Quantum Criticality in Polar Materials I : New Perspectives on - - PowerPoint PPT Presentation

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)

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Aren’t quantum fluctuations only important at T = 0 ? What does quantum critical mean?

“The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.” Sidney Coleman

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Simple Harmonic Oscillator (1d)

Variance hx2i = Ω K ⇢ 1 e

Ω T 1

+ 1 2

• (~ = 1, kB = 1)

0 < T < Ω

Thermal-Quantum Fluctuations

Ω < T

Thermal (classical) Fluctuations

hx2i ⇠ T K

hx2i 1 2 Ω K

Classical Fluctuations Quantum Fluctuations

Ω

T

Pure Quantum Fluctuations

T = 0

hx2i = Ω 2K

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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

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What about the specific heat ? In diamond effects of quantum fluctuations present at room temperatures ! At a continuous phase transition

K ! 0 ) hx2i ! 1

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How to think about temperature near a quantum critical point?

• Temperature is NOT a tuning parameter
• Temperature is a boundary effect!

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Fluctuations are Purely Quantum up to this Time-scale

Back to the SHO ......

Im χω = π 2 ω χ δ(ω − Ω) (ω > 0)

hx2i = 2 π Z ∞ dω ⇢ nω + 1 2

• Im χω

Nyquist Theorem

hx2i = ⇢ nΩ + 1 2

• Ωχ

χ = 1 K(= Re χω=0)

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Generalization to all modes in the Brillouin Zone

hδφ2i = 2 π X

q

Z ∞ dω ⇢ nω + 1 2

• Im χqω

Scalar Order Parameter

φ = ¯ φ + δφ

Average

hδφi = 0

Im χqω = π 2 ω χq δ(ω − ωq) (ω > 0)

(propagating limit, simplest case)

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Generalization to all modes in the Brillouin Zone

hδφ2i = 2 π X

q

Z ∞ dω ⇢ nω + 1 2

• Im χqω

Our Focus:

hδφ2

T i

Strongly Temperature-Dependent Contribution Dominant in Determining Temperature-Dependence

• f Observable Properties

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ω q

T qT qBZ qz

(~ = 1, kB = 1)

qT ∝ T

1 z

qBZ < qT Purely Classical Fluctuations qBZ > qT Quantum Fluctuations Present

Dispersion and Important Wavevectors

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Generalization to all modes in the Brillouin Zone

hδφ2i = 2 π X

q

Z ∞ dω ⇢ nω + 1 2

• Im χqω

Focus: hδφ2

T i

hδφ2

T i ⇡ T

X

q<qBZ

χq (T ωq for q < qBZ)

hδφ2

T i ⇡ T

X

q<qT

χq (T ⌧ ωq for q < qT )

(χ−1

q

∝ κ2 + q2) (χ−1 ∝ κ2)

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Ferroelectrics: The Simplest Polar Materials

A FE is a material that has a spontaneous polarization that is switchable by an electric field

• f practical magnitude

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SrTiO3 - Almost a Ferroelectric

Ferroelectricity induced by Uniaxial Stress, Ca and O-18 Substitution

(Muller and Burkhard 79)

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SrTiO3 - Almost a Ferroelectric

Ferroelectricity induced by Uniaxial Stress, Ca and O-18 Substitution

(Rowley et al. 14)

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

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

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Important Role in (Classical) 
 Critical Phenomena

First Calculation of Logarithmic Corrections to Mean-Field Theory in d=d* !!

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

Why Study FE Quantum Criticality ? Quest for Universality Simplicity (Possible Applications)

Controlled Additional Degrees of Freedom (and maybe novel metals and exotic superconductors)

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χ−1 ∝ T 2

Simpler way to get this result ??

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• 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)

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Self-consistent Landau Approach

Minimization Observed moment requires fluctuation-averaging (due to coarse-graining over qT)

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κ2 ∝ X

q<qT

T κ2 + q2 ≈ T Z qT

κ

qd−1 q2 ≈ T qd−2

T

( 1 − ✓ κ qT ◆d−2)

χ−1 ∝ κ2 ∝ T

(d+z−2) z

Temptation... Most probable vs. average values...coarse-graining over qT!

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We can Fourier transform in the limit to obtain

When is this approach valid ? ✓ κ qT ◆2 ∝ T

(d+z−4) z

( 1 − ✓ κ qT ◆d−2) lim

T →0

✓ κ qT ◆ → 0 if deff ≡ d + z > 4

Ferroelectrics d = 3, z = 1

χ−1 ∝ κ2 ∝ T

(d+z−2) z

= T 2

Agrees with previous calculation by different methods

(log terms)

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Finite-Size Scaling in Space and T

• Space (near CCP) ξ ∼ t−ν

For L << ξ χ = χ(L)

χ ∼ t−γΦ L ξ ⇥ ∼ t−γΦ L t−ν ⇥ χ ∼ t−γ L ξ ⇥p ∼ t−γΦ L t−ν ⇥ γ

ν

∼ L

γ ν

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• Time

ξ ∼ g−ν − → ξτ ∼ g−zν Lτ =

• kBT

(near QCP)

χ ∼ g−γΦ Lτ ξτ ⇥ ∼ g−γΦ Lτ g−zν ⇥

χ ∼ g−γ Lτ ξτ ⇥p ∼ g−γΦ Lτ g−zν ⇥ γ

zν

∼ L

γ zν

τ

∼ T − γ

zν

For Lτ << ξτ χ = χ(Lτ)

χ−1 ∝ T 2

(here z = 1, ν = 1/2, γ = 1 → γ zν = 2)

(near FE-QCP)

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Gruneisen Ratio Near Classical Phase Transition

ΓCF E(T → Tc) ∝ (T − Tc)0

(supported by experiment)

Zhu et al (13)

Minimization + Fluctuation-Averaging

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Gruneisen Ratio Minimization + Fluctuation-Averaging

Zhu et al (13)

In the vicinity of a (d=3) FE-QCP

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Scaling Approach to the Gruneisen Ratio Γ = α cP = − 1 VmT ∂S/∂V ∂S/∂T

Γ = 1 g Φ ✓Lτ ξτ ◆ = 1 g Φ ✓ Lτ g−zν ◆ = ˜ Γ0L

1 zν

τ

= Γ0T − 1

zν

Dimensionally

[Γ] = 1 g

• Near a (FE)-QCP

Γ−1 ∝ T 2

(z = 1, ν = 1 2) Zhu et al (13)

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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

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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!

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Thoughts on n-doped STO Mott criterion for doped semiconductors

n

1 3

c a∗ B ≈ 0.26

✓ a∗

B =

✏~2 m∗e2 ◆

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ρ = ρ0 + AT 2 A = f(n)

Transport in n-doped STO

• C. Collignon, X. Lin, C.W. Richau, B. Fauque and K. Behina, Ann. Rev. Cond. Mat. Phys.

1025 (2019).

Origin of this Robust T-Dependence ??

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Drude Model

• S. Stemmer and J. Allen ROPP 81, 062502 (2018)

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Energy Scales

weak elector-electron interactions

n = 5.5 × 1017cm−3

TF ∼ 13K

TD ∼ 400K

Slow Electrons and Fast Phonons !

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

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• Sc. Reports 2016

α = −d(ln TC) d ln M = −10

(α = 0.5 BCS)

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!!

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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 ?

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Next Time: A Flavor for Two Current Research Projects When do metals close to polar quantum critical points develop strongly interacting novel phases ??

T

<φαφβ>

Polar

(a) (b)

FL NFL NFL

marginal 3D 3D 2D

pressure/doping/strain,etc. (i) (ii) (iii)

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