diff --git a/Manuscript/EPAWTFT.bbl b/Manuscript/EPAWTFT.bbl index 717c28a..1f8be8a 100644 --- a/Manuscript/EPAWTFT.bbl +++ b/Manuscript/EPAWTFT.bbl @@ -6,7 +6,7 @@ %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled -\begin{thebibliography}{180}% +\begin{thebibliography}{179}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} @@ -50,15 +50,6 @@ \providecommand \BibitemShut [1]{\csname bibitem#1\endcsname}% \let\auto@bib@innerbib\@empty % -\bibitem [{\citenamefont {Dirac}\ and\ \citenamefont - {Fowler}(1929)}]{Dirac_1929}% - \BibitemOpen - \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.~A.~M.}\ - \bibnamefont {Dirac}}\ and\ \bibinfo {author} {\bibfnamefont {R.~H.}\ - \bibnamefont {Fowler}},\ }\href {\doibase 10.1098/rspa.1929.0094} {\bibfield - {journal} {\bibinfo {journal} {Proc. R. Soc. Lond. A}\ }\textbf {\bibinfo - {volume} {123}},\ \bibinfo {pages} {714} (\bibinfo {year} - {1929})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Rayleigh}(1894)}]{RayleighBook}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~W.~S.}\ @@ -1441,9 +1432,9 @@ {G.~Rauhut}}\ and\ \bibinfo {author} {\bibfnamefont {H.-J.}\ \bibnamefont {Werner}},\ }\href {\doibase 10.1002/(SICI)1096-987X(199808)19:11<1241::AID-JCC4>3.0.CO;2-K} {\bibfield - {journal} {\bibinfo {journal} {J. Comp. Chem.}\ }\textbf {\bibinfo {volume} - {19}},\ \bibinfo {pages} {1241} (\bibinfo {year} {1998})}\BibitemShut - {NoStop}% + {journal} {\bibinfo {journal} {J. Comput. Chem.}\ }\textbf {\bibinfo + {volume} {19}},\ \bibinfo {pages} {1241} (\bibinfo {year} + {1998})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Sch{\"u}tz}\ \emph {et~al.}(1999)\citenamefont {Sch{\"u}tz}, \citenamefont {Hetzer},\ and\ \citenamefont {Werner}}]{Schutz_1999}% diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 107f2d8..5f7f01c 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -297,8 +297,9 @@ unless otherwise stated, atomic units will be used throughout. \end{subfigure} \caption{% Exact energies for the Hubbard dimer ($U=4t$) as functions of $\lambda$ on the real axis (\subref{subfig:FCI_real}) and in the complex plane (\subref{subfig:FCI_cplx}). - Only the interacting closed-shell singlets are plotted in the complex plane, becoming degenerate at the EP (black dot). - The contour followed around the EP in order to interchange states is also represented. + Only the interacting closed-shell singlets are shown in the complex plane, becoming degenerate at the EP (black dot). + Following a contour around the EP (black solid) interchanges the states, while a second rotation (black dashed) + returns the states to their original energies. \label{fig:FCI}} \end{figure*} @@ -357,6 +358,8 @@ These $\lambda$ values correspond to so-called EPs and connect the ground and ex Crucially, the energy surface becomes non-analytic at $\lambda_{\text{EP}}$ and a square-root singularity forms with two branch cuts running along the imaginary axis from $\lambda_{\text{EP}}$ to $\pm \i \infty$ (see Fig.~\ref{subfig:FCI_cplx}). On the real $\lambda$ axis, these EPs lead to the singlet avoided crossing at $\lambda = \Re(\lambda_{\text{EP}})$. The ``shape'' of this avoided crossing is related to the magnitude of $\Im(\lambda_{\text{EP}})$, with smaller values giving a ``sharper'' interaction. +\hugh{In the limit $U/t \to 0$, the two EPs converge at $\lep \to 0$ to create a conical intersection with +a gradient discontinuity on the real axis.} Remarkably, the existence of these square-root singularities means that following a complex contour around an EP in the complex $\lambda$ plane will interconvert the closed-shell ground and excited states (see Fig.~\ref{subfig:FCI_cplx}). This behaviour can be seen by expanding the radicand in Eq.~\eqref{eq:singletE} as a Taylor series around $\lambda_{\text{EP}}$ to give @@ -1174,16 +1177,21 @@ divergences caused by the MP critical point. When a Hamiltonian is parametrised by a variable such as $\lambda$, the existence of abrupt changes in the eigenstates as a function of $\lambda$ indicate the presence of a zero-temperature quantum phase transition (QPT).% \cite{Heiss_1988,Heiss_2002,Borisov_2015,Sindelka_2017,CarrBook,Vojta_2003,SachdevBook,GilmoreBook} -Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point. -\hugh{The existence of an EP \emph{on} the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}} +Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis. +\hugh{When these points converge on the real axis, they effectively ``annihilate'' each other and no longer behave as EPs. +Instead, they form a ``critical point'' singularity that resembles a conical intersection, and +the convergence of a pair of complex-conjugate EPs on the real axis is therefore diagnostic of a QPT.\cite{Cejnar_2005, Cejnar_2007a}} + Since the MP critical point corresponds to a singularity on the real $\lambda$ axis, it can immediately be recognised as a QPT with respect to varying the perturbation parameter $\lambda$. However, a conventional QPT can only occur in the thermodynamic limit, which here is analogous to the complete basis set limit.\cite{Kais_2006} -The MP critical point and corresponding $\beta$ singularities in a finite basis must therefore be modelled by pairs of EPs -that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005} +The MP critical point and corresponding $\beta$ singularities in a finite basis must therefore be modelled by pairs of +complex-conjugate EPs that tend towards the real axis, exactly as described by Sergeev \etal\cite{Sergeev_2005} In contrast, $\alpha$ singularities correspond to large avoided crossings that are indicative of low-lying excited states which share the symmetry of the ground state,\cite{Goodson_2004} and are thus not manifestations of a QPT. +\hugh{Notably, since the exact MP critical point corresponds to the interaction between a bound state +and the continuum, its functional form is more complicated than a conical intersection and remains an open question.} %======================================= \subsection{Critical Points in the Hubbard Dimer} diff --git a/Manuscript/fig1b.pdf b/Manuscript/fig1b.pdf index c1d358a..2cc179b 100644 Binary files a/Manuscript/fig1b.pdf and b/Manuscript/fig1b.pdf differ diff --git a/Notebooks/Fig1bAntoine.nb b/Notebooks/Fig1bAntoine.nb index fb7cf5b..296d105 100644 --- a/Notebooks/Fig1bAntoine.nb +++ b/Notebooks/Fig1bAntoine.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 182642, 3205] -NotebookOptionsPosition[ 181503, 3177] -NotebookOutlinePosition[ 181899, 3193] -CellTagsIndexPosition[ 181856, 3190] +NotebookDataLength[ 250367, 4340] +NotebookOptionsPosition[ 249242, 4313] +NotebookOutlinePosition[ 249580, 4328] +CellTagsIndexPosition[ 249537, 4325] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -74,7 +74,7 @@ Cell[BoxData[{ CellChangeTimes->{{3.8146263256101303`*^9, 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