Interaction Range, The Goldstone Theorem, and Long-Range Order in the Heisenberg Ferromagnet

1967 ◽  
Vol 156 (2) ◽  
pp. 630-631 ◽  
Author(s):  
Robert V. Lange
Keyword(s):  
Long Range ◽  
Range Order ◽  
Heisenberg Ferromagnet ◽  
Long Range Order ◽  
Goldstone Theorem ◽  
Interaction Range

Superfluidity and Superconductivity

2018 ◽  
Author(s):  
Norman J. Morgenstern Horing
Keyword(s):  
Magnetic Field ◽  
Wave Function ◽  
Wave Packet ◽  
Long Range ◽  
Cooper Pair ◽  
Bose Condensation ◽  
Range Order ◽  
Condensed State ◽  
Long Range Order ◽  
Coherent Wave

Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ‎ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function<ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.

Time-Resolved, Laser-Induced Phase Transformation in Aluminum

1984 ◽  
Vol 35 ◽  
Author(s):  
S. Williamson ◽  
G. Mourou ◽  
J.C.M. Li
Keyword(s):  
Point Defect ◽  
Long Range ◽  
Rapid Heating ◽  
Range Order ◽  
Nucleation Theory ◽  
Long Range Order ◽  
Time Resolved ◽  
Aluminum Films ◽  
Thin Aluminum ◽  
Initial Nucleus

ABSTRACTThe technique of picosecond electron diffraction is used to time resolve the laser-induced melting of thin aluminum films. It is observed that under rapid heating conditions, the long range order of the lattice subsists for lattice temperatures well above the equilibrium point, indicative of superheating. This superheating can be verified by directly measuring the lattice temperature. The collapse time of the long range order is measured and found to vary from 20 ps to several nanoseconds according to the degree of superheating. Two interpretations of the delayed melting are offered, based on the conventional nucleation and point defect theories. While the nucleation theory provides an initial nucleus size and concentration for melting to occur, the point defect theory offers a possible explanation for how the nuclei are originally formed.

scholarly journals Finite temperature off-diagonal long-range order for interacting bosons

2020 ◽  
Vol 102 (18) ◽  
Author(s):  
A. Colcelli ◽  
N. Defenu ◽  
G. Mussardo ◽  
A. Trombettoni
Keyword(s):  
Long Range ◽  
Finite Temperature ◽  
Range Order ◽  
Long Range Order
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