morning modifications to MP critical points

Manuscript/EPAWTFT.tex

@@ -1238,7 +1238,7 @@ eigenstates as a function of $\lambda$ indicate the presence of a zero-temperatu
 Meanwhile, as an avoided crossing becomes increasingly sharp, the corresponding EPs move increasingly close to the real axis, eventually forming a critical point.
 The existence of an EP \emph{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 within the perturbation theory approximation.
+recognised as a QPT \hugh{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
@@ -1270,11 +1270,11 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are
 		\subcaption{\label{subfig:rmp_ep_to_cp}}
     \end{subfigure}
 	\caption{%
-		Modelling the RMP critical point using the asymmetric Hubbard dimer.
+		RMP critical point using the asymmetric Hubbard dimer with $\epsilon = 2.5 U$.
 		(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
 		(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
 		real axis, giving a sharp avoided crossing on the real axis (solid).
-		\titou{vertical axe label wrong in b.}
+		(\subref{subfig:rmp_ep_to_cp}) Convergence of the ground-state EP onto the real axis in the exact limit $t \rightarrow 0$.
 	\label{fig:RMP_cp}}
 \end{figure*}
 %------------------------------------------------------------------%
@@ -1284,19 +1284,20 @@ The simplified site basis of the Hubbard dimer makes explicilty modelling the io
 Instead, we can use an asymmetric version of the Hubbard dimer \cite{Carrascal_2015,Carrascal_2018} 
 where we consider one of the sites as a ``ghost atom'' that acts as a 
 destination for ionised electrons being originally localised on the other site.
-To mathematically model this scenario, in this asymmetric Hubbard dimer, we introduce a one-electron potential $-\epsilon$ on the left site to 
+To mathematically model this scenario in this asymmetric Hubbard dimer, we introduce a one-electron potential $-\epsilon$ on the left site to 
 represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$.
 The reference Slater determinant for a doubly-occupied atom can be represented using the RHF
-orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$, which corresponds to strictly localise the two electrons on the left site.
+orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$, 
+which corresponds to strictly localising the two electrons on the left site.
 %and energy
 %\begin{equation}
 %   E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
 %\end{equation}
-With this representation, the parametrised \hugh{atomic} RMP Hamiltonian becomes
+With this representation, the parametrised \hugh{asymmetric} RMP Hamiltonian becomes
 \begin{widetext}
 \begin{equation}
 \label{eq:H_RMP}
-\hugh{\bH_\text{atom}\qty(\lambda)} = 
+\hugh{\bH_\text{asym}\qty(\lambda)} = 
 \begin{pmatrix}
     2(U-\epsilon) - \lambda U &	-\lambda t				 &	-\lambda t	            &	0	        \\
     -\lambda t				  &	(U-\epsilon) - \lambda U &	0		                &	-\lambda t	\\
@@ -1338,7 +1339,7 @@ Molecules containing these atoms are therefore often class $\beta$ systems with
 a divergent RMP series due to the MP critical point. \cite{Goodson_2004,Sergeev_2006} 
 
 % EXACT VERSUS APPROXIMATE
-The critical point in the exact case $t=0$ lies on the \titou{negative} real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines), 
+The critical point in the exact case $t=0$ lies on the negative real $\lambda$ axis (Fig.~\ref{subfig:rmp_cp}: dashed lines), 
 mirroring the behaviour of a quantum phase transition.\cite{Kais_2006}
 However, in practical calculations performed with a finite basis set, the critical point is modelled as a cluster
 of branch points close to the real axis.
@@ -1346,45 +1347,53 @@ The use of a finite basis can be modelled in the asymmetric dimer by making the
 idealised destination for the ionised electrons with a non-zero (yet small) hopping term $t$.
 Taking the value $t=0.1$ (Fig.~\ref{subfig:rmp_cp}: solid lines), the critical point becomes a
 sharp avoided crossing with a complex-conjugate pair of EPs close to the real axis (Fig.~\ref{subfig:rmp_cp_surf}).
-In the limit $t \to 0$, these EPs approach the real axis, mirroring Sergeev's discussion on finite basis
+In the limit $t \to 0$, these EPs approach the real axis (Fig.~\ref{subfig:rmp_ep_to_cp}),
+mirroring Sergeev's discussion on finite basis
 set representations of the MP critical point.\cite{Sergeev_2006}
 
 %------------------------------------------------------------------%
 % Figure on the UMP critical point
 %------------------------------------------------------------------%
 \begin{figure*}[t]
-\begin{subfigure}{0.49\textwidth}
-    \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point}
-\subcaption{\label{subfig:ump_cp}}
-\end{subfigure}
-\begin{subfigure}{0.49\textwidth}
-\includegraphics[height=0.65\textwidth]{ump_ep_to_cp}
-\subcaption{\label{subfig:ump_ep_to_cp}}
-\end{subfigure}
+	\begin{subfigure}{0.32\textwidth}
+        \includegraphics[height=0.75\textwidth,trim={0pt 5pt -10pt 15pt},clip]{ump_cp}	
+		\subcaption{\label{subfig:ump_cp}}
+    \end{subfigure}
+    %
+    \begin{subfigure}{0.32\textwidth}
+	\includegraphics[height=0.75\textwidth]{ump_cp_surf}
+		\subcaption{\label{subfig:ump_cp_surf}}
+    \end{subfigure}
+    %
+    \begin{subfigure}{0.32\textwidth}
+	\includegraphics[height=0.75\textwidth]{ump_ep_to_cp}	
+		\subcaption{\label{subfig:ump_ep_to_cp}}
+    \end{subfigure}
+%    \includegraphics[height=0.65\textwidth,trim={0pt 5pt 0pt 15pt}, clip]{ump_critical_point}
 \caption{%
     The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$).
     (\subref{subfig:ump_cp}) As $U/t$ increases, the avoided crossing on the real $\lambda$ axis
     becomes increasingly sharp.
-    (\subref{subfig:ump_ep_to_cp}) The convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$
-    mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit.
+    (\subref{subfig:ump_cp_surf}) Complex energy surfaces for $U = 5t$.
+    (\subref{subfig:ump_ep_to_cp}) Convergence of the EPs at $\lep$ onto the real axis for $U/t \to \infty$.
+    %mirrors the formation of the RMP critical point and other QPTs in the complete basis set limit.
 \label{fig:UMP_cp}}
 
 \end{figure*}
 %------------------------------------------------------------------%
 
 % RELATIONSHIP BETWEEN QPT AND UMP
-\titou{Returning to the symmetric Hubbard dimer?}
-In Sec.~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated
-Hubbard dimer was due to a complex-conjugate pair of EPs just outside the radius of convergence.
+Returning to the symmetric Hubbard dimer, we showed in Sec.~\ref{sec:spin_cont} that the slow
+convergence of the strongly correlated UMP series
+was due to a complex-conjugate pair of EPs just outside the radius of convergence.
 These EPs have positive real components and small imaginary components (see Fig.~\ref{fig:UMP}), suggesting a potential 
 connection to MP critical points and QPTs (see Sec.~\ref{sec:MP_critical_point}).
 For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's 
 Hamiltonian displayed in Eq.~\eqref{eq:HamiltonianStillinger}, while the explicit electron-electron interaction
 becomes increasingly repulsive.
-\titou{Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent 
+Closed--shell critical points along the positive real $\lambda$ axis then represent 
 points where the two-electron repulsion overcomes the attractive HF potential 
-and a single electron dissociates from the molecule.\cite{Sergeev_2006}}
-\titou{T2: I'd like to discuss that with you.}
+and a single electron dissociates from the molecule (see Ref.~\onlinecite{Sergeev_2006})
 
 In contrast, symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
 Consider one of the two reference UHF solutions where the spin-up and spin-down electrons are localised on the left and right sites respectively. 
@@ -1394,10 +1403,9 @@ As $\lambda$ becomes greater than 1 and the HF potentials become attractive, the
 driving force for the electrons to swap sites.
 This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
 for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
-\titou{Note that, although this appears to be an avoided crossing between the ground and first-excited state, 
-the earlier excited-state avoided crossing means that the first-excited state qualitatively 
-represents the double excitation for $\lambda > 0.5.$}
-\titou{T2: I find it hard to understand. I'd like to discuss this as well.}
+While this appears to be an avoided crossing between the ground and first-excited state, 
+the presence of an earlier excited-state avoided crossing means that the first-excited state qualitatively 
+represents the reference double excitation for $\lambda > 0.5.$
 
 % SHARPNESS AND QPT
 The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
@@ -1407,15 +1415,13 @@ This delocalisation dampens the electron swapping process and leads to a ``shall
 that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
 As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
 term to make electron delocalisation less favourable.
-\titou{In other words, the electron localises on each site forming a so-called Wigner crystal.
-T2: is it worth saying again?}
+In other words, the electrons localise on individual sites to form a Wigner crystal.
 These effects create a stronger driving force for the electrons to swap sites until eventually this swapping
 occurs exactly at $\lambda = 1$.
 In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided 
 crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
 We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes
-a new type of MP critical point \titou{and represents a QPT in the perturbation approximation}.
-\titou{T2: what do you mean by this?} 
+a new type of MP critical point and \hugh{represents a QPT as the perturbation parameter $\lambda$ is varied.}
 Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the 
 radius of convergence (see Fig.~\ref{fig:RadConv}).