- #1
MathematicalPhysicist
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This question is more about the maths than the physics.
So I am reading the textbook by Bergersen and Plischke, and they get the following:
$$m= \tanh [ \beta (qJm+h)]$$
where ##m## is the magnetization, ##q## is the number of nearest neighbours of site ##0##, ##J## and ##h##are the coefficients in the Hamiltonian: ##H = -J\sum_{<ij>} \sigma_i \sigma_j -h \sum_i \sigma_i##;
For the question, they write that ##m(h,T) ## satisfies: ##m(h,T) = -m(-h,T)##;
but I tried to showed this and I didn't succeed.
Here's my attempt:
$$-m(-h,T) = -\tanh [ \beta(-qJm(-h,T)-h) ] = m(-h,T)$$
I used the fact that ##\tanh(-x) = -\tanh(x)##.
Am I wrong?
Did they mean something else here?
Thanks.
So I am reading the textbook by Bergersen and Plischke, and they get the following:
$$m= \tanh [ \beta (qJm+h)]$$
where ##m## is the magnetization, ##q## is the number of nearest neighbours of site ##0##, ##J## and ##h##are the coefficients in the Hamiltonian: ##H = -J\sum_{<ij>} \sigma_i \sigma_j -h \sum_i \sigma_i##;
For the question, they write that ##m(h,T) ## satisfies: ##m(h,T) = -m(-h,T)##;
but I tried to showed this and I didn't succeed.
Here's my attempt:
$$-m(-h,T) = -\tanh [ \beta(-qJm(-h,T)-h) ] = m(-h,T)$$
I used the fact that ##\tanh(-x) = -\tanh(x)##.
Am I wrong?
Did they mean something else here?
Thanks.