Content-Type: multipart/mixed; boundary="-------------0009181837889" This is a multi-part message in MIME format. ---------------0009181837889 Content-Type: text/plain; name="00-369.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-369.comments" 2000 MSC: 82B10, 82B24, 82D40. To appear in the proceedings of the XIII International Congress of Mathematical Physics, London, July 18-24, 2000 ---------------0009181837889 Content-Type: text/plain; name="00-369.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-369.keywords" interfaces, Falicov-Kimball model, Heisenberg model, XXZ model, droplet states, 111 interface ---------------0009181837889 Content-Type: application/x-tex; name="london2000BN.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="london2000BN.tex" %----------------------------------------------------------------------- % Beginning of article.tex %----------------------------------------------------------------------- % % AMS-LaTeX 1.2 sample file for generic proceedings. % \documentclass{gen-p-l} \usepackage[dvips]{graphics} \hyphenation{states} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} % \newcommand{\Rl}{\mathbb{R}} \newcommand{\Nl}{\mathbb{N}} \newcommand{\Ir}{\mathbb{Z}} \newcommand{\Cx}{\mathbb{C}} \newcommand{\A}{\mathcal{A}} \newcommand{\HH}{\mathcal{H}} \newcommand{\KK}{\mathcal{K}} \newcommand{\N}{\mathcal{N}} \newcommand{\Proj}{{\rm Proj}} \newcommand{\Span}{{\rm span}} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} \newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} \newcommand{\ket}[1]{\lvert#1\rangle} \newenvironment{alphlist}{% \def\theenumi{\alph{enumi}}\def\theenumii{\roman{enumii}}% \def\labelenumi{(\theenumi)}\def\labelenumii{(\theenumii)}% \begin{enumerate}% }{% \end{enumerate}} % Blank box placeholder for figures (to avoid requiring any % particular graphics capabilities for printing this document). \newcommand{\blankbox}[2]{% \parbox{\columnwidth}{\centering % Set fboxsep to 0 so that the actual size of the box will match the % given measurements more closely. \setlength{\fboxsep}{0pt}% \fbox{\raisebox{0pt}[#2]{\hspace{#1}}}% }% } \begin{document} \title{Interfaces and droplets in quantum lattice models} \author{Bruno Nachtergaele} \address{Department of Mathematics, University of California, Davis, Davis, CA 95616-8633} \email{bxn@math.ucdavis.edu} \thanks{This material is based on work supported by the National Science Foundation under Grant No. DMS0070774. \newline\indent Copyright \copyright\ 2000 Bruno Nachtergaele. Reproduction of this article in its entirety, by any means, is permitted for non-commercial purposes.} \subjclass{82B10, 82B24, 82D40} \date{August 16, 2000.} \begin{abstract} This paper is a short review of recent results on interface states in the Falicov-Kimball model and the ferromagnetic XXZ Heisenberg model. More specifically, we discuss the following topics: 1) The existence of interfaces in quantum lattice models that can be considered as perturbations of classical models. 2) The rigidity of the 111 interface in the three-dimensional Falicov-Kimball model at sufficiently low temperatures. 3) The low-lying excitations and the scaling of the gap in the 111 interface ground state in the ferromagnetic XXZ Heisenberg model in three dimensions. 4) The existence of droplet states in the XXZ chain and their properties. \end{abstract} \maketitle \section*{Introduction} Domain walls or interfaces play an important role in many transport phenomena and are essential for the understanding of dynamical processes such as magnetic ordering and magnetization reversal. In many interesting situations quantum mechanics is important to correctly describe the stability, fluctuations, and excitations of magnetic interfaces, and we expect that this is even more true for a reliable description of the dynamics. Equilibrium states describing interfaces in the Ising model are well-known since the seminal work by Dobrushin \cite{Dob}, and many interesting results on interfaces in classical models have been proved since then. In comparison, there have been very few results for quantum models. In the last few years however, some progress has been made and it is the purpose of this paper to review the main results that have been obtained. In the next four sections we will discuss the following results: \begin{enumerate} \item Existence of interfaces in quantum lattice models that can be considered as perturbations of classical models. \item Rigidity of the 111 interface in the Falicov-Kimball model. \item Gapless excitations of the 111 interface in the XXZ Heisenberg model. \item Droplet states in the XXZ chain. \end{enumerate} \section{Perturbations of classical models} Often, quantum lattice models can be thought of as perturbations of a classical spin system when some parameter in the Hamiltonian is not too large (or too small). Under certain conditions its behavior can then be described by a suitable perturbation theory. The following result is due to Borgs, Chayes, and Fr\"ohlich \cite{BCF}. In order to simplify the presentation we do not describe the result in its most general form here. Let $n$ be a fixed integer $\geq 2$. For every finite subset $\Lambda\subset\Ir^d$, let the Hilbert space for the system in volume $\Lambda$ be given by $$ \HH_\Lambda = \bigotimes_{x\in \Lambda}\HH_x,\quad \HH_x = \Cx^n, \mbox{ for all } x . $$ Consider translation invariant local Hamiltonians of the form \begin{equation} H_\Lambda=\sum_{x,y\in\Lambda,\abs{x-y}=1}h_{xy} + \lambda \sum_{A\subset\Lambda}V_A \label{BCFmodel}\end{equation} where $h_{xy}$ is a translation invariant isotropic nearest neighbor interaction that is diagonal in a tensor product basis labeled by ``classical configurations'', i.e., there is a function $h(\sigma_0, \sigma_1)$, such that $h_{xy}\ket{\sigma}=h(\sigma_x,\sigma_y)\ket{\sigma}$. Here, $\sigma=\{\sigma_x\}_{x\in\Lambda}, 1\leq\sigma_x\leq n$, and $\ket{\sigma}=\bigotimes_{x\in\Lambda}\ket{\sigma_x}\in\HH_\Lambda$, and $\{\ket{1},\ldots,\ket{n}\}$ is a basis of $\Cx^n$. Assume that this classical nearest neighbor interaction has exactly two ground state, of the form $\ket{\sigma_0}\otimes\ket{\sigma_0}$. The quantum perturbation given by the $V_A$ is also assumed to be translation invariant and sufficiently short range in the sense that, for some $r\geq 0$, sufficiently large, $$ \sum_{A\subset\Ir^d,\, 0\in A} \Vert V_A\Vert e^{r D(A)}<+\infty $$ where $\Vert\,\cdot\,\Vert$ is the usual operator norm, and $D(A)$ is the size of the smallest tree subgraph of $\Ir^d$ containing the vertices of $A$. \begin{theorem}[Borgs-Chayes-Fr\"ohlich \cite{BCF}]\label{thm:BCF} Under the assumptions described above, and if $d\geq 3$, the model defined by (\ref{BCFmodel}) has Gibbs states with a rigid interface in the principle coordinate directions, provided that the inverse temperature $\beta$ is sufficiently large and $\beta\lambda$ is sufficiently small. \end{theorem} The proof of this theorem relies on recently developed techniques to derive Pirogov-Sinai type results for quantum lattice models \cite{DFF, BKU}. The next two results are examples of interesting situations where perturbation theory as described above, does not apply. They also illustrate two features that distinguish quantum models from their classical counterparts: 1) in a system that has classically several equivalent energy minimizing configurations, ground state selection will typically occur in the quantum model in the same way as tunneling in a symmetric double well potential leads to a unique ground state, and 2) finite-state models can have continuous symmetries and gapless energy spectra. \section{Rigidity of the 111 interface in the Falicov-Kimball model} It is expected that the three-dimensional Ising model does not have Gibbs states with an interface in the 111 direction, due to a large ground state degeneracy for the obvious boundary conditions that would favor such an interface. A proof of the Solid-on-Solid version of this statement was given by Kenyon \cite{Keny}. In contrast, in the Falicov-Kimball and Heisenberg XXZ models, in dimensions $\geq 2$, this degeneracy is lifted by the ``quantum'' terms, i.e., ground state selection occurs. Our next result is that, for the half-filled, neutral Falicov-Kimball model in three dimensions the 111 interface is rigid at sufficiently low temperatures \cite{DMN}. The Falicov-Kimball model consists of two subsystems living on the same lattice $\Ir^d$: ``ions'', described by Ising type variables $W(x)\in\{0,1\}$, and ``electrons'', described by fermion operators $c^+_x,c_x$. The ions and electrons interact via an on-site Coulomb term as in the Hubbard model. The local Hamiltonians are \begin{equation} H_\Lambda=-\sum_{\{x,y\}\subset\Lambda,\vert x-y\vert =1} c^+_x c_y +2U\sum_{x\in\Lambda}W(x)c^+_x c_x \label{FKmodel}\end{equation} We only consider the half-filled neutral case, i.e., the number of ions and the number of electrons are both taken to be equal to half the number of lattice sites and we take $U>0$. Kennedy and Lieb proved that, for $d\geq 2$, this model has an Ising type phase transition, with two low-temperature phases dominated by the two checkerboard configurations for the ions \cite{KL}. In order to study the 111 interface in the three-dimensional model, we integrate out the fermion degrees of freedom using the methods of Messager and Miracle-Sol\'e \cite{MM}, and obtain an effective Hamiltonian for the ions in terms of the the Ising variables $s_x=(-1)^{\vert x\vert}(2W(x)-1)$: $$ H^{\rm eff}_\Lambda(\{s_x\})= -J(U,\beta)\sum_{\{x,y\}\cap\Lambda\neq\emptyset,\vert x-y\vert=1}s_xs_y +\sum_{B\cap\Lambda\neq\emptyset,\vert B\vert \geq 2} R_B(U) $$ where, and $00$, and where $n(B)$ is the length of the shortest closed path in the lattice visiting all sites in $B$ at least once, and where $R_B$ satisfies a bound of the form $$ \abs{R_B(U)} \leq \mbox{constant}\times\left(\frac{\rm constant}{U} \right)^{n(B)-1}. $$ Analysis of the terms with $n(B)=4$, indicates that ground state selection indeed occurs for the model with 111 boundary conditions. The following theorem shows that the 111 interface is stable against thermal fluctuations at sufficiently low temperatures. \begin{theorem}[Datta-Messager-Nachtergaele \cite{DMN}] There exist constants $U_0$, $\! C$, $c$, and $D$, such that for all $U> U_0$, and $\beta> DU^3$, there exists a Gibbs state $\omega_{111}$ of the half-filled neutral three-dimensional Falicov-Kimball model defined by (\ref{FKmodel}) with the following properties: \begin{eqnarray*} \omega_{111}(s_x)\geq \phantom{-}1-Ce^{-c\beta/U^3}&\mbox{if}& x_1+x_2+x_3\geq 1\\ \omega_{111}(s_x)\leq -1+Ce^{-c\beta/U^3}&\mbox{if}& x_1+x_2+x_3\leq -1 \end{eqnarray*} \end{theorem} The Falicov-Kimball model also exhibits phase separation, i.e., spontaneous formation of domains of different phases, in the ground state for certain intervals of densities \cite{Kend, KH}. \section{Gapless excitations of the 111 interface in the XXZ Heisenberg model} The spin 1/2 XXZ Heisenberg model defined by \begin{equation} H_\Lambda=-\sum_{x,y\in\Lambda,\vert x-y\vert =1} \frac{1}{\Delta}(S_x^{1} S_y^{1} + S_x^{2} S_y^{2})+ S_x^{3} S_y^{3} \label{XXZmodel}\end{equation} with $\Delta > 1$, can also be regarded as a perturbation of the Ising model. Kennedy proved that this model has a phase transition for all $d\geq 2$, $\Delta >1$ \cite{Kend}. Theorem \ref{thm:BCF} applies and interface states in the principal coordinate directions exist for $d\geq 3$, and $\beta$ and $\beta/\Delta$ large enough. One can also show that for $d\geq 2$, ground state selection occurs for suitable conditions that favor a diagonal (11, 111, etc.) interface. The model differs from the Falicov-Kimball model, however, in one important aspect: the existence of a diagonal interface is accompanied by the breaking of a continuous symmetry (rotations about the third axis), and consequently has a gapless spectrum above the ground states. These gapless excitations were first described by Koma and Nachtergaele in the two-dimensional case \cite{KN2, KN3}. Matsui proved their existence in all dimensions $\geq 2$ \cite{Mat}. I conjecture that the XXZ model has a 111 interface Gibbs state in three dimensions at low temperatures. This cannot be proved using the standard expansion techniques, which all rely on the existence of a gap. Therefore, it is reasonable to first try to understand the gapless excitations of the model. The third component of the total spin or, equivalently, the number of down spins, is a conserved quantity of the XXZ model. Therefore, it is natural to consider the interface problem and the excitation spectrum in the canonical ensemble where this quantity is fixed. The following theorem holds in the canonical ensemble. An equivalence-of-ensembles result is an essential part of its proof. \begin{theorem}[Bolina-Contucci-Nachtergaele-Starr \cite{BCNS1}] Let $070, $$ where $\delta(q,\nu)$ is an exponent between $0$ and $1/2$, depending on the filling factor $\nu$ of the interface plane and the parameter $q \in(0,1)$. \end{theorem} See \cite{BCN, BCNS1, BCNS2} for more details and related results. \section{Droplet states in the XXZ chain} The last result we would like to present is concerned with droplet states rather than interfaces. One does not expect that a localized droplet in a translation invariant system can be described by a stationary state, i.e., an eigenvector, of the Hamiltonian. On the other hand, it seems obvious that for a ferromagnetic model such as the XXZ Heisenberg Hamiltonian, any state of a given finite energy in a sufficiently large volume should be a superposition of states that predominantly consist of a small number of domains each described by one of the ground states. We have obtained an explicit result of this kind in the one-dimensional case. Let $H^{++}_{[1,L]}$ be the following Hamiltonian with ``+'' boundary fields for a chain of $L$ spins: \begin{equation} H^{++}_{[1,L]}=-\sum_{x=1}^L \frac{1}{\Delta}(S_x^{1} S_{x+1}^{1} + S_x^{2} S_{x+1}^{2})+ (S_x^{3} S_{x+1}^{3} - \frac{1}{4}) -A(\Delta)(S^3_1+S^3_L), \label{dropletmodel}\end{equation} where $S^i_x$ are the standard spin-1/2 matrices and $A(\Delta)= \frac{1}{2}\sqrt{1-\Delta^{-2}}$. $H^{++}_{[1,L]}$ commutes with $\sum_{x=1}^LS_n^3$, and therefore the subspaces $\HH_{L,n}$ of $\HH_{[1,L]}$ consisting of all states with a fixed number, $n$, of down spins, are invariant. The ground state in each of these invariant subspaces is a delocalized droplet of $n$ down spins, i.e., this ground state is dominated by configurations where all $n$ down spins are in a subinterval of length $n + O(1)$, but all such intervals are roughly equally probable. Thus, we expect that the droplet of size $n$ behaves as a particle freely moving in an interval of length $L-n+1$. Our results on the spectrum show that this particle has mass of order $q^{-n}$, where $q\in(0,1)$ is related to the anisotropy by $\Delta=(q+q^{-1})/2$. To state the precise result, let $\lambda_{L,n}(1) \leq \lambda_{L,n}(2) \leq \dots $ denote the eigenvalues of $H^{++}_{[1,L]}$ restricted to $\HH_{L,n}$, and let $\psi^{++}_{L,n}(1), \psi^{++}_{L,n}(2), \dots $ be the corresponding eigenvectors. It is convenient, though not necessary, to describe the spectral projection corresponding to the $L-n+1$ lowest eigenvalues rather than the individual eigenvectors. Define $$ \HH^{k}_{L,n} = \Span \{ \psi^{++}_{L,n}(j) : 1 \leq j \leq k\}\, . $$ and denote by $\Proj(\HH^{k}_{L,n})$ the corresponding orthogonal projection. We have no exact expressions for the this spectral projection for finite $n$, but we can describe it with an error of no more than order $q^{n/2}$ in the operator norm. To state the result we need to introduce the ``droplet'' subspace of $\HH_{L,n}$. We start with the definition of the kink and antikink states, which are ground state of the model with a different choice of boundary fields \cite{ASW, GW}. For every interval $[a,b]\subset\Ir$, and $n=0,1, \ldots, b-a+1$, the kink and antikink states, $\psi^{+-}_{[a,b]}(n)$ and $\psi^{-+}_{[a,b]}(n)$ respectively, are given by the following expressions: \begin{eqnarray*} \psi^{+-}_{[a,b]}(n) = \sum_{a \leq x_1 < \dots < x_n \leq b} q^{\sum_{k=1}^n (b+1-x_k)} \left( \prod_{k=1}^n S_{x_k}^- \right) \ket{\uparrow \dots \uparrow}_{[a,b]} \\ \psi^{-+}_{[a,b]}(n) = \sum_{a \leq x_1 < \dots < x_n \leq b} q^{\sum_{k=1}^n (x_k+1-a)} \left( \prod_{k=1}^n S_{x_k}^- \right) \ket{\uparrow \dots \uparrow}_{[a,b]} \end{eqnarray*} We now define the ``droplet'' states, describing a localized droplet of size $n$ centered at $x$: for $n\geq 0$ and $\floor{n/2} \leq x \leq L - \ceil{n/2}$ define \begin{equation} \xi_{L,n}(x) \ = \psi^{+-}_{[1,x]}(\floor{n/2}) \otimes \psi^{-+}_{[x+1,L]}(\ceil{n/2})\, . \end{equation} Here, for any real number $x$, $\floor{x}$ is the greatest integer $\leq x$, and $\ceil{x}$ is the least integer $\geq x$. The droplet subspace is now defined as the span of the droplet states at all possible positions: \begin{equation} \KK_{L,n} = \Span \{ \xi_{L,n}(x) : \, \floor{n/2}\leq x \leq L-\ceil{n/2} \}\, . \label{KLn}\end{equation} Note that $\dim \KK_{L,n} = L-n+1$, which is the number of subintervals of length $n$ of an interval of length $L$. The orthogonal projection onto $\KK_{L,n} $ is denoted by $\Proj(\KK_{L,n})$. \begin{theorem}[Nachtergaele-Starr \cite{NS}] \hfill\vspace{-1mm} \begin{alphlist} \item The first $L-n+1$ eigenvalues of $H^{++}_{[1,L]}$ restricted to $\HH_{L,n}$ are all approximately equal to $A(\Delta)$. More precisely we have the following: $$ \lambda_{L,n}(1),\dots, \lambda_{L,n}(L-n+1) \in [A(\Delta) - O(q^n),A(\Delta) + O(q^n)]\, . $$ \item For sufficiently large $n$, the first $L-n+1$ eigenvalues are separated from the rest of spectrum by a gap of approximate magnitude $\gamma=1-\Delta^{-1}$: $$ \lambda_{L,n}(L-n+2) \geq A(\Delta) + \gamma - O(n^{-1/4}). $$ \item The spectral subspace $\HH^{L-n+1}_{L,n}$ spanned by the first $L-n+1$ eigenvectors, for sufficiently large $n$, is close to the space $\KK_{L,n}$ defined in (\ref{KLn}). More precisely we have the the following bound on the norm of the difference of the corresponding projections: $$ \|\Proj(\KK_{L,n}) - \Proj(\HH^{L-n+1}_{L,n})\| = O(q^{n/2})\, . $$ \end{alphlist} \end{theorem} The gap of approximately $\gamma=1-\Delta^{-1}$ shown in part (b) of the Theorem is the energy required to overturn one spin in either the interior or the exterior of the droplet \cite{KN1, KN4}. This energy is less than the energy required to create an additional droplet at infinite distance, which is $2A(\Delta)$, i.e., droplets attract. Part (c) of the Theorem means that the eigenvectors can be approximated by a superposition of the states $\xi_{L,n}(x)$, which in turn are given by an explicit formula, with increasing accuracy as $n\to\infty$. Note that the Theorem also implies that, for any sequence of states with energies converging to $A(\Delta)$, we must have that the distances of these states to the subspaces $\KK_{L,n}$ converges to zero, i.e., in the limit the states contain exactly one droplet, which may either be localized or not. \begin{figure}[tb] \resizebox{12truecm}{8truecm}{\includegraphics{dropspec2.eps}} \parbox{11truecm}{\caption{\baselineskip=5 pt\small \label{fig:dropspec} Spectrum for $H^{++}_{[1,12]}$ when $\Delta = 2.125$ ($q=1/4$). The (trivial) $n=0$ state is not shown. Its energy is $-A(\Delta)$.} } \end{figure} Figure \ref{fig:dropspec} illustrates these features of the spectrum for $L=12$ and $\Delta=2.125$. The multiplet of droplet states is clearly visible, as well as the gap above it. \bibliographystyle{amsalpha} \begin{thebibliography}{99} \bibitem{ASW} F.C. Alcaraz, S.R. Salinas, and W.F. Wreszinski, \textit{Anisotropic ferromagnetic quantum domains}, Phys. Rev, Lett. {\bf 75} (1995), 930--933. \bibitem{BCN} O. Bolina, P. Contucci, and B. Nachtergaele, \textit{Path integral representation for interface states of the anisotropic Heisenberg model}, to appear in Rev. Math. Phys., \texttt{arXiv:math-ph/9908004}. \bibitem{BCNS1} O. Bolina, P. Contucci, B. Nachtergaele, and S. Starr, \textit{Finite-volume excitations of the 111 interface in the quantum XXZ model}, Comm. Math. Phys. {\bf 212} (2000), 63--91, \texttt{arXiv:math-ph/9908018}. \bibitem{BCNS2} O. Bolina, P. Contucci, B. Nachtergaele, and S. 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Poincar\'e, Probab. Statist., \textbf{33} (1997), 591--618. \bibitem{KN1} T. Koma and B. Nachtergaele, \textit{The spectral gap of the ferromagnetic XXZ chain}, Lett. Math. Phys. \textbf{40} (1997), 1--16. \bibitem{KN2} T. Koma and B. Nachtergaele, \textit{Low-lying spectrum of quantum interfaces}, Abstracts of the AMS, \textbf{17} (1996), 146, and unpublished notes. \bibitem{KN3} T. Koma and B. Nachtergaele, \textit{Interface states of quantum lattice models}, In Matsui, T. (eds.) \textit{Recent Trends in Infinite Dimensional Non-Commutative Analysis}. RIMS Kokyuroku \# 1035, Kyoto, 1998, 133--144. \bibitem{KN4} T. Koma and B. Nachtergaele, \textit{The complete set of ground states of the ferromagnetic XXZ chains}, Adv. Theor. Math. Phys., \textbf{2} (1998), 533--558, \texttt{arXiv:cond-mat/9709208} \bibitem{Mat} T. Matsui, \textit{On the spectra of the kink for ferromagnetic $XXZ$ models}, Lett. Math. Phys. {\bf 42} (1997), 229--239. \bibitem{MM} A. Messager and S. Miracle-Sol\'e, \textit{Low temperature states in the Falikov-Kimbal model}, Rev. Math. Phys. {\bf{8}} (1996) 271--299. \bibitem{NS} B. Nachtergaele and S. Starr, \textit{Droplet states of the XXZ Heisenberg chain}, \texttt{arXiv:math-ph/0009002}. \end{thebibliography} \end{document} %----------------------------------------------------------------------- % End of article.tex %----------------------------------------------------------------------- ---------------0009181837889 Content-Type: application/postscript; name="dropspec2.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dropspec2.eps" %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: /home/plane/s/sstarr/m/dropspec2.eps %%CreationDate: 08/23/ 0 18:46:25 %%DocumentNeededFonts: Helvetica %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 61 200 537 612 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp 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lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def %%EndProlog %%BeginSetup MathWorks begin 0 cap end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 61 200 537 612 MathWorks begin bpage %%EndPageSetup %%BeginObject: obj1 bplot /dpi2point 12 def portraitMode 0216 7344 csm 527 -2 5711 4946 MR c np 93 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6920 5190 PR 6 w gs 899 389 5363 4229 MR c np 0 sg 2213 0 899 4617 2 MP stroke 0 -4228 899 4617 2 MP stroke gr 0 sg gs 899 389 5363 4229 MR c np 85 0 1112 4430 2 MP stroke 85 0 1112 4409 2 MP stroke 85 0 1112 4374 2 MP stroke 85 0 1112 4327 2 MP stroke 85 0 1112 4271 2 MP stroke 85 0 1112 4208 2 MP stroke 85 0 1112 4141 2 MP stroke 85 0 1112 4076 2 MP stroke 85 0 1112 4015 2 MP stroke 85 0 1112 3964 2 MP stroke 85 0 1112 3924 2 MP stroke 85 0 1112 3899 2 MP stroke 85 0 1282 4291 2 MP stroke 85 0 1282 4284 2 MP stroke 85 0 1282 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2868 2 MP stroke 86 0 1622 2868 2 MP stroke 86 0 1622 2864 2 MP stroke 86 0 1622 2857 2 MP stroke 86 0 1622 2853 2 MP stroke 86 0 1622 2852 2 MP stroke 86 0 1622 2850 2 MP stroke 86 0 1622 2846 2 MP stroke 86 0 1622 2846 2 MP stroke 86 0 1622 2839 2 MP stroke 86 0 1622 2838 2 MP stroke 86 0 1622 2836 2 MP stroke 86 0 1622 2834 2 MP stroke 86 0 1622 2833 2 MP stroke 86 0 1622 2831 2 MP stroke 86 0 1622 2829 2 MP stroke 86 0 1622 2826 2 MP stroke 86 0 1622 2821 2 MP stroke 86 0 1622 2820 2 MP stroke 86 0 1622 2817 2 MP stroke 86 0 1622 2810 2 MP stroke 86 0 1622 2803 2 MP stroke 86 0 1622 2802 2 MP stroke 86 0 1622 2799 2 MP stroke 86 0 1622 2794 2 MP stroke 86 0 1622 2791 2 MP stroke 86 0 1622 2786 2 MP stroke 86 0 1622 2784 2 MP stroke 86 0 1622 2783 2 MP stroke 86 0 1622 2774 2 MP stroke 86 0 1622 2772 2 MP stroke 86 0 1622 2772 2 MP stroke 86 0 1622 2768 2 MP stroke 86 0 1622 2767 2 MP stroke 86 0 1622 2765 2 MP stroke 86 0 1622 2762 2 MP stroke 86 0 1622 2757 2 MP stroke 86 0 1622 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MP stroke 85 0 2133 3018 2 MP stroke 85 0 2133 3010 2 MP stroke 85 0 2133 3009 2 MP stroke 85 0 2133 3007 2 MP stroke 85 0 2133 3006 2 MP stroke 85 0 2133 3004 2 MP stroke 85 0 2133 3002 2 MP stroke 85 0 2133 3000 2 MP stroke 85 0 2133 2998 2 MP stroke 85 0 2133 2996 2 MP stroke 85 0 2133 2985 2 MP stroke 85 0 2133 2984 2 MP stroke 85 0 2133 2984 2 MP stroke 85 0 2133 2984 2 MP stroke 85 0 2133 2984 2 MP stroke 85 0 2133 2975 2 MP stroke 85 0 2133 2971 2 MP stroke 85 0 2133 2969 2 MP stroke 85 0 2133 2968 2 MP stroke 85 0 2133 2967 2 MP stroke 85 0 2133 2966 2 MP stroke 85 0 2133 2964 2 MP stroke 85 0 2133 2962 2 MP stroke 85 0 2133 2958 2 MP stroke 85 0 2133 2957 2 MP stroke 85 0 2133 2957 2 MP stroke 85 0 2133 2956 2 MP stroke 85 0 2133 2951 2 MP stroke 85 0 2133 2951 2 MP stroke 85 0 2133 2950 2 MP stroke 85 0 2133 2950 2 MP stroke 85 0 2133 2948 2 MP stroke 85 0 2133 2944 2 MP stroke 85 0 2133 2943 2 MP stroke 85 0 2133 2943 2 MP stroke 85 0 2133 2943 2 MP stroke 85 0 2133 2942 2 MP stroke 85 0 2133 2942 2 MP stroke 85 0 2133 2942 2 MP stroke 85 0 2133 2942 2 MP stroke 85 0 2133 2941 2 MP stroke 85 0 2133 2940 2 MP stroke 85 0 2133 2938 2 MP stroke 85 0 2133 2931 2 MP stroke 85 0 2133 2928 2 MP stroke 85 0 2133 2926 2 MP stroke 85 0 2133 2926 2 MP stroke 85 0 2133 2925 2 MP stroke 85 0 2133 2923 2 MP stroke 85 0 2133 2921 2 MP stroke 85 0 2133 2920 2 MP stroke 85 0 2133 2919 2 MP stroke 85 0 2133 2918 2 MP stroke 85 0 2133 2916 2 MP stroke 85 0 2133 2916 2 MP stroke 85 0 2133 2915 2 MP stroke 85 0 2133 2912 2 MP stroke 85 0 2133 2910 2 MP stroke 85 0 2133 2908 2 MP stroke 85 0 2133 2906 2 MP stroke 85 0 2133 2906 2 MP stroke 85 0 2133 2904 2 MP stroke 85 0 2133 2904 2 MP stroke 85 0 2133 2904 2 MP stroke 85 0 2133 2904 2 MP stroke 85 0 2133 2903 2 MP stroke 85 0 2133 2902 2 MP stroke 85 0 2133 2897 2 MP stroke 85 0 2133 2889 2 MP stroke 85 0 2133 2888 2 MP stroke 85 0 2133 2883 2 MP stroke 85 0 2133 2881 2 MP stroke 85 0 2133 2879 2 MP stroke 85 0 2133 2878 2 MP stroke 85 0 2133 2875 2 MP stroke 85 0 2133 2874 2 MP stroke 85 0 2133 2873 2 MP stroke 85 0 2133 2873 2 MP stroke 85 0 2133 2873 2 MP stroke 85 0 2133 2873 2 MP stroke 85 0 2133 2871 2 MP stroke 85 0 2133 2871 2 MP stroke 85 0 2133 2870 2 MP stroke 85 0 2133 2870 2 MP stroke 85 0 2133 2870 2 MP stroke 85 0 2133 2868 2 MP stroke 85 0 2133 2863 2 MP stroke 85 0 2133 2860 2 MP stroke 85 0 2133 2858 2 MP stroke 85 0 2133 2847 2 MP stroke 85 0 2133 2845 2 MP stroke 85 0 2133 2843 2 MP stroke 85 0 2133 2842 2 MP stroke 85 0 2133 2841 2 MP stroke 85 0 2133 2835 2 MP stroke 85 0 2133 2833 2 MP stroke 85 0 2133 2832 2 MP stroke 85 0 2133 2832 2 MP stroke 85 0 2133 2831 2 MP stroke 85 0 2133 2829 2 MP stroke 85 0 2133 2823 2 MP stroke 85 0 2133 2820 2 MP stroke 85 0 2133 2820 2 MP stroke 85 0 2133 2820 2 MP stroke 85 0 2133 2820 2 MP stroke 85 0 2133 2819 2 MP stroke 85 0 2133 2818 2 MP stroke 85 0 2133 2818 2 MP stroke 85 0 2133 2813 2 MP stroke 85 0 2133 2796 2 MP stroke 85 0 2133 2796 2 MP stroke 85 0 2133 2794 2 MP stroke 85 0 2133 2793 2 MP stroke 85 0 2133 2793 2 MP stroke 85 0 2133 2793 2 MP stroke 85 0 2133 2789 2 MP stroke 85 0 2133 2787 2 MP stroke 85 0 2133 2786 2 MP stroke 85 0 2133 2786 2 MP stroke 85 0 2133 2785 2 MP stroke 85 0 2133 2773 2 MP stroke 85 0 2133 2772 2 MP stroke 85 0 2133 2771 2 MP stroke 85 0 2133 2758 2 MP stroke 85 0 2133 2757 2 MP stroke 85 0 2133 2755 2 MP stroke 85 0 2133 2753 2 MP stroke 85 0 2133 2753 2 MP stroke 85 0 2133 2751 2 MP stroke 85 0 2133 2748 2 MP stroke 85 0 2133 2746 2 MP stroke 85 0 2133 2745 2 MP stroke 85 0 2133 2745 2 MP stroke 85 0 2133 2740 2 MP stroke 85 0 2133 2740 2 MP stroke 85 0 2133 2739 2 MP stroke 85 0 2133 2739 2 MP stroke 85 0 2133 2734 2 MP stroke 85 0 2133 2731 2 MP stroke 85 0 2133 2719 2 MP stroke 85 0 2133 2716 2 MP stroke 85 0 2133 2713 2 MP stroke 85 0 2133 2710 2 MP stroke 85 0 2133 2706 2 MP stroke 85 0 2133 2706 2 MP stroke 85 0 2133 2705 2 MP stroke 85 0 2133 2705 2 MP stroke 85 0 2133 2698 2 MP stroke 85 0 2133 2697 2 MP stroke 85 0 2133 2696 2 MP stroke 85 0 2133 2695 2 MP stroke 85 0 2133 2687 2 MP stroke 85 0 2133 2686 2 MP stroke 85 0 2133 2686 2 MP stroke 85 0 2133 2684 2 MP stroke 85 0 2133 2683 2 MP stroke 85 0 2133 2672 2 MP stroke 85 0 2133 2671 2 MP stroke 85 0 2133 2670 2 MP stroke 85 0 2133 2670 2 MP stroke 85 0 2133 2670 2 MP stroke 85 0 2133 2669 2 MP stroke 85 0 2133 2669 2 MP stroke 85 0 2133 2668 2 MP stroke 85 0 2133 2662 2 MP stroke 85 0 2133 2661 2 MP stroke 85 0 2133 2661 2 MP stroke 85 0 2133 2660 2 MP stroke 85 0 2133 2658 2 MP stroke 85 0 2133 2657 2 MP stroke 85 0 2133 2656 2 MP stroke 85 0 2133 2655 2 MP stroke 85 0 2133 2654 2 MP stroke 85 0 2133 2637 2 MP stroke 85 0 2133 2636 2 MP stroke 85 0 2133 2636 2 MP stroke 85 0 2133 2635 2 MP stroke 85 0 2133 2635 2 MP stroke 85 0 2133 2633 2 MP stroke 85 0 2133 2631 2 MP stroke 85 0 2133 2627 2 MP stroke 85 0 2133 2615 2 MP stroke 85 0 2133 2613 2 MP stroke 85 0 2133 2611 2 MP stroke 85 0 2133 2610 2 MP stroke 85 0 2133 2610 2 MP stroke 85 0 2133 2608 2 MP stroke 85 0 2133 2608 2 MP stroke 85 0 2133 2607 2 MP stroke 85 0 2133 2606 2 MP stroke 85 0 2133 2604 2 MP stroke 85 0 2133 2597 2 MP stroke 85 0 2133 2597 2 MP stroke 85 0 2133 2595 2 MP stroke 85 0 2133 2594 2 MP stroke 85 0 2133 2592 2 MP stroke 85 0 2133 2583 2 MP stroke 85 0 2133 2582 2 MP stroke 85 0 2133 2582 2 MP stroke 85 0 2133 2581 2 MP stroke 85 0 2133 2580 2 MP stroke 85 0 2133 2579 2 MP stroke 85 0 2133 2578 2 MP stroke 85 0 2133 2577 2 MP stroke 85 0 2133 2576 2 MP stroke 85 0 2133 2574 2 MP stroke 85 0 2133 2572 2 MP stroke 85 0 2133 2572 2 MP stroke 85 0 2133 2572 2 MP stroke 85 0 2133 2566 2 MP stroke 85 0 2133 2564 2 MP stroke 85 0 2133 2561 2 MP stroke 85 0 2133 2560 2 MP stroke 85 0 2133 2556 2 MP stroke 85 0 2133 2556 2 MP stroke 85 0 2133 2551 2 MP stroke 85 0 2133 2549 2 MP stroke 85 0 2133 2549 2 MP stroke 85 0 2133 2548 2 MP stroke 85 0 2133 2544 2 MP stroke 85 0 2133 2537 2 MP stroke 85 0 2133 2534 2 MP stroke 85 0 2133 2529 2 MP stroke 85 0 2133 2527 2 MP stroke 85 0 2133 2525 2 MP stroke 85 0 2133 2524 2 MP stroke 85 0 2133 2522 2 MP stroke 85 0 2133 2520 2 MP stroke 85 0 2133 2519 2 MP stroke 85 0 2133 2518 2 MP stroke 85 0 2133 2518 2 MP stroke 85 0 2133 2518 2 MP stroke 85 0 2133 2517 2 MP stroke 85 0 2133 2516 2 MP stroke 85 0 2133 2515 2 MP stroke 85 0 2133 2514 2 MP stroke 85 0 2133 2514 2 MP stroke 85 0 2133 2512 2 MP stroke 85 0 2133 2512 2 MP stroke 85 0 2133 2511 2 MP stroke 85 0 2133 2511 2 MP stroke 85 0 2133 2510 2 MP stroke 85 0 2133 2508 2 MP stroke 85 0 2133 2507 2 MP stroke 85 0 2133 2493 2 MP stroke 85 0 2133 2492 2 MP stroke 85 0 2133 2491 2 MP stroke 85 0 2133 2489 2 MP stroke 85 0 2133 2484 2 MP stroke 85 0 2133 2482 2 MP stroke 85 0 2133 2482 2 MP stroke 85 0 2133 2482 2 MP stroke 85 0 2133 2481 2 MP stroke 85 0 2133 2477 2 MP stroke 85 0 2133 2477 2 MP stroke 85 0 2133 2474 2 MP stroke 85 0 2133 2472 2 MP stroke 85 0 2133 2471 2 MP stroke 85 0 2133 2467 2 MP stroke 85 0 2133 2459 2 MP stroke 85 0 2133 2458 2 MP stroke 85 0 2133 2456 2 MP stroke 85 0 2133 2456 2 MP stroke 85 0 2133 2447 2 MP stroke 85 0 2133 2447 2 MP stroke 85 0 2133 2443 2 MP stroke 85 0 2133 2440 2 MP stroke 85 0 2133 2435 2 MP stroke 85 0 2133 2435 2 MP stroke 85 0 2133 2434 2 MP stroke 85 0 2133 2433 2 MP stroke 85 0 2133 2433 2 MP stroke 85 0 2133 2431 2 MP stroke 85 0 2133 2431 2 MP stroke 85 0 2133 2429 2 MP stroke 85 0 2133 2429 2 MP stroke 85 0 2133 2428 2 MP stroke 85 0 2133 2425 2 MP stroke 85 0 2133 2422 2 MP stroke 85 0 2133 2421 2 MP stroke 85 0 2133 2419 2 MP stroke 85 0 2133 2417 2 MP stroke 85 0 2133 2413 2 MP stroke 85 0 2133 2412 2 MP stroke 85 0 2133 2406 2 MP stroke 85 0 2133 2403 2 MP stroke 85 0 2133 2396 2 MP stroke 85 0 2133 2395 2 MP stroke 85 0 2133 2394 2 MP stroke 85 0 2133 2392 2 MP stroke 85 0 2133 2381 2 MP stroke 85 0 2133 2380 2 MP stroke 85 0 2133 2376 2 MP stroke 85 0 2133 2375 2 MP stroke 85 0 2133 2374 2 MP stroke 85 0 2133 2374 2 MP stroke 85 0 2133 2373 2 MP stroke 85 0 2133 2372 2 MP stroke 85 0 2133 2371 2 MP stroke 85 0 2133 2370 2 MP stroke 85 0 2133 2368 2 MP stroke 85 0 2133 2365 2 MP stroke 85 0 2133 2356 2 MP stroke 85 0 2133 2355 2 MP stroke 85 0 2133 2354 2 MP stroke 85 0 2133 2352 2 MP stroke 85 0 2133 2347 2 MP stroke 85 0 2133 2347 2 MP stroke 85 0 2133 2347 2 MP stroke 85 0 2133 2347 2 MP stroke 85 0 2133 2346 2 MP stroke 85 0 2133 2344 2 MP stroke 85 0 2133 2342 2 MP stroke 85 0 2133 2338 2 MP stroke 85 0 2133 2338 2 MP stroke 85 0 2133 2337 2 MP stroke 85 0 2133 2335 2 MP stroke 85 0 2133 2334 2 MP stroke 85 0 2133 2333 2 MP stroke 85 0 2133 2332 2 MP stroke 85 0 2133 2330 2 MP stroke 85 0 2133 2323 2 MP stroke 85 0 2133 2319 2 MP stroke 85 0 2133 2317 2 MP stroke 85 0 2133 2316 2 MP stroke 85 0 2133 2315 2 MP stroke 85 0 2133 2314 2 MP stroke 85 0 2133 2311 2 MP stroke 85 0 2133 2308 2 MP stroke 85 0 2133 2302 2 MP stroke 85 0 2133 2294 2 MP stroke 85 0 2133 2292 2 MP stroke 85 0 2133 2290 2 MP stroke 85 0 2133 2287 2 MP stroke 85 0 2133 2286 2 MP stroke 85 0 2133 2284 2 MP stroke 85 0 2133 2280 2 MP stroke 85 0 2133 2275 2 MP stroke 85 0 2133 2275 2 MP stroke 85 0 2133 2267 2 MP stroke 85 0 2133 2263 2 MP stroke 85 0 2133 2260 2 MP stroke 85 0 2133 2259 2 MP stroke 85 0 2133 2257 2 MP stroke 85 0 2133 2256 2 MP stroke 85 0 2133 2256 2 MP stroke 85 0 2133 2255 2 MP stroke 85 0 2133 2254 2 MP stroke 85 0 2133 2254 2 MP stroke 85 0 2133 2253 2 MP stroke 85 0 2133 2253 2 MP stroke 85 0 2133 2253 2 MP stroke 85 0 2133 2250 2 MP stroke 85 0 2133 2248 2 MP stroke 85 0 2133 2247 2 MP stroke 85 0 2133 2244 2 MP stroke 85 0 2133 2238 2 MP stroke 85 0 2133 2233 2 MP stroke 85 0 2133 2231 2 MP stroke 85 0 2133 2229 2 MP stroke 85 0 2133 2223 2 MP stroke 85 0 2133 2210 2 MP stroke 85 0 2133 2208 2 MP stroke 85 0 2133 2208 2 MP stroke 85 0 2133 2207 2 MP stroke 85 0 2133 2199 2 MP stroke 85 0 2133 2197 2 MP stroke 85 0 2133 2196 2 MP stroke 85 0 2133 2195 2 MP stroke 85 0 2133 2194 2 MP stroke 85 0 2133 2193 2 MP stroke 85 0 2133 2193 2 MP stroke 85 0 2133 2192 2 MP stroke 85 0 2133 2191 2 MP stroke 85 0 2133 2188 2 MP stroke 85 0 2133 2182 2 MP stroke 85 0 2133 2178 2 MP stroke 85 0 2133 2174 2 MP stroke 85 0 2133 2172 2 MP stroke 85 0 2133 2170 2 MP stroke 85 0 2133 2169 2 MP stroke 85 0 2133 2162 2 MP stroke 85 0 2133 2158 2 MP stroke 85 0 2133 2156 2 MP stroke 85 0 2133 2155 2 MP stroke 85 0 2133 2153 2 MP stroke 85 0 2133 2151 2 MP stroke 85 0 2133 2145 2 MP stroke 85 0 2133 2129 2 MP stroke 85 0 2133 2125 2 MP stroke 85 0 2133 2118 2 MP stroke 85 0 2133 2118 2 MP stroke 85 0 2133 2116 2 MP stroke 85 0 2133 2115 2 MP stroke 85 0 2133 2115 2 MP stroke 85 0 2133 2113 2 MP stroke 85 0 2133 2112 2 MP stroke 85 0 2133 2110 2 MP stroke 85 0 2133 2108 2 MP stroke 85 0 2133 2108 2 MP stroke 85 0 2133 2108 2 MP stroke 85 0 2133 2084 2 MP stroke 85 0 2133 2081 2 MP stroke 85 0 2133 2080 2 MP stroke 85 0 2133 2078 2 MP stroke 85 0 2133 2077 2 MP stroke 85 0 2133 2076 2 MP stroke 85 0 2133 2076 2 MP stroke 85 0 2133 2071 2 MP stroke 85 0 2133 2066 2 MP stroke 85 0 2133 2052 2 MP stroke 85 0 2133 2047 2 MP stroke 85 0 2133 2037 2 MP stroke 85 0 2133 2029 2 MP stroke 85 0 2133 2027 2 MP stroke 85 0 2133 2025 2 MP stroke 85 0 2133 2020 2 MP stroke 85 0 2133 2018 2 MP stroke 85 0 2133 2016 2 MP stroke 85 0 2133 2015 2 MP stroke 85 0 2133 2014 2 MP stroke 85 0 2133 2009 2 MP stroke 85 0 2133 2004 2 MP stroke 85 0 2133 1991 2 MP stroke 85 0 2133 1989 2 MP stroke 85 0 2133 1987 2 MP stroke 85 0 2133 1980 2 MP stroke 85 0 2133 1976 2 MP stroke 85 0 2133 1973 2 MP stroke 85 0 2133 1967 2 MP stroke 85 0 2133 1937 2 MP stroke 85 0 2133 1930 2 MP stroke 85 0 2133 1929 2 MP stroke 85 0 2133 1928 2 MP stroke 85 0 2133 1926 2 MP stroke 85 0 2133 1922 2 MP stroke 85 0 2133 1914 2 MP stroke 85 0 2133 1908 2 MP stroke 85 0 2133 1904 2 MP stroke 85 0 2133 1897 2 MP stroke 85 0 2133 1894 2 MP stroke 85 0 2133 1889 2 MP stroke 85 0 2133 1887 2 MP stroke 85 0 2133 1880 2 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stroke 85 0 2303 2842 2 MP stroke 85 0 2303 2834 2 MP stroke 85 0 2303 2833 2 MP stroke 85 0 2303 2831 2 MP stroke 85 0 2303 2831 2 MP stroke 85 0 2303 2821 2 MP stroke 85 0 2303 2820 2 MP stroke 85 0 2303 2820 2 MP stroke 85 0 2303 2820 2 MP stroke 85 0 2303 2820 2 MP stroke 85 0 2303 2816 2 MP stroke 85 0 2303 2793 2 MP stroke 85 0 2303 2793 2 MP stroke 85 0 2303 2793 2 MP stroke 85 0 2303 2792 2 MP stroke 85 0 2303 2786 2 MP stroke 85 0 2303 2786 2 MP stroke 85 0 2303 2786 2 MP stroke 85 0 2303 2772 2 MP stroke 85 0 2303 2771 2 MP stroke 85 0 2303 2756 2 MP stroke 85 0 2303 2755 2 MP stroke 85 0 2303 2755 2 MP stroke 85 0 2303 2749 2 MP stroke 85 0 2303 2747 2 MP stroke 85 0 2303 2746 2 MP stroke 85 0 2303 2740 2 MP stroke 85 0 2303 2740 2 MP stroke 85 0 2303 2739 2 MP stroke 85 0 2303 2733 2 MP stroke 85 0 2303 2716 2 MP stroke 85 0 2303 2712 2 MP stroke 85 0 2303 2706 2 MP stroke 85 0 2303 2706 2 MP stroke 85 0 2303 2706 2 MP stroke 85 0 2303 2697 2 MP stroke 85 0 2303 2696 2 MP 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stroke 85 0 2303 2262 2 MP stroke 85 0 2303 2257 2 MP stroke 85 0 2303 2256 2 MP stroke 85 0 2303 2255 2 MP stroke 85 0 2303 2254 2 MP stroke 85 0 2303 2254 2 MP stroke 85 0 2303 2254 2 MP stroke 85 0 2303 2248 2 MP stroke 85 0 2303 2248 2 MP stroke 85 0 2303 2248 2 MP stroke 85 0 2303 2236 2 MP stroke 85 0 2303 2228 2 MP stroke 85 0 2303 2209 2 MP stroke 85 0 2303 2208 2 MP stroke 85 0 2303 2203 2 MP stroke 85 0 2303 2195 2 MP stroke 85 0 2303 2194 2 MP stroke 85 0 2303 2194 2 MP stroke 85 0 2303 2194 2 MP stroke 85 0 2303 2191 2 MP stroke 85 0 2303 2187 2 MP stroke 85 0 2303 2176 2 MP stroke 85 0 2303 2171 2 MP stroke 85 0 2303 2170 2 MP stroke 85 0 2303 2157 2 MP stroke 85 0 2303 2157 2 MP stroke 85 0 2303 2156 2 MP stroke 85 0 2303 2148 2 MP stroke 85 0 2303 2121 2 MP stroke 85 0 2303 2117 2 MP stroke 85 0 2303 2116 2 MP stroke 85 0 2303 2112 2 MP stroke 85 0 2303 2111 2 MP stroke 85 0 2303 2110 2 MP stroke 85 0 2303 2110 2 MP stroke 85 0 2303 2081 2 MP stroke 85 0 2303 2079 2 MP stroke 85 0 2303 2078 2 MP stroke 85 0 2303 2078 2 MP stroke 85 0 2303 2069 2 MP stroke 85 0 2303 2041 2 MP stroke 85 0 2303 2027 2 MP stroke 85 0 2303 2027 2 MP stroke 85 0 2303 2018 2 MP stroke 85 0 2303 2017 2 MP stroke 85 0 2303 2012 2 MP stroke 85 0 2303 1989 2 MP stroke 85 0 2303 1988 2 MP stroke 85 0 2303 1978 2 MP stroke 85 0 2303 1970 2 MP stroke 85 0 2303 1928 2 MP stroke 85 0 2303 1927 2 MP stroke 85 0 2303 1926 2 MP stroke 85 0 2303 1912 2 MP stroke 85 0 2303 1896 2 MP stroke 85 0 2303 1884 2 MP stroke 85 0 2303 1869 2 MP stroke 85 0 2303 1849 2 MP stroke 85 0 2303 1849 2 MP stroke 85 0 2303 1829 2 MP stroke 85 0 2303 1825 2 MP stroke 85 0 2303 1817 2 MP stroke 85 0 2303 1786 2 MP stroke 85 0 2303 1782 2 MP stroke 85 0 2303 1742 2 MP stroke 85 0 2303 1725 2 MP stroke 85 0 2303 1715 2 MP stroke 85 0 2303 1699 2 MP stroke 85 0 2303 1683 2 MP stroke 85 0 2303 1642 2 MP stroke 85 0 2303 1632 2 MP stroke 85 0 2303 1615 2 MP stroke 85 0 2303 1600 2 MP stroke 85 0 2303 1542 2 MP stroke 85 0 2303 1531 2 MP stroke 85 0 2303 1527 2 MP stroke 85 0 2303 1499 2 MP stroke 85 0 2303 1443 2 MP stroke 85 0 2303 1430 2 MP stroke 85 0 2303 1342 2 MP stroke 85 0 2303 1240 2 MP stroke 85 0 2474 4233 2 MP stroke 85 0 2474 4233 2 MP stroke 85 0 2474 4233 2 MP stroke 85 0 2474 4233 2 MP stroke 85 0 2474 3913 2 MP stroke 85 0 2474 3913 2 MP stroke 85 0 2474 3913 2 MP stroke 85 0 2474 3877 2 MP stroke 85 0 2474 3877 2 MP stroke 85 0 2474 3877 2 MP stroke 85 0 2474 3822 2 MP stroke 85 0 2474 3822 2 MP stroke 85 0 2474 3822 2 MP stroke 85 0 2474 3775 2 MP stroke 85 0 2474 3774 2 MP stroke 85 0 2474 3759 2 MP stroke 85 0 2474 3759 2 MP stroke 85 0 2474 3754 2 MP stroke 85 0 2474 3754 2 MP stroke 85 0 2474 3754 2 MP stroke 85 0 2474 3737 2 MP stroke 85 0 2474 3737 2 MP stroke 85 0 2474 3732 2 MP stroke 85 0 2474 3724 2 MP stroke 85 0 2474 3715 2 MP stroke 85 0 2474 3712 2 MP stroke 85 0 2474 3712 2 MP stroke 85 0 2474 3707 2 MP stroke 85 0 2474 3689 2 MP stroke 85 0 2474 3689 2 MP 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stroke 85 0 2644 3389 2 MP stroke 85 0 2644 3382 2 MP stroke 85 0 2644 3336 2 MP stroke 85 0 2644 3299 2 MP stroke 85 0 2644 3268 2 MP stroke 85 0 2644 3253 2 MP stroke 85 0 2644 3220 2 MP stroke 85 0 2644 3185 2 MP stroke 85 0 2644 3174 2 MP stroke 85 0 2644 3152 2 MP stroke 85 0 2644 3106 2 MP stroke 85 0 2644 3105 2 MP stroke 85 0 2644 3105 2 MP stroke 85 0 2644 3099 2 MP stroke 85 0 2644 3066 2 MP stroke 85 0 2644 3052 2 MP stroke 85 0 2644 3037 2 MP stroke 85 0 2644 3026 2 MP stroke 85 0 2644 3019 2 MP stroke 85 0 2644 2984 2 MP stroke 85 0 2644 2957 2 MP stroke 85 0 2644 2951 2 MP stroke 85 0 2644 2942 2 MP stroke 85 0 2644 2904 2 MP stroke 85 0 2644 2873 2 MP stroke 85 0 2644 2870 2 MP stroke 85 0 2644 2820 2 MP stroke 85 0 2644 2793 2 MP stroke 85 0 2644 2786 2 MP stroke 85 0 2644 2740 2 MP stroke 85 0 2644 2706 2 MP stroke 85 0 2644 2670 2 MP stroke 85 0 2644 2636 2 MP stroke 85 0 2644 2582 2 MP stroke 85 0 2814 4233 2 MP stroke 85 0 2814 4233 2 MP stroke 85 0 2814 3913 2 MP stroke 85 0 2814 3877 2 MP stroke 85 0 2814 3822 2 MP stroke 85 0 2814 3754 2 MP stroke 85 0 2814 3678 2 MP stroke 85 0 2814 3602 2 MP stroke 85 0 2814 3529 2 MP stroke 85 0 2814 3467 2 MP stroke 85 0 2814 3419 2 MP stroke 85 0 2814 3389 2 MP stroke 85 0 2984 4233 2 MP stroke gr %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 942 4787 mt (0) s gs 899 389 5363 4229 MR c np 0 -26 984 4617 2 MP stroke gr 1112 4787 mt (1) s gs 899 389 5363 4229 MR c np 0 -26 1154 4617 2 MP stroke gr 1296 4787 mt (2) s gs 899 389 5363 4229 MR c np 0 -26 1325 4617 2 MP stroke gr 1473 4787 mt (3) s gs 899 389 5363 4229 MR c np 0 -26 1495 4617 2 MP stroke gr 1636 4787 mt (4) s gs 899 389 5363 4229 MR c np 0 -26 1665 4617 2 MP stroke gr 1813 4787 mt (5) s gs 899 389 5363 4229 MR c np 0 -26 1835 4617 2 MP stroke gr 1983 4787 mt (6) s gs 899 389 5363 4229 MR c np 0 -26 2005 4617 2 MP stroke gr 2154 4778 mt (7) s gs 899 389 5363 4229 MR c np 0 -26 2176 4617 2 MP stroke gr 2324 4787 mt (8) s gs 899 389 5363 4229 MR c np 0 -26 2346 4617 2 MP stroke gr 2474 4787 mt (9) s gs 899 389 5363 4229 MR c np 0 -26 2516 4617 2 MP stroke gr 2630 4787 mt (10) s gs 899 389 5363 4229 MR c np 0 -26 2686 4617 2 MP stroke gr 2814 4787 mt (11) s gs 899 389 5363 4229 MR c np 0 -26 2857 4617 2 MP stroke gr 2984 4787 mt (12) s gs 899 389 5363 4229 MR c np 0 -26 3027 4617 2 MP stroke gr 814 4531 mt (0) s gs 899 389 5363 4229 MR c np 17 0 899 4489 2 MP stroke gr 814 4018 mt (2) s gs 899 389 5363 4229 MR c np 17 0 899 3976 2 MP stroke gr 814 3506 mt (4) s gs 899 389 5363 4229 MR c np 17 0 899 3464 2 MP stroke gr 814 2993 mt (6) s gs 899 389 5363 4229 MR c np 17 0 899 2951 2 MP stroke gr 814 2481 mt (8) s gs 899 389 5363 4229 MR c np 17 0 899 2439 2 MP stroke gr 780 1968 mt (10) s gs 899 389 5363 4229 MR c np 17 0 899 1926 2 MP stroke gr 780 1456 mt (12) s gs 899 389 5363 4229 MR c np 17 0 899 1414 2 MP stroke gr 780 943 mt (14) s gs 899 389 5363 4229 MR c np 17 0 899 901 2 MP stroke gr 780 431 mt (16) s gs 899 389 5363 4229 MR c np 17 0 899 389 2 MP stroke 85 0 4176 3972 2 MP stroke 85 0 4176 1443 2 MP stroke 85 0 4346 4493 2 MP stroke 85 0 4346 4079 2 MP stroke 85 0 4346 3548 2 MP stroke 85 0 4346 2937 2 MP stroke 85 0 4346 2284 2 MP stroke 85 0 4346 1629 2 MP stroke 85 0 4346 1013 2 MP stroke 85 0 4346 478 2 MP stroke 85 0 4516 3042 2 MP stroke 85 0 4516 2963 2 MP stroke 85 0 4516 2842 2 MP stroke 85 0 4516 2689 2 MP stroke 85 0 4516 2517 2 MP stroke 85 0 4516 2338 2 MP stroke 85 0 4516 2167 2 MP stroke 85 0 4516 2014 2 MP stroke 85 0 4516 1893 2 MP stroke 85 0 4516 1815 2 MP stroke 86 0 4686 2590 2 MP stroke 86 0 4686 2567 2 MP stroke 86 0 4686 2531 2 MP stroke 86 0 4686 2486 2 MP stroke 86 0 4686 2438 2 MP stroke 86 0 4686 2390 2 MP stroke 86 0 4686 2346 2 MP stroke 86 0 4686 2310 2 MP stroke 86 0 4686 2286 2 MP stroke 85 0 4857 2476 2 MP stroke 85 0 4857 2469 2 MP stroke 85 0 4857 2458 2 MP stroke 85 0 4857 2446 2 MP stroke 85 0 4857 2432 2 MP stroke 85 0 4857 2419 2 MP stroke 85 0 4857 2409 2 MP stroke 85 0 4857 2401 2 MP stroke 85 0 5027 2448 2 MP stroke 85 0 5027 2446 2 MP stroke 85 0 5027 2443 2 MP stroke 85 0 5027 2439 2 MP stroke 85 0 5027 2435 2 MP stroke 85 0 5027 2432 2 MP stroke 85 0 5027 2430 2 MP stroke 85 0 5197 2441 2 MP stroke 85 0 5197 2440 2 MP stroke 85 0 5197 2439 2 MP stroke 85 0 5197 2438 2 MP stroke 85 0 5197 2437 2 MP stroke 85 0 5197 2437 2 MP stroke 85 0 5367 2439 2 MP stroke 85 0 5367 2439 2 MP stroke 85 0 5367 2439 2 MP stroke 85 0 5367 2439 2 MP stroke 85 0 5367 2438 2 MP stroke 85 0 5538 2439 2 MP stroke 85 0 5538 2439 2 MP stroke 85 0 5538 2439 2 MP stroke 85 0 5538 2439 2 MP stroke 85 0 5708 2439 2 MP stroke 85 0 5708 2439 2 MP stroke 85 0 5708 2439 2 MP stroke 85 0 5878 2439 2 MP stroke 85 0 5878 2439 2 MP stroke 85 0 6048 2439 2 MP stroke gr 4031 4787 mt (0) s gs 899 389 5363 4229 MR c np 0 -26 4048 4617 2 MP stroke gr 4201 4787 mt (1) s gs 899 389 5363 4229 MR c np 0 -26 4218 4617 2 MP stroke gr 4372 4787 mt (2) s gs 899 389 5363 4229 MR c np 0 -26 4389 4617 2 MP stroke gr 4542 4787 mt (3) s gs 899 389 5363 4229 MR c np 0 -26 4559 4617 2 MP stroke gr 4712 4787 mt (4) s gs 899 389 5363 4229 MR c np 0 -26 4729 4617 2 MP stroke gr 4882 4787 mt (5) s gs 899 389 5363 4229 MR c np 0 -26 4899 4617 2 MP stroke gr 5052 4787 mt (6) s gs 899 389 5363 4229 MR c np 0 -26 5069 4617 2 MP stroke gr 5223 4787 mt (7) s gs 899 389 5363 4229 MR c np 0 -26 5240 4617 2 MP stroke gr 5393 4787 mt (8) s gs 899 389 5363 4229 MR c np 0 -26 5410 4617 2 MP stroke gr 5529 4787 mt (9) s gs 899 389 5363 4229 MR c np 0 -26 5580 4617 2 MP stroke gr 5699 4787 mt (10) s gs 899 389 5363 4229 MR c np 0 -26 5750 4617 2 MP stroke gr 5869 4787 mt (11) s gs 899 389 5363 4229 MR c np 0 -26 5921 4617 2 MP stroke gr 6040 4787 mt (12) s gs 899 389 5363 4229 MR c np 0 -26 6091 4617 2 MP stroke 2213 0 3963 4617 2 MP stroke gr 3759 4531 mt (0.8) s gs 899 389 5363 4229 MR c np 17 0 3963 4489 2 MP stroke gr 3759 3506 mt (0.9) s gs 899 389 5363 4229 MR c np 17 0 3963 3464 2 MP stroke gr 3759 2481 mt (1) s gs 899 389 5363 4229 MR c np 17 0 3963 2439 2 MP stroke gr 3759 1456 mt (1.1) s gs 899 389 5363 4229 MR c np 17 0 3963 1414 2 MP stroke gr 3759 431 mt (1.2) s gs 899 389 5363 4229 MR c np 17 0 3963 389 2 MP stroke 0 -4484 3963 4617 2 MP stroke 171 0 340 -3792 170 0 3027 4181 4 MP stroke 171 0 340 205 170 0 3027 4284 4 MP stroke gr 2176 4915 mt (n) s 5240 4915 mt (n) s 653 2439 mt -90 rotate (E/A\() s 90 rotate %%IncludeResource: font Symbol /Symbol /ISOLatin1Encoding 120 FMSR 653 2206 mt -90 rotate (D) s 90 rotate %%IncludeResource: font Helvetica /Helvetica /ISOLatin1Encoding 120 FMSR 653 2133 mt -90 rotate (\)) s 90 rotate 4670 157 mt ("multiplicity of ground states") s 3997 303 mt (0) s 4167 303 mt (2) s 4337 303 mt (7) s 4508 303 mt (10) s 4678 303 mt (9) s 4848 303 mt (8) s 5018 303 mt (7) s 5189 303 mt (6) s 5359 303 mt (5) s 5529 303 mt (4) s 5699 303 mt (3) s 5869 303 mt (2) s 6040 303 mt (1) s end eplot %%EndObject epage 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