\section{Introduction}
In this paper, we further develop a technique from \cite{RY} and apply it to study the Kobayashi Conjecture, $0$-cycles on hypersurfaces of general type, and Seshadri constants of very general hypersurfaces. The idea of the technique is to translate results about very general points on very general hypersurfaces to results about arbitrary points on very general hypersurfaces.

Our first application is to hyperbolicity. Recall that a complex variety is Brody hyperbolic if it admits no holomorphic maps from $\mathbb{C}$.

\begin{conjecture}[Kobayashi Conjecture]
\label{conj-Kobayashi}
A very general hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ is Brody hyperbolic if $d$ is sufficiently large. Moreover, the complement $\mathbb{P}^n \setminus X$ is also Brody hyperbolic for large enough $d$.
\end{conjecture}

First conjectured in 1970 \cite{K}, the Kobayashi Conjecture has been the subject of intense study, especially in recent years \cite{Siu, Deng, Brotbeck, D2}. The suspected optimal bound for $d$ is approximately $d \geq 2n-1$. However, the best current bound is for $d$ greater than about $(en)^{2n+2}$ \cite{D2}. A related conjecture is the Green-Griffiths-Lang Conjecture.

\begin{conjecture}[Green-Griffiths-Lang Conjecture]
\label{conj-GGL}
If $X$ is a variety of general type, then there is a proper subvariety $Y \subset X$ containing all the entire curves of $X$.
\end{conjecture}

The Green-Griffiths-Lang Conjecture says that holomorphic images of $\mathbb{C}$ under nonconstant maps do not pass through a general point of $X$. Conjecture \ref{conj-GGL} is well-studied for general hypersurfaces, and it is a natural result to prove on the way to proving Conjecture \ref{conj-Kobayashi}. We provide a new proof of the Kobayashi Conjecture using previous results on the Green-Griffiths-Lang Conjecture.

\begin{theorem}
A general hypersurface in $\mathbb{P}^n$ of degree $d$ admits no nonconstant holomorphic maps from $\mathbb{C}$ for $d \geq d_{2n-3}$, where $d_2 = 286, d_3 = 7316$ and
\[ d_n = \left\lfloor \frac{n^4}{3} (n \log(n \log(24n)))^n \right\rfloor .\]
\end{theorem}

Our proof appears to be substantially simpler than the previous proofs (compare with \cite{Siu, Deng, Brotbeck, D2}), and can be adapted in a straightforward way as others use jet bundles to obtain better bounds for Conjecture \ref{conj-GGL}. Unfortunately, the bound of about $(2n \log(n\log(n)))^{2n+1}$ that we obtain is slightly worse than Demailly's bound of $(en)^{2n+2}$. However, assuming the optimal result on the Green-Griffiths-Lang Conjecture, our technique allows us to prove the conjectured bound of $d \geq 2n-1$ for the Kobayashi Conjecture.

The Kobayashi Conjecture for complements of hypersurfaces has also been studied by several authors \cite{Dar, BD}. Using results of Darondeau \cite{Dar} along with our Grassmannian technique, we are able to prove the Kobayashi Conjecture for complements as well.

\begin{theorem}
If $X$ is a general hypersurface in $\mathbb{P}^n$ of degree at least $d_{2n}$, where $d_n = (5n)^2 n^n$, then $\mathbb{P}^n \setminus X$ is Brody hyperbolic.
\end{theorem}

Our bound of about $100 \cdot 2^n n^{2n+2}$ is slightly worse than Brotbeck and Deng's bound of about $e^3 n^{2n+6}$ \cite{BD}, but our proof is substantially shorter.

Our second application concerns the Chow equivalence of points on very general complete intersections. Chen, Lewis, and Sheng \cite{CLS} make the following conjecture, which is inspired by work of Voisin \cite{voisinChow, V2,V3}.

\begin{conjecture}
\label{conj-CLS}
Let $X \subset \mathbb{P}^n$ be a very general complete intersection of multidegree $(d_1, \dots, d_k)$. Then for every $p \in X$, the space of points of $X$ rationally equivalent to $p$ has dimension at most $2n-k-\sum_{i=1}^k d_i$. If $2n-k-\sum_{i=1}^k d_i < 0$, we understand this to mean that $p$ is equivalent to no other points of $X$.
\end{conjecture}

If this Conjecture holds, then the result is sharp \cite{CLS}. Voisin \cite{voisinChow, V2, V3} proves Conjecture \ref{conj-CLS} for hypersurfaces in the case $2n-d-1 < -1$. Chen, Lewis, and Sheng \cite{CLS} extend the result to $2n-d-1 = -1$, and also prove the analog of Voisin's bound for complete intersections. Both papers use fairly involved Hodge theory arguments. Roitman \cite{R1,R2} proves the $2n-k-\sum_{i=1}^k d_i = n-2$ case. Using Roitman's result, we prove all but the $2n-k-\sum_{i=1}^k d_i = -1$ case of Conjecture \ref{conj-CLS}, and in this case we prove the result holds with the exception of possibly countably many points.

\begin{theorem}
\label{thm-chow}
If $X \subset \mathbb{P}^n$ is a very general complete intersection of multidegree $(d_1, \dots, d_k)$, then no two points of $X$ are rationally Chow equivalent if $2n-k-\sum_{i=1}^k d_i < -1$. If $2n-k-\sum_{i=1}^k d_i = -1$, then the set of points rationally equivalent to another point of $X$ is a countable union of points. If $2n-k-\sum_{i=1}^k d_i \geq 0$, then the space of points of $X$ rationally equivalent to a fixed point $p \in X$ has dimension at most $2n-k-\sum_{i=1}^k d_i$ in $X$.
\end{theorem}

Together with Chen, Lewis and Sheng's result, this completely resolves Conjecture \ref{conj-CLS} in the case of hypersurfaces. Our method appears substantially simpler than the previous work of Voisin \cite{voisinChow, V2, V3} and Chen, Lewis, and Sheng \cite{CLS}, although in the case of hypersurfaces, we do not recover the full strength of Chen, Lewis, and Sheng's result.

The third result relates to Seshadri constants. Let $\epsilon(p,X)$ be the Seshadri constant of $X$ at the point $p$, defined to be the infimum of $\frac{\deg C}{\operatorname{mult}_p C}$ over all curves $C$ in $X$ passing through $p$. Let $\epsilon(X)$ be the Seshadri constant of $X$, defined to be the infimum of the $\epsilon(p,X)$ as $p$ varies over the hypersurface.

\begin{theorem}
\label{thm-seshadri}
Let $r > 0$ be a real number. If for a very general hypersurface $X_0 \subset \mathbb{P}^{2n-1}$ of degree $d$ the Seshadri constant $\epsilon(p,X_0)$ of $X_0$ at a general point $p$ is at least $r$, then for a very general $X \subset \mathbb{P}^n$ of degree $d$, the Seshadri constant $\epsilon(X)$ of $X$ is at least $r$.
\end{theorem}

The layout of the paper is as follows. In Section \ref{sec-technique}, we lay out our general technique, and immediately use it to prove Theorem \ref{thm-seshadri}. In Section \ref{sec-hyperbolicity}, we discuss how to use the results of Section \ref{sec-technique} to prove hyperbolicity results. In Section \ref{sec-cycles}, we discuss how to prove Theorem \ref{thm-chow}.

\subsection*{Acknowledgements}
We would like to thank Xi Chen, Izzet Coskun, Jean-Pierre Demailly, Mihai P\u{a}un, Chris Skalit, and Matthew Woolf for helpful discussions and comments.

\section{The Technique}
\label{sec-technique}
We set some notation. Let $B$ be the moduli space of complete intersections of multidegree $(d_1, \dots, d_k)$ in $\mathbb{P}^{n+k}$ and $\mathcal{U}_{n,\underline{d}} \subset \mathbb{P}^{n+k} \times B$ be the variety of pairs $([p],[X])$ with $[X] \in B$ and $p \in X$. We refer to elements of $\mathcal{U}_{n,\underline{d}}$ as pointed complete intersections. When we talk about the codimension of a countable union of subvarieties of $\mathcal{U}_{n,\underline{d}}$, we mean the minimum of the codimensions of each component.

We need the following result from \cite{RY}.
\begin{proposition}
\label{GrassmanProp}
Let $C \subset \mathbb{G}(r-1,m)$ be a nonempty family of $r-1$-planes of codimension $\epsilon > 0$, and let $B \subset \mathbb{G}(r,m)$ be the space of $r$ planes that contain some $r-1$-plane $c$ with $[c] \in C$. Then $\operatorname{codim}(B \subset \mathbb{G}(r,m)) \leq \epsilon -1$.
\end{proposition}
\begin{proof}
For the reader's convenience, we sketch the proof. Consider the incidence-correspondence $I = \{ ([b],[c]) | \: [b] \in B, [c] \in C \} \subset \mathbb{G}(r,m) \times \mathbb{G}(r-1,m)$. The fibers of $\pi_2$ over $C \subset \mathbb{G}(r-1,m)$ are all $\mathbb{P}^{m-r}$'s, while for a general $[b] \in B$, the fiber $\pi_1^{-1}([b])$ has codimension at least $1$ in the $\mathbb{P}^r$ of $r-1$-planes contained in $b$ (since otherwise it can be shown that $C = \mathbb{G}(r,m)$). The result follows by a dimension count.
\end{proof}

We need a few other notions for the proof. A parameterized $r$-plane in $\mathbb{P}^m$ is a degree one injective map $\Lambda: \mathbb{P}^r \to \mathbb{P}^m$. Let $G_{r,m,p}$ be the space of parameterized $r$-planes in $\mathbb{P}^m$ whose images pass through $p$. If $(p,X)$ is a pointed hypersurface in $\mathbb{P}^m$, a parameterized $r$-plane section of $(p,X)$ is a pair $(\Lambda^{-1}(p), \Lambda^{-1}(X)) =: \phi^* (p,X)$, where $\Lambda: \mathbb{P}^r \to \mathbb{P}^m$ is a parameterized $r$-plane whose image does not lie entirely in $X$. We say that $\Lambda: \mathbb{P}^r \to \mathbb{P}^m$ contains $\Lambda': \mathbb{P}^{r-1} \to \mathbb{P}^m$ if $\Lambda(\mathbb{P}^r)$ contains $\Lambda'(\mathbb{P}^{r-1})$.

\begin{corollary}
\label{GrassmanCor}
If $C \subset G_{r-1,m,p}$ is a nonempty subvariety of codimension $\epsilon > 0$ and $B \subset G_{r,m,p}$ is the subvariety of parameterized $r$-planes that contain some $r-1$-plane $[c] \in C$, then $\operatorname{codim}(B \subset G_{r,m,p}) \leq \epsilon -1$.
\end{corollary}

Let $\mathcal{X}_{n,\underline{d}} \subset \mathcal{U}_{n,\underline{d}}$ be an open subset. For instance, $\mathcal{X}_{n,\underline{d}}$ might be equal to $\mathcal{U}_{n,\underline{d}}$ or it might be the universal complete intersection over the space of smooth complete intersections. Our main technical tool is the following.

\begin{theorem}
\label{thm-technicaltool}
Suppose we have an integer $m$ and for each $n \leq n_0$ we have $Z_{n,d} \subset \mathcal{X}_{n,\underline{d}}$ a countable union of locally closed varieties satisfying: \begin{enumerate}
\item If $(p,X) \in Z_{n,\underline{d}}$ and is a parameterized hyperplane section of $(p',X')$, then $(p',X') \in Z_{n+1,\underline{d}}$.
\item $Z_{m-1,\underline{d}}$ has codimension at least $1$ in $\mathcal{X}_{m-1,d}$.
\end{enumerate}
Then the codimension of $Z_{m-c,\underline{d}}$ in $\mathcal{X}_{m-c,\underline{d}}$ is at least $c$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm-technicaltool}]
We adopt the method from \cite{RY}. We prove that for a very general point $(p_0,X_0)$ of a component of $Z_{m-c,\underline{d}}$, there is a variety $\mathcal{F}_{m-c}$ and a map $\phi: \mathcal{F}_{m-c} \to \mathcal{U}_{m-c,\underline{d}}$ with $(p_0,X_0) \in \phi(\mathcal{F}_{m-c})$ and $\operatorname{codim}(\phi^{-1}(Z_{m-c,\underline{d}}) \subset \mathcal{F}_{m-c}) \geq c$. This suffices to prove the result.

So, let $(p_0, X_0)$ be a general point of a component of $Z_{m-c,\underline{d}}$, and let $(p_1,X_1) \in \mathcal{X}_{m-1,\underline{d}}$ be very general, so that $(p_1,X_1)$ is not in the closure of any component of $Z_{m-1,\underline{d}}$ by hypothesis 2. Choose $(p,Y) \in \mathcal{X}_{N,\underline{d}}$ for some sufficiently large $N$ such that $(p_0,X_0)$ and $(p_1,X_1)$ are parameterized linear sections of $(p,Y)$. Then for all $n < N$, let $\mathcal{F}_{n}$ be the space of parameterized $n$-planes in $\mathbb{P}^N$ passing through $p$ such that for $\Lambda \in \mathcal{F}_{n}$, $\Lambda^*(p,Y)$ is in $\mathcal{X}_{n,\underline{d}}$. This means that $\mathcal{F}_n$ is an open subset of $G_{n,N,p}$. Let $\phi: \mathcal{F}_{n} \to \mathcal{X}_{n,\underline{d}}$ be the map sending $\Lambda: \mathbb{P}^n \to \mathbb{P}^N$ to $\Lambda^*(p,Y)$.

We prove that $\operatorname{codim}(\phi^{-1}(Z_{m-c,\underline{d}}) \subset \mathcal{F}_{m-c}) \geq c$ by induction on $c$. For the $c=1$ case, we see by construction that $\phi^{-1}(Z_{m-1,\underline{d}})$ has codimension at least $1$ in $\mathcal{F}_{m-1}$, since $(p_1,X_1)$ is a parameterized $m-1$-plane section of $(p,Y)$ but is not in the closure of any component of $Z_{m-1,\underline{d}}$. Now suppose we know that $\operatorname{codim}(\phi^{-1}(Z_{m-c,\underline{d}}) \subset \mathcal{F}_{m-c}) \geq c$. We use Corollary \ref{GrassmanCor} with $C$ equal to $\phi^{-1}(Z_{m-c-1,\underline{d}})$. By hypothesis 1, we see that $B$ is contained in $\phi^{-1}(Z_{m-c,\underline{d}})$. It follows from this that
\[ c \leq \operatorname{codim}( \phi^{-1}(Z_{m-c,\underline{d}}) \subset \mathcal{F}_{m-c})) \leq \operatorname{codim}(B \subset \mathcal{F}_{m-c}) \]
\[ \leq \operatorname{codim}(C \subset \mathcal{F}_{m-c-1}) -1 .  \]
Rearranging, we see that
\[ \operatorname{codim}( \phi^{-1}(Z_{m-c-1,\underline{d}}) \subset \mathcal{F}_{m-c-1}) = \operatorname{codim}(C \subset \mathcal{F}_{m-c-1}) \geq c+1 . \]
The result follows.
\end{proof}

As an immediate application, we prove Theorem \ref{thm-seshadri}.

\begin{proof}[Proof of Theorem \ref{thm-seshadri}]
Let $r$ be given. Let $Z_{m,d} \subset \mathcal{U}_{m,d}$ be the set of pairs $(p,X)$ where $\epsilon(p,X) < r$. We apply Theorem \ref{thm-technicaltool} to $Z_{m,d}$. We see that $Z_{m,d}$ is a countable union of algebraic varieties, and by hypothesis, $Z_{2n-1,d} \subset \mathcal{U}_{2n-1,d}$ has codimension at least $1$. Now suppose that $(p_0,X_0) \in Z_{m,d}$. Then there is some curve $C$ in $X_0$ with $\frac{\deg C}{\operatorname{mult}_{p_0} C} < r$. Thus, for any $X$ containing $p$, we see that the Seshadri constant of $X$ at $p$ is at most $\frac{\deg C}{\operatorname{mult}_{p_0} C} < r$. This shows that the $Z_{m,d}$ satisfy the conditions of Theorem \ref{thm-technicaltool}, which shows that $Z_{n,d} \subset \mathcal{U}_{n,d}$ has codimension at least $n$. By dimension reasons, this means that $\mathcal{U}_{n,d}$ cannot dominate the space of hypersurfaces, so the result follows.
\end{proof}

\section{Hyperbolicity}
\label{sec-hyperbolicity}
Let $\mathcal{X}_{n,d}$ be the universal hypersurface in $\mathbb{P}^n$ over the open subset $U$ in the moduli space of all degree $d$ hypersurfaces in $\mathbb{P}^n$ consisting of all smooth hypersurfaces. Many people have developed a technique for restricting the entire curves contained in a fiber of the map $\pi_2: \mathcal{X}_{n,d} \to U$. See the article of Demailly for a detailed description of some of this work \cite{D2}. For a variety $X$, let $\operatorname{ev}: J_k(X) \to X$ be the space of $k$-jets of $X$. Then, if $X \subset \mathbb{P}^n$ is a smooth degree $d$ hypersurface, there is a vector bundle $E_{k,m}^{GG} T_X^*$ whose sections act on $J_k(X)$. Global sections of $E_{k,m}^{GG} T_X^* \otimes \mathcal{O}(-H)$ vanish on the $k$-jets of entire curves. This means that sections of $E_{k,m}^{GG} T_X^* \otimes \mathcal{O}(-H)$ cut out a closed subvariety $S'_{k,m}(X) \subset J_k(X)$ such that any entire curve is contained in $\operatorname{ev}(S'_{k,m}(X))$. In fact, it can be shown that any entire curve is contained in the closure of $\operatorname{ev}(S_{k,m}(X))$, where $S_{k,m}(X) \subset J_k(X)$ is $S'_{k,m}(X)$ minus the space of singular $k$-jets.

The construction is functorial. In particular, if $V$ is the relative tangent bundle of the map $\pi_2$, there is a vector bundle $E_{k,m}^{GG} V^*$ whose restriction to each fiber of $\pi_2$ is $E_{k,m}^{GG} T_{X}^*$. Let $\mathcal{Y}_{n,d} \subset \mathcal{X}_{n,d}$ be the locus of $(p,X) \in \mathcal{X}_{n,d}$ such that $p \in \operatorname{ev}(S_{k,m}(X))$. Then by functoriality, $\mathcal{Y}_{n,d}$ is a finite union of locally closed varieties.

\begin{theorem}
\label{thm-Hyperbol}
Suppose that $\mathcal{Y}_{r-1,d} \subset \mathcal{X}_{r-1,d}$ is codimension at least 1. Then $\mathcal{Y}_{r-c,d} \subset \mathcal{X}_{r-c,d}$ is codimension at least $c$. In particular, if $\mathcal{Y}_{2n-3, d}$ is codimension at least $1$ in $\mathcal{X}_{2n-3,d}$ and $d \geq 2n-1$, then a very general $X \subset \mathbb{P}^n$ of degree $d_n$ is hyperbolic.
\end{theorem}
\begin{proof}
 We check that $\mathcal{Y}_{r-1,d}$ satisfies both conditions of Theorem \ref{thm-technicaltool}. Condition 2 is a hypothesis. Condition 1 follows by the functoriality of Demailly's construction. Namely, if $(p,X_0)$ is a parameterized linear section of $(p,X)$, then the natural map $X_0 \to X$ induces a pullback map on sections
 \[ H^0(E_{k,m}^{GG} T_X^* \otimes \mathcal{O}(-H)) \to H^0(E_{k,m}^{GG} T_{X_0}^* \otimes \mathcal{O}(-H)) , \]
compatible with the natural inclusion of jets $J_k(X_0) \to J_k(X)$. In particular, if some section $s$ of $H^0(E_{k,m}^{GG} T_X^* \otimes \mathcal{O}(-H))$ takes a nonzero value on a jet $\alpha(j)$, where $j \in J_k(X_0)$ and $\alpha:J_k(X_0) \to J_k(X)$ is the natural inclusion, then the restriction of $s$ to $X_0$ takes a nonzero value on the original jet $j \in J_k(X_0)$. Thus, if $X_0$ has a nonsingular $k$-jet at $p$ which is annihilated by every section in $H^0(E_{k,m}^{GG} T_{X_0}^* \otimes \mathcal{O}(-H))$, $X$ has such a $k$-jet as well.

To see the second statement, observe that by Theorem \ref{thm-technicaltool}, $\mathcal{Y}_{n,d}$ has codimension in $\mathcal{X}_{n,d}$ at least $2n-3-n+1 = n-2$. It follows that a general $X$ of degree $d$ in $\mathbb{P}^n$ satisfies that the image of any entire curve is contained in an algebraic curve. Since $d \geq 2n-1$, by a theorem of Voisin \cite{V2, V3}, any algebraic curve in $X$ is of general type. The result follows.
\end{proof}

The current best bound for the Green-Griffiths-Lang Conjecture is from Demailly \cite{D1,D2}. The version we use comes out of Demailly's proof.

\begin{theorem}[\cite{D2}, Section 10]
If $k$ and $m$ are sufficiently large, we have that $\mathcal{Y}_{n,d} \subset \mathcal{X}_{n,d}$ is codimension at least $1$ for $d \geq d_n$, where $d_2 = 286, d_3 = 7316$ and
\[ d_n = \left\lfloor \frac{n^4}{3} (n \log(n \log(24n)))^n \right\rfloor .\]
\end{theorem}

Using this bound, we obtain the following.

\begin{corollary}
The Kobayashi Conjecture holds hypersurfaces when $d \geq d_{2n-3}$, where $d_2 = 286, d_3 = 7316$ and
\[ d_n = \left\lfloor \frac{n^4}{3} (n \log(n \log(24n)))^n \right\rfloor .\]
\end{corollary}

This bound of about $(2n \log (n\log n))^{2n+1}$ is slightly worse than the best current bound for the Kobayashi Conjecture from \cite{D2}, which is about $(en)^{2n+2}$. However, our technique is strong enough to allow us to prove the optimal bound from the Kobayashi Conjecture, provided one could prove the optimal result for the Green-Griffiths-Lang Conjecture.

\begin{corollary}
If $\mathcal{Y}_{d-2,d}$ has codimension at least $1$ in $\mathcal{X}_{d-2,d}$ (as we would expect from the Green-Griffiths-Lang Conjecture), then a very general hypersurface of degree $d \geq 2n-1$ in $\mathbb{P}^n$ is hyperbolic.
\end{corollary}
\begin{proof}
We apply Theorem \ref{thm-Hyperbol}. We know that if $\mathcal{Y}_{2n-3, d}$ is codimension at least $1$ in $\mathcal{X}_{2n-3,d}$, then the Kobayashi Conjecture holds for hypersurfaces in $\mathbb{P}^n$ of degree $d$. We apply this result with $d = 2n-1$.
\end{proof}

Work has also been done on the hyperbolicity of complements of hypersurfaces in $\mathbb{P}^n$. There are similar jet bundle techniques in this case. Given a variety $Z$, and ample line bundle $A$, a subsheaf $V \subset T_Z$ and a subvariety $X \subset Z$, one can construct the vector bundles $E_{k,m} V^*(\log X)$, sections of which act on $k$-jets of $Z \setminus X$. It can be shown that any section of $H^0(E_{k,m} V^*(\log X) \otimes A^{*})$ must vanish on the $k$-jet of any entire curve in $Z \setminus X$. Then sections of $H^0(E_{k,m} V^*(\log X) \otimes A^{*})$ cut out a subvariety $S_{k,m}$ in the locus of nonsingular $k$-jets on $Z \setminus X$ such that any entire curve lies in the closure of $\operatorname{ev}(S_{k,m})$. Darondeau \cite{Dar} studies these objects for hypersurfaces in $\mathbb{P}^n$. Namely, he proves the following.

\begin{theorem}[Darondeau]
\label{thm-Darondeau}
Let $d \geq (5n)^2 n^n$, let $U_{n,d}$ be the space of smooth degree $d$ hypersurfaces in $\mathbb{P}^n$ and consider the space $\mathbb{P}^n \times U_{n,d}$, with divisor $\mathcal{X}_{n,d}$ corresponding to the universal hypersurface. Let $V = T_{\pi_2}$ be the relative tangent space of projection of $\mathbb{P}^n \times U_{n,d}$ onto $U_{n,d}$, and consider the locus $S_{k,m}$ cut out by sections in $H^0(E_{k,m} V^*(\log \mathcal{X}_{n,d}) \otimes \mathcal{O}(-H))$. Then $\operatorname{ev}(S_{k,m})$ is codimension at least $2$ in $\mathbb{P}^n \times U_{n,d}$.
\end{theorem}

Using our technique, we obtain the following effective form of the Kobayashi Conjecture.

\begin{theorem}
Let $Z_{n,d} \subset \mathbb{P}^n \times U_{n,d} \setminus \mathcal{X}_{n,d}$ be the locus of pairs $(p,X)$ where $p \in (\mathbb{P}^n \setminus X) \setminus \operatorname{ev}(S_{k,m})$. Then if $Z_{r-1,d}$ has codimension at least $1$ in $(\mathbb{P}^{r-1} \times U_{r-1,d}) \setminus \mathcal{X}_{r-1,d}$, we have that $Z_{r-c,d}$ has codimension at least $c$ in $(\mathbb{P}^{r-c} \times U_{r-c,d}) \setminus \mathcal{X}_{r-c,d}$.
\end{theorem}
\begin{proof}
The proof is very similar in spirit to the proof of Theorem \ref{thm-technicaltool}, but we give a new proof here for completeness. Let $(p, X_0)$ be a general point of a component of $Z_{r-c,d}$. We will find a family $\phi: \mathcal{F}_{r-c} \to \mathbb{P}^{r-c} \times U_{r-c,d} \setminus \mathcal{X}_{r-c,d}$ with $\phi^{-1}(Z_{r-c})$ having codimension at least $c$ in $\mathcal{F}_{r-c}$.

Let $(p, X_1) \in \mathbb{P}^{r-1} \times U_{r-1,d} \setminus \mathcal{X}_{r-1,d}$ be very general so that $(p,X_1)$ is not in the closure of any component of $Z_{r-1,d}$. Let $(p,Y)$ be a hypersurface in some high dimensional projective space $\mathbb{P}^N$ together with a point $p \in \mathbb{P}^N \setminus Y$ such that $(p,X_0)$ and $(p,X_1)$ are both parameterized linear sections of $(p,Y)$. Let $\mathcal{F}_{r-c}$ be the space of parameterized linear spaces in $\mathbb{P}^N$ passing through $p$. Then we have a natural map $\phi: \mathcal{F}_{r-c} \to \mathbb{P}^{r-c} \times U_{r-c,d} \setminus \mathcal{X}_{r-c,d}$ sending the parameterized $r-c$ plane $\Lambda$ to $\Lambda^* (p,Y)$. By construction, the image of $\phi$ will certainly contain the point $(p,X_0)$, and since $(p,X_1)$ is very general, $\phi^{-1}(Z_{r-1,d})$ will have codimension at least $1$ in $\mathcal{F}_{r-1}$. Observe that if $(p,Y_0) \in Z_{r-c-1,d}$ is a linear section of $(p,Y_1)$, then $(p,Y_1) \in Z_{r-c,d}$, since if there is a nonsingular jet $j$ at $p$ such that all the sections of $E_{k,m} T_{\mathbb{P}^{r-c-1}}^*(\log Y_0) \otimes \mathcal{O}(-H)$ vanish on $j$, then certainly all the sections of $E_{k,m} T_{\mathbb{P}^{r-c}}^*(\log Y_1) \otimes \mathcal{O}(-H)$ will vanish on $j$. By repeated application of Corollary \ref{GrassmanCor}, we see that the codimension of $\phi^{-1}(Z_{r-c,d})$ in $\mathcal{F}_{r-c,d}$ is at least $c$. This concludes the proof.
\end{proof}

It follows that if $Z_{n,d} \subset \mathbb{P}^n \times U_{n,d} \setminus \mathcal{X}_{n,d}$ has codimension at least $1$ for $d \geq d_n$, then a very general hypersurface complement in $\mathbb{P}^n$ is Brody hyperbolic when $d \geq d_{2n-1}$. Using this together with Darondeau's result, we obtain the following corollary, weakening the bound a bit for brevity.

\begin{corollary}
Let $d \geq (10n)^2 (2n)^{2n}$. Then if $X \subset \mathbb{P}^n$ is very general, the complement $\mathbb{P}^n \setminus X$ is Brody hyperbolic.
\end{corollary}

\section{$0$-cycles}
\label{sec-cycles}
Let $R_{\mathbb{P}^1, X, p} = \{ q \in X | Nq \sim Np \text{ for some integer $N$} \}$, where the relation $\sim$ means Chow equivalent. The goal of this section is to prove all but the $2n - \sum_{i=1}^k (d_i - 1) = -1$ case of the following conjecture of Chen, Lewis and Sheng \cite{CLS}.

\begin{conjecture}
\label{mainConj}
Let $X \subset \mathbb{P}^{n}$ be a very general complete intersection of multidegree $(d_1, \dots, d_k)$. Then for every $p \in X$, $\operatorname{dim} R_{\mathbb{P}^1, X, p} \leq 2n -k - \sum_{i=1}^k d_i$.
\end{conjecture}

Here, we adopt the convention that $\operatorname{dim} R_{\mathbb{P}^1, X, p}$ is negative if $R_{\mathbb{P}^1, X, p} = \{p \}$. Together with the main result of \cite{CLS}, this completely resolves Conjecture \ref{mainConj} in the case of hypersurfaces.

Chen, Lewis and Sheng consider the more general notion of $\Gamma$ equivalence, although we are unable to prove the $\Gamma$ equivalence version here. The special case $\sum_i d_i = n+1$ is a theorem of Roitman \cite{R1, R2} and the case $2n -k - \sum_{i=1}^k d_i \leq -2$ is a theorem of Chen, Lewis and Sheng \cite{CLS} building on work of Voisin \cite{voisinChow, V2, V3}, who proves the result only for hypersurfaces. Chen, Lewis and Sheng prove Conjecture \ref{mainConj} for hypersurfaces and for arbitrary $\Gamma$ in the boundary case $2n - k - \sum_{i=1}^k d_i = -1$ in \cite{CLS}. The case $2n - k - \sum_{i=1}^k d_i = -1$ appears to be the most difficult, and is the only one we cannot completely resolve with our technique.

We provide an independent proof of all but the $2n - k - \sum_{i=1}^k d_i = -1$ case of Conjecture \ref{mainConj}. Aside from Roitman's result, this is the first result we are aware of addressing the case $2n - k - \sum_{i=1}^k d_i \geq 0$. We rely on the result of Roitman in our proof, but not the results of Voisin \cite{voisinChow, V2} or Chen, Lewis, and Sheng \cite{CLS}.

Let $E_{n,\underline{d}} \subset \mathcal{U}_{n,\underline{d}}$ be the set of $(p,X)$ such that $R_{\mathbb{P}^1, X, p}$ has dimension at least $1$. Let $G_{n,\underline{d}} \subset \mathcal{U}_{n,\underline{d}}$ be the set of $(p,X)$ such that $R_{\mathbb{P}^1,X,p}$ is not equal to $\{p\}$. Both $E_{n,\underline{d}}$ and $G_{n,\underline{d}}$ are countable unions of closed subvarieties of $\mathcal{U}_{n,\underline{d}}$. When we talk about the codimension of $E_{n,\underline{d}}$ or $G_{n,\underline{d}}$ in $\mathcal{U}_{n,\underline{d}}$, we mean the minimum of the codimensions of each component. We prove Conjecture \ref{mainConj} by proving the following theorem.

\begin{theorem}
\label{mainThm}
The codimension of $E_{n,\underline{d}}$ in $\mathcal{U}_{n,\underline{d}}$ is at least $-n + \sum_i d_i$ and the codimension of $G_{n,\underline{d}}$ in $\mathcal{U}_{n,\underline{d}}$ is at least $-n-1 + \sum_i d_i$.
\end{theorem}

\begin{corollary}
Conjecture \ref{mainConj} holds for $2n- k - \sum_{i=1}^k d_i \neq -1$. In the special case $2n-k- \sum_{i=1}^k d_i = -1$, the space of $p \in X$ Chow-equivalent to some other point of $X$ has dimension $0$ (i.e., is a countable union of points) but might not be empty as Conjecture \ref{mainConj} predicts.
\end{corollary}
\begin{proof}
First we consider the case $2n - k - \sum_i d_i \geq 0$. Let $\pi_1: \mathcal{U}_{n,\underline{d}} \to B$ be the projection map. If $\pi_1|_{E_{n,\underline{d}}}$ is not dominant, then the result holds trivially. Thus, we may assume that the very general fiber of $\pi_1|_{E_{n,\underline{d}}}$ has dimension $n - k - \operatorname{codim}(E_{n,\underline{d}} \subset \mathcal{U}_{n,\underline{d}})$. If the bound on $E_{n,\underline{d}}$ from Theorem \ref{mainThm} holds, then the space of points $p$ of $X$ with positive-dimensional $R_{\mathbb{P}^1,X,p}$ has dimension at most $2n - k - \sum_i d_i$, which implies Conjecture \ref{mainConj} in the case $2n - k - \sum_i d_i \geq 0$.

Now we consider the situation for $2n - k - \sum_i d_i \leq -1$. Conjecture \ref{mainConj} states that $\pi_1|_{G_{n,\underline{d}}}$ is not dominant for this range. By Theorem \ref{mainThm}, we see that the dimension of $G_{n,\underline{d}}$ is less than the dimension of $B$ if $2n - k - \sum_i d_i \leq -2$, proving Conjecture \ref{mainConj}. In the case $2n - k-\sum_i d_i = -1$, the dimension of $G_{n,\underline{d}}$ is at most that of $B$, which shows that there are at most finitely many points of $X$ equivalent to another point of $X$. This proves the result.
\end{proof}

Our technique would prove Conjecture \ref{mainConj} in all cases if we knew that $G_{n,\underline{d}}$ had codimension $-n + \sum_{i=1}^k d_i$ in $\mathcal{U}_{n,\underline{d}}$. However, this is not true for Calabi-Yau hypersurfaces.

\begin{proposition}
A general point of a very general Calabi-Yau hypersurface $X$ is rationally equivalent to at least one other point of the hypersurface.
\end{proposition}
\begin{proof}
Let $X$ be a very general Calabi-Yau hypersurface. Then we claim that a general point of $X$ is Chow equivalent to another point of $X$. To see this, observe that any point $p$ of $X$ has finitely many lines meeting $X$ to order $d-1$ at $p$. Such a line meets $X$ in a single other point. Moreover, every point of $X$ has a line passing through it that meets $X$ at another point of $X$ with multiplicity $d-1$. Thus, let $q_1$ be a general point of $X$, let $\ell_1$ be a line through $p$ meeting $X$ at a second point, $p$ to order $d-1$, and $\ell_2$ be a different line meeting $X$ at $p$ to order $d-1$. Let $q_2$ be the residual intersections of $\ell_2$ with $X$. Then $q_1 \sim q_2$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{mainThm}]
First consider the bound on $E_{n,\underline{d}}$. By Roitman's Theorem plus the fact that $R_{\mathbb{P}^1,X,p}$ is a countable union of closed varieties, we see that $E_{-1+\sum_{i=1}^k d_i,\underline{d}}$ has codimension at least one in $\mathcal{U}_{-1 + \sum_{i=1}^k d_i,\underline{d}}$. We note that if $p \sim q$ as points of $Y$, and $Y \subset Y'$, then $p \sim q$ as points of $Y'$ as well. The rest of the result follows from Theorem \ref{thm-technicaltool} using $Z_{n,\underline{d}} = E_{n,\underline{d}}$.

Now consider $G_{n,\underline{d}}$. From Roitman's theorem, it follows that a very general point of a Calabi-Yau complete intersection $X$ is equivalent to at most countably many other points of $X$. Thus, a very general hyperplane section of such an $X$ satisfies the property that the very general point is equivalent to no other points of $X$. From this, we see that $G_{-2+\sum_{i=1}^k d_i,\underline{d}}$ has codimension at least $1$ in $\mathcal{U}_{-2+\sum_{i=1}^k d_i,\underline{d}}$. Together with Theorem \ref{thm-technicaltool}, this implies the result.
\end{proof}

\bibliographystyle{plain}


\section{Introduction}

Over the last decade there has emerged growing interest in the  so-called warm dense matter (WDM), which is of key importance for the description of, e.g., astrophysical systems \cite{knudson,militzer}, laser-excited solids \cite{ernst}, and inertial confinement fusion targets \cite{nora,schmit,hurricane3}.
The WDM regime is characterized by the simultaneous occurence of strong (moderate) correlations of ions (electrons), thermal effects as well as quantum effects of the electrons. In dimensionless units, typical parameters are the Brueckner parameter $r_s=\overline{r}/a_\textnormal{B}$ and the reduced temperature $\theta=k_\textnormal{B}T/E_\textnormal{F}$, both being of the order of unity (more generally in the range $0.1 \dots 10$). Here $\overline{r}$ and $a_\textnormal{B}$ denote the mean interparticle distance and the Bohr radius, respectively. A third relevant parameter is the classical coupling parameter of the ionic component, $\Gamma_i = Z_i^2e^2/\overline{r}k_BT$ which is often larger than unity indicating that the ionic component is far from an ideal gas.
This makes the theoretical description of this peculiar state of matter particularly challenging, as there is no small parameter to perform an expansion around.

In the ground state, there exists a large toolkit of approaches that allow for the accurate description of manifold physical systems, the most successful of which arguably being Kohn-Sham density functional theory (DFT), e.g. \cite{ks,dft_review}.
The basic idea of DFT is to map the complicated and computationally demanding quantum many-body problem onto an effective single-particle problem. This would be exact if the correct exchange-correlation functional of the system of interest was available which is, of course, not the case. In practice, therefore, one has to use an approximation. The foundation of the great success of DFT has been the local density approximation (LDA), i.e., the usage of the exchange-correlation energy $E_{xc}$ of the uniform electron gas (UEG) with the same density as the more complicated system of interest. Accurate data for $E_{xc}$ of the UEG was obtained by Ceperley and Alder \cite{alder} using a Quantum Monte Carlo (QMC) method, from which Perdew and Zunger \cite{perdew} constructed a simple parametrization with respect to density, $E_{xc}(r_s)$, that is still used to this day.

However, the accurate description of warm dense matter requires to extend DFT to finite temperature. This has been realized long ago  by Mermin \cite{mermin} who used a superposition of excited states weighted with their thermal occupation probability. A strict approach to the thermodynamic properties of this system also requires an appropriate finite-temperature extension of the LDA, in particular replacement of the ground state energy functionals
by free energies, i.e. $E \to f = E - TS$. This means, a finite-temperature version of the LDA requires accurate parametrizations of the  {\em exchange correlation free energy} with respect to temperature and density \cite{karasiev2,dharma, dharma_cpp15, gga,burke,burke2}, i.e., $f_{xc}(r_s,\theta)$, even though in some cases the entropic correction maybe small. This seemingly benign task, however, turns out to be far from trivial because accurate date for the free energy are much more involved than the ground state results. While for the ground state reliable QMC data are known for a long time, until recently \cite{tim2,tim_prl,dornheim,dornheim2,groth,dornheim3,dornheim_cpp,malone,malone2,dornheim_prl,dornheim_pop_17}, the notorious fermion sign problem \cite{loh,troyer} has prevented reliable QMC simulations in the warm dense regime.
Therefore, during the recent four decades many theoretical approaches to $f_{xc}(r_s,\theta)$ have been developed that lead to a variety of parametrizations, for an overview on early works, see e.g.~\cite{kraeft-book, dewitt-tribute} . Some of them have gained high popularity and they were successfully applied in many fields, even though there accuracy has remained poorly tested. It is the purpose of the present article to present such a quantitative comparison of earlier models to new simulation results.

In Sec.~\ref{sec:free}, we introduce a selection of such functions. First, we analyze the purely analytical expression presented  by Ebeling and co-workers, e.g. \cite{ebeling1}. Next, we study  functional fits to linear response data based on static local field correction schemes that were suggested by Singwi, Tosi, Land, and Sj\"olander (STLS) \cite{stls_original} (Sec.~\ref{sub:ichimaru}) and Vashishta and Singwi (VS) \cite{vs_original} (Sec.~\ref{sub:vs}). As a fourth example we  consider the quantum-classical mapping developed by Dharma-wardana and Perrot (PDW) \cite{pdw_prl,pdw}  (Sec.~\ref{sub:pdw}).
Finally, we consider the recent parametrization by Karasiev \textit{et al.}~(KSDT) \cite{karasiev} (Sec.~\ref{sub:ksdt}) that is based on the QMC data by Brown \textit{et al.}~that became available recently~\cite{brown}. However, those data have a limited accuracy due to (i) the usage of the fixed node approximation \cite{node} and (ii) an inappropriate finite-size correction (see \cite{dornheim_prl}) giving rise to systematic errors in the free energy results as we will show below.
In Sec.~\ref{sec:results}, we compare all aforementioned parametrizations of $f_{xc}$ to the new, accurate QMC data by Dornheim \textit{et al.}~\cite{dornheim_prl} that are free from any systematic bias, and, hence, allow us to gauge the accuracy of models. A particular emphasis is laid on the warm dense matter regime.

\section{\label{sec:free}Free Energy Parametrizations}

\subsection{\label{sub:ebeling}Ebeling's Pad\'e formulae}
The idea to produce an analytical formula for the thermodynamic quantities that connect known analyitical limits via a smooth Pad\'e approximant is due to Ebeling, Kraeft, Richert and co-workers \cite{ebeling_richert81, ebeling_ebeling_richert82, ebeling_richert85_1, ebeling_richert85_2}. These approximations have been quite influential in the description of nonideal plasmas and electron-hole plasmas in the 1980s and 1990s receiving, in part, a substantial amount of citations.
These approximations have been improved continuously in the following years, and we, therefore, only discuss the more recent versions, cf.~\cite{ebeling1,ebeling2} and references therein.

Ebeling {\em et al.} used Rydberg atomic units and introduced a reduced thermal density
\begin{equation}
\overline{n} = n\Lambda^3 = 6\sqrt{\pi}r_s^{-3}\tau^{-3/2} \quad ,
\end{equation}
with the usual thermal wavelength $\Lambda$ and $\tau=k_\textnormal{B}T/\textnormal{Ry}$ being the temperature in energy units.
The Pad\'e approximation for $f_{xc}$ then reads \cite{ebeling1}
\begin{eqnarray}
f_{xc}^\textnormal{Ebeling,Ry}(r_s,\tau) = - \frac{ f_0(\tau) \overline{n}^{1/2} + f_3(\tau) \overline{n} + f_2 \overline{n}^2\epsilon^\textnormal{Ry}(r_s) }{ 1 + f_1(\tau) \overline{n}^{1/2} + f_2 \overline{n}^2 } \quad , \label{eq:ebeling_Ry}
\end{eqnarray}
with the coefficients
\begin{eqnarray}
f_0(\tau) = \frac{2}{3}\left(\frac{\tau}{\pi}\right)^{1/4}\ ,\quad f_1(\tau) = \frac{1}{8f_0(\tau)}\sqrt{2}(1+\textnormal{log}(2))\ ,\quad f_2 = 3 \ ,\quad f_3(\tau) = \frac{1}{4}\left( \frac{\tau}{\pi}\right)^{1/2}\ ,
\end{eqnarray}
and the ground state parametrization for the exchange correlation energy
\begin{equation}
\epsilon^\textnormal{Ry}(r_s) = \frac{0.9163}{r_s} + 0.1244\ \textnormal{log}\left( 1 + \frac{ 2.117 r_s^{-1/2}}{1+0.3008\sqrt{r_s}}\right) \quad .
\end{equation}
To achieve a better comparability with the other formulas discussed below, we re-express Eq.~(\ref{eq:ebeling_Ry}) in Hartree atomic units as a function of $r_s$ and the reduced temperature $\theta=k_\textnormal{B}T/E_\textnormal{F}$:
\begin{eqnarray}
\label{eq:ebeling}f_{xc}^\textnormal{Ebeling,Ha}(r_s,\theta) &=& -\frac{1}{2} \frac{ Ar_s^{-1/2}\theta^{-1/2} + B r_s^{-1}\theta^{-1} + C\theta^{-3}\epsilon^\textnormal{Ry}(r_s) }{ 1 + D\theta^{-1}r_s^{1/2}+ C \theta^{-3}} \quad , \quad \textnormal{with} \\
A &=& \frac{2}{3\sqrt{\pi}}\left(\frac{8}{3}\right)^{1/2}\left(\frac{4}{9\pi}\right)^{-1/6}\ , \quad B = \frac{2}{3\pi}\left( \frac{4}{9\pi} \right)^{-1/3}\ , \quad C=\frac{64}{3\pi}\ , \quad \\ \nonumber D &=& \frac{ (1+\textnormal{log}(2))\sqrt{3} }{4}\left( \frac{4}{9\pi}\right)^{1/6}\ .
\end{eqnarray}
Evidently, Eq.~(\ref{eq:ebeling}) incorporates the correct ground state limit
\begin{equation}
\lim_{\theta \to 0} f_{xc}^\textnormal{Ebeling,Ha}(r_s,\theta) = -\frac{1}{2}\epsilon^\textnormal{Ry}(r_s) \quad ,
\end{equation}
where the pre-factor $1/2$ is due to the conversion between Rydberg and Hartree units. Similarly, in the high-temperature limit the well-known Debye-H\"uckel result is recovered, e.g.~\cite{dewitt}:
\begin{eqnarray}\label{eq:DBH}
\lim_{\theta \to \infty} f_{xc}^\textnormal{Ebeling,Ha}(r_s,\theta) = -\frac{1}{2} A\ r_s^{-1/2} \theta^{-1/2}
= - \frac{1}{\sqrt{3}}r_s^{-3/2}T^{-1/2} \quad .
\end{eqnarray}
Results for the warm dense UEG computed from these formulas are included in the figures below. For Pad\'e approximations to the UEG at strong coupling in the quasi-classical regime, see, e.g., Ref.~\cite{stolz_ebeling}.

\subsection{\label{sub:ichimaru}Parametrization by Ichimaru and co-workers}
In the mid-eighties, Tanaka, Ichimaru and co-workers \cite{stls85,stls} extended the original STLS scheme \cite{stls_original} for the static local field corrections to finite temperature and numerically obtained the interaction energy $V$ (per particle) of the UEG via integration of the static structure factor $S(k)$,
\begin{equation}\label{eq:V}
V = \frac{1}{2} \int_{k<\infty} \frac{ \textnormal{d}\mathbf{k} }{ (2\pi)^3 } [S(\mathbf{k}) - 1]\frac{4\pi}{\mathbf{k}^2} \quad ,
\end{equation}
for $70$ parameter combinations with $\theta=0.1,1,5$ and $r_s\sim10^{-3},\dots,74$.
Subsequently, there has been introduced a parametrization for $V$ as a function of $r_s$ and $\theta$ \cite{ichimaru_rev_2,ichimaru_rev}
\begin{equation}\label{eq:ichi_V}
V(r_s,\theta) = - \frac{1}{r_s} \frac{ a_\textnormal{HF}(\theta) + \sqrt{2}\lambda r_s^{1/2} \textnormal{tanh}(\theta^{-1/2}) B(\theta) + 2\lambda^2r_sC(\theta)E(\theta)\textnormal{tanh}(\theta^{-1}) }{1+ \sqrt{2}\lambda r_s^{1/2}D(\theta)\textnormal{tanh}(\theta^{-1/2}) + 2\lambda^2 r_s E(\theta)}\ ,
\end{equation}
with the definitions
\begin{eqnarray}\label{eq:ichi_definitions}
a_\textnormal{HF}(\theta) &=& 0.610887\ \textnormal{tanh}\left(\theta^{-1}\right) \frac{ 0.75 + 3.4363\theta^2 -0.09227\theta^3 +  1.7035\theta^4 }{ 1+ 8.31051\theta^2 + 5.1105\theta^4 }\\
B(\theta) &=& \frac{  x_1 + x_2\theta^2 + x_3 \theta^4 }{ 1 + x_4\theta^2 + x_5\theta^4 }\ , \quad
C(\theta) = x_6 + x_7 \textnormal{exp}\left( -\theta^{-1} \right) \ , \\
D(\theta) &=& \frac{ x_8 + x_9\theta^2 + x_{10}\theta^4 }{ 1 + x_{11}\theta^2 + x_{12}\theta^4} \ , \quad
E(\theta) = \frac{x_{13}+x_{14}\theta^2+x_{15}\theta^4}{1+x_{16}\theta^2 + x_{17}\theta^4} \quad .
\end{eqnarray}
In addition to the exact limits for $\theta\to 0$ and $\theta\to\infty$, the parametrization from Eq.~(\ref{eq:ichi_V}) also approaches the well-known Hartree-Fock limit for high density,
\begin{equation}
\lim_{r_s\to0}V(r_s,\theta) = - \frac{ a_\textnormal{HF}(\theta)}{r_s} \quad ,
\end{equation}
which has been parameterized  by Perrot and Dharma-wardana \cite{pdw84}, see Eq.~(\ref{eq:ichi_definitions}).
Naturally, the free parameters $x_i$, $i=1,\dots,17$ have been determined by a fit of Eq.~(\ref{eq:ichi_V}) to the STLS data for $V$ and the resulting values are listed in Tab.~\ref{tab:ichi}.
From the interaction energy $V(r_s,\theta)$, the free exchange-correlation energy is obtained by integration
\begin{equation}\label{eq:fxc}
f_{xc}(r_s,\theta) = \frac{1}{r_s^2}\int_0^{r_s} \textnormal{d}\overline{r}_s\ \overline{r}_s  V(\overline{r}_s,\theta) \quad .
\end{equation}
Plugging in the expression for $V(r_s,\theta)$ from Eq.~(\ref{eq:ichi_V}) into (\ref{eq:fxc}) gives the final parametrization for $f_{xc}(r_s,\theta)$
\begin{eqnarray}\label{eq:ichi_fxc}
f_{xc}(r_s,\theta) = &-& \frac{1}{r_s}\frac{c(\theta)}{e(\theta)} \\ \nonumber
&-& \frac{ \theta}{2 e(\theta) r_s^2\lambda^2 } \left[ \left( a_\textnormal{HF}(\theta)-\frac{c(\theta)}{e(\theta)}\right)
-\frac{d(\theta)}{e(\theta)}\left( b(\theta) - \frac{ c(\theta)d(\theta)}{e(\theta)}\right)\right] \\ \nonumber
&\times&\textnormal{log}\left|  \frac{ 2 e(\theta) \lambda^2 r_s }{ \theta} + \sqrt{2}d(\theta)\lambda r_s^{1/2} \theta^{-1/2} +1 \right| \\ \nonumber
&-& \frac{\sqrt{2}}{e(\theta)}\left( b(\theta) - \frac{ c(\theta)d(\theta) }{e(\theta)}\right) \frac{ \theta^{1/2} }{r_s^{1/2}\lambda}\\ \nonumber
&+& \frac{ \theta }{ r_s^2\lambda^2 e(\theta) \sqrt{4e(\theta)-d^2(\theta)}}\left[ d(\theta)\left(a_\textnormal{HF}(\theta)-\frac{c(\theta)}{e(\theta)}\right)\right. \\ \nonumber &+& \left. \left(2-\frac{d^2(\theta)}{e(\theta)}\right)\left(b(\theta)-\frac{c(\theta)d(\theta)}{e(\theta)}\right)\right] \\ \nonumber
&\times&\left[ \textnormal{atan}\left( \frac{ 2^{3/2} e(\theta) \lambda r_s^{1/2} \theta^{-1/2} + d(\theta) }{ \sqrt{4e(\theta)-d^2(\theta)} } \right) - \textnormal{atan}\left( \frac{ d(\theta) }{ \sqrt{4e(\theta)-d^2(\theta)}}\right)\right] \ ,
\end{eqnarray}
with the abbreviations
\begin{eqnarray}
b(\theta) &=& \theta^{1/2}\ \textnormal{tanh}\left( \theta^{-1/2} \right) B(\theta) \ , \quad
c(\theta) = C(\theta)e(\theta), \\ \nonumber
d(\theta) &=& \theta^{1/2}\ \textnormal{tanh}\left( \theta^{-1/2} \right) D(\theta) \ , \quad
e(\theta) = \theta\ \textnormal{tanh}\left( \theta^{-1} \right) E(\theta) \quad .
\end{eqnarray}

\begin{table}
\centering
\begin{tabular}{ c c c c c  }
$x_1$ & $x_2$ & $x_3$ &
$x_4$ &
$x_5$  \\ \hline
$3.4130800\times10^{-1}$ &
$1.2070873\times10$ &
$1.148889\times10^{0}$ &
$1.0495346\times10$ &
$1.326623\times10^0$\vspace{0.2cm} \\
$x_6$ & $x_7$ & $x_8$ &
$x_9$ &
$x_{10}$  \\ \hline
$8.72496\times10^{-1}$ &
$2.5248\times10^{-2}$ &
$6.14925\times10^{-1}$ &
$1.6996055\times10$ &
$1.489056\times10^0$ \vspace{0.2cm}\\
$x_{11}$ &
$x_{12}$ &
$x_{13}$ &
    $x_{14}$ &
    $x_{15}$ \\ \hline
    $1.010935\times10$ &
$1.22184\times10^0$ &
$5.39409\times10^{-1}$ &
$2.522206\times10^{0}$ &
$1.78484\times10^{-1}$ \vspace{0.2cm} \\
$x_{16}$ &
$x_{17}$ \\ \hline
$2.555501\times10^{0}$ &
$1.46319\times10^{-1}$

  \end{tabular}
\caption{Fit parameters by Ichimaru \cite{ichimaru_rev} for the $f_{xc}(r_s,\theta)$ parametrization from Eq.~(\ref{eq:ichi_fxc}), fitted to STLS data \cite{stls}.}
\label{tab:ichi}
\end{table}

\subsection{\label{sub:vs}Vashishta-Singwi parametrization}
Despite the overall good performance of STLS in the ground state \cite{bohm}, it has long been known that this scheme does not fulfill the compressibility sum-rule (CSR, see, e.g., Ref.~\cite{stls2} for a detailed discussion). To overcome this obstacle, Vashishta and Singwi \cite{vs_original} have introduced modified local field corrections (VS), where the CSR it automatically fulfilled. This idea had been extended in an approximate way to finite temperature by Stolzmann and R\"osler \cite{stolzmann}, until more recently Sjostrom and Dufty \cite{stls2} obtained an exhaustive data set of results that are exact within the VS framework.

As already explained in the previous section for the STLS data, they have first calculated the static structure factor $S(k)$, computed the interaction energy $V$ by integration (Eq.~(\ref{eq:V}), fitted the parametrization from Eq.~(\ref{eq:ichi_V}) to this data, and thereby obtained the desired parametrization of $f_{xc}(r_s,\theta)$ as given in Eq.~(\ref{eq:ichi_fxc}) (albeit with the new fit parameters listed in Tab.~\ref{tab:vs}).

\begin{table}
\centering
\begin{tabular}{ c c c c c  }
$x_1$ & $x_2$ & $x_3$ &
$x_4$ &
$x_5$  \\ \hline
$1.8871493\times10^{-1}$ &
$1.0684788\times10$ &
$1.1088191\times10^{2}$ &
$1.8015380\times10$ &
$1.2803540\times10^2$\vspace{0.2cm} \\
$x_6$ & $x_7$ & $x_8$ &
$x_9$ &
$x_{10}$  \\ \hline
$8.3331352\times10^{-1}$ &
$-1.1179213\times10^{-1}$ &
$6.1492503\times10^{-1}$ &
$1.6428929\times10$ &
$2.5963096\times10$ \vspace{0.2cm}\\
$x_{11}$ &
$x_{12}$ &
$x_{13}$ &
    $x_{14}$ &
    $x_{15}$ \\ \hline
    $1.0905162\times10$ &
$2.9942171\times10$ &
$5.3940898\times10^{-1}$ &
$5.8869626\times10^{4}$ &
$3.1165052\times10^{3}$ \vspace{0.2cm} \\
$x_{16}$ &
$x_{17}$ \\ \hline
$3.8887108\times10^{4}$ &
$2.1774472\times10^{3}$

  \end{tabular}
\caption{Fit parameters by Sjostrom and Dufty \cite{stls2} for the $f_{xc}(r_s,\theta)$ parametrization from Eq.~(\ref{eq:ichi_fxc}), fitted to VS data.}
\label{tab:vs}
\end{table}

\subsection{\label{sub:pdw}Perrot-Dharma-wardana parametrization}
Dharma-wardana and Perrot \cite{pdw_prl,pdw} have introduced an independent, completely different idea. In particular, they employ a \textit{classical mapping} such that the correlation energy of the electron gas at $T=0$ (that has long been known from QMC calculations~\cite{alder,perdew}) is exactly recovered by the simulation of a classical system at an effective ``quantum temperature'' $T_q$. However, due to the lack of accurate data at finite $T$, an exact mapping had not been possible, and the authors introduced a modified temperature $T_c$, where they assumed an interpolation between the exactly known ground state and classical (high $T$) regimes, $T_{c}=\sqrt{T^2+T^2_q}$. Naturally, at warm dense matter conditions this constitutes a largely uncontrolled approximation.

To obtain the desired parametrization for $f_{xc}$, extensive simulations of the UEG in the range of $r_s=1,\dots,10$ and $\theta=0,\dots,10$ were performed. These have been used as input for a fit with the functional form
\begin{eqnarray}\label{eq:pdw}
f_{xc}(r_s,\theta) &=& \frac{ \epsilon(r_s) - P_1(r_s,\theta) }{ P_2(r_s,\theta) }, \\ \nonumber
P_1(r_s,\theta) &=& \left(A_2(r_s)u_1(r_s) + A_3(r_s)u_2(r_s)\right) \theta^2 Q^2(r_s) + A_2(r_s)u_2(r_s)\theta^{5/2}Q^{5/2}(r_s), \\ \nonumber
P_2(r_s,\theta) &=& 1 + A_1(r_s)\theta^2 Q^2(r_s) + A_3(r_s)\theta^{5/2}Q^{5/2}(r_s) + A_2(r_s)\theta^3Q^3(r_s), \\ \nonumber
Q(r_s) &=& \left( 2 r_s^2 \lambda^2 \right)^{-1}\ , \quad n(r_s) = \frac{3}{4\pi r_s^3}\ , \quad u_1(r_s) = \frac{\pi n(r_s)}{2}\ , \quad u_2(r_s) = \frac{2\sqrt{\pi n(r_s)}}{3}, \\ \nonumber
A_k(r_s) &=& \textnormal{exp}\left( \frac{ y_k(r_s) + \beta_k(r_s)z_k(r_s) }{ 1 + \beta_k(r_s) }
\right)\ , \quad \beta_k(r_s) = \textnormal{exp}\left( 5(r_s - r_k)
\right), \\ \nonumber
y_k(r_s) &=& \nu_k\ \textnormal{log}(r_s) + \frac{a_{1,k} + b_{1,k}r_s + c_{1,k}r_s^2  }{ 1 + r_s^2/5} \ , \quad
z_k(r_s) = r_s \frac{ a_{2,k} + b_{2,k}r_s }{ 1 + c_{2,k} r_s^2 } \quad ,
\end{eqnarray}
which becomes exact for $\theta\to 0$ and $\theta\to\infty$, but is limited to the accuracy of the classical mapping data in between. Further, it does not include the exact Hartree-Fock limit for $r_s\to0$, so that it cannot reasonably be used for $r_s<1$.
For completeness, we mention that a functional form similar to Eq.~(\ref{eq:pdw}) has recently been used by Brown \textit{et al.}~\cite{brown3} for a fit to their RPIMC data \cite{brown}.

\begin{table}
\centering
\begin{tabular}{ l c c c c c c c c  }
$k$ & $a_{1,k}$ & $b_{1,k}$ & $c_{1,k}$ & $a_{2,k}$ & $b_{2,k}$ & $c_{2,k}$ & $\nu_k$ & $r_k$ \\ \hline
1 &
5.6304 &
-2.2308 &
1.7624 &
2.6083 &
1.2782 &
0.16625 &
1.5 &
4.4467 \\
2 &
5.2901 &
-2.0512 &
1.6185 &
-15.076 &
24.929 &
2.0261 &
3 &
4.5581 \\
3 &
3.6854 &
-1.5385 &
1.2629 &
2.4071 &
0.78293 &
0.095869 &
3 &
4.3909 \\

\end{tabular}
\caption{Fit parameters by Perrot and Dharma-wardana \cite{pdw} for the $f_{xc}(r_s,\theta)$ parametrization from Eq.~(\ref{eq:pdw}).}
\label{tab:pdw}
\end{table}
Similar ideas of quantum-classical mappings were recently investigated by Dufty and Dutta, see e.g. \cite{dufty_dutta1, dufty_dutta2}.

\subsection{\label{sub:ksdt}Parametrization by Karasiev \textit{et al.}}
Karasiev and co-workers \cite{karasiev} (KSDT) utilized as the functional form for $f_{xc}$ an expression similar to Eq.~(\ref{eq:ichi_V}), which Ichimaru and co-workers \cite{ichimaru_rev_2,ichimaru_rev} suggested for the interaction energy:
\begin{eqnarray}\label{eq:ksdt}
f_{xc}(r_s,\theta) &=& - \frac{1}{r_s} \frac{ a_\textnormal{HF}(\theta) + b(\theta) r_s^{1/2} + c(\theta) r_s }{ 1 + d(\theta)r_s^{1/2} + e(\theta)r_s }, \\ \nonumber
b(\theta) &=& \textnormal{tanh}\left( \theta^{-1/2} \right) \frac{ b_1 + b_2\theta^2+b_3\theta^4}{ 1 + b_4\theta^2 + \sqrt{1.5}\lambda^{-1}b_3\theta^4 } \ , \quad
c(\theta) = \left[ c_1 + c_2\ \textnormal{exp}\left( -\frac{c_3}{\theta} \right)\right] e(\theta), \\ \nonumber
d(\theta) &=& \textnormal{tanh}\left( \theta^{-1/2} \right) \frac{ d_1 + d_2\theta^2+d_3\theta^4}{ 1 + d_4\theta^2 + d_5\theta^4 } \ , \quad
e(\theta) = \textnormal{tanh}\left( \theta^{-1} \right) \frac{ e_1 + e_2\theta^2+e_3\theta^4}{ 1 + e_4\theta^2 + e_5\theta^4 } \ . \quad
\end{eqnarray}
Further, instead of fitting to the interaction energy $V$, they used the relation
\begin{equation}\label{eq:exc}
E_{xc}(r_s,\theta) = f_{xc}(r_s,\theta) -\left. \theta \frac{\partial f_{xc}(r_s,\theta)}{\partial\theta}\right|_{r_s}
\end{equation}
and fitted the rhs.~of Eq.~(\ref{eq:exc}) to the recently published RPIMC data for the exchange correlation energy $E_{xc}$ by Brown \textit{et al.}~\cite{brown} that are available for the parameters $r_s=1,\dots,40$ and $\theta=0.0625,\dots,8$.

\begin{table}
\centering
\begin{tabular}{ c c c c c c c  }

  $b_1$ & $b_2$ & $b_3$ & $b_4$ & $c_1$ & $c_2$ & $c_3$   \\
    \hline
  0.283997 &
  48.932154 &
  0.370919 &
  61.095357 &
  0.870089 &
  0.193077 &
  2.414644 \vspace{0.2cm} \\

    $d_1$ & $d_2$ & $d_3$ & $d_4$ & $d_5$ & $e_1$ & $e_2$  \\
           \hline
  0.579824 &
  94.537454 &
  97.839603 &
  59.939999 &
  24.388037 &
  0.212036 &
  16.731249 \vspace{0.2cm}
  \\

     $e_3$ & $e_4$ & $e_5$ &
  & & & \\  \hline
     28.485792 &
  34.028876 &
  17.235515 &
  & & & \\

  \end{tabular}
\caption{Fit parameters by Karasiev \textit{et al.}~\cite{karasiev} for the $f_{xc}(r_s,\theta)$ parametrization from Eq.~(\ref{eq:ksdt}).}
\label{tab:KSDT}
\end{table}

\section{Results\label{sec:results}}
In this section we analyze the behavior of the analytical approximations for the exchange-correlation free energies that were summarized above by comparison to our recent simulation results that cover the entire relevant density range for temperatures $\Theta \ge 0.5$. These data
have an unprecedented accuracy of the order of $0.1\%$, for details, see Refs.~\cite{dornheim_prl, dornheim_pop_17}.

\subsection{\label{sub:temperature}Temperature dependence}

 \begin{figure}[t]
 \includegraphics[width=0.5\textwidth]{Contrib_fxc_rs1_theta.pdf} \includegraphics[width=0.5\textwidth]{Contrib_fxc_rs6_theta.pdf}~\hspace{-3mm}\vspace*{-0.6cm}\\
  \includegraphics[width=0.5\textwidth]{Contrib_delta_fxc_rs1_theta.pdf}
    \includegraphics[width=0.5\textwidth]{Contrib_delta_fxc_rs6_theta.pdf}~\hspace{-3mm}
 \caption{Temperature dependence of $f_{xc}$ at fixed density $r_s=1$ (left) and $r_s=6$ (right). Top: QMC data (symbols) taken from Dornheim \textit{et al.}\cite{dornheim_prl}, a parametrization of RPIMC data by Karasiev \textit{et al.}\cite{karasiev} (KSDT), a semi-analytic Pad\'e approximation by Ebeling \cite{ebeling1}, a parametrization fitted to STLS and VS data by Ichimaru \cite{ichimaru_rev} and Sjostrom and Dufty \cite{stls2}, respectively, and a fit to classical mapping data by Perrot and Dharma-wardana \cite{pdw} (PDW).   Bottom: Relative deviation to the QMC data.
  \label{fig:theta} }
 \end{figure}

In Fig.~\ref{fig:theta}, we show the temperature dependence of the  exchange-correlation free energy as a function of the reduced temperature $\theta$ for two densities that are relevant for contemporary warm dense matter research, namely $r_s=1$ (left) and $r_s=6$ (right).
For both cases, all depicted parametrizations reproduce the correct classical limit for large $\theta$  [c.f.~Eq.~(\ref{eq:DBH})] and four of them (Ebeling, KSDT, STLS and PDW) are in excellent agreement for the ground state as well.
For completeness, we note that the small differences between KSDT and Ebeling and PDW are due to different ground state QMC input data. In particular, Karasiev \textit{et al.}~used more recent QMC results by Spink \textit{et al.}\cite{spink}, although in the context of WDM research the deviations to older parametrizations are negligible. Further, we note that the STLS parametrization reproduces the STLS data for $\theta=0$ that, however, are in good agreement with the exact QMC results as well. The VS-parametrization, on the other hand, does not incorporate any ground state limit and, consequently, the behavior of $f^\textnormal{VS}_\textnormal{xc}(r_s,\theta)$ becomes unreasonable below $\theta=0.0625$.
Similarly, the lowest temperature (despite the ground state limit) included in the fit for $f^\textnormal{PDW}_\textnormal{xc}(r_s,\theta)$ is $\theta=0.25$ and the rather unsmooth connection between this point and $\theta=0$ does not appear to be trustworthy as well.

Let us now check the accuracy of the different models at intermediate, WDM temperatures. As a reference, we use the recent, accurate QMC results for the macroscopic UEG by Dornheim \textit{et al.}~\cite{dornheim_prl}, i.e., the red squares. For $r_s=1$, the semi-analytic expression by Ebeling (blue) exhibits the largest deviations exceeding $\Delta f_{xc}/f_{xc}= 25\%$ for $\theta\sim 1$.
For lower density, $r_s=6$, the Ebeling parametrization is significantly more accurate although here, too, appear deviations of $\Delta f_{xc}/f_{xc}\sim 10\%$ to the exact data at intermediate temperature. Therefore, this parametrization produces reliable data in the two limiting cases of zero and high temperature, but is less accurate in between.

Next consider the STLS curve (black). It is in very good agreement with the QMC data, and the error does not exceed $\Delta f_{xc}/f_{xc} = 4\%$ over the entire $\theta$-range for both depicted $r_s$ values. The largest deviations appear for intermediate temperatures as well.

Third, we consider the VS-model (yellow line). For $r_s=1$, the VS-parametrization  by Sjostrom and Dufty \cite{stls2} exhibits the same trends as the STLS curve albeit with larger deviations, $\Delta f_{xc}/f_{xc}> 5\%$. Further, for $r_s=6$, $f_{xc}^\textnormal{VS}$ exhibits much larger deviations to the exact result and the error attains $\Delta f_{xc}/f_{xc}\approx 8\%$. Evidently, the constraint to automatically fulfill the CSR does not improve the accuracy of other quantities, in particular the interaction energy $V$ (which was used as an input for the parametrization, see Sec.~\ref{sub:vs}) or the static structure factor $S(k)$ itself.

Fourth, the parametrization based on the classical mapping (PDW, light blue) exhibits somewhat  opposite trends as compared to Ebeling, STLS, and VS and predicts a too large exchange correlation free energy for all $\theta$.
The magnitude of the deviations is comparable to VS and does not exceed $\Delta f_{xc}/f_{xc}=5\%$.

Finally, we consider the recent parametrization by Karasiev \textit{et al.}~(KSDT, green) \cite{karasiev} that is based on RPIMC results \cite{brown}.
For $r_s=6$, there is excellent agreement with the new reference QMC data with a maximum deviation of $\Delta f_{xc}/f_{xc}\sim 1\%$ for $\theta=4$. This is, in principle, expected since the main source of error for their input data, i.e., the nodal error and the insufficient finite-size correction, are less important for larger $r_s$. However, for $r_s=1$ there appear significantly larger deviations exceeding $\Delta f_{xc}/f_{xc}= 5\%$ at high temperature. In fact, for $r_s=1$ and the largest considered temperature, $\theta=8$, the KSDT parametrization exhibits the largest deviations of all depicted parametrizations.

\subsection{\label{sub:density}Density dependence}

 \begin{figure}[t]
 \includegraphics[width=0.5\textwidth]{Contrib_fxc_theta0p5_rs.pdf} \includegraphics[width=0.5\textwidth]{Contrib_fxc_theta4_rs.pdf}~\hspace{-3mm}\vspace*{-0.6cm}\\
  \includegraphics[width=0.5\textwidth]{Contrib_delta_fxc_theta0p5_rs.pdf}
    \includegraphics[width=0.5\textwidth]{Contrib_delta_fxc_theta4_rs.pdf}~\hspace{-3mm}
 \caption{Density dependence of $f_{xc}$ at fixed temperature $\theta=0.5$ (left) and $\theta=4$ (right). Top: QMC data taken from Dornheim \textit{et al.}\cite{dornheim_prl}, a parametrization of RPIMC data by Karasiev \textit{et al.}\cite{karasiev} (KSDT), a semi-analytic Pad\'e approximation by Ebeling \cite{ebeling1}, a parametrization fitted to STLS and VS data by Ichimaru \cite{ichimaru_rev} and Sjostrom and Dufty \cite{stls2}, respectively, and a fit to classical mapping data by Perrot and Dharma-wardana \cite{pdw} (PDW).   Bottom: Relative deviation to the QMC data.
  \label{fig:density}}
 \end{figure}

As a complement to Sec.~\ref{sub:temperature}, in Fig.~\ref{fig:density} we investigate in more detail the density dependence of the different parametrizations for two relevant temperatures, $\theta=0.5$ (left) and $\theta=4$ (right).

Most notably the Ebeling and PDW parametrizations do not include the correct high density ($r_s\to0$) limit, i.e.~Eq.~(\ref{eq:ichi_definitions}), and therefore are not reliable for $r_s<1$.
For $\theta=0.5$, $f_{xc}^\textnormal{Ebeling}$ is in qualitative agreement with the correct results, but the deviations rapidly increase with density and exceed $\Delta f_{xc}/f_{xc} = 10\%$, for $r_s=1$. At higher temperature, $\theta=4$, the situation is worse, and the Ebeling parametrization shows systematic deviations over the entire density range.
The STLS fit displays a similarly impressive agreement with the exact data as for the $\theta$-dependence (c.f.~Fig.~\ref{fig:theta}), and the deviations do not exceed $\Delta f_{xc}/f_{xc}\sim 3\%$ for both depicted $\theta$-values.
On the other hand, the VS results are again significantly less accurate than STLS although the deviation remains below $\Delta f_{xc}/f_{xc}=8\%$ for both temperatures. Further, we notice that the largest deviations occur for $r_s \geq 2$, i.e., towards stronger coupling, which is expected since here the pair distribution function exhibits unphysical negative values at short distance, e.g.~\cite{stls2}.
Again, the incorporation of the CSR has not improved the quality of the interaction energy or the structure factor compared to STLS.
The classical mapping data (PDW) does exhibit deviations not exceeding $\Delta f_{xc}/f_{xc} =5\%$ for $r_s\geq1$, i.e., in the range where numerical data have been incorporated into the fit. Overall, the quality of this parametrization is comparable to the VS-curve although the relative deviation appears to be almost constant with respect to the density. This is not surprising as the approximation has not been conducted with respect to coupling (the effective classical system is solved with the hypernetted chain method, which is expected to be accurate in this regime) but, instead, in the interpolation of the effective temperature $T_c$. Further, we notice a peculiar non-smooth and almost oscillatory behavior of $f^\textnormal{PDW}_{xc}$ around $r_s=5$ that is more pronounced for $\theta=0.5$ and the origin of which remains unclear.
Finally, we again consider the KSDT-fit based on the RPIMC data by Brown \textit{et al.}~\cite{brown} (a similar analysis for more temperatures can be found in Ref.~\cite{dornheim_prl}).
For $\theta=0.5$, this parametrization is in excellent agreement with the reference QMC data and the deviations are in the sub-percent regime over the entire depicted $r_s$-range.
However, for larger temperatures there appear significant errors that, at $\theta=4$, attain a maximum of $\Delta f_{xc}/f_{xc}\sim 10\%$ for $r_s=0.1$, i.e., at parameters where STLS, VS, and PDW are in very good agreement with the reference QMC data. Interestingly, these deviations only vanish for $r_s \leq 10^{-4}$.
Naturally, the inaccuracies of the KSDT-fit are a direct consequence of the systematic errors of the input data and the lack of accurate simulation data for $r_s<1$, prior to Ref.~\cite{dornheim_prl}.

\section{Discussion}

In summary, we have compared five different parametrizations of the  exchange-correlation free energy of the unpolarized UEG to the recent QMC data by Dornheim \textit{et al.}~\cite{dornheim_prl} and, thereby, have been able to gauge their accuracy with respect to $\theta$ and $r_s$ over large parts of the warm dense matter regime. We underline that all these parametrizations are highly valuable, the main merit being their easy and flexible use and rapid evaluation. At the same time, an unbiased evaluation of their accuracy had not been done and appears highly important, as this allows to constrain the field of applicability of these models and to indicate directions for future improvements.

Summarizing our findings,
we have observed that the semi-analytic parametrization by Ebeling \cite{ebeling1} is mostly reliable in the high- and zero temperature limits, but exhibits substantial deviations in between.
The STLS-fit given by Ichimaru an co-workers \cite{ichimaru_rev_2,ichimaru_rev}, on the other hand, exhibits a surprisingly high accuracy for all investigated $r_s$-$\theta$-combinations with a typical relative systematic error of $\sim2\%$. The more recent Vashishta-Singwi (VS) results \cite{stls2} that automatically fulfill the compressibility sum-rule display a qualitatively similar behavior but are significantly less accurate everywhere.
The classical mapping suggested by Perrot and Dharma-wardana \cite{pdw} constitutes an approximation rather with respect to temperature than to coupling strength and, consequently, exhibits different trends. In particular, we have found that the relative systematic error is nearly independent of $r_s$, but decreases with increasing $\theta$ and eventually vanishes for $\theta\to\infty$. Overall, the accuracy of the PDW-parametrization is comparable to VS and, hence, inferior to STLS. Finally, the more recent fit by Karasiev \textit{et al.}~\cite{karasiev} to RPIMC data \cite{brown} is accurate for large $r_s$ and low temperature, where the input data is not too biased by the inappropriate treatment of finite-size errors in the underlying RPIMC results.
For higher temperatures (where the exchange correlation free energy constitutes only a small fraction of the total free energy) there occur relative deviations of up to $\sim10\%$.

Thus we conclude that an accurate parametrization of the exchange-correlation free energy that is valid for all $r_s$-$\theta$-combinations is presently not available. However, the recent QMC data by Dornheim \textit{et al.}~\cite{dornheim_prl} most certainly constitute a promising basis for the construction of such a functional. Further, thermal DFT calculations in the local spin-density approximation require a parametrization of $f_{xc}$ also as a function of the spin-polarization $\xi = (N_\uparrow - N_\downarrow) / (N_\uparrow + N_\downarrow)$, i.e., $f_{xc}(r_s,\theta,\xi)$ for all warm dense matter parameters. Obviously, this will require to extend the QMC simulations beyond the unpolarized case, $\xi\in(0,1]$ and, in addition, reliable data for $\theta<0.5$ are indispensable. This work is presently under way.
We also note that the quality of the currently available KSDT fit for $f_{xc}(r_s,\theta,\xi)$ remains to be tested for $\xi> 0$. The accuracy of this parametrization is limited by (i) the quality of the RPIMC data (for the spin-polarized UEG ($\xi=1$) they are afflicted with a substantially larger nodal error than for the unpolarized case that we considered in the present paper, see Ref.~\cite{groth}) and (ii) by the quality of the PDW-results \cite{pdw} that have been included as the only input to the KSDT-fit for $0<\xi<1$ at finite $\theta$.
Therefore, we conclude that the construction of a new, accurate function $f_{xc}(r_s,\theta,\xi)$ is still of high importance for thermal DFT and semi-analytical models, for comparisons with experiments, but also for explicitly time-dependent approaches such as time-dependent DFT and quantum hydrodynamics \cite{manfredi, michta}.

\begin{acknowledgement}
SG and TD contributed equally to this work.
 We acknowledge helpful comments from A. F\"orster on the Pad\'e formulas of Ebeling \textit{et al.}~and from Fionn~D.~Malone
 This work was supported by the Deutsche Forschungsgemeinschaft via project BO1366-10 and via SFB TR-24 project A9 as well as grant shp00015 for CPU time at the Norddeutscher Verbund f\"ur Hoch- und H\"ochstleistungsrechnen (HLRN).
\end{acknowledgement}