\section{Introduction}
\input{./sections/introduction}

\section{The chiral Gross-Neveu model}
\input{./sections/model}

\section{Correlators and related quantities}
\input{./sections/results}

\newpage
\acknowledgments

We are indebted to Laurin Pannullo, Marc Wagner, Marc Winstel and 
Andreas Wipf for numerous enlightening discussions and previous
collaborations on inhomogeneous condensates. We furthermore thank 
Björn Wellegehausen for providing the simulation code base used in 
this work. This work has been funded by the Deutsche 
Forschungsgemeinschaft (DFG) under Grant No. 406116891 within the 
Research Training Group RTG 2522/1. The numerical simulations were 
performed on resources of the Friedrich Schiller University of Jena 
supported in part by the DFG grants INST 275/334-1 FUGG and INST 
275/363-1 FUGG. 

\bibliographystyle{JHEP}


\section{Phase diagrams}

We study phase diagrams in the $(\mu,T)$ plane using
\eqref{eq:cshort} and \eqref{eq:clong}. In order to quantify finite-volume effects 
we performed an infinite-volume extrapolation at fixed lattice spacing and show the 
results in Fig.~\ref{fig:pd_infinite_volume}. 

First of all, there is a region where  $C_{\mathrm{short}}>0$ and thus homogeneous 
configurations dominate, i.e. the red region in Fig.~\ref{fig:pd_infinite_volume}. 
Its extent in the $\mu$-direction shrinks in the infinite-volume limit, which can 
be understood by recalling that the chiral spiral's wavelength is inversely 
proportional to $\mu$. This means that for small but non-vanishing chemical potential 
smaller lattices are simply not large enough to fit in a full wavelength, thus 
leading to a suppression of inhomogeneous configurations. The large extent of the 
red region in the $T$-direction on the smallest lattice can also be seen to be a 
finite-size effect.

We furthermore observe that the boundary between the "short-ranged" 
($C_{\mathrm{short}}\approx0$) and "inhomogeneous" ($C_{\mathrm{short}}<0$) regions remains essentially unchanged 
in the infinite-volume extrapolation. This is actually expected because the
appearance of $C_{\mathrm{short}}<0$ only depends on the two scales introduced by the temperature
and the chemical potential respectively, neither of which changes in the infinite-volume limit. 
More precisely, negative values of $C_{\mathrm{short}}$ occur whenever the preferred
wavelength of the oscillatory (quasi-)condensates is smaller than the thermally
induced finite correlation length. From their analytically known values in an
expansion in $1/N_{\mathrm{f}}$ (see \cite{LMW21}), it is easy to anticipate the shape of this
boundary -- at least on a qualitative level.
In contrast, $C_{\mathrm{long}}$ becomes smaller and smaller in the infinite-volume limit since it
probes the largest scales on which we do not expect correlations.
This is indeed strong evidence against spontaneous symmetry breaking
and fundamentally differs from what was observed in \cite{LPW20}.

Finally, we study discretization effects via a continuum extrapolation at roughly 
fixed physical volume in Fig.~\ref{fig:pd_continuum}. We find remnant structural order
for three strongly decreasing lattice spacings. However, due to subtleties in the
scale-setting procedure (see \cite{LMW21}) we refrain from drawing 
definite conclusions at this point and instead hope to revisit the continuum limit
in the future. In related models in higher dimensions the continuum limit turned out
to be crucial for removing misleading lattice artifacts \cite{BKW21}. We do emphasize,
however, that our results are consistent with the
persistence of remnant inhomogeneities as $a$ appproaches zero -- in accordance with
the observation that in lower dimensions the UV limit usually tends to be less problematic.

\begin{figure}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_63_infVol}
\end{subfigure}
\hspace{-2.3cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_127_infVol}
\end{subfigure}
\hspace{-2.8cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_255_infVol}
\end{subfigure}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_63_infVol}
\end{subfigure}
\hspace{.11cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_127_infVol}
\end{subfigure}
\hspace{-.4cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_255_infVol}
\end{subfigure}
\caption{From left to right: infinite-volume extrapolation of the $(\mu,T)$ phase 
	diagram at fixed lattice spacing; top row: using $C_{\mathrm{short}}$ 
	(Eq.~\eqref{eq:cshort}, taken from \cite{LMW21}) ; bottom row: using 
	$C_{\mathrm{long}}$ (Eq.~\eqref{eq:clong}).}
\label{fig:pd_infinite_volume}
\end{figure}

\begin{figure}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_63_cont}
\end{subfigure}
\hspace{-2.3cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_127_cont}
\end{subfigure}
\hspace{-2.8cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_2_255_cont}
\end{subfigure}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_63_cont}
\end{subfigure}
\hspace{.11cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_127_cont}
\end{subfigure}
\hspace{-.4cm}
\begin{subfigure}{.333\linewidth}
	\includegraphics[scale=0.33]{pd_alt_2_255_cont}
\end{subfigure}
\caption{From left to right: continuum extrapolation of the $(\mu,T)$ phase 
	diagram at roughly fixed physical volume; top row: using $C_{\mathrm{short}}$ 
	(Eq.~\eqref{eq:cshort}, taken from \cite{LMW21}); bottom row: using 
	$C_{\mathrm{long}}$ (Eq.~\eqref{eq:clong}).}
\label{fig:pd_continuum}
\end{figure}

\clearpage
