comments for Hugh

Manuscript/EPAWTFT.tex

@@ -1268,29 +1268,20 @@ states which share the symmetry of the ground state,\cite{Goodson_2004} and are
 		(\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.}
 	\label{fig:RMP_cp}}
 \end{figure*}
 %------------------------------------------------------------------%
 
 % INTRODUCING THE MODEL
 The simplified site basis of the Hubbard dimer makes explicilty modelling the ionisation continuum impossible.
-Instead, we can use an asymmetric Hubbard dimer \cite{Carrascal_2015,Carrascal_2018} to consider one site as a ``ghost atom'' that acts as a 
-destination for ionised electrons.
-In this asymmetric model, we introduce a one-electron potential $-\epsilon$ on the left site to 
+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 
 represent the attraction between the electrons and the model ``atomic'' nucleus [see Eq.~\eqref{eq:H_FCI}], where we define $\epsilon > 0$.
-%The exact Hamiltonian [Eq.~\eqref{eq:H_FCI}] then becomes
-%\begin{equation}
-%\label{eq:H_FCI_Asymm}
-%\bH = 
-%\begin{pmatrix}
-%	U-2\epsilon & -t        & -t        &  0 \\
-%   -t           & -\epsilon &  0        & -t \\
-%   -t           &  0        & -\epsilon & -t \\
-%    0           & -t        & -t        &  U \\
-%\end{pmatrix}.
-%\end{equation}
 The reference Slater determinant for a doubly-occupied atom can be represented using the RHF
-orbitals [see Eq.~\eqref{eq:RHF_orbs}] with $\theta_{\alpha}^{\text{RHF}} = \theta_{\beta}^{\text{RHF}} = 0$.
+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.
 %and energy
 %\begin{equation}
 %   E_\text{HF}(0, 0) = \frac{1}{2} (2 U - 4 \epsilon).
@@ -1299,7 +1290,7 @@ With this representation, the parametrised RMP Hamiltonian becomes
 \begin{widetext}
 \begin{equation}
 \label{eq:H_RMP}
-\bH_\text{RMP}\qty(\lambda) = 
+\Tilde{\bH}_\text{RMP}\qty(\lambda) = 
 \begin{pmatrix}
     2(U-\epsilon) - \lambda U &	-\lambda t				 &	-\lambda t	            &	0	        \\
     -\lambda t				  &	(U-\epsilon) - \lambda U &	0		                &	-\lambda t	\\
@@ -1308,17 +1299,16 @@ With this representation, the parametrised RMP Hamiltonian becomes
 \end{pmatrix}.
 \end{equation}
 \end{widetext}
-\titou{Shall we change the symbol $\bH_\text{RMP}\qty(\lambda)$ to avoid confusion? Maybe $\Tilde{\bH}_\text{RMP}\qty(\lambda)$?}
 
 % DERIVING BEHAVIOUR OF THE CRITICAL SITE
-For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish with $t=0$.
+For the ghost site to perfectly represent ionised electrons, the hopping term between the two sites must vanish (\ie, $t=0$).
 This limit corresponds to the dissociative regime in the asymmetric Hubbard dimer as discussed in Ref.~\onlinecite{Carrascal_2018}, 
 and the RMP energies become
 \begin{subequations}
 \begin{align}
-    E_{-} &= 2U - 2 \epsilon - U \lambda,
+    E_{-} &= 2(U - \epsilon) - \lambda U,
     \\
-    E_{\text{S}} &= U - \epsilon - U \lambda,
+    E_{\text{S}} &= (U - \epsilon) - \lambda U,
     \\ 
     E_{+} &= U \lambda,
 \end{align}
@@ -1376,19 +1366,21 @@ set representations of the MP critical point.\cite{Sergeev_2006}
 %------------------------------------------------------------------%
 
 % 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.
-These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential 
+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.
-Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent 
+\titou{Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent 
 points where the two-electron repulsion overcomes the attractive HF potential and an electron
-are successively expelled from the molecule.\cite{Sergeev_2006}
+are successively expelled from the molecule.\cite{Sergeev_2006}}
+\titou{T2: I'd like to discuss that with you.}
 
 Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
-Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
+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.  
 The spin-up HF potential will then be a repulsive interaction from the spin-down electron 
 density that is centred around the right site (and vice-versa).
@@ -1396,9 +1388,10 @@ 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}).
-Note that, although this appears to be an avoided crossing between the ground and first-excited state, 
+\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 $\lambda > 0.5.$
+represents the double excitation for $\lambda > 0.5.$}
+\titou{T2: I find it hard to understand. I'd like to discuss this as well.}
 
 % SHARPNESS AND QPT
 The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
@@ -1408,12 +1401,15 @@ 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?}
 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 and represents a QPT in the perturbation approximation. 
+a new type of MP critical point \titou{and represents a QPT in the perturbation approximation}.
+\titou{T2: what do you mean by this?} 
 Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the 
 radius of convergence (see Fig.~\ref{fig:RadConv}).