(* 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[ 29779505, 500705] NotebookOptionsPosition[ 29753689, 500284] NotebookOutlinePosition[ 29754217, 500305] CellTagsIndexPosition[ 29754134, 500300] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[TextData[StyleBox["Hubbard Model in the Complex Plane", "Title"]], \ "Section", CellChangeTimes->{{3.814254200255485*^9, 3.814254207621756*^9}, { 3.815632353710492*^9, 3.8156323539889936`*^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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3.814269873696*^9}},ExpressionUUID->"682f5d84-4017-4e58-bb21-1f874ccd820f"], Cell[TextData[{ StyleBox["This important question can be paraphrased as \ \[OpenCurlyDoubleQuote]does the ground-state EP ever fall within the radius \ of convergence?\[CloseCurlyDoubleQuote]. \nTo answer this, we can consider \ the fixed FCI point on the corresponding Riemann surface\n\nFirstly the \ nature of the symmetry-breaking in UHF will always give an energy that is an \ increasingly accurate estimate of the exact energy. \nThis is because UHF can \ capture the strong correlation effects by 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"], Cell[BoxData[""], "Input",ExpressionUUID->"8c5e9104-07c8-4b0b-813c-b366f0d1a1a6"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["6) MP Critical Point ", "Section", CellChangeTimes->{{3.815576093982725*^9, 3.815576100894288*^9}, { 3.815595706655005*^9, 3.81559570686423*^9}, {3.81563242847005*^9, 3.815632428584237*^9}},ExpressionUUID->"9c3afb5c-1e35-4825-b17a-\ e135a14e0764"], Cell[TextData[{ "In this section, we will consider how the ideas behind the critical point \ are manifested in the Hubbard dimer.\n\n", StyleBox["1) Auto-ionisation in the RMP case:", FontWeight->"Bold"], "\nWe can model the mass RMP auto-ionisation event (negative \[Lambda]) \ using an asymmetric Hubbard dimer, since the site with lower\nenergy behaves \ as a ghost atom for the ionised electrons.\n\n", StyleBox["2) Relationship to symmetry-breaking in UMP case:\n", FontWeight->"Bold"], "The physical driving forces behind the MP critical point also allows us to \ rationalise why we get an exceptional point so\nclose to the radius of \ convergence in the UMP case, behaving as a class \[Alpha] singularity." }], "Text", CellChangeTimes->{{3.8155761035431232`*^9, 3.815576303053629*^9}, { 3.8156315852434*^9, 3.81563160940511*^9}},ExpressionUUID->"04b0d44a-4577-4793-9a56-\ e8536fb3d90b"], Cell[CellGroupData[{ Cell["RMP Critical Point", "Subsection", CellChangeTimes->{{3.815576317214039*^9, 3.815576336862337*^9}, 3.815578636442253*^9},ExpressionUUID->"c40150dd-e149-4f12-a4d0-\ e4c531089cf5"], Cell[CellGroupData[{ Cell["Establishing a model", "Subsubsection", CellChangeTimes->{{3.815592459994959*^9, 3.815592462467548*^9}},ExpressionUUID->"f11320a2-702e-4503-aa20-\ 5735decc161f"], Cell[TextData[{ "To model the auto-ionisation in the Hubbard dimer, we need to turn one of \ the sites into a ghost atom that will act as\na destination for ionised \ electrons. We do this by considering an asymmetric Hubbard dimer where the \ \[OpenCurlyDoubleQuote]atomic\[CloseCurlyDoubleQuote] site\nhas a negative \ diagonal term in the one-electron Hamiltonian, representing the nuclear \ attraction. This term is already\nencoded in the ", ButtonBox["Initialisation", BaseStyle->"Hyperlink", ButtonData->"Initialisation"], " section such that the core Hamiltonian is" }], "Text", CellChangeTimes->{{3.8155763587548847`*^9, 3.815576480537758*^9}, { 3.815592484384218*^9, 3.8155925190831547`*^9}, {3.815631720767021*^9, 3.81563172076747*^9}},ExpressionUUID->"69c30843-e141-4480-b777-\ ce904f036bee"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Hc", "//", "MatrixForm"}]], "Input", CellChangeTimes->{{3.815576483178186*^9, 3.81557648828689*^9}}, CellLabel-> "In[255]:=",ExpressionUUID->"001cd2bb-86c2-48a5-a6d7-90fa5209e57f"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ { RowBox[{"-", "\[Delta]\[Epsilon]"}], RowBox[{"-", "t"}]}, { RowBox[{"-", "t"}], "0"} }, GridBoxAlignment->{"Columns" -> {{Center}}, "Rows" -> {{Baseline}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.7]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, 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True}, LabelStyle -> {FontFamily -> "Times"}, LegendLayout -> "Column"]], { 0.2, 0.85}, Identity]]& ], AutoDelete->True, Editable->True, SelectWithContents->False, Selectable->True]], "Output", CellChangeTimes->{ 3.815719015296961*^9, {3.8157190487731857`*^9, 3.8157192264845543`*^9}, 3.815719424821185*^9, 3.815719455307868*^9, 3.815719500179942*^9, 3.815719554842126*^9, 3.815719585464944*^9, 3.815719622347924*^9, { 3.815719657659305*^9, 3.815719679150237*^9}}, CellLabel->"Out[37]=",ExpressionUUID->"861d191d-1e9f-4e4e-ad30-789f20f9ae47"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Summary", "Subsubsection", CellChangeTimes->{{3.815579051714787*^9, 3.815579052978105*^9}},ExpressionUUID->"45a7f018-32e4-49ba-a17a-\ 3f8ae3dcf613"], Cell[TextData[{ "(1) We have shown that the RMP critical point on the negative axis can be \ modelled by considering the asymmetric Hubbard dimer with t = 0.\n\n(2) The \ position of the critical point is controlled by the ratio \[Delta]\[Epsilon] \ / U. If the magnitude of the ratio becomes greater than one, then the series \ diverges.\n We can interpret large U as strong electron repulsion \ effects in electron dense molecules, such as ", Cell[BoxData[ FormBox[ SuperscriptBox["F", "-"], TraditionalForm]],ExpressionUUID-> "7c32ed58-ea5b-4b4e-9b3a-86f80ff8c2e4"], ". A small \[Delta]\[Epsilon] is also likely to correspond to \n \ strong nuclear screening by the core and valence electrons. Both of these \ factors are common in atoms on the right-hand-side of periodic table, \n \ eg. F, O, Ne, etc, as well as negatively-charged species, and so we recover \ the class \[Beta] system classification.\n \n(3) The effect of a finite \ basis is modelled by a non-zero value of t, with the corresponding EPs \ tending towards the real axis as t \[Rule] 0.\n\nThis example is therefore \ both an interesting illustration of the critical point, but also reinforces \ that the Hubbard dimer as a useful model\nfor understanding the behaviour of \ perturbation theory." }], "Text", CellChangeTimes->{{3.8155790557363462`*^9, 3.815579282342352*^9}, { 3.81559315161896*^9, 3.815593452578453*^9}, {3.815631978387048*^9, 3.8156320497735367`*^9}},ExpressionUUID->"51bfad00-5d13-4354-8b0f-\ 04f9429a3a3a"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["UMP Symmetry-Breaking.", "Subsection", CellChangeTimes->{{3.815576328659938*^9, 3.81557633547749*^9}, { 3.815632452400516*^9, 3.815632454167923*^9}},ExpressionUUID->"268b8332-fe42-41c6-b19e-\ 73e3ea0061a2"], Cell[CellGroupData[{ Cell["Establishing the model", "Subsubsection", CellChangeTimes->{{3.815594190960177*^9, 3.815594214596055*^9}},ExpressionUUID->"0711f2a5-af44-471c-9d75-\ b33afbef5ac9"], Cell[TextData[{ "The ground-state EP observed in the UMP series also has a small imaginary \ part and falls close to the real axis. \nAs the correlation strength \ increases, this EP moves closer to the real axis and closer to the radius of \ convergence at \n\[Lambda] = 1. So can we understand this using the arguments \ related to the critical point?\n\n", StyleBox["Closed-shell case:", FontWeight->"Bold"], "\nThe work by Stillinger builds an argument around the HF potential ", Cell[BoxData[ FormBox[ SubscriptBox["v", RowBox[{"HF", " "}]], TraditionalForm]],ExpressionUUID-> "c0d53961-e5a3-47d7-9ab2-cafe5ef0533a"], "which, by itself, is repulsive and concentrated around the \noccupied \ orbitals. As \[Lambda] becomes increasingly positive along the real axis, \ this HF potential becomes attractive for \[Lambda] > 1.\nHowever, the \ explicit two-electron becomes increasingly repulsive for larger positive \ \[Lambda], until eventually single electrons are\nsuccessively expelled from \ the molecule.\n\n", StyleBox["Effect of symmetry-breaking:\n", FontWeight->"Bold"], "Things become more subtle for the symmetry-broken case because we must now \ consider the \[Alpha] and \[Beta] HF potentials.\nWhen UHF symmetry breaking \ occurs in the Hubbard dimer, with the \[Alpha] electron localising on the \ left site, the \[Alpha] HF potential\nwill then be a repulsive interaction \ localised around the \[Beta] electron, so on the right site. The same is true \ for the \[Beta] HF potential.\nTherefore, as \[Lambda] becomes greater than 1 \ and the HF potentials become attractive, there is a driving force for the \ \[Alpha] and \[Beta] \nelectrons to swap sites. If both electrons swap at the \ same time, then we get a double excitation. Therefore, we expect an \navoided \ crossing as \[Lambda] is increased beyond 1. \n\nThe \ \[OpenCurlyDoubleQuote]sharpness\[CloseCurlyDoubleQuote] of this avoided \ crossing is controlled by the strength of the electron-electron repulsion... \ see below." }], "Text", CellChangeTimes->{{3.8155792905298233`*^9, 3.815579294892961*^9}, 3.8155919483729067`*^9, {3.8155935100837927`*^9, 3.815593620497604*^9}, { 3.8155937026993217`*^9, 3.815593854410417*^9}, {3.815593887617949*^9, 3.815594185471031*^9}, {3.8155942211450863`*^9, 3.815594413870212*^9}, { 3.815594479599547*^9, 3.8155945782441387`*^9}, 3.815594794044093*^9, 3.815594885133697*^9, {3.81563208679291*^9, 3.815632132522955*^9}},ExpressionUUID->"9a2b2ecf-b7ba-4757-b65a-\ 1fbee2be0810"] }, Open ]], Cell[CellGroupData[{ Cell["UMP Energies", "Subsubsection", CellChangeTimes->{{3.8155946662513313`*^9, 3.815594672900529*^9}},ExpressionUUID->"1201fa36-0b33-4107-ad70-\ 618c08d2803a"], Cell["Borrowed from above...", "Text", CellChangeTimes->{{3.815594678525896*^9, 3.815594682272575*^9}},ExpressionUUID->"0977c876-e080-4a08-abb0-\ 07a03779af24"], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Epsilon]UMP", "1"], "[", RowBox[{"U_", ",", "t_", ",", "\[Lambda]_"}], "]"}], "=", FractionBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "-", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", SuperscriptBox["\[Lambda]", "2"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", " ", SuperscriptBox["U", "4"]}], "-", RowBox[{"8", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}], "-", RowBox[{"2", " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "+", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", SuperscriptBox["\[Lambda]", "2"]}]}], ")"}], " ", "#1"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", SuperscriptBox["U", "2"]}], "+", RowBox[{"2", " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}]}], ")"}], " ", SuperscriptBox["#1", "2"]}], "+", SuperscriptBox["#1", "3"]}], "&"}], ",", "1"}], "]"}], "U"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Epsilon]UMP", "2"], "[", RowBox[{"U_", ",", "t_", ",", "\[Lambda]_"}], "]"}], "=", FractionBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "-", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", SuperscriptBox["\[Lambda]", "2"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", " ", SuperscriptBox["U", "4"]}], "-", RowBox[{"8", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}], "-", RowBox[{"2", " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "+", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", SuperscriptBox["\[Lambda]", "2"]}]}], ")"}], " ", "#1"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", SuperscriptBox["U", "2"]}], "+", RowBox[{"2", " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}]}], ")"}], " ", SuperscriptBox["#1", "2"]}], "+", SuperscriptBox["#1", "3"]}], "&"}], ",", "2"}], "]"}], "U"]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ SubscriptBox["\[Epsilon]UMP", "3"], "[", RowBox[{"U_", ",", "t_", ",", "\[Lambda]_"}], "]"}], "=", FractionBox[ RowBox[{"Root", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "-", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "4"], " ", SuperscriptBox["\[Lambda]", "2"]}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"2", " ", SuperscriptBox["U", "4"]}], "-", RowBox[{"8", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}], "-", RowBox[{"2", " ", SuperscriptBox["U", "4"], " ", "\[Lambda]"}], "+", RowBox[{"4", " ", SuperscriptBox["t", "2"], " ", SuperscriptBox["U", "2"], " ", SuperscriptBox["\[Lambda]", "2"]}]}], ")"}], " ", "#1"}], "+", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"-", "3"}], " ", SuperscriptBox["U", "2"]}], "+", RowBox[{"2", " ", SuperscriptBox["U", "2"], " ", "\[Lambda]"}]}], ")"}], " ", SuperscriptBox["#1", "2"]}], "+", SuperscriptBox["#1", "3"]}], "&"}], ",", "3"}], "]"}], "U"]}], ";"}]}], "Input", CellLabel->"In[74]:=",ExpressionUUID->"09d6f124-bb11-412e-bec6-81715f49c632"] }, Open ]], Cell[CellGroupData[{ Cell["Small U/t = 3", "Subsubsection", CellChangeTimes->{{3.815594563091886*^9, 3.8155945654717093`*^9}, { 3.815594773186194*^9, 3.815594778423319*^9}, {3.815595445556635*^9, 3.815595445991561*^9}},ExpressionUUID->"31dad6b7-333f-4866-80b6-\ 257aa54c247e"], Cell["\<\ The HF potentials will be weak and the electrons will slowly delocalise over \ both sites as \[Lambda] increases. 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The key feature here is \ the fact that the\nHF potential is different for the \[Alpha] and \[Beta] \ electrons, and by extension the potential felt by an electron is ", StyleBox["not ", FontWeight->"Bold"], "strictly localised around\nthat electrons orbitals. This is completely \ different to the symmetric structure in the closed-shell case. Because of \ these different \npotentials, there can be a large driving force for spatial \ rearrangement occurring as soon as the HF potential becomes positive (at \ \[Lambda]=1). \nThis sudden change in electron distribution is made more \ extreme if the HF potential and electron repulsion dominate over the \n\ one-electron terms, which also corresponds to the regime of strong wave \ function symmetry breaking. \n\nWe have therefore provided a physical \ motivation behind why this EP can behave like a QPT for strong symmetry \ breaking. \nThis supports the conclusions in Antoine\[CloseCurlyQuote]s \ report that the symmetry-broken EP behaves as a class \[Beta] singularity." }], "Text", CellChangeTimes->{{3.8155957260444117`*^9, 3.815595732835033*^9}, { 3.815631070831706*^9, 3.815631400252574*^9}, {3.815631533980709*^9, 3.81563156311534*^9}, {3.815632283939207*^9, 3.815632287773448*^9}},ExpressionUUID->"bed7772a-c117-450e-940d-\ c8bf700e9310"], Cell[BoxData[ RowBox[{ 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