Added discussion on UMP relationship to critical point
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Hugh Burton
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@ -117,12 +117,14 @@
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\newcommand{\bbR}{\mathbb{R}}
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\newcommand{\bbC}{\mathbb{C}}
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% Making life easier
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\newcommand{\Lup}{\mathcal{L}^{\uparrow}}
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\newcommand{\Ldown}{\mathcal{L}^{\downarrow}}
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\newcommand{\Lsi}{\mathcal{L}^{\sigma}}
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\newcommand{\Rup}{\mathcal{R}^{\uparrow}}
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\newcommand{\Rdown}{\mathcal{R}^{\downarrow}}
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\newcommand{\Rsi}{\mathcal{R}^{\sigma}}
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\newcommand{\vhf}{v_{\text{HF}}}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.}
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@ -840,6 +842,7 @@ gradient discontinuities or spurious minima.
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%==========================================%
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\subsection{Spin-Contamination in the Hubbard Dimer}
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\label{sec:spin_cont}
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%==========================================%
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%%% FIG 2 %%%
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@ -1362,83 +1365,75 @@ set representations of the MP critical point.\cite{Sergeev_2006}
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% Figure on the UMP critical point
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%------------------------------------------------------------------%
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\begin{figure*}[t]
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{ump_critical_point}
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\subcaption{\label{subfig:ump_cp}}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{ump_ep_to_cp}
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\subcaption{\label{subfig:ump_ep_to_cp}}
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\end{subfigure}
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\caption{%
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\hugh{Modelling the RMP critical point using the asymmetric Hubbard dimer.
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(\subref{subfig:rmp_cp}) Exact critical points with $t=0$ occur on the negative real $\lambda$ axis (dashed).
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(\subref{subfig:rmp_cp_surf}) Modelling a finite basis using $t=0.1$ yields complex-conjugate EPs close to the
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real axis, giving a sharp avoided crossing on the real axis (solid).
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}
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\label{fig:RMP_cp}}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{ump_critical_point}
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\subcaption{\label{subfig:ump_cp}}
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\end{subfigure}
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\begin{subfigure}{0.49\textwidth}
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\includegraphics[height=0.65\textwidth]{ump_ep_to_cp}
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\subcaption{\label{subfig:ump_ep_to_cp}}
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\end{subfigure}
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\caption{%
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\hugh{%
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The UMP ground-state EP becomes a critical point in the strong correlation limit (large $U/t$).
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(\subref{subfig:ump_cp}) As the $U/t$ increases, the avoided crossing on the real $\lambda$ axis
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becomes increasingly sharp.
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(\subref{subfig:ump_ep_to_cp}) The convergence of the EPs onto the real axis for $U/t \rightarrow \infty$
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mirrors the formation of the RMP critical point and other QPTs in the infinite basis set limit.
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}
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\label{fig:UMP_cp}}
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\end{figure*}
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%------------------------------------------------------------------%
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% RELATIONSHIP BETWEEN QPT AND UMP
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\hughDraft{%
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The ground-state EP observed in the UMP series also has a small imaginary part and falls close to the real axis.
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As the correlation strength increases, this EP moves closer to the real axis and closer to the radius of convergence at
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$\lambda = 1$. So can we understand this using the arguments related to the critical point?
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Closed-shell case:
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The work by Stillinger builds an argument around the HF potential $v_{\text{HF}}$ which, by itself,
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is repulsive and concentrated around the
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occupied orbitals. As $\lambda$ becomes increasingly positive along the real axis, this HF potential
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becomes attractive for $\lambda > 1$.
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However, the explicit two-electron becomes increasingly repulsive for larger positive $\lambda$,
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until eventually single electrons are successively expelled from the molecule.
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Effect of symmetry-breaking:
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Things become more subtle for the symmetry-broken case because we must now consider the $\alpha$ and $\beta$ HF potentials.
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When UHF symmetry breaking occurs in the Hubbard dimer, with the $\alpha$ electron localising on the left site,
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the $\alpha$ HF potential
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will then be a repulsive interaction localised around the $\beta$ electron, so on the right site.
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The same is true for the $\beta$ HF potential.
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Therefore, as $\lambda$ becomes greater than 1 and the HF potentials become attractive,
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there is a driving force for the $\alpha$ and $\beta$.
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electrons to swap sites. If both electrons swap at the same time, then we get a double excitation. Therefore, we expect an
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avoided crossing as $\lambda$ is increased beyond 1.
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The "sharpness" of this avoided crossing is controlled by the strength of the electron-electron repulsion... see below.
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\hugh{%
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In Section~\ref{sec:spin_cont} we showed that the slow convergence of the UMP series for the strongly correlated
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Hubbard dimer was due to a complex-conjguate pair of EPs just outside the radius of convergence.
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These EPs have positive real components and small imaginary components (Fig.~\ref{fig:UMP}), suggesting a potential
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connection to MP critical points and QPTs.
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For $\lambda>1$, the HF potential becomes an attractive component in Stillinger's
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Hamiltonian [Eq.~\eqref{eq:HamiltonianStillinger}], while the explicit electron-electron interaction
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becomes increasingly repulsive.
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Critical points along the positive real $\lambda$ axis for closed-shell molecules then represent
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points were the two-electron repulsion overcomes the attractive HF potential and an electron
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are successively expelled from the molecule.\cite{Sergeev_2006}
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}
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\hughDraft{%
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For small $U/t$, the HF potentials will be weak and the electrons will slowly delocalise over
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both sites as $\lambda$ increases. The UHF reference itself already has some degree of delocalisation over
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both sites as we are only just beyond the CFP. This leads to a
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"shallow" avoided crossing, and the corresponding EPs corresponding with have non-zero imaginary parts.
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At larger $U/t$, the HF potentials will becomes increasingly repulsive and the "swapping" driving force will become stronger.
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Furthermore, the strong on-site repulsion will make delocalisation over both sites increasingly unfavourable.
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We therefore see that the electrons swap sites almost instantaneously as $\lambda$ passes through 1, as this is the threshold point
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where the HF potential becomes attractive. This very rapid change in the nature of the state corresponds to a "sharp" avoided
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crossing, with EPs close to the real axis.
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Note that, although this appears to be an avoided crossing with the first-excited state,
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by the time we have reached $\lambda \approx 1$,
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we are beyond the excited-state avoided crossing and thus the first-excited state qualitatively corresponds to a double
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excitation from the reference. This matches our expectation of both electrons swapping sites.
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Okay we can't actually do $U/t = \infty$, but we can make it very large. With such a strong attractive HF potential and such strong
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on-site repulsion, the electrons swapping process will occur exactly at $\lambda=1$. In this limit, the change is so sudden that it
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now corresponds to a QPT in the perturbation approximation, and again the EPs fall on the real axis.
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\hugh{%
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Symmetry-breaking in the UMP reference creates different HF potentials for the spin-up and spin-down electrons.
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Consider the reference UHF solution where the spin-up and spin-down electrons are localised on the left and
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right sites respectively.
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The spin-up HF potential will then be a repulsive interaction from the spin-down electron
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density that is centred around the right site (and vice-versa).
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As $\lambda$ becomes greater than 1 and the HF potentials become attractive, there will be a sudden
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driving force for the electrons to swap sites.
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This swapping process can also be represented as a double excitation, and thus an avoided crossing will occur
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for $\lambda \geq 1$ (Fig.~\ref{subfig:ump_cp}).
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Note that, although this appears to be an avoided crossing between the ground and first-excited state,
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the earlier excited-state avoided crossing means that the first-excited state qualitatively
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represents the double excitation for $\lambda > 0.5.$
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}
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\hughDraft{%
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By applying the separation of the Hamiltionian proposed by Stillinger, we have been able to rationalise why the UMP ground-state
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occurs so close to radius of convergence and increasingly close to the real axis for large $U/t$. The key feature here is the fact that the
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HF potential is different for the $\alpha$ and $\beta$ electrons, and by extension the potential felt by
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an electron is not strictly localised around
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that electrons orbitals. This is completely different to the symmetric structure in the closed-shell case. Because of these different
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potentials, there can be a large driving force for spatial rearrangement occurring as soon as the HF potential becomes positive
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(at $\lambda=1$).
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This sudden change in electron distribution is made more extreme if the HF potential and electron repulsion dominate over the
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one-electron terms, which also corresponds to the regime of strong wave function symmetry breaking.
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We have therefore provided a physical motivation behind why this EP can behave like a QPT for strong symmetry breaking.
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This supports the conclusions in Antoine's report that the symmetry-broken EP behaves as a class $\beta$ singularity.
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% SHARPNESS AND QPT
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\hugh{%
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The ``sharpness'' of the avoided crossing is controlled by the correlation strength $U/t$.
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For small $U/t$, the HF potentials will be weak and the electrons will delocalise over the two sites,
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both in the UHF reference and as $\lambda$ increases.
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This delocalisataion dampens the electron swapping process and leads to a ``shallow'' avoided crossing
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that corresponds to EPs with non-zero imaginary components (solid lines in Fig.~\ref{subfig:ump_cp}).
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As $U/t$ becomes larger, the HF potentials become stronger and the on-site repulsion dominates the hopping
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term to make electron delocalisation less favourable.
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These effects create a stronger driving force for the electrons to swap sites until eventually this swapping
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occurs exactly at $\lambda = 1$.
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In this limit, the ground-state EPs approach the real axis (Fig.~\ref{subfig:ump_ep_to_cp}) and the avoided
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crossing creates a gradient discontinuity in the ground-state energy (dashed lines in Fig.~\ref{subfig:ump_cp}).
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We therefore find that, in the strong correlation limit, the symmetry-broken ground-state EP becomes becomes
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a new type of MP critical and represents a QPT in the perturbation approximation.
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Furthermore, this argument explains why the dominant UMP singularity lies so close, but always outside, the
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radius of convergence (see Fig.~\ref{fig:RadConv}).
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}
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%%====================================================
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%\subsection{The physics of quantum phase transitions}
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%%====================================================
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