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Im working on a numerical method for the Ising model. I'm asked to calculate both the absolute magnetizetion and the specific heat capacity: $$c = \frac{\beta^2}{N} \left( \langle H^2 \rangle - \langle H \rangle^2 \right)$$

$$m_{\text{abs}} = \frac{1}{N} \langle \lvert \sum_{x} s_x \rvert \rangle$$

When plotting the resulting values against $\beta$ (inverse temperature of the system) I get normal results in contrast to other studies, but I have negative Heat Capacity values for low temperatures, which I havent seen anywhere else. Is there any problem or my simulations are correct?enter image description hereenter image description here

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    $\begingroup$ There is definitely a problem with your simulation or calculations. $\langle H^2\rangle - \langle H \rangle^2$ cannot be negative. $\endgroup$
    – Gec
    Commented Jan 28 at 13:18
  • $\begingroup$ Have you check finite size effects? Is it 2D model? $\endgroup$
    – Artem Alexandrov
    Commented Jan 28 at 13:36
  • $\begingroup$ @ArtemAlexandrov This cannot be finite size effects. See the comment by Gec. $\endgroup$
    – Tobias Fünke
    Commented Jan 28 at 14:21
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    $\begingroup$ To elaborate on @Gec's point, the covariance inequality requires that the expectation value of $H$ satisfies $E(H^2) \geq [E(H)]^2$. $\endgroup$
    – Michael Seifert
    Commented Jan 28 at 14:37
  • $\begingroup$ @MichaelSeifert It's actually even more basic than that: a variance is always nonnegative since ${\rm Var}(X) = E(X^2)-E(X)^2 = E\bigl((X-E(X))^2\bigr) \geq 0$. $\endgroup$
    – Yvan Velenik
    Commented Jan 28 at 17:15

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