From 7d75809697c8cd7abd1d155f27bd5d2d7ed509a4 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 30 Nov 2020 22:43:30 +0100 Subject: [PATCH] comments for Hugh --- Manuscript/EPAWTFT.tex | 50 +++++++++++++++++++----------------------- 1 file changed, 23 insertions(+), 27 deletions(-) diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 9845ad9..af32d9a 100644 --- a/Manuscript/EPAWTFT.tex +++ b/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}).