\section*{Changes}



	



	\section{Introduction}
	
	
	In recent years, there has been considerable interest in understanding how statistical mechanics
	emerges from the quantum dynamics of isolated many-body systems.  Such considerations invariably
	require a correspondence between energy, a quantity well-defined in quantum mechanics, and
	temperature, which is necessary for a statistical-mechanical description.  Since temperature is not
	a priori defined in quantum mechanics, assigning temperatures to energies is a nontrivial issue.
	
	Ideas regarding thermalization in isolated quantum systems, e.g., the eigenstate thermalization
	hypothesis (ETH) \cite{Deutsch_PRA1991, Srednicki_PRE1994, Srednicki_1996, Srednicki_1999,
		Rigol_thermalization_Nat2008, AlessioRigol_Chaos_AdvPhys2016,
		Reimann_NJP2015,Deutsch_RepProgPhys2018, Mori_Ikeda_Ueda_thermalizationreview_JPB2018} and its
	extensions or variants are often tested or verified using numerical ``exact diagonalization''
	calculations \cite{ Rigol_thermalization_Nat2008, Rigol_Quenches_PRA2009,
		Rigol_thermalization_PRL2009, BiroliKollathLauchli_FluctationsAndThermalization_PRL2010,
		SantosRigol_Thermalization_PRE2010, SantosRigol_LocalizationAndThermalization_PRE2010,
		RigolSantos_Chaos_PRA10, Roux_quantumquenches_PRA2010, NeunhahnMarquardt_thermalization_PRE2012,
		RigolSrednicki_Thermalization_2012, SantosPolkovnikovRigol_typicality_PRE2012,
		SantosPolkovnikovRigol_typicality_PRE2012, GenwayHoLee_ThermalizationSmallHubbard_PRA2012,
		Steinigeweg_etal_EthThermalization_PRE2013, KimIkedaHuse_TestETH_PRE2014,
		BeugelingHaque_ThermalizationScaling_PRE2014, SorgVidmarHeidrichMeisner_thermalization_PRA14,
		Steinigeweg_etal_ETHLimits_PRL2014, FratusSrednicki_EstateThermalization_PRE2015,
		BeugelingMoessnerHaque_matrixelements_PRE2015, NandkishoreHuse_MBL_AnnRev2015,
		JohriNandkishoreBhatt_MBL_PRL2015, MondainiFratusSrednickiRigol_ETH2D_PRE2016,
		ChandranSchulzBurnell_AnyonETH_PRB2016, LuitzBarLev_AnomalousThermalization_PRL2016,
		AlessioRigol_Chaos_AdvPhys2016, MondainiRigol_ETH2DII_PRE2017, Dymarsky_Lashkari_Liu_PRE2018,
		GarrisonGrover_SingleEstate_PRX2018, YoshizawaIyodaSagawa_NumericalETH_PRL2018,
		KhaymovichHaqueMcClarty_Behemoths_PRL2019, Vidmar_HeidrichMeisner_ETHpolaron_PRB2019,
		MierzejewskiVidmar_ETH_PRL2020, BrenesPappalardiGoold_EntanglementETH_PRL2020,
		LeBlondRigol_EstateThermalization_PRE2020, BrenesRigol_etal_ETHPerturbed_PRL2020, Noh_ETH_PRE2021,
		Sugimoto_Hamazaki_Ueda_ETH_localRMT_PRL2021,  Nakerst_Haque_ETHclassicallimit_PRE2021, 
		Richter_Steinigeweg_Dymarsky_Gemmer_PRL2022, SugimotoHamazakiUeda_ETHLongRange_PRL2022}.  As a
	result, quantum systems with Hilbert space dimensions between $\sim10^3$ and $\sim10^5$ have
	acquired particular relevance.  It is therefore important to ask how meaningful various definitions
	of thermodynamic quantities like temperature or entropy are for finite systems, particularly systems
	of sizes typically treated by full numerical diagonalization.  In this work, we critically examine
	different ways of calculating entropy from the energy eigenvalues of finite systems, and compare the
	temperature derived from the entropy with the so-called `canonical' temperature.
	
	The most common definition of temperature for finite isolated quantum systems is the canonical
	temperature.  This is obtained for any energy  $E$ by inverting the canonical equation 
	\begin{equation}\label{eq:canonical_energy}
		E = \langle H \rangle \ = \ \frac{\tr( e^{-\beta H}H)}{\tr( e^{-\beta H})} 
		\ = \ \frac{ \sum_{n} e^{-\beta E_{n}}E_{n} }{\sum_{n} e^{-\beta E_{n}}}. 
	\end{equation}
	If the eigenvalues $\{E_n\}$ of a system are known, then this relationship provides a map
	between energy and the canonical temperature  $T_c=(k_B\beta_{c})^{-1}$.  (Here $k_B$ is the Boltzmann constant.)  The relationship \eqref{eq:canonical_energy} originates in statistical
	mechanics from the context of a system with a bath.  However, it is widely used in the study of
	thermalization of isolated (bath-less) quantum systems, to obtain an energy-temperature
	correspondence \cite{Rigol_thermalization_Nat2008,Rigol_thermalization_PRL2009,
		Rigol_Quenches_PRA2009,RigolSantos_Chaos_PRA10,
		SantosRigol_Thermalization_PRE2010,Roux_quantumquenches_PRA2010,
		RigolSrednicki_Thermalization_2012,SantosPolkovnikovRigol_typicality_PRE2012,
		NeunhahnMarquardt_thermalization_PRE2012,RigolSrednicki_FDT_PRL2013,
		SorgVidmarHeidrichMeisner_thermalization_PRA14,NandkishoreHuse_MBL_AnnRev2015,
		FratusSrednicki_EstateThermalization_PRE2015,Essler_Quench_JoSM2016,
		Greinergroup_thermalization_Science2016,AlessioRigol_Chaos_AdvPhys2016,
		GarrisonGrover_SingleEstate_PRX2018,LuGrover_RenyiEntropy_PRE2019,
		SekiYunoki_thermal_PRR2020,Noh_ETH_PRE2021}.  
	
	An alternate way to define temperature is to use the thermodynamic relation
	\cite{Reif_StatPhys,Reichl_StatPhys,huang_statisticalmech,RKPathria_StatPhys,LandauLifshitz_StatPhys,kardar_statphysp_2007}
	\begin{equation}\label{eq:entropy_temp}
		T = \left( \pd{E}{S} \right)_{X_i} = \left( \pd{S}{E} \right)_{X_i}^{-1}.
	\end{equation}
	Here $S$ is the (thermal) entropy, and the subscript $X_i$ denotes the system parameters that should be
	held constant.  For an isolated (i.e. microcanonical) quantum system, defining the entropy $S(E)$
	at a particular energy $E$ involves counting the number of eigenstates (`microstates') within an
	energy window $\dE$ around that energy $E$.  This raises the question of how to choose the width
	$\dE$, or possibly avoiding an explicit choice of $\dE$ and instead estimating the density of
	states.  In the large-size (thermodynamic) limit, these choices can be shown to become unimportant.
	This paper aims to explore the consequences of these choices for finite sizes, focusing
	on systems of Hilbert space dimensions $\sim10^4$ typical for full numerical diagonalization
	studies.  
	
	Motivated by the analysis of Ref.~\cite{Gurarie_Equivalence_AJoP2007}, we consider three choices for
	defining the entropy.  In each case, we compare the resulting temperature obtained using Eq.\
	\eqref{eq:entropy_temp} with the canonical temperature obtained directly form inverting Eq.\
	\eqref{eq:canonical_energy}.  
	
	

	\begin{figure
		\includegraphics[width=0.98\linewidth]{dE_illustration.png}
		\caption{Illustration of two ways of choosing the energy window $\dE$ used to
			define the microcanonical entropy.  The top figure shows the obvious choice: the width $\dE$ is
			the same at all energies.  The lower schematic illustrates an energy-dependent width:
			$\dE(E)\propto\sqrt{T_c^2 C_c}$, where the temperature $T_c$ is the canonical temperature
			corresponding to energy $E$ and  $C_c$ is the heat capacity at $T=T_c$.  The window is shown at
			three energy values $E_a$, $E_b$, and $E_c$, which are in the regions of low temperature, infinite
			temperature, and low negative temperature.  Each vertical tick marks an eigenvalue. 
			\label{fig:dE_illustration}
		}
	\end{figure}

	
	
	First, we consider an arbitrarily chosen but constant (energy-independent) window, as illustrated
	in Figure \ref{fig:dE_illustration}(top).  This is the most obvious choice, but turns out to be far from
	optimal --- the resulting temperature deviates strongly at finite sizes from the canonical
	temperature.  
	
	Second, noting that the leading correction to the large-size limit can be made to disappear by
	choosing $\dE\propto\sqrt{T_c^2C_c}$ \cite{Gurarie_Equivalence_AJoP2007}, we examine the result of
	using such an energy-dependent window width.  (Here $C_c$ is the heat capacity.)  Such an
	energy-dependent window is illustrated in Figure \ref{fig:dE_illustration}(bottom).  We show that this
	choice works extremely well for the sizes of interest, modulo some caveats regarding the
	proportionality constant.
	
	Finally, we can formulate the microcanonical entropy in terms of the density of states (d.o.s.)
	without reference to a specific energy window $\dE$, and use standard numerical estimation
	procedures for the d.o.s., via approximations to the integrated d.o.s.\ or cumulative spectral
	function.  We show that this choice is sub-optimal, similar to the choice of an energy-independent
	$\dE$.  We analyze the reason for the strong finite-size mismatch in this case.
	
	This article is organized as follows.  Section \ref{sec:preliminaries} recounts the definitions and
	saddle-point expressions which lead to the three choices for calculating the microcanonical entropy.
	We also provide details (\ref{sec:numerics_Hamiltonians}) of the quantum many-body systems
	that we use for numerical exploration --- we provide results for multiple systems with different
	geometries, to ensure that the resulting conclusions are not artifacts of a particular lattice or
	Hamiltonian.  In the next three sections, we describe the procedures for, and the results of the
	three ways of calculating entropy.  First, Section \ref{sec:const_dE} describes using a
	constant-width energy window $\dE$.  Second, Section \ref{sec:energy_dependent_dE} describes using
	an energy-dependent $\dE$ designed to cancel out the explicit $\dE$-dependence of the resultant
	temperature, following/extending the suggestion of Ref.~\cite{Gurarie_Equivalence_AJoP2007}.  Third,
	Section \ref{sec:Omega} outlines using a smoothened cumulative d.o.s.\ $\Omega(E)$ to
	calculate the d.o.s. $g(E)$ thereby bypassing the need to explicitly select a $\dE$.  In this Section, we also present (\ref{sec:omega_explain_deviation}) explanations for the deviations seen in this third method.  Section
	\ref{sec:concl} provides concluding discussion and some context.
	
	
	



	
	
	
	\section{Preliminaries}\label{sec:preliminaries}
	

	
	
	We first recall the standard definition of the microcanonical entropy and highlight the roles of the
	density of states $g(E)$ and of the energy window $\dE$ (subsection \ref{sec:entropy_dos_window}).
	We then recall (\ref{sec:saddlepoint}) the saddle-point formulation often used to show the
	equivalence of microcanonical and canonical ensembles in the large-size limit
	\cite{Gurarie_Equivalence_AJoP2007}, and extend beyond the leading order in order to analyze the
	effect of $\dE$ at finite sizes.  Subsection \ref{sec:numerics_Hamiltonians} describes the quantum
	systems to be used in subsequent sections to provide numerical examples.
	



	
	\subsection{Entropy, density of states,  and the energy window} \label{sec:entropy_dos_window}
	
	The microcanonical entropy $S(E)$ of a system at energy $E$ is  \cite{Gurarie_Equivalence_AJoP2007,
		Reichl_StatPhys, huang_statisticalmech, RKPathria_StatPhys, LandauLifshitz_StatPhys, kardar_statphysp_2007} 
	\begin{equation}\label{eq:entropy_def}
		S(E) = k_B \ln\Gamma(E).
	\end{equation}
	where $\Gamma(E)$ is the statistical weight, which is the number
	of microstates at energy $E$.  For a quantum system, microstates are to be interpreted as
	eigenstates. For a quantum system with a discrete spectrum, counting the number of eigenstates is
	problematic because at a particular energy there is usually zero or one eigenstate, or perhaps a
	handful if there are degeneracies.  Thus, $S(E)$ would be $=-\infty$ for all values of energy except
	at a countable number of discrete energy values.  This issue is
	usually resolved \cite{Gurarie_Equivalence_AJoP2007, 
		RKPathria_StatPhys, LandauLifshitz_StatPhys} by taking
	$\Gamma(E)$ to be the number of eigenstates in an energy window $\dE$ around $E$, rather than the
	number of eigenstates exactly at energy $E$.  Thus we define 
	\begin{equation}\label{eq:dos}
		\Gamma(E) = g(E)\dE = \dE\sum_{n}\delta(E-E_n),
	\end{equation} 
	where the sum is understood to to include all the eigenvalues of the system Hamiltonian that lie in the window, i.e., all $E_n$ satisfying $E_n\in(E-\frac{\dE}{2},E+\frac{\dE}{2})$  
	%
	\footnote{It is also common to use the window $[E,E+dE)$. We choose to work with the window
		$[E-\dE/2,E+\dE/2]$. This choice presumably does not have significant effects.}.
	
	
	Here $g(E)$ is the density of states --- the number of many-body eigenstates per unit energy
	interval.  Although defined as a sum over delta functions, the density of states can be thought of
	as a smooth function of energy over energy scales much larger than the typical level spacing.  In
	numerical work, this is often achieved by broadening the delta functions into Gaussians or
	Lorentzians of finite width \cite{Bruus_dAuriac_2DHubbard_levelstatistics_PRB1997,
		Molina_Relano_Retamosa_misleading_PRE2002, Duine_MacDonald_vortex_PRA2010,
		SorgVidmarHeidrichMeisner_thermalization_PRA14, Shchadilova_Ribeiro_Haque_PRL2014,
		Wehling_excitons_PRB2017, Pixley_etal_disorderedWeyl_PRB2017, LiuLu_GrapheneNanoribbons_PRR2022,
		HonerkampKennesNag_multiWeylSemimetals_PRB2022}.)  Alternatively, we can define $\Omega(E)$ as the
	number of eigenstates with energy less than $E$, i.e., the integrated density of states.  Fitting a
	smooth function to the staircase form of $\Omega(E)$, one can obtain a smooth density of states as
	the derivative; $g(E)= \Omega'(E)$.
	
	The entropy now depends on an energy window $\dE$; we are thus faced with choosing an appropriate $\dE$.  The purpose of introducing a finite energy width was to smooth out the discreteness of the energy spectrum; thus $\dE$ should be large enough to include a large number of eigenstates.  On the other hand, we want $\dE$ to be sufficiently small so that the density of states (regarded as a smooth function of energy) does not vary appreciably within the window, i.e., $\dE$ should be much smaller than the scale of the bandwidth of the system.   Other than these general principles, we have the freedom to choose $\dE$, and the entropy $S$ will in general depend on the choice.  
	
	As we will explain next (\ref{sec:saddlepoint}), the sub-leading contributions to the entropy, which
	contain $\dE$, vanish in the large system size limit. So, for infinite systems, it will generally
	not matter what $\dE$ is, but for finite quantum systems, the choice can drastically affect the
	entropy, and resultant temperature.  The purpose of this paper is to investigate and clarify the
	effect of this choice for finite systems whose Hilbert space sizes makes them accessible to full
	numerical diagonalization.
	
	
	
	\subsection{Saddle point expressions} \label{sec:saddlepoint}
	
	To understand the role of $\dE$, it is helpful to express the entropy as an integral over (complex)
	inverse temperature and perform saddle-point approximations \cite{Gurarie_Equivalence_AJoP2007,
		Reif_StatPhys, huang_statisticalmech}, extending the order beyond what is necessary in the
	thermodynamic limit, to account for finite sizes.
	
	Replacing the delta function in Eq.\ \eqref{eq:dos} with
	$\delta(x) = \int_{-\infty}^{\infty}(2\pi)^{-1}e^{i\beta x}d\beta$, and defining the free energy as
	$F(\beta) = -\beta^{-1}\ln(\sum_n e^{-\beta E_n})$, we can write the entropy as
	\begin{equation}\label{eq:entropy_free_def}
		e^{S/k_B} = \Gamma(E) = \dE \int_{-i\infty}^{i\infty} \frac{d\beta}{2\pi} e^{\beta (E-F(\beta))} .
	\end{equation}
	To apply the saddle-point approximation, one first finds the critical point of the exponent  $h(\beta) =
	\beta(E-F(\beta))$. The condition is 
	\begin{equation}\label{eq:saddlepoint_condition}
		E-F(\beta)-\beta\pd{F}{\beta}=0 \ ,
	\end{equation}
	which is equivalent to Eq.\ \eqref{eq:canonical_energy} defining the canonical temperature.  Thus
	the saddle point is at $\beta=\beta_c$, the canonical inverse temperature.  The leading order
	saddle-point approximation is thus
	\begin{equation}\label{eq:saddlepoint_leading}
		\frac{S}{k_B} = \beta_c E- \beta_c F(\beta_c)
	\end{equation}
	with $\beta_c$ being the solution of Eq.\ \eqref{eq:saddlepoint_condition} or Eq.\
	\eqref{eq:canonical_energy}.  This matches the standard thermodynamic relation, and is consistent
	with the energy derivative of $S$ being the inverse temperature $\beta_c$.  
	
	To examine the effect of $\dE$, one needs to extend the calculation to the next order.  One expands
	$h(\beta)$ as a Taylor series about $\beta_c$, up to second order, as the first order term is zero
	by definition.  Introducing the heat capacity $C = \pd{E}{T} = -T\pds{F}{T}$,
	%

	%
	one obtains \cite{Gurarie_Equivalence_AJoP2007}
	\begin{equation}\label{eq:entropy_sub}
		e^{S/k_B} \approx \frac{\dE}{2\pi} e^{\beta_c(E-F(\beta_c))}\int_{-\infty}^{\infty} dy e^{-k_B
			T_{c}^{2}C_c y^2/2}.
	\end{equation}
	where $T_{c}$ and $C_{c}$ are the values at the saddle point $\beta_c$.  Evaluating the  Gaussian
	integral, one obtains 
	\begin{multline} \label{eq:entropy_final}
		\frac{S}{k_B} =  \beta_c E- \beta_S F(\beta_c)  + \ln\dE -  \ln\sqrt{2\pi k_{B}T_{c}^{2}C_{c}} . 
	\end{multline}
	Here $\beta_c$ is determined by the energy, and hence so are $T_{c}$ and $C_{c}$.  The two leading
	terms are both extensive in system size.  The first correction term $\ln\dE$ is sub-extensive as long
	as $\dE$ is not chosen to grow exponentially or faster with system size.  The second correction term
	grows logarithmically with system size, as the heat capacity is extensive.  Thus, at large enough
	sizes, the leading-order saddle-point approximation suffices, and the choice of energy window $\dE$
	plays no role.
	
	At finite sizes, however, the two correction terms need to be considered. From Eq.\ \eqref{eq:entropy_final}, we see that choosing $\dE$ to be an energy-independent constant (the most obvious choice, explored in Section \ref{sec:const_dE}) would leave an energy-dependent correction term, causing deviations from the canonical temperature. We also see (as pointed out in Ref.\ \cite{Gurarie_Equivalence_AJoP2007} and explored below in Section \ref{sec:energy_dependent_dE}) that the energy-dependence of the correction terms could be canceled by a judicious choice of energy-dependence for the width $\dE$.  
	

	
	\subsection{Hamiltonians and numerics}  \label{sec:numerics_Hamiltonians}
	
	To investigate the effect of different choices for defining the microcanonical entropy in finite systems, we use several spin-1/2 lattice systems, consisting of $L$ spins with $N$ spin-ups. The spins interact via \textit{XXZ} interactions, which have a $U(1)$ symmetry conserving $N$. We check all results and show data for 1D, 2D, and fully-connected geometries, to demonstrate that our results are very general and not particular to any model. In the case of the 1D chain, we also include magnetic fields in the $z$ and $x$ directions; the latter break the $U(1)$ symmetry. The Hilbert space dimension is $D=\binom{L}{N}$ when $N$ is conserved and $D=2^L$ otherwise. 
	
	The system parameters are always chosen such that the level spacing statistics match that expected of chaotic quantum systems. This ensures that there are no complications due to proximity to integrability or localization.
	
	
	\textit{Staggered field XXZ chain}: Starting with the open-boundary \textit{XXZ} chain (anisotropic Heisenberg chain)
	\begin{align} \label{eq:Ham_chain}
		H_H = J \sum_{j=1}^{L-1} (S^{x}_{j} S^{x}_{j+1} + S^{y}_{j} S^{y}_{j+1}) + \Delta \sum_{j=1}^{L} S^{z}_{j} S^{z}_{j+1}
	\end{align} 
	we introduce magnetic fields in both the transverse ($x$) and longitudinal ($z$) directions. To remove symmetries in the model we stagger the two fields along the even and odd sites respectively, and in addition, we break the staggered pattern at the start of the chain by inserting $x$ and $z$ fields on the first and second sites respectively :
	\begin{align}
		H_S = H_H + \sum_{j} h_z S^{z}_{2j+1} + \sum_{j}h_x S^{x}_{2j} +h_{x}S^{x}_{1} + h_{z}S^{z}_{2}. 
	\end{align}
	
	\textit{Square Lattice}: This model is a two dimensional square lattice with open boundary
	conditions and \textit{XXZ} interactions:
	\begin{align}  \label{eq:Ham_square}
		H_{\text{SQ}} = \sum_{\langle j,k \rangle} J_{jk}(S^{x}_{j} S^{x}_{k} + S^{y}_{j} S^{y}_{k}) +  \Delta_{jk} S^{z}_{j} S^{z}_{k}
	\end{align} 
	where $\langle\rangle$ means that we restrict the summation to nearest-neighbor pairs.  In order to
	remove any symmetries, we draw the values $J_{jk}$, $\Delta_{jk}$ from the uniform distribution
	$[0,1]$:
	
	\textit{Fully connected Lattice}: This model consists of a network of spins, where each spin $j$ is
	connected to every other spin $k$:
	\begin{align}  \label{eq:Ham_FC}
		H_{\text{FC}} = \sum_{j}  \sum_{k\neq j} J_{jk}(S^{x}_{j} S^{x}_{k} + S^{y}_{j} S^{y}_{k}) +  \Delta_{jk} S^{z}_{j} S^{z}_{k}
	\end{align}  
	We draw the values $J_{jk}$, $\Delta_{jk}$ from the uniform distribution $[-0.2,0.2]$ and
	$[-0.1,0.1]$ respectively.
	
	
	\textit{Units}: For numerical work, we use $k_B=1$.  We express energies in units of spin
	couplings.  Thus,  if $J=1$ then when we plot energies they are actually $E/J$, whereas if the
	couplings have other values, then the energy is to be thought of as the energy divided by a
	hypothetical coupling equal to $1$.  In this way, the units of all energies, temperatures, and
	entropies in the figures are determined.
	
	

	\section{Constant width window}\label{sec:const_dE}
	

	\begin{figure
		\includegraphics[width=0.98\linewidth]{ThSandB_dEconst_var_2DXXZ_Jr1_Dr1_L20N4_p6_ex25.png}
		\includegraphics[width=0.96\linewidth]{constdE_RMSvsD_1DL10to16_J1D0_95_stagh_hz0_5_hx0_5l.png}
		\caption{\textbf{(A)} Microcanonical entropy $S(E)$ calculated with constant (energy-independent)
			$\dE$.  Evaluated only at the eigenenergies $E=E_n$.  \textbf{(B)} Resultant inverse temperature
			$\beta(E)$ obtained as derivative of 6th-order polynomial fitted to $S(E)$ data.  25 eigenstates
			from both ends of the spectrum (shown in gray in \textbf{(A)}) were excluded from the fit.  \textbf{(A,B)}
			$5\times$4 square \textit{XXZ} lattice, Eq.\ \eqref{eq:Ham_square}, with $L=5\times4$,
			$N=4$. \textbf{(C):} Root-mean-square (RMS) distance between $\beta(E)$ and the canonical
			temperature, versus Hilbert space dimension $D=2^L$, with $\dE = 0.5$.  Spin chain with $L$ spins,
			Eq.\ \eqref{eq:Ham_chain}, $J=1$, $\Delta=0.95$, $h_z = h_x = 0.5$.  Units used in this and all
			subsequent figures are explained in subsection \ref{sec:numerics_Hamiltonians}.  }
		\label{fig:S_const_dE}
	\end{figure}

	
	
	
	We start by discussing the obvious choice for defining entropy --- using a constant
	energy-independent width $\dE$. Eq.\ \eqref{eq:entropy_final} implies that an energy-independent
	$\dE$ acts as an energy-independent shift of the entropy, and hence does not affect the temperature.
	However, the last term in Eq.\ \eqref{eq:entropy_final} is energy-dependent, hence the temperature
	will differ from the canonical temperature which corresponds to the leading (first two) terms. This
	deviation should vanish in the large-size limit, but the extent of this deviation for finite sizes
	is not a priori clear.
	%
	We show below using explicit numerical examples that using an energy-dependent width is a poor
	choice for the system sizes under study here (sizes accessible to exact diagonalization), as the
	resulting temperature deviates significantly from the canonical temperature.
	
	Examples of entropy, found by counting the number of eigenvalues within the window
	$[E-\dE/2,E+\dE/2]$, are shown in Figure \ref{fig:S_const_dE}(A) for an open-boundary square
	spin-$\frac{1}{2}$ lattice with random \textit{XXZ} interactions.   Here and in later figures, we
	plot one point for each eigenvalue, so that there is at least one eigenvalue in each window --- the
	minimum value of entropy is $\ln1=0$ and we thus avoid the possibility of obtaining entropy
	$\ln0=-\infty$.  
	
	To calculate the temperature from the entropy, we fit a polynomial to the $S(E)$ points and then
	take the derivative of the polynomial; results are shown in  Figure
	\ref{fig:S_const_dE}(B).
	
	Increasing $\dE$ by a factor approximately increases the number of eigenvalues in each window by
	that factor, so that the $S(E)$ curve undergoes an approximately constant upward shift.  This leaves
	the temperature $[S'(E)]^{-1}$ approximately unchanged.  Accordingly, for moderate $\dE$, the
	calculated $\beta(E)$ curves are robust to changes in $\dE$.  For the largest $\dE (=4.5)$ shown,
	this argument does not work as the window is significant compared to the variation scale of the
	density of states $g(E)$, so that $g(E)$ cannot be considered constant within
	each window.  This leads to a markedly different $\beta(E)$ curve for the very large $\dE$ case.  This effect is not captured by the saddle-point analysis or Eq.\ \eqref{eq:entropy_final}.  
	
	Even for moderate $\dE$, the deviation from the canonical $\beta_C(E)$ curve is considerable. This
	shows that using a constant energy window width is not a feasible approach to defining entropy for
	finite quantum systems having Hilbert space dimensions $\sim{}O(10^4)$. The results of Figure
	\ref{fig:S_const_dE}(B) provide a visual presentation of the extent of the discrepancy for such sizes. In Figure
	\ref{fig:S_const_dE}(C) we show the root-mean-square deviation of the inverse temperatures obtained
	with a constant $\dE$ from the canonical $\beta_C(E)$ values, as a function of Hilbert space
	dimension. (We show data for a spin chain, for which finite size scaling is numerically more
	convenient than for a square lattice.) The deviation decreases with increasing size, as expected, but
	very slowly.
	
	A remark on the edges of the spectrum is in order.  As there are only a few eigenvalues in each
	window, the values of entropy are noticeably discrete, $\ln n$ where $n$ is a small integer.  For a
	regime with such discrete behavior, statistical-mechanical considerations are not meaningful --- we
	expect entropy to be a continuous function of energy.  We therefore omit the edge points from the
	polynomial fit to $S(E)$.  In Figure \ref{fig:S_const_dE} we have omitted 25 eigenstates from each edge of the spectrum.  This is approximately the energy region for
	which $S(E)\lesssim\ln10$ for the $\dE=0.5$ case.  Later in the paper, the same criterion will be
	used to exclude the spectral edges.
	
	



	\section{Energy dependent window}\label{sec:energy_dependent_dE}
	
	We now describe using an energy-dependent $\dE\propto\sqrt{T_c^2C_c}$.
	
	
	\subsection{Rationale}
	
	In Eq.\ \eqref{eq:entropy_final}, the leading-order terms (first two terms) yield the canonical
	temperature upon differentiation by energy, but there are two correction terms, $\ln\dE$ and
	$-\ln\sqrt{2\pi{k_B}T_cC_c}$.  Ref.\ \cite{Gurarie_Equivalence_AJoP2007} suggests choosing the two
	terms to be equal, so that the corrections vanish and the canonical temperature is recovered.
	
	We note, however, that the correction terms need not cancel exactly --- they will not affect the
	temperature $[S'(E)]^{-1}$ as long as they sum to an energy-dependent constant.  Thus we could
	choose
	\begin{equation}\label{eq:eps_def}
		\dE(E) \equiv \alpha^{-1}\sqrt{2\pi k_{B}T_{c}^{2}C_{c}}.
	\end{equation} 
	with $\alpha$ being some constant.  The window $\dE$ is then energy-dependent because $T_c$ and
	$C_c$ are energy-dependent.  The proposal of Ref.\ \cite{Gurarie_Equivalence_AJoP2007} corresponds
	to $\alpha=1$.  
	
	The energy-dependent $\dE$ can also be motivated by the physical idea that the energy window should be determined by the fluctuation of energy in the canonical ensemble, as suggested, e.g., in Refs.~\cite{Polkovnikov_DEntropy_AoP2011,SantosPolkovnikovRigol_Entropy_PRL2011}.  It turns out \cite{huang_statisticalmech} that the distribution in energy in the canonical ensemble has width $\propto\sqrt{T^2C}$.  Thus, choosing the energy uncertainty as the energy window leads to the same prescription as Eq.\ \eqref{eq:eps_def}.
	
	Labeling the number of states in an energy-dependent energy window $\dE(E)$ as $\tilde{\Gamma}(E)$,
	the entropy is obtained as 
	\begin{equation}\label{eq:S_epsilon}
		\tilde{S}(E) = k_B \ln \left( \tilde{\Gamma}(E) \right) .
	\end{equation}
	Differentiation gives the corresponding inverse temperature $\tilde{\beta}$.   In the following
	subsection we will describe doing this numerically, and discuss the results obtained.  
	
	
	We do not know of a principle guiding the choice of $\alpha$.  We will show that, for the sizes of
	primary interest to us, $\alpha$ needs to be larger than $1$.  The reason is that for $\alpha=1$,
	the energy window $\dE(E)$ turns out to be too broad, exceeding the energy scale at which the
	density of states varies, leading to poor results similar to the large constant $\dE$ case in Figure
	\ref{fig:S_const_dE}.  For larger system sizes, the acceptable range of $\alpha$ broadens,
	compatible with our expectation that the choice of $\dE$ should be less important for larger sizes.
	
	


	\subsection{Numerical calculation and results}\label{sec:fin_sys_res}
	
	
	In order to obtain $\dE(E)$ as defined in Eq.\ \eqref{eq:eps_def}, we first need to calculate the heat capacity $C_c$ and the canonical temperature.  We start by numerically computing the energy $E$ as a function of $\beta$ (or of $T$) via \eqref{eq:canonical_energy}, using the numerically calculated eigenvalues of the Hamiltonian $H$.  This leaves us with a (numerical approximation to a) smooth injective function $E(T)$.  As usual, inverting this numerically provides the canonical temperature $T_c$ as a function of $E$.  In addition, estimating the derivative of $E(T)$ provides us with the heat capacity $C = \partial E/ \partial T$, which we evaluate at the canonical value $T_c$.  Figure \ref{fig:de_fit}(A) shows an example of $C_c$ as a function of energy, calculated in this way.  The function is non-monotonic, going to zero at both zero and infinite temperatures.
	


	
	

	\begin{figure
		\includegraphics[width=0.95\linewidth]{CandDE_2DXXZ_L20N4_Jr1_Dr1.png}
		\caption{\textbf{(A):} Specific heat $C = \partial E/ \partial T$.   \textbf{(B):} $\dE(E) =
			\alpha^{-1}\sqrt{2\pi k_B T_{S}^{2}C_{S}}$, for $\alpha=5$, numerically computed.  Dashed curve: 12-th order
			polynomial fit, used to avoid the spurious divergence.  Data for  $5\times$4 square \textit{XXZ}
			lattice, Eq.\ \eqref{eq:Ham_square}, with  $N=4$.} 
		\label{fig:de_fit}
	\end{figure}

	
	
	We can now calculate $\dE(E)$ via \eqref{eq:eps_def} for all energies.  Figure \ref{fig:de_fit}(B) shows an example. 
	
	At the center of the spectrum, $T_c\to\infty$ and $C_c\to0$, leading to a finite $\dE(E)\propto\sqrt{T_c^2C_c}$.  Due to the numerical discreteness of computed $E$ and $T$ functions, the computed $\dE(E)$ acquires a spurious divergence at this point.  This effect can be confined to a smaller energy region by using a finer grid of $T$ (or $E$) values.  We avoid the effect by fitting a polynomial, excluding points within the direct vicinity of the discontinuity.  An example  is shown in  Figure \ref{fig:de_fit}(B).  The fitted polynomial is then used as $\dE(E)$.  
	
	
	

	\begin{figure
		\hspace{0.1\linewidth}
		\includegraphics[width=0.3\linewidth]{Square_tikzit.pdf}
	
		\hspace{0.15\linewidth}
		\includegraphics[width=0.3\linewidth]{LatticeA2A_tikzit.pdf}\\
	
		\includegraphics[width=0.999\linewidth]{ThSandB_Epsilon_twice_2DXXZ_Jr1_Dr1_L20N4x5_2Dfullcon_Jr02Dr01_L16N5x4_p6_exlog10_alph1.png}
		\includegraphics[width=0.999\linewidth]{RMSvsAlpha_2DL20and28_N4and5_Jr1Dr1L_alph1.png}
		\caption{Using an energy-dependent window, Eq.\ \eqref{eq:eps_def}.  Left panels \textbf{(A,B)} and
			right panels \textbf{(C,D)} show data for two different systems.  \textbf{(A,C):} Microcanonical
			entropy calculated with $\dE(E) = \alpha^{-1}\sqrt{2\pi k_B T_{S}^{2}C_{S}}$ along with 6th-order
			polynomial fit.  Fit excludes gray points.  \textbf{(B,D)} Resultant temperature
			$\tilde{\beta}(E)$, with two different values of $\alpha$ in each case, compared with canonical
			temperature $\beta_C(E)$.  \textbf{(A,B)} Data for $5\times$4 square \textit{XXZ} lattice, Eq.\
			\eqref{eq:Ham_square}, with $N=4$.  \textbf{(C,D)} Fully connected lattice, , Eq.\
			\eqref{eq:Ham_FC}, with $L=16$, $N=5$.  \textbf{(E)} RMS distance between $\tilde{\beta}(E)$ and
			$\beta_C(E)$ plotted against $\alpha$ for two square lattices, Eq.\ \eqref{eq:Ham_square}, a
			$5\times4$ lattice with $N=4$, and a $7\times4$ lattice with $N=5$.  
		}\label{fig:S_eps}
	\end{figure}

	
	
	We are now equipped to compute the entropy using an energy dependent energy window, $\dE(E) = \alpha^{-1}\sqrt{2\pi k_B T_{c}^{2}C_{c}}$.
	In Figure \ref{fig:S_eps}(A,C), we show the numerically computed entropy $\tilde{S}$. The data shown is for a chaotic square lattice of spins-1/2's with \textit{XXZ}-like connections between nearest neighbor spins, and also for a fully connected lattice with \textit{XXZ}-like connections between every pair of sites.  We used $\alpha=5$ and $\alpha=4$ in the two cases.   
	
	As before, we numerically fit a polynomial to our entropy data, excluding values at the edge of the spectrum, and then take its derivative to obtain a temperature. The resultant inverse temperature $\tilde{\beta}$ are shown in Figures \ref{fig:S_eps} (B,D).  For  $\alpha\sim5$, the temperature matches the canonical temperature remarkably well. 
	
	We found that the procedure provides an excellent match between $\tilde{\beta}$ and $\beta_c$ when $\alpha$ is larger than some minimum value, as shown by the vanishing root-mean-square deviation in Figure \ref{fig:S_eps}(E).  Beyond this minimal $\alpha$ the exact choice is somewhat arbitrary, as long as the window is not made ultra-small.  Figure \ref{fig:S_eps}(E) shows that this minimal value (the value of $\alpha$ at which the deviation drops to zero) is smaller for larger systems.   This is consistent with the expectation that the exact choice of the window $\dE$ is increasingly irrelevant as the system size is increased.  
	
	To summarize: this numerical analysis shows that using an energy-dependent window is a very successful strategy for defining entropy in a finite-size system.  We observed good agreement
	between the resultant temperature and the canonical temperature for all systems we checked,
	including various 1D chaotic spin chains, and a handful of 2D lattices, including that presented in
	Figure \ref{fig:S_eps}.
	

	


	



	\section{Using the integrated density of states}\label{sec:Omega}
	
	\subsection{Formulation: avoiding $\dE$}
	
	An alternative to using an explicit energy window is to use the expression for  the entropy  in terms of the
	(integrated) density of states:
	\begin{multline}\label{eq:entropy_from_dos}
		S = k_B \ln\Gamma(E) = k_B \ln\bigg(g(E)\dE\bigg) \\ 
		= k_B \ln g(E) + k_B \ln\dE
		\\ =   k_B \ln(\pd{\Omega(E)}{E}) + k_B \ln\dE  .
	\end{multline}
	Here
	\begin{equation}
		\Omega(E) = \sum_{n=1}^{D} \Theta(E-E_n)
	\end{equation}
	is the number of eigenstates $\ket{E_n}$ with energy less than $E$, i.e., the integrated d.o.s. or
	cumulative d.o.s., also known as the cumulative level density
	\cite{Graef_etal_stadium_PRL1992, Relano_Dukelsky_Retamosa_integrable_PRE2004, Le_etal_heliumatoms_PRA2005,
		Santhanam_Bandyopadhyay_PRL2005, Mulhall_PRC2009, Wang_Gong_butterflyFloquet_PRE2010,
		Enciso_etal_HaldaneShastry_PRE2010, Jain_Samajdar_billiards_RMP2017,
		Ormand_Brown_nuclear_Lanczos_PRC2020, Corps_Molina_Relano_Thoulessenergy_MBL_PRB2020,
		Magner_etal_semiclassical_PRC2021, Corps_Relano_longrange_PRE2021} or the cumulative spectral
	function 
	\cite{Ciliberti_Grigera_localization_levelspacing_PRE2004, Gusso_Rego_nanoelectromechanical_PRB2006,
		SantosRigol_Thermalization_PRE2010, Frye_Hutson_approachchaos_PRA2016,
		Dettmann_Knight_randomgraphs_EPL2017, Abuelenin_unfolding_PhysicaA2018,
		Chaudhuri_etal_tensormodels_PRD2018, Vidmar_HeidrichMeisner_ETHpolaron_PRB2019,
		vonSmekal_2color_staggeredquarks_PRD2019, Corps_Molina_Relano_Thoulessenergy_MBL_PRB2020,
		Kourehpaz_TypicalityChaos_arxiv2021, Lydzba_Sowinski_fewfermions_PRA2022}.  The density of states
	increases exponentially with the 
	system size; hence the first term in Eq.~\eqref{eq:entropy_from_dos} is extensive.  On the other
	hand, $\ln\dE$ is presumably either constant or at most weakly increasing with system size.  Thus,
	at large enough system size the second term can be neglected, so that the entropy is approximated by
	the first term:
	\begin{equation}\label{eq:def-OmegaEntropy}
		S \approx S_\Omega  = k_B \ln g(E) = k_B \ln(\pd{\Omega(E)}{E}) .
	\end{equation}
	One can thus use a continuous approximation for $\Omega(E)$ to define a continuous $S_\Omega(E)$, without choosing an explicit energy window $\dE$.    
	


	
	We will show in the next subsection that using Eq.~\ref{eq:def-OmegaEntropy} leads to noticeable
	deviations of the resulting temperature from the canonical temperature at finite sizes.  We discuss
	the reason for this deviation in subsection \ref{sec:omega_explain_deviation}.  
	
	

	


	
	

	
	
	
	



	\subsection{Numerical results using integrated density of states}\label{sec:omega_res}
	

	\begin{figure
		\includegraphics[width=0.98\linewidth]{OmegaANDdOmega_fit_2DXXZ_L28N5_Jr1Dr1_tanh10_ins_Pe.png}
		\caption{\label{fig:omega_fit}
			\textbf{Upper:} Cumulative density of states $\Omega(E)$ (normalized by Hilbert space
			dimension),   fitted by a  polynomial  $f(E)$ 
			of degree 10. Inset: zoom.  \textbf{Lower:}
			$\Omega'(E)/D$ from derivative of fitted function, compared with histogram (distribution) of
			eigenvalues, 
			$P(E_n)$.  Data for $7\times4$ square \textit{XXZ} lattice, Eq.\ \eqref{eq:Ham_square}, with $N=5$.  
		}
	\end{figure}

	
	From the numerically calculated eigenvalue spectrum of the system Hamiltonian, the integrated
	d.o.s.\ $\Omega(E)$ can be obtained as a function of $E$.  It is a series of steps with constant
	integer values between eigenvalues and a step to the next integer at every eigenvalue.  In order to
	obtain a derivative of this non-smooth function, we will fit an analytic function to the computed
	$\Omega(E)$ data, and then simply take its derivative.  Fitting an analytic function to $\Omega(E)$
	is common practice in the unfolding procedure utilized for computing level spacing statistics
	\cite{Haake1991QuantumSO,Guhr_RMTinQM_PR1998, Molina_Relano_Retamosa_misleading_PRE2002,
		Abul_Unfolding_PhysicaA2014, Abuelenin_unfolding_PhysicaA2018}.
	
	We found that using a function of the form 
	\begin{equation}
		\frac{D}{2} \big[ 1+\tanh(f(E)) \big]
	\end{equation}
	works remarkably well to fit the numerical $\Omega(E)$ data, where $f(E)$ is some polynomial in $E$.
	If $f(E)$ is a monotonically increasing polynomial, then the form of the function automatically
	imposes the correct low-energy and high-energy behavior of the smoothened $\Omega(E)$.  This
	functional form was inspired by its use in \cite{Chaudhuri_etal_tensormodels_PRD2018} to unfold a
	many-body spectrum.  Once the fitting function $f(E)$ is determined, the density of states is
	obtained as the derivative: 
	$g(E) = \Omega'(E) = \frac{D}{2}\sech^2(f(E))f'(E)$.
	
	An example of numerically calculated $\Omega(E)$ and the fitted function are shown in Figure
	\ref{fig:omega_fit} (top).  The resulting derivative $\Omega'(E)$ is compared with the normalized
	histogram of eigenvalues.  In both panels, we normalize the functions by $D$, so that the function
	itself is plotted between $0$ and $1$, and its derivative can be directly compared with the
	normalized eigenvalue distribution $P(E_n)$.
	
	
	With $\Omega'(E)$ in hand, we are now equipped to compute $S_{\Omega}$ and the resultant temperature
	$\beta_{\Omega}$.  $S_\Omega$ is  the logarithm of $g(E) = \Omega'(E)$, and $\beta_{\Omega}$ is
	thus the derivative of this. For the analytical approximation to $\Omega'(E)$ above, one obtains
	\begin{equation}
		\beta_{\Omega}(E) = \frac{f''(E)}{f'(E)} - 2f'(E)\cdot\tanh(f(E)).
	\end{equation}  
	
	Examples of the computed entropy $S_{\Omega}$ are shown in Figure \ref{fig:S_omega}(A,C). The
	numerical results shown are for a 1D spin chain, and a square lattice, both with nearest neighbor
	\textit{XXZ}-like connections and open boundary conditions. The resultant inverse temperature
	$\beta_{\Omega}$ are shown in panels (B,D).  While  $\beta_{\Omega}$ has the correct overall form,
	the deviations from the canonical curves are clearly visible. 
	
	The extent of the deviation is shown quantitatively in Figure \ref{fig:S_omega}(E) by plotting the
	root-mean-square (RMS) distance between the canonical and resultant temperature, versus Hilbert
	space dimension $D$, for a $L$-site spin chain ($D=2^L$).  The deviation decreases very slowly ---
	it would take very large system sizes for the two temperatures to be considered `close'.
	
	

	\begin{figure
		\hspace{0.12\linewidth}
		\includegraphics[width=0.3\linewidth]{Lattice_tikzit.pdf}
		\hspace{0.1\linewidth}
		\includegraphics[width=0.3\linewidth]{Square_tikzit.pdf}\\
		\includegraphics[width=0.99\linewidth]{ThSandB_OmegaAn_tanh10_double_XXZ_J1D0_95_L12_staghzhx0_5_and_2DXXZ_Jr1Dr1_L20N4_log10.png}
		\includegraphics[width=0.97\linewidth]{RMSvsD_1DL10to16_J1D0_95_stagh_hz0_5_hx0_5l.png}
		\caption{  \label{fig:S_omega}
			Entropy and temperature using a continuous approximation to the cumulative density of states. 
			\textbf{(A,C)} $S_{\Omega} = \ln\Omega'(E)$. 
			\textbf{(B,D)} Resultant inverse temperature $\beta_{\Omega} = \partial_E S_{\Omega}$ compared with
			the canonical temperature $\beta_C$. 
			\textbf{(A,B)} \textit{XXZ} chain, Eq.\ \eqref{eq:Ham_chain},  with $L=12$, $J=1$, $\Delta=0.95$,
			$h_z = h_x = 0.5$.  
			\textbf{(C,D)} $5\times4$ square \textit{XXZ} lattice, Eq.\ \eqref{eq:Ham_square},  with $N=4$.
			\textbf{(E)} RMS distance between $\beta_{\Omega}(E)$ and $\beta_C(E)$, versus $D$ $(2^L)$, for the
			spin chain, Eq.\ \eqref{eq:Ham_chain}, with the same parameters as in \textbf{(A,B)} and variable $L$.
		}
	\end{figure}

	
	
	\subsection{Explaining the finite-size deviation} \label{sec:omega_explain_deviation}
	
	We discuss two ways of understanding the finite-size deviation of the temperature obtained using
	$S_{\Omega}$ as defined in Eq.\ \eqref{eq:def-OmegaEntropy}.
	
	First, using Eq.\ \eqref{eq:def-OmegaEntropy} for the entropy means omitting the $\ln\dE$ term from
	the definition, Eq.\ \eqref{eq:entropy_from_dos}, which can be written as 
	\begin{equation}\label{eq:S_Somega}
		S = S_{\Omega} + k_B\ln\dE . 
	\end{equation}
	Since we avoided making an explicit choice for $\dE$, it is not obvious what the effect of dropping
	the second term is, but we can analyze different cases:
	\begin{itemize}
		\item If we consider $\dE$ to be energy-independent, then $S_{\Omega}$ will lead to the same
		temperature as obtained from the full microcanonical entropy $S$.  However, we know from Section
		\ref{sec:const_dE} that $S$ obtained using an energy-independent $\dE$ leads to considerable
		finite-size deviations in the temperature.
		\item On the other hand, if $\dE$ were to be energy-dependent, e.g., if it were designed to cancel
		the sub-leading deviations from the canonical ensemble as in Section \ref{sec:energy_dependent_dE},
		then $S_{\Omega}'(E)$ will differ from $S'(E)$, and we would again get finite-size deviations.
	\end{itemize}
	Thus, it appears that
	$S_{\Omega}'(E)$ can be expected to deviate from $\beta_c$ in either case.  
	
	



	




	
	Second, we note that Eq.\ \eqref{eq:def-OmegaEntropy} applies the logarithm to a dimensionful
	quantity, which strictly speaking is not allowed.  For consistency, one needs to multiply the
	argument of the logarithm in Eq.~\eqref{eq:def-OmegaEntropy} by a quantity having the dimensions of
	energy; let us call this quantity $\epsilon$. Then Eq.\ \eqref{eq:def-OmegaEntropy} should really
	have the form
	\begin{equation}
		S_{\Omega}=\ln\left[g(E)\epsilon\right] = \ln\left[\Omega'(E)\epsilon\right] . 
	\end{equation}
	One can then carry out the same saddle point approximation as previously performed (subsection
	\ref{sec:saddlepoint}) with the original definition of $S$, leading to
	\begin{equation}\label{eq:omega_saddle}
		\frac{S_{\Omega}}{k_B} \approx \beta_c E- \beta_c F(\beta_c)) + \ln\epsilon -\ln\sqrt{2\pi k_{B}T_{c}^{2}C_{c}}.
	\end{equation}
	Unless $\epsilon$ is carefully chosen (as we did for $\dE$ in Section
	\ref{sec:energy_dependent_dE}), the last term will lead to finite-size deviations.  The procedure in
	this section avoids specifying $\dE$ and ignores the need for the quantity $\epsilon$.  Thus, one
	would expect deviations due to the $-\ln\sqrt{T_{c}^{2}C_{c}}$ term, essentially of the same type as
	that encountered in \ref{sec:const_dE} when using an explicit energy-independent value of $\dE$.
	
	
	\section{Concluding Discussion \& Context}  \label{sec:concl}
	
	The equivalence of microcanonical and canonical ensembles emerges in the infinite-size limit.
	Since statistical mechanics in isolated systems is being intensively
	discussed through finite-size examples obtained by numerical diagonalization, it is important to
	understand deviations from ensemble equivalence in systems of such sizes.  In this work, we
	contribute to this question by investigating and comparing various ways of computing the
	microcanonical entropy, and then comparing the resultant temperatures (via Eq.\ \eqref{eq:entropy_temp})
	to the canonical temperature (Eq.\ \eqref{eq:canonical_energy}).
	
	The microcanonical entropy $S(E)$ is defined as in Eqs.\ \eqref{eq:entropy_def} and \eqref{eq:dos}.
	Inspired by the discussion of Ref.\ \cite{Gurarie_Equivalence_AJoP2007}, we explored three ways of
	calculating $S(E)$ numerically: in terms of a constant-width energy window (Section
	\ref{sec:const_dE}), using an energy-dependent window with the energy-dependence designed to cancel
	sub-leading terms (Section \ref{sec:energy_dependent_dE}), and using an approximation that avoids the
	energy window altogether (Section \ref{sec:Omega}).
	
	We have demonstrated that the energy-dependent window,
	$\dE(E) = \alpha^{-1}\sqrt{2\pi k_B T_{c}^{2}C_{c}}$, works extremely well for the sizes under
	consideration.  The use of the energy-dependent window was suggested by Ref.\
	\cite{Gurarie_Equivalence_AJoP2007} with $\alpha=1$, designed to exactly cancel the sub-leading terms
	in Eq.\ \eqref{eq:entropy_final}.  We have shown (Figure \ref{fig:S_eps}) that, for the sizes under
	question, a larger $\alpha$ is required, because for $\alpha=1$ the windows exceed the
	energy scale of variation of the density of states.  A consequence of this result is that, since we
	have no further criterion for fixing $\alpha$, the microcanonical entropy is only defined up to an
	arbitrary additive constant.  The constant is sub-extensive (and thus unimportant in the
	infinite-size limit), and an additive constant does not affect the temperature.  Nevertheless, the
	arbitrariness is unsatisfactory. 
	In fact, a pleasing aspect of the suggestion of Ref.\ \cite{Gurarie_Equivalence_AJoP2007} was a reasonable criterion for defining entropy without such an arbitrary constant.  For the size ranges under consideration in this work,
	the $\alpha=1$ prescription is not usable, so we are forced to accept an arbitrary shift. 
	
	The other two procedures (energy-independent $\dE$ and $S\approx S_\Omega$) both lead to very noticeable
	finite-size deviations.  The deviations in the constant-$\dE$ case are expected from the sub-leading
	corrections.  For the procedure in Section \ref{sec:Omega} using  $S\approx
	k_B\ln(\pd{\Omega(E)}{E})$, we have argued that the deviations are essentially of the same type as
	that in the constant-$\dE$ case.
	
	Numerical calculations of microcanonical entropy and/or temperature have appeared in the recent
	literature \cite{SantosPolkovnikovRigol_Entropy_PRL2011,Kourehpaz_TypicalityChaos_arxiv2021,
		RussomannoFavaHeyl_Entropy_PRB2021, SilvaGoold_Temperature_PRA2022}.  Refs.\
	\cite{Kourehpaz_TypicalityChaos_arxiv2021, SilvaGoold_Temperature_PRA2022} have used the
	$S\approx S_\Omega$ approximation.  The resulting temperature shown in Ref.\
	\cite{Kourehpaz_TypicalityChaos_arxiv2021} (Figure 4) appears to have similar deviations from the
	canonical temperature as we have presented and analyzed in Section \ref{sec:Omega}.  The density of
	states is approximated in Ref.\ \cite{Kourehpaz_TypicalityChaos_arxiv2021} via a polynomial fit to
	the cumulative spectral function and in Ref.\ \cite{SilvaGoold_Temperature_PRA2022} via the kernel
	polynomial method \cite{WeisseFehske_KernelMethod_RMP2006,CiracBanuls_PRL2020_spectralanalysis}.  In
	Ref.~\cite{SantosPolkovnikovRigol_Entropy_PRL2011}, the microcanonical entropy has been calculated
	with $\dE$ ``determined by the energy uncertainty'' --- this is presumably equivalent to the
	energy-dependent window we explored in Section \ref{sec:energy_dependent_dE}.
	
	For finite quantum systems, the entanglement entropy of a subsystem is often discussed as
	representing the thermal entropy \cite{Deutsch_Entropy_IOP2010, Deutsch_Sharma_PRE2013,
		GarrisonGrover_SingleEstate_PRX2018, LuGrover_RenyiEntropy_PRE2019, SekiYunoki_thermal_PRR2020}.
	The reason is that, in a chaotic (thermalizing) quantum system, it is expected that the reduced
	density matrix of subsystems smaller than half the system should resemble thermal density matrices
	\cite{LindenPopescu_ThermEq_PRE2009,Muller2015, Greinergroup_thermalization_Science2016,
		Dymarsky_Lashkari_Liu_PRE2018, GarrisonGrover_SingleEstate_PRX2018, SekiYunoki_thermal_PRR2020,
		CBurkeNakerstHaque_Temperature_arxiv21}.  It is amusing to note that the entanglement entropy is
	obtained from eigenstates, whereas the entropies and temperatures studied in this work are derived from eigenenergies.  Ref.\ \cite{GarrisonGrover_SingleEstate_PRX2018} describes the finite-size behavior of the deviation of the entanglement entropy from the canonical entropy for a particular
	spin chain.  The behavior of the temperature derived using Eq.\ \eqref{eq:entropy_temp} from the
	entanglement entropy (interpreted as the entropy) would be interesting to examine in future work.
	
	It has been argued that entropy for finite-size systems should be defined as $S\sim \ln\Omega(E)$
	\cite{PearsonEtAl_GibbsEntropy_PRA1985, TalknerHanggiMorillo_MicrocanFluctuation_PRE2008,
		DunkelHilbert_NegativeT_Nat2014, HilbertHanggi_ThermoIsolated_PRE2014,
		Campisi_MicroEntropy_PRE2015, Hanggi_Temperature_RoyalSoc2016}, instead of the more common
	$S\sim \ln\Omega'(E)$ examined here.   This approach avoids the appearance of negative
	temperatures even in systems with finite Hilbert space dimension.  
	It may be interesting to ask how the deviations from ensemble equivalence behave under this
	alternate definition. 
	

	



	\section*{Acknowledgments}
	\noindent\textit{Acknowledgments.-} PCB thanks Maynooth University (National University of Ireland,
	Maynooth) for funding provided via the John \& Pat Hume Scholarship.  This work was in part
	supported by the Deutsche Forschungsgemeinschaft under grants SFB 1143 (project-id 247310070).  The
	authors acknowledge the Irish Centre for High-End Computing (ICHEC) for the provision of
	computational facilities.  The authors are grateful for useful discussions with Tarun Grover, Goran
	Nakerst, and Paul Watts.

\normalem
