Quit

A Notebook for arXiv:0803.3295

Building an AdS/CFT superconductor

Prepared by C. P. Herzog, February 12, 2009

The equations (6),  (7), and (13) from the paper.

psieomoriginal = ψ''[r] + (f0'[r]/f0[r] + 2/r) ψ'[r] + (ϕ[r]^2/f0[r]^2) ψ[r] + (2/f0[r]) ψ[r]

phieomoriginal= ϕ''[r] + (2/r) ϕ'[r] - (2 ψ[r]^2/f0[r])ϕ[r]

Axeomoriginal[ω_] = Ax''[r] + (f0'[r]/f0[r]) Ax'[r] + (ω^2/f0[r]^2 - 2 ψ[r]^2/f0[r]) Ax[r]

f0rel = {f0-> Function[r, r^2 - 1/r]}

I find it more convenient to work in terms of the coordinate z=1/r.  I also prefer to work with a function f = 1-(z^3) instead of f0 = (1-(z^3))/(z^2).  Finally, I will rescale ψ = z psi so that psi = a + b z + ... near the boundary z=0.

ftest[z_] = 1-z^3

myeom = {psieom = Collect[(psieomoriginal /.{ψ-> Function[r, psi[1/r]/r], ϕ-> Function[r,phi[1/r]], Ax-> Function[r, Ax[1/r]], f0-> Function[r, r^2 f[1/r]]}/. r-> 1/z /. f-> ftest)/z^4, {ψ''[z], ψ'[z], ψ[z]}, Simplify]  ,phieom = Collect[(phieomoriginal /.{ψ-> Function[r, psi[1/r]/r], ϕ-> Function[r,phi[1/r]], Ax-> Function[r, Ax[1/r]], f0-> Function[r, r^2 f[1/r]]}/. r-> 1/z /. f->ftest)/z^4, {ϕ''[z], ϕ'[z], ϕ[z]}, Simplify],Axeom[ω_]= Collect[(Axeomoriginal[ω] /.{ψ-> Function[r, psi[1/r]/r], ϕ-> Function[r,phi[1/r]], Ax-> Function[r, Ax[1/r]], f0-> Function[r, r^2 f[1/r]]}/. r-> 1/z/. f->ftest)/z^4, {Ax''[z], Ax'[z], Ax[z]}, Simplify] }

We develop a power series for ϕ and ψ near the horizon z=1.

Note that ϕ must vanish at the horizon in order that (Subscript[A, μ]) have a well defined norm, (Subscript[A, μ]) (Subscript[A, ν]) ((g )^(μν))< ∞.

phitest[z_] = aa(z-1)  + bb(z-1) ^2 + cc (z-1) ^3 +dd(z-1)^4

For psi on the other hand, we insist only that psi < ∞.

psitest[z_] =(AA+BB (z-1) + CC(z-1) ^2+DD(z-1) ^3 + EE(z-1)^4 )

myphipsiseries = Simplify[Series[Simplify[({psieom, phieom} ) /. {phi->phitest, psi->psitest}], {z,1,3}]]

myphipsirels = Solve[Join[Table[SeriesCoefficient[myphipsiseries[[1]], i]==0, {i,-1,2}],Table[SeriesCoefficient[myphipsiseries[[2]], i]==0, {i,0,2}]], {bb,cc,dd,BB,CC,DD,EE}][[1]]

psiic[ aa_, AA_][z_] = Evaluate[psitest[z] /. myphipsirels]

phiic[aa_,AA_][z_] =Evaluate[ phitest[z] /.myphipsirels]

Here is the heart of the notebook, a call to NDSolve to solve numerically these two differential equations.

As the differential equation is singular near the horizon, we will use these power series expansions to initialize the numerical differential equation solver a small distance "small" away from the horizon.  The numerical differential equation solver also has trouble near the boundary z=0.  Thus we will integrate to a distance "delta" away from z=0.

small = 1/100000

delta = 1/1000000

The following is admittedly kind of dumb.  We should really just make one call to NDSolve, but my computer is fast enough it doesn't matter.

mypsi[AA_?NumberQ, aa_?NumberQ] :=  psi /. NDSolve[{psieom==0, phieom==0, phi[1-small] == phiic[ aa,AA][1-small] ,
psi[1-small] == psiic[ aa,AA][1-small] ,
phi'[1-small] == phiic[ aa,AA]'[1-small] ,
psi'[1-small] == psiic[ aa,AA]'[1-small] }, {phi, psi},{ z, 1- small,delta}, WorkingPrecision->20][[1]]

myphi[AA_?NumberQ, aa_?NumberQ] :=  phi /. NDSolve[{psieom==0, phieom==0, phi[1-small] == phiic[ aa,AA][1-small] ,
psi[1-small] == psiic[ aa,AA][1-small] ,
phi'[1-small] == phiic[ aa,AA]'[1-small] ,
psi'[1-small] == psiic[ aa,AA]'[1-small] }, {phi, psi},{ z, 1- small,delta}, WorkingPrecision->20][[1]]

I also need a series expansion near the boundary z=0.

psizero[z_] = A1 + B1 z + C1 z^2 + D1 z^3

phizero[z_] = A2 + B2 z + C2 z^2 + D2 z^3

myzeroseries = Series[{psieom/z, phieom} /. {psi->psizero, phi->phizero, f-> ftest}, {z,0,1}]

Solve[Join[Table[SeriesCoefficient[myzeroseries[[1]], i], {i,0,1}],Table[SeriesCoefficient[myzeroseries[[2]], i], {i,0,1}]]==0, {C1,C2, D1,D2}][[1]]

I'll just use the series to order (z^2), setting D1=D2=0.

temprels = Solve[{ phizero[z]==phi0, phizero'[z]==phip,psizero[z]==psi0, psizero'[z]==psip} /.{C2->(A1^2) A2,D2->0,C1->-((A1 (A2^2))/2),D1->0}, {A1, A2,  B1, B2}][[1]];

The idea is to propagate the solution out to a distance delta from the boundary and then use this series solution to compute the values A1, A2, B1, and B2 which have physical meaning as an expectation value and source for the scalar and a chemical potential and charge density for the gauge field.

bryvalues = Series[{A1,B1,A2,B2} /. temprels, {z,0,2}]

myfalloffs[AA_, aa_] := Normal[bryvalues] /. {z->delta, psi0-> mypsi[AA,aa][delta], psip-> mypsi[AA,aa]'[delta],phi0->myphi[AA,aa][delta], phip-> myphi[AA,aa]'[delta]};

myfalloffs[1/50, 406/100]

A command to look for a solution where B1=0, i.e. a scalar of conformal dimension one.

mapitone[ xseed_, yseed_] := Join[{xx->xseed},FindRoot[myfalloffs[xseed,yy][[2]]==0, {yy,yseed}, WorkingPrecision->20]]

A command to look for a solution where A1=0, i.e. a scalar of conformal dimension two.

mapittwo[ xseed_, yseed_] := Join[{xx->xseed},FindRoot[myfalloffs[xseed,yy][[1]]==0, {yy,yseed}, WorkingPrecision->20]]

mapitone[1/10000, 1]

mapittwo[1/10000, 4]

I used commands like the following to find the solutions with the right boundary condition for the scalar.  In this data structure, xx and yy refer to the horizon values for psi and phi respectively.  Thus by looking for a solution with a small value of xx, we are looking for a solution near the phase transition.  The next four numbers in an item in the list describe the near boundary behavior of psi and phi.

y0 =11/10;mysweep = Table[{{temprels = mapitone[ii, y0], y0 = yy /. temprels }[[1]], myfalloffs[xx,yy]/. temprels}, {ii,1/100, 1/10,1/100}];InputForm[Chop[mysweep], NumberMarks->False]

y0 =4;mysweep = Table[{{temprels = mapittwo[ii, y0], y0 = yy /. temprels }[[1]], myfalloffs[xx,yy]/. temprels}, {ii,1/100, 1/10,1/100}];InputForm[Chop[mysweep], NumberMarks->False]

Data for the solutions

xx and yy are the constants giving the horizon behavior of psi and phi.  The following set of four numbers give the boundary behavior of the fields.

dimonesweep = {{{xx->1/10000,yy->1.12043889119510991063},{0.000130315716227508739042,0,-1.12043889843628472562,1.12043891088417666}},{{xx->1/5000,yy->1.12043766313440755112},{0.000260631425398699882435,0,-1.120437692099176688,1.12043774189071211}},{{xx->3/10000,yy->1.12043715950211139551},{0.00039094735371692611689,0,-1.12043722467282604709,1.120437336703965201}},{{xx->1/2500,yy->1.1204383581629267182},{0.00052126321241103892799,0,-1.12043847402199272512,1.120438673188632524}},{{xx->1/2000,yy->1.12043801061117345985},{0.00065157954284247875964,0,-1.12043819164043977299,1.120438502839070558}},{{xx->3/5000,yy->1.12043796181456177665},{0.00078189546145633255056,0,-1.12043822249673861307,1.120438670622809639}},{{xx->7/10000,yy->1.12043762430070819839},{0.00091221137730928662926,0,-1.12043797911809856255,1.120438589067399683}},{{xx->1/1250,yy->1.12043749131314523539},{0.00104252729716885104175,0,-1.12043795474810022206,1.120438751416594661}},{{xx->9/10000,yy->1.12043742530667802618},{0.00117284322312673176254,0,-1.12043801184155254172,1.120439020125184722}},{{xx->1/1000,yy->1.12043715761485207791},{0.00130315914036450189735,0,-1.12043788173190249446,1.120439126526321856}},{{xx->1/500,yy->1.12043548169784682243},{0.00260631849966171721994,0,-1.12043837816364749506,1.120443357341593366}},{{xx->3/1000,yy->1.1204330167445732368},{0.0039094784388731893433,0,-1.12043953378675270483,1.120450736939605501}},{{xx->1/250,yy->1.12042967262941909338},{0.0052126391296664820275,0,-1.12044125846648530804,1.120461175188611266}},{{xx->1/200,yy->1.1204253557473098296},{0.0065158008306924269237,0,-1.12044345858680835284,1.120474578476666029}},{{xx->3/500,yy->1.12042018377384148385},{0.0078189638049388672101,0,-1.12044625180871766226,1.120491064472841829}},{{xx->7/1000,yy->1.12041400934580006376},{0.009122128271459129202,0,-1.1204494907506141423,1.120510485799981005}},{{xx->1/125,yy->1.12040692331409500826},{0.0104252945812367651368,0,-1.12045326624259863749,1.120532933299555796}},{{xx->9/1000,yy->1.12039891979803671945},{0.0117284628831747260228,0,-1.1204575723794248143,1.120558401075577825}},{{xx->1/100,yy->1.12038994941614570512},{0.0130316334914845403812,0,-1.12046235975156058733,1.120586839728098528}},{{xx->1/50,yy->1.12024850424263952185},{0.0260635219050933215791,0,-1.12053812923598571797,1.121036055839203786}},{{xx->3/100,yy->1.12001290825758601925},{0.039095918998004095583,0,-1.12066450318131381936,1.1217848630797973}},{{xx->1/25,yy->1.11968317330358100938},{0.052129079337674902393,0,-1.12084141170848991796,1.122833224963956355}},{{xx->1/20,yy->1.11925937737374263607},{0.065163256662464964077,0,-1.12106881841641278493,1.124181151729514106}},{{xx->3/50,yy->1.11874163036027509509},{0.078198704465277758064,0,-1.12134668620011704952,1.125828666143914949}},{{xx->7/100,yy->1.11813007086186424226},{0.091235675727179034641,0,-1.12167497408173772184,1.127775800254328124}},{{xx->2/25,yy->1.11742485452851989931},{0.104274422477319895109,0,-1.12205362562152434042,1.130022583606975041}},{{xx->9/100,yy->1.11662617125957554127},{0.117315196694507859769,0,-1.12248258616190880166,1.13256906058130615}},{{xx->1/10,yy->1.11573422660422469723},{0.130358248896834924027,0,-1.12296178433748317483,1.135415271512116147}},{{xx->3/20,yy->1.1098846103501463796},{0.195616381363158521621,0,-1.12610842125043313433,1.154144200835539553}},{{xx->1/5,yy->1.10174872769647512795},{0.260968396293486804418,0,-1.13049629789952934767,1.180375560180743401}},{{xx->1/4,yy->1.09137960881022997275},{0.32644405452473453159,0,-1.13610812871220975777,1.214119146113768747}},{{xx->3/10,yy->1.07884464489026174014},{0.3920718237552698508,0,-1.14292187663837579199,1.255386073077493827}},{{xx->7/20,yy->1.06422498293536162294},{0.45787858330936194333,0,-1.15091089330252939112,1.304187720481336909}},{{xx->2/5,yy->1.04761479739272013528},{0.52388934883280604497,0,-1.16004411171714642558,1.360534533250495017}},{{xx->9/20,yy->1.02912039389263980374},{0.59012701909090335226,0,-1.17028625075880057623,1.424434668537999312}},{{xx->1/2,yy->1.00885920456891087256},{0.65661215447842545345,0,-1.18159808729952018071,1.495892594956008433}},{{xx->11/20,yy->0.986958634276734401709},{0.72336278590898846867,0,-1.19393675699071994333,1.574907645896192098}},{{xx->3/5,yy->0.963554773454148968385},{0.79039426392979884077,0,-1.2072560863689365255,1.661472587171007005}},{{xx->13/20,yy->0.93879101118420364858},{0.8577191480788181631,0,-1.2215069780120982676,1.755572270753762935}},{{xx->7/10,yy->0.912816540061000773638},{0.92534714128465354445,0,-1.23663782306468509426,1.85718240252416619}},{{xx->3/4,yy->0.885784789926179315989},{0.99328507249045837662,0,-1.25259494528626945934,1.96626848341766576}},{{xx->4/5,yy->0.85785180677726722204},{1.06153692525766454602,0,-1.26932307108589585805,2.082784958240429593}},{{xx->17/20,yy->0.82917460956846186887},{1.13010391478719378228,0,-1.28676581480550947049,2.20667460747362698}},{{xx->9/10,yy->0.799909554798448089209},{1.19898461007213367786,0,-1.30486617239146291584,2.33786820610512784}},{{xx->19/20,yy->0.770210744163412343769},{1.26817509729700838303,0,-1.3235670142896994429,2.47628445802192336}},{{xx->1,yy->0.740228500283488245978},{1.33766917978577627605,0,-1.34281156755147971193,2.62183021267314051}},{{xx->11/10,yy->0.679987737974656699883},{1.47753333650378575189,0,-1.38270923269032186454,2.93388149904868117}},{{xx->6/5,yy->0.620263868459910194138},{1.61849113316761637443,0,-1.42412880953535533882,3.27306405697326708}},{{xx->13/10,yy->0.561997776627018636795},{1.76043660043680643919,0,-1.46667137587268422015,3.63828394766999838}},{{xx->7/5,yy->0.505972952896035743026},{1.90325241272368193255,0,-1.50997810665117097933,4.02835258952756369}},{{xx->3/2,yy->0.452805789407482696517},{2.04681775954558027774,0,-1.55373553104808851977,4.4420312191878814}},{{xx->8/5,yy->0.402946853101821375254},{2.19101483316663493451,0,-1.59767749881756666787,4.8780734055607134}},{{xx->17/10,yy->0.35669114259411917411},{2.33573363916339809968,0,-1.64158433014957718709,5.33526176907208551}},{{xx->9/5,yy->0.314194792338208941294},{2.48087510903907437843,0,-1.68527981931207636275,5.81243664501694537}},{{xx->19/10,yy->0.275495615029562472395},{2.62635269551260532402,0,-1.72862681740853950497,6.30851594841825168}},{{xx->2,yy->0.240535138376206727701},{2.7720927498240265888,0,-1.77152207499777826245,6.82250674906179888}},{{xx->21/10,yy->0.209180327830243562685},{2.9180340172255008616,0,-1.81389089292053625797,7.35350980427598191}},{{xx->11/5,yy->0.18124364873204649841},{3.06412656957590633493,0,-1.85568198320814696478,7.90071866827322453}},{{xx->23/10,yy->0.156500738928779193561},{3.21033043662720092111,0,-1.89686279149774767311,8.46341498893567487}},{{xx->12/5,yy->0.134705295382788373263},{3.3566141394234888424,0,-1.93741541857034667579,9.04096147303791173}},{{xx->5/2,yy->0.115601146883158960346},{3.5029532582172869555,0,-1.97733317248546126444,9.63279363578194752}},{{xx->13/5,yy->0.0989316726853383085809},{3.6493291206762478166,0,-2.01661773960019122272,10.23841124094847816}},{{xx->27/10,yy->0.0844468457294509491614},{3.7957276504738448654,0,-2.0552769122566582804,10.85736998777724188}},{{xx->14/5,yy->0.0719082529857888512412},{3.9421383899307100738,0,-2.09332279860154728898,11.48927379913860996}},{{xx->29/10,yy->0.0610924345703091748753},{4.0885536921681477507,0,-2.13077044060198126867,12.1337679167835781}},{{xx->3,yy->0.0517928814974309334203},{4.2349680670428294618,0,-2.16763676209742098354,12.79053285129521982}},{{xx->31/10,yy->0.0438209838778947509557},{4.3813776604948332599,0,-2.20393978076845950376,13.45927917290160695}},{{xx->16/5,yy->0.0370061982827673052665},{4.5277798461937929471,0,-2.23969802910978865342,14.13974309507497556}},{{xx->33/10,yy->0.0311956363947966240215},{4.6741729093695327689,0,-2.27493013519236017167,14.83168275688341946}},{{xx->17/5,yy->0.0262532599911390954576},{4.8205558049124272278,0,-2.30965452624760499753,15.5348751182611357}},{{xx->7/2,yy->0.022058821284595718502},{4.9669279728554456525,0,-2.34388922348686181285,16.24911336705209262}},{{xx->18/5,yy->0.0185066404655040058578},{5.1132892009278695894,0,-2.37765170764243944585,16.9742047781089772}},{{xx->37/10,yy->0.0155043257769224419706},{5.2596395205793179234,0,-2.41095883315066667154,17.70996891874444973}},{{xx->19/5,yy->0.0129714801163141961282},{5.4059791314736748479,0,-2.44382678021469040662,18.45623616399715633}},{{xx->39/10,yy->0.0108384356712201390936},{5.5523083442718037609,0,-2.47627103295971462239,19.21284644850150761}},{{xx->4,yy->0.00904505166140285732389},{5.6986275412375060964,0,-2.5083063762329260917,19.97964822730492204}},{{xx->41/10,yy->0.00753959397317830576567},{5.844937146029073029,0,-2.53994690628295083437,20.75649759780884956}},{{xx->21/5,yy->0.006277703390198815926},{5.9912376016966718389,0,-2.57120604634594304603,21.54325752511751252}},{{xx->43/10,yy->0.00522145392356614589339},{6.137529359119052527,0,-2.60209657770738398659,22.33979727793403263}},{{xx->22/5,yy->0.00433850562916901459355},{6.2838128634249113522,0,-2.63263065745010230804,23.1459917771967902}},{{xx->9/2,yy->0.00360135660062840266041},{6.4300885487981754344,0,-2.66281985514991015076,23.96172119719151813}},{{xx->23/5,yy->0.00298666960325476390709},{6.5763568338600981929,0,-2.6926751802348379011,24.78687052711547779}},{{xx->47/10,yy->0.00247468682293093158085},{6.7226181182974432097,0,-2.72220710979395336828,25.6213291850599363}},{{xx->24/5,yy->0.00204871262898958926397},{6.8688727836518212725,0,-2.75142562508352062696,26.4649907802669451}},{{xx->49/10,yy->0.00169466718272483509685},{7.0151211905090184211,0,-2.78034023286086633861,27.31775275320559299}},{{xx->5,yy->0.00140069792296264504519},{7.1613636771573605185,0,-2.80895998776787569608,28.1795160612721272}},{{xx->51/10,yy->0.00115684236682746395628},{7.3076005647838939377,0,-2.83729353191634867439,29.05018512252702282}},{{xx->26/5,yy->0.000954742106258895528933},{7.4538321537760481103,0,-2.86534910817401240118,29.92966747468775947}},{{xx->53/10,yy->0.00078739367622109173808},{7.6000587265250886508,0,-2.89313458747863245291,30.8178736402188594}},{{xx->27/5,yy->0.000648936366321531018307},{7.74628054796642353,0,-2.92065748879316240776,31.7147169443152024}},{{xx->11/2,yy->0.000534477482506556749842},{7.8924978659336818741,0,-2.94792499915116701471,32.6201133503204012}},{{xx->28/5,yy->0.000439927889220144790772},{8.0387109140568488375,0,-2.97494399499261346442,33.5339813569279527}},{{xx->57/10,yy->0.000361882391784534747525},{8.1849199107290660553,0,-3.00172105679323626601,34.4562418247254878}},{{xx->29/5,yy->0.000297507806571249070161},{8.3311250599329249381,0,-3.02826248601261237907,35.3868178522821578}},{{xx->59/10,yy->0.000244445539995994051757},{8.4773265540107677805,0,-3.05457432255050529631,36.3256347042078695}},{{xx->6,yy->0.00020073636864074339765},{8.6235245738448025581,0,-3.08066235919426109238,37.272619700490212}},{{xx->61/10,yy->0.000164754782157367032666},{8.7697192874880423518,0,-3.10653215105604035204,38.2277020663330744}},{{xx->31/5,yy->0.00013515320057503456167},{8.9159108544489645497,0,-3.13218903346827553717,39.1908129138006018}},{{xx->63/10,yy->0.000110814706858798172563},{9.0620994239640116551,0,-3.15763813143092355445,40.1618851238361233}},{{xx->32/5,yy->0.0000908150482132544129821},{9.2082851361825592482,0,-3.18288436917441903705,41.1408532526482603}},{{xx->13/2,yy->0.0000743902251843899894425},{9.3544681243657589717,0,-3.20793248630967323838,42.127653535926722}},{{xx->33/5,yy->0.0000609080317484719071143},{9.500648511570420457,0,-3.23278703923029937283,43.1222236993779679}},{{xx->67/10,yy->0.0000498471808434101367411},{9.6468264156054801625,0,-3.25745241651461819562,44.124502991969239}},{{xx->34/5,yy->0.0000407774899109065805762},{9.7930019468349909519,0,-3.2819328446103935442,45.1344320794662757}},{{xx->69/10,yy->0.0000333441091516057555605},{9.9391752095329405599,0,-3.3062323974361854375,46.1519530093726799}},{{xx->7,yy->0.0000272544238158547150851},{10.0853463027229940361,0,-3.3303550021166942851,47.1770091397904352}},{{xx->71/10,yy->0.000022268313621594453074},{10.2315153181442380313,0,-3.354304449373517866,48.209545121538348}},{{xx->36/5,yy->0.0000181874649908223932713},{10.3776823439029333099,0,-3.378084394710524654,49.2495067812912715}},{{xx->73/10,yy->0.0000148489832945368087082},{10.5238474627463121642,0,-3.4016983699854519004,50.2968411528838886}},{{xx->37/5,yy->0.0000121187563697028023247},{10.6700107549814770593,0,-3.4251497867130118415,51.351496413210469}},{{xx->15/2,yy->9.8871644405085804283*^-6},{10.8161722924763193469,0,-3.4484419416116075949,52.4134217873332964}},{{xx->38/5,yy->8.06364165901395357843*^-6},{10.9623321483481780982,0,-3.4715780234114725951,53.4825676138383982}},{{xx->77/10,yy->6.57428778921789165612*^-6},{11.1084903870583410367,0,-3.4945611151117870911,54.5588851828857462}},{{xx->39/5,yy->5.35819348183049946642*^-6},{11.2546470746213272062,0,-3.5173942021173591209,55.642326837809653}},{{xx->79/10,yy->4.36571938847147715716*^-6},{11.4008022681490283834,0,-3.5400801714819265239,56.7328457747605215}},{{xx->8,yy->3.55595065603078450483*^-6},{11.5469560279721111585,0,-3.5626218240297404009,57.830396233930217}},{{xx->81/10,yy->2.89540425376565604092*^-6},{11.6931084065246800132,0,-3.5850218687404496464,58.934933219958303}},{{xx->41/5,yy->2.35705045253135343771*^-6},{11.8392594557542364995,0,-3.6072829341459143319,60.046412667865506}},{{xx->83/10,yy->1.9180969163720061871*^-6},{11.9854092251994929978,0,-3.6294075681312863299,61.1647913234653404}},{{xx->42/5,yy->1.56050434255480193078*^-6},{12.1315577616932513781,0,-3.6513982418416686406,62.2900267400515176}},{{xx->17/2,yy->1.26926092914959529609*^-6},{12.2777051095280557924,0,-3.6732573532077098118,63.422077245500072}},{{xx->43/5,yy->1.0320502982097278925*^-6},{12.4238513111056180406,0,-3.6949872302575722422,64.560901932537308}},{{xx->87/10,yy->8.38957989989072420021*^-7},{12.569996407381346755,0,-3.7165901330996082664,65.706460627426078}},{{xx->44/5,yy->6.81818465890297218114*^-7},{12.7161404370499066394,0,-3.7380682568156005394,66.8587138567502994}},{{xx->89/10,yy->5.53972213205786526132*^-7},{12.8622834367093935747,0,-3.7594237340733032368,68.017622838521216}},{{xx->9,yy->4.4998493102572332803*^-7},{13.0084254416585669517,0,-3.7806586385828482967,69.1831494763768209}},{{xx->91/10,yy->3.65430173682113891105*^-7},{13.1545664854599296787,0,-3.8017749851219266971,70.3552562945471826}},{{xx->46/5,yy->2.96696382561450100221*^-7},{13.3007066008269488089,0,-3.82277473544470285,71.5339065114359339}},{{xx->93/10,yy->2.40831104004951594339*^-7},{13.4468458177139232662,0,-3.8436597950407567753,72.7190638770958667}},{{xx->47/5,yy->1.9543863673595002036*^-7},{13.5929841658214262591,0,-3.864432020480988088,73.910692800079762}},{{xx->19/2,yy->1.58568487395086000976*^-7},{13.7391216736314032341,0,-3.8850932176259232624,75.1087582406710832}},{{xx->48/5,yy->1.28625797734521025067*^-7},{13.8852583680197790375,0,-3.9056451457533519927,76.3132257440176137}},{{xx->97/10,yy->1.04314110647598661118*^-7},{14.0313942747031037759,0,-3.926089517708978286,77.524061391309993}},{{xx->49/5,yy->8.45786435648186770107*^-8},{14.1775294184033345378,0,-3.9464280024345832734,78.7412318042593675}},{{xx->99/10,yy->6.85697664480901243258*^-8},{14.3236638246516371965,0,-3.9666622265741062436,79.9647041494475982}},{{xx->10,yy->5.55765104182883446691*^-8},{14.4697975134194413551,0,-3.9867937746576710598,81.1944460441123687}},{{xx->101/10,yy->4.50365695831887261941*^-8},{14.6159305077127366715,0,-4.0068241918280967744,82.4304256369418475}},{{xx->51/5,yy->3.64883928243565601842*^-8},{14.7620628287111123622,0,-4.0267549864828462552,83.6726115807565322}},{{xx->103/10,yy->2.95571665406009783969*^-8},{14.9081944964445691511,0,-4.0465876276086401119,84.920972953080178}},{{xx->52/5,yy->2.39381473492549508673*^-8},{15.0543255303090457811,0,-4.0663235503158769397,86.1754793221785155}},{{xx->21/2,yy->1.93836461005052279252*^-8},{15.2004559488672415748,0,-4.0859641544636538093,87.436100691977748}},{{xx->53/5,yy->1.56930280318422743503*^-8},{15.3465857701631583625,1.097471*^-10,-4.1055108065573519698,88.702807504058736}},{{xx->107/10,yy->1.2703132802921968784*^-8},{15.4927150114414797241,2.606144*^-10,-4.1249648409475882742,89.9755706279013757}},{{xx->54/5,yy->1.02809393811130602492*^-8},{15.6388436891353312356,0,-4.1443275604441125494,91.2543613410885614}},{{xx->109/10,yy->8.31921149167257521143*^-9},{15.7849718192524661797,0,-4.163600238102317307,92.539151338067672}},{{xx->11,yy->6.73069247199647777033*^-9},{15.9310994171475716338,0,-4.1827841174779910964,93.829912703475262}},{{xx->111/10,yy->5.44461446960131347987*^-9},{16.0772264976420354738,0,-4.2018804139969655172,95.1266179125319633}},{{xx->56/5,yy->4.40357166957430398361*^-9},{16.2233530749068840026,0,-4.2208903149487187687,96.4292398069093023}},{{xx->113/10,yy->3.56102500395318708418*^-9},{16.3694791627674667597,2.53921*^-10,-4.2398149821452657961,97.7377516168093421}},{{xx->57/5,yy->2.87924362878143893393*^-9},{16.515604774469825409,6.872387*^-10,-4.2586555511627117916,99.0521269252999311}},{{xx->23/2,yy->2.32764055568115472294*^-9},{16.6617299227459917998,0,-4.2774131318025383572,100.3723396507492136}},{{xx->58/5,yy->1.88143458006431179294*^-9},{16.8078546198496637746,1.31066*^-10,-4.2960888108699767431,101.69836408380598}},{{xx->117/10,yy->1.52054276500624202503*^-9},{16.9539788778886604635,4.498231*^-10,-4.3146836520520104231,103.0301748576419559}},{{xx->59/5,yy->1.2286997826456566805*^-9},{17.1001027081297176527,1.5529726*^-9,-4.3331986947443189803,104.3677469064981649}},{{xx->119/10,yy->9.92728681517488435357*^-10},{17.2462261216593958252,2.51099*^-10,-4.3516349559348732799,105.7110554734676452}},{{xx->12,yy->8.01965512257300848049*^-10},{17.3923491294187408503,1.071262*^-9,-4.3699934343616430578,107.0600762017933921}},{{xx->121/10,yy->6.47770760352693360925*^-10},{17.5384717414295532614,4.533291*^-9,-4.3882751056166360513,108.4147849808930515}},{{xx->61/5,yy->5.23153628593002940958*^-10},{17.6845939681271315427,1.61701*^-9,-4.4064809225321183232,109.775157945329433}},{{xx->123/10,yy->4.2245370722624748301*^-10},{17.8307158186098014961,8.1211077*^-9,-4.4246118258822835029,111.1411717122187014}},{{xx->62/5,yy->3.41097816557872637238*^-10},{17.976837302951339047,4.9894587*^-9,-4.4426687269806208785,112.5128029334541994}},{{xx->25/2,yy->2.75368588642754457062*^-10},{18.1229584294931231252,4.245651*^-9,-4.460652527200101024,113.890028748973224}},{{xx->63/5,yy->2.22276719957859547576*^-10},{18.2690792072783188069,4.908051*^-9,-4.4785641071624900709,115.2728265231800133}},{{xx->127/10,yy->1.79397049834503260468*^-10},{18.4151996446541657646,7.493838*^-9,-4.4964043295438930281,116.661173879395729}},{{xx->64/5,yy->1.44770377197504554536*^-10},{18.5613197498308529036,1.4520134*^-8,-4.5141740412325732336,118.0550487359681566}},{{xx->129/10,yy->1.16811827604795154567*^-10},{18.7074395304596215613,3.4444645*^-8,-4.5318740738219709222,119.4544293103202362}}}

dimtwosweep = {{{xx->1/10000,yy->4.06371791161361771577},{0,0.00029454834161098045743,-4.0637179294789956915,4.06371794246387482}},{{xx->1/5000,yy->4.06371610547587405215},{0,0.0005890957331777792548,-4.0637161769373781813,4.06371622887685376}},{{xx->3/10000,yy->4.06371477912395701747},{0,0.0008836432887214580919,-4.0637149399123453151,4.06371505677611097}},{{xx->1/2500,yy->4.06371414231176647187},{0,0.0011781901335398945155,-4.0637144281576300638,4.06371463591519831}},{{xx->1/2000,yy->4.06371293873327733417},{0,0.0014727371686979156572,-4.0637133853677826191,4.06371370998888469}},{{xx->3/5000,yy->4.06371289437853410678},{0,0.0017672842661492103787,-4.0637135375321916263,4.06371400498655206}},{{xx->7/10000,yy->4.06371277826659286119},{0,0.0020618318623562112089,-4.0637136536702930893,4.0637142899277354}},{{xx->1/1250,yy->4.06371272299352859178},{0,0.0023563789738664994839,-4.0637138663779347827,4.06371469740807028}},{{xx->9/10000,yy->4.06371259770163053382},{0,0.0026509263738963338364,-4.0637140447975576994,4.06371509657012326}},{{xx->1/1000,yy->4.0637124204607407019},{0,0.002945473811755635813,-4.063714206998943259,4.06371550548362159}},{{xx->1/500,yy->4.06371008463696222097},{0,0.005890950602600205146,-4.0637172307888294139,4.06372242473166045}},{{xx->3/1000,yy->4.06370498044428019333},{0,0.008836433465603483079,-4.0637210592826252924,4.06373274566053686}},{{xx->1/250,yy->4.0636982275707010067},{0,0.011781925259836462082,-4.0637268121682080256,4.06374758797411367}},{{xx->1/200,yy->4.06368954637173854684},{0,0.014727429366570265124,-4.0637342097957383824,4.06376667203850279}},{{xx->3/500,yy->4.06367891340818991726},{0,0.017672949088112199159,-4.0637432287206112121,4.06378997442968282}},{{xx->7/1000,yy->4.06366637881805605663},{0,0.020618487123203849947,-4.0637539190746035533,4.06381754530823877}},{{xx->1/125,yy->4.0636519053755810756},{0,0.023564046751232205298,-4.0637662436239009697,4.06384934747098327}},{{xx->9/1000,yy->4.06363550118744317747},{0,0.026509631159405860934,-4.063780210466580209,4.06388538905165852}},{{xx->1/100,yy->4.06361716974371018416},{0,0.029455243191902686316,-4.0637958230817192792,4.06392567357183878}},{{xx->1/50,yy->4.06332775553708644135},{0,0.0589135671447686892,-4.0640423629745752569,4.06456178925639976}},{{xx->3/100,yy->4.06284539824071780241},{0,0.08837805515420927446,-4.0644532427438997259,4.06562204304293188}},{{xx->1/25,yy->4.06217018321341240887},{0,0.11785179153945347585,-4.0650285178908532943,4.06710661206500656}},{{xx->1/20,yy->4.06130214818139720576},{0,0.14733786104864236976,-4.065768184641939731,4.06901566278485476}},{{xx->3/50,yy->4.06024135638698136848},{0,0.17683935036585520533,-4.0666722525526485661,4.07134942363884159}},{{xx->7/100,yy->4.05898793111037096999},{0,0.20635934829550600155,-4.0677407795499595351,4.07410822034025779}},{{xx->2/25,yy->4.05754195637198630718},{0,0.23590094870900247731,-4.0689737720119599407,4.07729237546828158}},{{xx->9/100,yy->4.05590357871011301668},{0,0.26546724629561103419,-4.0703712870401096666,4.08090231141127801}},{{xx->1/10,yy->4.05407294498995457448},{0,0.29506134142371051644,-4.0719333700112487181,4.08493848763818518}},{{xx->3/20,yy->4.04204218790418546005},{0,0.44355761574714731604,-4.082214154302694182,4.11153273985639664}},{{xx->1/5,yy->4.0252367485351082014},{0,0.5932169586230489302,-4.0966186207759052042,4.14888278197254733}},{{xx->1/4,yy->4.00369446197627242145},{0,0.7444342811527135925,-4.1151580463785334493,4.19710703346071978}},{{xx->3/10,yy->3.97746418578231425712},{0,0.8976106302848189133,-4.1378466992141044268,4.25635810275953917}},{{xx->7/20,yy->3.94660600833753650362},{0,1.0531546654189168924,-4.1647017284839542616,4.32682309912011532}},{{xx->2/5,yy->3.91119145477050887285},{0,1.2114840823830119836,-4.1957429695723009042,4.40872392140831923}},{{xx->9/20,yy->3.87130370581367302652},{0,1.373026978487790965,-4.2309927010886502536,4.50231754984611137}},{{xx->1/2,yy->3.82703786265919170337},{0,1.5382231363864297253,-4.2704753818734729854,4.60789635003805961}},{{xx->11/20,yy->3.77850119684358334952},{0,1.7075252156446776497,-4.3142173147316051281,4.72578830612386947}},{{xx->3/5,yy->3.72581340865244321904},{0,1.881399825736681834,-4.3622462678639204198,4.8563571909612562}},{{xx->13/20,yy->3.6691068743329856462},{0,2.0603284462666285284,-4.4145910279198896006,5.00000259971149267}},{{xx->7/10,yy->3.60852686669740394858},{0,2.2448081887233802802,-4.4712809116139582936,5.15715986566833409}},{{xx->3/4,yy->3.54423172908719857447},{0,2.4353523414167339761,-4.5323451881805013865,5.32829972280787647}},{{xx->4/5,yy->3.47639299462957181718},{0,2.6324906689510757275,-4.5978124451505576378,5.5139277298387866}},{{xx->17/20,yy->3.40519543468500130104},{0,2.8367694651187030982,-4.667709903987634281,5.71458341849045767}},{{xx->9/10,yy->3.33083698766046464999},{0,3.0487512508748134895,-4.7420626303701019283,5.93083900600637265}},{{xx->19/20,yy->3.25352859207357233391},{0,3.2690141554172366649,-4.820892721455629457,6.16329775627048659}},{{xx->1,yy->3.17349385910716553754},{0,3.4981508937837159018,-4.9042184082830669477,6.41259182195723188}},{{xx->11/10,yy->3.00620012148922383182},{0,3.9854805989321282357,-5.0844045689116611979,6.96434256751224522}},{{xx->6/5,yy->2.83097501854544387801},{0,4.5157080802250010649,-5.2826535772638139383,7.59161166708477382}},{{xx->13/10,yy->2.64998895372854271389},{0,5.0938957123454145969,-5.4988744866785602588,8.30015327557723892}},{{xx->7/5,yy->2.46551769580294875589},{0,5.725090615427100618,-5.7328237935779384332,9.09578969466230274}},{{xx->3/2,yy->2.27988268081381846391},{0,6.414191903193260595,-5.9840818943481128919,9.98421739470316673}},{{xx->8/5,yy->2.09537985802402459272},{0,7.165805158038517226,-6.2520389563513869627,10.9707980609005044}},{{xx->17/10,yy->1.91420203212869462721},{0,7.984098438756305366,-6.5358935962861605637,12.0603542681865417}},{{xx->9/5,yy->1.73836160945089735961},{0,8.872676510327662559,-6.8346662702226746254,13.2569923610802651}},{{xx->19/10,yy->1.56962163504273827834},{0,9.834489060055682346,-7.1472271898104105884,14.563973936624724}},{{xx->2,yy->1.40944239865993948932},{0,10.871784287364283778,-7.4723361737753372139,15.98365110428154059}},{{xx->21/10,yy->1.25894884058165069145},{0,11.986112183149310022,-7.808689914439478643,17.5174709815722711}},{{xx->11/5,yy->1.11892085524958669215},{0,13.178373954098123078,-8.1549710780889185137,19.1660438945019902}},{{xx->23/10,yy->0.98980528079984659378},{0,14.448907322856826446,-8.5098938699124907254,20.9292607740393758}},{{xx->12/5,yy->0.871745677529805881943},{0,15.797593742839725365,-8.8722418842536786928,22.8064400826109887}},{{xx->5/2,yy->0.76462439222868665084},{0,17.223972984422041414,-9.240895852098402773,24.7964840839409825}},{{xx->13/5,yy->0.668111079110995822151},{0,18.727352735436562639,-9.6148507136394580181,26.8980273744037031}},{{xx->27/10,yy->0.581712511990462900973},{0,20.306904609576575113,-9.9932229282035125831,29.109565918078709}},{{xx->14/5,yy->0.504819709994971603315},{0,21.961742017909527319,-10.3752498486240986917,31.4295605162877903}},{{xx->29/10,yy->0.436749796261827184095},{0,23.690978655561290747,-10.7602832987950104809,33.8565132872427718}},{{xx->3,yy->0.376781287979916834717},{0,25.493769080109752181,-11.1477794949776277426,36.3890192266334917}},{{xx->31/10,yy->0.32418246078914092343},{0,27.36933397706482517,-11.5372870715076278719,39.02579679776429}},{{xx->16/5,yy->0.278233138297502069851},{0,29.31697343172445635,-11.928434564647393142,41.7657022319293881}},{{xx->33/10,yy->0.238240606156653838144},{0,31.336071394249151346,-12.3209183064265840897,44.6077318720582495}},{{xx->17/5,yy->0.203550564328793556948},{0,33.426094104704691092,-12.714491286320110802,47.5510166696498947}},{{xx->7/2,yy->0.173554018797566146973},{0,35.586584782372222946,-13.108953271699035881,50.5948118371376295}},{{xx->18/5,yy->0.147690991762839228325},{0,37.817156248163463847,-13.5041422974128855562,53.7384840354133001}},{{xx->37/10,yy->0.125451801436056395099},{0,40.117482649420769546,-13.8999274562165443295,56.9814977360530937}},{{xx->19/5,yy->0.106376569888022212128},{0,42.487291216688046571,-14.2962029264337459748,60.3234017568377846}},{{xx->39/10,yy->0.090053439097017464664},{0,44.926354396086523854,-14.6928830717212917122,63.7638169438392019}},{{xx->4,yy->0.0761159706883698926616},{0,47.434482755073247864,-15.0898984463065438605,67.3024248066742756}},{{xx->41/10,yy->0.0642400076329610368917},{0,50.011518666135375293,-15.4871925887846636075,70.9389577599570002}},{{xx->21/5,yy->0.0541402602724918120197},{0,52.65733095034754055,-15.8847194416974374181,74.6731906089820204}},{{xx->43/10,yy->0.0455667955421396311983},{0,55.37181027564024619,-16.2824412882749694371,78.5049333209570587}},{{xx->22/5,yy->0.0383015602660726484261},{0,58.15486520767091847,-16.6803271242345061744,82.4340251055133384}},{{xx->9/2,yy->0.0321550284691045128242},{0,61.00641925138737254,-17.0783513588162143151,86.460329427530869}},{{xx->23/5,yy->0.0269630367729484435556},{0,63.92640786089428624,-17.4764927463015743865,90.583729588366811}},{{xx->47/10,yy->0.0225838353897541443653},{0,66.91477657673901677,-17.8747335939953782732,94.8041256202672152}},{{xx->24/5,yy->0.0188953858164677728163},{0,69.97147918332150117,-18.2730591015771028326,99.121431454797131}},{{xx->49/10,yy->0.0157929018242090973706},{0,73.09647617578702975,-18.6714568104848556211,103.535572443268805}},{{xx->5,yy->0.013186636239423160725},{0,76.28973372240880624,-19.0699162055476842487,108.0464835798566695}},{{xx->51/10,yy->0.0109999019203652391581},{0,79.55122264078715389,-19.4684283720657373884,112.6541080586623562}},{{xx->26/5,yy->0.00916731393026877113775},{0,82.88091761110724834,-19.8669857241949211929,117.3583959359856074}},{{xx->53/10,yy->0.00763323233898531697856},{0,86.27879657561482874,-20.2655817723406704942,122.1593029887530039}},{{xx->27/5,yy->0.00635042409964503251992},{0,89.74484031252352809,-20.6642109633239717657,127.056790141408026}},{{xx->11/2,yy->0.00527883211675502496662},{0,93.27903176752237799,-21.0628685052068680154,132.0508223042556146}},{{xx->28/5,yy->0.00438456522780647447924},{0,96.8813560526182445,-21.4615502933888730781,137.141368459164027}},{{xx->57/10,yy->0.00363898913480437830974},{0,100.55179974606381459,-21.8602527334890114458,142.328400229634856}},{{xx->29/5,yy->0.00301794901807089834308},{0,104.29035102940868901,-22.2589727392220015202,147.611892344282096}},{{xx->59/10,yy->0.00250109831980575976683},{0,108.09699926495793828,-22.6577076249809760514,152.991821969970827}},{{xx->6,yy->0.00207132207628791235493},{0,111.97173496795937125,-23.0564550506791873287,158.4681681985202666}},{{xx->61/10,yy->0.00171424261258336333155},{0,115.91454949140313577,-23.4552129637877839292,164.040912070615036}},{{xx->31/5,yy->0.00141779694226098805812},{0,119.92543513473231533,-23.8539795893246362898,169.71003636647684}},{{xx->63/10,yy->0.00117187686195076429385},{0,124.00438483913428131,-24.2527533629818422634,175.475525222303774}},{{xx->32/5,yy->0.000968019615461428272752},{0,128.15139226242612012,-24.6515329219695610048,181.337364175323468}},{{xx->13/2,yy->0.000799151030346430022068},{0,132.36645146406268226,-25.0503170550575413658,187.295539676136986}},{{xx->33/5,yy->0.000659361823957796006346},{0,136.64955728248032784,-25.4491047254712143042,193.350039644713892}},{{xx->67/10,yy->0.000543721304840961869827},{0,141.0007046060026156,-25.847895010714921849,199.50085270977978}},{{xx->34/5,yy->0.000448119664531294379764},{0,145.419889187959516,-26.2466870886800105731,205.74796816136954}},{{xx->69/10,yy->0.000369132819044062594417},{0,149.90710677275438566,-26.6454802612264247366,212.091376446039281}},{{xx->7,yy->0.000303912066388793504642},{0,154.46235370169044314,-27.0442739037268936166,218.531068546993168}},{{xx->71/10,yy->0.000250091211528939705457},{0,159.08562632880174472,-27.4430674686812992856,225.06703603469983}},{{xx->36/5,yy->0.000205703308182023411606},{0,163.7769214146884453,-27.8418604637279563388,231.699270846997112}},{{xx->73/10,yy->0.000169112272883966544972},{0,168.53623615338751901,-28.2406524872281125658,238.427765994648096}},{{xx->37/5,yy->0.000138967351957903496241},{0,173.36356755039188139,-28.6394431629082154069,245.252514556448901}},{{xx->15/2,yy->0.000114144964474092849929},{0,178.25891317401349123,-29.0382321771732592008,252.173510305454879}},{{xx->38/5,yy->0.000093715411942963882828},{0,183.22227044503796637,-29.4370192419430355028,259.190746968400728}},{{xx->77/10,yy->0.0000769101005296297639485},{0,188.25363726406117043,-29.835804121169616464,266.304219226471351}},{{xx->39/5,yy->0.0000630925371901777279217},{0,193.35301148063514387,-30.2345866033022155207,273.513921758394038}},{{xx->79/10,yy->0.0000517360655309242261127},{0,198.52039119437173292,-30.633366511165467008,280.819849709096235}},{{xx->8,yy->0.0000424073757693117537976},{0,203.75577453682429666,-31.0321436914075854913,288.221998492477578}},{{xx->81/10,yy->0.0000347474693132131666907},{0,209.05915977442252163,-31.4309180007259682306,295.720363558959899}},{{xx->41/5,yy->0.0000284603883679333622903},{0,214.43054529391722285,-31.8296893291114687087,303.314940803571792}},{{xx->83/10,yy->0.0000233026291725925181349},{0,219.86992971450025314,-32.2284575890288979963,311.005726531835662}},{{xx->42/5,yy->0.0000190727505160341561619},{0,225.37731147064563924,-32.6272227011547660657,318.792717028085627}},{{xx->17/2,yy->0.0000156052201212448734589},{0,230.95268925933851767,-33.025984599102197076,326.675908773744654}},{{xx->43/5,yy->0.0000127637964121058271458},{0,236.5960618479961231,-33.424743236659917649,334.655298667503358}},{{xx->87/10,yy->0.0000104362794671435330196},{0,242.30742807909910604,-33.823498562430426665,342.730883336390154}},{{xx->44/5,yy->8.5304488388624016685*^-6},{0,248.08678682623896995,-34.222250561529607382,350.902660160229683}},{{xx->89/10,yy->6.97022174960425213661*^-6},{0,253.93413708111171119,-34.62099920805084179,359.170626285069577}},{{xx->9,yy->5.69377542995780354868*^-6},{0,259.8494777498354889,-35.019744486768894725,367.53477899781611}}}

Reproducing Figure 1

An approximation for the critical temperature, measured in units of Sqrt[ρ], for the dimension one case.

myTc1 = 3/4/Pi/Sqrt[dimonesweep[[1]][[2]][[4]]]

figure1a = ListLinePlot[Table[{3/4/Pi/Sqrt[dimonesweep[[i]][[2]][[4]]]/myTc1, Sqrt[2]dimonesweep[[i]][[2]][[1]]/Sqrt[dimonesweep[[i]][[2]][[4]]]/myTc1},{i,1,Length[dimonesweep]}], FrameLabel-> {T/(Subscript[T, c]), "<(Subscript[O, 1])>"/(Subscript[T, c])}, Frame->True, RotateLabel->False]

An approximation for the critical temperature for the dimension two case.

myTc2= 3/4/Pi/Sqrt[dimtwosweep[[1]][[2]][[4]]]

figure1b = ListLinePlot[Table[{3/4/Pi/Sqrt[dimtwosweep[[i]][[2]][[4]]]/myTc2, Sqrt[dimtwosweep[[i]][[2]][[2]]/dimtwosweep[[i]][[2]][[4]]Sqrt[2]]/myTc2},{i,1,Length[dimtwosweep]}], FrameLabel-> {T/(Subscript[T, c]),Sqrt[ "<(Subscript[O, 2])>"]/(Subscript[T, c])}, Frame->True, RotateLabel->False]

The conductivity

We develop a series solution for the Axeom near the horizon.

Atest[z_] = (1-z)^(I ω/3)(1 + aa(1-z)+ bb(1-z)^2 + cc(1-z)^3)

myAeqseries = Simplify[Series[Simplify[Axeom[ω]/(1-z)^(I  ω/3) /. Ax-> Atest], {z,1,3}]];

myAeqrels = Simplify[Solve[Table[SeriesCoefficient[myAeqseries, i]==0, {i,0,2}], {aa,bb,cc}]][[1]];

Aic[ ω_, psi_][z_] = Atest[z] /. myAeqrels;

We use this series solution to give NDSolve initial conditions.  We also do a little reformatting of the data structures (probably for unnecessary reasons).

realAic[ω_, aa_, AA_][z_] = Aic[ω, psiic[aa,AA]][z];

Aeq[ω_, psi_] = Axeom[ω]

The heart of the notebook as concerns conductivity calculations.

myAx[ω_, AA_, aa_] := Ax /. NDSolve[{Aeq[ω,mypsi[AA,aa]]==0, Ax[(1-small)]==realAic[ω,aa,AA][1-small],Ax'[1-small] == realAic[ω,aa,AA]'[1-small]}, Ax, {z,(1-small), delta}][[1]]

We'll be sloppy here and evaluate the conductivity by looking at z=delta rather than matching to a series expansion near the boundary.

mycond[ω_,AA_, aa_] := I myAx[ω, AA,aa]'[delta]/myAx[ω, AA,aa][delta]/ω

Reproducing Figure 2

Note the temperatures used in this replot of figure 2 are different than the ones used in the paper.

figure2curve1[i_] := ListLinePlot[Table[{ω, Re[mycond[ω  3/4/Pi, xx /. dimonesweep[[i]][[1]],yy /. dimonesweep[[i]][[1]]] ]}, {ω,2,200, 2}], PlotRange->{{0,200},{0,1.2}}, Frame->True, RotateLabel->False, FrameLabel->{ω/T, Re[σ]}]

Length[dimonesweep]

The T/Tc ratios for the plot.

dimoneTs = {3/4/Pi/Sqrt[dimonesweep[[1]][[2]][[4]]]/myTc1, 3/4/Pi/Sqrt[dimonesweep[[50]][[2]][[4]]]/myTc1,3/4/Pi/Sqrt[dimonesweep[[100]][[2]][[4]]]/myTc1,3/4/Pi/Sqrt[dimonesweep[[165]][[2]][[4]]]/myTc1}

figure2a = Show[{figure2curve1[1],figure2curve1[50],figure2curve1[100], figure2curve1[165]}]

figure2curve2[i_] := ListLinePlot[Table[{ω, Re[mycond[ω  3/4/Pi, xx /. dimtwosweep[[i]][[1]],yy /. dimtwosweep[[i]][[1]]] ]}, {ω,2,200, 2}], PlotRange->{{0,200},{0,1.2}}, Frame->True, RotateLabel->False, FrameLabel->{ω/T, Re[σ]}]

Length[dimtwosweep]

dimoneTs = {3/4/Pi/Sqrt[dimtwosweep[[1]][[2]][[4]]]/myTc2, 3/4/Pi/Sqrt[dimtwosweep[[40]][[2]][[4]]]/myTc2,3/4/Pi/Sqrt[dimtwosweep[[80]][[2]][[4]]]/myTc2,3/4/Pi/Sqrt[dimtwosweep[[126]][[2]][[4]]]/myTc2}

figure2b = Show[{figure2curve2[1],figure2curve2[40],figure2curve2[80], figure2curve2[126]}]

Reproducing Figure 3

figure3curve1[i_] := ListLinePlot[Table[{ω, Re[mycond[ω  dimonesweep[[i]][[2]][[1]] Sqrt[2], xx /. dimonesweep[[i]][[1]],yy /. dimonesweep[[i]][[1]]] ]}, {ω,0.01,2.5, 0.02}], PlotRange->{{0,2.5},{0,1.2}}, Frame->True, RotateLabel->False, FrameLabel->{ω/"<(Subscript[O, 1])>", Re[σ]}]

dimoneTs = {3/4/Pi/Sqrt[dimonesweep[[1]][[2]][[4]]]/myTc1, 3/4/Pi/Sqrt[dimonesweep[[75]][[2]][[4]]]/myTc1,3/4/Pi/Sqrt[dimonesweep[[100]][[2]][[4]]]/myTc1,3/4/Pi/Sqrt[dimonesweep[[165]][[2]][[4]]]/myTc1}

figure3a = Show[{figure3curve1[1],figure3curve1[75],figure3curve1[100], figure3curve1[165]}]

figure3curve2[i_] := ListLinePlot[Table[{ω, Re[mycond[ω  Sqrt[ dimtwosweep[[i]][[2]][[2]] Sqrt[2]], xx /. dimtwosweep[[i]][[1]],yy /. dimtwosweep[[i]][[1]]] ]}, {ω,0.01,2.5, 0.02}], PlotRange->{{0,2.5},{0,1.2}}, Frame->True, RotateLabel->False, FrameLabel->{ω/Sqrt["<(Subscript[O, 2])>"], Re[σ]}]

Length[dimtwosweep]

dimtwoTs = {3/4/Pi/Sqrt[dimtwosweep[[1]][[2]][[4]]]/myTc2, 3/4/Pi/Sqrt[dimtwosweep[[80]][[2]][[4]]]/myTc2,3/4/Pi/Sqrt[dimtwosweep[[100]][[2]][[4]]]/myTc2,3/4/Pi/Sqrt[dimtwosweep[[126]][[2]][[4]]]/myTc2}

figure3b = Show[{figure3curve2[1],figure3curve2[80],figure3curve2[100], figure3curve2[126]}]