\documentclass[prl,showpacs,twocolumn,amsmath,amssymb,floatfix,superscriptaddress]{revtex4-1}

\usepackage{graphicx}
\usepackage{pstricks,pst-node,pst-text,pst-3d}
\usepackage{color}

\graphicspath{{Figures_local/}}

\newcommand{\pdagger}{{\phantom{\dagger}}}
\newcommand{\dt}{\Delta\tau}
\newcommand{\reff}[1]{Fig.\ \ref{fig:#1}}
\newcommand{\reffl}[1]{Figure\ \ref{fig:#1}}
\newcommand{\refq}[1]{(\ref{eq:#1})}
\newcommand{\myparagraph}[1]{{\it #1} -- }


\newcommand{\neel}{N\'{e}el}
\newcommand{\TN}{T_{\text{N}}}
\newcommand{\TND}{T_{\text{N}}^{\text{DMFT}}}
\newcommand{\old}[1]{{\green[{\bf old:} #1]}}
\newcommand{\nb}[1]{{\red[#1]}}
\newcommand{\eg}[1]{{\blue[#1]}}
\newcommand{\NNcorr}{\langle\hat{\boldsymbol{\sigma}}_i\cdot\hat{\boldsymbol{\sigma}}_j\rangle}

\setlength\dbltextfloatsep{14pt plus 2pt minus 4pt }
\setlength\textfloatsep{14pt plus 2pt minus 4pt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}

	\title{Double occupancy as a universal probe for antiferromagnetic correlations\\
	       and entropy in cold fermions on optical lattices}

\author{E.~V.~Gorelik}
\affiliation{Institute of Physics, Johannes Gutenberg University, Mainz, Germany}
\author{D.~Rost}
\affiliation{Institute of Physics, Johannes Gutenberg University, Mainz, Germany}
\author{T.~Paiva}
\affiliation{Instituto de Fisica, Universidade Federal do Rio de Janeiro, Brazil}
\author{R.~Scalettar}
\affiliation{Department of Physics, UC Davis, USA}
\author{A.~Kl\"umper}
\affiliation{University of Wuppertal, Wuppertal, Germany}
\author{N.~Bl\"umer}
\affiliation{Institute of Physics, Johannes Gutenberg University, Mainz, Germany}

\date{\today}
  \begin{abstract}
     % We study the dimensional dependence of  the double occupancy $D$ using dynamical mean-field theory, direct quantum Monte Carlo (in dimensions $d=2,3$), and Bethe Ansatz (in $d=1$), and establish $D$ as a local probe of antiferromagnetic (AF) correlations. As a function of entropy $s=S/(N k_{\text{B}})$, $D$ is nearly universal with respect to dimension; the minimum in $D(s)$ approaches $s\approx \log(2)$ at strong coupling. Long-range order appears hardly relevant for the current search of AF signatures in cold fermions. Thus, experimentalists need not achieve $s<\log(2)/2$ and should consider lower dimensions, for which the AF effects are larger.

We demonstrate that the signatures of antiferromagnetic (AF) correlations in the double occupancy $D$ persist in all dimensions down to $d=1$, therefore establishing $D$ as a local probe of AF correlations. As a function of entropy $s=S/(N k_{\text{B}})$, $D$ is nearly universal with respect to dimension; the minimum in $D(s)$ approaches $s\approx \log(2)$ at strong coupling, also marking the applicability limit of spin models. Long-range order appears hardly relevant for the current search of AF signatures in cold fermions. Thus, experimentalists need not achieve $s<\log(2)/2$ and should consider lower dimensions, for which the AF effects are larger. \nb{$D$ measurable! unique AF signature almost within current $s$ reach, new: weak coupling, cubic $T_N$ nonuniversal (sc vs.\ bcc) }
%The relevance of long-range AF order for the current search of AF signatures in cold fermions is revisited: experimentalists need not achieve $s<\log(2)/2$ and should consider lower dimensions, for which the AF effects are larger.
  \end{abstract}
%  \begin{abstract}
%  	We verify signatures of antiferromagnetic (AF) correlations in the double occupancy $D$ [Gorelik et al., PRL {\bf 105}, 065301 (2010)] and study their dimensional dependence using direct quantum Monte Carlo in dimensions $d=2,3$ and Bethe Ansatz in $d=1$. We find quantitative agreement with dynamical mean-field theory (DMFT) in the cubic case and qualitative agreement down to $d=1$. As a function of entropy $s=S/(N k_{\text{B}})$, $D$ is nearly universal with respect to $d$; the minimum in $D(s)$ approaches $s\approx \log(2)$ at strong coupling. Long-range order appears hardly relevant for the current search of AF signatures in cold fermions. Thus, experimentalists need not achieve $s<\log(2)/2$ and should consider lower dimensions, for which the AF effects are larger.
%  \end{abstract}
  \pacs{67.85.-d, 03.75.Ss, 71.10.Fd, 75.10.-b}
  \maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Figures/D_S_vs_T_U15etc}
\caption{(Color online)
Hypercubic lattice ($1\le d\le 3$) at strong coupling:
a) $D(T)$ as estimated from  DMFT ($d=3$, circles), QMC ($d=2$, $3$, squares/diamonds), and BA ($d=1$, dash-dotted line). 
b) Corresponding estimates of  entropy per particle $s=S/N$. All interactions correspond approximately to the ground state Mott transition at $U/(\sqrt{Z}t)\approx 6$. 
}\label{fig:3d_SvsT}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Figures/D_vs_S_U15etc}
\caption{(Color online)
Hypercubic lattice ($1\le d\le 3$) at strong coupling: 
Double occupancy as a function of entropy per particle. In all cases, 
the minimum of the double occupancy corresponds to $s\approx \log(2)$ (dotted line). 
The shaded area indicates the nonmagnetic contribution to $D$. 
Inset: the effect of interaction strength on the double occupancy for $d=1$.
}\label{fig:dim_DvsS}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Figures/Eall_NN_NNN_s_U15etc}
\caption{(Color online)
Hypercubic lattice ($1\le d\le 3$) at strong coupling: further observables as 
a function of entropy per particle. 
a) Rescaled total and kinetic energies as a function of entropy per particle.
b) Spin-spin correlations $\langle S_i \cdot S_{i+\delta}\rangle$ 
for the nearest neighbours ($\delta_{d=1}=1$, $\delta_{d=2}=(1,0)$ , $\delta_{d=3}=(1,0,0)$) 
and for the next-nearest neighbours ($\delta_{d=1}=2$, $\delta_{d=2}=(1,1)$, $\delta_{d=3}=(1,1,0)$).
Inset: long range order spin correlations.
}\label{fig:dim_ENNvsS}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\includegraphics[width=\columnwidth]{Figures/D_Ekin_cubic_s_DMFT_QMC_sN}
\caption{(Color online)
Cubic lattice ($d=3$): 
Comparison of DMFT (circles) and direct QMC (diamonds) results for the entropy 
dependence of the a) double occupancy $D(s)$, and b) kinetic energy $E_{kin}$. 
Inset: The  \neel\ entropy per particle $s_N(U)$, 
estimated from DMFT (circles) and QMC (diamonds). This lines show critical entropies 
for the Heisenberg model ($s^{\text{Heisenberg}}_N=\frac{1}{2}\log 2$) 
and Weiss mean-field theory ($s^{\text{Weiss}}_N=\log 2$).
%Arrows: \neel\ temperature $T_N$ (QMC estimates \cite{Staudt00}). 
}\label{fig:3d_D_Ekin}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Bibliography
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace{-3ex}
\begin{thebibliography}{16}

\vspace{-2ex}
\bibitem{Tokura03} 
Y.\ Tokura, 
Phys.\ Today {\bf 56}, 50 (2003).

\bibitem{Dagotto05} 
E.\ Dagotto, 
Science {\bf 309}, 257 (2005).

\bibitem{AnisimovBook2010} 
V.\ Anisimov and Y.\ Izyumov, 
{\em Electronic Structure of Strongly Correlated Materials}, 
Springer Series in Solid-State Sciences, Vol. 163 (Springer, Berlin, 2010).

\bibitem{Hofstetter_PRL02}
W. Hofstetter, J. I. Cirac, P. Zoller, E. Demler, and M. D. Lukin
Phys.\ Rev.\ Lett. {\bf  89}, 220407 (2002).

\bibitem{Zoller05} 
D.\ Jaksch and P.\ Zoller, 
Ann.\ Phys.\ (NY) {\bf 315}, 52 (2005).

\bibitem{Esslinger10} 
T.\ Esslinger, 
Ann.\ Rev.\ Cond.\ Matt.\ Phys.\ {\bf 1}, 129 (2010).

\bibitem{Koehl05} 
M.\ K\"ohl, H.\ Moritz, Th.\ St\"oferle, K.\ G\"unter,  and T.\ Esslinger, 
Phys.\ Rev.\ Lett.\ {\bf 94}, 080403 (2005). 

\bibitem{Bloch08_ferm} 
U.\ Schneider {\it et al.},
Science {\bf 322}, 1520 (2008).

\bibitem{Esslinger08_Nature} 
R.\ J\"{o}rdens, N.\ Strohmaier, K.\ G\"{u}nter, H.\ Moritz, and T.\ Esslinger, 
Nature {\bf 455}, 204 (2008).

\bibitem{Santos10}
M. Colom\'e-Tatch\'e, C. Klempt, L. Santos,  and T. Vekua,
arXiv:1009.2606.

\bibitem{Greif10} 
D.\ Greif, L.\ Tarruell, T.\ Uehlinger, R.\ J\"ordens, and T.\ Esslinger,
Phys.\ Rev.\ Lett.\ {\bf 106}, 145302 (2011).

\bibitem{Joerdens10}  
R.\ J\"ordens {\it et al.}, 
Phys.\ Rev.\ Lett.\ {\bf 104}, 180401 (2010).

\bibitem{FWerner05} 
F.\ Werner, O.\ Parcollet, A.\ Georges, and S.\ R.\ Hassan, 
Phys.\ Rev.\ Lett.\ {\bf 95}, 056401 (2005).

\bibitem{Wessel10} 
S.\ Wessel, 
Phys.\ Rev.\ B {\bf 81}, 052405 (2010).

\bibitem{Kollath06}
C.\ Kollath, A.\ Iucci, I.\ P.\ McCulloch, and T.\ Giamarchi, 
Phys.\ Rev.\ A {\bf 74}, 041604(R) (2006).

\bibitem{Trotzky10}
S.\ Trotzky, Yu-Ao Chen, U.\ Schnorrberger, P.\ Cheinet, and I.\ Bloch, 
Phys.\ Rev.\ Lett.\ {\bf 105}, 265303 (2010).

\bibitem{Corcovilos10}
%T.\ A.\ Corcovilos, S.\ K.\ Baur, J.\ M.\ Hitchcock, E.\ J.\ Mueller, and R.\ G.\ Hulet, Phys.\ Rev.\ A {\bf 81}, 013415 (2010).
T.\ Corcovilos, S.\ Baur, J.\ Hitchcock, E.\  Mueller, and R.\  Hulet, Phys.\ Rev.\ A {\bf 81}, 013415 (2010).

\bibitem{fn:length}
Current experimental trap geometries ($\sim 10^5$ fermions) imply AF cores smaller than $40\times 40\times 20$ lattice sites, i.e. each AF site will ``feel'' a boundary within 10 sites.

\bibitem{Takahashi77}
M.\ Takahashi, 
J.\ Phys.\ C {\bf 10}, 1289-7301 (1977).

\bibitem{Gorelik_PRL10}
E.\ V.\ Gorelik, I.\ Titvinidze, W.\ Hofstetter, M.\ Snoek, and N.\ Bl\"umer, 
Phys. Rev. Lett. {\bf 105}, 065301 (2010).

\bibitem{Fuchs11}
S.\ Fuchs {\em et al.},
Phys.\ Rev.\ Lett.\ {\bf 106}, 030401 (2011).

\bibitem{Maier_RMP_2005}
T.\ Maier, M.\ Jarrell, T.\ Pruschke, and M.\ Hettler, 
Rev. Mod. Phys. {\bf 77}, 1027 (2005).

\bibitem{De_Leo_PRA_2011}
L.\ De Leo, J.\ Bernier, C.\ Kollath, A.\ Georges, and V.\ W. Scarola,
Phys. Rev. A {\bf 83}, 023606 (2011).

\bibitem{Hirsch86} J.\  Hirsch and R.\  Fye, Phys.\ Rev.\ Lett.\ {\bf 56}, 2521 (1986).

\bibitem{Blankenbecler81} 
R. Blankenbecler, D. J. Scalapino, and R. L. Sugar, 
Phys. Rev. D {\bf 24}, 2278 (1981).

\bibitem{Bluemer05} 
N.\ Bl\"umer and E. Kalinowski, 
Physica B {\bf 359}, 648 (2005).

\bibitem{Bluemer07}  
N.\ Bl\"umer, 
Phys.\ Rev.\ B {\bf 76}, 205120 (2007).

\bibitem{BA}
G.~J\"uttner, A.\ Kl\"umper, and J.\ Suzuki, 
Nucl.\ Phys.\ B {\bf 522}, 471 (1998).

\bibitem{Kent05} 
P.\ R.\ C.\ Kent, M.\ Jarrell, T.\ A.\ Maier, and Th.\ Pruschke, 
Phys. Rev. B {\bf 72}, 060411(R) (2005).

\bibitem{Staudt00} 
R.\ Staudt, M.\ Dzierzawa, and A.\ Muramatsu, 
Eur.\ Phys.\ J.\ B {\bf 17}, 411 (2000).

\bibitem{Paiva10}
Th. Paiva, R. Scalettar, M. Randeria, and N. Trivedi, 
Phys.\ Rev.\ Lett.\ {\bf 104}, 066406 (2010).

\bibitem{fn:supplement}
See Supplemental Material at \dots\ for an analysis of finite-size and Trotter errors.

\bibitem{fn:scale}
All scales are set by the root mean square energy $\langle \epsilon^2\rangle_{U=0}^{1/2}=\sqrt{Z} t$ (for coordination number $Z$).
% not by the band edges ($\sim\! Z$). 

%\bibitem{fn:2d}
%Deviations for $d=2$ in Figs.\ \ref{fig:3d_SvsT} and \ref{fig:dim_DvsS} are due to a slightly too low interaction 
%$U/t=12$ instead of the scaling value $15/\sqrt{6/4}\approx 12.25$.
%$U/t=12<15/\sqrt{6/4}\approx 12.25$.\nb{rm}

\bibitem{Oitmaa94} 
J.\ Oitmaa, C.\ J.\ Hamer, and Z.\ Weihong, 
Phys.\ Rev.\ B {\bf 50}, 3877 (1994).

\bibitem{Weihong91} 
Z.\ Weihong, J.\ Oitmaa, and C.\ J.\ Hamer, 
Phys.\ Rev.\ B {\bf 43}, 8321 (1991).

\bibitem{Sandvik97} 
A.\ W.\ Sandvik, 
Phys.\ Rev.\ B {\bf 56}, 11678 (1997).

\bibitem{Frustration} 
E.\ V.\ Gorelik and N.\ Bl\"umer, in preparation.

\bibitem{Snoek_NJP08} 
M.\ Snoek, I.\ Titvinidze, C.\ T\"oke, K.\ Byczuk, and W.\ Hofstetter, 
New J.\ Phys. {\bf 10}, 093008 (2008).

\bibitem{Helmes_PRL08} 
R.\ W.\ Helmes, T.\ A.\ Costi, and A.\ Rosch, 
Phys.\ Rev.\ Lett. {\bf 100}, 056403 (2008).

\bibitem{Bluemer11CCP} 
N.\ Bl\"umer and E.\ V.\ Gorelik, 
Comp.\ Phys.\ Comm.\ {\bf 118}, 115 (2011).

\end{thebibliography}

\end{document}

