Obtaining an expression for spontaneous magnetization in 1D Ising model with H = 0 from the beginning

Question

Obtaining an expression for spontaneous magnetization in 1D Ising model with    H  =  0  from the beginning

Firetto8w · Accepted Answer

You can just compute the expected value of       M    N    =      ∑          i      =      1        N        σ    i   with respect to the Gibbs measure:  ⟨      M    N        ⟩          N      ,      T        =      ∑          i      =      1        N    ⟨      σ    i        ⟩          N      ,      T        ,where   ⟨  ⋅      ⟩          N      ,      T       represents expectation with respect to the Gibbs measure for a system of N spins at temperature T (and   H  =  0).Of course, if one uses free or periodic boundary condition, then the expectation is 0, at all temperatures, by symmetry. In order to get something nontrivial, let us assume + boundary condition (that is, assume that the first and last spins interact each with a boundary spin with fixed value 1).The expectation of       σ    i   can be computed easily either through the transfer matrix, or using a high-temperature expansion. The latter is shorter, since only two graphs contribute to both the numerators and the denominator: setting   x  =  tanh  ⁡  (  J      /    k  T  )  ⟨      σ    i        ⟩          N      ,      T        =                    x        i            +              x                  N          +          1          −          i                            1      +              x                  N          +          1                      .Therefore,  ⟨      M    N        ⟩          N      ,      T        =  2                    x                  N          +          1                    −      x                      x                  N          +          2                    −              x                  N          +          1                    +      x      −      1        .In particular,      lim          T      →      0            1    N    ⟨      M    N        ⟩          N      ,      T        =  1.

Obtaining an expression for spontaneous magnetization in 1D Ising model with H=0 from the beginning

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