(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 12.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 3756398, 64944] NotebookOptionsPosition[ 3744844, 64750] NotebookOutlinePosition[ 3745183, 64765] CellTagsIndexPosition[ 3745140, 64762] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Hubbard Model in Complex Plane", "Section", CellChangeTimes->{{3.814254200255485*^9, 3.814254207621756*^9}},ExpressionUUID->"71fa1d4e-529b-47cd-949e-\ c419f26d916b"], Cell[TextData[{ StyleBox["Contributors: ", FontWeight->"Bold"], "Hugh G. A. 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(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}, {-536, Automatic}}, 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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