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StyleBox[" \[PlusMinus] i 4t / U\nCFP: 1 = 2t / U\n\nSo we see \ that the magnitudes of each exceptional point are all closely related.\nIs it \ possible that each of these exceptional point is dictated by the fundamental \ exceptional point in the FCI energy?\nOr is this an artefact of our \ incredible simple model?", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.814264982175564*^9, 3.8142651334405727`*^9}, { 3.814265279765139*^9, 3.814265402285018*^9}, {3.814266218490487*^9, 3.814266218614222*^9}, {3.814380174186116*^9, 3.8143802102073708`*^9}},ExpressionUUID->"07e56b25-3c15-4626-9e46-\ 6c77981db19f"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["UMP", "Section", CellChangeTimes->{{3.814266241518415*^9, 3.8142662431040993`*^9}, { 3.814271804132291*^9, 3.814271804468823*^9}},ExpressionUUID->"3854b603-a008-4f0e-b49e-\ 08c3ef01c6e9"], Cell[TextData[StyleBox["We can now consider the MP expansion from the \ symmetry-broken UHF reference state.\nThe optimal rotation 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3.8143800276140223`*^9}},ExpressionUUID->"88abadb0-1d05-421c-b4b3-\ 14793c0012a1"], Cell[TextData[StyleBox["In this section, we plot the Riemann surfaces for the \ UMP expansions at the two U/t ratios considered above.", FontColor->RGBColor[1, 0, 0]]], "Text", CellChangeTimes->{{3.8142684330360317`*^9, 3.814268452731592*^9}, { 3.814268496797098*^9, 3.814268577899151*^9}, {3.814380016478723*^9, 3.814380060553014*^9}, {3.814380118556081*^9, 3.814380140302573*^9}},ExpressionUUID->"0a1530fa-55e8-44ea-8d84-\ 88d4b1fd62fa"], Cell[CellGroupData[{ Cell["U = 7t", "Subsubsection", CellChangeTimes->{{3.814268543123382*^9, 3.814268547049837*^9}, { 3.814268615042334*^9, 3.8142686417938538`*^9}},ExpressionUUID->"7622ddd0-a3ff-4d4b-b73e-\ 3215ff17337a"], Cell[TextData[StyleBox["Here we are well towards the strong correlation \ regime, where it is known that UMP will be slowly convergent but RMP \ diverges.\nWe see a single exceptional point on the ground-state surface \ which falls just outside (maybe on?) the radius of convergence. \nAn \ exceptional point this close to the radius of convergence gives an \ increasingly slow convergence of the UMP series, but it will converge \ evenutally!\n\nNote that the ground-state energy is remarkably flat since the \ UHF energy is a pretty good estimate. \nMost of the UMP expansion is actually \ correcting the spin-contamination in the wave function.\n\nOn the other hand, \ there is an exceptional point on the excited energy surface that is well \ within the radius of convergence.\nWe can therefore say that the use of a \ symmetry-broken UHF wave function can retain a convergent ground-state \ perturbation series\nat the expense of a divergent excited-state perturbation \ series. 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spatially separating the two \ electrons.\nAt the fixed point \[Lambda]=1, the energies of the states will \ always be the FCI energy, and this is independent of the reference orbitals.\n\ \nIf the ground-state is a good FCI estimate, it will cross from \[Lambda]=0 \ to 1 in a \[OpenCurlyDoubleQuote]flat\[CloseCurlyDoubleQuote] fashion. This \ makes it far less likely to interact with \na higher energy state, so it is \ less likely to encounter an exceptional point. Equally, the first excited \ state will tend towards the \nexact first-excited singlet FCI energy. It is \ possible for these states to cross on the real-\[Lambda] line, but any \ crossing must occur\nbeyond \[Lambda]=1 because otherwise they would not \ connect to the correct fixed point.\n\n", FontColor->RGBColor[1, 0, 0]], StyleBox["So what happens to the RMP exceptional point?", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\n\nIn the RMP case, the convergence is controlled by an \ exceptional point between the ground- and doubly-excited RHF configurations.\n\ These two states interact strongly in the \[Lambda] expansion in the large \ U/t limit because the ground RHF state is heavily \ncontaminated by the 3rd \ FCI singlet state. It is this strong interaction that causes the exceptional \ point, and ultimately\nthe divergence. \n\nIn the UMP case, this \ \[OpenCurlyDoubleQuote]energetic\[CloseCurlyDoubleQuote] contamination from \ the 3rd FCI state is reduced (removed?) by replacing it with triplet \ contamination.\nThe strong interaction is then shifted from the ground \ configuration to the \ \[OpenCurlyDoubleQuote]open-shell\[CloseCurlyDoubleQuote] configurations in \ the UHF framework, \nsince the 3rd FCI singlet state contributions are \ rotated into the virtual orbitals.\nThis explains why we see the severe \ excited-state exceptional point in the UMP Riemann surface.\nIn a way, we can \ think of this as the \ \[OpenCurlyDoubleQuote]open-shell\[CloseCurlyDoubleQuote] configurations \ \[OpenCurlyDoubleQuote]shielding\[CloseCurlyDoubleQuote] the ground-state \ from the strong interaction (and\nexceptional point), hence we have more \ robust convergence in ground-state, but less robust convergence in \ excited-state.", FontColor->RGBColor[1, 0, 0]] }], "Text", CellChangeTimes->{{3.8142698778026943`*^9, 3.814270002740933*^9}, { 3.8142701039070168`*^9, 3.8142701163949423`*^9}, {3.81427015808629*^9, 3.814270160550354*^9}, {3.8142702008455763`*^9, 3.814270503562158*^9}, { 3.814270542836125*^9, 3.814270975747213*^9}},ExpressionUUID->"9b51c24b-654d-48dc-878d-\ 1415e41df166"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[TextData[StyleBox["Summary", "Section"]], "Section", CellChangeTimes->{{3.8142698234669323`*^9, 3.8142698246701813`*^9}},ExpressionUUID->"555b6565-ed9c-41db-8747-\ 3eab70a65674"], Cell[TextData[{ StyleBox["In this notebook, we have investigated the convergence of the RMP \ and UMP series in the Hubbard dimer through visualisation \non Riemann \ surfaces. \n\n", FontColor->RGBColor[1, 0, 0]], StyleBox["Our main conclusions:", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\n1) Exact FCI problem has intrinsic exceptional points in the U/t \ complex plane, corresponding to an avoided crossing on the real axis.\n2) \ When RMP converges, it does so fast. Both the ground- and excited-states have \ the same radius of convergence as they are controlled\n by the same \ exceptional point. However, this exceptional point is prone to moving within \ the radius of convergence, causing divergence.\n3) Allowing the orbitals to \ break symmetry in UMP gives a more robust convergence for ground-state by \ removing energetic contamination\n in the reference at the expense of \ adding triplet contamination. This convergence is at the expense of a \ massively divergent excited-state\n perturbation series.\n \n", FontColor->RGBColor[1, 0, 0]], StyleBox["Moving Forwards...", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["\nOne of the more significant ideas could be the effect of orbital \ optimisation on the radius of convergence. In the UHF case we saw that \n\ ground-state orbital optimisation massively improved the robustness of the \ ground-state MP expansion. ", FontColor->RGBColor[1, 0, 0]], StyleBox["Would excited-state\norbital optimisation would do the same for \ excited-state MP expansions at the expense of ground-state convergence? \n", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], StyleBox["If so, this would provide reassurance in the computation of \ dynamic correlation for state-specific excited states, and would suggest\n\ that we should expect reliable behaviour in orbital-optimised coupled-cluster \ expansions.", FontColor->RGBColor[1, 0, 0]], "\n" }], "Text", CellChangeTimes->{{3.814269827673455*^9, 3.814269857755002*^9}, { 3.814271151755376*^9, 3.814271392564*^9}, {3.814271434020362*^9, 3.8142716695547*^9}},ExpressionUUID->"c8622b61-f1b5-4100-9c90-7559df35a50a"] }, Closed]] }, WindowSize->{1440, 851}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, FrontEndVersion->"12.0 for Mac OS X x86 (64-bit) (April 8, 2019)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) 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