text:
\section*{Abstract}
{\bf

We reveal an intriguing anomaly in the temperature dependence of the specific heat of a one-dimensional Bose gas.~The observed peak holds for arbitrary interaction and remembers a superfluid-to-normal phase transition in higher dimensions, but phase transitions are not allowed in one dimension.~The presence of the anomaly signals a region of unpopulated states which behaves as an energy gap and is located below the hole branch in the excitation spectrum.~The anomaly temperature is found to be of the same order of the energy of the maximum of the hole branch.~We rely on the Bethe Ansatz to obtain the specific heat exactly and provide interpretations of the analytically tractable limits.~The dynamic structure factor is computed with the Path Integral Monte Carlo method for the first time.~We notice that at temperatures similar to the anomaly threshold, the energy of the thermal fluctuations become comparable with the maximal hole energy, leading to a qualitative change in the structure of excitations.~This excitation pattern experiences the breakdown of the
quasiparticle description for any value of the interaction strength at the
anomaly, similarly to any superfluid phase transition at the critical
temperature.~We provide indications for future observations and how the hole anomaly can be employed for in-situ thermometry, identifying different collisional regimes and understanding other anomalies in atomic, solid-state, electronic and spin-chain systems.
}

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\section{Introduction}
\label{Sec:intro}

In a number of classical and quantum many-body systems, the specific heat shows a non-monotonic behavior with temperature, often referred to as an {\em Anomaly} and which occurs for different reasons.~A first example is provided by the onset of a thermal second-order phase transition where the specific heat shows a sharp peak located at the critical temperature~\cite{Landau2013}.~This anomaly appears in very different transitions: normal/superconductor~\cite{Tinkham2004}, normal/superfluid in helium~\cite{Ceperley1995, Lipa2003} and in strongly-interacting ultracold atomic Fermi gases~\cite{Ku2012, Hou2013}. By definition, the specific heat provides information on the variation of the internal energy in the system due to a change of temperature. As a consequence, the temperature dependence of the specific heat and the anomaly can be then explained by a specific structure of the excitation energy spectrum of the system. In particular, the anomaly often reveals the presence of unpopulated states in the spectrum of very different kinds of systems.~This is the case of the two-level model with the energy gap $\Delta$ in which the specific heat experiences a peak whose value is of the order of $N k_B$, where $N$ is the number of atoms, and which is located at the anomaly temperature $k_B T_A \approx 0.4 \Delta$.~This effect is known as {\em Schottky Anomaly}~\cite{Tari2003}.~Similar phenomena can be observed in solid-state and lattice systems as long as the thermal energy is comparable with the gap and for which a small temperature increase induces a significant change in the specific heat.~At higher temperatures, the levels are instead evenly populated resulting in lower specific heat.~Schottky-like anomalies emerge in systems of very different nature:~Bose-Hubbard model \cite{Rizzatti2020}, metals \cite{Lucas2017}, crystals \cite{He2009}, compounds \cite{Raju1992}, quantum ferrimagnets \cite{Nakanishi2002} and in spin systems \cite{Harris1998, Brambleby2017, Jurcisinova2018} even simulating the anomaly in black holes \cite{Dinsmore2020}.~An external magnetic field changes $\Delta$ and then $T_A$, and the shape of the peak in the specific heat, as observed \cite{Adhikari2019}.

Another class of systems that exhibit an anomaly is restricted to one spatial dimensionality.~We focus here on the paradigmatic one-dimensional (1D) Bose gas with contact repulsive interactions and which exhibits a complicated continuous spectrum \cite{Lieb1963II}.~At low momenta, the linear phononic dispersion determines the low-temperature thermodynamics.~An excellent description for this low-energy regime is provided by the Luttinger Liquid theory which is valid for any interaction strength~\cite{DeRosi2017}.~As the temperature increases, higher momenta get explored and the deviation of the spectrum from the phononic behavior becomes relevant \cite{Meinert2015, Fabbri2015}, resulting in a continuous structure delimited by two branches of elementary excitations.~The upper Lieb I and lower Lieb II branches are associated with the particle and hole excitations, respectively.~In particular, Lieb II branch does not have any counterpart for bosonic ensembles in higher spatial dimensions.~The existence of a peak in the temperature dependence of the specific heat in a 1D Bose gas is known from the thermal Bethe-Ansatz solution \cite{Yang1969} which provides an exact numerical result for this integrable system although it does not give any physical explanation for the anomaly.~The anomaly cannot be either interpreted in terms of a thermal phase transition which does not occur because of the 1D geometry~\cite{Landau2013}.~In addition, the complicated structure of the spectrum has not permitted so far an easy interpretation of its effects on the corresponding behavior of thermodynamic quantities like the specific heat.~We will show that the presence of the anomaly is related to an important change in the structure of the excitations which is exactly described by the dynamic structure factor (DSF).~Previous studies on the DSF of a 1D Bose gas have relied on the Bethe-Ansatz method~\cite{Panfil2014} and interpreted the excitations in terms of particles and holes \cite{DeNardis2018}.~In the strongly-repulsive regime, the DSF has been obtained by using the Bose-Fermi mapping \cite{Cherny2006, Lang2015} and by performing a perturbative expansion on the inverse interaction strength \cite{Granet2020}.~However, most of the results were limited at temperatures lower than the anomaly value $T_A$ and do not provide then any insight into the peak of the specific heat.

In this work, we report the presence of a peak in the temperature-dependence of the specific heat at constant volume for any finite interaction strength of a 1D Bose gas.~We solve numerically the thermal Bethe-Ansatz equations within the Yang-Yang theory, which give an exact result of the specific heat at all temperatures $T$ and interaction strengths~\cite{Yang1969}.~Then, we investigate analytically different tractable regimes, with the use of several perturbative theories holding at low and high temperature, and weak and strong interactions.~We demonstrate that the peak can be interpreted in terms of a novel anomaly effect, never reported so far in ultracold gases, and which provides a fundamental insight into the importance of the intrinsic features of the complicated spectrum.~This work contains both a qualitative physical interpretation for the anomaly in terms of the presence of unpopulated states in the spectrum which simulate an energy gap as well as a quantitative explanation based on the behavior of the DSF in a wide range of temperatures.~Most importantly, both descriptions hold for any value of the interaction strength.~We show that the new kind of anomaly shares the thermodynamic features of the Schottky analogue as in both cases the maximum value of the specific heat is of the order of $N k_B$ and the anomaly temperature $T_A$ can be expressed as
\begin{equation}
\label{Eq:prop law}
k_B T_A\left(\gamma\right) \sim \Delta\left(\gamma\right)
\end{equation}
as being proportional and of the same order of the ``energy gap" $\Delta$. We have made explicit the dependence on the interaction strength $\gamma$ characterizing the new anomaly and which plays the role of the magnetic field in the Schottky analogue.~The origins of the two anomalies are related to the spectral properties and, in particular, to the presence of unpopulated states.~While in solid-state systems $\Delta$ is provided by the energy gap between the two lowest levels in the spectrum, in a 1D Bose gas we interpret $\Delta$ as the energy of the maximum of the Lieb II hole-like branch below which unpopulated states are present at zero temperature.~We refer to this phenomenon then as \textit{Hole Anomaly}.~The hole energy $\Delta$ behaves as an energy gap at the thermodynamic level, by inducing the anomaly in the temperature-dependence of the specific heat.~In order to quantify the structure of the excitations, we calculate the DSF for a wide range of temperatures and interaction strengths using the ab-initio Path Integral Monte Carlo numerical method.~Our results show that the anomaly temperature $T_A$ determines a critical threshold at which a significant change in the structure of the DSF occurs.~While for temperature below $T_A$ the description of excitations in terms of quasiparticles holds, at higher temperatures it fails, as it occurs in any superfluid phase transition around the critical temperature, which is indeed forbidden in 1D systems.~The breakdown of the quasiparticle description is due to the thermal shift of the DSF peak in frequency which is quantified to be of the order of the hole energy $\Delta$ around the anomaly temperature.~These results for the DSF are general as they are observed to be valid for several values of the interaction strength.

\section{Model}
\label{Sec:Model}
A 1D gas of $N$ bosons with contact repulsive interactions is described by the Hamiltonian
\begin{equation}
\label{Eq:H}
H = - \frac{\hbar^2}{2m}\sum_{i = 1}^N \frac{\partial^2}{\partial x_i^2} + g\sum_{i > j}^N\delta(x_i - x_j),
\end{equation}
where $m$ is the atom mass, $g = - 2 \hbar^2/(m a)>0$ is the 1D coupling constant, and $a<0$ is the 1D $s$-wave scattering length.~The dimensionless interaction strength $\gamma = - 2/(n a)$ depends on the gas parameter $n a$, with $n=N/L$ the linear density and $L$ the length of the system.~There is a continuous interaction crossover which encompasses different quantum degeneracy regimes.~In the Gross-Pitaevskii (GP) regime of weak repulsion $\gamma \ll 1$ and of high density $n |a| \gg 1$ the gas admits a mean-field description \cite{Pitaevskii2016}.~In the Tonks-Girardeau (TG) regime of very strong repulsion $\gamma \gg 1$ and low density $n |a| \ll 1$, bosons become impenetrable and the system wavefunction can be mapped onto that of an ideal Fermi gas (IFG), resulting in identical thermodynamic behavior \cite{Girardeau1960}.

The Lieb-Liniger model describes the system at zero temperature, where the ground-state energy $E_0$, chemical potential $\mu_0 = (\partial E_0/\partial N)_{a, L}$ and sound velocity \\ $v = \sqrt{n/m (\partial \mu_0/\partial n)_{a}}$ can be exactly calculated from the Bethe-Ansatz method as a function of $\gamma$~\cite{Lieb1963, Lieb1963II, Pitaevskii2016}. The sound velocity smoothly changes from the mean-field $v_{\rm GP} =\sqrt{g n/m}$ to the Fermi value $v_F = \hbar \pi n/m$ in the TG regime.

\section{Specific Heat}
\label{Sec:heat}

Within the canonical ensemble, the thermodynamics of a 1D Bose gas is captured by the Helmholtz free energy $A = E - TS$, with $E$ the internal energy and $S = - \left(\partial A / \partial T \right)_{a, N, L}$ the entropy. The chemical potential is defined as $\mu = \left(\partial A / \partial N \right)_{T, a, L}$ and the specific heat at constant volume (or length $L$, in 1D) is
\begin{equation}
\label{Eq:specific heat}
C = \left(  \partial E/ \partial T\right)_{a, N, L} .
\end{equation}

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\columnwidth,angle=0,clip=]{heat}
\caption{
\label{fig:heat}
Specific heat at constant length per particle vs temperature in Fermi units $\tau = T/T_F$ and in the thermodynamic limit. The symbols denote numerical thermal Bethe-Ansatz results for several interaction strengths $\gamma$. Lines correspond to the analytical theories. The ideal Bose gas (IBG), Hartree-Fock (HF) and ideal Fermi gas (IFG) descriptions are independent on the value of $\gamma$.
}
\end{center}
\end{figure}

Figure~\ref{fig:heat} shows with symbols the specific heat per particle as a function of the temperature $T$ and for characteristic values of the interaction strength $\gamma$, obtained from the thermal Bethe-Ansatz (TBA) equations.~In this figure, the temperature $\tau = T/T_F$ is rescaled by the Fermi value $T_F$ provided by the corresponding energy $E_F = k_B T_F = \hbar^2 \pi^2 n^2/(2 m)$.~The peak in the specific heat is more defined and located at higher temperature $T_A$ by approaching the fermionized TG regime of large $\gamma$.~We provide below the understanding of dominant effects in the regimes of the specific heat which may be treated analytically and which show an excellent agreement with TBA in Fig.~\ref{fig:heat}.

For weak interactions $\gamma \ll 1$, a reliable description is provided by Hartree-Fock (HF) theory, which yields the chemical potential $\mu_{\rm HF} = \mu_{\rm IBG} + 2 g n$ \cite{Pitaevskii2016}, where $\mu_{\rm IBG}$ is the corresponding value of the ideal Bose gas (IBG). The thermodynamic quantities depend then on the coupling constant $g$ only through the contribution at zero temperature, and the specific heat, Eq.~\eqref{Eq:specific heat}, is the same as that of the IBG.
The low-$T$ expansion of the equation of state in terms of the effective fugacity close to unity $\tilde{z} = e^{\beta \left( \mu_{\rm HF} - 2 g n \right)} \approx 1$ \cite{Gradshteyn2014} gives (see Appendix \ref{SubSec:Low-T HF} for the complete derivation)
\begin{equation}
\label{Eq:HF low T}
\frac{C_{\rm HF}}{N k_B} \approx \frac{3}{8} \frac{\zeta\left(3/2\right)}{\zeta\left(1/2\right)}M\left(\tau\right)\left[ 1- \frac{\pi M\left(\tau\right) D\left(\tau\right)}{\zeta\left(1/2\right)\zeta\left(3/2\right)}  \right]
\end{equation}
where $M = \sqrt{\pi \tau} \zeta\left(1/2 \right)$, $\zeta\left(x \right)$ is the Riemann zeta function and \\ $D = \left( 3 M^2 - 18 M + 32 \right) \left( M - 2  \right)^{-3}$. Equation~\eqref{Eq:HF low T} agrees with TBA for $T \lesssim T_d$ where $T_d = T_F/\pi^2$ is the quantum degeneracy temperature. At high temperatures $T \gg T_d$, the virial expansion for $\tilde{z} \ll 1$ provides (see Appendix \ref{SubSec:virial HF})
\begin{equation}
\label{Eq:HF virial}
\frac{C_{\rm HF}}{N k_B} = \frac{1}{2} \left[ 1 - \frac{n \lambda}{4 \sqrt{2}} - \frac{2\sqrt{3} - 5}{16 \sqrt{2}} \left( n \lambda  \right)^3 + O\left( n \lambda  \right)^5      \right]
\end{equation}
where $\lambda = \sqrt{2 \pi \hbar^2/\left(m k_B T  \right)}$ is the thermal wavelength.
Eq.~\eqref{Eq:HF virial} is an expansion for $n \lambda \ll 1$.

In the weakly-interacting regime at low temperatures $T/T_d \ll \sqrt{\gamma} \ll 1$~\cite{Kheruntsyan2003}, the gas behaves as a quasicondensate and its thermodynamics can be described by the Bogoliubov (BG) theory in terms of a gas of non-interacting bosonic quasi-particles~\cite{Pitaevskii2016, DeRosi2019}.~The specific heat per particle is (see Appendix \ref{Sec:BG})
\begin{equation}
\label{Eq:C BG}
\frac{C_{\rm BG}}{N k_B} = \frac{1}{n \left(k_B T\right)^2} \int_{-\infty}^{+\infty} \frac{dp}{2 \pi \hbar} \epsilon^2\left(p\right)\frac{e^{\beta \epsilon\left(p\right)}}{\left[e^{\beta \epsilon\left(p\right)} - 1 \right]^2} ,
\end{equation}
where $\beta = \left(k_B T  \right)^{-1}$ and $\epsilon(p) = \sqrt{p^2 v^2 + \left[p^2/(2 m)\right]^2}$ is the BG spectrum at zero temperature \cite{Lieb1963,Lieb1963II}, which depends on $\gamma$ through the sound velocity $v$.~Within the Luttinger Liquid (LL) theory of very low temperatures, one considers only the phononic contribution to the BG dispersion, $\epsilon(p) \approx v |p|$, and obtains the universal result $C_{\rm LL}/\left(N k_B\right) = \pi k_B T/ \left(3 n \hbar v\right)$ \cite{Jiang2015} which is valid for the whole interaction crossover~\cite{DeRosi2017}.~Our finding for the LL regime corrects a misprint in Ref.~\cite{Pitaevskii2016}.~The next-to-leading term in the low-$p$ expansion of the BG spectrum, $\epsilon(p) \approx v|p| \left[ 1 + p^2/(8 m^2 v^2)\right]$, allows to calculate the first correction beyond Luttinger Liquid \cite{DeRosi2019}
$ C_{\rm BG} = C_{\rm LL} [ 1 - 3 \pi^2  ( k_B T)^2/ (  10 m^2 v^4) + O(T^4)]  $ (see Appendix \ref{SubSec:BG low T}).

For $T/T_d \gg \max\left(1, \gamma^2\right)$, one enters into the decoherent classical (DC) regime where both phase and density fluctuations are large and the gas approaches the IBG behavior at high temperature \cite{Kheruntsyan2003}.~The specific heat per particle is (see Appendix \ref{Sec:DC})
\begin{equation}
\label{Eq:C DC}
\frac{C_{\rm DC}}{N k_B} \approx \frac{1}{2} - \left(1 + \frac{1}{2\sqrt{2}} \right) \frac{n \lambda}{4 \pi} - \frac{3\gamma^2}{32 \pi \sqrt{2}}  \left(n \lambda\right)^3 .
\end{equation}

For strong interactions $\gamma^2 \gtrsim \max\left(\pi^2, T/T_d\right)$, the thermodynamics can be understood via the hard-core (HC) model \cite{Motta2016, DeRosi2019}.~The specific heat is then obtained from that of an ideal Fermi gas, subtracting from the system size $L$ a ``negative excluded volume'' $N a$, where the diameter of the HC is equal to the scattering length $a < 0$: \\ $C_{\rm HC}(L) = C_{\rm IFG}(L \to \hat{L}\equiv L - N a)$.~Following the Sommerfeld expansion of the IFG specific heat \cite{Ashcroft1976} holding for $\tau \ll 1$ and reported in Appendix \ref{SubSec:Sommerfeld IFG}, we get the low-temperature expansion
\begin{equation}
\label{Eq:C HC Sommerfeld}
\frac{C_{\rm HC}}{N k_B} = \frac{\pi^2}{6} \hat{\tau} \left[     1 + \frac{2}{5} \pi^2 \hat{\tau}^2 + \frac{35}{36} \pi^4 \hat{\tau}^4 + O \left( \hat{\tau}^6   \right)\right] ,
\end{equation}
where $\hat{\tau} = k_B T/\hat{E}_F$.~The effective Fermi energy $\hat{E}_F = \hbar^2 \pi^2 \hat{n}^2/(2 m)$ depends on the rescaled density $\hat{n} = n/(1 - an)$ which considers the HC correction and it is valid for $n |a| \ll 1$.~At high temperature $\pi^2 < T/T_d \lesssim \gamma^2$, we apply the virial expansion to an IFG (see Appendix \ref{SubSec:virial IFG}), and we derive the high-$T$ behavior of the specific heat per particle:
\begin{equation}
\label{Eq:C HC virial}
\frac{C_{\rm HC}}{N k_B} = \frac{1}{2} \left[ 1 + \frac{\hat{n} \lambda}{4 \sqrt{2}} -\frac{2\sqrt{3}-5}{16 \sqrt{2}}\left(  \hat{n} \lambda \right)^3 + O\left(   \hat{n} \lambda\right)^4     \right] + B ,
\end{equation}
where the shift $B = - \left(2\sqrt{2} +1  \right)/\left[4 \sqrt{2 \pi}  \left(\gamma + 2\right) \right]$ corresponds to the $O\left(n \lambda\right)$-term of the DC regime, Eq.~\eqref{Eq:C DC}, calculated at the upper temperature bound of the HC approximation $T = \gamma^2 T_d$ and with the density replaced by its rescaled value $n \to \hat{n}$. The shift ensures the continuous crossover between the virial HC and the DC regimes~\cite{Kheruntsyan2003}. Differently from the virial expansions, Eqs.~\eqref{Eq:HF virial} and~\eqref{Eq:C HC virial}, where the thermal wavelength is much larger than the absolute value of the scattering length $\lambda \gg |a|$, in the DC regime, Eq.~\eqref{Eq:C DC}, $\lambda < |a|$. The TG regime ($\gamma = +\infty$) which has the same thermodynamic properties of an IFG, is then recovered from the HC model when $a = 0$, but it is not connected with the DC regime at high $T$. In fact, by approaching the TG limit, while the validity range of temperature of the virial HC theory gets broader $T/T_d < +\infty$ and $B \to 0$, the DC regime disappears as its condition $T/T_d \gg \gamma^2$ is not longer satisfied, see Fig.~\ref{fig:heat}.

\section{Hole Anomaly}
\label{Sec:Anomaly}

At a microscopic level, the underlying mechanism responsible for the appearance of the anomaly in the specific heat is represented with a sketch in Fig.~\ref{fig:sketch}.~It portrays the main features of the dynamic structure factor of a 1D Bose gas for temperature smaller and close to the anomaly threshold $T_A$.~At zero temperature, the DSF exhibits a continuous structure which is delimited by the Lieb I and II branches \cite{Imambekov2012} corresponding to the particle and hole excitation dispersion, respectively, for any interaction strength $\gamma$.~In the spectrum, there are then states which are not populated and some of them are located below the lower Lieb II hole branch.~The thermal fluctuations smear the borders of the DSF by an amount of energy equal to $k_B T$.~At low temperature $T < T_A$, the quasiparticle picture of the excitations is valid.~At small momenta, quasiparticles are provided by phonons with linear dispersion $\omega(k) = v|k|$ where $\omega$ is the frequency and $k$ is the wavenumber.~At high temperature $T > T_A$, it is no longer possible to identify a single dominant excitation and the quasiparticle picture breaks down.~The temperature $T_A$ where this occurs can be estimated by the value of the ``gap'' $\Delta$ at the inflection point of the hole branch located at the Fermi wavenumber $k_F = \pi n$, so that $ k_BT_A \sim \Delta$, see also Eq.~\eqref{Eq:prop law}.~The quantity $\Delta$ corresponds then to the energy of the maximum of the lower Lieb II branch.~Up to the same level of accuracy, the value of $\Delta$ can be approximated by the typical energy of phonons calculated at the inflection point $\Delta \propto v \hbar k_F$ \cite{Pitaevskii2016, Lang2017}, so that we obtain
\begin{equation}
\label{Eq:T_A}
k_B T_A \sim v \hbar  k_F ,
\end{equation}
which has been derived from a microscopic description.

\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth,angle=0,clip=]{key}
\caption{
\label{fig:sketch}
Sketch of the dynamic structure factor at a temperature below (left) and around (right) the value of the Hole Anomaly. Upper particle-like Lieb~I and lower hole-like Lieb~II branches are reported with solid curves.~Dashed line denotes the linear phononic spectrum $\omega(k) = v |k|$ where $\omega$ is the frequency, $v$ is the sound velocity and $k$ is the wavenumber.~The hole excitation responsible for the anomaly is located at the Fermi wavenumber $k_F$ and its energy is equal to $\Delta$.~The dynamic structure factor at zero temperature is reported with the blue shaded region with vertical lines.~Its thermal contribution is instead denoted with red shading.
}
\end{center}
\end{figure}

The spectrum at zero temperature can be calculated from the exact Bethe-Ansatz method~\cite{Lieb1963II}. For any value of the interaction strength $\gamma$, the relevant hole excitation for the anomaly is always located at the maximum of the Lieb II branch at $k_F$, while its energy value $\Delta$ changes in the interaction crossover.~We report in Fig.~\ref{fig:max} the comparison between the hole energy $\Delta$ and the temperature $T_A$ of the peak in the specific heat, to test the validity of the proposed microscopic anomaly mechanism.~The hole energy $\Delta$ has been calculated with Bethe Ansatz~\cite{Gaudin1971,Reichert2019II}.~We also show the energy scales at which the anomaly occurs for small and large values of $\gamma$. Such scales correspond to the upper-temperature bounds of the validity of the BG, Eq.~\eqref{Eq:C BG}, and Sommerfeld HC, Eq.~\eqref{Eq:C HC Sommerfeld}, theories, the latter of which has been shifted by the degeneracy energy $k_B T_d$.
The temperature ranges of the validity of the BG and Sommerfeld HC approaches are $T/T_d \ll \sqrt{\gamma}$ and $k_B T \ll \hat{E}_F $, respectively.
We finally report half of the phononic energy calculated at the Fermi wavenumber $v \hbar k_F/2$.~This provides a universal energy scale for the hole excitation $\Delta \approx v \hbar k_F/2$ for any $\gamma$ \cite{Lang2017}. Overall, a good agreement between $\Delta$ and $T_A$ is found (keeping in mind that their relation is through a coefficient of the order of unity that depends on $\gamma$) even if $\gamma$ is changed by more than four orders of magnitude. This fully supports the hole scenario for the anomaly, described by Eq.~\eqref{Eq:prop law}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\columnwidth,angle=0,clip=]{max}
\caption{
\label{fig:max}
Hole energy $\Delta$ and anomaly temperature $T_A$ in Fermi units vs interaction strength $\gamma$. Circles denote numerical Bethe-Ansatz results for $\Delta$. Triangles represent $T_A$ estimated from the maxima in the specific heat shown in Fig.~\ref{fig:heat}. Dot-dashed lines correspond to the upper-temperature bounds of the BG and Sommerfeld HC theories, for small and large values of $\gamma$ respectively.~Dashed line shows the phononic energy scale at the Fermi wavenumber $k_F$.
}
\end{center}
\end{figure}

This important result solves the open problem of relating the features of the microscopic continuous excitation spectrum with the thermodynamic behavior of a 1D Bose gas.~In fact, similarly to the Schottky anomaly in the two-level model, the new hole anomaly emerges from the presence of unpopulated states in the spectrum.~The close analogy to the Schottky anomaly is further reinforced if one associates the energy $\Delta$ of the maximum of the hole branch with the energy gap in the two-level system.~Indeed, both phenomena share a similar proportionality relation for the anomaly temperature $k_BT_A \sim \Delta$, see also Eq.~\eqref{Eq:prop law}, where $T_A$ and $\Delta$ are of the same order in both cases.~While in the Schottky anomaly all parameters appearing in the above proportionality relation can be changed by applying an external magnetic field, in a 1D Bose gas they all depend on the interaction strength $\gamma$ controlled by the magnetic Fano-Feshbach resonances in experiments  \cite{Meinert2015}. By approaching the strongly-correlated Tonks-Girardeau regime with very large interaction strength $\gamma$, $k_B T_A \approx \Delta$, Fig.~\ref{fig:max}.~The reduction of the discrepancy between the anomaly temperature $T_A$ and the hole energy $\Delta$ for strong interactions is explained with the presence of just one physical energy set by the Fermi value $E_F = k_B T_F = \hbar^2 \pi^2 n^2/(2 m)$ in the fermionized Tonks-Girardeau regime and which provides the common limit for $T_A$ and $\Delta$ for large $\gamma$. Finally, the monotonic increasing behavior of the ``energy gap'' $\Delta$ with $\gamma$ in Fig.~\ref{fig:max} determines the anomaly peak in the specific heat getting more defined and located at higher temperatures $T_A$ by increasing the interaction strength, see Fig.~\ref{fig:heat}.

\subsection{Chemical Potential}

We argue here that the anomaly is a universal property of all 1D atomic gases, even if they are described by a different Hamiltonian from the one we have considered, Eq.~\eqref{Eq:H}.~The non-monotonic dependences with temperature might be found not only in the specific heat but also in other interesting thermodynamic quantities like the chemical potential, see Fig.~\ref{fig:thermo}.~Indeed, at temperatures much larger than the interaction energy, one can approximately consider the gas as being ideal.~The chemical potential is negative and it is a decreasing function for both ideal Bose and Fermi gases in the limit of high temperature.~As it was demonstrated by some of the authors of the present paper in Ref.~\cite{DeRosi2017}, for a wide class of 1D systems with a gapless spectrum, the presence of the phononic excitations whose dispersion is linear $v |p|$ leads to a $T^2$-increase in the chemical potential in the Luttinger Liquid regime of low temperatures.~As the chemical potential is an increasing and a decreasing function at low and high temperatures, respectively, it must exhibit a maximum.~Within the Luttinger Liquid formalism, the Fermi energy $E_F = k_B T_F$ is a relevant scale for both fermionic and bosonic gases with the same density $n$ and atomic mass $m$.~The characteristic anomaly temperature at which the maximum of the chemical potential appears can be estimated by setting the thermal $T^2$-correction equal to the dominant contribution $\mu_0/E_F \approx (T/T_F)^2$, provided by the chemical potential at zero temperature and which can be expressed in terms of the sound velocity $\mu_0 \propto m v^2$ for any value of the interaction strength.~By combining the above approximate conditions one exactly recovers Eq.~\eqref{Eq:T_A}, representing the characteristic temperature of the anomaly in the specific heat. In fact, even if the values of the anomaly temperature $T_A$ are different in the chemical potential and in the specific heat, they are of the same order and both can be then well approximated by Eq.~\eqref{Eq:T_A} which has been obtained here from thermodynamic considerations rather than a microscopic description.

\begin{figure}[h!]
\begin{center}
\subfloat[]{\includegraphics[width=0.8\columnwidth,angle=0,clip=]{C2}} \\
\vspace{-1.5cm}
\subfloat[]{\includegraphics[width=0.8\columnwidth,angle=0,clip=]{mu}}
\caption{Specific heat at constant length per particle (upper panel) and chemical potential rescaled by its value at zero temperature $\mu_0$ (lower panel) vs temperature in Fermi units $\tau = T/T_F$. The symbols denote numerical thermal Bethe-Ansatz results for several interaction strengths $\gamma$. Solid line corresponds to the ideal Fermi gas (IFG) finding. Vertical lines represent the hole energy $\Delta$ and they are reported in an increasing order of $\gamma$
from low (left) to high (right) values.
}
\label{fig:thermo}
\end{center}
\end{figure}

For sake of clarity, Fig.~\ref{fig:thermo} shows with symbols the specific heat per particle and the chemical potential as a function of the temperature $T$ and for characteristic values of the interaction strength $\gamma$, obtained from the thermal Bethe-Ansatz equations.~Solid line represents the ideal Fermi gas result. Vertical lines denote the hole energy $\Delta$ for different values of $\gamma$. We notice that both the specific heat and the chemical potential exhibit a maximum for any value of the interaction strength $\gamma$. In addition, for both these thermodynamic quantities, the discrepancy between the anomaly temperature $T_A$, corresponding to the position of the maximum, and the hole energy $\Delta$ decreases by approaching the fermionized TG regime of large $\gamma$, as depicted also in Fig.~\ref{fig:max}.
The chemical potential as a function of temperature has been previously calculated with thermal Bethe-Ansatz method in the strongly-interacting regime \cite{Lang2015} and for different values of the interaction strength \cite{DeRosi2017, DeRosi2019}.

\section{Dynamic Structure Factor}
\label{Sec:DSF}

Quantitative information on the excitation spectrum of the system is provided by the dynamic structure factor $S(k,\omega)$ which describes the dynamic response with frequency $\omega$ of a quantum many-body system to a weak density perturbation with wavenumber $k$. It is defined by the Fourier transform of the real-time $t$ density-density correlation function
\begin{equation}
\label{Eq:DynamicStructureFactor}
S(k,\omega) = \frac{1}{N} \int_{-\infty}^{+\infty} \frac{dt}{2 \pi}  e^{i \omega t} \textrm{Tr} \left[n_T \rho_{-k}(t) \rho_k(0) \right],
\end{equation}
where $\rho_k\left(t\right) =  e^{i H t/\hbar} \rho_k e^{-i H t/\hbar}$ is the time evolution, following the Hamiltonian $H$ of Eq.~\eqref{Eq:H}, of the density perturbation operator $\rho_k  = \sum_{i = 1}^N e^{-ikx_i}$. The thermal density matrix $n_T = e^{-\beta H}/Z$, where $Z = \textrm{Tr} \left(e^{-\beta H}\right)$ is the partition function, allows for the calculation of the expectation value of any quantum observable $O$: $\langle O \rangle = \textrm{Tr} \left(n_T O\right)$.

\begin{figure}[h!]
	\begin{center}
		\includegraphics[width=0.8\columnwidth,angle=0,clip=]{spectrum_gamma1}
\caption{
\label{fig:spectrum_gamma1}
Dynamic structure factor for the interaction strength $\gamma = 1$. Solid line shows the energy of the hole branch at zero temperature calculated with exact Bethe Ansatz and its value for $k=k_F$ gives its maximum energy $\Delta$. Path Integral Monte Carlo numerical results are for:~i) the dynamic structure factor which is represented with the heatmap in units of the inverse of the Fermi frequency $\omega_F = E_F/\hbar$ and for different temperatures $\tau = T/T_F$;~ii) the single-mode frequency $\omega_{\rm SM}$ is denoted by dots whose sizes are larger than the error bars. Wavenumber $k$ is in units of the Fermi value $k_F$.}
\end{center}
\end{figure}

We have calculated the DSF for different temperatures $T$ and interaction strengths $\gamma$ by employing the Path Integral Monte Carlo (PIMC) technique, see Sec. \ref{Sec:PIMC} and Appendix \ref{Sec:DSF gamma 10}.~We consider here the results for the intermediate regime $\gamma = 1$, which cannot be handled with perturbative theories in Fig.~\ref{fig:heat}.~However, the behavior of the DSF described below is the same for different values of $\gamma$, see Appendix  \ref{Sec:DSF gamma 10}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\columnwidth,angle=0,clip=]{CrossoverT_gamma1}
\caption{
\label{fig:crossover_g1}
Path Integral Monte Carlo results of the dynamic structure factor vs frequency $\omega$ for two values of the wavenumber $k$ (left and right column) and interaction strength $\gamma = 1$. Each row corresponds to different temperatures $\tau = T/T_F$. Vertical line denotes the single-mode frequency $\omega_{\rm SM}$. Colors represent the regimes below ($\tau \leq 0.114$), around ($\tau \approx 0.173$) and above ($\tau \geq 0.347$) the anomaly.
}
\end{center}
\end{figure}

In Fig.~\ref{fig:spectrum_gamma1} we provide characteristic examples of the DSF at temperatures below, around and above the anomaly value $T_A$. For $T < T_A$, the DSF resembles its behavior at zero temperature~\cite{Panfil2014}.~At small wavenumber $k$, the thermal excitations stay in the Luttinger Liquid regime and the DSF exhibits a sharp peak located at a frequency $\omega$ which satisfies the linear phononic law $\omega(k) = v|k|$. As $k$ increases, the DSF structure gets broader, but it remains centered at a $\omega > 0$ value, while the $\omega < 0$ contributions are yet negligible. For all $k$ values, the peak of the DSF is then well approximated by the Feynman relation, for which the full excitation spectrum is described in terms of a coherent single-mode (SM) quasi-particle \cite{Feynman54}:
\begin{equation}
\label{Eq:Feynman}
\hbar \omega_{\rm SM}(k) = \hbar^2 k^2/\left[2 m S(k)\right] \ ,
\end{equation}
where $S(k) = \int_{-\infty}^{+\infty} d\omega S(k,\omega)$ is the static structure factor. In Fig.~\ref{fig:spectrum_gamma1}, we show for comparison the single-mode frequency $\omega_{\rm SM}$ obtained within the Feynman approximation based on PIMC results for $S(k)$. As the temperature $T$ is increased, the zero-temperature approximation no longer provides a reliable description for the excitations of the gas. At $T \approx T_A$, the DSF does not show anymore the phononic behavior at small $k$, but it exhibits instead a signal around $\omega = 0$. At large $k$, there is a broad distribution centered at $\omega_{\rm SM}$. For $T > T_A$, the DSF shows a very broad signal with the maximum centered at $\omega = 0$ for any value of $k$. At temperatures above the anomaly, the dynamics of the system is not then described by a coherent single excitation with a finite frequency.

We finally notice that the breakdown of the quasiparticle picture, Eq.~\eqref{Eq:Feynman}, at the anomaly temperature $T_A$ is due to the thermal shift in the values of the frequency of the structure of the DSF. By comparing the position of the peak (red color) of the DSF in the first and the central panel in Fig.~\ref{fig:spectrum_gamma1}, its shift is estimated to be of the order of the hole energy $\Delta$ at $T \approx T_A$.
The same observation may be made for any value of the interaction strength, see also Fig.~\ref{fig:Spectrum_g10}. The exact results of the DSF based on the ab-initio PIMC method fully support then the validity of Eq.~\eqref{Eq:prop law} reported in Fig.~\ref{fig:max} and the intuitive explanation of the hole anomaly mechanism that we provided in Fig.~\ref{fig:sketch} and based on the presence of the region below the hole branch in the spectrum which effectively simulates the behavior of an energy gap at the thermodynamic level.

To quantify the DSF in the temperature crossover, in Fig.~\ref{fig:crossover_g1} we report $S(k,\omega)$ as a function of frequency $\omega$ and temperature $T$ and for fixed values of the wavenumber $k$. This crossover is explained in terms of the interplay between quantum correlations (i.e. interaction effects) and thermal fluctuations in the dynamics of the excitations of the system. At temperatures below that of the anomaly $T < T_A$, quantum correlations are more important than thermal fluctuations which are then treated as a small perturbation. Above the anomaly threshold $T>T_A$, the DSF is characterized by a broad incoherent component and thermal fluctuations dominate over quantum effects. In the non-trivial intermediate regime around the anomaly $T \approx T_A$, where simple analytical theories cannot be applied, our results show the coexistence of quantum correlations and thermal fluctuations whose contributions are comparable.

It is worth noticing that the superfluid-to-normal phase transition in Bose systems, in two and three spatial dimensions, shows the same trend in the DSF when crossing the critical temperature $T_c$: the quasi-particle collective excitation turns to a broad thermal response for $T > T_c$. Therefore the present 1D case, showing also a finite peak in the specific heat, resembles the critical behavior of Bose systems at higher dimensionalities.

\section{Path Integral Monte Carlo Method}
\label{Sec:PIMC}

The PIMC method relies on the description of an ensemble of $N$ quantum atoms in terms of a set of $N$ classical polymers, each of them reproducing the quantum delocalization of one particle~\cite{Ceperley1995}. In this way, the thermal average $\langle O \rangle$ of the quantum observable $O$ is expressed as a multidimensional integral which can be efficiently computed with a Monte Carlo algorithm. The PIMC makes use of the convolution property of the propagator which admits an analytical approximation only for small imaginary times. We calculate such approximation by choosing a pair-product scheme which is based on the exact solution of the two-body problem for the Hamiltonian in Eq.~\eqref{Eq:H} \cite{Gaveau1986,Yan2015}.~In order to recover the Bose statistics of the indistinguishable quantum particles, we sample stochastically the permutations among the atoms with the worm algorithm~\cite{Boninsegni2006}.~PIMC results have been carefully checked by benchmarking the expectation values for the energy per particle and the isothermal compressibility against the exact TBA calculation in Appendix \ref{Sec:benchmarkPIMC}.

The PIMC method is exact for the calculation of the static properties like the energy~\cite{Ceperley1995}, but it only allows for an indirect estimation of the dynamic properties, such as the DSF $S(k,\omega)$, Eq.~\eqref{Eq:DynamicStructureFactor}. In order to compute $S(k,\omega)$, one has to recover the correlation function in the imaginary time  $\varepsilon$:~$F(k,\varepsilon) =  \textrm{Tr}\left[n_T \rho_{-k}(-i\varepsilon) \rho_k(0) \right]/N$ which is related to the DSF via a Laplace transform $F(k,\varepsilon) = \int_{-\infty}^{+\infty} d\omega S(k,\omega) \left[e^{-\hbar\omega \varepsilon} + e^{-\hbar\omega \left(\beta-\varepsilon\right)}\right]$. The inversion of the Laplace transform is a mathematically ill-conditioned problem and make unfeasible a precise reconstruction of the DSF from PIMC data of the $F(k,\varepsilon)$, which are unavoidably affected by statistical uncertainties. Yet, several numerical approaches have been presented in literature to tackle this problem \cite{Boninsegni1996, Sandvik1998, Mishchenko2000, Reichman2009, Vitali2010, Roggero2013, Ferre2016}. These methods are able to recover the main features of the DSF, like the frequencies and the spectral weight of the main peaks, and have provided insightful results in the study of several quantum many-body systems \cite{Boninsegni1998, Rossi2012, Saccani2012, Rota2013, Bertaina2016}. We reconstruct the DSF by employing a simulated annealing procedure which minimizes the $\chi^2$-deviation between the expectation value of $F(k,\varepsilon)$, obtained from a guess of $S(k,\omega)$, and the PIMC data \cite{MacKeow1997,Ferre2016}.

\section{Conclusions}

In this work, we provide a detailed description of the new hole anomaly in 1D
Bose gases evidenced by a non-monotonous behavior in thermodynamic quantities
such as the chemical potential and specific heat as a function of
temperature.~We argue that the presence of the anomaly is due to the region of
unpopulated states located below the lower hole branch in the excitation
spectrum at zero temperature and which behaves as an energy gap from the
thermodynamic point of view.~We show that the anomaly temperature $T_A$ and the
energy $\Delta$ of the maximum of the hole branch are functions of the
interaction strength $\gamma$ and they are proportional and of the same order:
$k_B T_A\left(\gamma\right) \sim \Delta\left(\gamma\right)$.~Another excellent
energy scale for the hole anomaly is provided by phonons at the Fermi momentum,
which ensures the high control over both $T_A$ and $\Delta$ by finely tuning
$\gamma$ through the sound velocity.~We provide an additional simple
characterization of the anomaly in terms of a qualitative change in the
structure of the excitations, governed by a single quasiparticle mode at low
temperatures which gets suppressed by thermal fluctuations at higher
temperatures.
The breakdown of the quasiparticle picture is due to the thermal shift in
frequency
of the structure of the excitations which has been shown to be of the order of
the hole energy $\Delta$ around the anomaly temperature $T_A$.~Our description
for the hole anomaly is valid for any value of the interaction strength $\gamma$
and solves the open problem of relating the effects of the complicated
continuous spectrum to the thermodynamic behavior of a 1D Bose gas.~The anomaly
is a reminiscence of a phase transition, that is not allowed in 1D systems, and
signals a change of quantum regime.

The exact results of the specific heat as a function of temperature and interaction strength $\gamma$ were obtained with the Bethe-Ansatz method and the main features were described analytically.~Beyond the validity of different analytical limits, novel quantum regimes emerge.~We computed the dynamic structure factor with the ab-initio Path Integral Monte Carlo technique. We carefully characterized then the behavior of the dynamic response with temperature, by showing that quantum correlation and thermal fluctuation contributions are comparable in the anomaly regime for any $\gamma$. The Path Integral Monte Carlo method has been applied to a 1D Bose gas with contact interactions for the first time in our work. Our calculations extend to a wide range of temperatures compared to previous studies which were restricted only to very low temperature.

We show that an anomaly is always present in the chemical potential of any 1D
atomic gas
of both bosonic and fermionic nature. A similar proportionality
relation $k_B T_A\left(\gamma\right) \sim \Delta\left(\gamma\right)$ applies to
the Schottky anomaly in the two-level model where the dependence on the
interaction strength $\gamma$ is replaced by the magnetic field and $\Delta$ has
to be interpreted as the energy gap.~The two-level model is a low-temperature
approximation of any discrete spectrum which is found in many different
solid-state systems where the dependence of the Schottky anomaly on the magnetic
field has been indeed observed.~A hole-like anomaly, whose position and
structure change with an external magnetic field, has been also detected in the
temperature-dependence of the specific heat in 1D spin chains \cite{Dender1997,
Hammar1999}.~The continuous excitation spectrum of this system, which exhibits
a twofold particle-hole nature similar to the case of the 1D Bose gas, has
been experimentally measured  showing an excellent agreement with the exact
calculation based on the Bethe-Ansatz \cite{Lake2013, Mourigal2013}.~A similar
anomaly may be also found in 1D electronic systems with  particle-hole
spectrum \cite{Barak2010}.
A  tantalizing possibility is that
the hole anomaly could be employed as a quantum simulator as it provides an
in-depth understanding of diverse anomalies in other more complicated many-body
systems.~The new anomaly shares typical properties of a quantum
simulator as its future observation is feasible and it can be achieved
in clean experimental atomic settings where precise control and broad
tunability of the interaction strength $\gamma$ are possible.~In addition, the
applicability of several exact methods to the 1D Bose gas provides fundamental
insights into the problem.~This concept of quantum simulation
\cite{Altman2021} at the thermodynamic level  aims at
the understanding of strongly-correlated systems, the development of innovative
materials, and the emergence of new quantum technologies.

At the experimental level, the temperature-dependence of the specific heat
has been observed in a three-dimensional strongly-interacting Fermi gas, where
the detection of the peak allowed for a precise measurement of the critical
temperature of the superfluid phase transition \cite{Ku2012}. The chemical
potential as a function of temperature and interaction strength $\gamma$ has
been measured in a 1D Bose gas,  resulting in an excellent agreement with
thermal Bethe-Ansatz solution \cite{Salces-Carcoba2018}. The employed
experimental technique in both measurements is the in-situ absorption imaging
whose signal-to-noise ratio has been recently enhanced \cite{Hans2021}. In 1D,
three-body losses are strongly reduced \cite{Tolra2004}, and the spatial uniform
density can be achieved with a flatbox potential \cite{Rauer2018}. All these
premises make the observation of the novel hole anomaly in a 1D Bose gas
particularly appealing for current experimental settings
\cite{Salces-Carcoba2018}. At fixed interaction strength $\gamma$, the detection
of the anomaly may be employed as a precise in-situ temperature probe. The
$\gamma$-dependence of the anomaly can be observed by applying Fano-Feshbach
resonances \cite{Chin2010, Meinert2015}. The dynamic structure factor of a 1D
Bose gas has been probed using Bragg spectroscopy \cite{Meinert2015,
Fabbri2015}, but at temperatures smaller than the anomaly threshold.

Looking forward, in harmonically trapped 1D Bose gases, the specific heat
determines the time-dependence of the temperature in hydrodynamic breathing
modes \cite{DeRosi2015}.~Hole anomaly may play then a key role in the
hydrodynamic-collisionless transition which has been predicted with the
evolution of breathing modes due to an increase of temperature
\cite{DeRosi2016}. This intriguing idea may explain the mismatch between the
predictions and the measurements of the breathing mode frequencies through the
crossover between the regimes of the quasicondensate and the ideal Bose gas
\cite{Fang2014}. The HC description, which is valid close to the
strongly-correlated TG regime where the anomaly effect is enhanced, is expected
to hold even in very different systems at low density. The anomaly should be
well visible then even for positive scattering length $a > 0$ in:~i) short-range
interacting systems like the metastable super Tonks-Girardeau gas
\cite{Astrakharchik2005, Haller2009}, ii) finite-range interacting ensembles of
dipolar \cite{Arkhipov2005, Girardeau2012} and Rydberg atoms
\cite{Osychenko2011}, 1D bosonic $^4$He (liquid) \cite{Bertaina2016} and 1D
fermionic $^3$He (gas) \cite{Astrakharchik2014}. Our findings can also be
extended to 1D liquids in bosonic mixtures \cite{Petrov2016, Parisi2019} at
finite temperature \cite{DeRosi2021}.

\section*{Acknowledgements}
The authors gratefully acknowledge G. Lang and J.-S. Caux for the insightful suggestions.

\paragraph{Author contributions}

G. D. R., G. E. A. and J. B. devised the initial concepts and theory. Analytical results were derived by G. D. R. Monte Carlo numerical simulations were implemented by R. R. Bethe-Ansatz exact calculations were performed by G. E. A. The manuscript was written by G. D. R. with suggestions from G. E. A., R. R. and J. B. The project was supervised by J. B. and  G. E. A.

\paragraph{Funding information}

G. D. R.'s received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk\l odowska-Curie grant agreement {\it UltraLiquid} No. 797684 and with the grant IJC2020-043542-I funded by \\ MCIN/AEI/10.13039/501100011033 and by "European Union NextGenerationEU/PRTR". G. D. R., G. E. A. and J. B. were partially supported by grant PID2020-113565GB-C21 funded by MCIN/AEI/10.13039/501100011033, the Spanish MINECO (FIS2017-84114-C2-1-P), and the
Secretaria d'Universitats i Recerca del Departament d'Empresa i Coneixement de la Generalitat de Catalunya
 within the ERDF Operational Program of Catalunya (project QuantumCat, Ref. 001-P-001644).

\paragraph{Data and materials availability}
Data corresponding to the ideal Bose and Fermi gas in Fig. \ref{fig:heat} may be found available online at \href{https://upcommons.upc.edu/handle/2117/353128}{https://upcommons.upc.edu/handle/2117/353128}. All the other data needed to evaluate the conclusions are present in the main text and/or the
Appendices. Additional data related to this work
may be requested to the authors (giulia.de.rosi@upc.edu).

\begin{appendix}

\section{Main Quantities}
From the Helmholtz free energy $A = E - T S$, where $E$ is the internal energy and $T$ the temperature, we calculate the entropy
\begin{equation}
\label{Eq:S}
S = - \left(\partial A/\partial T  \right)_{a, N, L} ,
\end{equation}
and the pressure
\begin{equation}
\label{Eq:P}
P = - \left(\partial A/\partial L \right)_{T, a, N} = n \left(\mu - A/N \right) ,
\end{equation}
where $\mu$ is the chemical potential.
The Tan's contact parameter \cite{DeRosi2019} is defined in systems with zero-range interactions
\begin{equation}
\label{Eq:Tan}
\mathcal{C} = \frac{4 m}{\hbar^2}\left( \frac{\partial A}{\partial a}  \right)_{T, N, L} = \frac{4 n N}{a^2} g_2\left(0\right),
\end{equation}
and it is proportional to the normalized pair correlation function \cite{Pitaevskii2016} at zero relative distances $x = 0$
\begin{equation}
\label{Eq:g2 def}
g_2 \left(x = x_1 - x_2 \right) = \frac{\langle  \hat{\psi}^+\left(x_2\right)   \hat{\psi}^+\left(x_1\right) \hat{\psi}\left(x_1\right)\hat{\psi}\left(x_2\right) \rangle}{n^2}
\end{equation}
where $\hat{\psi}\left(x \right)$ is the field operator.

\section{Hartree-Fock theory for a weakly-interacting Bose gas}
\label{Sec:HF}

In this Appendix, we provide details about the low-and the high-temperature expansions within the Hartree-Fock approximation.

The equation of state for a 1D weakly-interacting Bose gas with density $n$ and pressure $P$ can be obtained from
 \begin{equation}
 \label{Eq:EOS HF}
 \begin{cases}
 n \lambda = g_{1/2}(\tilde{z}) \\
 P \lambda = g n^2 \lambda + k_B T g_{3/2}(\tilde{z}),
 \end{cases}
  \end{equation}
where $\tilde{z} = e^{\beta(\mu_{\rm HF} - 2 g n)}$ is the effective fugacity within the Hartree-Fock (HF) theory \cite{Pitaevskii2016}, $\lambda = \sqrt{2\pi\hbar^2/(m k_B T)}$ is the thermal wavelength and the Bose functions are
\begin{equation}
\label{Eq:gBose}
g_\nu\left( z  \right) = \frac{1}{\Gamma\left(\nu \right)} \int_{0}^{+\infty} dx \frac{x^{\nu - 1}}{z^{-1} e^x - 1} ,
\end{equation}
where $\Gamma\left(\nu\right)$ is the Euler gamma function.

\subsection{Low-temperature expansion}
\label{SubSec:Low-T HF}

The Bose functions can be approximated for $\tilde{z} = e^{-\alpha} \approx 1$, with small and positive $\alpha$ \cite{Gradshteyn2014}
\begin{equation}
\label{Eq:g z 1}
g_\nu(e^{-\alpha}) = \Gamma(1 - \nu)\alpha^{\nu - 1} + \sum_{i = 0}^{+\infty} (-1)^i \frac{\zeta(\nu - i)}{i!} \alpha^{i}
\end{equation}
where $\zeta(x)$ is the Riemann zeta function.

By inverting the expression for $n$ in Eq.~\eqref{Eq:EOS HF} where we employ Eq.~\eqref{Eq:g z 1}, we obtain the following approximation for the effective fugacity
\begin{equation}
\label{Eq:z IBG low T}
\tilde{z} \approx e^{-\pi \left[ \frac{\sqrt{\pi \tau}}{\sqrt{\pi \tau}\zeta\left(1/2\right) - 2}\right]^{2}} ,
\end{equation}
where $\tau = k_B T/E_F$, and $E_F = k_B T_F = \hbar^2 \pi^2 n^2/\left( 2 m \right)$ is the Fermi energy.
By making use of the definition of $\tilde{z}$ in Eq.~\eqref{Eq:z IBG low T} and by considering only the real solution, we find the low-temperature behavior of the chemical potential:
\begin{equation}
\label{Eq:mu HF low T}
\mu_{\rm HF} \approx 2 g n - E_F \left[\frac{\pi \tau}{\sqrt{\pi \tau} \zeta\left(1/2\right) - 2}    \right]^2 .
\end{equation}
By combining Eqs.~\eqref{Eq:g z 1} - \eqref{Eq:z IBG low T} in the equation of the pressure $P$, given by Eq.~\eqref{Eq:EOS HF}, we obtain
\begin{equation}
\label{Eq:P HF low T}
P_{\rm HF} \approx g n^2 + \frac{\sqrt{\pi}}{2} n E_F \zeta\left(3/2\right)\tau^{3/2} \left\{1 - \frac{\pi^{3/2} \sqrt{\tau} }{\zeta\left(3/2\right)}\frac{\left[3 \sqrt{\pi \tau}   \zeta\left(1/2\right) - 4  \right]}{\left[ \sqrt{\pi \tau} \zeta\left(1/2\right) - 2\right]^2} \right\}.
\end{equation}
From Eqs.~\eqref{Eq:P} and \eqref{Eq:mu HF low T}-\eqref{Eq:P HF low T}, we calculate the low-temperature expansion of the free energy per particle
\begin{equation}
\label{Eq:A HF low T}
\frac{A_{\rm HF}}{N} \approx g n - \frac{\sqrt{\pi}}{2} E_F \zeta\left(3/2\right) \tau^{3/2} \left\{1 - \frac{3 \pi^{3/2} \sqrt{\tau}}{\zeta\left(3/2\right) \left[ \sqrt{\pi \tau} \zeta\left(1/2\right) - 2 \right]}        \right\}
\end{equation}
from which we get the corresponding entropy per particle, Eq.~\eqref{Eq:S}
\begin{equation}
\label{Eq:S HF low T}
\frac{S_{\rm HF}}{N k_B} \approx \frac{3 }{4}\zeta\left( 3/2\right)\sqrt{\pi \tau} \left\{1 - \frac{\pi^{3/2} \sqrt{\tau}}{\zeta\left(3/2\right)} \frac{\left[3\sqrt{\pi \tau} \zeta\left(1/2\right) - 8   \right]}{\left[\sqrt{\pi \tau} \zeta\left(1/2\right) - 2  \right]^2} \right\} ,
\end{equation}
and the energy per particle
\begin{equation}
\label{Eq:E HF low T}
\frac{E_{\rm HF}}{N} \approx g n + \frac{\sqrt{\pi}}{4} E_F \zeta(3/2)\tau^{3/2}\left\{1 - \frac{3 \pi^{3/2} \sqrt{\tau}}{\zeta(3/2)} \frac{\left[   \sqrt{\pi \tau} \zeta(1/2) - 4 \right]}{\left[  \sqrt{\pi \tau} \zeta(1/2) - 2 \right]^2} \right\} .
\end{equation}
We notice that in the HF approximation, all the thermodynamic quantities depend on the interactions only through their contribution at zero temperature. The Tan's contact per particle, Eq.~\eqref{Eq:Tan}, is then independent on temperature:
\begin{equation}
\label{Eq:Tan HF low T}
\frac{\mathcal{C_{\rm HF}}}{N} = 2 n^3 \gamma^2
\end{equation}
and we obtain the pair correlation function of an ideal Bose gas $g_2\left(0\right)_{\rm HF} = 2$.

\subsection{High-temperature virial expansion}
\label{SubSec:virial HF}

The Bose functions, Eq.~\eqref{Eq:gBose}, admit the series representation in terms of small effective fugacity $\tilde{z} \ll 1$: $g_\nu (\tilde{z}) = \sum _{i = 1}^{+\infty} \tilde{z}^i/i^\nu$.
By inverting the expression for $n$ in Eq.~\eqref{Eq:EOS HF} and by expanding for $n \lambda \ll 1$, we get
\begin{equation}
\label{Eq:virial z HF}
\tilde{z}= n \lambda - \frac{(n \lambda)^2}{\sqrt{2}}  + \frac{\sqrt{3} - 1}{\sqrt{3}} (n \lambda)^3 +O[(n\lambda)^4].
\end{equation}
By making use of the definition of $\tilde{z}$ in Eq.~\eqref{Eq:virial z HF} and an expansion for $n \lambda \ll 1$, we obtain the virial expansion of the chemical potential:
\begin{equation}
\label{Eq:virial mu BG full}
\mu_{\rm HF} = k_B T \Bigl[\ln\left(n \lambda \right) - \frac{n \lambda}{\sqrt{2}} + \frac{3\sqrt{3} - 4}{4\sqrt{3}} \left(n \lambda \right)^2  - \frac{2\sqrt{3} - 5}{6\sqrt{2}} \left(n \lambda  \right)^3 +O[(n\lambda)^4]\Bigr] + 2 g n.
\end{equation}
From the equation of $P$, Eq.~\eqref{Eq:EOS HF}, we derive the high-temperature behavior of the pressure:
\begin{equation}
\label{Eq:virial P BG full}
P_{\rm HF} = n k_B T \Bigl[ 1 - \frac{n \lambda}{2\sqrt{2}} + \frac{3\sqrt{3} - 4}{6\sqrt{3}}  \left( n \lambda \right)^2- \frac{5}{4}\frac{\left( 2\sqrt{3} -5 \right)}{6\sqrt{2}} \left( n \lambda \right)^3+O[(n\lambda)^4]\Bigr] + g n^2 .
\end{equation}
The expansion of the free energy per particle is
\begin{equation}
\label{Eq:A virial HF}
\frac{A_{\rm HF}}{N} =  k_B T  \Bigl[\ln\left(n \lambda \right) - 1 - \frac{n \lambda}{2\sqrt{2}} + \frac{3\sqrt{3} - 4}{4\sqrt{3}} \frac{\left(   n \lambda\right)^2}{3} + \frac{2\sqrt{3} - 5}{6\sqrt{2}} \frac{\left(n \lambda\right)^3}{4}
 +O[(n\lambda)^4] \Bigr] + g n ,
\end{equation}
which has been previously derived at the order $O[(n\lambda)^2]$ \cite{DeRosi2019}.
The entropy and the energy per particle are, respectively
\begin{equation}
\label{Eq:S virial HF}
\frac{S_{\rm HF}}{N k_B} = - \left[  \ln\left( n \lambda  \right) - \frac{3}{2} - \frac{n \lambda}{4\sqrt{2}}  - \frac{2\sqrt{3} - 5}{6\sqrt{2}} \frac{\left(  n \lambda\right)^3}{8} +O[(n\lambda)^5]  \right],
\end{equation}
\begin{equation}
\label{Eq:E virial HF}
\frac{E_{\rm HF}}{N} = \frac{1}{2}k_B T \left[1 - \frac{n \lambda}{2\sqrt{2}} + \frac{3\sqrt{3} - 4}{6\sqrt{3}} \left(n \lambda\right)^2 + \frac{2\sqrt{3} - 5}{8\sqrt{2}} \left(n \lambda\right)^3 +O[(n\lambda)^4]  \right] + g n.
\end{equation}

\section{Bogoliubov theory for a weakly-interacting Bose gas at low temperature}
\label{Sec:BG}

We report here the calculation of the low-temperature expansion of the Bogoliubov theory.

The thermodynamics of a quasicondensate can be described in terms of a gas of noninteracting bosonic quasiparticles \cite{Pitaevskii2016, DeRosi2019}, by applying the Bogoliubov (BG) theory, whose free energy per particle is
\begin{equation}
\label{Eq:A BG}
\frac{A_{\rm BG}}{N} = \frac{E_0}{N} + \frac{k_BT}{n}\int_{-\infty}^{+\infty}\frac{dp}{2\pi \hbar}  \ln \left[1 - e^{-\beta \epsilon(p)} \right],
\end{equation}
where $E_0$ is the ground-state energy obtained within the Lieb-Liniger theory at zero temperature \cite{Lieb1963}, $\epsilon(p) = \sqrt{p^2 v^2 + \left[p^2/(2 m)\right]^2}$ is the $T = 0$ BG spectrum \cite{Lieb1963,Lieb1963II} and $\beta = \left(k_B T  \right)^{-1}$.~From Eq.~\eqref{Eq:A BG}, we can calculate the complete thermodynamics of the system, such as the entropy per particle, Eq.~\eqref{Eq:S}
\begin{equation}
\label{Eq:S BG}
\frac{S_{\rm BG}}{N k_B} =\frac{1}{n} \int_{-\infty}^{+\infty} \frac{dp}{2 \pi \hbar} \left[ \frac{\beta \epsilon(p)}{e^{\beta \epsilon(p)} - 1} - \ln\left(1 - e^{-\beta \epsilon(p)}   	\right)\right] \ ,
\end{equation}
and the energy per particle
\begin{equation}
\label{Eq:E BG}
\frac{E_{\rm BG}}{N} = \frac{E_0}{N} + \frac{1}{n} \int_{-\infty}^{+\infty} \frac{dp}{2 \pi \hbar} \frac{\epsilon(p)}{e^{\beta \epsilon(p)} - 1} .
\end{equation}
The chemical potential, the pressure and the Tan's contact parameter have been derived in Ref.~\cite{DeRosi2019}.

\subsection{Low-temperature expansion from non-linear Bogoliubov spectrum}
\label{SubSec:BG low T}
The low-momentum expansion of the Bogoliubov dispersion relation \\ $x = \epsilon(p) = v|p| \left[ 1 + p^2/(8 m^2 v^2)\right]> 0$ can be inverted to get the real and positive solution $p$. Hence, for the free energy per particle, Eq.~\eqref{Eq:A BG}, we obtain the integral:
\begin{equation}
\label{Eq:integral A BG small p}
\int_{0}^{+\infty} dx
\frac{1}{v \left[1 + \frac{3 p^2(x)}{8\left(m v \right)^2}\right]} \ln\left(1 - e^{-\frac{x}{k_B T}} \right) \approx - \frac{\pi^2}{6}\frac{\left(k_B T \right)}{v} + \frac{\pi^4}{120 } \frac{(k_B T)^3}{m^2 v^5},
\end{equation}
where the analytic result may be found expanding the integrand for $|p| \ll m v$, which is valid at low temperatures.~We get the low-$T$ behavior of the free energy per particle, within the Bogoliubov theory \cite{DeRosi2019}:
\begin{equation}
\label{Eq:A BG small p}
\frac{A_{\rm BG}}{N} = \frac{E_0}{N} - \frac{\pi}{6} \frac{\left(k_B T\right)^2}{\hbar n v} \left[   1 - \frac{\pi^2}{20} \frac{\left(k_B T  \right)^2}{m^2 v^4} + O(T^4)\right],
\end{equation}
from which we calculate the entropy and the energy per particle
\begin{equation}
\label{Eq:S BLL BG}
\frac{S_{\rm BG}}{N k_B} = \frac{\pi}{3} \frac{k_B T }{\hbar n v} \left[1 - \frac{\pi^2}{10} \frac{\left( k_B T  \right)^2}{m^2 v^4} + O\left(   T^4\right)   \right]\ ,
\end{equation}
\begin{equation}
\label{Eq:E BLL BG}
\frac{E_{\rm BG}}{N} = \frac{E_0}{N} + \frac{\pi}{6} \frac{\left( k_B T \right)^2}{\hbar n v}  \left[  1 - \frac{3\pi^2 }{20} \frac{\left( k_B T \right)^2}{m^2 v^4} + O\left(T^4  \right) \right] \ .
\end{equation}
The Tan's contact per particle, Eq.~\eqref{Eq:Tan}, is \cite{DeRosi2019}
\begin{equation}
\label{Eq:Tan BG}
\frac{\mathcal{C}_{\rm BG}}{N} = \frac{\mathcal{C}_{0}}{N} +  \frac{\bar{\mathcal{C}}}{N} \frac{\left(k_B T\right)^2}{m^2 v^4} \left[ 1 - \frac{\pi^2}{4} \frac{\left(k_B T\right)^2}{m^2 v^4} + O\left(T^4  \right) \right]
\end{equation}
where the factor $\bar{\mathcal{C}} = \pi m^3 v^2 N \gamma^2(\partial v/\partial \gamma)_n /(3 \hbar^3)$ depends on the sound velocity $v = \sqrt{g n/m}$ and the value at zero temperature $\mathcal{C}_0/N = n^3 \gamma^2$ has to be compared with that of the HF theory, Eq.~\eqref{Eq:Tan HF low T}.~The pair correlation function approaches the unity corresponding to the value of the coherent regime:
\begin{equation}
\label{Eq:g2 BG}
g_2\left(0\right)_{\rm BG} = 1 + \frac{\pi^5}{24}\frac{\tau^2}{\gamma^{3/2}}\left[1 - \frac{\pi^6}{16} \frac{\tau^2}{\gamma^2} + O\left(\tau^4  \right)  \right] ,
\end{equation}
where $\tau = T/T_F$.~Our finding, Eq.~\eqref{Eq:g2 BG}, agrees with the result at the mean-field level of Ref.~\cite{Kheruntsyan2003}, and we derived the additional $O\left(T^4\right)$ correction.

\section{Decoherent Classical regime}
\label{Sec:DC}

We derive in the following the thermodynamic quantities in the decoherent classical regime.

The pair correlation function in the decoherent classical (DC) regime is close to the HF value $g_2\left(0\right)_{\rm HF} = 2$ \cite{Kheruntsyan2003}
\begin{equation}
\label{Eq:g2 DC}
g_2\left(0\right)_{\rm DC} \approx g_2\left(0\right)_{\rm HF} - \frac{\gamma}{\sqrt{2}}n \lambda
\end{equation}
from which we calculate the Tan's contact per particle, Eq.~\eqref{Eq:Tan}
\begin{equation}
\label{Eq:Tan DC}
\frac{\mathcal{C}_{\rm DC}}{N} \approx \frac{\mathcal{C}_{\rm HF}}{N}  \left(1 - \frac{\gamma}{2 \sqrt{2}} n \lambda\right)
\end{equation}
where we have used Eq.~\eqref{Eq:Tan HF low T}.
By integrating the above equation with $\lambda < |a|$ and $a < 0$, we find the free energy per particle
\begin{multline}
\label{Eq:A DC}
\frac{A_{\rm DC}}{N} = k_B T \left[\ln\left(n \lambda\right) - 1  \right] + \frac{1}{N} \int_a^\lambda da \frac{\partial A_{\rm DC}}{\partial a} \\ \approx k_B T \left[\ln\left(n \lambda\right) - 1  \right] - \frac{\hbar^2 n^2 \gamma}{m} \left[ 1 + \left( 2 + \frac{1}{\sqrt{2}} \right)\frac{1}{\gamma n \lambda} - \frac{\gamma}{4 \sqrt{2}} n \lambda        \right]
\end{multline}
where the constant of the integration has been chosen equal to the leading classical gas value whose prefactor is provided by the thermal energy $k_B T$.
The entropy, Eq.~\eqref{Eq:S}, and the energy per particle are, respectively
\begin{equation}
\label{Eq:S DC}
\frac{S_{\rm DC}}{N k_B} \approx - \left[ \ln\left(n \lambda \right) -   \frac{3}{2}  \right] + \left(1 + \frac{1}{2\sqrt{2}}    \right) \frac{n \lambda}{2 \pi} + \frac{\gamma^2}{16 \pi\sqrt{2}} \left( n \lambda  \right)^3  ,
\end{equation}
\begin{equation}
\label{Eq:E DC}
\frac{E_{\rm DC}}{N} \approx \frac{1}{2}k_B T  \left[ 1 -  \left( 1 + \frac{1}{2\sqrt{2}}   \right) \frac{n \lambda}{\pi} \right] -   \frac{\hbar^2 n^2  \gamma}{m} \left( 1- \frac{3 \gamma}{8 \sqrt{2}} n \lambda \right) .
\end{equation}

\section{Ideal Fermi gas}
\label{Sec:IFG}

In this Appendix, we provide details about the derivation of the Sommerfeld and the virial expansions of an ideal Fermi gas.

The equation of state of a 1D ideal Fermi gas (IFG) with density $n$ and pressure $P$ can be found from
\begin{equation}
 \label{Eq:EOS IFG}
 \begin{cases}
 n \lambda = f_{1/2}(z) \ , \\
 P \lambda = k_B T f_{3/2}(z),
\end{cases}
  \end{equation}
where we have defined the fugacity $z = e^{\mu/(k_B T)}$ and the Fermi functions
\begin{equation}
\label{Eq:fFermi}
f_\nu\left( z  \right) = \frac{1}{\Gamma\left(\nu \right)} \int_{0}^{+\infty} dx \frac{x^{\nu - 1}}{z^{-1} e^x + 1} ,
\end{equation}
where $\Gamma(\nu)$ is the Euler Gamma function.

  \subsection{Low-temperature Sommerfeld expansion}
\label{SubSec:Sommerfeld IFG}

The Sommerfeld expansion \cite{Ashcroft1976} allows for the calculation of integrals of the form:
\begin{equation}
\label{Eq:Sommerfeld}
\int_{0}^{+\infty} d\epsilon H(\epsilon)f(\epsilon) = \int_{0}^\mu d\epsilon H(\epsilon) + \sum_{i = 1}^{+\infty} a_i (k_B T)^{2i} \frac{d^{2i-1}}{d\epsilon^{2i-1}} H(\epsilon)|_{\epsilon = \mu},
\end{equation}
where
\begin{equation}
 f(\epsilon) = \frac{1}{e^{\frac{\epsilon - \mu}{k_B T}} + 1}
 \end{equation}
  is the Fermi-Dirac distribution.~We consider the 1D density of states of the IFG:
\begin{equation}
\label{Eq:Hepsilon}
 H(\epsilon) = \frac{1}{2\sqrt{E_F \epsilon}} ,
\end{equation}
where $E_F = k_B T_F = \hbar^2 \pi^2 n^2/\left(2 m\right)$ is the Fermi energy. In Eq.~\eqref{Eq:Sommerfeld}, we have introduced
\begin{equation}
 a_i = \left(2 - \frac{1}{2^{2(i - 1)}}  \right) \zeta(2i) ,
\end{equation}
where $\zeta\left(i\right)$ is the Riemann zeta function.

At very low temperature, the chemical potential of the IFG approaches the Fermi energy and we set then $\mu \to E_F (1+ \delta)$ with $0\leq\delta \ll 1$. We take into account the Sommerfeld expansion, Eq.~\eqref{Eq:Sommerfeld}, up to the $O(\delta^3)$-order, corresponding to the integer $i = 3$, and we impose the normalization condition $\int_{0}^{+\infty} d\epsilon H(\epsilon) f(\epsilon) = 1$. We solve the resulting equation for the real solution $\delta$ and we expand in series, finding the chemical potential
\begin{equation}
\label{Eq:mu IFG Sommerfeld}
\mu_{\rm IFG} = E_F \left( 1 + \frac{\pi^2}{12} \tau^2 + \frac{\pi^4}{36} \tau^4 + \frac{7\pi^6}{144} \tau^6 + O(\tau^8)   \right) ,
\end{equation}
with $\tau = k_B T/E_F$.

The Sommerfeld expansion, Eq.~\eqref{Eq:Sommerfeld}, enables one to get the low-temperature behavior $k_B T \ll \mu$ of the Fermi functions
$f_\nu\left(e^{\frac{\mu}{k_B T}}\right) \approx \frac{1}{\nu \Gamma(\nu)} \left(\frac{\mu}{k_B T}\right)^\nu \left[ 1 + \frac{\pi^2}{6}  \nu\left(\nu - 1 \right)\left( \frac{k_B T}{\mu} \right)^2   \right]$.
By using the latter expression in Eq.~\eqref{Eq:EOS IFG}, one recovers Eq.~\eqref{Eq:mu IFG Sommerfeld}, and obtains the pressure:
\begin{equation}
\label{Eq:P IFG Sommerfeld}
P_{\rm IFG} = \frac{2}{3} n E_F \left[1 + \frac{\pi^2}{4} \tau^2 + \frac{\pi^4}{20} \tau^4 + \frac{35 \pi^6}{432}  \tau^6 + O(\tau^8) \right].
\end{equation}
From Eqs.~\eqref{Eq:mu IFG Sommerfeld}-\eqref{Eq:P IFG Sommerfeld} and \eqref{Eq:P}, we calculate the low-temperature expansion of the free energy per particle:
\begin{equation}
\label{Eq:A Sommerfeld IFG}
\frac{A_{\rm IFG}}{N}= \frac{E_F}{3} \left(1 - \frac{\pi^2}{4}\tau^2 - \frac{\pi^4}{60} \tau^4 - \frac{7 \pi^6}{432} \tau^6 + O(\tau^8)   \right) ,
\end{equation}
the entropy per particle, Eq.~\eqref{Eq:S}
\begin{equation}
\label{Eq:S Sommerfeld IFG}
\frac{S_{\rm IFG}}{N k_B} = \frac{\pi^2}{6} \tau \left( 1 + \frac{2 \pi^2}{15} \tau^2 + \frac{7 \pi^4}{36} \tau^4 + O\left (\tau^6  \right)\right) ,
\end{equation}
the energy per particle
\begin{equation}
\label{Eq:E Sommerfeld IFG}
\frac{E_{\rm IFG}}{N} = \frac{E_F}{3} \left( 1 + \frac{ \pi^2}{4} \tau^2 + \frac{\pi^4}{20} \tau^4 + \frac{35\pi^6}{432}\tau^6 + O\left (\tau^8  \right)\right) ,
\end{equation}
and the specific heat per particle, Eq.~\eqref{Eq:specific heat}, which corresponds to Eq.~\eqref{Eq:C HC Sommerfeld} with zero scattering length $a = 0$.

The Tan's contact per particle, Eq.~\eqref{Eq:Tan}, is \cite{DeRosi2019}:
\begin{equation}
\label{Eq:Tan IFG}
\frac{\mathcal{C}_{\rm IFG}}{N} = \frac{4 m}{\hbar^2} P_{\rm IFG}
\end{equation}
where $P_{\rm IFG}$ is given by Eq.~\eqref{Eq:P IFG Sommerfeld}.
The pair correlation function approaches zero in the IFG limit $\gamma \to \infty$:
\begin{equation}
\label{Eq:g2 IFG}
g_2\left(0\right)_{\rm IFG} = \frac{4}{3} \frac{\pi^2}{\gamma^2} \left[1 + \frac{\pi^2}{4} \tau^2 + \frac{\pi^4}{20} \tau^4 + \frac{35 \pi^6}{432}  \tau^6 + O(\tau^8) \right]
\end{equation}
and it agrees with the finding of Ref.~\cite{Kheruntsyan2003} at the order $O\left(\tau^2\right)$, but we have derived higher thermal corrections.

\subsection{High-temperature virial expansion}
\label{SubSec:virial IFG}

The Fermi functions, Eq.~\eqref{Eq:fFermi}, can be approximated as $  f_\nu(z) = \sum_{i = 1}^{+\infty} (-1)^{i-1} z^i/i^\nu$ for small fugacity $z \ll 1$.
By inverting the equation for the density $n$, Eq.~\eqref{Eq:EOS IFG}, and by expanding for $n \lambda \ll 1$, we calculate
\begin{equation}
\label{Eq:virial z IFG}
 z(n \lambda) = n \lambda + \frac{(n \lambda)^2 }{\sqrt{2}} +  \frac{\sqrt{3} - 1}{\sqrt{3}} (n\lambda)^3 + O[(n \lambda)^4],
\end{equation}
from which, employing the definition of $z$ and an additional expansion for $n \lambda \ll 1$, we derive the chemical potential:
\begin{equation}
\label{Eq:virial mu IFG}
\mu_{\rm IFG}  = k_B T \left[\ln(n \lambda) + \frac{n \lambda}{\sqrt{2}}  + \frac{3\sqrt{3} - 4}{4\sqrt{3}}(n\lambda)^2 + \frac{2 \sqrt{3} - 5}{6\sqrt{2}}(n\lambda)^3 + O[(n \lambda)^4] \right].
\end{equation}
By considering Eq.~\eqref{Eq:virial z IFG} in the equation of $P$, Eq.~\eqref{Eq:EOS IFG}, we find the high-temperature behavior of the pressure
\begin{equation}
\label{Eq:virial P IFG}
P_{\rm IFG} = n k_B T \left[ 1 + \frac{n \lambda}{2\sqrt{2}} + \frac{3\sqrt{3} - 4}{6\sqrt{3}}\left( n \lambda \right)^2 + \frac{2\sqrt{3} - 5}{8\sqrt{2}} \left(n \lambda \right)^3  + O[(n \lambda)^4] \right],
\end{equation}
and the virial expansion of the free energy per particle:
\begin{equation}
\label{Eq:virial A IFG}
\frac{A_{\rm IFG}}{N} =  k_B T \left[\ln\left(n \lambda\right) - 1 + \frac{n \lambda}{2 \sqrt{2}} + \frac{3\sqrt{3} - 4}{6\sqrt{3}}\frac{(n \lambda)^2}{2} + \frac{2\sqrt{3} - 5}{6\sqrt{2}}\frac{(n \lambda)^3 }{4}  + O[(n \lambda)^4]  \right].
\end{equation}
The entropy per particle is:
\begin{equation}
\label{Eq:S virial IFG}
\frac{S_{\rm IFG}}{N k_B} = - \left[\ln \left(n \lambda  \right)  - \frac{3}{2} + \frac{n  \lambda}{4\sqrt{2}} - \frac{2 \sqrt{3} - 5}{6 \sqrt{2}} \frac{\left(  n \lambda \right)^3}{8} + O\left( n \lambda\right)^5\right]
\end{equation}
and the energy per particle  becomes
\begin{equation}
\label{Eq:E virial HC}
\frac{E_{\rm IFG}}{N} = \frac{k_B T}{2}
\left[ 1 +\frac{n \lambda}{2 \sqrt{2}} + \frac{3\sqrt{3} - 4}{6\sqrt{3}} \left(  n\lambda \right)^2 + \frac{2\sqrt{3} - 5}{8\sqrt{2}}\left(  n \lambda \right)^3  + O\left( n \lambda\right)^4\right]
\end{equation}
from which we derive the virial expansion of the specific heat which corresponds to Eq.~\eqref{Eq:C HC virial} with $a = 0$.
The Tan's contact is given by Eq.~\eqref{Eq:Tan IFG} where the pressure is now provided by Eq.~\eqref{Eq:virial P IFG}. The pair correlation function, Eq.~\eqref{Eq:Tan}, is
\begin{equation}
\label{Eq:g2 IFG virial}
g_2\left(0\right)_{\rm IFG} = \frac{2 \pi^2}{\gamma^2} \frac{P_{\rm IFG}}{n E_F}
\end{equation}
whose leading term recovers the result of Ref.~\cite{Kheruntsyan2003}.

\section{Benchmark of the Path Integral Monte Carlo Results}
\label{Sec:benchmarkPIMC}

We present here a benchmark for the expectation values of the internal energy $E$ and the isothermal compressibility of a 1D Bose gas at finite temperature which have been calculated with numerical Path Integral Monte Carlo (PIMC) method.~Such benchmark is based on the comparison with exact Thermal Bethe-Ansatz (TBA) findings.

PIMC technique provides exact results for the static properties at finite temperature \cite{Ceperley1995}.~For a Bose system, the expectation value of an observable $O$ is expressed as a multidimensional integral which is computed via Monte-Carlo sampling of the coordinates $R$:
\begin{equation}
\label{Eq:average}
\langle O \rangle =  \textrm{Tr} \left(n_T O \right) = \frac{1}{Z N!} \sum_\mathcal{P} \int dR \ G(R,\mathcal{P} R;\beta) O(R) .
\end{equation}
In Eq.~\eqref{Eq:average} we have introduced the thermal density matrix $n_T = e^{-\beta H}/Z$ where $Z = \textrm{Tr} \left(e^{-\beta H}\right)$ is the partition function, $\beta = \left(k_B T  \right)^{-1}$ is the inverse temperature and $H$ is the Hamiltonian, Eq.~\eqref{Eq:H}.~In the last equality, we have considered the coordinate representation $G(R_1,R_2;\beta) = \langle R_2 | e^{-\beta H} | R_1\rangle$ and $O(R)= \langle R|O|R\rangle$ where $R_i = \{x_{1,i}, x_{2,i} \ldots, x_{N,i} \}$ is a set of the coordinates of the $N$ atoms of the system.~$G(R_1,R_2;\beta)$ is the Green function  propagator describing the evolution in the imaginary time $\beta$ from the initial $R_1$ to the final $R_2$ configuration.~The configuration $\mathcal{P} R$ appearing in Eq.~\eqref{Eq:average} is obtained by applying a permutation $\mathcal{P}$ of the particle labels to the initial configuration $R$ and the sum over the $N!$ permutations allows to take into account the quantum statistics of the identical bosonic atoms.

The key aspect of the Path Integral formalism is the convolution property of the propagator:
\begin{equation}
\label{Eq:convolution}
G(R_1,R_3;\beta_1+\beta_2) = \int \, dR_2 G(R_1,R_2;\beta_1) G(R_2,R_3;\beta_2) ,
\end{equation}
which can be easily generalized to a series of intermediate steps $R_2 \ldots R_M$ defining a path with $M$ configurations and with total time $\beta = \varepsilon M$, where $\varepsilon$ is the time step.~For a finite value of $M$, the path is discretized with time.~In the opposite case of very large $M$, the path becomes continuous and
$\varepsilon$ approaches zero, corresponding to the limit of high temperatures $T$.~In the latter classical limit, the propagator $G$ admits an analytical approximation where the quantum effects of the non-commutativity between the kinetic and interaction potential operators in the Hamiltonian $H$ are neglected.~The thermal expectation value, Eq.~\eqref{Eq:average}, can be then approximated as
\begin{equation}
\label{Eq:QuantumAverage}
\langle O \rangle \simeq \frac{1}{Z N!} \sum_\mathcal{P} \int  dR_1 \ldots dR_M O(R_1) \prod_{i = 1}^M G(R_i,R_{i+1};\varepsilon)
\end{equation}
where we require the boundary condition $R_{M+1} = \mathcal{P} R_1$.~The quantity $p(R_1,\ldots,R_M) =  \prod_{i=1}^M G(R_i,R_{i+1};\varepsilon)$ is a probability distribution as it is positive definite and its integral over the space of configurations is equal to the unity.~The PIMC approach allows for the evaluation of the integral in Eq.~\eqref{Eq:QuantumAverage} with a stochastic sampling of the $N \times M$ degrees of freedom according to $p(R_1,\ldots,R_M)$.~Eq.~\eqref{Eq:QuantumAverage} becomes exact in the limit $M \to \infty$ where the imaginary time $\varepsilon$ is small and the analytical approximation for the propagator $G$ is accurate, by allowing for an exact calculation of the thermal average $\langle O \rangle$ with PIMC method.~For this reason, it is important to optimize the number of the convolution terms $M$ in Eq.~\eqref{Eq:QuantumAverage} with a proper benchmark of the PIMC results.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.8\columnwidth]{CompareEnergy.pdf}
\caption{Energy per particle $E/N$ vs temperature for different values of the interaction strength $\gamma$.~Symbols denote the Path Integral Monte Carlo (PIMC) results.~Lines correspond to the Thermal Bethe-Ansatz (TBA) findings.~In the Inset, we report the ratio of the PIMC vs TBA results.~Energy and temperature $\tau = T/T_F$ are normalized to the corresponding Fermi values defined by $E_F = k_B T_F$. }
\label{fig:CompareEnergy}
\end{center}
\end{figure}

We have calculated the energy per particle $E/N$ with $E = \langle H \rangle$ as a function of temperature and for different values of the interaction strength $\gamma$.~The comparison of such PIMC results with the corresponding TBA ones, Fig.~\ref{fig:CompareEnergy}, provides the estimate of the optimal value for $M$. The excellent PIMC-TBA agreement witnesses the reliability of our numerical findings.

\begin{figure}[h!]
\begin{center}
	\includegraphics[width=0.8\columnwidth]{CompareChi0.pdf}
\caption{Isothermal compressibility $\chi(0)$ vs temperature for different values of the interaction strength $\gamma$.~Symbols denote PIMC results.~Lines correspond to the TBA findings.~In the Inset, we report the ratio of the PIMC vs TBA results.
Isothermal compressibility and temperature $\tau = T/T_F$ are normalized to the corresponding Fermi values $E_F = k_B T_F$.
}
\label{fig:CompareChi0}
\end{center}
\end{figure}

In order to test the accuracy of the PIMC results for the imaginary-time correlation function $F(k,\varepsilon)$, from which we recover the dynamic structure factor, we have done a similar study for the isothermal compressibility
\begin{equation}
\label{Eq:compressibility}
\chi(0) = \left( \frac{\partial n}{\partial P}  \right)_{T, a, N} \ ,
\end{equation}
where the pressure $P$ is defined by Eq.~\eqref{Eq:P}.~The isothermal compressibility corresponds to the zero-wavenumber limit of the static density response function $\chi(k)$ \cite{Pitaevskii2016}, which is related to $F(k,\varepsilon)$:
\begin{equation}
\label{Eq:chiPIMC}
\chi(k) =  \int_0^{\beta} d\varepsilon\,  F(k,\varepsilon) \ .
\end{equation}
We calculate $\chi(k)$ with PIMC procedure from Eq.~\eqref{Eq:chiPIMC} and we extrapolate $\chi(0)$ from the behavior at smallest $k$.~In Fig.~\ref{fig:CompareChi0}, we compare the PIMC $\chi(0)$ results with the TBA isothermal compressibility evaluated from Eq.~\eqref{Eq:compressibility} as a function of temperature and for different values of the interaction strength $\gamma$.

\section{Dynamic Structure Factor for different interaction strengths}
\label{Sec:DSF gamma 10}

\begin{figure}[h!]
\begin{center}
	\includegraphics[width=0.8\columnwidth]{Spectrum_gamma10.pdf}
\caption{
Dynamic structure factor for the interaction strength $\gamma = 10$.
Solid line shows the energy of Lieb II branch at zero temperature calculated with exact Bethe-Ansatz and its value for $k=k_F$ gives its maximum energy $\Delta$. Path Integral Monte Carlo numerical results are for:~i) the dynamic structure factor which is represented with the heatmap in units of the inverse of the Fermi frequency $\omega_F = E_F/\hbar$ and for different temperatures $\tau = T/T_F$;~ii) the single-mode frequency $\omega_{\rm SM}$ is denoted by dots whose sizes are larger than the error bars. Wavenumber $k$ is in units of the Fermi value $k_F$.
}
\label{fig:Spectrum_g10}
\end{center}
\end{figure}

\begin{figure}[h!]
\begin{center}
	\includegraphics[width=0.8\textwidth]{CrossoverT_gamma10.pdf}
\caption{
Path Integral Monte Carlo results of the dynamic structure factor vs frequency $\omega$ for two values of the wavenumber $k$ (left and right column) and interaction strength $\gamma = 10$. Each row corresponds to different temperatures $\tau = T/T_F$. Vertical line denotes the single-mode frequency $\omega_{\rm SM}$. Colors represent the regimes below ($\tau \leq 0.4$), around ($\tau \approx 0.699$) and above ($\tau \geq 1.062$)  the anomaly.
}
\label{fig:Crossover_g10}
\end{center}
\end{figure}

We report here our PIMC results for the dynamic structure factor (DSF) $S(k, \omega)$ for the interaction strength value $\gamma = 10$.
For strong interactions, the DSF distribution gets broader in frequency $\omega$ \cite{Panfil2014}, see Figs.~\ref{fig:Spectrum_g10}-\ref{fig:Crossover_g10} which have to be compared with the corresponding ones for $\gamma = 1$ in Figs.~\ref{fig:spectrum_gamma1}-\ref{fig:crossover_g1}.

\end{appendix}

\nolinenumbers

\bibliographystyle{SciPost_bibstyle}

redpajama_set_name: RedPajamaArXiv


text:
Tag Archives: Salmon Arm Silverbacks
Sampson signs on with Marian University
Salmon Arm Silverbacks forward Jack Sampson has announced his commitment to the Marian University Sabres for the 2019-20 season. The [...]
Hirose nets first NHL goal
Fresh off signing an entry-level deal after his junior season at Michigan State University, former Salmon Arm Silverbacks standout Taro [...]
3 alums named Hobey finalists
Three BCHL graduates have been announced as top-10 finalists for the 2019 Hobey Baker Award, the NCAA announced today. Two Salmon Arm Si [...]
Hirose turns pro with Red Wings
The latest, but certainly not least, of the burst of NCAA free agent signings this week was that of Salmon Arm Silverbacks alum and [...]
Interior Division playoff preview
No. 1 Penticton Vees vs. No. 8 Cowichan Valley Capitals Regular season head-to-head: (2 games) The Vees won both matchups in regulat [...]
Wilson signs on with AIC
Salmon Arm Silverbacks forward Justin Wilson has played his entire hockey career in the province of B.C. up until this point. The BCHL v [...]
WJC crucible paying off for Verboon
Tested in the highest-profile Junior hockey tournament in the world, Matthew Verboon has emerged on the other side a more confident, and [...]
Silverbacks vs. Rivermen
Tonight - Salmon Arm Silverbacks (17-14-4-0) at Langley Rivermen (16-18-1-0) 7 p.m. at George Preston Arena in Langley Presented by th [...]
Verboon signs on with Colgate
After playing his youth hockey in his home country of Switzerland, including international duty at last year's U-18 World Junior Champio [...]

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: distanceTo method gives no result, returns always 0.0km I'm trying to get the distance between two locations,
but i always get the result of 0.0km.
i have tried 2 versions, but both of them give me the result of 0.0 km.

*

*Version with LatLng

*Version with location
This version is with the location
private fun onLocationchanged(location:Location):Float {

        if(mStartLocation == null)
        {
            mStartLocation  = mCurrentLocation
            mEndLocation = mCurrentLocation
        }
        else {
            mEndLocation = mCurrentLocation
        }
        mCurrentLocation = location;
        val distance = mStartLocation.distanceTo(mEndLocation)
        tvDistanzKm.text = distance.toString()
        return distance
        //Testing...No Difference
        val currentLocation = mStartLocation.latitude
        tvStart.text = currentLocation.toString()
        val endLocation = mEndLocation.latitude
        tvEnd.text = endLocation.toString()
    }

This version is with LatLang
fun getLaufDistanz():Float {

if(mLatLngStart == null) {
    mLatLngStart = mLatLngCurrent
    mLatLngEnde = mLatLngCurrent
}else{
    mLatLngEnde = mLatLngCurrent
}
val laufStartPunkt = Location("")
laufStartPunkt.latitude = mLatLngCurrent.latitude
laufStartPunkt.longitude = mLatLngCurrent.longitude

val laufEndPunkt = Location("")
laufEndPunkt.latitude = mLatLngCurrent.latitude
laufEndPunkt.longitude = mLatLngCurrent.longitude

val laufDistanzInMeter = laufStartPunkt.distanceTo(laufEndPunkt)
tvDistanzKm.text = laufDistanzInMeter.toString()
return laufDistanzInMeter

}
This is my callback
      //Result is always 0.0km
                    val distance = mStartLocation.distanceTo(mEndLocation)
                    tvDistanzKm.text = distance.toString()

                    //Testing...No Difference
                  //  mStartLocation == mEndLocation ???????
                    val currentLocation = mStartLocation.latitude
                    tvStart.text = currentLocation.toString()
                    val endLocation = mEndLocation.latitude
                    tvEnd.text = endLocation.toString()

                    //Testing the latitude/longitude in TextView
                    tvLatitude.text = location.latitude.toString()
                    tvLongitude.text = location.longitude.toString()

                    if (location != null) {
                          if (location.hasSpeed()) {
                            val getkmh = (location.speed * 3.6)
                            tvTempo.text = getkmh.toString()
                            tvSpeed.text = location.speed.toString()
                        } else {
                            tvTempo.text = "0.00"
                            tvSpeed.text = "0.00"
                        }
                        if (location.hasAccuracy()) {
                            tvAccuracy.text = location.accuracy.toString()
                        } else {
                            tvAccuracy.text = "schlechte Ortung"
                        }
                        if (location.hasAltitude()) {
                            tvAltitude.text = location.altitude.toString()
                        } else {
                            tvAltitude.text = "Nix Höhe"
                            Toast.makeText(
                                applicationContext,
                                "keine Höhe erkannt",
                                Toast.LENGTH_SHORT
                            )
                                .show()
                        }
                         startLocationUpdates()
                        updatePolyline()
                        buttons()
                        onLocationchanged(location)
                        getLaufDistanz()
                    }

i dont know where the problem is, so, would you please help me.
thank you.
i've tried it in different ways, but always with no result.
i found no solution online , also not on stackoverflow.
i'm new to Kotlin.
thank you

A: You're setting both mStartLocation and mEndLocation to the same object so the distance should be 0.0 between them. See your logic here:
    if(mStartLocation == null)
    {
        mStartLocation  = mCurrentLocation
        mEndLocation = mCurrentLocation
    }
    else {
        mEndLocation = mCurrentLocation
    }
    //after above lines, both mEndLocation and mStartLocation point at the same object
    mCurrentLocation = location; // this will have no effect on next line
    val distance = mStartLocation.distanceTo(mEndLocation) //doesn't use "location"

In your code above the most recent location passed into this function isn't even used as part of the formula. I think it should be as simple as this:
var previousLocation: Location? = null
var totalDistance = 0f

private fun onLocationchanged(location:Location): Float {
    val distance = previousLocation?.distanceTo(location) ?: 0f
    previousLocation = location
    totalDistance += distance
    tvDistanzKm.text = totalDistance.toString()
    return totalDistance
}

A: Location to Km or M -
private fun distanceLoc(location1: Location, location2: Location): String {
        val theta = location1.longitude - location2.longitude
        var dist = (sin(degToRad(location1.latitude))
                * sin(degToRad(location2.latitude))
                * cos(degToRad(location1.latitude))
                * cos(degToRad(location2.latitude))
                * cos(degToRad(theta)))
        dist = acos(dist)
        dist = radToDeg(dist)
        dist *= 60.0 * 1.1515
        var distStr = "$dist Km."
        if (dist < 0) {
            dist *= 1000
            distStr = "$dist Meter"
        }
        return distStr
    }

    private fun radToDeg(rad: Double): Double {
        return rad * 180 / Math.PI
    }

    private fun degToRad(deg: Double): Double {
        return deg * Math.PI / 180.0
    }

redpajama_set_name: RedPajamaStackExchange


Training (or "learning") and development is a command and individual responsibility. Success in this area is fundamental to success in our lives, our work and our public service as Companions. Training and development create new knowledge, skills, abilities and confidence.
Knowing what benefits we want from learning helps motivate us to design our own training plans and programs. One of the biggest benefits is continuous learning--whether or not one is in a formal training program.
We're most familiar with formal and other-directed forms of learning and development that include a strong focus on systematic structure. However, the most common forms of learning and development are informal and self-directed.
Chapter Action Plan (CAP) "How To"


Defending champion Roger Federer today ruthlessly ended Tim Henman's ever-fading Wimbledon dreams for another year.
Federer simply outclassed a shell-shocked Henman, cruising to a 6-4 6-0 6-2 victory in just one hour 24 minutes on Centre Court.
The British number three won just five points in the second set as he bowed out in the second round for the second year in succession.
It was Federer's 43rd successive win on grass and surely nothing will stop him from claiming a fourth straight title next week, joining Bjorn Borg and Pete Sampras as the only players to do so in the open era.
Henman had been looking forward to playing a match at the All England Club without the usual massive burden of expectation, and was rated a 10-1 shot for victory despite a 6-4 winning record against Federer.
That included a quarter-final victory here in 2001, but since then Federer has won three straight titles in SW19 and his first-round win over Richard Gasquet surpassed Borg's record of 41 straight wins on grass.
The first two games went with serve before an errant forehand from Henman gave Federer two break points. The British number three attempted to serve and volley but a good return forced Henman to place his backhand wide.
Superb returning from Henman created two break points in the eighth game but Federer stood firm to take a one-set lead.
There were glimpses of the shots which took Henman to four Wimbledon semi-finals, but all too often they came in desperate defence.
One such backhand winner saved a break point in the opening game of the second set, but Federer promptly converted the next and broke Henman again - partly thanks to an astounding forehand winner which had Henman staring in disbelief - to race into a 4-0 lead.
Henman briefly queried a close line call as he lost his serve for a third time to fall 5-0 behind, but his heart was clearly not in it and Federer looked keen to put him out of his misery.
A thumping ace gave Federer the set 6-0 in just 22 minutes as the Centre Court crowd attempted a Mexican wave to try to amuse themselves.
Henman had won just five points in that second set whitewash and Federer showed no signs of letting up, despite being friends with his opponent.
Two more breaks of serve put the Swiss star 4-0 ahead, although Henman had the dubious satisfaction of at least getting to deuce on Federer's serve.
Henman finally held serve in the fifth game to avoid the ignominy of another whitewash - ending a run of 11 straight games for Federer - but that was merely delaying the inevitable, Federer serving out to seal a resounding victory.


text:
Fort Myers All Destinations
Explore Fort Myers
Fort Myers Tours
All Fort Myers Tours
Lakes Park Tours & Tickets
Burroughs Home & Gardens Tours & Tickets
Calusa Nature Center and Planetarium Tours & Tickets
Sanibel Island Tours & Tickets
Edison and Ford Winter Estates Tours & Tickets
Six Mile Cypress Slough Preserve Tours & Tickets
Rookery Bay National Estuarine Research Reserve Tours & Tickets
How to Spend 3 Days in Fort Myers
Everglades Tours from Fort Myers
Edison and Ford Winter Estates
Rookery Bay National Estuarine Research Reserve
Zoomers Amusement Park
J.N. Ding Darling National Wildlife Refuge
Six Mile Cypress Slough Preserve
Burroughs Home & Gardens
Calusa Nature Center and Planetarium
Step back into old Florida, and explore the estates of two of the United States' most prominent men. Registered as National Historic sites, the buildings and beautifully landscaped lawns of the Edison and Ford Winter Estates tell stories of early Florida as well as the lives and times of the inventors.
Rookery Bay National Estuarine Research Reserve—commonly just called Rookery Bay Reserve—is a mangrove estuary in Naples, Florida. It is an important ecological area as it is one of the last undisturbed mangrove estuaries in North America. Rookery Bay Reserve covers 110,000 acres and is home to over 150 wildlife species. People are able to visit certain sections of Rookery Bay Reserve on guided tours.
During a boat tour, you'll board a six-person boat that takes you slowly through the reserve. Your guide will point out animal species you come across, which may include birds, deer, raccoons, tortoises and even bottle-nosed dolphins. Another way to tour the reserve is on a guided kayak excursion where you'll be taken through backwater bays within the mangrove forest. Just like with the boat tour you'll get the chance to learn about the animals and vegetation you see from a knowledgeable guide.
Before or after your guided tour, visit Rookery Bay Reserve's Environmental Learning Center. You'll learn more about the reserve through hands-on exhibits, a large onsite aquarium and films in its auditorium.
A noncaptive, natural refuge for the endangered Florida manatee, Fort Myers' Manatee Park is one of the top places in the area to spot the aquatic mammal. You'll also find walking paths, kayak and canoe rentals, playgrounds, and more.
Manatee Park Tours & Tickets
A far cry from the busy beaches of Miami or Fort Lauderdale, this 7-mile (11-kilometer) expanse of pristine Florida coastline has managed to preserve its sleepy charm. Beachgoers can enjoy the sprawling sky and white sand dunes of the Gulf Coast, all while staying in a nearby resort town that offers a relaxed vibe and upscale amenities.
Fort Myers Beach Tours & Tickets
Sanibel Island is a tropical retreat known for its quiet shores and its warm, shallow water. Beach goers can enjoy local seafood restaurants, swaying palm trees, and an abundance of sea shells, while nature lovers flock to the J.N. Ding Darling National Wildlife Refuge to see alligators, herons, and other wildlife.
From roller coasters and bumper boats to miniature golf, Zoomers amusement park in Fort Myers is a prime destination for family-friendly fun. Enjoy a variety of outdoor rides and indoor games, including air hockey and video games. Guests can pay as they go or opt for an all-access pass.
Zoomers Amusement Park Tours & Tickets
Spend the day surrounded by wildlife one of the best spots for bird-watching in Florida. This 6,300-acre (2,550-hectare) refuge, which occupies a third of Sanibel Island, includes a wildlife drive, tram tour, picnic spots, and a scavenger hunt. The wetland swamp is also part of the largest undeveloped mangrove ecosystem in the United States.
J.N. Ding Darling National Wildlife Refuge Tours & Tickets
Few places show off the biodiversity of Florida's waterways better than Six Mile Cypress Slough Preserve. Here, 3,500 acres (1,416 hectares) of prime Florida wetland and upland ecosystems—created as rainwater drains into Estero Bay—provide a habitat for alligators, wading birds, wild pigs, otters, butterflies, and other native fauna.
Walk back in time at the Burroughs Home & Gardens, a 1901 Georgian Colonial Revival mansion and one of the oldest homes in Fort Myers, Florida. Sprawling 6,000 square feet along the riverfront, the historic property features bay and stained glass windows, grand winding staircase, a widow's walk, water features, and a spacious veranda.
The magnificent abundance of Florida's wildlife need not be hard to find. Within the beachy and sunlit corridors of Fort Myers lie the Calusa Nature Center and Planetarium, and it's here that you'll be able to escape the trappings of the modern industrial era and revert back to blissful nature. Walk one of several boarded walkways and take in the pine and cypress swamp or venture in to the conservancy and approach bald eagles, rare owls, wild skunks, domesticated pythons and more. The attached planetarium is always exploring the skies and holds showings daily. An added bonus are the planetarium's live reptile and mammal presentations (including alligator and snake feedings) as part of the daily show.
More Things to Do in Fort Myers
IMAG History and Science Center
IMAG History and Science Center teems with interactive fun and gives both youngsters and their parents a chance to participate in hands-on demonstrations about the wonders of nature. Feel the power of a hurricane, run through a rainstorm, observe rays and sharks, or be a weather forecaster for the day in this family-friendly museum.
Lakes Park
Lakes Park, also known asLakes Regional Park, is an ideal spot for getting a healthy dose of exercise and fresh air in Fort Myers, Florida. Located on the bay with more than 150 acres (61 hectares) of man-made lakes, this park offers birdwatching, a miniature railroad, playgrounds, gardens, and 2.5 miles (4 kilometers) of trails for hiking and biking.
Southwest Florida Museum of History
Nature and history meet and come alive in the Southwest Florida Museum of History. Starting with the area's first Paleolithic settlers (think 12-foot giant sloth), come see in vivid detail how the southwest Florida region was once covered in swamp and dominated by terrible dinosaurs only to make way for tenacious settlers and modern railroads. Housed in a former Atlantic Coastline Railroad depot, the Southwest Florida Museum of History is a fun way to take a trip back in time—giving visitors a historical perspective on this beautiful south Florida city.
More Tours in Fort Myers
Things to do near Fort Myers
Recommended for Fort Myers


According to Liz Moskow, culinary director at Sterling-Rice Group, it's very possible that chocolate cake will become a regular breakfast food because of all the health benefits it offers.
The Denver food expert recently told Food Business News that a Syracuse University study found that dark chocolate improves cognitive functions like abstract reasoning, memory, and the ability to focus — all of which make your workday much easier. Moskow also mentioned another study from Tel Aviv University which found that eating dessert in the morning can even help you lose weight, as our metabolisms are most active during that time of the day.
That said, if your breakfast desserts contain a ton of fat and sugar, you won't reap any sweet benefits. Don't worry, though — we've got you covered. Here are five deliciously healthy chocolate recipes.
1. There's no need to give up Nutella when you can make your own healthy homemade version!
2. It's pretty much impossible not to drool over this amazing chocolate banana bread.
3. Not only are these double chocolate muffins low on calories, but they're chock-full of protein!
4. If you want an especially decadent breakfast without the guilt, this dark chocolate avocado mousse is the only way to go.
5. And of course, you can have your dark chocolate zucchini cake and eat it in the morning, too!
Get ready for a new revolution, chocolate lovers. We'll never have to feel shame about starting the day with chocolately goodness ever again!