# TAP free energy and the p-spin spherical spin glass

+ 2 like - 0 dislike
70 views

Consider the $p$-spin spherical spin glass model with Hamiltonian $$H_{N,p}(\sigma)=\frac{1}{{N}^{\frac{(p-1)}{2}}} \sum \limits_{i_1,...i_p} J_{i_1,...i_p} \sigma_{i_1} \sigma_{i_2} .. \sigma_{i_p}$$ where $$\sigma = (\sigma_1,...,\sigma_{i_N}) \in S^{N-1}(\sqrt{N}) \subseteq \mathbb{R}^{N}$$ the Euclidean sphere with radius $\sqrt{N}$.

I am a mathematician, so my physics knowledge is very limited.

I am trying to understand why is the TAP functional, given by $$f_{TAP}(q)(\sigma) = 2^{-\frac{1}{2}}q^{\frac{p}{2}}\frac{1}{N}H_{N,p}(\sigma) + B(q)$$ where $$B(q)= -\frac{1}{2\beta} \log(1-q) -\frac{\beta}{4} (1+(p-1)q^{p} - pq^{p-1} ) ,$$ important in the study of p-spin spherical spin glass. A description in words or some  comments would be enough for now, as a start. I have also found a few papers concerning the $f_{TAP}$ functional but they seem too advanced for me.

Could someone give me some help with this?

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.