1
$\begingroup$

I am integrating the following Gaussian over all possible matrix elements $J_{ij}$: $$ I=\int \exp{\left\{-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji} \right\}} \left (\prod_{ij}\mathrm{d}J_{ij} \right)$$

How can I deal with the $\sum_{ij}J_{ij}J_{ji}$ terms? The fact that I am integrating over matrix elements confuses me. Any help or advice is always appreciated, thanks !

$\endgroup$

1 Answer 1

3
$\begingroup$

Decompose the sum over $i,j$ as $$-a\sum_{ij}J_{ij}^2+b\sum_{ij}J_{ij}+c\sum_{ij}J_{ij}J_{ji}=$$ $$\qquad\qquad=\sum_{i}\left[(c-a)J_{ii}^2+bJ_{ii}\right]+\sum_{i<j}\left[-a(J_{ij}^2+J_{ji}^2)+b(J_{ij}+J_{ji})+2cJ_{ij}J_{ji}\right]$$ $$\qquad\qquad\equiv\sum_{i} A_i+\sum_{i<j} B_{ij}.$$ Then perform the Gaussian integrals separately for each term in the sum (assuming $c<a$ and $c^2/a<a$), $$I=\left(\prod_{i=1}^N\int e^{A_i}dJ_{ii}\right)\left(\prod_{i<j=1}^{N}\int\int e^{B_{ij}}dJ_{ij}dJ_{ji}\right)$$ $$\qquad\qquad=\left(\frac{\sqrt{\pi } e^{\frac{b^2}{4 a-4 c}}}{\sqrt{a-c}}\right)^N\left(\frac{\pi e^{\frac{b^2}{2 a-2 c}}}{\sqrt{a^2-c^2}}\right)^{N(N-1)/2}=\pi^{N^2/2}e^{\frac{N^2b^2}{4a-4c}}(a-c)^{-N/2}(a^2-c^2)^{-N(N-1)/2}.$$

$\endgroup$
2
  • $\begingroup$ Thank you, it makes a lot sense now. If $b\equiv b_{ij}$ depends on $i$ and $j$, does the second integral become : $\propto e^{\frac{a(b_{ij}^2+b_{ji}^2)+2c(b_{ij}b_{ji})}{4(a^2-c^2)}}$ ? $\endgroup$
    – Matt
    Jan 10, 2020 at 11:29
  • 1
    $\begingroup$ yes, that is correct. $\endgroup$ Jan 10, 2020 at 11:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.