# High Temperature Expansions and Cumulants

In this paper the authors perform a high-temperature expansion of the correlation functions for a Heisenberg model on a lattice. Starting from

$$\left<\mathbf{S}_i\cdot\mathbf{S}_j\right>_\beta = \mathrm{Tr}(\mathbf{S}_i\cdot\mathbf{S}_j\,e^{-\beta H})/\mathrm{Tr}\, (e^{-\beta H})$$

They write

expanding in powers of $$\beta$$ and formally dividing out the denominator gives a cumulant expansion

$$\left<\mathbf{S}_i\cdot\mathbf{S}_j\right>_\beta = \sum_{m=0}^\infty \frac{(-\beta)^m}{m!} \left<\mathbf{S}_i\cdot\mathbf{S}_j H^m\right>_c$$

where $$\left<\cdots\right>_c$$ is a cumulant average

$$\left_c = \left< ABC \cdots \right>_0 - \sum \left< \cdots \right>_0 \left< \cdots \right>_0 + 2! \sum \left< \cdots \right>_0\left< \cdots \right>_0\left< \cdots \right>_0 - 3! \left< \cdots \right>_0\left< \cdots \right>_0\left< \cdots \right>_0\left< \cdots \right>_0 + \cdots$$

and $$\left< \cdots \right>_0$$ is an average (at $$\beta = 0$$) over all orientations of the spin operators contained within the brackets. The sums, which are multiplied by $$(-1)^{n-1}(n-1)!$$ are over all ways that the product of the operators $$\mathbf{S}_i\cdot\mathbf{S}_j$$ and $$H$$ can be distributed into $$n$$ averages.

I am familiar with the concept of cumulant expansions, such as done in the diffuse interacting gas model and path integrals, but I would like to derive for myself the above statements and understand their connection to both cumulants of probability distributions as well as the types of expansions performed when dealing with interacting systems. Those expansions generally split the Hamiltonian (or Lagrangian) into a non-interacting (Gaussian) part and an interacting part which can be treated as a perturbation to the non-interacting distribution. In this case, we are rather considering a sort of perturbation about the $$\beta=0$$ distribution (which is also that of non-interacting degrees of freedom), but I don't fully see how to flesh out that analogy.

One resource I have looked at is the book "Series Expansion Methods" by Oitmaa, Hamer, and Zheng. They don't treat the high-T expansions in the way I am looking for, but in Appendix 6 they define a moment $$\left<\,\,\right>$$ as the average of a set of variables, and then define the cumulant $$\left[\,\,\,\right]$$ as follows

$$\left<\alpha\cdots\zeta\right>=\sum_{P}\left[\alpha\cdots\beta\right]\left[\gamma\cdots\zeta\right]$$

where the sum goes over all possible partitions $$P$$ of the set of variables, or symbolically

$$\left<\,\,\right> = \sum_{n=1}^\infty \frac{1}{n!}\left[\,\,\,\right]^n$$

For example,

\begin{align*} \left<\alpha\right>&=[\alpha]\\ \left<\alpha\beta\right>&=[\alpha\beta]+[\alpha][\beta] \\ \left<\alpha\beta\gamma\right>&=[\alpha\beta\gamma]+[\alpha\beta][\gamma]+[\beta\gamma][\alpha] + [\gamma\alpha][\beta]+[\alpha][\beta][\gamma] \end{align*}

[which] can be inverted to give symbolically

$$\left[\,\,\,\right] = \sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n}\left<\,\,\right>^n$$ for example \begin{align*} [\alpha]&=\left<\alpha\right>\\ [\alpha\beta]&=\left<\alpha\beta\right>-\left<\alpha\right>\left<\beta\right>\\ [\alpha\beta\gamma]&=\left<\alpha\beta\gamma\right>-\left<\alpha\beta\right>\left<\gamma\right>-\left<\beta\gamma\right>\left<\alpha\right> - \left<\gamma\alpha\right>\left<\beta\right>+2\left<\alpha\right>\left<\beta\right>\left<\gamma\right> \end{align*}

These "symbolic" equations are quite mystifying to me. I see how to make the inversion by identifying the first as $$e^{[\,]}-1$$ and obtaining the second by expanding $$\ln(1+\left<\,\,\right>)$$, but I don't understand how to interpret either of these equations to write down the corresponding expressions as given, since it is not clear what $$[\,\,\,]^n$$ and $$\left<\,\,\right>^n$$ mean. It is also unclear precisely how this relates to the $$\beta=0$$ expansion above (I see the mathematical relation in the definition of the cumulants, but not how to get from the expectation value to the cumulant expansion).

I am looking to understand

(1) How to obtain the high-temperature expansion of Harris, Kallin, and Berlinsky,

(2) How to understand the "symbolic" equations of Oitmaa, Hamer, and Zheng, and

(3) The relationship between cumulant, high-T, and perturbative expansions for interacting systems such as in QFT (preferably some literature resources where I can dig deeper into this)