ g Φ g Φ Γ 0 L z ν z ν τ ξ τ g − z ν Γ − 1 ∝ T 2 ( z = 1 , ν = 1 � 30 2)

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Why Study Ferroelectric Quantum Criticality? Quest for Universality in Quantum Criticality Simple Examples: Few Degrees of Freedom and Non-Dissipative Dynamics Reside in marginal dimension allowing for detailed interplay between experiment and theory Additional Degrees of Freedom (e.g. Spin and Charge) can be added Systematically � 33

Open Questions for Future Research Specific FE/PE materials for Study at low T Add Spin: A Multiferroic QCP Add Charge: An Exotic Metal and an Unexpected Superconductor! � 34

Thoughts on n-doped STO Mott criterion for doped semiconductors 1 ✏ ~ 2 ✓ ◆ c a ∗ 3 B ≈ 0 . 26 a ∗ n B = m ∗ e 2 � 35

Transport in n-doped STO ρ = ρ 0 + AT 2 A = f ( n ) Origin of this Robust T-Dependence ?? C. Collignon, X. Lin, C.W. Richau, B. Fauque and K. Behina, Ann. Rev. Cond. Mat. Phys. � 36 1025 (2019).

Drude Model � 37 S. Stemmer and J. Allen ROPP 81, 062502 (2018)

Energy Scales weak elector-electron interactions T F ∼ 13 K n = 5 . 5 × 10 17 cm − 3 T D ∼ 400 K Slow Electrons and Fast Phonons ! � 38

(S. Rowley et al., arXiv:1801:08121) � 39

Sc. Reports 2016 α = − d (ln T C ) ( α = 0 . 5 BCS) = − 10 d ln M Quantum critical fluctuations enhance superconductivity ??! Soft TO phonons (very weak coupling to charge density) Plasmons (much fine-tuning required) Wanted: How to get (s-wave) Cooper pairing without retardation!! � 40

Next Time: A Flavor for Two Current Research Projects Can quantum fluctuations “toughen” a system against macroscopic instabilities resulting in a line of classical first-order transitions ending in a quantum critical point ? � 41

Next Time: A Flavor for Two Current Research Projects (a) (b) FL T (i) <φ α φ β > 3D NFL (ii) Polar 2D marginal (iii) NFL 3D pressure/doping/strain,etc. When do metals close to polar quantum critical points develop strongly interacting novel phases ?? � 42

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