The study of warm dense matter, widely existing in nature and laboratories, is challenging due to the interplay of quantum and classical fluctuations. We develop a variational diagrammatic Monte Carlo method and determine the exchange-correlation kernel $K_{\mathrm{XC}}(q;T)$ of a warm dense electron gas for a broad range of temperature $T$ and momentum $q$. We observe several interesting physics, including the $T$-dependent evolution of the hump structure and the large-$q$ tail and the emergence of a scaling relation. Particularly, by deriving an analytical form for $q\to \infty$, we obtain large-$q$ tails of $K_{\mathrm{XC}}$ with high precision. It is shown that the $K_{\mathrm{XC}}$ data can be reliably transformed into real space, which can be directly used in density-functional-theory calculations of real warm dense matter.

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• Warm-dense matter
• Diagrammatic methods
• Monte Carlo methods