Quantum Criticality in Polar Materials II : A Flavor for Two - PowerPoint PPT Presentation

Quantum Criticality in Polar Materials II : A Flavor for Two Current Research Projects P . Chandra (Rutgers) How can systems that have classical first-order transitions display quantum criticality ?? Can metals near polar quantum critical points host novel strongly correlated phases ?? Many more questions for future research ! 1

Interplay of Quantum Criticality with First Order Phase Transitions? P. Coleman (Rutgers) M. Continentino (CBPF) G. Lonzarich (Cambridge) Quantum Annealed Criticality arXiv 1805.11771 2

Interplay of Quantum Criticality with First Order Phase Transitions? P. Coleman (Rutgers) M. Continentino (CBPF) G. Lonzarich (Cambridge) Quantum Critical Endpoints Grigera et al. Science (2001) Brando et al. RMP (2016) 3

Classical and Quantum Phase Transitions ! 4

Quantum Critical Endpoints u > 0 ˜ 2nd order u < 0 ˜ 1st order ∆ u ˜ = + u − u 0 ! 5 Brando et al. RMP (2016) Grigera et al. Science (2001)

Experimental Motivation: Ferroelectrics McWhan et al., J.Phys. C (1985) Jona and Shirane, FE Crystals (1962) Classically First-Order ! 6

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) ! 7

Quantum Criticality with Classical First-Order Transitions ? PC, Lonzarich, Rowley and Scott, ROPP (2017) 8

(Classical) Larkin-Pikin Mechanism (A. I. Larkin and S. Pikin, Sov. Phys. JETP 29, 891 (1969)) Interaction of strain with fluctuating critical order parameter Diverging Specific Heat in a Clamped System Coupling of the uniform strain to the energy density 1st Order Transition in the Unclamped System LP Criterion for 1st Order Transition Macroscopic Instability of the ✓ dT c ◆ 2 Critical Point κ < ∆ C V T c d ln V κ − 1 = K − 1 − ( K + 4 κ ∼ K c 2 3 µ ) − 1 L c 2 Discontinuous Phase Transition T Shear Strain Crucial ! 9 Generalization for the Quantum Case ???

Overview of the Classical Larkin-Pikin Mechanism Simplest case: Isotropic elasticity and scalar order parameter ( ~ x ) Compressible system where order parameter is coupled to the volumetric strain S [ ψ , u ] = S A + S B + S I = 1 Z d 3 x ( L A [ ψ ] + L B [ u ] + L I [ ψ , e ]) . T L A [ ψ , a, b ] = 1 2( ∂ µ ψ ) 2 + a 2 ψ 2 + b 4! ψ 4 , Physics of the Order Parameter a ∝ T − T c and b > 0 T c Sole Contribution for the Clamped Case ! 10

Overview of the Classical Larkin-Pikin Mechanism Simplest case: Isotropic elasticity and scalar order parameter ( ~ x ) Compressible system where order parameter is coupled to the volumetric strain S [ ψ , u ] = S A + S B + S I = 1 Z d 3 x ( L A [ ψ ] + L B [ u ] + L I [ ψ , e ]) . T L B [ u ] = 1 ✓ K − 2 ◆ � e 2 ll + 2 µe 2 3 µ − σ ab e ab ab 2 Describes Elastic Degrees of Freedom u a ( ~ x ) σ ab External Stress Local Atomic Displacement x ) = 1 ✓ @ u a ◆ + @ u b e ab ( ~ Strain Tensor 2 @ x b @ x a e ll ( x ) = Tr[ e ( ~ x )] ! 11 Volumetric Strain

Overview of the Classical Larkin-Pikin Mechanism Simplest case: Isotropic elasticity and scalar order parameter ( ~ x ) Compressible system where order parameter is coupled to the volumetric strain S [ ψ , u ] = S A + S B + S I = 1 Z d 3 x ( L A [ ψ ] + L B [ u ] + L I [ ψ , e ]) . T L I [ ψ , e ] = λ e ll ψ 2 Interaction between the Volumetric Strain and the Squared Amplitude of the Order Parameter ✓ dT c ◆ Coupling Constant Associated with λ = the Strain-Dependence of T c d ln V ψ 2 “Energy Density” of the Order Parameter ! 12

Overview of the Classical Larkin-Pikin Mechanism Simplest case: Isotropic elasticity and scalar order parameter ( ~ x ) Compressible system where order parameter is coupled to the volumetric strain S [ ψ , u ] = S A + S B + S I = 1 Z d 3 x ( L A [ ψ ] + L B [ u ] + L I [ ψ , e ]) . T Physics of the Order Parameter Strain -“Energy Density” Coupling Describes Elastic Degrees of Freedom Key Idea: Integrate out Gaussian Elastic Degrees of Freedom Z Z Z D [ u ] e − S [ ψ ,u ] D [ ψ ] e − S [ ψ ] Z = D [ ψ ] Z = − → ! 13

Elastic Degrees of Freedom Gaussian, but Integration Must be Performed Carefully Special Role of Boundary Normal Modes ( λ ∼ L ) (Wavelength Comparable to System Size) “Elastic Anomaly”: Integration over Boundary Modes Generates a Non-Local Order Parameter Interaction in the Bulk Action Destroys Locality of Original Theory and Paradoxically is Independent of Detailed Boundary Conditions (as a Bulk Term in the Action) ! 14

Elastic Degrees of Freedom Gaussian, but Integration Must be Performed Carefully In a system with Periodic Boundary Conditions (Larkin-Pikin choice) x ) = e ab + 1 i X q )] e i ~ q · ~ x , e ab ( ~ 2[ q a u b ( ~ q ) + q b u a ( ~ V ~ q 6 =0 Boundary Mode Fluctuating Atomic Displacements { a, b } ∈ [1 , 3] V = L 3 u a ( q ) u a ( x ) Fourier Transform of q = 2 ⇡ l, m, n L ( l, m, n ) ~ Integers ! 15

Elastic Degrees of Freedom Gaussian, but Integration Must be Performed Carefully In a system with Periodic Boundary Conditions (Larkin-Pikin choice) x ) = e ab + 1 i X q )] e i ~ q · ~ x , e ab ( ~ 2[ q a u b ( ~ q ) + q b u a ( ~ V ~ q 6 =0 Boundary Mode Fluctuating Atomic Displacements Formally solid forms a 3-torus I I Burger’s vector of the e ab ( x ) dx b = e ab dx b = b a enclosed defects Boundary Modes of the Strain have a Topological Character ! 16

Integration over the Strain Fields Correction to the Action of the Order Parameter S [ ψ ] = S A [ ψ , a, b ] + ∆ S [ ψ ] where Z e − ∆ S [ ψ ] = D [ u ] e − ( S B + S I ) ! 17

The resulting action ✓ 1 ◆  1 S [ ψ ] = S A [ ψ , a, b ⇤ ] − λ 2 V � 1 Z Z d 3 x 0 ψ 2 ( x ) ψ 2 ( x 0 ) d 3 x K − K + 4 2 T V 2 3 µ 12 λ 2 b ∗ = b − K + 4 3 µ Distance-Independent Interaction Between the Energy Densities of O ( λ 2 ) Perturbative the Order Parameter Renormalization of the Short-Range Interaction This term drives a non-perturbative first order transition at arbitrarily small coupling λ ! 18

µ 6 = 0 The resulting action Prefactor Only Nonzero for Finite Shear Modulus (Solids but not Liquids) κ − 1 ✓ 1 ◆  1 S [ ψ ] = S A [ ψ , a, b ⇤ ] − λ 2 V � 1 Z Z d 3 x 0 ψ 2 ( x ) ψ 2 ( x 0 ) d 3 x K − K + 4 2 T V 2 3 µ Distance-Independent Interaction Subtly from q = 0 Strain Between the Energy Densities of finite q elastic the Order Parameter fluctuations. Only Residual Present This term drives a non-perturbative repulsion due for the first order transition at arbitrarily to “boson small coupling λ Clamped hole” in the System longitudinal interactions Present for Clamped System ! 19

The resulting action ✓ 1 ◆  1 S [ ψ ] = S A [ ψ , a, b ⇤ ] − λ 2 V � 1 Z Z d 3 x 0 ψ 2 ( x ) ψ 2 ( x 0 ) d 3 x K − K + 4 2 T V 2 3 µ  1 � Z S [ ψ ] = S A − λ 2 V Ψ 2 ≡ d 3 x ψ 2 ( x ) 2 T κ ( Ψ 2 ) 2 V Volume Average of the Energy Density Intensive Variable ✓ 1 ◆ h ( δ Ψ 2 ) i ⇠ O V δ Ψ 2 = Ψ 2 � h Ψ 2 i ! 20

The resulting action ✓ 1 ◆  1 S [ ψ ] = S A [ ψ , a, b ⇤ ] − λ 2 V � 1 Z Z d 3 x 0 ψ 2 ( x ) ψ 2 ( x 0 ) d 3 x K − K + 4 2 T V 2 3 µ S [ ψ ] = S A − λ 2 V 2 T κ ( Ψ 2 ) 2 Ψ 2 � 2 = � 2 = 2 Ψ 2 h Ψ 2 i � h Ψ 2 i 2 + O (1 /V ) � � h Ψ 2 i + δ Ψ 2 ) ! 21

Set of Self-Consistent Equations L A ( ψ , a ) � λ 2 + λ 2 V S [ ψ ] = 1  � Z d 3 x κ h Ψ 2 i ψ 2 ( x ) 2 κ h Ψ 2 i 2 T d ψ Ψ 2 e − S A [ ψ ] R h Ψ 2 i = . d ψ e − S A [ ψ ] R φ = � λ h Ψ 2 i Introduce Auxiliary “Strain” Variable κ Z ˜ F ( φ ) D ψ e − S [ ψ , φ ] e − T = S [ ψ , φ ] = 1 Z L A ( ψ , a ) + λφψ 2 + κ h 2 φ 2 i d 3 x T Self-Consistency Imposed by Stationarity of the Free Energy ∂ ˜ F [ φ ] λ h Ψ 2 i + κφ ⇥ ⇤ = 0 = ) V = 0 . ∂φ ! 22

Set of Self-Consistent Equations L A ( ψ , a ) � λ 2 + λ 2 V S [ ψ ] = 1  � Z d 3 x κ h Ψ 2 i ψ 2 ( x ) 2 κ h Ψ 2 i 2 T d ψ Ψ 2 e − S A [ ψ ] R h Ψ 2 i = . d ψ e − S A [ ψ ] R φ = � λ h Ψ 2 i Introduce Auxiliary “Strain” Variable κ Z ˜ F ( φ ) D ψ e − S [ ψ , φ ] e − T = S [ ψ , φ ] = 1 Z L A ( ψ , a ) + λφψ 2 + κ h 2 φ 2 i d 3 x T κ = κ − ∆ κ ˜ Integration out of order parameter fluctuations φ Integration out of elasticity variable ! 23

Review of the Original Larkin-Pikin argument S [ ψ , φ ] = 1 d 3 x [ L A ( ψ , a + 2 λφ )] + κ V Z 2 φ 2 T a → x = a + 2 λφ Free energy of the clamped system Z e − F ( a ) D [ ψ ] e − S A [ ψ ,a ] = T Free energy of the unclamped system F [ φ , a ] = F [ x ] + κ V ˜ 2 φ 2 Shift of tuning parameter due to energy x = a + 2 λφ fluctuations ! 24

∂ x = h Ψ 2 i ∂ F 1 V 2 φ = � λ h Ψ 2 i = � 2 λ ✓ ∂ F ◆ ⌘ � 2 λ V κ F 0 [ x ] κ V κ ∂ x f ≡ 2 λ f ≡ 2 λ ˜ ˜ F , V κ F V κ Two equations describing the unclamped system f = f [ x ] + λ ( f 0 [ x ]) 2 ˜ a = x + 2 λ f 0 [ x ] ✓ ◆ a ∝ T − T c ≡ t T c that must be solved self-consistently ! 25

Continuous Transition in the Clamped System f ∝ − | t | 2 − α ( α > 0) t = x + 2 λ f 0 [ x ] 1st Order Phase Transition for the Unclamped System !! = x − 2 λ (2 − α ) | x | 1 � α sgn( x ) ∼ f Non-Monotonic t t 1st x ! 26