ω 2 α 2 ω 1 α 1 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 1
Berry Curvature, Spin, and Anomalous Velocity Michael Stone Institute for Condensed Matter Theory University of Illinois Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 2
Talk based on: Motivation M.A.Stephanov, Y.Yin, Chiral Kinetic Theory , Phys. Rev. Lett. 109 162001 (2012). Our Work MS, V.Dwivedi, A Classical Version of the Non-Abelian Gauge Anomaly Phys. Rev. D88 045012 (2013). V.Dwivedi, MS, Classical chiral kinetic theory and anomalies in even space-time dimensions , J. Phys. A 47 025401 (2014). MS, V.Dwivedi, T.Zhou, Berry Phase, Lorentz Covariance, and Anomalous Velocity for Dirac and Weyl Particles , arXiv:1406.0354 Also important J.Y.Chen, D.T.Son, M.A.Stephanov, H.U.Yee, Y.Yin, Lorentz Invariance in Chiral Kinetic Theory , arXiv:1404.5963 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 3
Bruno Zumino 1923-2014 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 4
Outline Covariant Berry Connection 1 Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski Relativistic Mechanics of Spinning Particles 2 Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor Massless Case 3 A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations Conclusions 4 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 5
Covariant Berry Connection Outline Covariant Berry Connection 1 Anomalous Velocity WKB and Berry Berry,Thomas, and Pauli-Lubanski Relativistic Mechanics of Spinning Particles 2 Mathisson-Papatrou-Dixon equations Anomalous velocity Meaning of Conditions on Spin Tensor Massless Case 3 A Gauge Invariance? Wigner Translations Physical Meaning of Wigner Translations Conclusions 4 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 6
Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle ⇒ anomalous velocity: − ∂ε ( k , x ) ˙ k = + e ( ˙ x × B ) , ∂ x ∂ε ( k , x ) − ( ˙ x ˙ = k × Ω ) . ∂ k Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 7
Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle ⇒ anomalous velocity: − ∂ε ( k , x ) ˙ k = + e ( ˙ x × B ) , ∂ x ∂ε ( k , x ) − ( ˙ x ˙ = k × Ω ) . ∂ k � Many applications! � Want to use for Dirac and Weyl particles � Can we make these equations covariant? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 7
Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle ⇒ anomalous velocity: − ∂ε ( k , x ) ˙ k = + e ( ˙ x × B ) , ∂ x ∂ε ( k , x ) − ( ˙ x ˙ = k × Ω ) . ∂ k � Many applications! � Want to use for Dirac and Weyl particles � Can we make these equations covariant? ˙ x × B ) → ˙ x ν , k = e ( E + ˙ k µ = eF µν ˙ µ = 0 , 1 , 2 , 3 . � Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 7
Covariant Berry Connection Anomalous Velocity Anomalous Velocity Luttinger, Blount, Niu, and others show that a Berry phase in the equations of motion of a Bloch quasiparticle ⇒ anomalous velocity: − ∂ε ( k , x ) ˙ k = + e ( ˙ x × B ) , ∂ x ∂ε ( k , x ) − ( ˙ x ˙ = k × Ω ) . ∂ k � Many applications! � Want to use for Dirac and Weyl particles � Can we make these equations covariant? ˙ x × B ) → ˙ x ν , k = e ( E + ˙ k µ = eF µν ˙ µ = 0 , 1 , 2 , 3 . � x = v ε − ( ˙ x i = v i,ε + Ω ij ˙ k j , ˙ k × Ω ) → ˙ i = 1 , 2 , 3 . ? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 7
Covariant Berry Connection WKB and Berry Covariant WKB for Dirac Look for WKB solution of Dirac equation ( i � γ µ ( ∂ µ + ieA µ / � ) − m ) ψ = 0 . as ψ ( x ) = a ( x ) e − iϕ ( x ) / � , a = a 0 + � a 1 + � 2 a 2 + . . . , where a 0 ( x ) = u α ( k ( x )) C α ( x ) and u α ( k ) (and later v α ( k ) ) are solutions to ( γ µ k µ − m ) u α ( k ) = 0 ( γ µ k µ + m ) v α ( k ) = 0 covariantly normalized so that u α u β = δ αβ = − ¯ ¯ v α v β Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 8
Covariant Berry Connection WKB and Berry Spin Transport Equation Plug WKB solution into Dirac. Find that ∂V µ � � V µ ∂ ∂x µ + 1 � + ie � 2 mS µν αβ F µν − i a αβ,ν ˙ k ν C β ( x ) = 0 . δ αβ ∂x µ 2 where ie 2 mS µν αβ F µν gives Larmor precession, and ∂u β a αβ,ν = i ¯ u α ∂k ν , ν = 0 , 1 , 2 , 3 is an unconventional, but covariant Berry connection. Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 9
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Covariant Berry Curvature Matrix-valued connection form ∂u β a αβ,ν dk ν = i ¯ ∂k ν dk ν . u α Curvature form F = d a − i a 2 . Use Dirac equation to find 1 2 m 2 ( S µν ) αβ dk µ ∧ dk ν , F αβ = where � i � ( S µν ) αβ = ¯ u α 4[ γ µ , γ ν ] u β = i ¯ u α σ µν u β . Note that Dirac ⇒ k µ S µν = 0 . Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 10
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Pauli-Lubanski Tensor Use mass-shell condition E 2 ≡ k 2 0 = k 2 + m 2 to eliminate k 0 and find that 1 � S ij − k i k j � dk i ∧ dk j , F αβ = E S 0 j − S i 0 2 m 2 E αβ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 11
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Pauli-Lubanski Tensor Use mass-shell condition E 2 ≡ k 2 0 = k 2 + m 2 to eliminate k 0 and find that 1 � S ij − k i k j � dk i ∧ dk j , F αβ = E S 0 j − S i 0 2 m 2 E αβ Expression in parentheses is a skew-symmetric tensor generalization of the Pauli-Lubanski vector Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 11
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have � 1 � �� 1 ( k · σ ) k F = σ + · ( d k × d k ) . 2 m 2 γ 2 m 2 (1 + γ ) Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 12
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have � 1 � �� 1 ( k · σ ) k F = σ + · ( d k × d k ) . 2 m 2 γ 2 m 2 (1 + γ ) What does this mean this physically? Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 12
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have � 1 � �� 1 ( k · σ ) k F = σ + · ( d k × d k ) . 2 m 2 γ 2 m 2 (1 + γ ) What does this mean this physically? Look at connection 1 � σ � a αβ,i ˙ m 2 (1 + γ )( k × ˙ k i = k ) · 2 αβ γ 2 � σ � 1 + γ ( β × ˙ = β ) · αβ , β = k /E = k /mγ 2 � σ � = − ω Thomas · αβ . 2 Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 12
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Berry versus Llewellyn Thomas Explicitly, in 3+1 dimensions we have � 1 � �� 1 ( k · σ ) k F = σ + · ( d k × d k ) . 2 m 2 γ 2 m 2 (1 + γ ) What does this mean this physically? Look at connection 1 � σ � a αβ,i ˙ m 2 (1 + γ )( k × ˙ k i = k ) · 2 αβ γ 2 � σ � 1 + γ ( β × ˙ = β ) · αβ , β = k /E = k /mγ 2 � σ � = − ω Thomas · αβ . 2 Covariant Berry-transport is Thomas precession Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 12
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Nishina, Thomas, Hund Yoshio Nishina, Llewellyn Thomas, Friedrich Hund Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 13
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Thomas versus Lobachevsky Thomas precession is parallel transport on the positive-energy mass-shell: Z P −R R X Q Embedding of three-dimensional Lobachevsky space into four-dimensional Minkowski space. The arrow shows the sterographic parametrization of the e ball x 2 1 + x 2 2 + x 2 3 < R 2 . embedded space by the Poincar´ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 14
Covariant Berry Connection Berry,Thomas, and Pauli-Lubanski Non-covariant WKB With u † α u β = δ αβ = v † α v β , have � ∂ � � � ∂t + v · ∇ + 1 C β ( x , t ) = 0 , δ αβ 2div v + N αβ with � � � � � � e 1 σ + 1 ( k · σ ) k − i A αβ,i ˙ k i , N αβ = − i B · mγ 2 m 2 2 γ + 1 αβ ∂u β A αβ,i = iu † ∂k i , i = 1 , 2 , 3 α and �� � 1 σ + 1 ( k · σ ) k � F αβ = − · ( d k × d k ) . 2 m 2 γ 3 m 2 γ + 1 αβ Michael Stone (ICMT Illinois) Spin and Velocity ESI Vienna, August 11th 2014 15
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