Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 1 Collective Effects in Equilibrium and Nonequilibrium Physics Website: http://cncs.bnu.edu.cn/mccross/Course/ Caltech Mirror: http://haides.caltech.edu/BNU/ Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 2 Today’s Lecture Superfluids and superconductors • What are superfluidity and superconductivity? • Review of phase dynamics • Description in terms of a macroscopic phase • Supercurrents that flow for ever • Josephson effect • Four sounds Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 3 The Amazing World of Superfluidity and Superconductivity • Electric currents in loops that flow for ever (measured for ∼ decade) • Beakers of fluid that empty them- selves • Fluids that flow without resistance through tiny holes • Flow in surface films less than an atomic layer thick • Flow driven by temperature differ- ences (fountain effect) Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 4 History of Superfluidity and Superconductivity 1908 Liquefaction of 4He by Kamerlingh Onnes 1911 Discovery of superconductivity by Onnes (resistance drops to zero) 1933 Meissner effect: superconductors expel magnetic field 1937 Discovery of superfluidity in 4He by Allen and Misener 1938 Connection of superfluidity with Bose-Einstein condensation by London 1955 Feynman’s theory of quantized vortices 1956 Onsager and Penrose identify the broken symmetry in superfluidity ODLRO 1957 BCS theory of superconductivity 1962 Josephson effect 1973 Discovery of superfluidity in 3He at 2mK by Osheroff, Lee, and Richardson 1986 Discovery of high-Tc superconductors by Bednorz and Müller 1995- Study of superfluidity in ultracold trapped dilute gases Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 5 Review of Phase Dynamics with a Conserved Quantity Rotational symmetry in the XY plane (angle 2) Sz The XY and Z components of the spin have different prop- erties: S⊥ P −1 Θ Sz = hsizi is a conserved quantity i inP −1 S⊥ = hsi⊥i is the XY order parameter i in Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 5 Review of Phase Dynamics with a Conserved Quantity Rotational symmetry in the XY plane (angle 2) Sz The XY and Z components of the spin have different prop- erties: S⊥ P −1 Θ Sz = hsizi is a conserved quantity i inP −1 S⊥ = hsi⊥i is the XY order parameter i in Sz and 2 are canonically conjugate variables, so that with the free energy Z " # 1 S2 F = dd x K (∇2)2 + z − S b 2 2χ z z we get δF S˙ =− giving S˙ =−∇·j with j =−K ∇2 z δ2 z Sz Sz δ ˙ F ˙ −1 2 = giving 2 = χ (Sz − χbz) δSz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 6 Spin Current ˙ =−∇· Sz jSz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 6 Spin Current ˙ =−∇· Sz jSz This is a conservation law with a current jSz of the conserved quantity Sz given by a phase gradient =− ∇2 jSz K Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 6 Spin Current ˙ =−∇· Sz jSz This is a conservation law with a current jSz of the conserved quantity Sz given by a phase gradient =− ∇2 jSz K For example Θ(r) L Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) • No dynamics in full thermodynamic equilibrium: Sz = χb0z Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) • No dynamics in full thermodynamic equilibrium: Sz = χb0z • Add an additional external field b1z = γ B1z ˙ 2 =−b1z =−γ B1z the usual precession of a magnetic moment in an applied field (Larmor precession). Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) • No dynamics in full thermodynamic equilibrium: Sz = χb0z • Add an additional external field b1z = γ B1z ˙ 2 =−b1z =−γ B1z the usual precession of a magnetic moment in an applied field (Larmor precession). • Note that this is an equilibrium state: Sz 6= χ(b0z + b1z) but is a conserved quantity Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) • No dynamics in full thermodynamic equilibrium: Sz = χb0z • Add an additional external field b1z = γ B1z ˙ 2 =−b1z =−γ B1z the usual precession of a magnetic moment in an applied field (Larmor precession). • Note that this is an equilibrium state: Sz 6= χ(b0z + b1z) but is a conserved quantity— no approximations Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 7 Phase Dynamics ˙ −1 2 = χ (Sz − χbz) • No dynamics in full thermodynamic equilibrium: Sz = χb0z • Add an additional external field b1z = γ B1z ˙ 2 =−b1z =−γ B1z the usual precession of a magnetic moment in an applied field (Larmor precession). • Note that this is an equilibrium state: Sz 6= χ(b0z + b1z) but is a conserved quantity— no approximations • For formal proof see Halperin and Saslow, Phys. Rev. B 16, 2154 (1977), Appendix: “the Larmor precession theorem” Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 8 Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 8 Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points • generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable 1 ε = K (∇2)2 2 • thermodynamic identity dε = Tds+ µzdsz + 8 · d(∇2) with 8 = K ∇2 • equilibrium phase dynamics (Larmor precession theorem) ˙ 2 = µz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 8 Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points • generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable 1 ε = K (∇2)2 2 • thermodynamic identity dε = Tds+ µzdsz + 8 · d(∇2) with 8 = K ∇2 • equilibrium phase dynamics (Larmor precession theorem) ˙ 2 = µz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 8 Hydrodynamic Approach Hydrodynamics: a formal derivation of long wavelength dynamics of conserved quantities and broken symmetry variables in a thermodynamic approach Starting points • generalized rigidity: extra contribution to the energy density from gradients of the broken symmetry variable 1 ε = K (∇2)2 2 • thermodynamic identity dε = Tds+ µzdsz + 8 · d(∇2) with 8 = K ∇2 • equilibrium phase dynamics (Larmor precession theorem) ˙ 2 = µz Derive • dynamical equations for conserved quantities and broken symmetry variables for slowly varying disturbances Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 9 Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µzdsz + 8 · d(∇2) • conjugate fields are ∂ε ∂ε µ = and 8 = z ∂s ∂∇2 z s,∇2 s,sz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 9 Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µzdsz + 8 · d(∇2) • conjugate fields are ∂ε ∂ε µ = and 8 = z ∂s ∂∇2 z s,∇2 s,sz Or with the free energy density f = ε − Ts df =−sdT + µzdsz + 8 · d(∇2) • conjugate fields are ∂ f ∂ f µ = and 8 = z ∂s ∂∇2 z T,∇2 T,sz Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 9 Rigidity and the Thermodynamic Identity In terms of the energy density dε = Tds + µzdsz + 8 · d(∇2) • conjugate fields are ∂ε ∂ε µ = and 8 = z ∂s ∂∇2 z s,∇2 s,sz Or with the free energy density f = ε − Ts df =−sdT + µzdsz + 8 · d(∇2) • conjugate fields are ∂ f ∂ f µ = and 8 = z ∂s ∂∇2 z T,∇2 T,sz These give −1 µz = χ (Sz − χbz) and 8 = K ∇2 Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 10 Entropy Production Tds = dε − µzdsz − 8 · d(∇2) Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 10 Entropy Production Tds = dε − µzdsz − 8 · d(∇2) • Form time derivative of entropy density ε µ 8 (∇2) ds = 1 d − z dsz − · d dt T dt T dt T dt Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 10 Entropy Production Tds = dε − µzdsz − 8 · d(∇2) • Form time derivative of entropy density ε µ 8 (∇2) ds = 1 d − z dsz − · d dt T dt T dt T dt ε • Conservation laws and dynamics of broken symmetry variable (j , jsz unknown) ds 1 ε µ 8 =− ∇·j + z ∇·jsz − ·∇µ dt T T T z Back Forward Collective Effects in Equilibrium and Nonequilibrium Physics: Lecture 4, April 7, 2006 10