Higgs mechanism

  it can describe systems where the equations of motion or the Lagrangian obey symmetries,  
  but the lowest-energy vacuum solutions do not exhibit that same symmetry.
  One important consequence of the distinction between true symmetries and gauge symmetries, 
  is that the spontaneous breaking of a gauge symmetry does not give rise to characteristic               
  massless Nambu–Goldstone physical modes, but only massive modes, 
  like the plasma mode in a superconductor, or the Higgs mode observed in particle physics.
  in which a condensate of Cooper pairs of electrons spontaneously breaks the U(1) gauge 
  symmetry associated with light and electromagnetism
  Spontaneously breaking of a continuous symmetry is inevitably accompanied by gapless (meaning    
  that these modes do not cost any energy to excite) Nambu–Goldstone modes associated with slow 
  long-wavelength fluctuations of the order parameter.
  Mermin and Wagner, states that, at finite temperature, thermally activated fluctuations of 
  Nambu–Goldstone modes destroy the long-range order, and prevent spontaneous symmetry breaking 
  in one- and two-dimensional systems.
   quantum fluctuations of the order parameter prevent most types of continuous symmetry 
   breaking in one-dimensional systems even at zero temperature (an important exception is 
   ferromagnets, whose order parameter, magnetization, is an exactly conserved quantity and does 
   not have any quantum fluctuations)
   Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in
   models exhibiting spontaneous breakdown of continuous symmetries.
    the Mermin–Wagner theorem： continuous symmetries cannot be spontaneously broken at finite 
    temperature in systems with sufficiently short-range interactions in dimensions d ≤ 2