diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgjsm" "b/data_all_eng_slimpj/shuffled/split2/finalzzgjsm" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgjsm" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nJamison~\\cite{Jamison1983} initiated the study of the mean subtree order of a tree. A number of extensions of this mean to other (connected) graphs have recently been considered:\n\\begin{itemize}\n\\item the mean order of the sub-$k$-trees of a $k$-tree~\\cite{StephensOellermann2018},\n\\item the mean order of the subtrees (i.e., minimally connected subgraphs) of a graph~\\cite{ChinGordonMacpheeVincent2018}, and\n\\item the mean order of the connected induced subgraphs of a graph~\\cite{KroekerMolOellermann2018}.\n\\end{itemize}\nFor a tree $T$, all of these means equal the mean subtree order of $T$. However, for connected graphs in general, the last two means have rather different behaviour.\n\nIn this article, we continue the study of the average order of the connected induced subgraphs of a graph $G$, called the {\\em mean connected induced subgraph (CIS) order} of $G$. An in-depth study of the mean CIS order of cographs was undertaken in~\\cite{KroekerMolOellermann2018}, where the connected cographs of order $n$ having largest and smallest mean CIS order were determined (both the maximum and minimum values tend to $n\/2$ asymptotically). Here, we focus on the mean CIS order of block graphs, i.e., graphs for which every block is complete. We extend several of Jamison's results~\\cite{Jamison1983} on the mean subtree order of trees to the more general setting of the mean CIS order of connected block graphs (note that every tree is a connected block graph).\n\nIn particular, Jamison~\\cite{Jamison1983} demonstrated that among all trees of order $n$, the path $P_n$ has minimum mean subtree order (or equivalently, mean CIS order). Our main result is that the path $P_n$ has minimum mean CIS order among all connected block graphs of order $n$. This supports the conjecture of Kroeker, Mol, and Oellermann~\\cite{KroekerMolOellermann2018} that the path $P_n$ has minimum mean CIS order among all connected graphs of order $n$, which has been verified for all $n\\leq 9$.\n\nA key tool in the proof of our main result is an extension of the ``local-global mean inequality'' proven by Jamison~\\cite{Jamison1983} for trees. For a given tree $T$, and every vertex $v$ of $T$, Jamison demonstrated that the mean order of all connected induced subgraphs of $T$ containing $v$ (i.e., the ``local'' mean CIS order of $T$ at $v$) is at least as large as the mean CIS order of $T$ (i.e., the ``global'' mean CIS order of $T$). It is known that this inequality between local and global mean CIS orders does not extend to all connected graphs (at least not at every vertex)~\\cite{KroekerMolOellermann2018}. However, we note that it was recently proven, in a more general context, that every graph with nonempty edge set contains at least one vertex at which the local mean CIS order is larger than the global mean CIS order (apply~\\cite[Theorem 3.1]{AMW} to the collection of vertex sets that induce connected subgraphs of $G$). In other words, while the local-global mean inequality does not necessarily hold at \\emph{every} vertex of a connected graph $G$, it must hold at \\emph{some} vertex of $G$. In this article, we demonstrate that the local-global mean inequality does hold at every vertex of a connected block graph. This fact is essential to the proofs of the three key lemmas used to establish our main result. \n\nWe now give a brief description of the layout of the article. In Section~\\ref{Preliminaries}, we provide notation and preliminaries that will be used throughout the article. In Section~\\ref{Main_Result}, we state three key lemmas (the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stetching Lemma), and we use them to prove our main result. We then describe an interesting connection between the mean CIS order of block graphs and the mean sub-$k$-tree order of $k$-trees, and explain the implications of our main result in this setting. In Section~\\ref{LocalGlobalSection}, we prove the local-global mean inequality for the mean CIS order of block graphs. In Section~\\ref{KeyLemmas}, we prove the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma. We conclude with some open problems.\n\t\n\\section{Notation and Preliminaries}\\label{Preliminaries}\n\nFor a graph $G$, the vertex and edges sets of $G$ are denoted by $V(G)$ and $E(G)$, respectively. The \\emph{order} of $G$ is $|V(G)|$ and the \\emph{size} of $G$ is $|E(G)|$. For $U\\subseteq V(G)$, the subgraph of $G$ induced by $U$ is denoted $G[U]$. The (open) neighbourhood of a vertex $v$ of $G$ is denoted $N_G(v)$.\n\nLet $G$ be a graph of order $n$. Let $\\mathcal{C}_G$ denote the collection of connected induced subgraphs of $G$. The \\emph{CIS polynomial of $G$} is given by\n\\[\n\\Phi_G(x)=\\sum_{H\\in \\mathcal{C}_G}x^{|V(H)|}=\\sum_{i=1}^na_ix^i,\n\\]\nwhere $a_i$ is the number of connected induced subgraphs of $G$ of order $i$ for each $i\\in\\{1,\\dots,n\\}$.\nOne easily verifies that $\\Phi_G(1)$ is the total number of connected induced subgraphs of $G$, and that $\\Phi'_G(1)$ is the sum of the orders of all connected induced subgraphs of $G$. Throughout, we use the shorthand notation $N_G=\\Phi_G(1)$ and $W_G=\\Phi'_G(1)$. The \\emph{mean CIS order of $G$}, denoted $M_G$, is given by\n\\[\nM_G=\\frac{\\Phi'_G(1)}{\\Phi_G(1)}=\\frac{W_G}{N_G}.\n\\]\nFor a vertex $v\\in V(G)$, let $\\mathcal{C}_{G,v}$ denote the collection of connected induced subgraphs of $G$ containing $v$. The \\textit{local CIS polynomial of $G$ at $v$} is given by\n\\[\n\\Phi_{G, v}(x)=\\sum_{H\\in\\mathcal{C}_{G,v}}x^{|V(H)|}=\\sum_{i=1}^nb_ix^i,\n\\]\nwhere $b_i$ is the number of connected induced subgraphs of $G$ of order $i$ containing $v$ for each $i\\in\\{1,\\dots,n\\}.$ So $N_{G,v}=\\Phi_{G,v}(1)$ denotes the total number of connected induced subgraphs of $G$ containing $v$, and $W_{G,v}=\\Phi'_{G,v}(1)$ denotes the sum of the orders of all connected induced subgraphs of $G$ containing $v$. The \\emph{local mean CIS order of $G$}, denoted $M_{G,v}$, is given by\n\\[\nM_{G,v}=\\frac{\\Phi'_{G,v}(1)}{\\Phi_{G,v}(1)}=\\frac{W_{G,v}}{N_{G,v}}.\n\\]\n\nThe next lemma gives a recursion for the local mean CIS order of a block graph at a cut vertex $v$. It holds trivially if $v$ is not a cut vertex.\n\n\\begin{lemma}\\label{LocalSum}\nLet $G$ be a block graph with vertex $v$, and let $H_1,\\dots, H_k$ be the components of $G-v$. For $i\\in\\{1,\\dots,k\\},$ let $G_i=G[V(H_i)\\cup \\{v\\}]$. Then\n\\[\nM_{G,v}=\\left[\\sum_{i=1}^k M_{G_i,v}\\right]-(k-1).\n\\]\nFurther, we have\n\\[\nM_{G,v}\\geq M_{G_i,v}\n\\]\nfor all $i\\in\\{1,\\dots,k\\}.$\n\\end{lemma}\n\n\\begin{proof}\nBy a straightforward counting argument,\n\\[\n\\Phi_{G,v}(x)=\\tfrac{1}{x^{k-1}}\\prod_{i=1}^k\\Phi_{G_i,v}(x).\n\\]\nTaking the natural logarithm on both sides and differentiating with respect to $x$, we obtain\n\\[\n\\frac{\\Phi'_{G,v}(x)}{\\Phi_{G,v}(x)}=\\left[\\sum_{i=1}^k\\frac{\\Phi'_{G_i,v}(x)}{\\Phi_{G_i,v}(x)}\\right]-\\frac{k-1}{x}.\n\\]\nSubstituting $x=1$ yields\n\\[\nM_{G,v}=\\left[\\sum_{i=1}^k M_{G_i,v}\\right]-(k-1).\n\\]\nSince $M_{G_i,v}\\geq 1$ for all $i\\in\\{1,\\dots,k\\},$ it follows that $M_{G,v}\\geq M_{G_i,v}$ for all $i\\in\\{1,\\dots,k\\}$.\n\\end{proof}\n\nWe extend the notion of the local mean CIS order of a graph $G$ in two natural ways. For a subset $U$ of $V(G)$, we let $M_{G,U}$ denote the mean order of all connected induced subgraphs of $G$ containing every vertex of $U$. We let $\\Phi_{G,U}(x)$ denote the corresponding generating polynomial. We let $M^*_{G,U}$ denote the mean order of all connected induced subgraphs of $G$ containing at least one vertex of $U$. We let $\\Phi^*_{G,U}(x)$ denote the corresponding generating polynomial, and $N^*_{G,U}=\\Phi^*_{G,U}(1)$ and $W^*_{G,U}(x)=\\Phi^{*'}_{G,U}(1).$ Note that if $U$ contains only a single vertex $u$, then $M_{G,U}=M^*_{G,U}=M_{G,u}.$\n\n\\begin{lemma}\\label{LocalBlockLemma}\nLet $G$ be a block graph, and let $U=\\{u_1,\\dots,u_k\\}$ be the vertex set of a single block $B$ of $G$. For each $i\\in\\{1,\\dots,k\\},$ let $G_i$ be the connected component of $G-E(B)$ containing $u_i$. Then\n\\begin{align}\\label{LocalAtBlock}\nM^*_{G,U}=\\frac{N^*_{G,U}+1}{N^*_{G,U}}\\sum_{i=1}^k \\frac{W_{G_i,u_i}}{N_{G_i,u_i}+1}.\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy a straightforward counting argument,\n\\[\n\\Phi^*_{G,U}(x)+1=\\prod_{i=1}^k \\left[ 1+\\Phi_{G_i,u_i}(x)\\right].\n\\]\nTaking the natural logarithm on both sides and differentiating, we obtain\n\\[\n\\frac{\\Phi^{*'}_{G,U}(x)}{\\Phi^*_{G,U}(x)+1}=\\sum_{i=1}^k\\frac{\\Phi'_{G_i,u_i}(x)}{\\Phi_{G_i,u_i}(x)+1}.\n\\]\nSubstituting $x=1$ gives\n\\[\n\\frac{W^*_{G,U}}{N^*_{G,U}+1}=\\sum_{i=1}^k\\frac{W_{G_i,u_i}}{N_{G_i,u_i}+1}.\n\\]\nMultiplying both sides by $\\frac{N^*_{G,U}+1}{N^*_{G,U}}$ and noting that $M^*_{G,U}=\\frac{W^*_{G,U}}{N^*_{G,U}},$ we obtain~(\\ref{LocalAtBlock}).\n\\end{proof}\n\nA key idea that we use in many of our arguments states that if the connected induced subgraphs of a graph $G$ can be partitioned into two or more sets, then $M_G$ is a convex combination (or weighted average) of the mean orders of each of the sets in the partition. This tool was used by Jamison~\\cite[Lemma 3.8]{Jamison1983} for the mean subtree order of a tree. For example, since every connected induced subgraph of $G$ either contains a given vertex $v$, or does not contain $v$, we can write\n\\[\n\\Phi_G(x)=\\Phi_{G,v}(x)+\\Phi_{G-v}(x).\n\\]\nIt follows that $M_G$ is a convex combination of $M_{G,v}$ and $M_{G-v}$. Another useful application of this principle is to disconnected graphs. If $G$ is a disconnected graph with components $G_1,\\dots,G_k$, then $M_G$ is a convex combination of $M_{G_1},\\dots,M_{G_k}$. It follows that $\\min\\{M_{G_i}\\}\\leq M_G\\leq \\max\\{M_{G_i}\\}.$\n\n\\section{Proof of the Main Result}\\label{Main_Result}\n\nIn this section, we show that among all block graphs of order $n$, the path has minimum mean CIS order. We use three key lemmas, namely the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma, which are proven in Section~\\ref{KeyLemmas}. We state these lemmas here, and provide illustrations depicting how they are used in the proof of the main result (see Figure~\\ref{LemmaPictures}). By \\emph{gluing} two vertices from disjoint graphs, we mean the process of identifying these two vertices.\n\n\n\\medskip\n\n\\noindent{\\bf The Vertex Gluing Lemma} (Lemma \\ref{vertex_gluing})\\\\\n\\textit{Let $H$ be a connected block graph of order at least $2$ having vertex $v$. Fix a natural number $n\\geq 3$. Let $P:u_1\\dots u_n$ be a path of order $n$. For $s \\in \\{1, \\dots, n\\}$, let $G_s$ be the block graph obtained from the disjoint union of $P_n$ and $H$ by gluing $v$ to $u_s$. If $1\\leq i1$, and suppose that the statement holds for all connected block graphs of order at least $2$, with less than $k$ blocks. Since $v$ is not a cut vertex, it is contained in only one block $B$ of $G$, and $N_G(v)=V(B)-v$. Since we can partition the connected induced subgraphs of $G-v$ into all those that contain at least one neighbour of $v$ and all those that do not, we can write $M_{G-v}$ as a convex combination of $\\mu_{G,v}$ and $M_{G-B}$. So it suffices to show that $\\mu_{G,v}\\geq M_{G-B}.$\n\t\nLet $U=V(B)-v=\\{v_1,v_2,\\dots, v_k\\}$. Note that $U$ is either a singleton, or induces a block in $G-v$. For each $i\\in\\{1,\\dots,k\\}$, let $G_i$ be the connected component of $(G-v)-E(B)$ containing $v_i$. By Lemma~\\ref{LocalBlockLemma},\n\\[\n\\mu_{G,v}=M^*_{G-v,U}=\\frac{N^*_{G-v,U}+1}{N^*_{G-v,U}}\\sum_{i=1}^k \\frac{W_{G_i,v_i}}{N_{G_i,v_i}+1}>\\sum_{i=1}^k \\frac{W_{G_i,v_i}}{N_{G_i,v_i}+1}.\n\\]\n\t\nNow $G-B$ may be a disconnected graph, so $M_{G-B}$ is at most $M_H$, where $H$ is a connected component of $G-B$ of largest mean CIS order. In particular, $H$ is a subgraph of $G_i-v_i$ for some $i\\in\\{1,\\dots,k\\}$. Without loss of generality, suppose $H$ is a subgraph of $G_1-v_1$. Let $H_1=G[V(H)\\cup\\{v_1\\}]$. By Lemma~\\ref{LocalSum}, we have\n\\[\nM_{G_1,v_1}\\geq M_{H_1,v_1},\n\\]\nand together with $W_{G_1,v_1}\\geq W_{H_1,v_1},$ which clearly holds since every connected induced subgraph of $H_1$ containing $v_1$ is a connected induced subgraph of $G_1$ containing $v_1$, this implies\n\\[\n\\frac{W_{G_1,v_1}}{1+N_{G_1,v_1}}\\geq \\frac{W_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nAltogether, we have\n\\[\n\\mu_{G,v}>\\frac{W_{G_1,v_1}}{1+N_{G_1,v_1}}\\geq \\frac{W_{H_1,v_1}}{1+N_{H_1,v_1}}=\\frac{\\mu_{H_1,v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nNote that $H_1$ has order at least $2$, and fewer than $k$ blocks, and that $v_1$ is a non-cut vertex of $H_1$. Hence, by the induction hypothesis, applied to $H_1$ at $v_1$,\n\\[\n\\mu_{G,v}> \\frac{M_{H_1-v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}.\n\\]\nFinally,\n\\[\n\\frac{M_{H_1-v_1}(N_{H_1,v_1}-1)+N_{H_1,v_1}}{1+N_{H_1,v_1}}\\geq M_{H_1-v_1}\n\\]\nis equivalent to\n\\[\nW_{H_1-v_1}\\leq \\frac{N_{H_1,v_1}N_{H_1-v_1}}{2},\n\\]\nwhich holds, by Lemma \\ref{WeightBound}. So we have shown that\n\\[\n\\mu_{G,v}> M_{H_1-v_1}=M_H\\geq M_{G-B},\n\\]\nand it follows, from our earlier observation, that $\\mu_{G,v}> M_{G-v}.$\n\\end{proof}\n\n\\begin{theorem}[The Local-Global Mean Inequality]\\label{LocalGlobal}\nLet $G$ be a connected block graph.\n\\begin{enumerate}\n\\item \\label{CorA} If $v$ is a vertex of $G$, then $M_{G}\\leq M_{G,v}$.\n\\item \\label{CorB} If $B$ is a block of $G$, then $M_{G}\\leq M^*_{G,B}$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFor \\ref{CorA}, let $G'$ be the graph obtained from $G$ by adding a new leaf vertex $u$ to $v$. Note that $\\mu_{G',u}=M_{G,v}.$ Thus, by Lemma~\\ref{MuLem},\n\\[\nM_{G,v}=\\mu_{G',u}\\geq M_{G'-u}=M_{G}.\n\\]\n\nFor \\ref{CorB}, let $G'$ be the graph obtained from $G$ by adding a new vertex $u$ and joining it to all vertices of $B$. Note that $\\mu_{G',u}=M^*_{G,B}.$ Thus by Lemma~\\ref{MuLem},\n\\[\nM^*_{G,B}=\\mu_{G',u}\\geq M_{G'-u}=M_G. \\qedhere\n\\]\n\\end{proof}\n\n\\section{The Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma} \\label{KeyLemmas}\n\nIn this section, we prove the Vertex Gluing Lemma, the Edge Gluing Lemma, and the Stretching Lemma. The Vertex Gluing Lemma extends the Gluing Lemma of~\\cite{MolOellermann2017} from trees to connected block graphs, and the proof is very similar once the local-global mean inequality is established for connected block graphs. The proof is included in Appendix~\\ref{Appendix} for completeness.\n\n\\begin{lemma}[The Vertex Gluing Lemma] \\label{vertex_gluing}\nLet $H$ be a connected block graph of order at least $2$ having vertex $v$. Fix a natural number $n\\geq 3$. Let $P:u_1\\dots u_n$ be a path of order $n$. For $s \\in \\{1, \\dots, n\\}$, let $G_s$ be the block graph obtained from the disjoint union of $P_n$ and $H$ by gluing $v$ to $u_s$. If $1\\leq i0$ is obvious. The inequality $W_{F,u}-N_{F,u}-N_{F-u}=W_{F,u}-N_F\\geq 0$ follows by Lemma~\\ref{WeiNum}, and the inequality $N_{F-u}+1-N_{F,u}\\geq 0$ follows by Lemma~\\ref{NumCom}. The inequality $(N_{F, u}-1)\\frac{n(n-1)(n-2)}{12}>0$ is immediate since $N_{F,u}\\geq 2$ and $n\\geq 4$.\nFinally,\n\\begin{align}\\label{MuIneq}\nW_{F, u}N_{F-u}-N_{F, u}N_{F-u}-N_{F, u}W_{F-u}+W_{F-u}\\geq 0 \\ \\ \\ &\\Longleftrightarrow \\ \\ \\ \\frac{W_{F,u}-N_{F,u}}{N_{F,u}-1}\\geq \\frac{W_{F-u}}{N_{F-u}}\\nonumber \\\\\n&\\Longleftrightarrow \\ \\ \\ \\mu_{F,u}\\geq M_{F-u},\n\\end{align}\nand (\\ref{MuIneq}) holds by Lemma \\ref{MuLem}. Therefore, the function $\\frac{d}{ds}M_{G_s}$ is strictly positive on the interval $\\left[1, \\frac{n}{2}\\right)$, and we conclude that $M_{G_1}0$ for all $s\\in [1,n-1]$. By the quotient rule, $\\tfrac{d}{ds}\\left[M_{G_s}\\right]$ has the same sign as\n\\begin{align}\\label{derivative}\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right],\n\\end{align}\nso it suffices to show that this expression is positive for all $s\\in[1,n-1]$.\n\nLet $s\\in[1,n-1]$. We first derive expressions for $N_{G_s}$, $W_{G_s}$, $\\tfrac{d}{ds}[N_{G_s}]$, and $\\tfrac{d}{ds}[W_{G_s}]$. We have\n\\begin{align*}\n\\Phi_{G_s}(x)&=\\frac{\\Phi_{F_s,v}(x)\\Phi_{H,u}(x)}{x}+\\Phi_{F_s-v}(x)+\\Phi_{H-u}(x)\\\\\n&=(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}x^i\\right)\\left[1+\\Phi_{H,u}(x)\\right]+\\Phi_{H-u}(x)+\\Phi_{P_{n-s-1}}(x)-1.\n\\end{align*}\nSubstituting $x=1$ yields\n\\begin{align}\\label{NG}\nN_{G_s}=2^{s-1}(n-s+1)(N_{H,u}+1)+N_{H-u}+\\tbinom{n-s}{2}-1.\n\\end{align}\nWe also find\n\\begin{align*}\n\\Phi'_{G_s}(x)&=\\left[(s-1)(1+x)^{s-2}\\left(\\sum_{i=0}^{n-s}x^i\\right)+(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}ix^{i-1}\\right)\\right]\\left[1+\\Phi_{H,u}(x)\\right]\\\\\n& \\ \\ \\ +(1+x)^{s-1}\\left(\\sum_{i=0}^{n-s}x^i\\right)\\left[\\Phi'_{H,u}(x)\\right]+\\Phi'_{H-u}(x)+\\Phi'_{P_{n-s-1}}(x).\n\\end{align*}\nSubstituting $x=1$ and simplifying yields\n\\begin{align}\nW_{G_s}&=\\left[(s-1)2^{s-2}(n-s+1)+2^{s-1}\\tbinom{n-s+1}{2}\\right]\\left[N_{H,u}+1\\right]\\nonumber\\\\\n& \\ \\ \\ +2^{s-1}(n-s+1)W_{H,u}+W_{H-u}+\\tbinom{n-s+1}{3}\\nonumber \\\\\n&=2^{s-2}(n-s+1)\\left[(n-1)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]+W_{H-u}+\\tbinom{n-s+1}{3}.\\label{WG}\n\\end{align}\nDifferentiating (\\ref{NG}) and (\\ref{WG}) with respect to $s$, and letting $L=\\ln(2)$ for ease of reading, we find\n\\begin{align}\n\\tfrac{d}{ds}\\left[N_{G_s}\\right]&=2^{s-1}L\\left(n-s+1-\\tfrac{1}{L}\\right)(N_{H,u}+1)-\\tfrac{2(n-s)-1}{2}, \\mbox{ and }\\label{NGprime}\\\\\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]&=2^{s-2}L\\left(n-s+1-\\tfrac{1}{L}\\right)\\left[\\left(n-1\\right)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]-\\tfrac{3(n-s)^2-1}{6}\\label{WGprime}\n\\end{align}\n\nFor convenience, we let $t=n-s$ and rewrite (\\ref{NG}), (\\ref{WG}), (\\ref{NGprime}), and (\\ref{WGprime}) below. Since $s\\in[1,n-1]$, we have $t\\in[1,n-1]$ as well.\n\\begin{align*}\nN_{G_s}&=2^{s-1}\\left(t+1\\right)(N_{H,u}+1)+N_{H-u}+ \\tbinom{t}{2}-1,\\\\\nW_{G_s}&=2^{s-2}(t+1)\\left[(n-1)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]+W_{H-u}+\\tbinom{t+1}{3},\\\\\n\\tfrac{d}{ds}[N_{G_s}]&=2^{s-1}L\\left(t+1-\\tfrac{1}{L}\\right)(N_{H,u}+1)-\\tfrac{2t-1}{2}, \\mbox{ and }\\\\\n\\tfrac{d}{ds}[W_{G_s}]&=2^{s-2}L\\left(t+1-\\tfrac{1}{L}\\right)\\left[\\left(n-1\\right)\\left(N_{H,u}+1\\right)+2W_{H,u}\\right]-\\tfrac{3t^2-1}{6}.\n\\end{align*}\nBy substituting these expressions into (\\ref{derivative}), expanding, and regrouping (and confirming with a computer algebra system), we find\n\\begin{align*}\n\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]\n&=\\sum_{i=1}^5 E_i,\n\\end{align*}\nwhere\n\\begin{align*}\nE_1&=2^{s-2}\\left(Lt+L-1\\right)\\left[(N_{H,u}+1)N_{H-u}-2W_{H-u}\\right],\\\\\nE_2&=2^{s-1}\\left(Lt+L-1\\right)\\left[W_{H,u}N_{H-u}-N_{H,u}W_{H-u}\\right],\\\\\nE_3&=2^{s-3}(t+1)^2\\left[(N_{H,u}+1)\\left((n-1)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right)+2W_{H,u}(Lt-2L+1)\\right],\\\\\nE_4&=-\\tfrac{1}{12}(t+1)^2(t^2-4t+2), \\mbox{ and }\\\\\nE_5&=2^{s-2}\\left(Lt+L-1\\right)(n-2)(N_{H,u}+1)N_{H-u}+\\tfrac{2t-1}{2}W_{H-u}-\\tfrac{3t^2-1}{6}N_{H-u}.\n\\end{align*}\n\nWe now show that $\\sum_{i=1}^5 E_i>0$. We first demonstrate that $E_1>0$ and $E_2\\geq 0$. Note that since $t\\geq 1$, the factor $Lt+L-1\\geq 2L-1>0$. By Lemma~\\ref{WeightBound}, $N_{H,u}N_{H-u}\\geq 2W_{H-u}$. Therefore,\n\\[\nE_1\\geq 2^{s-2}(Lt+L-1)N_{H-u}>0.\n\\]\nBy Theorem~\\ref{LocalGlobal}\\ref{CorA}, $M_{H,u}\\geq M_{H-u}$, or equivalently $W_{H,u}N_{H-u}\\geq W_{H-u}N_{H,u}$. It follows immediately that $E_2\\geq 0$.\n\nNext we show that $E_3+E_4> 0$. We begin by bounding $E_3$. Since $H$ has order at least $2$, we have $W_{H,u}\\geq N_{H,u}+1$. This gives\n\\[\nE_3\\geq 2^{s-3}(t+1)^2(N_{H,u}+1)\\left[(n+1)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right]\n\\]\nNow we use the fact that $t\\leq n-1$, or equivalently $n+1\\geq t+2$. This gives\n\\begin{align*}\nE_3&\\geq 2^{s-3}(t+1)^2(N_{H,u}+1)\\left[(t+2)(Lt-2L+1)-\\tfrac{2}{3}(Lt^2+(2-L)t-1)\\right]\\\\\n&=2^{s-3}(t+1)^2(N_{H,u}+1)\\tfrac{1}{3}\\left[Lt^2+(2L-1)t+8-12L\\right]\n\\end{align*}\nFinally, since $H$ has order at least $2$, we have $N_{H,u}\\geq 2$. Applying this inequality along with $s\\geq 1$ gives\n\\[\nE_3\\geq \\tfrac{1}{4}(t+1)^2\\left[Lt^2+(2L-1)t+8-12L\\right].\n\\]\nTherefore,\n\\begin{align*}\nE_3+E_4&\\geq \\tfrac{1}{4}(t+1)^2\\left[Lt^2+(2L-1)t+8-12L\\right]-\\tfrac{(t+1)^2(t^2-4t+2)}{12}\\\\\n&=\\tfrac{1}{12}(t+1)^2\\left[(3L-1)t^2+(6L+1)t+22-36L\\right].\n\\end{align*}\nRecalling that $L=\\ln(2)$, one can verify that the quadratic in the square brackets of this last expression is positive for all $t\\geq 1$. Thus, we conclude that $E_3+E_4>0$.\n\nFinally, we show that $E_5>0$. We use the inequalities $W_{H-u}\\geq N_{H-u}$, $N_{H,u}\\geq 2$, $s\\geq 1$, and $n-2\\geq t-1$.\n\\begin{align*}\nE_5&\\geq \\tfrac{3}{2}\\left(Lt+L-1\\right)(t-1)N_{H-u}+\\tfrac{2t-1}{2}N_{H-u}-\\tfrac{3t^2-1}{6}N_{H-u}\\\\\n&=\\tfrac{1}{6}\\left[(9L-3)t^2-(9L+3)t+7\\right]\n\\end{align*}\nRecalling that $L=\\ln(2)$, it is straightforward to verify that this last expression is strictly positive for all $t$, and hence $E_5>0$.\n\nWe have shown that $E_1>0,$ $E_2\\geq 0$, $E_3+E_4>0$, and $E_5>0$ for all $s\\in[1,n-1]$. It follows that $\\tfrac{d}{ds}\\left[W_{G_s}\\right]N_{G_s}-W_{G_s}\\tfrac{d}{ds}\\left[N_{G_s}\\right]=\\sum_{i=1}^5 E_i>0$ for all $s\\in[1,n-1]$, as desired.\n\\end{proof}\t\n\t\n\t\n\n\\section{Concluding Remarks}\n\nIn this article, we demonstrated that among all connected block graphs of order $n$, the path has smallest mean CIS order. This extends Jamison's result: {\\em Among all trees of order $n$, the path has smallest mean subtree order}. Moreover, our main result lends support to the conjecture made in \\cite{KroekerMolOellermann2018}: {\\em Among all connected graphs of order $n$, the path has minimum mean CIS order}.\n\n\\medskip\n\nThe problem of determining the structure of those block graphs, of a given order, with maximum mean CIS order remains open. It was conjectured by Jamison \\cite{Jamison1983} that a tree with maximum mean subtree order among all trees of order $n$, called an {\\em optimal} tree of order $n$, is a caterpillar. This is known as Jamison's Caterpillar Conjecture. This conjecture has been verified for all $n\\leq 24$ (see~\\cite{Jamison1983, MolOellermann2017}). Mol and Oellermann~\\cite{MolOellermann2017} made some progress on describing the structure of optimal trees. They proved that in any optimal tree of order $n$, every leaf is adjacent with a vertex of degree at least $3$, and that the number of leaves in an optimal tree of order $n$ is $\\mathrm{O}(\\log_2n)$ (moreover, the number of leaves is $\\Theta(\\log_2n)$ if Jamison's Caterpillar Conjecture is true).\n\nTurning to block graphs, for $n\\in\\{3,4\\}$, the complete graph has maximum mean CIS order among all block graphs of order $n$. We have verified that for $5\\leq n\\leq 11$, the block graph of order $n$ with maximum mean CIS order is a tree (more specifically, a caterpillar). We make the following conjecture, which strengthens Jamison's Caterpillar Conjecture.\n\n\\begin{conjecture}\nFor $n\\geq 5$, if $G$ has maximum mean CIS order among all block graphs of order $n$, then $G$ is a caterpillar.\n\\end{conjecture}\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe ongoing transition to 'green' energy urges us to develop advanced electrochemical devices for energy conversion and storage and, consequently, increases the need for a deeper theoretical understanding of such systems. The electrode\/solution interface, which is an essential element of electrochemical devices, is especially interesting but presents a challenge for researchers. At the charged electrode\/electrolyte interface the concentration of ions may be many times higher than the bulk solution value. That, of course, strongly affects the rate of the electrochemical reactions involving ionic species. Thus, tuning the electric double layer (EDL) composition may be of use in a number of electrochemical devices. For instance, in alkali-metal batteries (including Li-ion ones) the so-called solid-electrolyte interphase (SEI) layer is formed at the surface of both the anode and the cathode active materials. SEI consists of insoluble and partially soluble reduction products of electrolyte components and has a great impact on battery performance \\cite{peled2017sei}. Particularly, the SEI structure affects the electron transfer kinetics and the battery cycle life. Controlling the EDL composition would be an extra tool for managing the SEI formation process. In an alkali-salt electrolyte the interface structure can be altered by introducing organic cations that would behave differently inside the EDL due to the factors arising from its complex molecular structure, e.g. steric limitations, dipole moment of the functional groups and specific interaction with the surface such as $\\pi-\\pi$ stacking. The remarkable modifiability of the organic cation implies that a compound can be designed to efficiently alter the EDL structure. Furthermore, the mixed electrolyte composition including both simple (alkali) and organic cations is already becoming more common as researchers and engineers are exploring the benefits of ionic liquids when applied to batteries, fuel cells, supercapacitors, solar cells, {\\sl etc}. \\cite{fedorov2014ionic}. Thus, it is also becoming more relevant to make a theoretical description of the EDL incorporating both simple and organic cations. \n\n\nAnother bright example confirming the importance of the EDL structure is $Li$-air ($Li-O_2$) battery which is a promising electrochemical storage technology that hypothetically can provide several times higher specific energy than that of conventional $Li$-ion batteries \\cite{aurbach2016advances}. Unfortunately, the cell capacity values achieved in the experiments so far are way below the theoretical expectations mostly due to the electrode surface passivation by the discharge product itself, i.e. $Li_2O_2$. There are two possible pathways of $Li_2O_2$ formation during the discharge process \\cite{johnson2014role}. Relatively large micron-size $Li_2O_2$ particles which do not consume much of the electrode surface can be formed as a result of the chemical disproportionation reaction of $LiO_2$ intermediate in the bulk electrolyte \\cite{zhai2013disproportionation, mitchell2013mechanisms}. Alternatively, it is possible to grow a thin $Li_2 O_2$ film through $Li O_2$ electrochemical reduction \\cite{wen2013situ, johnson2014role} at the surface with the consumption of extra $Li^{+}$ ions. The predominance of the second surface-mediated pathway leads to complete electrode surface passivation (and cell \"death\") while most of the available space inside the porous cathode is still unfilled with the discharge product. That drastically decreases the cell specific capacity. Exclusion of $Li^{+}$ ions from the EDL could inhibit the undesired surface-mediated pathway and slow down the passivation process thus enabling the accumulation of a larger amount of the discharge product inside the porous electrode. Introduction of organic cations into the electrolyte solution would be an easy practical way of controlling the EDL structure.\n\nFrom the physics point of view, such organic cations should be attracted by the cathode more strongly than $Li^{+}$ ions. \nThus, the greater attractive force acting on each of the organic cations must cause the lithium cations to be expelled from the cathode surface at its sufficiently large surface charge density. One of the possible ways to make the attraction of the organic cations stronger is using of an organic salt with highly polar cations. It can be, for instance, some room temperature ionic liquid, whose organic cations usually possess a rather high dipole moment or electronic polarizability. As is well known, in an inhomogeneous electric field a polar particle suffers a dielectrophoretic force (see, for instance, \\cite{jones1979dielectrophoretic}) directed to the highest electric field. In our case, the electric field inhomogeneity takes place due to the electrostatic screening of the electrode charge by the mobile electrolyte ions, so that the maximal electric field is reached on the electrode surface. Therefore, the occurrence of a dipole moment on the organic cation should lead to its higher attraction to the electrode with respect to $Li^{+}$. The latter will provoke the expelling of lithium cations from the electrode surface due to the excluded volume interactions and their complete replacement with the electrochemically inactive organic cations at a sufficiently large electrode surface charge density (see Fig. \\ref{Schematic}). \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{FIG_0.PNG}}\n\\caption{Schematic representation of the electrode\/solution interface in mixed electrolyte containing simple (red circle) and molecular (large red circle with a small blue one) cations: bulk solution \u2013 dipole moments of molecular cations are disordered, equimolar cations concentration; EDL \u2013 dipoles are highly ordered according to electric field, molecular cations dominate.}\n\\label{Schematic}\n\\end{figure}\n\nIn order to understand whether it is possible to replace (at least partially) the lithium cations with organic dipolar cations on the cathode surface at physically reasonable parameters of the system, such as surface charge density, dielectric permittivity of the solvent, dipole moment of the cation, and ionic concentrations in the bulk, it would be useful to have a simple analytical model of a two-component electrolyte solution with one type of anions and two types of cations on the negatively charged metal electrode. Within such a theory, one of the cations would be a simple structureless particle with a charge $+e$, whereas the other one -- a particle possessing not only a charge $+e$, but also a certain permanent dipole moment $p$. A self-consistent field (SCF) theory, based on the modified Poisson-Boltzmann equation for the electrostatic potential and taking into account the steric interactions of ions and polarity of one type of cations could be such a theory. Despite the fact that there are a lot of SCF theories of electrolyte solutions taking into account polarizable\/polar additives \\cite{coalson1996statistical,abrashkin2007dipolar,gongadze2011langevin,frydel2011polarizable,budkov2018nonlocal,budkov2019statistical,Sin2017,Misra2013,Hatlo2012,Ben-Yaakov2011,Wei1993,Das2012,Jiang2014,gongadze2011generalized,budkov2016theory,budkov2018theory,budkov2015modified,nakayama2015differential,lopez2018diffuse,lopez2018multiionic}, to the best of our knowledge, up to present, the issues raised above have not been systematically discussed in the literature. Therefore, below we will formulate such a SCF theory and study in its framework the possibility, in principle, to exclude simple cations (which mimic the $Li^{+}$ ions) from the cathode surface by adding some dipolar cations (which mimic organic cations) to the electrolyte solution. In order to confirm our theoretical observations, we will compare the obtained results with those obtained by molecular dynamics simulation. \n\n\n\\section{Theory}\nLet us consider an interface between a flat charged electrode with a surface charge density $\\sigma$ and an electrolyte solution with one type of monovalent anions and two types of monovalent cations. Let us also assume that the cations of the first type (simple cations) carry an electric charge $e$, whereas the cations of the second type (molecular cations) in addition to the same charge possess a permanent dipole moment $p$. We start from the grand thermodynamic potential of the electrolyte solution in the local density approximation, which can be written in the form \\cite{budkov2016theory,budkov2018theory}\n\\begin{equation}\n\\label{Grand_pot}\n\\Omega=\\int\\limits_{0}^{\\infty}\\left(-\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2}{2}+ \\rho_{c}\\psi-c_{+}^{(2)}\\Psi +f-\\mu_{+}^{(1)}c_{+}^{(1)}-\\mu_{+}^{(2)}c_{+}^{(2)}-\\mu_{-}c_{-}\\right)dz,\n\\end{equation} \nwhere $f=f(T,c_{+}^{(1)},c_{+}^{(2)},c_{-})$ is the Helmholtz free energy density of a reference system (a system without electrostatic interactions) and the auxiliary function\n\\begin{equation}\n\\Psi(z)=k_{B}T\\ln\\frac{\\sinh{\\beta p\\mathcal{E}(z)}}{\\beta p\\mathcal{E}(z)},\n\\end{equation}\nis introduced. $\\mathcal{E}(z)=-\\psi^{\\prime}(z)$ is the local electric field and $\\rho_{c}(z)=e\\left(c_{+}^{(1)}(z)+c_{+}^{(2)}(z)-c_{-}(z)\\right)$ is the local charge density of the ions; $T$ is the temperature and $k_B$ is the Boltzmann constant. Using the local Legendre transformation \\cite{maggs2016general,budkov2016theory,budkov2018theory,mceldrew2018theory}, we can rewrite the grand thermodynamic potential in the form\n\\begin{equation}\n\\label{Grand_pot_2}\n\\Omega=-\\int\\limits_{0}^{\\infty}dz\\left(\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2(z)}{2}+P(T,\\bar{\\mu}_{+}^{(1)},\\bar{\\mu}_{+}^{(2)},\\bar{\\mu}_{-})\\right),\n\\end{equation}\nwhere the local chemical potentials of the species\n\\begin{equation}\n\\bar{\\mu}_{+}^{(1)}=\\mu_{+}^{(1)}-e\\psi,~\\bar{\\mu}_{+}^{(2)}=\\mu_{+}^{(2)}-e\\psi+\\Psi,\n\\end{equation}\n\\begin{equation}\n\\bar{\\mu}_{-}=\\mu_{-}+e\\psi\n\\end{equation}\nare introduced. A variation of functional (\\ref{Grand_pot_2}) with respect to the potential $\\psi(z)$ leads to the following self-consistent field equation \\cite{budkov2016theory}:\n\\begin{equation}\n\\frac{d}{dz}\\left(\\epsilon(z)\\psi^{\\prime}(z)\\right)=-e\\left(\\bar{c}_{+}^{(1)}(z)+\\bar{c}_{+}^{(2)}(z)-\\bar{c}_{-}(z)\\right), \\end{equation}\nwhere\n\\begin{equation}\n\\bar{c}_{+}^{(1,2)}=\\frac{\\partial{P}}{\\partial{\\bar{\\mu}}_{+}^{(1,2)}}~,\\bar{c}_{-}=\\frac{\\partial{P}}{\\partial{\\bar{\\mu}}_{-}},\n\\end{equation}\nare the local equilibrium concentrations of the ions and\n\\begin{equation}\n\\epsilon(z)=\\varepsilon\\varepsilon_0+\\frac{p^2}{k_{B}T}\\frac{L(\\beta p\\mathcal{E})}{\\beta p\\mathcal{E}}\\bar{c}_{+}^{(2)}(z)\n\\end{equation}\nis the local dielectric permittivity; $L(x)=\\coth{x}-x^{-1}$ is the Langevin function. The first integral of the self-consistent field equation determining the condition of the solution mechanical equilibrium \\cite{budkov2016theory,budkov2015modified} takes the form\n\\begin{equation}\n\\label{mech_eq_cond}\n-\\frac{\\varepsilon\\varepsilon_0\\mathcal{E}^2}{2}-p\\mathcal{E}\\bar{c}_{+}^{(2)}L(\\beta p\\mathcal{E})+P(T,\\bar{\\mu}_{+}^{(1)},\\bar{\\mu}_{+}^{(2)},\\bar{\\mu}_{-})=P(T,{\\mu}_{+}^{(1)},{\\mu}_{+}^{(2)},{\\mu}_{-}).\n\\end{equation}\nSolving equation (\\ref{mech_eq_cond}) with respect to $\\mathcal{E}$ at different $\\psi$, we obtain the implicit function $\\mathcal{E}=\\mathcal{E}(\\psi)$. In order to obtain the potential profile $\\psi(z)$, it is necessary to solve the ordinary first-order differential equation $\\psi^{\\prime}(z)=-\\mathcal{E}(\\psi)$ with the initial condition $\\psi(0)=\\psi_0(\\sigma)$, where $\\psi_0(\\sigma)$ is the surface potential of the electrode corresponding to the fixed surface charge density $\\sigma$. In order to obtain the function $\\psi_0(\\sigma)$, it is necessary to use the boundary condition $\\epsilon_s\\mathcal{E}_0=\\sigma$,\nwhere the local dielectric permittivity at the electrode is $\\epsilon_s=\\epsilon(0)$ and the electric field at the electrode $\\mathcal{E}_{0}=\\mathcal{E}(0)$ are introduced. The latter is an implicit function of the surface potential $\\psi_0$, determined by eq. (\\ref{mech_eq_cond}).\n\n\nAs a reference system, let us consider three-component symmetric lattice gas model \\cite{maggs2016general,budkov2016theory,kornyshev2007double} with the equation of state\n\\begin{equation}\nP(T,{\\mu}_{+}^{(1)},{\\mu}_{+}^{(2)},{\\mu}_{-})=\\frac{k_B T}{v}\\ln\\left(1+e^{\\beta\\mu_{+}^{(1)}}+e^{\\beta\\mu_{+}^{(2)}}+e^{\\beta\\mu_{-}}\\right), \n\\end{equation}\nwhere $\\beta=(k_{B}T)^{-1}$. The local concentrations of the ions within the three-component lattice gas model are determined by the following relations\n\\begin{equation}\n\\bar{c}_{+}^{(1)}(z)v=\\frac{e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}},\n\\end{equation}\n\\begin{equation}\n\\bar{c}_{+}^{(2)}(z)v=\\frac{e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}},\n\\end{equation}\n\\begin{equation}\n\\bar{c}_{-}(z)v=\\frac{e^{\\beta(\\mu_{-}+e\\psi(z))}}{1+e^{\\beta(\\mu_{+}^{(1)}-e\\psi(z))}+e^{\\beta(\\mu_{+}^{(2)}-e\\psi(z)+\\Psi(z))}+e^{\\beta(\\mu_{-}+e\\psi(z))}}.\n\\end{equation}\n\nUsing the electrical neutrality condition for the bulk solution $c_{-,b}=c_{+,b}^{(1)}+c_{+,b}^{(2)}$ and \nintroducing the bulk concentrations of electrolytes $c_{1}=c_{+,b}^{(1)}$ and $c_{2}=c_{+,b}^{(2)}$, we obtain the relations for the species chemical potentials\n\\begin{equation}\n\\mu_{+}^{(1,2)}=k_{B}T\\ln\\left(\\frac{c_{1,2}v}{1-2v(c_{1}+c_{2})}\\right),~\\mu_{-}=k_{B}T\\ln\\left(\\frac{(c_{1}+c_{2})v}{1-2v(c_{1}+c_{2})}\\right).\n\\end{equation}\n\\section{Computational details: Molecular dynamics simulations}\nAll the molecular dynamics simulations were performed with the help of the LAMMPS \\cite{plimpton1993fast} simulation package. The cutoff distance for the electrostatic and van der Waals interactions was set to $2~nm$. The long range electrostatic interactions were calculated using the pppm scheme \\cite{hockney1988computer}. The simulations we carried out in an $NVT$-ensemble at $300~K$ using a Langevin thermostat (dump parameter $100~fs$) wan integration time-step of $1~fs$. Periodic boundary conditions were applied along the $x$ and $y$ directions, while the unwanted interactions between the replicas along the $z$ direction were eliminated by using special algorithms provided within the LAMMPS package \n\\cite{yeh1999ewald,ballenegger2009simulations}.\nThe simulation cell contained two parallel plates (parallel to $xy$ plane) constructed from frozen atoms ordered according to the hexagonal $2-d$ lattice with the parameter $a = 0.293~nm$ (that resembles Au(111) surface topology). The plate size was $58.6 \\times 60.90$ $\\AA$ ($xy$) with an $8~nm$ distance (along the $z$ axis) between them. The plates were charged oppositely by setting the required atomic charges. The space between the 'electrode plates' was filled with ions. The solvent effect was taken into account implicitly by setting the relative dielectric permittivity to 40 (close to that of commonly used organic solvents such as acetonitrile, dimethylsulfoxide, {\\sl etc.}). The solvent viscosity effect was simulated by a Langevin thermostat. The van der Waals interactions were described by the Lennard-Jones 6-12 potential with the depth $\\varepsilon=0.1~Kcal\/mol$ and the repulsion distance equal to $0.3~nm$ and $0.38~nm$ for the ions and 'electrode' atoms, respectively. The simple cations (and anions) were represented by individual atoms with the charge $+1e$ ($-1e$) and mass $10~g\/mol$. The organic cations were designed in a Drude-particle fashion with a constant dipole moment. It was composed of a Drude-core (mass $9.6~g\/mol$, charge $+3e$) and a pseudo electron (mass $0.4~g\/mol$, charge $-2e$). The distance between the core and the 'electron' was kept fixed during the simulation at the value of $0.4~\\AA$ or $0.2~\\AA$ that yield the dipole moment about $4~D$ ($3.84$) and $2~D$ ($1.92$), respectively. The pseudo-electrons were thermalized separately \\cite{jiang2011high} at $T=1~K$ temperature by a special Langevin thermostat with a dump parameter of $20~fs$. \n\nAt higher surface charge values the EDL capacity turned out to exceed the number of ions between the plates at the desired concentration ($1.0$ or $0.5~M$ of the cation-anion pairs). Therefore, prior to the main simulation an iterative procedure was employed to continuously add (or remove, in case of overshoot) ion pairs with cations of the required type to achieve the target bulk concentration value. \\footnote{Here 'bulk' concentration is attributed to the central $60~\\AA$ thick slab.} After the EDL was completely saturated and the target bulk concentrations were reached, the refiling procedure was stopped. The following productive run was performed to obtain the $1~ns$ long trajectory. \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_1AB.png}}\n\\caption{The concentration profiles of the simple cations calculated within the mean field theory at different values of the cathode surface charge density. The concentration profile of the molecular cations is also shown only for one case. The data are shown for the effective size $v^{1\/3}=3.7~\\AA$, temperature $T=300~K$, bulk concentrations of the cations $c_{1}=c_{2}=0.5~M$, and two dipole moments of the molecular cations $p=2~D$ (A) and $p=4~D$ (B).}\n\\label{Profiles_1}\n\\end{figure}\n\n\n\\section{Results and discussions}\nFirst, we discuss the cation concentration profiles near the electrode surface, which can be predicted by the formulated above mean field theory at different values of dipole moment of the molecular cation $p$ and the cathode surface charge density $\\sigma <0$. Let us consider the equimolar electrolyte compostion, i.e. $c_{1}=c_{2}=0.5~M$ (total $1~M$ salt concentration). Such value is within the typical range ($0.1-1.0~M$) for the real electrochemical devices discussed in the introductory section. We also assume that the dielectric permittivity $\\varepsilon=40$ that approximately corresponds to the value of common organic solvents such as dimethylsulfoxide or acetonitrile. Figure \\ref{Profiles_1} (A,B) shows the theoretical concentration profiles of the simple cations calculated for two values of the dipole moments of molecular cations at different values of the cathode surface charge density $|\\sigma|$. As is seen, at a sufficiently small $|\\sigma|$, the concentration of the simple cations monotonically decreases together with the distance from the electrode, so that its maximal value is attained at the electrode surface (at $z=0$). Moreover, higher absolute value of the surface charge density $|\\sigma|$ causes the increase in the local concentration of the simple cations on the electrode. However, when the $|\\sigma|$ exceeds a certain threshold value, the local concentration monotonically decreases, while the concentration profiles exhibit a pronounced maximum at a certain distance $z>0$. This maximum can be shifted to higher distances by a further increase in the surface charge density. As is seen in Fig. \\ref{Profiles_1} (B), the described effects become stronger at a larger dipole moment of the molecular cations. Moreover, at a rather large dipole moment of the molecular cations the maxima on the concentration profiles become sharper. Fig. \\ref{Profiles_1} (B) also shows the concentration profile of the molecular cations only for one case. As is seen, at $|\\sigma|=86~\\mu C\/cm^2$ the simple cations are depleted near the electrode surface, while the concentration of the molecular cations reaches the maximal value there. The latter clearly demonstrates the replacement of the simple cations by the molecular cations. \n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_2.png}}\n\\caption{The concentration profiles of the simple cations obtained by the MD simulations for the different cathode surface charge density. The data are shown for $p=4~D$.}\n\\label{Profiles_2}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_3.png}}\n\\caption{The local concentration of the simple cations on the electrode as a function of the cathode surface charge density calculated within the mean field approximation. The local concentration of molecular cations is also shown for only one case. The data are shown for different dipole moments and concentrations of the simple and molecular cations in the bulk.}\n\\label{surf_conc_1}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\center{\\includegraphics[width=0.8\\linewidth]{Fig_4.png}}\n\\caption{The local concentration of the simple cations on the electrode as a function of the cathode surface charge density obtained by the MD simulations. The local concentration of molecular cations is also shown for only one case. The data are shown for dipole moment $p=4~D$ of the molecular cation and different bulk concentrations of all cations.}\n\\label{surf_conc_2}\n\\end{figure}\n\nSuch a behavior can be interpreted as follows. At a rather small $|\\sigma|$ there is a lot of free space for both types of the cations near the electrode surface, a higher surface charge density increases the local concentration of the simple cations on the electrode, while the concentration profiles are described by the monotonically decreasing functions of the distance. In this case, the maximum of concentration is reached on the electrode surface. At a sufficiently strong surface charge density the molecular cations are more strongly attracted to the electrode than the simple cations due to the occurrence of an additional dielectrophoretic force \\cite{jones1979dielectrophoretic,budkov2018theory,budkov2016theory} acting on each dipolar cation. Therefore, at a rather large $|\\sigma|$ the presence of the molecular cations near the electrode is more thermodynamically favorable, so that the simple cations are expelled from the $\"$near-surface$\"$ layer due to the steric interactions. The latter explains the occurrence of a pronounced maximum on the concentration profiles of the simple cations, which can be shifted by the increase in the surface charge density (see Figs. \\ref{Profiles_1} (A,B)). The shift in the concentration maximum due to the increase in the surface charge density can also be explained by the simple cations exclusion the layers nearest to the electrode and their replacement with molecular cations. We would like to note that similar replacement from the cathode surface of alkaline cations by another alkaline cations due to a difference in their excess polarizabilities in aqueous medium was discussed within the mean-field theory in paper \\cite{lopez2018diffuse,lopez2018multiionic}.\n\nIn order to confirm our theoretical findings, we performed MD simulations (for the details of the simulation setup, see the previous section). Fig. \\ref{Profiles_2} shows the concentration profiles of the simple cations obtained by means of the MD simulations for the case $p=4~D$. The first concentration peak (which can be considered as a local concentration at the electrode surface) grows as the surface charge density increases up to about 65 $\\mu C\/cm^2$. A further increase in the electrode surface charge density leads to a decrease in the first concentration peak and simultaneous growth in the second one. Thus, the simple cations are expelled into the second ionic layer. The difference in comparison with the theoretical model is that the simple cations are pushed away from the surface not continuously (along the z coordinate) but layer by layer. Nevertheless, the general behavior of the concentration profiles predicted by the theoretical model is in qualitative agreement with that obtained by MD simulations. \n\nLet us now analyze the local concentration of the simple cations at the electrode surface as a function of the surface charge density for several electrolyte compositions. The results obtained within our mean field theory are presented in Fig. \\ref{surf_conc_1}. The grey dotted line represents the surface concentration of the molecular cations. Comparing it with the simple cations concentration in the same case (black line) one can see how the surface ionic composition depends on the electrode charge density. Considering the equimolar compositions ($c_{1}\/c_{2}=0.5\/0.5$, i.e. $1~M$ total and $c_{1}\/c_{2}=0.25\/0.25$, i.e. $0.5~M$ total), it should be noted that the surface concentration of the simple (as well as molecular) cations weakly depends on the total bulk electrolyte concentration. The higher dipole moment of the molecular cations leads to a lower surface concentration of the simple cations, as the dielectrophoretic force is stronger in that case and the molecular cations are more strongly attracted to the cathode surface. The decline from the equimolar compositions in case of the higher bulk concentration of the simple cations ($c_{1}\/c_{2}=0.5\/0.25$, i.e. $0.75~M$ total) leads to higher surface concentration of simple cations, as there are less molecular cations to compete for the place within the EDL. However, the first ionic layer still consists mostly of molecular cations. It should be also noted that the position of the maximum of the function $c_{surf}=c_{surf}(\\sigma)$ depends on the dipole moment $p$, but does not depend on the bulk electrolyte composition. Fig. \\ref{surf_conc_2} presents the surface concentration dependences for $p = 4~D$ obtained by means of MD simulations. Although there are some quantitative differences, the qualitative behavior of the system is in good agreement with the theoretical predictions.\n\n\nWe would like to note that in the present theoretical mean field model as well as in the MD simulations we modelled the solvent as a continuous dielectric medium with constant dielectric permittivity. Thereby, we neglected the effects of solvent polarization causing a reduction in the dielectric permittivity near the charged electrode relative to the bulk solution \\cite{yeh1999dielectric,gongadze2011langevin,gongadze2011generalized}. It is clear that accounting for this effect will not change qualitatively the simple cations exclusion effects, but can still significantly change the region of the surface charge densities, where this effect may take place. We neglected also an influence of ions on the dielectric permittivity of solution (so-called, dielectric decrement) \\cite{Ben-Yaakov2011}. Moreover, in this study we did not take into account the short-range specific interactions between the ionic species and the electrode \\cite{budkov2018theory,goodwin2017mean}. However, in this study we aimed to investigate the possibility, in principle, of excluding simple cations from the charged cathode by adding an organic salt with dipolar cations to the electrolyte solution at the simplest level of the theoretical model. The reasonable values of the surface charge densities, bulk concentrations of the electrolytes, and dipole moment of the organic cations, predicted by our mean field theory and qualitative agreement of the theoretical predictions with the MD simulations support the correctness of the obtained results.\n\n\n\\section{Conclusions}\nIn this work, we have formulated a mean field theory of the flat electric double layer on the interface between a metallic cathode and an electrolyte solution with two types of monovalent cations. One type of the cations is described as structureless charged particles (simple cation, e.g. alkali ion), whereas another one \u2013 as charged particle carrying polar groups with a considerable dipole moment (molecular cations). In practice the latter can be an organic cation (e.g. quaternary ammonium $NR_4^{+}$) with an attached acceptor group, such as trifluoromethyl ($-CF_3$), nitrile ($-CN$), carbonyl ($-COOR$), sulfonyl ($-SO_3 R$), {\\sl etc}. We have shown that at a sufficiently high absolute value of the cathode surface charge density (about $30~\\mu C\/cm^2$), the simple cations are expelled from the cathode surface by the molecular cations due to the stronger attraction of the latter to the electrode. We have shown that the stronger attractive force is realized due to the occurrence of the dielectrophoretic force acting on each molecular cation in the inhomogeneous electrostatic field. We have confirmed our theoretical predictions by molecular dynamics simulations. The reported study estimates the strength of the cation replacement effect for various sets of parameters, which can be further used for experiment design. \n\n\n\\begin{acknowledgement}\nThe reported study was partially supported by the RFBR according to research project no. 18-31-20015. The MD simulation part of the study carried out by Artem Sergeev was supported by Russian Science Foundation grant no. 19-43-04112.\n\\end{acknowledgement}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn the standard model (SM), the weak phase of $\\bs$-$\\bsbar$ mixing,\n$\\beta_s$, is $\\approx 0$. Thus, if its value is measured to be\nnonzero, this is a clear sign of new physics (NP). Indeed, experiments\nhave already started measuring $\\beta_s$ in $\\Bsdecay$. The results of\nthe CDF \\cite{CDF} and D\\O\\ \\cite{D0} collaborations hint at NP, but\nthe errors are very large. On the other hand, the LHCb collaboration\n\\cite{LHCb_betas} finds a central value for $\\beta_s$ which is\nconsistent with zero: $\\beta_s = (-0.03 \\pm 2.89~({\\rm stat}) \\pm\n0.77~({\\rm syst}))^\\circ$, implying that, if NP is present in\n$\\bs$-$\\bsbar$ mixing, its effect is small.\n\nIn the $\\bd$ system, the phase of $\\bd$-$\\bdbar$ mixing, $\\beta$, was\nfirst measured in the ``golden mode'' $\\bd\\to J\/\\psi \\ks$, and\nsubsequently in many other modes such as $\\btos$ penguin decays\n(e.g.\\ $\\bd\\to \\phi \\ks$), ${\\bar b} \\to {\\bar c}c{\\bar d}$ decays\n(e.g.\\ $\\bd\\to J\/\\psi \\pi^0$), etc. In the same vein, it is important\nto measure $\\beta_s$ in many different decay modes.\n\nOne process which is potentially a good candidate for measuring\n$\\beta_s$ is the pure $\\btos$ penguin decay $\\bskk$. Its amplitude can\nbe written\n\\beq\n{\\cal A}_s = V_{ub}^* V_{us} P'_{uc} + V_{tb}^* V_{ts} P'_{tc} ~.\n\\eeq\nNow, we know that $|V_{ub}^* V_{us}|$ and $|V_{tb}^* V_{ts}|$ are\n$O(\\lambda^4)$ and $O(\\lambda^2)$, respectively, where $\\lambda=0.23$\nis the sine of the Cabibbo angle. This suggests that the $V_{ub}^*\nV_{us} P'_{uc}$ term is possibly negligible compared to $V_{tb}^*\nV_{ts} P'_{tc}$. If this is justified, then there is essentially only\none decay amplitude, and $\\beta_s$ can be cleanly extracted from the\nindirect CP asymmetry in $\\bskk$.\n\nThe difficulty is that it is not completely clear whether $V_{ub}^*\nV_{us} P'_{uc}$ is, in fact, negligible. This term has a different\nweak phase than that of $V_{tb}^* V_{ts} P'_{tc}$, so that its\ninclusion will ``pollute'' the extraction of $\\beta_s$. That is, if it\ncontributes significantly to the amplitude, the value of $\\beta_s$\nmeasured in $\\bskk$ will deviate from the true value of $\\beta_s$, and\nthis theoretical error is directly related to the relative size of the\ntwo terms.\n\nThis issue has been examined by Ciuchini, Pierini and Silvestrini\n(CPS) in Ref.~\\cite{CPS}. In order to get a handle on the size of\n$P'_{uc}$, CPS proceeded as follows. They considered the\nU-spin-conjugate decay, $\\bdkk$, focusing specifically on $\\bd \\to\nK^{*0} {\\bar K}^{*0}$. This is a pure $\\btod$ penguin decay, whose\namplitude is\n\\beq\n{\\cal A}_d = V_{ub}^* V_{ud} P_{uc} + V_{tb}^* V_{td} P_{tc} ~.\n\\eeq\nIf one takes the values for the CKM matrix elements, including the\nweak phases, from independent measurements, then this amplitude\ndepends only on three unknown parameters: the magnitudes of $P_{uc}$\nand $P_{tc}$, and their relative strong phase. But there are three\nexperimental measurements one can make of this decay -- the branching\nratio, the direct CP asymmetry, and the indirect (mixing-induced) CP\nasymmetry. It is therefore possible to solve for all the unknown\nparameters. In particular, one can obtain $|P_{uc}|$. This quantity\ncan be related to $|P'_{uc}|$ by an SU(3)-breaking factor. Now, in\n2007, when Ref.~\\cite{CPS} was written, there were no experimental\nmeasurements of $B_{d,s}^0 \\to K^{*0} {\\bar K}^{*0}$. Instead, CPS\nassumed values for these measurements, inspired by QCD factorization\n(QCDf) \\cite{BBNS}. They found that, even allowing for 100\\% SU(3)\nbreaking, the value of $|P'_{uc}|$ is such that the error on $\\beta_s$\ndue to the inclusion of a nonzero $V_{ub}^* V_{us} P'_{uc}$ term is\nless than $1^\\circ$. This inspired CPS to dub $\\bskk$ the {\\it golden\n channel} for measuring $\\beta_s$.\n\nIn this paper, we re-examine the method of CPS. In particular, we want\nto establish to what extent CPS's conclusion is dependent on the\nvalues chosen for the $\\bdkk$ experimental observables. As we will\nsee, the CPS result holds for a significant subset of the input\nvalues. However, it also fails for other choices of the inputs -- the\nerror on $\\beta_s$ due to the presence of the $V_{ub}^* V_{us}\nP'_{uc}$ term can be as large as $18^\\circ$. It is therefore not\ncorrect to say that the $V_{ub}^* V_{us} P'_{uc}$ term has little\neffect, i.e.\\ that $\\beta_s$ can always be measured cleanly in\n$\\bskk$. On the other hand, it is true that $|P_{uc}|$ can be\nextracted from $\\bdkk$. This can then be used to obtain information\nabout $|P'_{uc}|$ if the SU(3)-breaking factor were known reasonably\naccurately. We discuss different ways, both experimental and\ntheoretical, of learning about the size of the SU(3) breaking.\n\nIn Sec.~2, we examine $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$, and\nshow how the $\\bd$ decay can be used to obtain information about the\n$\\bs$ decay. We allow for all values of the observables in the $\\bd$\ndecay, and compute the theoretical error on $\\beta_s$, allowing for\n100\\% SU(3) breaking. It turns out that this error can be\nsubstantial. In Sec.~3, we discuss ways, both experimental and\ntheoretical, of determining the SU(3) breaking. If this breaking is\nknown with reasonable accuracy, this greatly reduces the theoretical\nerror on $\\beta_s$, and allows this mixing quantity to be extracted\nfrom $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$ decays. We conclude in\nSec.~4.\n\n\\section{\\boldmath $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$}\n\n\\subsection{\\boldmath $\\bskk$}\n\\label{bskkSec}\n\n$\\bskk$ is a pure $\\btos$ penguin decay. That is, its amplitude\nreceives contributions only from gluonic and electroweak penguin (EWP)\ndiagrams. There are three contributing amplitudes, one for each of\nthe internal quarks $u$, $c$ and $t$ (the EWP diagram contributes only\nto $P'_t$):\n\\bea\n{\\cal A}_s &=& \\lambda^{(s)}_u P'_u + \\lambda^{(s)}_c P'_c + \\lambda^{(s)}_t P'_t \\nn\\\\\n &=& |\\lambda^{(s)}_u| e^{i\\gamma} P'_{uc} - |\\lambda^{(s)}_t| P'_{tc} ~,\n\\label{Bsamp}\n\\eea\nwhere $\\lambda^{(q')}_q \\equiv V_{qb}^* V_{qq'}$. (As this is a\n$\\btos$ transition, the diagrams are written with primes.) In the\nsecond line, we have used the unitarity of the\nCabibbo-Kobayashi-Maskawa (CKM) matrix ($\\lambda^{(s)}_u +\n\\lambda^{(s)}_c + \\lambda^{(s)}_t = 0$) to eliminate the $c$-quark\ncontribution: $P'_{uc} \\equiv P'_u - P'_c$, $P'_{tc} \\equiv P'_t -\nP'_c$. Also, above we have explicitly written the weak-phase\ndependence (including the minus sign from $V_{ts}$ in\n$\\lambda^{(s)}_t$), while $P'_{uc}$ and $P'_{tc}$ contain strong\nphases. (The phase information in the CKM matrix is conventionally\nparametrized in terms of the unitarity triangle, in which the interior\n(CP-violating) angles are known as $\\alpha$, $\\beta$ and $\\gamma$\n\\cite{pdg}.) The amplitude ${\\bar {\\cal A}}_s$ describing the\nCP-conjugate decay $\\bsbar\\to K^{(*)0} {\\bar K}^{(*)0}$ can be\nobtained from the above by changing the signs of the weak phases (in\nthis case, $\\gamma$).\n\nThere are three measurements which can be made of $\\bskk$: the\nbranching ratio, and the direct and indirect CP-violating asymmetries.\nThese yield the three observables\n\\bea\n\\label{'observables}\nX' & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_s|^2 + |{\\bar{\\cal A}}_s|^2 \\right) ~, \\nn \\\\\nY' & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_s|^2 - |{\\bar{\\cal A}}_s|^2 \\right) ~, \\nn\\\\\nZ'_I & \\equiv & {\\rm Im}\\left( e^{-2i \\beta_s} {\\cal A}^*_s {\\bar {\\cal A}}_s \\right) ~.\n\\eea\nAssuming one takes the values for $|\\lambda^{(s)}_u|$,\n$|\\lambda^{(s)}_t|$ and $\\gamma$ from independent measurements, ${\\cal\n A}_s$ then depends only on the magnitudes of $P'_{uc}$ and\n$P'_{tc}$, and their relative strong phase $\\delta'$. With $\\beta_s$,\nthis makes a total of four unknown parameters. These cannot be\ndetermined from only three observables -- additional input is needed.\n\nNote that, if $\\lambda^{(s)}_u P'_{uc}$ were negligible, we would only\nhave two unknowns -- $|P'_{tc}|$ and $\\beta_s$. These could be\ndetermined from the measurements of $X'$ and $Z'_I$ ($Y'$ would\nvanish). This demonstrates that if one extracts $\\beta_s$ from $Z'_I$\nassuming that $\\lambda^{(s)}_u P'_{uc}$ is negligible, and it is not,\nthen one will obtain an incorrect value for $\\beta_s$. The size of\nthis error is directly related to the size of $\\lambda^{(s)}_u\nP'_{uc}$. Here, the possibility of an error is particularly\nimportant. Since $\\beta_s \\approx 0$ in the SM, a nonzero measured\nvalue of $\\beta_s$ would indicate NP\\footnote{Note that, if there is\n an indication of NP, we will know that it is in $\\btos$\n transitions. However, we will not know if $\\bs$-$\\bsbar$ mixing\n and\/or the $\\btos$ penguin amplitude is affected.}. It is therefore\ncrucial to have this theoretical uncertainty under control.\n\n\\subsection{\\boldmath $\\bdkk$}\n\nIn order to deal with the $P'_{uc}$ problem in $\\bskk$, in\nRef.~\\cite{CPS}, CPS use its U-spin-conjugate decay $\\bdkk$ . This is\na pure $\\btod$ penguin decay, whose amplitude can be written\n\\beq\n{\\cal A}_d = |\\lambda^{(d)}_u| e^{i\\gamma} P_{uc} + |\\lambda^{(d)}_t| e^{-i\\beta} P_{tc} ~.\n\\eeq\nAs with ${\\cal A}_s$, we take the values for the magnitudes and weak\nphases of the CKM matrix elements from independent measurements. This\nleaves three unknown parameters in ${\\cal A}_d$: the magnitudes of\n$P_{uc}$ and $P_{tc}$, and their relative strong phase $\\delta$. And,\nas with $\\bskk$, there are three measurements which can be made of\n$\\bdkk$: the branching ratio, and the direct and indirect CP-violating\nasymmetries. Given an equal number of observables and unknowns, we can\nsolve for $|P_{uc}|$, $|P_{tc}|$ and $\\delta$.\n\nThe key point is that $|P_{uc}|$ and $|P'_{uc}|$ are equal under\nflavor SU(3) symmetry. Thus, given a value for $|P_{uc}|$ and a value\n(or range) for the SU(3)-breaking factor, one obtains the value (or\nrange) of $|P'_{uc}|$. With this, one can extract the true value (or\nrange) of $\\beta_s$ from the $\\bskk$ experimental data.\n\nNow, CPS focused mainly on the decays $B_{d,s}^0 \\to K^{*0} {\\bar\n K}^{*0}$. As mentioned in the introduction, there were no\nexperimental measurements of these decays when their paper was\nwritten, so it was necessary to assume experimental values in order to\nextract $|P_{uc}|$. CPS chose values roughly based on the QCDf\ncalculation of Ref.~\\cite{BRY}. They found that the value of\n$|P_{uc}|$ is such that, even allowing for 100\\% SU(3) breaking,\n$|\\lambda^{(s)}_u P'_{uc}|$ is indeed small. The upshot is that the\ntheoretical uncertainty in the extraction of $\\beta_s$ is less than\n$1^\\circ$.\n\nThere are several reasons not to take this result at face value.\nFirst, although QCDf has been very successful at describing the\n$B$-decay data, it is still a model. Indeed, the predictions and\nexplanations of other models of QCD -- perturbative QCD \\cite{pQCD}\n(pQCD) and SCET \\cite{SCET}, for example -- do not always agree with\nthose of QCDf. Second, QCDf assumes that factorization holds to\nleading order for all $B$ decays. However, $B^0_{d,s} \\to K^{(*)0}\n{\\bar K}^{(*)0}$ are penguin decays, and it has been argued that\nnon-factorizable effects are important for such decays. It may be that\nsub-leading effects in QCDf are, in fact, important for $\\bd \\to\nK^{*0} {\\bar K}^{*0}$. We therefore re-examine the CPS method taking a\nmore model-independent approach.\n\n\\subsection{\\boldmath Theoretical Uncertainty on $\\beta_s$}\n\nIn this subsection, we generalize the CPS method. First, we consider\nall final states in $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$. Second,\nwe scan over a large range of experimental input values.\n\nWe proceed as follows. The three experimental measurements of the\n$\\bd$ decay correspond to the three observables\n\\bea\n\\label{XYZdefs}\nX & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_d|^2 + |{\\bar{\\cal A}}_d|^2 \\right) \\nn\\\\\n&& \\hskip0.4truecm \n=~|\\lambda^{(d)}_u|^2 |P_{uc}|^2 + |\\lambda^{(d)}_t|^2 |P_{tc}|^2 - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\cos\\delta \\cos\\alpha ~, \\nn \\\\\nY & \\equiv & \\frac{1}{2} \\left( |{\\cal A}_d|^2 - |{\\bar{\\cal A}}_d|^2 \\right) =\n- 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\sin\\delta \\sin\\alpha ~, \\\\\nZ_I & \\equiv & {\\rm Im}\\left( e^{-2i \\beta} {\\cal A}_d^* {\\bar {\\cal A}}_d \\right)\n= |\\lambda^{(d)}_u|^2 |P_{uc}|^2 \\sin 2\\alpha - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t||P_{uc}| |P_{tc}| \\cos\\delta \\sin\\alpha ~. \\nn\n\\eea\nIt is useful to define a fourth observable:\n\\bea\n\\label{ZRdef}\nZ_R & \\equiv & {\\rm Re}\\left( e^{-2i \\beta} {\\cal A}_d^* {\\bar {\\cal A}}_d \\right) \\\\\n&& \\hskip0.4truecm \n=~|\\lambda^{(d)}_u|^2 |P_{uc}|^2 \\cos 2\\alpha + |\\lambda^{(d)}_t|^2 |P_{tc}|^2 - 2 |\\lambda^{(d)}_u||\\lambda^{(d)}_t| |P_{uc}| |P_{tc}| \\cos\\delta \\cos\\alpha ~. \\nn\n\\eea\nThe quantity $Z_R$ is not independent of the other three observables:\n\\beq\nZ_R^2 = X^2 - Y^2 - Z_I^2 ~.\n\\label{eq:ZR}\n\\eeq\nThus, one can obtain $Z_R$ from measurements of $X$, $Y$ and $Z_I$, up\nto a sign ambiguity.\n\n$X$, $Y$ and $Z_I$ are related to the branching ratio\n($\\mathcal{B}_d$), the direct CP asymmetry ($C_d$) and the indirect CP\nasymmetry ($S_d$) of $\\bdkk$ as follows:\n\\beq\nX = \\kappa_d \\mathcal{B}_d ~~,~~~~ \nY = \\kappa_d \\mathcal{B}_d C_d ~~,~~~~\nZ_I = \\kappa_d \\mathcal{B}_d S_d ~,\n\\eeq\nwhere\n\\beq\n\\kappa_d = \\frac{8 \\pi m_{B_d}^2}{\\tau_d p_c} ~.\n\\eeq\nIn the above, $m_{B_d}$ and $\\tau_d$ are the mass and the lifetime of\nthe decaying $\\bd$ meson, respectively, and $p_c$ is the momentum of\nthe final-state mesons in the rest frame of the $\\bd$. \n\n{}From Eqs.~(\\ref{XYZdefs}) and (\\ref{ZRdef}), the quantity $|P_{uc}|$\ncan then be written in terms of the observables as\n\\beq\n|P_{uc}|^2 = \\frac{1}{|\\lambda^{(d)}_u|^2} \\, \\frac{Z_R - X}{\\cos 2\\alpha - 1}\n= \\frac{\\kappa_d \\mathcal{B}_d}{|\\lambda^{(d)}_u|^2} \\, \\frac{\\pm \\sqrt{1-C_d^2-S_d^2} - 1}{\\cos 2\\alpha - 1} ~.\n\\label{eq:Puc2}\n\\eeq\nThe value of $\\alpha$ is not known exactly, but we know from\nindependent measurements that it is approximately $90^\\circ$. In what\nfollows, we fix $\\alpha$ to $90^\\circ$ for simplicity. Note that any\ndeviation of $\\alpha$ from this value decreases the denominator in\nEq.~(\\ref{eq:Puc2}), and thus makes $|P_{uc}|$ larger. The above\nexpression allows us to calculate $|P_{uc}|$ for a given set of\nobservables.\n\nOn the whole, the decays $\\bdkk$ have not yet been measured. One\nexception is $\\bd \\to K_S K_S$. From BaBar \\cite{Aubert:2006gm}, we\nhave\n\\beq\n\\mathcal{B}_d = (1.08 \\pm 0.28 \\pm 0.11) \\times 10^{-6} ~~,~~~~\nS_d = -1.28^{+0.80+0.11}_{-0.73-0.16} ~~,~~~~\nC_d = -0.40 \\pm 0.41 \\pm 0.06 ~,\n\\eeq\nwhile Belle finds \\cite{Nakahama:2007dg}\n\\beq\n\\mathcal{B}_d = (0.87^{+0.25}_{-0.20} \\pm 0.09) \\times 10^{-6} ~~,~~~~\nS_d = -0.38^{+0.69}_{-0.77} \\pm 0.09 ~~,~~~~\nC_d = 0.38 \\pm 0.38 \\pm 0.05 ~.\n\\eeq\nWe see that essentially all values of $\\sqrt{C_d^2 + S_d^2}$ are still\nexperimentally allowed.\n\nHere is an example of the calculation of $|P_{uc}|$ using\nEq.~(\\ref{eq:Puc2}). We take the QCDf-inspired central value of CPS\nfor the branching ratio ($\\mathcal{B}_d = 5 \\times\n10^{-7}$)\\footnote{In fact, the branching ratio for $\\bd \\to K^{*0}\n {\\bar K}^{*0}$ has been measured \\cite{BdK*K*bar}. The world average\n is $\\mathcal{B}_d = (8.1 \\pm 2.3) \\times 10^{-7}$ \\cite{hfag}. In\n order to make the generalization of the CPS method more direct, in\n our analysis we use the CPS value for $\\mathcal{B}_d$ (which differs\n from the experimental value by only a little more than $1\\sigma$).},\nand also take $0 \\le \\sqrt{C_d^2 + S_d^2} \\le 1$. We compute\n$|\\lambda^{(d)}_u|$ using values for the various quantities taken from\nthe Particle Data Group \\cite{pdg}. Including the errors on these\nquantities, we find that $|P_{uc}|$ can be as large as $1460 \\pm 170$\neV ($Z_R$ positive) or $2060 \\pm 240$ ($Z_R$ negative). For\ncomparison, $|P_{uc}|$ is only about 180 eV if the CP asymmetries are\nalso fixed at the QCDf-inspired central values of CPS. Note that the\ndiscrete ambiguity with $Z_R$ negative corresponds to the case for\nwhich $\\bd$ decays are dominated by $P_{uc}$. On the other hand, we\nnaively expect $P_{tc}$ to be larger. Still, even if this solution\nwere discarded, the results of our analysis below would not be changed\nfundamentally. The bottom line is that $|P_{uc}|$ can, in fact, be\nlarge in $\\bdkk$ decays.\n\nWe now return to $\\bskk$ decays. Even if $|P_{uc}|$ is large in $\\bd$\ndecays, because of the $|\\lambda^{(s)}_u|$ CKM suppression it is not\nclear whether or not $|\\lambda^{(s)}_u P'_{uc}|$ really plays a\nsignificant role in the $\\bs$ decays. In order to ascertain this, we\nproceed as follows. We apply the CPS method, but consider all\npossible values of the observables in both $\\bd$ and $\\bs$\ndecays\\footnote{In fact, the branching ratio for $\\bs \\to K^{*0} {\\bar\n K}^{*0}$ has been measured \\cite{BsK*K*bar}. Its value is $(2.81\n \\pm 0.46~({\\rm stat}) \\pm 0.45~({\\rm syst}) \\pm 0.34~(f_s\/f_d))\n \\times 10^{-5}$. The CPS value for $\\mathcal{B}_s$, which we use in\n our analysis, is $1.18 \\times 10^{-5}$.}. Thus, we use flavor SU(3)\nsymmetry to relate $|P_{uc}|$ and $|P'_{uc}|$, allowing for a 100\\%\nsymmetry breaking. In order to study the worst-case scenario (the\nlargest possible value of $|P'_{uc}|$ within 100\\% breaking), we fix\n$|P'_{uc}| = 2 |P_{uc}|$. Thus, for example, for the case where the\nbranching ratio $\\mathcal{B}_d$ is taken to be the QCDf-inspired\ncentral value of CPS, but the CP asymmetries take all possible values,\n$|P'_{uc}|$ can be as large as 2920 eV ($Z_R$ positive) or 4120 eV\n($Z_R$ negative).\n\nGiven the worst-case value of $|P'_{uc}|$, assuming the CKM phases to\nbe known, and fixing $\\beta_s = 0$ (in order to study the worst-case\nprediction in the SM), only two parameters are left unknown in the\n$\\bskk$ decay. These can be extracted from the branching ratio\n$\\mathcal{B}_s$ and the direct CP asymmetry $C_s$ (up to discrete\nambiguities, but this does not affect the following discussion). Once\nthis is done, all the theoretical parameters in $\\bskk$ are known, and\nwe can compute the time-dependent CP asymmetry $S_s$ and the effective\nphase $\\beta_s^{eff}$ ($\\arg{(\\bar\\mathcal{A}_s \/ \\mathcal{A}_s)}$).\nThus we get an evaluation of the (worst-case) theoretical uncertainty\nof $\\beta_s$ as extracted from the time-dependent CP asymmetry of\n$\\bskk$ decays.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig1.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n function of $|P'_{uc}|$ and the direct CP asymmetry $C_s$. The\n branching ratios are fixed to $\\mathcal{B}_d = 5 \\times 10^{-7}$ and\n $\\mathcal{B}_s = 11.8 \\times 10^{-6}$ (central values of CPS).}\n\\label{fig1}\n\\end{figure}\n\nWe now present figures showing the worst-case $\\beta_s^{eff}$ in\nvarious situations. The aim is to scan over the whole observable\nspace in order to ascertain how large $\\beta_s^{eff}$ can be within\nthe SM. In Fig.~\\ref{fig1}, we fix both branching ratios to the CPS\ncentral values, and give the worst-case value of $\\beta_s^{eff}$ as a\nfunction of $|P'_{uc}|$ and the direct CP asymmetry $C_s$. The\neffective phase is roughly proportional to $|P'_{uc}|$ and can be up\nto $10^\\circ$ in this restricted scenario. In Fig.~\\ref{fig2}, we\nrepeat the calculation but also allow the branching ratios to vary,\npresenting $\\beta_s^{eff}$ as a function of $C_s$ and the ratio of\nbranching ratios ($\\mathcal{B}_s\/\\mathcal{B}_d$). In this case, for\nthe central maximum value of $|P_{uc}|$, an effective phase of up to\n$12^\\circ$ ($Z_R$ positive) or $18^\\circ$ ($Z_R$ negative) is\nobtained. In Fig.~\\ref{fig3}, $\\beta_s^{eff}$ is given as a function\nof $\\sqrt{C_d^2+S_d^2}$ and $\\mathcal{B}_s\/\\mathcal{B}_d$.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig2A.eps}\n\t\t\\includegraphics[height=7.2cm]{fig2B.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n function of the direct CP asymmetry $C_s$ and the ratio of branching\n ratios ($\\mathcal{B}_s\/\\mathcal{B}_d$). The plot on the left\n (right) is for $Z_R$ positive (negative) in Eq.~(\\ref{eq:ZR}).}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[height=7.2cm]{fig3A.eps}\n\t\t\\includegraphics[height=7.2cm]{fig3B.eps}\n\\caption{Worst-case values of $\\beta_s^{eff}$ (in degrees) as a\n function of $\\sqrt{C_d^2+S_d^2}$ and the ratio of branching ratios\n ($\\mathcal{B}_s\/\\mathcal{B}_d$). The plot on the left (right) is\n for $Z_R$ positive (negative) in Eq.~(\\ref{eq:ZR}).}\n\\label{fig3}\n\\end{figure}\n\n{}From the above figures, it is clear that $\\beta_s^{eff}$ can be\nlarge within the SM, and that the conclusions of CPS hold only for\ncertain sets of values of the experimental inputs. Still, it is\ninteresting to note that a small theoretical error (say $\\beta_s^{eff}\n\\le 5^\\circ$) is found for a non-negligible subset of the input\nnumbers. The general behavior of solutions is as follows:\n\\begin{enumerate}\n\n\\item for $Z_R$ positive, $\\beta_s^{eff}$ is smaller for smaller\n values of $\\sqrt{C_d^2+S_d^2}$ (it's the opposite for $Z_R$\n negative),\n\n\\item $\\beta_s^{eff}$ is smaller for larger values of $|C_s|$ for\n fixed $|P'_{uc}|$,\n\n\\item $\\beta_s^{eff}$ is smaller for larger values of\n $\\mathcal{B}_s\/\\mathcal{B}_d$,\n\n\\item $\\beta_s^{eff}$ is smaller for smaller values of SU(3) breaking.\n\n\\end{enumerate}\nFor the first three points we cannot do anything -- the measurements\nof the observables are what they are. The fourth point can be\nunderstood as follows. The theoretical error $\\beta_s^{eff}$ is due to\nthe presence of a nonzero $P'_{uc}$ in ${\\cal A}_s$\n[Eq.~(\\ref{Bsamp})]. This error is roughly proportional to\n$|P'_{uc}|$, which is itself equal to the product of $|P_{uc}|$ and an\nSU(3)-breaking factor. For a given value of $|P_{uc}|$,\n$\\beta_s^{eff}$ is smaller if the SU(3)-breaking factor is\nsmaller. Thus, the assumption of CPS of 100\\% breaking often leads to\na large $\\beta_s^{eff}$. The precise knowledge of the SU(3) breaking\nbetween $|P_{uc}|$ and $|P'_{uc}|$ would therefore considerably reduce\nthe theoretical uncertainty on the extracted value of $\\beta_s$ using\nthis method. The determination of the size of SU(3) breaking is\ndiscussed in the next section.\n\n\\section{SU(3) Breaking}\n\nAs we have seen, the idea of obtaining information on $|P'_{uc}|$ by\nrelating it to $|P_{uc}|$ using flavor SU(3) is tenable. However, if\none simply takes an SU(3)-breaking factor of 100\\%, this can lead to a\ntheoretical error on the extraction of $\\beta_s$ of up to $18^\\circ$.\nThus, in order to use this method, a better determination of the size\nof SU(3) breaking must be found. In this section, we discuss ways,\nboth experimental and theoretical, of getting this information.\n\n\\subsection{Experimental Measurement of SU(3) Breaking}\n\n\\subsubsection{\\boldmath $B^0_{d,s} \\to K^{*0} {\\bar K}^{*0}$}\n\\label{BKKSU3break}\n\nThe decays $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$ really represent\nthree types of decay -- the final state can consist of $PP$, $PV$ or\n$VV$ mesons ($P$ is pseudoscalar, $V$ is vector). Now, the CPS method\napplies when the final state is a CP eigenstate. For $PP$ and $VV$\ndecays, this holds. However, $PV$ decays do not satisfy this\ncondition. Still, these decays can be used if the $K^{*0}$\/${\\bar\n K}^{*0}$ decays neutrally. That is, we have\n\\bea\nB^0 & \\to & \\frac{1}{\\sqrt{2}} \\left( K^0 {\\bar K}^{*0} + K^{*0} {\\bar K}^0 \\right) ~~~~~~ {\\hbox{(CP eigenstate)}} \\nn\\\\\n& \\to & K^0 {\\bar K}^0 \\pi^0 ~.\n\\label{PVdecay}\n\\eea\nOn the other hand, the CPS method cannot be used if the\n$K^{*0}$\/${\\bar K}^{*0}$ decays to charged particles. This is because,\nin this case, one cannot extract $|P_{uc}|$ from the $\\bd$ decay --\nthere are more theoretical unknowns than observables.\n\nThe SM value of SU(3) breaking can be found from any single pair of\ndecays -- $|P_{uc}|$ and $|P'_{uc}|$ can be extracted from the $\\bd$\nand $\\bs$ decays, respectively. (As we are interested in SU(3)\nbreaking in the SM, we set $\\beta_s$ to zero.) In principle, this\nvalue of SU(3) breaking ($|P'_{uc}|\/|P_{uc}|$) can then be used in a\ndifferent decay, and the method of the previous section can be\napplied. The problem here is that this approach is applicable only if\nthe SU(3) breaking in the two decays is expected to be\nsimilar. However, $PP$, $PV$ and $VV$ decays are all different\ndynamically, so that there is no a-priori reason to expect this to\nhold. For example, the decay of Eq.~(\\ref{PVdecay}) is very different\nfrom the $PP$ decay $B^0 \\to K^0 {\\bar K}^0$, and so the $PV$ and $PP$\nSU(3) breakings are not likely to be similar.\n\nThere is one exception, and it involves the $VV$ decays $B^0_{d,s} \\to\nK^{*0} {\\bar K}^{*0}$. Since the final-state particles are vector\nmesons, when the spin of these particles is taken into account, these\ndecays are in fact three separate decays, one for each polarization.\nThe polarizations are either longitudinal ($A_0$), or transverse to\ntheir directions of motion and parallel ($A_\\|$) or perpendicular\n($A_\\perp$) to one another. By performing an angular analysis of these\ndecays, the three polarization pieces can be separated. \n\nIt is also possible to express the polarization amplitudes using the\nhelicity formalism. Here, the transverse amplitudes are written as\n\\bea\nA_\\| &=& \\frac{1}{\\sqrt{2}} (A_+ + A_-) ~, \\nn\\\\\nA_\\perp &=& \\frac{1}{\\sqrt{2}} (A_+ - A_-) ~.\n\\eea\nHowever, in the SM, the helicity amplitudes obey the hierarchy\n\\cite{BRY,Kagan}\n\\beq\n\\left\\vert \\frac{A_+}{A_-} \\right\\vert = \\frac{\\Lambda_{QCD}}{m_b} ~.\n\\eeq\nThat is, in the heavy-quark limit, $A_+$ is negligible compared to\n$A_-$, so that $A_\\| = -A_\\perp$. Thus, one expects the SU(3) breaking\nfor the $\\|$ and $\\perp$ polarizations to be approximately equal.\nOne can therefore extract $|P_{uc}|$ and $|P'_{uc}|$ from the $\\bd$\nand $\\bs$ decays for one of the transverse polarizations, compute the\nSU(3) breaking ($|P'_{uc}|\/|P_{uc}|$), and apply this value of SU(3)\nbreaking to the other transverse polarization decay pair. In this way\nthe SU(3)-breaking factor can be measured experimentally, and can be\nused to determine the theoretical uncertainty in the extraction of\n$\\beta_s$.\n\n\\subsubsection{\\boldmath $B^+ \\to K^+ \\bar K^0$ and $B^+ \\to \\pi^+ K^0$}\n\nOther decays which can be used to measure SU(3) breaking are the\nU-spin-conjugate pair $B^+ \\to K^+ \\bar K^0$ and $B^+ \\to \\pi^+\nK^0$. While it is true that these do not involve $\\bs$ and $\\bd$\nmesons, both are pure penguin decays, just like $B^0_{d,s} \\to\nK^{(*)0} {\\bar K}^{(*)0}$. Restricting ourselves to the $PP$ final\nstates, we then expect that the SU(3) breaking in $B^+ \\to K^+ \\bar\nK^0$ and $B^+ \\to \\pi^+ K^0$ is similar (though not necessarily equal)\nto that in $\\bd \\to K^0 {\\bar K}^0$ and $\\bs \\to K^0 {\\bar K}^0$.\nThe measurement of SU(3) breaking can therefore be done using the\n$B^+$ decays and applied to the $\\bd$\/$\\bs$ decays.\n\nThere is a difference compared to the previous example. Since there\nare no indirect CP asymmetries in $B^+$ decays, one cannot measure\n$|P'_{uc}|\/|P_{uc}|$. The SU(3) breaking probed in $B^+ \\to K^+ \\bar\nK^0$ and $B^+ \\to \\pi^+ K^0$ is\n\\bea\n\\label{Y'\/Y}\n-\\frac{Y'}{Y} & = &\n\\frac{|\\lambda^{(s)}_u||\\lambda^{(s)}_t|}{|\\lambda^{(d)}_u||\\lambda^{(d)}_t|} \\,\n\\frac{\\sin{\\gamma}}{\\sin{\\alpha}} \\, \n\\frac{|P'_{uc}|}{|P_{uc}|} \\, \\frac{|P'_{tc}|}{|P_{tc}|} \\, \\frac{\\sin\\delta'}{\\sin\\delta} \\nn\\\\\n& = &\n\\frac{|P'_{uc}|}{|P_{uc}|} \\, \\frac{|P'_{tc}|}{|P_{tc}|} \\, \\frac{\\sin\\delta'}{\\sin\\delta} ~.\n\\eea\nIn the second line, all the CKM factors cancel due to the sine law\nassociated with the unitarity triangle. Thus, if the $B^+$ decay pair\nis used to measure the SU(3) breaking, the theoretical error in the\nextraction of $\\beta_s$ must be calculated relating $|P'_{uc}|\n|P'_{tc}| \\sin\\delta'$ of $\\bs \\to K^0 {\\bar K}^0$ to $|P_{uc}|\n|P_{tc}| \\sin\\delta$ of $\\bd \\to K^0 {\\bar K}^0$.\n\n\\subsubsection{Other SU(3) pairs}\n\nThere are many other pairs of decays that are related by U spin or\nSU(3): $\\bd \\to \\pi^+ \\pi^-$ and $\\bs \\to K^+ K^-$, $\\bd \\to \\pi^0K^0$\nand $\\bs \\to \\pi^0 \\kbar$, etc. A complete list of two- and three-body\ndecay pairs, as well as a discussion of the measurement of\nU-spin\/SU(3) breaking, is given in Ref.~\\cite{SU3break}. For some of\nthem we already have measurements of the breaking.\n\nFor example, consider the pair $\\bs \\to \\pi^+ K^-$ and $\\bd \\to \\pi^-\nK^+$. The measurement of the SU(3) breaking of Eq.~(\\ref{Y'\/Y}) gives\n\\cite{SU3break}\n\\beq\n-\\frac{Y'}{Y} = 0.92 \\pm 0.42 ~.\n\\eeq\nAlthough the error is still substantial, we see that the central value\nimplies small SU(3) breaking. The problem is that $\\bs \\to \\pi^+ K^-$\nand $\\bd \\to \\pi^- K^+$ are not pure-penguin decays, so that it is not\nclear how the above SU(3) breaking is related to that in $B^0_{d,s}\n\\to K^{(*)0} {\\bar K}^{(*)0}$, if at all. Still, if one measures the\nSU(3) breaking in several different decay pairs, it can give us a\nrough indication as to what to take for $B^0_{d,s} \\to K^{(*)0} {\\bar\n K}^{(*)0}$.\n\n\\subsection{Theoretical Input on SU(3) Breaking}\n\nConsider again the $\\bskk$ amplitude, Eq.~(\\ref{Bsamp}). If the\n$t$-quark contribution is eliminated using the unitarity of the CKM\nmatrix, we have\n\\beq\n{\\cal A}_s = T' \\lambda^{(s)}_u + P' \\lambda^{(s)}_c ~,\n\\eeq\nwhere $T' \\equiv P'_u - P'_t$, $P' \\equiv P'_c - P'_t$. Now, in QCDf\n$T'$ and $P'$ are calculated using a systematic expansion in\n$1\/m_b$. However, a potential problem occurs because the higher-order\npower-suppressed hadronic effects contain some chirally-enhanced\ninfrared (IR) divergences. In order to calculate these, one introduces\nan arbitrary infrared (IR) cutoff. The key observation here is that\nthe difference $T' - P'$ is free of these dangerous IR divergences\n\\cite{DMV}. And, although the calculation of various hadronic\nquantities in pQCD is different than in QCDf, the difference $T'-P'$\nis the same in both formulations. This also holds for $T - P$ in the\n$\\bdkk$ amplitude.\n\nSince $T' - P' = P'_{uc}$ and $T - P = P_{uc}$, this suggests that the\ncalculation of $|P'_{uc}|$ and $|P_{uc}|$ is under good control\ntheoretically. This allows us to calculate $|P'_{uc}|\/|P_{uc}|$, which\ngives us the theoretical prediction of SU(3) breaking. There are many\nquantities which enter into the calculation of $|P'_{uc}|$ and\n$|P_{uc}|$ -- the renormalization scale $\\mu$, the Gegenbauer\ncoefficients in the light-cone distributions, the quark masses,\netc.\\ -- and the errors on these quantities are quite large at\npresent. However, most of these quantities and their errors cancel in\nthe ratio $|P'_{uc}|\/|P_{uc}|$. For the various $B^0_{d,s} \\to\nK^{(*)0} {\\bar K}^{(*)0}$ decays we find\n\\bea\n\\label{SU3break}\nPP &:& \\frac{|P'_{uc}|}{|P_{uc}|} =\n\\frac{M_{B_s}^2 F_0^{\\bs \\to K}(M_K^2)}{M_{B_d}^2 F_0^{\\bd \\to K}(M_K^2)} = 0.86 \\pm 0.15 ~,\\nn\\\\\nPV &:& \\frac{|P'_{uc}|}{|P_{uc}|} = \\frac{M_{B_s}^2 F_+^{\\bs \\to K}(M_{K^*}^{2})}{M_{B_d}^2 F_+^{\\bd \\to K}(M_{K^*}^{2})} = 0.86 \\pm 0.15 ~,\\nn\\\\\nVP, VV_0 &:& \\frac{|P'_{uc}|}{|P_{uc}|} = \\frac{M_{B_s}^2 A_0^{\\bs \\to K^*}(M_{K^{(*)}}^{2})}{M_{B_d}^2 A_0^{\\bd \\to K^*}(M_{K^{(*)}}^{2})} = 0.87 \\pm 0.19 ~,\\\\\nVV_{\\|}, VV_{\\perp} &:& \\frac{|P'_{uc}|}{|P_{uc}|} = \\frac{M_{B_s} ( F_-^{\\bs \\to K^*}(M_{K^*}^{2}) \\pm F_+^{\\bs \\to K^*}(M_{K^*}^{2}))}{M_{B_d} (F_-^{\\bd \\to K^*}(M_{K^*}^{2}) \\pm F_+^{\\bd \\to K^*}(M_{K^*}^{2}))} = 0.79 \\pm 0.16~. \\nn\n\\eea\n\n\n\n\n\n\n\nAbove we have taken $F^{B \\to K^{(*)}}(M_{K^{(*)}}^2) \\simeq F^{B\\to\n K^{(*)}}(0)$ since the variation of the ratio of form factors over\nthis range of $q^2$ falls well within the errors of their calculation\n\\cite{BRY, LCSR}. For $PV$ and $VP$ decays, the spectator quark goes\nin the first meson. In the last expression, we have $F_+^{B \\to K^*} =\n0.00 \\pm 0.06$ \\cite{BRY}, so that one has the same SU(3) breaking for\nthe $\\|$ and $\\perp$ polarizations. As discussed in\nSec.~\\ref{BKKSU3break}, this is to be expected.\n\nNow, the QCDf calculation is to $O(\\alpha_s)$, and the above\nexpression indicates that, to this order, the SU(3)-breaking term is\nfactorizable. Thus, the theoretical prediction is fairly robust. On\nthe other hand, SCET says that there are long-distance contributions\nto $P'_c$ and $P_c$. Although this could introduce some uncertainty\ninto $|P'_{uc}|\/|P_{uc}|$, there might also be a partial cancellation\nin the ratio. Our point here is that, though one generally wants to\navoid theoretical input, since this is largely based on models, the\nSU(3) breaking in $|P'_{uc}|\/|P_{uc}|$ may be theoretically clean.\n\nIn Sec.~\\ref{BKKSU3break}, it was noted that the CPS method can be\nused when the final state in $B^0_{d,s} \\to K^{(*)0} {\\bar K}^{(*)0}$\nis a CP eigenstate. Thus, if one wishes to use the theoretical input\nof Eq.~(\\ref{SU3break}), one can simply apply it to $PP$ or $VV$\ndecays. However, this does not hold for $PV$ or $VP$ decays, which are\nnot CP eigenstates. Still, one can use the CPS method on the decay of\nEq.~(\\ref{PVdecay}), which is a linear combination of $PV$ and $VP$\nstates. And, since the theoretical $PV$ SU(3) breaking in\nEq.~(\\ref{SU3break}) is about equal to that of $VP$, this theoretical\ninput can be applied to the $PV + VP$ decay.\n\nFinally, in Sec.~\\ref{bskkSec} we noted that, in the presence of a\nnonzero $P'_{uc}$, one cannot cleanly extract $\\beta_s$ from $\\bskk$\n-- one needs additional input. In Ref.~\\cite{DMV2} it was the above\ntheoretical calculation of $|P'_{uc}|$ which was taken as the\ninput. We note that the method using $\\bskk$ and $\\bdkk$ is somewhat\nmore precise since most of the errors in the calculation of\n$|P'_{uc}|$ cancel in the SU(3)-breaking ratio of\nEq.~(\\ref{SU3break}).\n\n\\section{Conclusions}\n\nThe pure $\\btos$ penguin decay $\\bskk$ is potentially a good candidate\nfor measuring the $\\bs$-$\\bsbar$ mixing phase, $\\beta_s$. If its\namplitude were dominated by $V_{tb}^* V_{ts} P'_{tc}$, the indirect CP\nasymmetry would simply measure $\\beta_s$. Unfortunately, although the\nsecond contributing amplitude, $V_{ub}^* V_{us} P'_{uc}$, is expected\nto be small, it is not clear that it is completely negligible. A\nnonzero $V_{ub}^* V_{us} P'_{uc}$ can change the extracted value of\n$\\beta_s$ from its true value, i.e.\\ it can lead to a theoretical\nerror. Since the measurement of $\\beta_s$ is an important step in the\nsearch for new physics, the size of this theoretical error is\nimportant.\n\nThe size of $P'_{uc}$ has been examined by Ciuchini, Pierini and\nSilvestrini (CPS). They note that the amplitude $P_{uc}$ can be\nextracted from the U-spin-conjugate decay, $\\bdkk$, and can be related\nto $P'_{uc}$ by SU(3). They choose values for the $\\bdkk$ experimental\nobservables inspired by QCDf, allow for 100\\% SU(3) breaking, and\ncompute $P'_{uc}$. They find that the theoretical error on $\\beta_s$\nis very small, i.e.\\ that the presence of the $V_{ub}^* V_{us}\nP'_{uc}$ amplitude has little effect on the extraction of $\\beta_s$.\n\nIn this paper, we revisit the CPS method. In particular, we consider\nmost values of the $\\bdkk$ observables, still allowing for 100\\% SU(3)\nbreaking. We find that, although the theoretical error remains small\nfor a significant subset of these input values, it can be large for\nother values. We find that an error of up to $18^\\circ$ is possible,\nwhich makes the extraction of $\\beta_s$ from $\\bskk$ problematic.\n\nThis issue can be resolved if we knew the value of SU(3) breaking. We\ntherefore discuss different ways, both experimental and theoretical,\nof determining this quantity. From the experimental point of view, the\nsize of SU(3) breaking can be measured using a different $\\bd$\/$\\bs$\ndecay pair. We show that the $VV$ decay $B^0_{d,s} \\to K^{*0} {\\bar\n K}^{*0}$ or $B^+ \\to K^+ \\bar K^0$\/$B^+ \\to \\pi^+ K^0$ can be used\nin this regard. It is also possible to use theoretical input. Within\nQCDf, the SU(3)-breaking term is factorizable, and so the theoretical\nprediction for this quantity may be reasonably clean.\n\n\\bigskip\n\\noindent\n{\\bf Acknowledgments}: We thank Tim Gershon for helpful\ncommunications. This work was financially supported by NSERC of Canada\n(BB, DL), by the US-Egypt Joint Board on Scientific and Technological\nCo-operation award (Project ID: 1855) administered by the US\nDepartment of Agriculture and in part by the National Science\nFoundation under Grant No.\\ NSF PHY-1068052 (AD), and by FQRNT of\nQu\\'ebec (MI).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\nAfter formation of a young star and its circumstellar disk due to the gravitation collapse of the protostellar cloud the process of accretion of matter from the nearest surrounding the star can continue and have a form of the clumpy accretion. \\citet{1992PASP..104..479G} was probably the first who used this term. He tried to explain by such a way the observations of the strong extinction events observed in some young variables. Later this type of accretion has been considered by \\citet{1996ARA&A..34..207H} for explanation of the FUOR's phenomenon. This idea is quite popular up to now \\citep{2010ApJ...713.1143Z, 2013ApJ...764..141B, 2018MNRAS.474...88H}. The mechanism of formation of chondrule as a result of the clumpy accretion was discussed \\citet{1998Icar..134..137T}. Obviously, the fall of the clump on the disk should cause disturbances at the place of the fall. It is interesting to trace how this disturbance will develop and what structures on the disk can be caused by it.\n \nThe detection of various structures on images of protoplanetary disks is one of the most interesting results obtained with the ALMA interferometer \\citep[see, e.g.][]{2018A&A...619A.161C, 2018ApJ...869L..43H,2018ApJ...869L..42H, 2018ApJ...869L..50P}. Ring and spiral structures are most often observed. Less commonly, structures resembling highly elongated vortices are observed. A number of papers are devoted to theoretical studies of the formation of such structures. Their formation is associated with perturbations in the disks caused by the motion of the forming planets \\citep[e.g.,][]{2013A&A...549A..97R, 2015A&A...579A.106V, 2015ApJ...809...93D, 2016MNRAS.463L..22D, 2016ApJ...818...76J, 2018ApJ...866..110D}, the development of various kinds of instabilities in the disks ~\\citep[e.g.,][]{2015ApJ...815L..15B, 2015ApJ...806L...7Z, 2015ApJ...813L..14B, 2016ApJ...821...82O,2009ApJ...697.1269J, 2014ApJ...796...31B, 2014ApJ...794...55T, 2018A&A...609A..50D}, a large\nscale vertical magnetic field \\citep{2018MNRAS.477.1239S} or with the destruction of large bodies in collisions \\citep{2019ApJ...887L..15D,2020MNRAS.495..285N}. In all these papers, the source of disturbance is in the disk itself.\n\nIn our paper we discuss an alternative scenario for the formation of the observed structure. We investigate in the first time the dynamical response of circumstellar disk on the perturbation associated with the clumpy accretion events. Using the hydrodynamic simulations we calculate the disk images at $1$ mm and discuss the results in the context with interferometric observations of the protoplanetary disks with ALMA. \n\n\\section{Initial condition}\nA model consists of a young star of solar mass ($ M_{\\ast} =\nM_{\\odot} $) embedded in a gas disk with total mass is $ M_{disk}\n= 0.01M_{\\odot} $. At the beginning of simulation the disk matter was distributed\nazimuthally symmetrical within the radii $ r_{in} = 0.5$ and\n$r_{out} = 50$ AU. The initial density distribution of the disk\nis\n\\begin{equation}\n\\rho(r,z,0)=\\frac{\\Sigma_0}{\\sqrt{2\\pi}H(r)}\\frac{r_{in}}{r}e^{-\\frac{z^2}{2H^2(r)}},\n\\end{equation}\nwhere $\\Sigma_0$ is arbitrary scale parameter, which is determined by\ndisk mass. Hydrostatic scale height is $H(r)=\\sqrt{\\frac{\\kappa T_{mid}(r) r^3}{GM_{\\ast} \\mu m_H}}$, where $\\kappa$, $G$ and $m_H$ are\nthe Boltzmann constant, the gravitational constant and the mass of\na hydrogen atom and $\\mu=2.35$ is the mean molecular weight\n\\citep{1994A&A...286..149D}. Following~\\citet{1997ApJ...490..368C}\nwe determine the law of midplane temperature distribution\n$T_{mid}(r)=\\sqrt[4]{\\frac{\\gamma}{4}}\\sqrt{\\frac{R_{\\ast}}{r}}T_{\\ast}$,\nwhere $\\gamma=0.05$ \\citep{2004A&A...421.1075D}. The calculations were performed in the local thermodynamic equilibrium approximation $P(r,z,t)=c^2(r)\\rho(r,z,t)$, where $P$ is local pressure at the moment $t$ and $c$ is a sound speed. It was assumed that in the vertical direction along $z$ the disk is isothermal. The temperature of the star was assumed to be $T_{\\ast} = 5780$ K and star radius is $R_{\\ast}=R_{\\odot}$. The disk relaxed during $600$ years, and then the remnant of the fallen gas clump was added into it.\n\n\\subsection{Impulse approximation to the clump-disk collision}\n\nWhen a clump falls onto a disk, part of its kinetic energy is converted into thermal energy. An infrared spot should appear on the image of the disk where the clump fell. The thermal relaxation time in protoplanetary disks at the distance $\\geq20$ AU from the star is much less than the local orbital period \\citep{2017A&A...605A..30M}. Therefore, the remnant of the fallen clump quickly comes to thermodynamic equilibrium with the matter of the disk. It participates in the Keplerian motion of the matter, while maintaining the residual velocity component orthogonal to the plane of the disk. \n\nAt the initial moment of time, we assume that the remnant of the clump has already reached thermodynamic equilibrium with the matter of the disk. The remnant was generated as a density perturbation on the disk in the form of a disk segment bounded by radii $R_0$ and step $dR$ and distributed over the azimuthal angle $\\phi$ with the axis of symmetry along negative part of x-axis ($\\phi=30^{\\circ}$ for all models). The density of matter in the disturbance exceeded the local density of the disk by a factor of $\\displaystyle K=\\frac{\\Sigma_{cl}}{\\Sigma_d}$, where $\\Sigma_{cl}$ and $\\Sigma_d$ are local surface density of remnant and disk respectively. The remnant moves prograde. The particle velocity of the remnant is equal to $\\displaystyle V(R)=L\\cdot V_k(R)$, where $V_k(R)$ is Keplerian velocity at a given distance from the star and $L$ is parameter of the problem. The velocity vector had a residual inclination to the disk plane $\\displaystyle sin(I)=\\frac{V_z(R)}{V(R)}$ (Fig.~\\ref{fig:disk}). The residual angle of inclination of the remnant depends on the initial angle of the fall and the amount of kinetic energy that is spent on heating the disk in the region of fall. We considered a number of the possible options for the value and inclination of the velocity vector. The parameters of all calculated models are given in the Table~\\ref{tab:models}.\n\n\\begin{figure}[ht!]\n\\plotone{disk.eps}\n\\caption{\\normalsize Particle distribution at the initial moment of the remnant of the clump motion. Top is the projection onto the plane of the disk, bottom is a section along the axis $y$. \\label{fig:disk}}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n\\plotone{Fig2.eps}\n\\caption{\\normalsize \nThe average value of the $z$ coordinate of the particles in the cells of $R$, $\\phi$. The model parameters are $K=3$, $I=30^\\circ$ and $L=0.8$. The time in years is in the upper right corner of the pictures. \\label{fig:08inclImg}}\n\\end{figure*} \n\n\\begin{deluxetable*}{ccccccccc}\n\\tablenum{1}\n\\tablecaption{The models parameters \\label{tab:models}}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{L} & \\colhead{I} & \\colhead{K} & \\colhead{R} & \\colhead{dR} & \\colhead{$\\phi$} & \\colhead{Remnant mass} & \\nocolhead{} & \\colhead{Lifetime}\\\\\n\\colhead{Float} & \\colhead{Degrees} & \\colhead{Number} & \\colhead{AU} &\n\\colhead{AU} & \\colhead{Degrees} & \\colhead{Jupiter mass} & \\colhead{Sructures} & \\colhead{yrs}\n}\n\\decimalcolnumbers\n\\startdata\n0.8 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-hand spiral, Horseshoe & $> 600$ \\\\\n0.8 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-hand spiral, Horseshoe & $> 600$\\\\\n0.8 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, Faint two-arm spiral, Horseshoe & $> 600$\\\\\n0.8 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, Faint two-arm spiral, Horseshoe & $> 600$\\\\\n0.8 & 10 & 5 & 20 & 5 & 30 & 0.19 & Arc, One-hand spiral, Horseshoe & $> 600$\\\\\n\\hline\n1 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Multi Rings, Ring & $\\sim 1000$ \\\\\n1 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Ring & $> 600$\\\\\n1 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Faint two-arm spiral & $> 600$\\\\\n1 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Faint two-arm spiral & $> 600$\\\\\n1 & 5 & 5 & 20 & 5 & 30 & 0.19 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$\\\\\n1 & 5 & 10 & 20 & 5 & 30 & 0.38 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$\\\\\n\\hline\n1.2 & 5 & 3 & 20 & 5 & 30 & 0.11 & Arc, One-arm spiral, Multi Rings, Ring & $> 600$ \\\\\n1.2 & 10 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral & $> 600$\\\\\n1.2 & 20 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral & $> 600$\\\\\n1.2 & 30 & 3 & 20 & 5 & 30 & 0.11 & Arc, Bright two-arm spiral, Asymmetric ring & $\\sim 2000$\\\\\n1.2 & 30 & 1 & 20 & 5 & 30 & 0.04 & Arc, Bright two-arm spiral & $> 600$\\\\\n\\hline\n0.8 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc & $\\sim 100$\\\\\n1 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc & $\\sim 100$\\\\\n1.2 & 30 & 3 & 10 & 2 & 30 & 0.04 & Arc, Faint two-arm spiral, Multi Rings & $\\sim 450$\\\\\n\\enddata\n\\tablecomments{The column of ``Structures'' lists the types of observed asymmetries in the order of their appearance in the disk images. The column of ``Lifetimes'' is a long-lived structures lifetime. }\n\\end{deluxetable*}\n\n\\begin{figure*}[ht!]\n\\plotone{Fig1.eps}\n\\caption{\\normalsize The surface density multiplied by the distance from the center of mass ($\\Sigma R$). \nThe model parameters are $K=3$, $I=10^\\circ$ and $L=0.8$. The time in years is in the upper right corner of the pictures. \\label{fig:08sig}}\n\\end{figure*}\n\n\\begin{figure}[ht!]\n\\plotone{Fig3.eps}\n\\caption{\\normalsize \nThe average value of the $z$ coordinate of the particles in the cells of $R$, $\\phi$ along $x$ (left) and $y$ (right) axes after $600$ years. The model parameters are $K=3$, $L=0.8$ and the angles are in the upper right corner of the pictures. \\label{fig:08inclxy}}\n\\end{figure}\n\\begin{figure*}[h]\n\\plotone{Fig4.eps}\n\\caption{\\normalsize \nThe images at wavelength 1 mm. The color shows the flux of radiation ($F_\\nu$) multiplied by $R^2$ in conventional units. The model parameters are $K=3$, $I=10^\\circ$ and $L=0.8$. The time in years is in the upper right corner of the pictures. \\label{fig:08img}}\n\\end{figure*} \n\n\\section{Methods}\nThe evolution of the remnant in gas disk was simulated by\nthe SPH method (smooth particle hydrodynamics). The calculations\nwere performed using the code Gadget-2~\\citep{2001NewA....6...79S,\n2005MNRAS.364.1105S} modified by us \\citep{2016Ap.....59..449D}. In total,\nfrom $5\\cdot 10^5$ to $2.5\\cdot 10^6$ particles of the gas disk and from $5\\cdot 10^3$ to $2\\cdot 10^5$ particles of the perturbation were involved in the simulations. The calculations took into account the self-gravity of the disk. \n\nThe simulated region was divided by $200\\times30\\times90$ cells in spherical coordinates ($R,\\theta,\\phi$), in which the average values of the SPH particle density were determined. We assume that dust particles with a size of 1, 10, 100 microns and 1 mm are well mixed with the gas, and are distributed according to the law$\\frac{dn(s)}{ds}\\propto s^{-3.5}$, where n is the concentration and s is the size of the dust grain \\citep{1969JGR....74.2531D}. The dust to gas mass ratio in the disk was $0.01$ as the average in interstellar medium. The dust opacity is calculated using Mie theory for Magnesium-iron silicates~\\citep{1995A&A...300..503D}. RADMC-3D~\\citep{2012ascl.soft02015D} code was used for the 3-D radiative transfer calculations. \n\n\\begin{figure*}[ht!]\n\\plotone{Fig5.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08sig} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1$. \\label{fig:1sig}}\n\\end{figure*}\n\\begin{figure*}[ht!]\n\\plotone{Fig6.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1$. \\label{fig:1img}}\n\\end{figure*} \n\\begin{figure*}[ht!]\n\\plotone{Fig7.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $L=1$ after $600$ years. The angle of $I$ is in the upper right corner of the pictures. \\label{fig:img600}}\n\\end{figure*} \n \\begin{figure*}[ht!]\n\\plotone{Fig8.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08sig} \nfor model parameters $K=3$, $I=10^\\circ$ and $L=1.2$. \\label{fig:1.2sig}}\n\\end{figure*}\n\\begin{figure}[ht!]\n\\plotone{Fig9.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08inclxy} \nfor model parameters $K=3$, $L=1.2$.\\label{fig:12inclxy}}\n\\end{figure} \n\\begin{figure*}[ht!]\n\\plotone{Fig10.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=3$, $L=1.2$ after $600$ years. The angle of $I$ is in the upper right corner of the pictures. \\label{fig:img1_600}}\n\\end{figure*} \n\\begin{figure}[ht!]\n\\plotone{Fig11.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor model parameters $K=1$, $I=30^\\circ$, $L=1.2$ after $600$ years. \\label{fig:K1I30}}\n\\end{figure}\n\\begin{figure*}[ht!]\n\\plotone{Fig12.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor two models with $2.5\\cdot 10^6$ particles. The models parameters $K=3$, $I=5^\\circ$, $L=1$ (left) and $K=3$, $I=30^\\circ$, $L=1.2$ (right) at the time $1000$ years. \\label{fig:longImg}}\n\\end{figure*} \n\\begin{figure}[ht!]\n\\plotone{Fig13.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:longImg} for model with parameters $K=3$, $I=30^\\circ$, $L=1.2$ at the time $2000$ years. \\label{fig:im2000}}\n\\end{figure} \n\\begin{figure}[ht!]\n\\plotone{Fig14.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08inclxy} \nfor models with parameters $K=3$, $I=5^\\circ$, $L=1$ (top) and $K=3$, $I=30^\\circ$, $L=1.2$ (bottom). The time in years is in the upper right corner of the pictures. \\label{fig:longIncl}}\n\\end{figure} \n\\begin{figure*}[ht!]\n\\plotone{Fig15.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:08img} \nfor models with parameters $K=3$, $I=30^\\circ$, $R_0=10$ AU and $dR=2$ AU after $36.5$ years. The parameter of $L$ is in the upper right corner of the pictures. \\label{fig:closeImg}}\n\\end{figure*} \n\\begin{figure}[ht!]\n\\plotone{Fig16.eps}\n\\caption{\\normalsize The same as in Fig.~\\ref{fig:closeImg} \nfor models with parameters $K=3$, $I=30^\\circ$, $L=1.2$ for the time $400$ years. \\label{fig:close400}}\n\\end{figure} \n\\begin{figure}[ht!]\n\\plotone{Fig17.eps}\n\\caption{\\normalsize The average value of the $z$ along $y$ axes after $365$ (left) and $1460$ (right) years. The model parameters are $K=3$, $I=30$. The values of $L$ are in the upper right corner of the pictures. \\label{fig:yincl}}\n\\end{figure} \n\n\\section{Results} \nThe emergence of the remnant of the clump of matter in the disk leads to the propagation of density waves in horizontal and vertical directions. The strongest perturbations arise at large values of the parameters $K$, $L$, and $I$, as expected. Due to the differential rotation of the Keplerian disk the remnant stretches and transform during the time. A local increase in the surface density leads to the appearance of corresponding large-scale inhomogeneities in the disk images.\n\n\\subsection{Perturbations at large radii} \nFor the models discussed here, the initial position of the disturbing remnant of the clump was set equal to $R_0 = 20$ AU with a step of $dR = 5$ AU. Calculations have shown that the parameter $L$, which characterizes the kinetic energy of the falling clump, has the greatest influence on the type of disturbance in the disk. Therefore, we will sequentially discuss the three energy regimes considered in our models. \n\n\\subsubsection{Sub Keplerian perturbations}\nIn this case, when the parameter is $L = 0.8$, due to the differential rotation of the Keplerian disk the remnant stretches and transform to a piece of arc reembling a cyclonic vortex and then turn into a spiral during one revolution around the star ($\\sim 125$ years). Since the matter of the disk is involved in its motion the spiral dips and thickening of the density are visible in the disk (Fig.~\\ref{fig:08sig}). In the central part of the disk, the spiral splits into two branches. The spiral structure quickly ($\\sim 300$ years) twists into an asymmetric ring, which, scattering in the disk, retains the asymmetry until the end of the calculations ($600$ years). This asymmetry rotates with the disk. \n\nThe wave also passes in the vertical direction along the disk, which extends inside and to the edge of the disk. Perturbation twists the central plane of the disk. The maximum distortion of the disk plane occurs near the axis $y$, but does not coincide with its position. The inner parts of the disk inclines relative to the periphery. Over time, the radius of the outer boundary of the inclined area increases. It distorts 30 AU at the time of $600$ years (Fig.~\\ref{fig:08inclImg}). In this case, the tilt of the disk plane relative to the initial position is approximately equal to $0.2^\\circ$ at $I = 5^\\circ $ and $0.9^\\circ$ in the case of $I = 30^\\circ$. An increase in the angle of $I$ does not affect the speed of propagation of global perturbation to the edge of the disk (Fig.~\\ref{fig:08inclxy}). Increasing the initial density (parameter K) of the clump increases the vertical distortion of the disk. \n\nThe perturbations described above show themselves on the images of protoplanetary disks. The form of asymmetric structures in images corresponds to the perturbations of the density of the disk matter. However, on the periphery of the disk is visible a shadow from the matter above the disk plane. The asymmetric ring-shaped structure on the images has a horseshoe-shaped form (Fig.~\\ref{fig:08img}). Calculations have shown that the minimum value of $K$ in which this structure is visible on the images, equal to $3$ (about $0.1$ of Jupiter mass). An increase in the parameter $K$ to $5 $ increases the brightness of the structure and their lifetime, but its horseshoe form is preserved. \n\n\\subsubsection{Keplerian perturbations}\nFor this class of models of the phase of the disintegration of the clump remnant during the first orbit similar to the previous one (Fig.~\\ref{fig:1sig}). A piece of arc is converted to the one-hand spiral. It twisted into a symmetric ring-shaped structure during the next few revolutions ($\\sim 500$ years). \n\nAn increase of the radius of distortion of the central plane of the disk relative to the initial position is faster than in the previous case. It reaches $40$ AU at the time $600$ years. The maximum inclination angle is $0.2^\\circ$ in the case $I=5^\\circ$ and $0.8^\\circ$ at $I = 30^\\circ$. The direction of maximum distortion in the vertical direction is also close to the axis $y$, but does not coincide with it.\n\nThe visible asymmetry on the images of the disk are also appropriate for surface density (Fig.~\\ref{fig:1img}). In this case, the propagation of the density wave gives multi-lane image of the protoplanetary disk at a certain point in time (right image of Fig.~\\ref{fig:1img}). \nAn increase in the angle of $I$ affects the image of the ring-shaped structure. Two symmetric weakly pronounced spirals are visible on the images instead of the ring, if the value of $I\\geq20$ (Fig.~\\ref{fig:img600}). \n\n\\subsubsection{Super Keplerian perturbations}\nIn this case, the clump matter motion in the protoplanetary disk causes severe density perturbations and significantly distorts the disk in the vertical direction. As in previous cases, during one convolution of the clump, the vortex-like structure is stretched into a spiral, which is converted into two spirals during the next revolution (Fig.~\\ref{fig:1.2sig}). Each of the spirals is logarithmic, they are shifted by phase relative to each other by $180^\\circ$. The form of the spirals weakly depends on the angle of inclination of $I$. \n\nDisk distortion in the vertical direction differs from the models described above for this case. The periphery of the disk is distorted, and the inner parts of the disk deforms weaker. The waves are still propagating along the disk in a vertical direction at the time of 600 years as seen from Fig.~\\ref{fig:12inclxy}. In this case, the distortion of the inner region depends on the inclination angle and has the opposite character for $I<20$ and $I\\geq 20$. \n\nIt become seen noticeable asymmetry of the spirals on the periphery of the disk in images with an increase in the angle of inclination of $I$ (Fig.~\\ref{fig:img1_600}). \n\nFor this class of models, the case $K = 1$ (corresponds to the mass of the remnant $\\sim 12 M_{\\oplus}$) was considered. In this case, the perturbation is weaker; however, the two-arm spiral can also be identified in the disk images. It is more pronounced at a larger inclination angle $I$ (Fig.~\\ref{fig:K1I30}). \n\n\\subsubsection{Long-term dynamics} \nFor the models described above, the number of SPH particles was $5\\cdot 10^5$. These models have a lower resolution compared to the models, the calculations of which involved $2.5\\cdot 10^6$ particles. However, the calculations showed that at the initial phases of clump destruction, the images of disks obtained on the basis of models with a small number of particles show the same structures as for more accurate models. However, to study the long-term dynamics of the fallen clump remnant, higher resolution is required. We have calculated two limiting cases that correspond to the parameters that cause the minimum ($K=3,I=5,L=1$) and maximum ($K=3,I=30,L=1.2$) disturbance in the disk. \n\nOver time, density waves scattered and the all structure settles down to the plane of the disk. The ring of the first model stretches along the radius and loses brightness mixing with the matter of the protoplanetary disk with time. It is faintly noticeable at $1000$ years after the fall of the clump (Fig.~\\ref{fig:longImg} left). Than it is not visible against the background of the disk matter. For the second model spiral waves are still noticeable after $1000$ years (Fig.~\\ref{fig:longImg} right), but after $\\sim 2000$ years they completely disappear. An asymmetric ring structure can be seen on the disk by the time of 2000 years (Fig.~\\ref{fig:im2000}). \n\nThe charateristic time of the disk dynamic relaxation after the fall of the clump also depends on the place of its fall. For example, with $R_0 = 30$ AU and the same clump parameters as in the previous model, the lifetime of spirals and ring structures generated by its fall increases to $4 \\times 10^3$ years. At $R_0 = 50$ AU, the characteristic relaxation time of disturbances on the disk is even longer: $\\sim 10^4$ years.\n\nOn the Fig.~\\ref{fig:longIncl} it is seen that the plane of the disk settles to its original position over time. However, even 2000 years after the fall of the clump, a slight inclination of the disk plane near the axis $y$ remains. For the first model, it is equal to $\\sim 0.14^\\circ$, and for the second, it is $\\sim 0.72^\\circ$. \n\n\\subsection{Perturbations at small radii} \n\nIn this class of models the perturbation was located near the star at the distance $R_0 = 10$ AU with the step of $dR = 2$ AU. The clump had parameters $\\phi=30^\\circ$, $K=3$ and $I=30^\\circ$. The parameter $L$ was varied. The mass of the clump was about $13 M_{\\oplus}$. \n\nFig.~\\ref{fig:closeImg} shows images of the disk for three models with the L parameter $0.8$, $1$ and $1.2$ after one period ($36.5$ years). Bright dense structures and areas of shadow, which are caused by matter rising above the plane of the disk, are visible on the disk. But all structures are scattered after next period for models with $L \\leq 1$. For case $L=1.2$ waves propagate along the disk, which in time from 250 to 500 years can be seen in the images as a multi-lane structure (Fig.~\\ref{fig:close400}). \n\nIn the vertical direction, the disk is distorted in all considered cases. The maximum distortion is achieved near the $y$ axis as for models distant from the star. The Fig.~\\ref{fig:yincl} shows the average values of $z$ along the y-axis for two points in time $365$ and $1460$ years. One can see in all cases, the perturbation propagates outward from the inner part of the disk, tilting its central plane. With an increase in L, the final inclination of the disk increases, but remains within $0.5^\\circ$. \n\n\\section{Conclusion} \nCalculations have shown that at the initial stages of the remnant of the clump disintegration, the structures that are visible in the images of the protoplanetary disk are similar for the entire set of the models. However, the shape of the final long-lived structure primarily depends on the kinetic energy of the falling clump. \n\nDuring the first revolution of the center of the remnant (at the initial moment of time), an arc-like structure resembling a vortex is visible in the disk image. Similar structures are observed, for example, in objects HD 135344B \\citep{2018A&A...619A.161C} and HD 143006 \\citep{2018ApJ...869L..50P}. At the next stage of evolution, the image shows a tightly wound spiral, as, for example, in the case of object HD 163296 \\citep{2018ApJ...869L..42H}. In the case when the residual velocity of the remnant does not exceed the Keplerian velocity, a ring is a long-lived structure, which can also be asymmetric. In this case, at a certain moment in time, the passage of a wave over the disk can give a multi-lane structure in the image. The ring-shaped structure is visible in the images of a number of objects, for example, HD 169142 \\citep{2017A&A...600A..72F}, HD 97048 \\citep{2017A&A...597A..32V}, RU Lup, Elias 24, AS 209, GW Lup \\citep{2018ApJ...869L..42H}. In the case of a high kinetic energy of a clump, a two-armed spiral appears on the disk image, which, after several thousand years, transforms into an asymmetric ring. Two-arm spirals were obtained on images of objects Elias 27, IM Lup, WaOph 6 \\citep{2018ApJ...869L..43H}. The median age of these sources is $\\sim 1$ Myr~\\citep{2018ApJ...869L..42H}, and the youngest sources have estimated ages about a few tenths of Myr~\\citep{2018ApJ...869L..43H}.\n\nSince the velocity vector of the remnant has a residual inclination relative to the plane of the disk, it is distorted. However, over time, the inclination of the plane of the disk relative its original position decreases. The tilt angle at the end of calculations does not exceed $1^\\circ$. Probably, for a more noticeable change in the inclination of the disk, a large mass of the clump remnant is required. In this work, we were looking for the minimum mass of the remnant, which develops into a structure visible in the image of the protoplanetary disk. It turned out that in the case of a high-energy fall, the minimum mass of the remnant of the clump is $\\sim 10 M _{\\oplus}$. If the residual velocity of the remnant does not exceed Kepler one, then the minimum mass is $\\sim 0.1M_J$. \n\nRing-shaped structures formed from the material of the remnant of the fallen clump and the protoplanetary disk are long-living structures and can exist for more than $3000$ years. Thus with a sufficiently large mass of the falling clump, the evolution of the disturbance can lead to the formation of a planet on an inclined orbit. It should be noted that Toomre parameter in the disks under consideration is equal to $\\sim 40$ at a distance of $20$ AU. Consequently, in a more massive disk, the fall of a clump several times denser than the material of the disk can trigger the process of gravitational collapse and the formation of a planet.\n\nThe fall of the clump near the star can cause not only a FUOR flare, but also a strong increase in circumstellar extinction, leading to a deep and prolonged weakening of the optical brightness of the star and an increase in its infrared radiation. Such events were observed in three young objects: V1184 Tau~\\citep{2009AstL...35..114G}, RW Aur~\\citep{2015IBVS.6143....1S,2019A&A...625A..49K} and AA Tau~\\citep{2021AJ....161...61C}. \n\nLet us make a rough estimate of the additional mass of circumstellar matter, which can be added to the mass of the disk during the lifetime of a star due to episodic falls of clumps. Suppose that the average age of stars with protoplanetary disks $10^6$ years~\\citep{2018ApJ...869L..42H} and the average lifetime of one disturbance $3 \\times 10^3$ years. This will give the probability of observing one event $P_1 \\sim 3\\times 10^{-3}$. The probability close to unity will be obtained when $\\sim 3\\times 10^3$ clumps fall during the disk lifetime ($\\sim 1$~Myr). If the clump mass of $0.1 M_J$ is necessary to create a strong disturbance (see above), the disk mass will increase during this time by $0.03 M_\\odot$ at an average accretion rate on the disk of $3\\times 10^{-8} M_\\odot yr^{-1}$. Such an increase in mass is not critical for a typical disk mass of $0.01-0.2 M_\\odot$ and an accretion rate onto a young star of $\\sim 10^{-7}-10^{-8} M_\\odot yr^{-1}$.\n\nThus, the mechanism of the clumpy accretion in protoplanetary disks can explain the formation of the main types of structures identified in images of protoplanetary disks. In addition, one clump falling at an angle onto a protoplanetary disk can produce multi-lane bright ring-shaped structures. So far, we have demonstrated here the fundamental possibility of obtaining the observed structures on protoplanetary disks in the model of the clumpy accretion. Appreciate the complexity of this process, it needs a more detailed consideration, taking into account the thermal regime in the perturbed region of the disk. \n\n{\\bf Acknowledgments.} It is a pleasure to thank the referee for valuable and useful remarks. Authors acknowledge the support of Ministry of Science and Higher Education of the Russian Federation under the grant 075-15-2020-780 (N13.1902.21.0039).\n\n\\software{Gadget-2~\\citep{2001NewA....6...79S,\n2005MNRAS.364.1105S}, RADMC-3D~\\cite{2012ascl.soft02015D}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{I. Introduction}\n\nBose Einstein condensates (BECs) provide\n an excellent environment to study experimentally and theoretically\n a rich variety of macroscopic\nquantum phenomena. In a rotating single component condensate,\ntopological defects to the order parameter often manifest themselves as vortices that\ncorrespond to a zero of the order parameter with a circulation of\nthe phase. When they get numerous, these vortices arrange\nthemselves on a triangular lattice.\nIn fact, vortices were first\nobserved in two-component BEC's \\cite{matthews}. Since then two-component BECs and the topological excitations within have been experimentally realised in a number of configurations: a single isotope that is in two different\nhyperfine spin states \\cite{hall,ander,myatt,delannoy,madd,matthews,sch}, two different isotopes of the same\natom \\cite{papp} or isotopes of two different atoms \\cite{modugno,thall,ferrari}.\n\nWhen a two-component\ncondensate is under rotation, topological defects of both order\nparameters are created which lead to more exotic defects such as\nsingly or multiply quantised skyrmions\n \\cite{wh,sch,mertes,cooper,savage,ruo}. Analogy with the Skyrme model\n from particle physics is often invoked to represent the defects \\cite{manton,cho}. The\nsingly quantised skyrmions contain a vortex in one component which\nhas the effect of creating a corresponding density peak in the other\ncomponent. These singly quantised skyrmions are often referred to as\ncoreless vortices. Once numerous, these vortices and peaks arrange\nthemselves in (coreless vortex) lattices, that can be either\ntriangular or square. Other defects such as vortex sheets or giant\nskyrmions can also be observed \\cite{ktu,ktu1,ywzf}. The aim of this\npaper is to classify the type of defects according to the different\nparameters of the problem restricting ourselves to two-dimensions. While this\npaper will only concern itself with magnetically trapped\ntwo-component BEC's, there is active research into\nspinor condensates (for a recent review see\n\\cite{uk}).\n\n\n In the mean-field regime, the two-component Bose-Einstein condensate\nat zero temperature is described in terms of two wave functions (order parameters),\n$\\Psi_1$ and $\\Psi_2$, respectively representing component-1 and\ncomponent-2.\n The two-component condensate is placed into rotation about the $z$-axis with $\\bm{\\bar{\\Omega}}=\\bar{\\Omega}\\bm{\\hat{z}}$ where $\\bar{\\Omega}$ is the rotation frequency assumed to apply equally to both components. The Gross-Pitaevskii (GP) energy functional of the rotating two-component, two-dimensional BEC is then given by\n\\begin{equation}\n\\begin{split}\n\\label{enfn}\nE[\\Psi_1,\\Psi_2]=\\int \\sum_{k=1,2}\\Bigg(\\frac{\\hbar^2}{2m_k}|\\nabla\\Psi_k|^2+V_k(\\bm{r})|\\Psi_k|^2\\\\\n-\\hbar\\bar{\\Omega}\\Psi^*_k L_z \\Psi_k +\\frac{U_k}{2}|\\Psi_k|^4\\Bigg)+U_{12}|\\Psi_1|^2|\\Psi_2|^2\\quad d^2r\n\\end{split}\n\\end{equation}\nwhere $r^2=x^2+y^2$, and $V_k(\\bm{r})=m_k\\omega_k^2r^2\/2$ are the\nharmonic trapping potentials with trapping frequencies $\\omega_k$,\ncentered at the origin. Here $m_1$ and $m_2$ are the masses of the\nbosons in component-1 and component-2 respectively, and $\\hbar$ is\nPlanck's constant. The angular momentum is in the $z$-axis and is\ndefined as $L_z=i[\\bm{\\hat{z}}\\cdot(\\bm{r}\\times\\bm{p})]$ for linear\nmomentum $\\bm{p}$.\n The energy functional (\\ref{enfn})\n contains three interaction constants:\n $U_k$ representing the internal interactions in component $k$, and $U_{12}$\n representing the interactions between the two components.\n\n\n\n The time-dependent coupled Gross-Pitaevskii (GP) equations\n are obtained from a variational procedure, $i\\hbar\\partial\\Psi_k\/\\partial t=\\delta E\/\\delta\\Psi^*_k$,\n on the energy functional (\\ref{enfn}).\nLet $\\tilde{\\omega}=(\\omega_1+\\omega_2)\/2$ be the average of the\ntrapping frequencies of the two components and introduce the reduced mass $m_{12}$ such that\n$m_{12}^{-1}=m_1^{-1}+m_2^{-1}$. The GP energy and the coupled GP\nequations can be non-dimensionalised by choosing\n$\\tilde{\\omega}^{-1}$, $\\hbar\\tilde{\\omega}$ and\n$r_0=\\sqrt{\\hbar\/(2m_{12}\\tilde{\\omega})}$ as units of time, energy\nand length respectively. On defining the non-dimensional\nintracomponent coupling parameters $g_k=2U_km_{12}\/\\hbar^2$ and the\nintercomponent coupling parameter $g_{12}=2U_{12}m_{12}\/\\hbar^2$\n($\\equiv g_{21}$), the dimensionless coupled GP equations read\n \\begin{equation}\n \\begin{split}\n\\label{gp11}\ni\\frac{\\partial\\psi_k}{\\partial t}=&-\\frac{m_{12}}{m_k}\\left(\\nabla-i\\bm{A}_k\\right)^2\\psi_k+\\frac{m_k}{4m_{12}}\\left(\\frac{\\omega_k^2}{\\tilde{\\omega}^2}-\\Omega^2\\right)r^2\\psi_k\\\\\n&\\quad+g_k|\\psi_k|^2\\psi_k+g_{12}|\\psi_{3-k}|^2\\psi_k,\n\\end{split}\n\\end{equation}\nfor $\\Omega=\\bar{\\Omega}\/\\tilde{\\omega}$ and where\n\\begin{equation}\n \\bm{A}_k=\\frac{1}{2}\\frac{m_k}{m_{12}}\\bm{\\Omega}\\times\\bm{r},\n\\end{equation}\nfor $\\bm{\\Omega}=(0,0,\\Omega)$ and $\\bm{r}=(x,y,0)$. The ground\nstate of the energy (\\ref{enfn}) or the GP equations (\\ref{gp11}) is\ndetermined by preserving the normalisation condition which in this\npaper is either taken to be\n\\begin{equation}\n\\label{norm} \\int |\\psi_k|^2\\quad d^2r=N_k,\n\\end{equation}\nwhere $N_k$ is the total particle number of the $k$th-component, or\n\\begin{equation}\n\\int \\left (|\\psi_1|^2+|\\psi_2|^2\\right )\\quad d^2r=N_1+N_2.\n\\label{norm12}\n\\end{equation}\n\nThe following sections will only consider repulsive interactions so\nthat $g_1$, $g_2$ and $g_{12}$ are always non-negative.\nIn order to separate the regimes of interest, a non-dimensional\nparameter combining these $g_k$ and $g_{12}$ will appear:\n\\begin{equation}\\label{gamma12}\n \\Gamma_{12}=1-\\frac{g_{12}^2}{g_1g_2}.\n\\end{equation}\nFurthermore to simplify matters, we introduce ratio parameters $\\eta$ and $\\xi$ such that\n\\begin{equation}m_1=\\eta m_2\\hbox{ and }\\omega_1=\\xi\\omega_2\\hbox{ with\n}\\{\\eta,\\xi\\}>0.\\end{equation} The effective trapping potentials\n for each component coming from equation (\\ref{gp11}) are then,\nrespectively,\n\\begin{subequations}\n\\begin{align}\n V_1^{\\text{eff}}(r)=&(\\eta+1)\\left(\\frac{\\xi^2}{(\\xi+1)^2}-\\frac{1}{4}\\Omega^2\\right)r^2,\\\\\n V_2^{\\text{eff}}(r)=&\\frac{(\\eta+1)}{\\eta}\\left(\\frac{1}{(\\xi+1)^2}-\\frac{1}{4}\\Omega^2\\right)r^2.\n\\end{align}\n\\end{subequations}\nIt follows that the limiting rotational frequency for each component\nis $\\Omega_1^{\\text{lim}}=2\\xi\/(\\xi+1)$ and\n$\\Omega_2^{\\text{lim}}=2\/(\\xi+1)$, so that it is necessary to\nconsider a rotational frequency such that\n$0<\\Omega<\\Omega^{\\text{lim}}=\\min\\{{\\Omega_1^{\\text{lim}},\\Omega_2^{\\text{lim}}}\\}$.\n\n\n\n\n\n\n\n\n\nExperimentally, it is often the case that the $U_k$, $m_k$,\n$\\omega_k$ and $N_k$ are of the same order, so that much of the\ntheoretical and numerical analysis concerning two-component\ncondensates has assumed equality of these parameters.\n In the case where the\nintracomponent coupling strengths are equal, Kasamatsu et al.\n\\cite{ktu,ktu1} produced a numerical phase diagram in terms of the\nrotation and the intercomponent coupling strengths: they found\n phase separation regions with either vortex sheets\nor droplet behavior and regions of coexistence of the components\nwith coreless vortices, arranged in triangular or square lattices.\n\nIn this paper, we present a classification of the ground states and\nvarious types of topological defect when the intracomponent coupling\nstrengths are distinct. Depending on the magnitude of the various\nparameters the components can either coexist, spatially separate\nor exhibit symmetry breaking. A richer phenomenology of topological\ndefects is then found.\n\nMuch of the analysis carried out to investigate the ground states and topological defects will use a\n a nonlinear sigma model.\n It has been introduced previously in the literature \\cite{ktu,ktu3} for $\\eta=\\xi=1$ but\nwe will generalise this to the cases $\\eta$ and $\\xi$ different from\n1. This involves writing the total density\nas\n\\begin{equation}\\label{rhot}\n \\rho_T=|\\psi_1|^2+\\eta|\\psi_2|^2.\n\\end{equation}\n A normalised complex-valued spinor\n$\\bm{\\chi}=[{\\chi_1},{\\chi_2}]^T$ is introduced from which the\nwave functions are decomposed as $\\psi_1=\\sqrt{\\rho_T}\\chi_1$,\n$\\psi_2=\\sqrt{\\rho_T\/\\eta}\\chi_2$ where $|\\chi_1|^2+|\\chi_2|^2=1$.\n Then we define the spin density $\\bm{S}=\\bar{\\bm{\\chi}}\\bm{\\sigma}\\bm{\\chi}$ where\n$\\bm{\\sigma}$ are the Pauli matrices. This gives the components of $\\bm{S}$ as\n\\begin{subequations}\n \\label{seqs}\n\\begin{align}\n \\label{sx}\n S_x=&\\chi^*_1\\chi_2+\\chi_2^*\\chi_1,\\\\\n \\label{sy}\n S_y=&-i(\\chi^*_1\\chi_2-\\chi_2^*\\chi_1),\\\\\n S_z=&|\\chi_1|^2-|\\chi_2|^2,\n \\label{sz}\n\\end{align}\n\\end{subequations}\nwith $|\\bm{S}|^2 = 1$ everywhere. We will write the GP energy (\\ref{enfn}) in terms of\n $\\rho_T$ and $\\bm{S}$ that will allow us to derive information on\n the ground state of the system.\n\nThe paper is organised as follows. The different regions of the\n$\\Omega-\\Gamma_{12}$ phase diagrams are outlined in Sect. II with a\ndetailed analysis of the range of ground states and topological\ndefects. Then the non-linear $\\sigma$-model is developed in Sect.\nIII to analyse the ground states in terms of the total density. Lastly\nSect. IV takes the second normalisation condition (\\ref{norm12}) and presents an example $\\Omega-\\Gamma_{12}$ phase diagram.\n\n\\section*{II. The $\\Gamma_{12}-\\Omega$ Phase Diagram}\n\n\\subsection*{A. Numerical Parameter Sets}\n\n In this paper,\nthree different configurations are considered, two of which relate directly to experimental configurations.\n Firstly, we analyze a\n$^{87}$Rb-$^{87}$Rb mixture with one isotope in spin state $|F=2$,\n$m_f=1\\rangle$ and the other in state $|1$, $-1\\rangle$. The complex\nrelative motions between these two isotope components of rubidium\nwere considered experimentally by Hall et al \\cite{hall}. Here, the\nmasses are equal ($\\eta=1$) and the transverse trapping potentials\n are equal, $\\omega_1=\\omega_2$ ($\\xi=1$). The scattering lengths for each\ncomponent are $a_1=53.35\\AA$ and $a_2=56.65\\AA$.\n The intracomponent coupling strengths\n $U_k$ for a two-dimensional model are defined as\n$U_k={\\sqrt{8\\pi}\\hbar^2a_k}\/[{m_ka_{zk}}]$ ($k=1,2$),\nand the intercomponent coupling strength is defined as\n$U_{12}={\\sqrt{2\\pi}\\hbar^2a_{12}}\/[{m_{12}\\tilde{a}_z}]$,\nwhere\n $m_{12}$ is the reduced mass, given by $m_{12}^{-1}=m_1^{-1}+m_2^{-1}$,\n$a_k$, $a_{12}$ are the s-wave\n(radial) scattering lengths,\n$a_{zk}$ is the characteristic size of the condensate in the\n $z$ direction,\n \n\nand $\\tilde{a}_z=(a_{z1}+a_{z2})\/2$.\n In the experiment \\cite{hall}, the computations of approximate values of $a_{z1}$\n and $a_{z2}$ can be made using a Thomas-Fermi approximation in the $z$ direction\n as explained in \\cite{PS}, chapter 17. This relies on the assumption\n that $N_k a_k\/a_{zk}$ are large. Then, it is reasonable to describe the experiments through a two-dimensional model.\n For our simulations, we choose $g_1=0.0078$ and\n$g_2=0.0083$, which are consistent with the experimental data and we set $N_1=N_2=10^5$. We denote\nthis parameter set as `Experimental Set 1' (ES1).\n\nSecondly, we tackle a $^{41}$K-$^{87}$Rb mixture with both isotopes\nin spin state $|2$, $2\\rangle$, as was considered by Modugno et al\n\\cite{modugno}. Component-1 is identified with\nthe $^{41}$K isotope and component-2 with the $^{87}$Rb isotope. In the experiment,\n the masses are different ($\\eta\\sim 0.48$),\n but since $\\eta\\xi^2=1$, then $m_1 \\omega_1^2= m_2\\omega_2^2$\n so that both components experience the\nsame trapping potential.\nHere the scattering lengths for each component are $a_1=31.75\\AA$\nand $a_2=52.39\\AA$ and we choose intracomponent coupling strengths\nas $g_1=0.0067$ and $g_2=0.0063$. This set is denoted ES2.\n\n Lastly, in order to make an\nanalogy with previous theoretical studies, we consider the set\n(ES3) which contains a mixture chosen such that all the parameter\ngroups are equal, i.e. here $g_1=g_2=0.0078$ and $N_1=N_2=10^5$ with\nequality of the $m_k$ and\n$\\omega_k$ ($\\eta=\\xi=1$).\n\nIn each case, the features of the ground states will be explained as\n$g_{12}$ and $\\Omega$ are varied. While the value of the $g_k$ are\nnominally fixed, the value of the intercomponent coupling strength,\n$g_{12}$, can be altered by Feshbach resonance (see for instance\n\\cite{tojo,theis,bauer,kempen}), which allows for an\nextensive experimental range in $\\Gamma_{12}$, given by (\\ref{gamma12}) (keeping $\\Gamma_{12}\\le1$). Simulations\nare thus performed on the coupled GP equations (\\ref{gp11}) in\nimaginary time. In general, it is difficult to find the true minimizing energy state. But the use of various initial data converging to the same final state allows us to state that the true ground state will be of the same pattern as the one we exhibit.\n\n We would like to note that while we have chosen these particular parameters, we have conducted extensive numerical simulations over a range of different parameters sets and find the sets presented here illustrate well the possible ground states.\n\n\\subsection*{B. Classification of the Regions of the Phase Diagram}\n\nThe ground states can be classified according to\n\\begin{enumerate}\n\\item the symmetry properties\n\\item the properties of coexistence of the condensates or spatial\nseparation.\n\\end{enumerate}\n\nWhen there is no rotation ($\\Omega=0$), as illustrated\n in Fig. \\ref{gs}, the geometry of the ground state can either be\n \\begin{itemize}\\item two disks (coexistence of the components and symmetry preserving state)\n \\item a disk and an annulus (symmetry preserving with spatial\n separation of the components), which depends strongly on the fact that the $g_k$, $m_k$, $\\omega_k$ are\nnot equal\n \\item droplets (symmetry breaking and spatial separation).\n \\end{itemize}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{gs.eps}\n\\end{center}\n\\caption{Density (divided by $10^4$) plots along\n$y=0$ for component-1 (dashed lines), component-2 (dotted lines) and\nthe total density (solid lines) in which the components (a) coexist\nboth being disks ($\\Gamma_{12}=0.1$), (b) spatially separate to be a\ndisk and an annulus ($\\Gamma_{12}=0$) and (c) have symmetry broken\nto be two droplets ($\\Gamma_{12}=-0.3$). The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ with $\\eta=\\xi=1$ (set ES1). The\nangular velocity of rotation is $\\Omega=0$. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{gs}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{05.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2 (right column) in which\nthe components coexist and are both disks. The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1)\nand $\\Gamma_{12}=0.5$ (which gives $g_{12}=0.0057$). The angular\nvelocity of rotation is $\\Omega$ and it takes the values (a) $0.25$,\n(b) $0.5$, and (c) $0.75$. Vortices in one component have a\ncorresponding density peak in the other component (coreless\nvortices). In (c) the coreless vortex lattice is square. At these\nparameters the components are in region 1 of the phase diagram of\nFig. \\ref{sch_2}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{05}\n\\end{figure}\n\n\n\n\nUnder rotation, the different ground states\ncan be classified according to the parameters $\\Gamma_{12}$ and\n$\\Omega$. When the condensates are two disks, or a disk and annulus, defects may break some symmetry of the system as $\\Omega$ is increased. But the wave function retains some non trivial rotational symmetry. We will not refer to this as symmetry breaking since the bulk\n condensate keeps some symmetry. We will use the terminology symmetry breaking when the bulk completely breaks the symmetry of the system and is a droplet or has vortex sheets.\n We find that there are four distinct regions, determined by the geometry of the ground state.\n\n{\\em Region 1}: Both components are disk shaped, with no spatial separation. Above some critical velocity\n$\\Omega$, coreless vortices appear:\n a vortex in one component which has the effect of\ncreating a corresponding density peak in the other component. They\narrange themselves either on a triangular or on a square lattice.\nFigure \\ref{05} provides a typical example of the density profiles.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{03neg.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2 (right column) in which\nthe components have spatially separated. The parameters are\n$g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1)\n and $\\Gamma_{12}=-0.3$ (which gives\n$g_{12}=0.0092$). The angular velocity of rotation is $\\Omega$ and\nit takes the values (a) $0.1$, (b) $0.5$, and (c) $0.9$. In (a) the\ncomponents are rotating droplets (region 4 of Fig. \\ref{sch_2}), in\n(b) the components have spatially separated (but keep some symmetry) and have isolated density peaks (region 3 of Fig.\n\\ref{sch_2}) and in (c) there are vortex sheets present (region 2 of\nFig. \\ref{sch_2}). Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.} \\label{03neg}\n\\end{figure}\n\n{\\it Region 2}: Vortex sheets. Under the effect of strong rotation,\nthe components spatially separate and completely break the symmetry of the system. Nevertheless, they are approximately\n disk shaped with similar radii. Many vortices are nucleated that\narrange themselves into rows that can have various patterns: either\nthey can be striped or bent and are often disconnected from each\nother. A vortex sheet in one component corresponds to a region of\nmacroscopic density in the other component. These features can be\nseen in the density plots of Fig. \\ref{03neg}(c).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{13neg.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for\ncomponent-1 (left column) and component-2\n (right column). The parameters are $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with\n $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-1.3$ (which gives $g_{12}=0.0122$). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0$ and (b) $0.9$. In (a) the components are rotating droplets (region 4 of Fig. \\ref{sch_2}) while in (b) the components have spatially separated (but keep some symmetry) and there are no isolated density peaks (region 3 of Fig. \\ref{sch_2}). Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{13neg}\n\\end{figure}\n\n{\\em Region 3}: Spatial separation preserving some symmetry. Here\none component is a disk while the other component is an annulus and the disk\n fits within the annulus with a\nboundary layer region in which both components have microscopically\nsmall density as illustrated in Figure \\ref{03neg}(b) and \\ref{13neg}(b). Under rotation, the topological defects can\neither be coreless vortex lattices in the disk and (but not always)\ncorresponding isolated density peaks in the annulus, and\/or\n a giant skyrmion at the boundary interface between the two\ncomponents, as will be described later.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{drop.eps}\n\\end{center}\n\\caption{(Color online) A density plot for component-1 (left column) and component-2 (right column) showing examples of vortex nucleation in rotating droplets. The parameters are $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-9$ and $\\Omega=0.5$. At these parameters the components are in region 4 of the phase diagram of Fig. \\ref{sch_2}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{drop}\n\\end{figure}\n\n{\\it Region 4}: Rotating droplets. The components spatially separate\nand have broken symmetry such that the centres of each condensate\nare different and each component contains a single patch of density\n as illustrated in Figures \\ref{03neg}(a) and \\ref{13neg}(a). In the rotating droplets, vortices can appear provided the\nrotation is greater than some critical value. These features can be seen in the density plots\nof Fig. \\ref{drop}.\n\n\nThe $\\Omega-\\Gamma_{12}$ phase diagrams corresponding to the\nexperimental parameter sets introduced above (sets ES1, ES2 and ES3)\nare shown in Fig.'s \\ref{sch_2}, \\ref{sch_3} and \\ref{sch_1}. The\nlast one is of similar type as the one reported by Kasamatsu et al \\cite{ktu1}\n(there $g_1=g_2=4000$ and $N_1=N_2=1\/2$).\n\nNew features can be observed in Fig \\ref{sch_2} and \\ref{sch_3},\nsuch as isolated density peaks that eventually vanish as\n$\\Gamma_{12}$ is made more negative and the multiply quantised\nskyrmions at the interface between the disk component and the\nannular component in region 3. When $\\{\\eta,\\xi\\}\\neq1$ (Fig. \\ref{sch_3}), no region 2\nis found to exist. This absence is easily explained by two factors:\nthe onset of region 3 for (large) positive values of $\\Gamma_{12}$\nand the lack of vortices nucleated in component-1 (and to some\nextent in component-2).\n\nWe will now analyze in more detail some of\nthe features of regions 1, 3 and 4 of the phase diagrams. Vortex\nsheets (region 2) have been analysed (mainly in the case of equal $g_k$, $m_k$\nand $\\omega_k$) in \\cite{kt}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_2a.eps}\\\\\n\\includegraphics[scale=0.4]{sch_2b.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagrams for parameters\n$g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ with $\\eta=\\xi=1$ (set ES1)\nwith normalisation taken over the individual components (Eq.\n(\\ref{norm})). (a) Numerical simulations where triangles (squares) indicate\nthat the vortex lattice in both components is triangular (square) and diamonds that no vortices\nhave been nucleated. Filled triangles,\nsquares and diamonds are where the two components are disk-shaped\nand coexist. The empty triangles and empty diamonds\nare where the two components have spatially separated: one component\nis a disk and the other an annulus with a giant skyrmion at the\nboundary layer; those triangles with a dot in the centre\nrepresent the appearance of coreless vortices inside the annulus.\nThe crosses `x' are where the two components\nhave broken symmetry and are vortex sheets and the crosses `+' are rotating droplets.\n(b) A schematic representation of the numerical simulations. The\nsolid lines indicate the boundary between different identified\nregions (determined by the geometry of the ground state)\nand the dashed lines the boundary between triangular and\nsquare lattices in region 1 and the boundary between peaks and no\npeaks in region 3. The boundary between region 1 and the others can\nbe calculated analytically by Eq. (\\ref{spatsep}) to give\n$\\Gamma_{12}=0.008$. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_2}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_3a.eps}\\\\\n\\includegraphics[scale=0.4]{sch_3b.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagrams for parameters $g_1=0.0067$, $g_2=0.0063$, $N_1=N_2=10^4$ with $\\eta=0.48$ and $\\eta\\xi^2=1$ (set ES2) with normalisation taken over the individual components (Eq. (\\ref{norm})). (a) Numerical simulations where triangles indicate that the vortex lattice in component-2 is triangular, circles that component-2 contains rings of vortices and diamonds where no vortices have been nucleated. Filled triangles, circles and diamonds are where the two components are disk-shaped and coexist. The empty circles and empty diamonds are where the two components have spatially separated: one component is a disk and the other an annulus with a giant skyrmion at the boundary layer; those circles with a dot in the centre represent the appearance of coreless vortices inside the annulus. The crosses `+' are where the two components have broken symmetry and are rotating droplets.\n(b) A schematic representation of the numerical simulations.\nThe solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state).\nThe boundary between region 1 and region 3 is also calculated analytically by Eq. (\\ref{spatsep}) (dotted line).\nFor these parameters there is no region 2. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_3}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{sch_1.eps}\n\\end{center}\n\\caption{A schematic $\\Omega-\\Gamma_{12}$ phase diagram for $g_1=g_2\\equiv g=0.0078$, $N_1=N_2=10^5$ where $\\eta=\\xi=1$ (set ES3) and with the normalisation taken over the individual components (Eq. (\\ref{norm})). The solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state) and the dashed lines the boundary between triangular and square lattices. For these parameters there is no region 3. The unit of rotation is $\\tilde{\\omega}$.}\\label{sch_1}\n\\end{figure}\n\n\\subsection*{C. Analysis of the Features of the Phase Diagrams}\n\n\n\\subsubsection*{1. Square lattices}\n\nIt is a specific property of two components to stabilize square\nlattices. A requirement for the existence of square lattices is the\nnucleation of many vortices in both components, something not\npermitted in ES2 (Fig. \\ref{sch_3}), where only a small number of vortices\nare ever nucleated in component 1.\n\nSquare lattices generally occur at high rotational velocities and\nexamples have been observed experimentally \\cite{sch} and\nnumerically \\cite{wcbb,ktu,ktu1}. Mueller and Ho \\cite{mh} and later\nKe{\\c c}eli and \\\"Oktel \\cite{ko} have analysed the transition from\ntriangular to square lattices as $\\Gamma_{12}$ is varied in\ntwo-component condensates when $g_1\\sim g_2$ and $\\Omega$ is such that the condensate is in the lowest Landau\nlevel (LLL). Providing $\\Omega$ is large, according to \\cite{ko,mh}\nthe value of $\\alpha=\\sqrt{1-\\Gamma_{12}}$ determines whether the\nvortex lattices are triangular ($0<\\alpha<0.172$), in transition to\nbecoming square ($0.172<\\alpha<0.373$) or square ($\\alpha>0.373$).\nTherefore, assuming that, in the example parameters of $g_1=0.0078$,\n$g_2=0.0083$ and $\\eta=\\xi=1$ (set ES1), the $g_k$ are sufficiently close\nenough, the square lattice should first appear at $\\alpha=0.37$, or\nequivalently $\\Gamma_{12}=0.86$. Comparing this to the numerically\nobtained values, where the square lattice first appears for\n$0.8<\\Gamma_{12}<0.9$ in Fig. \\ref{sch_2}, the agreement seems good.\n It will be interesting to see how the\nanalysis of \\cite{ko,mh} has to be modified when $g_1$ becomes distant from $g_2$.\n\nNevertheless, the square lattices are present for lower rotational\nvelocities at which the LLL is not necessarily valid. In a\nforthcoming paper, using the nonlinear sigma model\npresented below, we hope to derive a vortex energy\n in terms\nof the positions of vortices. The ground state of this energy indeed\nleads to square lattices in some range of parameters (see also Section III.E). It turns out\nthat this vortex energy is similar to that of Barnett et al\n\\cite{barnett,barnett2}.\n\n\n\n\n\n\\subsubsection*{2. Rotating droplets}\n\n\n\nIn the parameter range of region 4, symmetry breaking with spatial\nseparation occurs. When the condensate is not under rotation, the\nground states of the two components will be `half ball'-like\nstructures, as can be seen in Fig. \\ref{13neg}(a) (if $g_1=g_2$ and\n$\\eta=\\xi=1$, then the two components are exactly half-balls). The\ndifference between the intercomponent strengths introduces a\ncurvature to the inner boundary of each component with the component\nthat corresponds to larger $g_k$ having positive curvature and the\nother component having negative curvature. This curvature depends on\nboth $\\Gamma_{12}$ and $\\Omega$. If $\\Gamma_{12}$ is held constant,\nthen the curvature increases as $\\Omega$ increases. When\n$\\Gamma_{12}$ goes to $-\\infty$, the droplets approach half-balls.\n\n\n\nVortex nucleation is also seen in region 4; see Fig. \\ref{drop}. In this figure there are four vortices in component-1 and three vortices in component-2. The number of vortices in each component will increase as $\\Omega$ is increased but this increase will also increase the curvature of the inner boundaries of the components, thus preventing the vortices aligning themselves into vortex sheets.\nExamples of the rotating droplet ground states can be seen in Fig. \\ref{13neg}(a), and further examples can be found in \\cite{kyt,ohberg,ohberg1,navarro,chris}.\n\n\n\n\\subsubsection*{3. Spatial separation preserving some symmetry: disk plus annulus}\n\n\n\n\n\n Region 3 is defined by the $\\Gamma_{12}$ and $\\Omega$ in which the\ncondensates have spatially separated components but still possess some symmetry about the origin: the disk component-1 sits securely within the annular\ncomponent-2 (see Fig.'s \\ref{03neg}(b) and \\ref{13neg}(b)).\nThere is a boundary layer evidenced where the outer edge\nof the disk overlaps the inner edge of the annulus.\n\n Let us describe the onset of region 3 from region 4, captured in Fig. \\ref{03neg} (sub-image (a) to (b) or (c) to (b)). For\n a particular $\\Gamma_{12}$, as $\\Omega$ is increased, then the curvature increases to such an extent that the components develop constant non-zero curvature, and are identified as a disk and an annulus.\nConversely, if $\\Omega$ is held constant and $\\Gamma_{12}$ is pushed to more negative values, then the curvature decreases.\n\nUnder rotation, defects can be observed: coreless vortices and giant\nskyrmions. Coreless vortices sit inside the disk while giant\n skyrmions are observed in the boundary layer.\n\n \\subsubsection*{4. Coreless Vortices in the disk plus annulus case}\n\n In the geometry of disk plus annulus, the vortices in the inside disk have corresponding isolated\ndensity peaks inside the annulus, hence in the microscopic density\nregions, as illustrated in Fig. \\ref{03neg}(b). Thus any vortex\nlattice structure produced in component-1 is replicated by a peak\nlattice structure in component-2, i.e. there is the continuing\npresence of coreless vortex lattices in the spatially separated\ncondensates \\cite{note2}.\n\n As the number of\nvortices in the disk increases with increasing angular\nvelocity, the number of density peaks inside the central hole of the\nannulus likewise increases. This has the effect that the central hole\nof the annulus can be masked at high angular velocities by the\nmultiple occurrence of the density peaks. A recent analytical\nunderstanding of the interaction between vortices and peaks\n has been provided by \\cite{kasa}. It may be extended to the case of the disk and annulus, for which the average density in the central hole is very small for one component.\n \nPushing $\\Gamma_{12}$ to lower values has the effect of reducing the\nsize of the boundary layer between the disk and annular components,\nbut also the isolated density peaks disappear; see Fig.\n\\ref{13neg}(b). A recent work has analysed, from an energy perspective, the preference for the ground state to contain or not contain density peaks \\cite{cate}.\n\n\n Figure \\ref{peaks} plots the maximum peak density of\nthe density peaks in component-2 as a function of $\\Gamma_{12}$ for\nthe parameters of Fig. \\ref{sch_2} and with $\\Omega=0.5$ and\n$\\Omega=0.75$. The disjoint region when $\\Omega=0.75$ for\n$-0.4<\\Gamma_{12}<0.1$ on Fig. \\ref{peaks} is due to the absence of\ndensity peaks in component-2 as a result of the appearance of vortex\nsheets (region 4). We see from Fig. \\ref{peaks},\n that the maximum peak density occurs when\n$\\Gamma_{12}\\sim0.01$ which is the value at which the components\nbegin to spatially separate (the transition between region 1 and\nregion 3) in set ES1. For $\\Gamma_{12}$ and $\\Omega$ that take\nvalues outside of region 1, the maximum density of the peaks\ndecreases linearly as $\\Gamma_{12}$ increases. The maximum density approaches microscopically small values for\n$\\Gamma_{12}\\sim-1.1$ when $\\Omega=0.5$ and for $\\Gamma_{12}\\sim-1$\nwhen $\\Omega=0.75$. An example of the transition can be seen in Fig.'s\n\\ref{03neg}(b) and \\ref{13neg}(b).\n\n\\begin{figure}\\begin{center}\n\\includegraphics[scale=0.5]{peaks.eps}\\end{center}\n\\caption{Plots of maximum peak density in component-2 against\n$\\Gamma_{12}$ for $\\Omega=0.5$ (dashed line) and $\\Omega=0.75$\n(solid line) when $g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ and\nwith $\\eta=\\xi=1$ (set ES1) . The disjoint region when $\\Omega=0.75$\nfor $-0.4<\\Gamma_{12}<0.1$ is due to the absence of isolated density\npeaks in component-2 as a result of the appearance of vortex sheets.\nDensity is measured in units of\n$r_0^{-2}$.}\\label{peaks}\\end{figure}\n\n\n\n\\subsubsection*{5. Giant skyrmion}\n\nA boundary layer between the overlap of component-1 and component-2\n is present for all values of $\\Gamma_{12}$ in region 3 but reduces\nin width as $\\Gamma_{12}$ becomes more negative (indeed the boundary\nlayer disappears only as $\\Gamma_{12}\\rightarrow-\\infty$). There are\nadditional topological defects that occur in the boundary layer that\nare not discernible on a traditional density plot. These\ntopological defects' presence can be observed in a phase profile,\nhowever a better visualisation is to use the pseudospin\nrepresentation (\\ref{rhot})-(\\ref{seqs}).\n One can plot the functions $S_x$, $S_y$ and\/or $S_z$\n which reveal the presence of all the topological defects -\n the coreless vortices (singly quantised skyrmions) that have already been visualised on the plots,\n and the new defect, a multiply quantised skyrmion.\n\nThe distinct nature of the two types of topological defect can be\nclearly seen in Fig. \\ref{topdef}(a,b) where an $S_x$ plot and an\n$(S_x,S_y)$ vectorial plot are shown respectively. A density plot of each component for these same parameters is seen in\nFig. \\ref{03neg}(b). The coreless vortices evident in Fig.\n\\ref{03neg}(b) are again clearly evident in Fig. \\ref{topdef}(a) and\n(b). A blow-up of the region close to one of (the three) coreless\nvortices in the $(S_x,S_y)$ vectorial plot, centred at $(1,-1)$ and\nexhibiting circular disgyration is shown in Fig. \\ref{topdef}(c).\nThe texture of $\\bm{S}$ can also exhibit cross- and radial-\ndisgyration \\cite{ohmi}. Conversely, the multiply quantised\nskyrmion, not present in the density plots of Fig.\n\\ref{03neg}(b), can clearly be seen in both Fig. \\ref{topdef}(a) and\n(b). The multiply quantised skyrmion forms a ring along the boundary\nlayer. A blow-up around $(0,-3.5)$ for the $(S_x,S_y)$ vectorial\nplot again shows the multiply quantised nature of this defect. A\nmultiply quantised skyrmion was evidenced by \\cite{ywzf} who termed\nit a giant skyrmion, a terminology retained in this paper.\n Increasing the rotation results in an increase in both the number of\ncoreless vortices and in the multiplicity of the giant skyrmion.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{topdef.eps}\n\\end{center}\n\\caption{(Color online) Plots of (a) $S_x$ and (b) a vectorial plot of $(S_x,S_y)$ for $\\Gamma_{12}=-0.3$ and $\\Omega=0.5$ when $g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$ and with $\\eta=\\xi=1$ (set ES1). The ring of skyrmions, at $r=r_s$, traces the boundary layer and the coreless vortices exist for $r0$\n or can be generalized and provide relevant information in the case $\\Gamma_{12}<0$.\n\n\\subsection*{A. Energy Functional Representation}\n\nWe write the energy functional of the two wave functions (Eq.\n(\\ref{enfn}), non-dimensionalised) as\n$E[\\psi_1,\\psi_2]=E_{\\text{KE}}+E_{\\text{PE}}+E_{\\text{I}}$ where\n\\begin{subequations}\n \\label{energyps}\n \\begin{align}\n \\begin{split} E_{\\text{KE}}&=\\frac{1}{\\eta+1}\\int\\left|\\left(\\nabla-\\frac{i}{2}(\\eta+1)\\bm{\\Omega}\\times\\bm{r}\\right)\\psi_1\\right|^2\\\\\n &\\quad+\\eta\\left|\\left(\\nabla-\\frac{i}{2\\eta}(\\eta+1)\\bm{\\Omega}\\times\\bm{r}\\right)\\psi_2\\right|^2\\quad d^2r,\n\\end{split}\\\\\n E_{\\text{PE}}&=\\int2j_1r^2|\\psi_1|^2+2j_2r^2|\\psi_2|^2\\quad d^2r,\\\\\n E_{\\text{I}}&=\\int\\frac{1}{2}g_1|\\psi_1|^4+\\frac{1}{2}g_2|\\psi_2|^4+g_{12}|\\psi_1|^2|\\psi_2|^2\\quad d^2r.\n\\end{align}\n\\end{subequations}\nwith\n\\begin{subequations}\n\\begin{align}\n j_1=&\\frac{1}{2}\\frac{(1+\\eta)\\xi^2}{(1+\\xi)^2}-\\frac{1}{8}(1+\\eta)\\Omega^2,\\\\ j_2=&\\frac{1}{2}\\frac{(1+\\eta)}{\\eta(1+\\xi)^2}-\\frac{1}{8\\eta}(1+\\eta)\\Omega^2.\n\\end{align}\n\\end{subequations}\nIt is assumed that the $g_k$, $m_k$, $\\omega_k$ and $N_k$ are distinct\nso that a weighted total density can be defined as (\\ref{rhot}) with\n $ \\psi_1=\\sqrt{\\rho_T}\\chi_1$ and $\n\\psi_2=\\sqrt{\\rho_T\/\\eta}\\chi_2$. The spin density vector $\\bm{S}$,\nwhich satisfies $|\\bm{S}|^2=1$ everywhere, has components given by\nEq.'s (\\ref{seqs}). We then have\n\\begin{equation}\n |\\psi_1|^2=\\frac{1}{2}\\rho_T(1+S_z),\\quad|\\psi_2|^2=\\frac{1}{2\\eta}\\rho_T(1-S_z).\n\\end{equation}\nWe introduce the phases $\\theta_1$ and $\\theta_2$ defined by $\\chi_1=|\\chi_1|\\exp(i\\theta_1)$ and $\\chi_2=|\\chi_2|\\exp(i\\theta_2)$.\n The energy functional is\nexpressed in terms of 4 variables: the total density $\\rho_T$, the component $S_z$\n and the angles $\\theta_1$, $\\theta_2$. We see that $E_{\\text{KE}}= E_{\\rho_T}+E_{\\text{Sz}}+E_{\\theta_1,\\theta_2}$ where\n\\begin{subequations}\n \\begin{align}\n E_{\\rho_T}=&\\int\\frac{1}{(\\eta+1)}(\\nabla\\sqrt{\\rho_T})^2\\quad d^2r,\\\\\n E_{\\text{Sz}}=&\\int \\frac{1}{4}\\frac{\\rho_T}{(\\eta+1)}\\frac{(\\nabla{{S_z}})^2}{(1-S_z^2)}\\quad d^2r,\\\\\n E_{\\theta_1,\\theta_2}=&\\int \\frac{1}{2}\\frac{\\rho_T}{(1+\\eta)}\\Bigg[(1+S_z)\\left(\\nabla\\theta_1-\\frac{1}{2}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\nonumber\\\\\n &\\quad+(1-S_z)\\left(\\nabla\\theta_2-\\frac{1}{2\\eta}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\Bigg]\\quad d^2r.\n \\end{align}\n\\end{subequations}\nThe other terms of the energy straightforwardly become\n\\begin{subequations}\n \\begin{eqnarray} E_{\\text{PE}}=&&\\int\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\quad d^2r,\\\\ E_{\\text{I}}=&&\\int\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)\\quad d^2r,\n \\end{eqnarray}\n\\end{subequations}\nwith\n\\begin{subequations}\n\\begin{align}\n\\label{c00}\n\\bar{c}_0=&\\frac{1}{4\\eta^2}(\\eta^2g_1+g_2+2\\eta g_{12}),\\\\\n \\bar{c}_1=&\\frac{1}{2\\eta^2}(\\eta^2g_1-g_2),\\\\\n\\label{c22}\n \\bar{c}_2=&\\frac{1}{4\\eta^2}(\\eta^2g_1+g_2-2\\eta g_{12}).\n\\end{align}\n\\end{subequations}\n\nThus the complete energy is\n\\begin{equation}\n \\label{sig}\n\\begin{split}\nE&=\\int \\frac{1}{(\\eta+1)}(\\nabla\\sqrt{\\rho_T})^2+\\frac{\\rho_T}{4(\\eta+1)}\\frac{(\\nabla{{S_z}})^2}{(1-S_z^2)}\\\\\n&+\\frac{\\rho_T}{2(1+\\eta)}\\times\\\\\n&\\Bigg[(1+S_z)\\left(\\nabla\\theta_1-\\frac{1}{2}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\\\\n&+(1-S_z)\\left(\\nabla\\theta_2-\\frac{1}{2\\eta}(1+\\eta)\\bm{\\Omega}\\times\\bm{r}\\right)^2\\Bigg]\\\\\n&+\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\\\\n&+\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)\\quad\nd^2r.\n\\end{split}\n\\end{equation}\nEnergy (\\ref{sig}) is subject to the constraints (\\ref{norm}) that can be rewritten\nas \\begin{subequations}\n \\label{nn2}\n\\begin{align}\n \\label{nn2a}\n \\int\\rho_T(1-S_z)\/2\\quad d^2r=&N_2\\eta,\\\\\n \\int\\rho_T(1+S_z)\/2\\quad d^2r=&N_1.\n \\label{nn2b}\n\\end{align}\n\\end{subequations}or equivalently\n\\begin{subequations}\n \\label{nn1}\n\\begin{align}\n \\label{nn1a}\n \\int\\rho_T\\quad d^2r=&N_1+N_2\\eta,\\\\\n \\int\\rho_TS_z\\quad d^2r=&N_1-N_2\\eta,\n \\label{nn1b}\n\\end{align}\n\\end{subequations}\nThe minimization of the energy under the constraints (\\ref{nn1})\nyields two coupled equations with two Lagrange multipliers, $\\mu$ and $\\lambda$. We will\nwrite them\n when the phase is constant (the gradient of the phases\n $\\theta_1$ and $\\theta_2$ can be ignored) and the effect of rotation is contained in an effective centrifugal dilation of the trapping potential included in $j_1,j_2$:\n\\begin{equation}\n \\begin{split}\n \\label{muu1}\n \\mu+\\lambda S_z=&-\\frac 1{(1+\\eta)}\\frac{\\Delta \\sqrt{\\rho_T}}{\\sqrt{\\rho_T}}\n +\\frac 1{4(1+\\eta)}\\frac{(\\nabla S_z)^2}{(1-S_z^2)}\\\\&\\quad+\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\\\\n&\\qquad+\\rho_T(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2),\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n \\begin{split}\n \\label{szeq1}\n \\lambda=&-\\frac 1{4(1+\\eta)}\\frac{\\Delta S_z+\\nabla \\rho_T \\cdot\\nabla S_z}{(1-S_z^2)}+\\frac 1{2(1+\\eta)}\\frac{S_z(\\nabla S_z)^2}{(1-S_z^2)}\\\\\n&\\quad +(j_1-j_2\/\\eta)r^2+\\left({\\bar{c}_1}+2\\bar{c}_2S_z\\right)\\rho_T\/2.\n\\end{split}\n\\end{equation}\n\nAs pointed out in \\cite{cooper}, in the general case where $\\eta$ and $\\xi$ are\n different from 1, we have seen from the expression $E_{\\theta_1,\\theta_2}$ that the effective velocity in each component is different. Nevertheless, in the case when $\\eta=\\xi=1$, it is possible to define an effective velocity shared by both components,\n\\begin{equation}\n v_{\\text{eff}}=\\frac{\\nabla\\Theta}{2}+\\frac{\\bm{R}S_z}{2(1-S_z^2)}\n\\end{equation} where $\\Theta=\\theta_1+\\theta_2$ and $\\bm{R}=S_y\\nabla S_x-S_x\\nabla\nS_y$. We note the identity\n\\begin{equation}\n \\frac{(\\nabla S_z)^2}{(1-S_z^2)}=(\\nabla \\bm{S})^2-\\frac{R^2}{(1-S_z^2)},\n\\end{equation} where $(\\nabla \\bm{S})^2=(\\nabla S_x)^2+(\\nabla S_y)^2+(\\nabla S_z)^2$.\nExpansion of the square in $E_{\\theta_1,\\theta_2}$,\n substituting in the $v_{\\text{eff}}$ and using the identity\nfrom above reduces the energy to the simple form found in \\cite{ktu,ktu3}.\n\\begin{equation}\n\\begin{split}\n\\label{sig2}\n E=&\\int\\frac{1}{2}(\\nabla\\sqrt{\\rho_T})^2+\\frac{\\rho_T}{8}(\\nabla\\bm{S})^2\\\\\n\n&\\quad+\\frac{\\rho_T}{2}\\left(v_{\\text{eff}}-\\bm{\\Omega}\\times\\bm{r}\\right)^2+\\frac{1}{2}r^2(1-\\Omega^2)\\rho_T\\\\\n&\\qquad+\\frac{\\rho_T^2}{2}\\left(c_0+c_1S_z+c_2S_z^2\\right)\\quad d^2r,\n\\end{split}\n\\end{equation}\nwhere ${c}_0$, ${c}_1$ and ${c}_2$ are equal to $\\bar{c}_0$, $\\bar{c}_1$ and $\\bar{c}_2$ with $\\eta$ set equal to unity.\n\n\\subsection*{B. A Thomas-Fermi approximation}\n\nThe typical Thomas-Fermi (TF) approximation to Eq.'s (\\ref{muu1})-(\\ref{szeq1}) is\nto assume that derivatives in $\\rho_T$ and $S_z$ are negligible in\nfront of the other terms. If we apply the TF approximation we then get\n\\begin{equation}\n \\begin{split}\n \\label{muu}\n \\mu+\\lambda S_z=&\\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\\\\n&\\quad+\\rho_T(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2),\n\\end{split}\n\\end{equation}\nand\n\\begin{equation}\n \\label{szeq}\n \\lambda=(j_1-j_2\/\\eta)r^2+\\frac{1}{2}\\left({\\bar{c}_1}+2\\bar{c}_2S_z\\right)\\rho_T.\n\\end{equation}\nThe TF energy is then\n \\begin{equation}\n \\label{sigTF}\n\\begin{split}\nE_{\\text{TF}}=&\\int \\left[(j_1+j_2\/\\eta)+(j_1-j_2\/\\eta)S_z\\right]r^2\\rho_T\\\\\n&\\quad+\\frac{\\rho_T^2}{2}\\left(\\bar{c}_0+\\bar{c}_1S_z+\\bar{c}_2S_z^2\\right)-\\mu\\rho_T\n-\\lambda \\rho_T S_z\\quad d^2r.\n\\end{split}\n\\end{equation}It is important to point out that the reduction of this quadratic form in $\\rho_T$ and\n$\\rho_T S_z$ yields\n\\begin{equation}\n \\label{sigTFred}\n\\begin{split}\nE_{\\text{TF}}=&\\int \\frac{\\bar c_2}2 \\left(\\rho_TS_z+\\frac{\\bar\nc_1}{2\\bar c_2}\\rho_T+\n\\frac 1{\\bar c_2}(j_1-j_2\/\\eta)r^2-\\frac{\\lambda}{\\bar c_2}\\right )^2\\\\\n&\\quad+\\frac 12 \\left(\\bar c_0-\\frac {\\bar c_1^2}{4\\bar c_2}\\right)\\times\\\\\n&\\qquad \\left[ \\rho_T-\\frac 1 {\\left(\\bar c_0-\\bar c_1^2\/4\\bar\nc_2\\right)}(\\mu -(j_1+j_2\/\\eta)r^2) \\right .\\\\\n &\\quad\\quad\\left . +\\frac 12\\frac{\\bar c_1}{(\\bar c_0\\bar c_2- \\bar c_1^2\/4)}(\\lambda\n-(j_1+j_2\/\\eta)r^2)\\right ]^2\\ d^2r\\\\\n&\\qquad\\quad+\\text{constant terms}.\n\\end{split}\n\\end{equation} The existence of a minimum for this quadratic form\nimplies that $\\bar c_2 \\geq 0$ and $\\bar c_0- \\bar c_1^2\/(4\\bar c_2)\n\\geq 0$. Since $\\bar c_0- \\bar c_1^2\/(4\\bar c_2)\n=g_1g_2\\Gamma_{12}\/(4\\eta^2\\bar c_2)$, this implies in particular\nthat $\\Gamma_{12}>0$.\n\nTherefore, if $\\Gamma_{12}>0$, (which implies $\\bar c_2>0$), a\nThomas Fermi approximation can be performed as has previously been considered,\n generally taking an approximation on the individual component\nwave functions $\\psi_1$ and $\\psi_2$\n\\cite{ohberg1,tripp,corro,hs,kyt,pb,riboli,jc}. Nevertheless, as we\nwill see below, if $\\bar c_2<0$ and $\\Gamma_{12}<0$, then $(\\bar c_0-\\frac {\\bar c_1^2}{4\\bar c_2})>0$\n and a TF\napproximation can still be performed on $\\rho_T$, provided we keep\ngradient terms in $S_z$.\n\n\n Multiplying (\\ref{szeq}) by $S_z$ and subtracting (\\ref{muu})\n leads to\n \\begin{equation}\n \\mu=(j_1+j_2\/\\eta)r^2+\\rho_T\\left(\\frac{\\bar{c}_1}2S_z+\\bar{c}_0\\right).\n\\end{equation}\n Simplification by $\\rho_T S_z$ with (\\ref{szeq}) yields, when\n $\\psi_1\\times\\psi_2\\neq0$,\n \\begin{equation}\n \\label{rhoty}\n\\rho_T=\\frac{a_3+a_4r^2}{g_1g_2\\Gamma_{12}},\n\\end{equation}\nand\n\\begin{equation}\n \\label{szform}\nS_z=\\frac{a_1+a_2r^2}{a_3+a_4r^2},\n\\end{equation}\nwhere\n\\begin{subequations}\n \\label{as}\n\\begin{align}\n &a_1=4\\eta^2(\\lambda\\bar{c}_0-\\mu\\bar{c}_1\/2),\\\\\n&a_2=\\eta g_1h_2-g_2h_1,\\\\\n&a_3=4\\eta^2(\\mu\\bar{c}_2-\\lambda\\bar{c}_1\/2),\\\\\n&a_4=-(\\eta g_1h_2+g_2h_1),\n\\end{align}\n\\end{subequations}\nand where we define\n\\begin{equation}\n h_k=2\\left(j_k-\\frac{g_{12}}{g_{3-k}}j_{3-k}\\right).\n\\end{equation}\nWhen only one component is present ($\\psi_1\\times\\psi_2=0$), this\nsimplifies to\n\\begin{equation}\\label{rhoonly}\n \\rho_T=\n\\begin{cases}\n\\frac{1}{g_1}\\left[\\mu+\\lambda-2j_1 r^2\\right]&\\hbox{ if } \\psi_{2}=0\\\\\n\\frac{\\eta}{g_2}\\left[\\eta(\\mu-\\lambda)-2j_2r^2\\right]&\\hbox{ if } \\psi_1=0\n\\end{cases}\n\\end{equation}\nsince $S_z=+1$ when $\\psi_2=0$ and $S_z=-1$ when $\\psi_1=0$.\n\n\n\nTo begin, we note from Eq. (\\ref{szform}) that\n$r^2=(a_1-a_3S_z)\/(a_4S_z-a_2)$, so that\n\\begin{equation}\\label{rhoSz}\n \\rho_T=\\frac{a_1a_4-a_2a_3}{g_1g_2\\Gamma_{12}(a_4S_z-a_2)}.\n\\end{equation}\nSince the intracomponent coupling strengths are chosen to be\ndistinct, the support of each component will not necessarily be\nequal. In order to\n lead the computations, we have to assume a geometry for the\n components: two disks, a disk and annulus or two half-balls (droplets)\n and a sign for $\\Gamma_{12}$.\n\n\\subsection*{C. $\\Gamma_{12}>0$}\n\n\\subsubsection*{1. Both components are disks}\n\nWe start when both components are disks. Without loss of\ngenerality, we can assume that the outer boundary of component-2 (at\n$r=R_2$) is larger than that of component-1 (at $r=R_1$). Therefore,\n$S_z=-1$ and (\\ref{rhoonly}) holds in the annulus, while (\\ref{rhoSz}) holds in the coexisting region, which is the inside disk. The integrals from Eq.'s (\\ref{nn2}) then\ngive\n\\begin{subequations}\n \\begin{equation}\n \\begin{split}\n \\label{szp1}\n \\frac{{N}_1}{2\\pi}=&\\int_0^{R_1}\\rho_T\\frac{(1+ S_z)}{2}\\quad rdr\\\\\n =&-\\frac{(a_1a_4-a_2a_3)^2}{4g_1g_2\\Gamma_{12}}\\int_{s_0}^{-1}\\frac{(1+ S_z)}{(a_4S_z-a_2)^3}\\quad dS_z,\n\\end{split}\n\\end{equation}\nwhere $s_0=S_z|_{r=0}=a_1\/a_3$ and\n \\begin{equation}\n \\begin{split}\n \\label{szp1b}\n \\frac{\\eta{N}_2}{2\\pi}=&\\int_0^{R_1}\\rho_T\\frac{(1- S_z)}{2}\\quad rdr+\\int_{R_1}^{R_2}\\rho_T\\quad rdr\\\\\n =&-\\frac{(a_1a_4-a_2a_3)^2}{4g_1g_2\\Gamma_{12}}\\int_{s_0}^{-1}\\frac{(1- S_z)}{(a_4S_z-a_2)^3}\\quad dS_z\\\\\n &\\quad-\\frac{g_2}{4\\eta j_2}\\int_{\\rho_d}^0\\rho_T\\quad d\\rho_T,\n\\end{split}\n\\end{equation}\n\\end{subequations}\nwith\n \\begin{equation}\n \\begin{split}\n \\rho_d=&\\frac{a_3+a_4r^2}{g_1g_2\\Gamma_{12}}\\Big|_{r=R_1}\\\\\n =&\\frac{a_2a_3-a_1a_4}{g_1g_2\\Gamma_{12}(a_2+a_4)},\n\\end{split}\n\\end{equation}\nsince $S_z=-1$ at $r=R_1$.\n\nCompletion of these integrals and noting that $\\{a_2+a_4,a_2-a_4\\}=\\{-2g_2h_1,2\\eta g_1h_2\\}$ gives\n\\begin{subequations}\n \\label{all}\n\\begin{equation}\n \\label{szp2}\n a_3^2=\n\\frac{8N_1g_1g_2^2\\Gamma_{12}h_1}{\\pi(s_0+1)^2}\\\\%\\qquad &k=1\\\\\n\\end{equation}\nwhich must necessarily be positive, i.e. $h_1\\Gamma_{12}>0$, and\n\\begin{equation}\n \\label{qq1}\n \\begin{split}\nN_1(a_2-s_0a_4)^2=&2\\eta^2N_2g_1g_2j_2h_1\\Gamma_{12}(1+s_0)^2\\\\\n&\\quad+2N_1(1+s_0)\\eta j_2g_1\\Gamma_{12}\\times\\\\\n&\\qquad[\\eta g_1h_2(1+s_0)+2g_2h_1(s_0-1)]\n\\end{split}\n\\end{equation}\n\\end{subequations}\nwhere in Eq. (\\ref{qq1}) the expression for $a_3$ from Eq.\n(\\ref{szp2}) has been substituted. This equation can always be\nsolved in terms of $s_0$ when $h_1 \\Gamma_{12}$ is positive since\nthe discriminant is equal to\n$8N_1\\eta^2g_2^2g_1j_2h_1\\Gamma_{12}(N_1g_{12}+N_2 g_2)$.\n We find that $$1+s_0=\\frac{2N_1g_2}{N_1g_2-\\eta N_1 g_{12}+\\sqrt{2N_1\\eta^2g_1j_2\\Gamma_{12}(N_1g_{12}+N_2 g_2)\/h_1}}.$$\n\nA similar calculation can be completed if the outer boundary of\ncomponent-2 is larger than that of component-1. This then gives that\n${h_2\\Gamma_{12}>0}$ (although expressions (\\ref{all}) change\nslightly).\n\n\n\n If an annulus develops in\n component $3-k$, it means that $s_0=1$ ($s_0=-1$) for $k=1$ ($k=2$).\n Inputting this choice into Eq. (\\ref{qq1}) gives\n\\begin{equation}\n \\begin{split}\n \\label{spatsep}\n&\\bar{g}_{12}=\\frac{N_kg_kj_{3-k}}{2(N_1j_1+N_2j_2)}\\\\\n&\\quad+\\frac{1}{2}\\left(\\left[\\frac{N_kg_kj_{3-k}}{N_1j_1+N_2j_2}\\right]^2+\\frac{4N_{3-k}g_1g_2j_{3-k}}{N_1j_1+N_2j_2}\\right)^{1\/2}\n \\end{split}\n\\end{equation}\nas the critical $g_{12}$ at which an annulus forms in\ncomponent-$\\{3-k\\}$. In the case of Fig. \\ref{sch_3}, the curve (\\ref{spatsep}) has been\nplotted in dashed lines and is close to the numerical curve.\n Notice that if $\\xi^2=1$ (equal trapping\nfrequencies for both components), then Eq. (\\ref{spatsep}) becomes\nindependent of $\\Omega$, as in the phase diagrams of\nFig. \\ref{sch_2}, where it yields\n $\\Gamma_{12}=0.008$.\n\nLet us point out that before the transition to the disk plus annulus takes place, there is\n a subregion of region 1, where there are 2 disks, but in one component the wave function has a local minimum\n at the origin. For instance, in the case of Fig. \\ref{sch_2}, it corresponds to $\\Gamma_1$ changing sign.\n\n\nAs a conclusion, in order for the ground state to be composed of two\ndisks, assuming that component-$k$ is the component with smaller\nsupport, we need that $h_k>0$, $\\Gamma_{12}>0$ and\n$g_{12}<\\bar{g}_{12}$. These three conditions can be summarised as\n\\begin{subequations}\n\\begin{align}\n\\label{con1a}\ng_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\frac{j_{3-k}}{j_{k}}g_{k},\\bar{g}_{12},\\sqrt{g_1g_2}\\right)\n\\hbox{ if } &h_1,\\ h_2>0,\\\\\n\\frac{j_{3-k}}{j_{k}}g_{k}<\ng_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\bar{g}_{12},\\sqrt{g_1g_2}\\right)\\hbox{\nif } &h_{3-k}<0.\n\\label{con1b}\n\\end{align}\n\\end{subequations}\n\n\\subsubsection*{2. A disk and an annular component}\n\nIn this case, we can assume a disk in component-1 and an annulus in component-2; there are three regions:\n an inner disk where only component-1 is present and (\\ref{rhoonly}) holds, an outer\n annulus where only component-2 is present and (\\ref{rhoonly}) holds and an inner\n annulus where both components coexist and (\\ref{rhoSz}) holds.\n\nIn order to use the TF approximation when one component is annular,\n computations similar to\n(\\ref{szp2})-(\\ref{qq1}) lead to $h_1h_2<0$ and\n$g_{12}>\\bar{g}_{12}$. This can be summarized (for an annulus in\ncomponent-$\\{3-k\\}$) as\n\\begin{equation}\n \\label{an}\n \\max\\left(\\frac{j_{3-k}}{j_{k}}g_{k},\\bar{g}_{12}\\right)< g_{12}<\\min\\left(\\frac{j_k}{j_{3-k}}g_{3-k},\\sqrt{g_1g_2}\\right),\n\\end{equation}\nwith $h_{3-k}<0$.\nEquation (\\ref{an}) also places the restriction that\n\\begin{align}\n g_k<&g_{3-k}\\left(\\frac{j_k}{j_{3-k}}\\right)^2\\nonumber\\\\\n =&g_{3-k}\\Lambda_k,\n \\label{toget}\n\\end{align}\nwhere $\\Lambda_k=({j_k}\/{j_{3-k}})^2$. Note that $\\Lambda_1\\Lambda_2=1$ such that an annulus develops in component-2 (-1) if $g_2 >$ ($<$) $g_1\\Lambda_2$.\n\n\n\n\\subsubsection*{3. Orders of Intracomponent Strengths and Special Cases}\n\n The effect that changing the order of the intracomponent\nstrengths and particle numbers has on $\\bar{g}_{12}$, and thus on\nthe phase diagrams, is now investigated. There are two cases to\nconsider, depending on the relative orders of the particle numbers.\nDrawing aid from experimental values, it is always expected that\n$\\min \\{N_1, N_2\\}\\gg \\max \\{ g_1,\\ g_2\\}$. Throughout it will be\nassumed that $g_2>\\Lambda_2 g_1$, so the annulus develops in\ncomponent-2 and that $j_1$ and $j_2$ are of order unity.\n\nIn the first case when the particle number of component-1 is much\ngreater than the particle number of component-2 ($N_1\\gg N_2$), it\nfollows that $\\bar{g}_{12}\\sim g_1j_2\/j_1$.\nThe boundary between region 1 and region 3 is then directly\ndependent on the value of the ratio $j_2g_1\/[j_1g_2]$. Notice that\nif $g_2\\gg g_1$, $\\Gamma_{12}$ evaluated at $g_{12}=\\bar{g}_{12}$\ntends to unity and as such an annulus would always be present in\ncomponent-2, whatever the value of $g_{12}$.\n\nConversely in the second case when the particle number of\ncomponent-1 is much smaller than the particle number of component-2\n($N_1\\ll N_2$), it follows that $\\bar{g}_{12}\\sim\\sqrt{g_1g_2}$\nwhich implies that the annulus will only develop near\n$\\Gamma_{12}=0$.\n\n\n\nWe can also look at some special cases - there are four that can be considered:\n\n{\\it Case (i)}. $\\Lambda_k g_{3-k}=g_k$. When $\\Lambda_2$\n(equivalently $\\Lambda_1$) is such that this equality is made, the\ntwo components are both disks and no annulus develops.\n\n{\\it Case (ii)}. $\\eta=\\xi=1$. Then $\\Lambda_k=1$\nand $R_2\\gtrless R_1\\iff N_2g_2\\Gamma_1\\gtrless N_1g_1\\Gamma_1$ when\nthere are two disks and the annulus develops in the component which\nhas the larger interaction strength.\n\n{\\it Case (iii)}. $\\eta=\\xi=1$ and $g_1=g_2\\equiv g$. When the\nintracomponent coupling strengths are equal, $\\bar{g}_{12}=g$ and\nthere are always two disks with $R_2\\gtrless R_1\\iff N_2\\gtrless\nN_1$.\n\n{\\it Case (iv)}. $\\eta=\\xi=1$, $g_1=g_2\\equiv g$ and $N_1=N_2\\equiv\nN$. When the particle numbers and intracomponent coupling strengths\nare equal, $\\bar{g}_{12}=g$ and there are always two disks with\n$R_2=R_1$. A detailed analysis of this case, explored numerically in\na phase diagram for all $\\Gamma_{12}$ and analytically in the TF\nlimit, was considered by \\cite{ktu1}.\n\n\\subsubsection*{4. Justification of the Thomas-Fermi approximation}\n\n The gradient terms in\n(\\ref{muu1})-(\\ref{szeq1}) can be neglected if $d_{hl}$,\n the characteristic length of variation of $\\rho_T$ and $S_z$ is much smaller than\n$d_c$, the characteristic size of the condensates. We have that\n$1\/d_{hl}^2$ is of order of $\\mu$ and $\\lambda$, which are of order\n$\\sqrt{N_kg_k}$, $\\sqrt{N_kg_{3-k}}$. Hence\n $d_{hl}$ is bounded above by the maximum\n of $(N_kg_k)^{-1\/4}$, $(N_kg_{3-k})^{-1\/4}$. From the expression of $a_3$,\nthe characteristic size of the condensate is of order the minimum of\n$(g_kN_k \\Gamma_{12})^{1\/4}$. Therefore, the Thomas Fermi approximation holds\n if $N_kg_k\\sqrt{\\Gamma_{12}}$, $N_kg_{3-k}\\sqrt{\\Gamma_{12}}$ are large. This requires\n the usual Thomas Fermi criterion that $N_kg_k$, $N_kg_{3-k}$ are large, but\n breaks down if $\\Gamma_{12}$ is too small.\n\n\n\nFor $\\Gamma_{12}=0$,\n the\n equations lead to spatial separation: either the radii of the disks tend to 0\n in the case of two disks or the outer radius of the inner disk tends\n to the inner radius of the annulus in the case of disk plus annulus\n so that $\\psi_1\\times\\psi_2=0$ everywhere. Another analysis\n has to be carried out to understand the region of coexistence,\n which is of small size and has strong gradients.\n\n\n\n\\subsection*{D. $\\Gamma_{12}<0$, beyond the TF approximation}\n\nFor negative $\\Gamma_{12}$, and if $N_kg_k$ are large, the Thomas Fermi\n approximation can be extended provided some model takes into account the small region where\nthe condensates coexist.\n\nIndeed, if we go back to (\\ref{sigTFred}), and have both $\\bar c_2<0$ and $\\Gamma_{12}<0$,\n then the coefficient in front of the second square is positive and the optimal situation\n is to have the square equal to 0, which leads to the inverted parabola (\\ref{rhoty}). On the other\n hand, the coefficient in front of the first square is negative, and the ground state involves\n derivatives in $S_z$ to compensate it. Under the assumption\n that the boundary layer where $S_z$ varies is small,\n we are going to derive a TF model with jump for $\\rho_T$. We will analyze it\n for the different geometries (disk plus annulus, droplets and vortex sheets)\n and show that it provides information consistent with the numerics.\n\n\n\n\n\n\\subsubsection*{1. Disk Plus Annulus}\n\n\n Assuming that the\nboundary layer is present only at some $r=r_s$, then $S_z=+1$ in the region in which component-2 is zero\n($r\\le r_s^-$) and $S_z=-1$ in the region in which component-1 is\nzero ($r\\ge r_s^+$). The\ntransition from $S_z=+1$ to $S_z=-1$ is not smooth, therefore\ncreating the jump in density.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{sig_1.eps}\n\\end{center}\n\\caption{Total density profiles (divided by $10^4$) obtained numerically (solid lines) and analytically (dashed lines) for two spatial separation cases with $g_1=0.0078$, $g_2=0.0083$, $N_1=N_2=10^5$, $\\eta=\\xi=1$ (set ES1) and $\\Gamma_{12}=-10$ and $\\Omega=0$: (a) annulus plus disk, analytical estimate coming from (\\ref{eann}) and (b) droplet, analytical estimate coming from (\\ref{edrop}). The inset in (a) shows the discontinuity of density at $r_s=3.63$. Distance is measured in units of $r_0$ and density in units of $r_0^{-2}$.}\\label{sig1}\n\\end{figure}\nTherefore, we are lead to minimise the integral\n\\begin{equation}\n \\label{eann}\n \\begin{split}\nI&=\\int_{B_{r_s}}2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2 \\quad d^2r\\\\\n&\\quad+\\int_{B_{R\\backslash r_s}}2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\quad d^2r\n\\end{split}\n\\end{equation}\nwith respect to $r_s$. Here $B_{r_s}$ is a ball of radius $r_s$ and $B_{R\\backslash r_s}$ is a torus with outer boundary at $r=R$ and inner boundary at $r=r_s$. Thus\n\\begin{equation}\n \\begin{split}\n&2\\pi r_s\\left[2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2\\right]\\Bigg|_{r=r_s^-}\\\\\n&\\qquad-2\\pi r_s\\left[2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\right]\\Bigg|_{r=r_s^+}=0,\n\\label{jumpp}\n\\end{split}\n\\end{equation}\nwhich implies that\n\\begin{subequations}\n \\label{rhopm}\n\\begin{align}\n\\rho^-=&\\frac{1}{g_1}\\left[\\mu_1-2j_1r^2\\right]\\quad r\\in B_{r_s}\\\\\n\\rho^+=&\\frac{\\eta}{g_2}\\left[\\mu_2-2j_2r^2\\right]\\quad r\\in B_{R\\backslash r_s}.\n\\end{align}\n\\end{subequations}\nThen using the normalisation conditions $\\int\\rho^-d^2r=N_1$ and $\\int\\rho^+d^2r=\\eta N_2$ we get an outer radius\n\\begin{subequations}\n \\begin{equation}\n R=\\left(r_s^2+\\sqrt{\\frac{g_2N_2}{\\pi j_2}}\\right)^{1\/2},\n \\end{equation}\nand chemical potentials\n\\begin{align}\n\\mu_1=&\\frac{N_1g_1}{\\pi r_s^2}+j_1r_s^2,\\\\\n\\mu_2=&2j_2\\left(r_s^2+\\sqrt{\\frac{g_2N_2}{\\pi j_2}}\\right).\n\\end{align}\n\\end{subequations}\nIt remains to find $r_s$. But, from (\\ref{jumpp}), it follows that\n\\begin{equation}\nr_s^2=\\frac{\\frac{4N_2j_2}{\\pi}-g_1\\left(\\frac{N_1}{\\pi r_s^2}-\\frac{j_1r_s^2}{g_1}\\right)^2}{4\\left[j_1\\left(\\frac{N_1}{\\pi r_s^2}-\\frac{j_1 r_s^2}{g_1}\\right)-2j_2\\sqrt{\\frac{N_2j_2}{\\pi g_2}}\\right]}.\n\\end{equation}\nThe above leaves a quartic equation in $r_s^2$, the solution of which can be found numerically.\n\nWe can then complete the energy calculation to give\n\\begin{equation}\n \\label{enda}\nI=\\frac{g_1N_1^2}{2\\pi r_s^2}+(N_1j_1+2N_2j_2)r_s^2-\\frac{\\pi j_1^2}{6g_1}r_s^6+\\frac 43\\left(\\frac{N_2^3g_2j_2}{\\pi}\\right)^{1\/2}.\n\\end{equation}\nA check plot of the density profile with the parameters $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ where $\\eta=\\xi=1$\nagrees well with this model\n where the boundary is calculated to be at $r_s=3.62$ (see Fig.\\ref{sig1}(a)).\n\n\n\\subsubsection*{2. Droplets}\n\nThis formalism can also be extended to study the droplet case. As before, assuming a thin boundary layer, one can allow for a jump in density. We thus need to minimise the Thomas-Fermi energy for two droplets described by $B_{1}$ and $B_{2}$ that exist in the regions $0\\le\\theta\\le\\alpha$ and $\\alpha\\le\\theta\\le2\\pi$ respectively. The energy is then\n\\begin{equation}\n \\label{edrop}\n \\begin{split}\nI&=\\alpha\\int_{0}^{R_1}2j_1r^2\\rho_T+\\frac{g_1}{2}\\rho_T^2 \\quad rdr\\\\\n&\\quad+(2\\pi-\\alpha)\\int_{0}^{R_2}2\\frac{j_2}{\\eta}r^2\\rho_T+\\frac{g_2}{2\\eta^2}\\rho_T^2\\quad rdr.\n\\end{split}\n\\end{equation}\nThe expressions for the density in each domain (\\ref{rhopm}) and the normalisation conditions allow completion of this integral to give\n\\begin{equation}\nI=\\frac{4\\sqrt{2}}{3}\\left[\\left(\\frac{N_1^3g_1j_1}{\\alpha}\\right)^{1\/2}+\\left(\\frac{N_2^3g_2j_2}{(2\\pi-\\alpha)}\\right)^{1\/2}\\right].\n\\end{equation}\n\nIt remains to find the optimum $\\alpha$. This is achieved through the condition $dI\/d\\alpha=0$ which gives\n\\begin{equation}\n \\begin{split}\n\\frac{(N_1^3g_1j_1)^{1\/2}}{\\alpha^{3\/2}}&=\\frac{(N_2^3g_2j_2)^{1\/2}}{(2\\pi-\\alpha)^{3\/2}}\\\\\n\\Rightarrow \\alpha&=2\\pi\\frac{\\bar{N}(\\bar{g}\\bar{j})^{1\/3}}{(1+\\bar{N}(\\bar{g}\\bar{j})^{1\/3})}\n\\end{split}\n\\end{equation}\nwhere we have set $\\bar{N}=N_1\/N_2$, $\\bar{g}=g_1\/g_2$ and $\\bar{j}=j_1\/j_2$. As expected, equality of the $N_k$, $g_k$ and setting $\\eta=\\xi=1$ ($\\Rightarrow j_1=j_2$) gives $\\alpha=\\pi$ and the condensate is then composed of two half-balls. Otherwise, a curvature is present. A check plot of the density profile with the parameters $g_1=0.0078$, $g_2=0.0083$ and $N_1=N_2=10^5$ where $\\eta=\\xi=1$ is given in Fig \\ref{sig1}(b).\n\nFinally we can note the energy for the droplets is\n\\begin{equation}\n\t\\label{endd}\n \\begin{split}\nI&=\\frac{4}{3\\sqrt{\\pi}}\\left(1+\\bar{N}(\\bar{g}\\bar{j})^{1\/3}\\right)^{-1\/2}\\times\\\\\n&\\quad\\left[(N_1^3g_1j_1)^{1\/2}+(N_1N_2^2(g_1g_2^2j_1j_2^2)^{1\/3})^{1\/2}\\right].\n\\end{split}\n\\end{equation}\nThis energy can be compared to the energy of the disk plus annulus (\\ref{enda}) to determine which is the optimum geometry.\nIndeed, in the numerical cases studied before, the droplets are preferred states for small $\\Omega$.\n\n\n\\subsubsection*{3. Regions of Vortex Sheets}\n\nIn the case of vortex sheets,\n we can assume that the global profile of the total density is TF-like, obeying Eq. (\\ref{rhoty}) in the bulk of the condensate. By working with the total density, we do not require any information on the vortex sheets themselves (and consequently $S_z$).\n\n\nWe thus take the form of $\\rho_T$ from Eq. (\\ref{rhoty}) from which we note that the outer boundary at $r=R$ satisfies $R=\\sqrt{-a_3\/a_4}$ and that completion of the normalisation condition ($\\ref{nn1a}$) gives\n\\begin{equation}\na_3=\\sqrt{-\\frac{2(N_1+\\eta N_2)g_1g_2\\Gamma_{12} a_4}{\\pi}}.\n\\end{equation}\nThis expression evidently requires $a_4\\Gamma_{12}<0$. We however expect the vortex sheets to be present only in the $\\Gamma_{12}<0$ domain, thus we can be more specific on the condition, namely that $a_4>0$, or\n\\begin{equation}\n g_{12}>\\frac{\\eta g_1j_2+g_2j_1}{(\\eta j_1+j_2)}.\n\\end{equation} We point out that this critical number corresponds to $\\Gamma_{12}=0$ in the cases studied above.\nWith $a_3$ as above and $a_4$ given by the parameters of the system, the density profile is then fully accessible.\n\n\n\\subsection*{E. Analysis of defects}\n\nThe advantage of the Thomas Fermi analysis of $\\rho_T$ in the nonlinear sigma\nmodel is that it allows for analysis of\n defects as a perturbation calculation when $N_k g_k$, $N_k g_{3-k}$ and $N_k g_{12}$ are large,\n in the spirit of what has been done for a single condensate in \\cite{ad,ar,cd,wei}\n or for two condensates in \\cite{wei2}.\n We do not need\n to analyze specifically the peaks because they are taken into account in the $S_z$ formulation.\n This has\nbeen attacked in a more complicated fashion in the case of a single coreless vortex in \\cite{ktu}.\n We will develop our ideas in a later work but let us point out that the starting point\n is the energy (\\ref{sig}) and if we call $p_i$ and $q_i$\n the location of the vortex in each component, the main terms coming from the vortex contributions lead to:\n \\begin{itemize}\\item Each vortex core of each component providing a kinetic energy term\n proportional to $\\rho_T(0) \\log d_{hl}\/(2(1+\\eta))$ where $d_{hl}$ is the healing length, the characteristic\n size of the vortex core, here of order $1\/\\sqrt {N_kg_k}$.\n \\item the rotational term providing a term in $-c\\Omega (\\rho_T(0))^2$, where $c$\n is a numerical constant. The balance of these two terms allows us to compute the critical velocity\n for the nucleation of the first vortex.\n \\item the kinetic energy then yielding a term in $-\\log |p_i-p_j|-\\log|q_i-q_j|$.\n \\item the rotation term yielding a term in $\\Omega (|p_i|^2+|q_i|^2)$.\n \\item the interaction term providing a term $\\rho_T^2 S_z^2$ for the perturbation of\n $S_z$ close to a coreless vortex. The ansatz can be made in several ways\n leading to an interaction term in $e^{-|p_i-q_i|^2}$.\n \\end{itemize}\n This leads to a point energy of the type\n $$\\sum_i a (|p_i|^2+|q_i|^2+be^{-|p_i-q_i|^2} )-\\sum_{i,j}(\\log|p_i-p_j|+\\log|q_i-q_j|)$$ where\n $a$ and $b$ are related to the parameters of the problem.\n The ground state of such a point energy leads to a square lattice for a sufficient number of points\n and for some range of $a$ and $b$.\n\n\n\\section*{IV. Phase Diagram Under Conserved Total Particle Number}\n\n In the experiment of Hall\net al \\cite{hall} a single component BEC of $^{87}$Rb in the\n$|1$,$-1\\rangle$ state was initially created, before a transfer of\nany desired fraction of the atoms from this $|1$,$-1\\rangle$ state\nto the $|2$,$1\\rangle$ state created the two-component BEC. Thus the\nratio of particle numbers $N_1\/N_2$ is controllable experimentally,\nwith the constraint that $N_1+N_2$ is constant (in the case of the\nexperiment of \\cite{hall}, $N_1+N_2=5\\times10^5$). As such,\nexperimentally, it is possible to keep the individual particle\nnumbers constant (as in the normalisation condition (\\ref{norm})) or\nto keep the total particle number $N_1+N_2$ constant, allowing $N_1$\nand $N_2$ to vary (as in (\\ref{norm12})). We produce an\n$\\Omega-\\Gamma_{12}$ phase diagram for the parameters $g_1=0.003$,\n$g_2=0.006$ and $\\eta=\\xi=1$ using\nthe normalisation condition given by (\\ref{norm12}) with $N_1+N_2=2.1\\times 10^5$.\nNote that if $g_1=g_2$, $m_1=m_2$ and\n$\\omega_1=\\omega_2$ ($\\eta=\\xi=1$), then the phase diagram would be\nidentical to that of Fig. \\ref{sch_1} (i.e. the normalisation\ncondition in this case would not be important). The $\\Omega-\\Gamma_{12}$ phase diagram is presented in Fig. \\ref{sketchv}. There are three distinct regions (determined by the geometry of the ground state):\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.4]{pp_midgv.eps}\\\\\n\\includegraphics[scale=0.4]{pp_midg2v.eps}\n\\end{center}\n\\caption{(Color online) $\\Omega-\\Gamma_{12}$ phase diagram for parameters $g_1=0.003$ and $g_2=0.006$\nwith $\\eta=\\xi=1$ using the normalisation condition given by (\\ref{norm12}). (a) Numerical simulations where triangles indicate that the lattice in both components is triangular, squares that the lattice in both components is square and diamonds where no vortices have been nucleated. Filled triangles, squares and diamonds are where the two components are disk-shaped and coexist, empty triangles, squares and diamonds are where only component-1 exists; those with a dot in the centre represent the appearance of coreless vortices in component-2 and those without a dot in the centre represent the complete disappearance of component-2. (b) A schematic representation of the numerical simulations.\nThe solid lines indicate the boundary between different identified regions (determined by the geometry of the ground state) and the dashed lines the boundary between triangular and square lattices. The unit of rotation is $\\tilde{\\omega}$.}\\label{sketchv}\n\\end{figure}\n\n{\\it Region 1}. In the first region, both components are disk-shaped. As before, the coreless vortices can either form a triangular or a square lattice depending on the values of $\\Gamma_{12}$ and $\\Omega$. Figure \\ref{7posv} shows this case for $\\Gamma_{12}=0.7$ and $\\Omega=0.65$ (where a triangular lattice is present) and $\\Omega=0.9$ (where a square lattice is present). In region 1, both components have the same radii.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{7posv_1.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for component-1 (left column) and component-2 (right column). The parameters are $g_1=0.003$ and $g_2=0.006$ with $\\eta=\\xi=1$ and $\\Gamma_{12}=0.7$ (which gives $g_{12}=0.0023$) and normalisation taken over the total density (Eq. (\\ref{norm12})). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0.65$ and (b) $0.9$. There is a triangular lattice in (a) and a square lattice in (b). At these parameters the components are in region 1 of the phase diagram Fig. \\ref{sketchv}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{7posv}\n\\end{figure}\n\n{\\it Region 2}. In the second region, only component-1 exists except for isolated density peaks that exist in component-2. These isolated density peaks occur at the same location as the vortices do in component-1, and are thus identical to the isolated coreless vortices described in detail in Sect. IV. Figure \\ref{3posv} shows this case $\\Gamma_{12}=0.3$ with $\\Omega=0.5$ and $\\Omega=0.9$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{3posv_1.eps}\n\\end{center}\n\\caption{(Color online) A series of density plots for component-1 (left column) and component-2 (right column). The parameters are $g_1=0.003$ and $g_2=0.006$ with $\\eta=\\xi=1$ and $\\Gamma_{12}=0.3$ (which gives $g_{12}=0.0035$) and normalisation taken over the total density (Eq. (\\ref{norm12})). The angular velocity of rotation is $\\Omega$ and it takes the values (a) $0.5$ and (b) $0.9$. At these parameters the components are in region 2 of the phase diagram Fig. \\ref{sketchv}. Distance is measured in units of $r_0$ and density in\nunits of $r_0^{-2}$.}\n\\label{3posv}\n\\end{figure}\n\n{\\it Region 3}. In the third region, only component-1 exists (the density peaks in component-2 that were present in region 2 are no longer present). Furthermore, only triangular vortex lattices are observed in this effective one component condensate (in component-1).\n\n\nComputations similar to Section III hold, except that now we have to\n set $\\lambda=0$. This leads to $a_1\/a_3=-\\bar c_1\/(2\\bar c_2)$. We see that this ratio (which\n is $S_z(0)$), reaches 1 or -1 when $\\bar c_1=\\pm2\\bar c_2$. In our numerical case, this leads to $\\bar \\Gamma_{12}=0.5$.\n We see clearly 3 regimes: $\\Gamma_{12}>\\bar\\Gamma_{12}$, where the condensates\nare 2 disks, $\\Gamma_{12}<0$, which is phase separation, in which case $S_z=1$ is the preferred state and\n $0<\\Gamma_{12}<\\bar \\Gamma_{12}$, in which case the TF approximation leads to a computation of $\\rho_T$ with\n coreless vortex lattices and variations in $S_z$ which improve the energy and lead to this intermediate state, still to be studied in more detail.\n\n\\section*{V. Conclusion}\n\nWe have presented phase diagrams of rotating two component condensates in terms of\n the angular velocity $\\Omega$ and a nondimensionnalized parameter related to\n the coupling strengths\n$\\Gamma_{12}=1-g_{12}^2\/(g_1g_2)$. We have analyzed\nthe various ground states and topological defects and have found\n four sets characterized by the\n symmetry preserving\/symmetry breaking,\n coexistence or spatial separation of the components.\n When the geometry of the ground\nstates is either two disks (coexistence of components, region 1) or a disk\nand an annulus (spatial separation keeping some symmetry, region 3), the topological\ndefects are coreless vortex lattices (with possible stabilization\nof the square lattice) or giant skyrmions at the boundary\ninterface between the two components. In the complete symmetry breaking case, we have found vortex\n sheets and droplets. The difference of masses or coupling strengths between\n the components can induce very different patterns.\n\nWe have introduced an energy (\\ref{sig}) related to the total density and a pseudo spin vector.\n The minimization in a generalized Thomas Fermi approximation provides a lot of information\n on the ground states for general masses, trapping frequencies and coupling strengths.\n Some parts of the phase diagrams can be justified rigorously, both in the case\n $\\Gamma_{12}>0$, which had been studied before, but also in the case $\\Gamma_{12}<0$\n with generalized models. This formulation of the energy\nshould bring in the future more information on the defects.\n\n\n\n\\section*{Acknowledgments}\n\nThe authors wish to thank Thierry Jolicoeur for useful discussions\nthat took place for the duration of this work. They are very grateful to the referee for his careful reading of the manuscript and his appropriate comments. We acknowledge\nsupport from the French ministry Grant ANR-BLAN-0238, VoLQuan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}