Magnetic impurity in a one-dimensional few-fermion system

We present a numerical analysis of spin-12 fermions in a one-dimensional harmonic potential in the presence of a magnetic point-like impurity at the center of the trap. The model represents a few-body analogue of a magnetic impurity in the vicinity of an s -wave superconductor. Already for a few particles we ﬁnd a ground-state level crossing between sectors with diﬀerent fermion parities. We interpret this crossing as a few-body precursor of a quantum phase transition, which occurs when the impurity ‘breaks’ a Cooper pair. This picture is further corroborated by analyzing density-density correlations in momentum space. Finally, we discuss how the system may be realized with existing cold-atoms platforms

gFI pewEody phse digrm for individul uto' vlues PP gFP grossover etween the 2 ↑ +2 ↓ nd 2 ↑ +3 ↓ setors PQ 1 Introduction untum phse trnsitions @9sA re trnsitions etween di'erent phses of mnyE ody quntum system t zero tempertureF sn the ground stte of the mnyEody system hnges qulittively s n externl ontrol prmeter is vriedF uh ontrol prmeter nD for exmpleD e n externl mgneti (eld orD in theoretil studiesD simply oupling onstnt in the rmiltoninF 9s ply n importnt role in quntum mnyE ody systemsF hey re typilly studied in the ontext of mrosopi numer of onstituents nd linked to the olletive ehvior of mny prtiles IF roweverD the qulittive ehvior of mny mesosopi systems with modest numer of prtilesD suh s tomi nulei P nd few old fermions QDRD n lso e understood using tools developed to study 9sF his implies the possiility to study the emergene of 9s from fewE ody dynmisD whih flls into rod lss of studies dedited to the soElled fewEody preursors of mnyEody phenomenD seeD eFgFD S!UF he trnsition from fewEody ehvior to mnyEody ehvior s funtion of the prtile numer hs een explored theoretilly in rod vriety of oneEdimensionl fermioni systems using ext digonliztion VD wonte grlo methods W!IID oupled luster expnsion IPD nd perturtion theory IQD see efsF SD T for reviewF por similr studies in higher sptil dimensionsD seeD eFgFD efsF QD IR!ITF he interest in this ottomEup pproh to mnyEody physis is driven in prtiulr y the existing P pigure IX Sketch of the transition driven by the magnetic impurity. ith inE resing the strength of the impurityEfermion intertionD GD level rossing etween the setors of di'erent fermioni prity @iFeFD odd vsF even prtile numerA oursF et the sme timeD the mgnetiztion of the ground stte hnges from S = 0 to S = 1 2 F he feE ture persists even in the presene of only hndful of prtiles nd onstitutes preursor of mnyEody F ultrold tomi setEups whose exquisite tunility nd ontrol dmits the reliztion of experiments with preisely determined smll numer of prtilesF uh setups hve een usedD for instneD to study the formtion of permi se nd piring gp in qusi oneE dimensionl fewEfermion systems IUDIVF enother exmple is given y the oservtion of fewEody ounterprt of the riggs mplitude mode ross the norml to super)uid phse trnsition in twoEdimensionl permi gs RF wotivted y these studiesD we explore the fewEody nlogue of lssil mgneti impurity in n sEwve superondutorF sn the mnyEody limitD this is wellEstudied prolem IWD PHD whih hs een found to host shrp from nonEmgneti totl spin S = 0 ground stte @ssuming tht the spin of the impurity is zeroA for wekly interting impurities to S = 1 2 ground stte t strong impurityEeletron intertions PI @see sketh in pigF I nd eppF hD whih onnet this limit to our fewEody modelAF his trnsitionD whih mnifests itself s rossing of the energy levels orresponding to the two ground sttesD is onneted to the soElled uEhiEusinov sttes @or simply hi sttesA ! exited sttes elow the threshold of the singleEprtile gp @soElled suEgp sttesA PP!PRF sn this workD we fous on the emergene of this in oneEdimensionl @IhA fewE ody system tht n e simulted using oldEtom setEupsF pei(llyD we investigte the fewEody setor of twoEomponent permi gs in Ih hrmoni trpF he stti mgneti impurity is relized s spinEseletive δEpotentil in the enter of the trpF vike in the mnyEody seD the ttrtive intertion etween prtiles fvors pir formtionD whih leds to fewEody nlogue of n energy gp IPD PSF he interply etween the timeE reverslEsymmetri pir formtion nd the mgneti impurity then drives the fewEody nlogue of the F por strong impurityEfermion ouplingsD it is energetilly fvorle to shield the impurity vi forming ound stte with one of the prtilesD therey hnging the fermioni prity of the ground stteF e remrk tht oth the impurityEfermion nd fermionEfermion intertions re essentil hereF his is in strk ontrst to the physis driven y glol mgneti (eldD whih reks pirs due to energetilly mismthed permi surfesD nd leds to level rossing even t vnishing fermionEfermion intertionsF hile Einitio results re not possile in the mnyEody limitD whih usully requires menE(eld nd T Emtrix pproximtions with vrying degrees of selfEonsisteny IWD PHD our fewEody model n e studied in numerilly ext mnnerF es numeril methodD Q we hoose the fullEon(gurtion intertion @pgsA method in trunted model spe @lso referred to s ext digonliztionAF o void exoritnt numeril ost tht ours for overly lrge model spesD we employ n e'etive twoEody intertion in trunted speD whih is inspired y the veeEuzuki method known in nuler physisD see efF PT for reviewF yne n (nd desription of the e'etive intertion for oldEtom systems in efF PUY its pplitions for prtiles in oneEdimensionl hrmoni trps nd rings re disussed in efsF PV!QHF yur pgs odeD written in the tuli lngugeD long with detiled desription of the method s well s n extensive enhmrk is ville in efF QIF es n lterntive numeril pproh we use the trnsorrelted method @gwA QPDQQ to generte enhmrk dt for vlidting the e'etive intertion resultsF he gw reE moves the wve funtion usp with similrity trnsformtion of the rmiltoninD whih improves the onvergene properties in (nite sis set expnsionF he trnsorrelted method for shortErnge intertions ws desried nd enhmrked for fewEfermion sysE tems in one sptil dimension in efF QR nd in three dimensions in efF QSF he reminder of this work is orgnized s followsX sn etF PD we present the model of interestF e lso rie)y disuss the numeril pprohesF e proeed with the min setion of this work in etF QD where we (rst review the emergene of piring in Ih trps nd susequently investigte the e'et of mgneti impurityF e relegte the disussion of tehnil detils s well s the disussion of dditionl dt to eppsF eD fD g nd hF purthermoreD we lulte experimentlly essile twoEody orreltion funtions tht exhiit signtures of the underlying physisF pinllyD we summrize our work nd propose n experimentl reliztion in etF RF 2 Model and numerical methods e re interested in n ensemle of few hrmonilly trpped twoEomponent fermionsD desried y the oneEdimensionl rmiltonin where N ↑ @N ↓ A is the numer of spinEup @spinEdownA fermionsD ω is the frequeny of the trpD m is the prtile mss nd g is the strength of intertionF x i (y j ) is the position of the ith spinEup @jth spinEdownA fermionF yur fous is on ttrtively interting prtilesD iFeFD g < 0D sine we re interested in the physis ssoited with piring of fermionsF xote tht the spin projetion of prtile is (xed ! stndrd ssumption for oldEtom systemsF he present study is limited to systems with N ↑ + N ↓ ≤ 9D whih n e relily ddressed using our implementtion of ext digonliztionF he term H imp ontins spinEdependent externl potentilD whih we use to model mgneti impurityF por simpliityD we employ spinEseletive δEpotentil in the enter of the trp suh tht where G σ enodes properties of the fermionEimpurity stteringF st is worth pointing out tht the shpe of the fermionEimpurity potentil is not of gret importne s long s the width of this potentil is smller thn ny other relevnt length sles of the prolemD whih re desried elowF sn the present workD for simpliityD we use the onvention G ↑ = −G ↓ ≡ G > 0 so tht ↑ prtiles re repelled nd ↓ prtiles ttrted y the entrl impurity with equl mgnitudeF yur results will lso qulittively desrie the sitution G ↑ = −G ↓ s long s the impurity ttrts only one spinEtype of the fermions so tht oundEstte my formF Length Scales of the Problem: he hrteristi length sle ssoited with the fermionEfermion intertion in iqF @IA is given y the oneEdimensionl sttering lengthD a 0 D whih is de(ned s a 0 = −2 2 /mgD seeD eFgFD TF imilrlyD one n de(ne length sle ssoited with the fermionEimpurity intertionF he hrteristi length sle for the trp is given y the hrmoni osilltor length ξ = /mωF por the reminder of this workD we shll use the hrmoni osilltor units in whih ξ = 1 nd ω = 1F he fourth relevnt length sle is onneted to the permi momentumD see lso susetion QFQF 2.1 Discussion of the model fefore proeeding with our nlysisD it is worthwhile to motivte the hoie of the rmilE tonin in iqF @IAF o this endD we (rst rell the ft tht the oneEdimensionl permi gs with shortErnge ttrtive intertions nd H imp = ω = 0 fetures n sEwve piring gp in the thermodynmi limitF roperties of the system n e lulted extly using the fethe nstz nd onneted to fg @frdeenEgooperEhrie'erA theory QTF yne omE ment is in order hereF hile fg theory ws originlly developed for extended systemsD the generl onepts of the theory of superondutivity my still e used to understnd fewEody systemsF sn prtiulrD the super)uid order prmeter n e stritly de(ned for fewE nd mnyEody systems through the eigenfuntion ssoited with the dominnt eigenvlue of the twoEody density mtrixD generlizing fg theory QUF ir orreltions re lerly expressed in momentum spe nd n e mesured in oldEtom setEups with quntum gs mirosopesF e reent rekthrough in this diretion is the oservtion of fewEody nlogues of gooper pirs in two dimensions with hndful of prtiles QVF sf H imp = 0D our system inludes spinEdependent externl potentil whih models mgneti impurityF his llows us to study the interply etween lolly roken timeE reversl symmetry nd sEwve piringF por N ↑ , N ↓ → ∞D it is known tht mgneti impurity in the viinity of n sEwve superondutor leds to the soElled hi sttes in the exittion spetrumF hese re energy levels within the piring gp indued y the lol reking of timeEreversl symmetry IWD PHF hese suEgp sttes re onventionE lly studied in the grndEnonil ensemle where they emerge s pir of exittions symmetrilly round the hemil potentilF sn the present workD we study fewEody systems with wellEde(ned numer of prtilesD thereforeD we re not le to see this ehvior expliitlyF roweverD we n study fewE ody nlogue of this physis y digonlizing the rmiltonin in two di'erent setors whih di'er in the fermion prityD nmely in the (N ↑ , N ↓ ) = (N, N ) nd (N, N + 1) setors referred to s S = 0 nd S = 1 2 D respetivelyF ithout the impurityD iFeFD if G = 0D the ground stte should e in the S = 0 setorD where the piring is strongestF roweverD if the fermionEimpurity intertion is strong enough to rek pirD then the S = 1 2 setor my host the lowest energy levelF his preursor of is skethed in pigF ID where lso the ound stte of one of the exess prtiles nd the impurity is indited s the reson for the energetilly fvorle on(gurtionF st is resonle to expet tht for lrge enough numer of prtiles our model reprodues the ehvior in the grndEnonil pitureF wenwhileD t smll numer of prtilesD our model o'ers systemti wy to pproh @menle lso to nlogue quntum simultionA s we shll disuss in the followingF S 2.2 Numerical methods o (nd the spetrum of the rmiltoninD we digonlize the rmiltonin in trunted rilert speF yur min tool is n e'etive intertion pproh in the hrmoni osilltor sisD whih we use to otin our desired fewEody phse digrmsF edditionllyD we vlidte our results without the mgneti impurity with the trnsorrelted methodF foth pprohes re rie)y desried elowF 2.2.1 Eective interaction approach o trunte the rilert spe with the e'etive intertion pprohD we keep only n b lowest eigensttes of the hrmoni osilltor sisF hile the truntion is neessry step to mke the prolem menle to numeril tretmentD it introdues is due to the disrded physil sttesF o mitigte this shortomingD n extrpoltion to the in(niteEsis limit is requiredF roweverD the numeril ost for preise extrpoltion rises omintorilly nd therefore eomes prolemti with more thn very few prtilesF sn order to minimize the numeril e'ort ! while still mintining ury ! we mke use of n e'etive intertion pproh known in the nulerEphysis ommunity in the ontext of the noEore shell model PUF he key step is to reple the re twoEody mtrix elements with e'etive vlues optimized for the pplied truntion shemeF his step essentilly onstitutes prtiulrly e'etive renormliztion proedure with regrd to the twoEody prolemX snsted of (xing only single prmeter @the oupling strength gAD the e'etiveEintertion pproh mounts to tuning ll intertion mtrix elements in order to mth the lowest prt of the energy spetrum to the nlyti solutionF his n e hieved y onstruting n e'etive intertion from the e'etive rmiltonin whose mtrix representtion reds s follows where E 1 , ..., E n re the n lowest twoEody eigenenergiesD whih n e lulted exE tly QWD RHY U is mtrix whose rows re formed y the orresponding eigenvetors projeted on the trunted rilert speF he e'etive potentil V eff = H eff − T @T is the kineti energy opertorA extly reprodues the in(niteEsis spetrum for the twoEody prolem lredy t (nite sis uto'F woreoverD nd this is the ruil numeril ene(tD the e'etive intertion sustntilly improves the onvergene properties of pgs lulE tions for N > 2F gonsequentlyD numeril vlues otined in smll trunted rilert spes my e muh more urte thn those otined for re intertionF he signi(ntly redued numeril e'ort llows us not only to proe lrger systems relily ut lso to sn heply the prmeter spe nd thus mp out fewEody phseEdigrmsF por detils of the method s well s on our implementtion we refer to efF QIF 2.2.2 Transcorrelated Method o hve dditionl enhmrk dt for the e'etive intertion pprohD we use potenE tilly more urteD ut more expensive gw for shortErnge intertions QRD QSF he δEfuntion intertionD equivlent to the fetheEeierls @fA oundry onditionD produes usp in the wve funtionD whih is di0ult to pture with sis set expnsionF o mitigte this prolemD we introdue tstrow ftor e τ D where τ is funtion of ll the prtile oordintes x i nd y i F he tstrow ftor inludes the usp tht stis(es the f oundry onditionsD nd is folded into the rmiltonin with similrity trnsformtioñ H = e −τ He τ . T his removes the usp from the wve funtionD whih gretly improves the onvergene with respet to sis set expnsion of the trnsorrelted rmiltoninHF sn prtiulrD for interting spinE 1 2 fermions in one dimensionD the onvergene of the groundEstte energy improves from n −1 b to n −3 b when expnding in trunted singleEprtile sis with n b plne wves QRF he downside of this pproh is tht it mkes the rmiltoninH nonErermitin nd more omplited to onstrutF tillD it n e digonlized with the widely vilE le ernoldi itertionD or in the se of lrger systemsD with fullEon(gurtion intertion quntum wonte grlo @pgswgA RID RPF por the results reported in this work we tightly (t ox of length L in relEspe to pture the relevnt prts of the ground stte wve funtionF e then use singleEprtile sis with n b plne wves to expndHD (nd the ground stte energyD nd inrese L nd n b until the desired ury is rehedF he results in this pper were produed with our tuli pkge Rimu.jl RQD whih inludes implementtions of the trnsorrelted rmiltonin onstrution nd pgswgF 3 Results sn this setionD we present our entrl results tht onern fewEody systems with mgE neti impurityF 3.1 Balanced systems without a magnetic impurity fefore ddressing systems with mgneti impurityD let us onsider lned system N ↑ = N ↓ with G = 0F yur im here is to highlight the preursor of piring gp in the fewEody limitD see lso the disussion in efF PSF ith G = 0D the rmiltonin is symmetri with respet to n exhnge of spin up nd spin down prtilesF sndeedD it is ler tht the exhnge {x i } → {y i } does not hnge the rmiltonin in iqF @IAF his implies tht if ψ(x 1 , ..., x N ↓ ; y 1 , ..., y N ↑ ) is n eigenstte of HD then ψ(y 1 , ..., y N ↑ ; x 1 , ..., x N ↓ ) is lso n eigenstte of HF vet us introdue the swp opertor T D whih performs the trnsformtion {x i } → {y i }F ine T nd H ommuteD then every eigenstte of H n e leled using the eigenvlues T = ±1 of T F st is worth noting tht in the spin lngugeD the opertor T is the spinE)ip opertorD whih enters the timeEreversl opertorF sn the limit N ↑ , N ↓ → ∞D the system fetures spin gp @seeD eFgFD QTAD iFeFD there is n energy di'erene etween the singlet ground stte nd the (rst triplet exited stteF his gp n e onneted to the energy di'erene etween the mnifolds with T = 1 nd T = −1D see efF PS for more detiled disussionF vet us illustrte the spinE)ip9 symmetry for the simplest lned systemD iFeFD for 1 ↑ +1 ↓D whih n e solved nlytilly QWD RHF he wve funtion for this system is written s ψ(x 1 , y 1 ) = φ cm (x 1 + y 1 ) φ rel (x 1 − y 1 ), @RA where φ cm nd φ rel desrie the enterEofEmss nd reltive motionD respetivelyF he enterEofEmss prt is lwys symmetri with respet to n exhnge of prtilesF sn the reltive prtD the exhnge of prtiles orresponds to prity opertionF he even funE tions φ rel orrespond to T = 1Y odd funtions hve T = −1F por g < 0 there is lwys n energy di'erene etween the lowest sttes in the T = 1 nd T = −1 mnifoldsD whih we write s 2∆ + 1F he prmeter ∆ here n e linked to fewEody nlogue of the piring gp tht ppers in mnyEody permi systemD see efF PS for more detiled

A B
pigure PX Spectrum and pairing gap. @eA he spetrum of the 2 ↑ +2 ↓ system s funtion of the intertion strength gF he @lueA solid urves show the sttes with T = 1Y the @redA dshed urves show the sttes with T = −1F he di'erene etween the lowest dshed urve nd the lowest solid urve is written s 2∆ + 1D where ∆ is the piring gpF @fA he piring gp s funtion of the intertion strength gF e lso present the dt from n erlier pgs lultion with re intertion @squresD lines re dded to guide n eyeA PSF @snsetA gonvergene of the gp s funtion of the sis uto' n b for re @lue symolsA nd e'etive @red symolsA intertions for N = 2 ↑ +2 ↓ t g = −1.5D inluding (t to the respetive dt @dshed urvesAF he horizontl dotted line is the result from PS otined with n b = 25F yur dt in the min plot re otined with n b = 13D for whih the errorErs re on the sle of the orresponding symolsF explntion 1 F st is worth noting tht the disussion ove pplies lso to systems without trpD in whih se the energy di'erene etween the mnifolds is simply given y the twoEody inding energyF e illustrte the fewEody nlogue of the piring gp for N ↑ + N ↓ = 2, 4, 6 in pigF PF sn pnel eD we show the lowElying energy spetrum of the 2 ↑ +2 ↓ system s funtion of the intertion strength g resolved for oth the positive @solid lue urvesA nd negtive @dshed red urvesA symmetry setors of the swp opertor T F sn pnel fD we show the piring gpD whih is determined from the level sping etween the lowest sttes in eh respetive setorF e ompre these results to the vlues of efF PS nd report good greement for ll ville intertion strengthsD modulo slight devition t lrger ouplingD whih is likely n rteft of the onvergene with the re intertion @see the inset of pigF PAF he results otined with the e'etive intertion hve een omputed y mximlly onsidering n b = 13 singleEprtile sttes whih ws enough to oserve onvergene essentilly indisernile t the sle of the (gureF por the present work it is extly the emergene of positive fewEody gp for g < 0 tht llows for the study of preursor of due to nonEtrivil interply etween prtile piring nd impurity stteringD whih we will disuss in the followingF st is worth noting tht the existene of negtive gp @∆ < 0A on the repulsive side is mere onsequene of the (nite numer of prtiles in the trp ! exittions etween even nd odd spinE)ip symmetry should e ville t no energy ost in the thermodyE nmi limitF por the emergene of nonEnlyti ehviorD iFeFD n expeted quntum phse trnsition t g = 0D one would hve to refully extrpolte to n in(nite numer of prE tilesD where the lowest sttes of oth setors eome degenerteF uh n extrpoltionD 1 In short, the additive term 1 in 2∆ + 1 accounts for the spacing between energy levels of a noninteracting system; the factor 2 in front of ∆ appears because the spin ip leaves two atoms unpaired not a single atom as in the standard denition of the gap. V howeverD is fr from trivil nd ertinly eyond the sope of the present work or ny ext digonliztion study @seeD eFgFD efF IP for disussion of the mtterAF 3.2 Two-and three-body systems with a magnetic impurity o provide si understnding out the role of the mgneti impurityD we disuss the systems 1 ↑ +1 ↓ nd 1 ↑ +2 ↓ in this susetionF sn generlD these systems re not solvle nlytilly @seeD eFgFD RRAD nd one hs to rely on numeril methodsF tillD s we show in eppF eD the limiting ses G → ∞ nd g → −∞ n e ddressedD providing some dditionl insight into the prolemF sn prtiulrD they show tht the groundEstte energy of the spin triplet9 1 ↑ +2 ↓ system n e lower thn the groundEstte energy of the spin singlet9 1 ↑ +1 ↓ system only if oth G nd |g| re su0iently lrgeF 3.2.1 Numerical results rereD we disuss our numeril results for smll systems with generl ouplings g nd GF o nlyze the potentil preursor of mnyEody D we fous on the energy di'erene whih fetures sign hnge when the prity of the groundEstte hngesF nless otherwise notedD we disuss results of the ext digonliztion method with e'etive potentilD see PFPFIF he presented groundEstte energies re otined y extrpoE lting to the in(niteEsis limit n b → ∞ ording to the funtionl form @see lso eppF fA . @SA rereD E ∞ denotes the extrpolted groundEstte energyY σ is the exponent of the onverE geneF hile σ ws found to e 0.5 for the onventionl re intertion RSD we empirilly found tht σ = 1.0 yields the most urte results for the e'etive intertion pproh used in this studyF o visulize the size of the extrpoltion e'etD we inlude results of di'erent extrpoltion shemes prmetrized y σ where pplileF e present ∆ 21 for g = 0 in pnel e of pigF QF he onstnt vlue of ∆ 21 my e strightforwrdly derived sine for g = 0 the system is olletion of nonEinterting fermions whose energy is sum of oneEody energiesF he vlue of ∆ 21 is given y the spinEdown prtile in the (rst exited stte of the hrmoni osilltorF his energy equls 3/2F st is independent of G QWD RHD due to the odd prity @nd hene node in the trp enterA of the (rst exited stteF sn pnel f of the sme (gureD we present ∆ 21 for the the opposite seD nmely G = 0 @iFeFD without the mgneti impurityA s funtion of the prtile intertion strength gF ine there is no reking of spinE)ip symmetry one would expet no rossover s funtion of intertion strengthD so tht the lned 1 ↑ +1 ↓ se remins the ground stte for ll ouplingsF woreoverD t g → −∞ the dimer is trnsprent for the fermion RTD thereforeD ∆ 21 is expeted to onverge to 1/2 in the limit |g| → ∞F es pprent from the (gureD these expettions re supported y our numeril dtF xoteD tht the urrent implementtion of the e'etive intertion pproh for this onE (gurtion is onstrined to the @lredy sustntilA intertion strength |g| 5D eyond whih the disrepny etween the vrious extrpoltion shemes eomes sizele @this re is mrked in gry in the (gureA nd eventully re too ontminted y (niteEsize efE fets to llow for relile sttementsF esults from the gw t (xed n b = 41 re presented in the sme pnel with yellow symols @nd dshed lineAF hese results show smooth onvergene to the expeted limiting vlue 1/2 for |g| → ∞F he gw dt vlidte

A B C
pigure QX Few-body systems with a magnetic impurity. nels show the di'erene etween the groundEstte energies of the 1 ↑ +2 ↓ nd 1 ↑ +1 ↓ systemsD ∆ 21 F @eA esults for vnishing fermionEfermion intertionsD g = 0F @fA esults for G = 0D iFeFD without mgneti impurityF rereD di'erent mrkers orrespond to three extrpoltion shemes @see iqF S nd eppF fAF he error rs show the ssoited (tting errorF he shded re mrks the prmeter regionD where the result strongly depends on the extrpoltion shemeF his prmeter region shll not e onsidered lterF he yellow disks re the results of the trnsorrelted method @gwAY the orresponding dshed urve is dded to guide the eyeF he ornge dshedEdotted line is our nlytil predition for the limit g → −∞F @gA qeneri se where G = 0 nd g = 0 @hereD g = −2.0AF e interpret the hnge in the sign of ∆ 21 s fewEody preursor of trnsition from the S = 0 to S = 1 2 ground stteF the e'etive intertion pproh nd our interpoltion sheme @see iqF @SAA with (xed exponent σ = 1F pinllyD in pnel g of pigF QD we illustrte the generi se when oth g nd G re (niteF he gpD ∆ 21 D is shown s funtion of G t (xed intertion strength g = −2F e oserve tht groundEstte level rossing ours when ∆ 21 = 0 @indited y the dotted gry lineAF por lrge vlues of GD the fermionEfermion piring pulls the energy of the 1 ↑ +2 ↓ system elow tht of the lned systemF ixittions etween the two setorsD whih involve the hnge of the fermioni prity @orD equivlentlyD hnge of prtile numerAD n e onsidered s fewEody ounterprt of the suEgp sttes tht our in the mnyEody limitF por ompletenessD we show results of di'erent extrpoltion shemes in pigF Q g s well s the dt otined with the gwF xote tht the point where ∆ 21 vnishes is @lmostA independent of the extrpoltion sheme for ll onsidered ouplingsD iFeFD ouplings outside the shded re in pnel pigF Q fF e omine the ove (ndings to produe the fewEody phse digrm9 in the g vsF G prmeter plneD see pigF RF he urve in the (gure is determined y the ondition ∆ 21 = 0F sn the re S = 1 2 9D the groundEstte energy of 1 ↑ +2 ↓ is elow tht of 1 ↑ +1 ↓F he opposite is true otherwiseF he three possile senrios illustrted in pnels eD fD nd g of pigF Q re indited y the lk rrowsD where the x9Ediretion of the ssoited pnel grees with the diretion of the rrowsF prom this nlysis it indeed eomes pprent tht oth ouplings g nd G need to e sustntil in order to enle groundEstte trnsition ndD heneD fewEody nlogue of F o estimte potentil systemti errorD we show our dt orresponding to σ = 0.5 nd 1 @sme olor sheme s in pigF QAD see lso eppF gF he phse digrm eyond the presented intertions g nd G requires more involved numeril nlysisF roweverD it will not hnge the tkewy pigure RX Few-body phase diagram with a magnetic impurity in the g − G planeF hi'erent mrkers orrespond to di'erent (t funtions s in pigF Q fF he three rrows shemtilly show the three ses disussed in pnels eD f nd g of pigF QF messge onveyed hereF pinllyD we note tht simple energy shift due to glol mgneti (eld @desried y Bσ z A my lso led to ∆ 21 < 0D howeverD the underlying physis would e very di'erentF sndeedD glol mgneti (eld would led to rossing of the energy levels regrdless of the ttrtive intertion strengthF ynly lol mgneti (eld tht we study here my led to sutle interply etween pir formtion nd spinEdependent sttering o' the impurityF 3.3 Approaching the many-body limit: scaling with particle number o fr we hve investigted the e'et of the mgneti impurity in the smllest possile system nd mpped out the fewEody phseEdigrm9 in the g − G plneF sn this setionD we lulte fewEody phseEdigrms9 for lrger systems @see lso eppF gAD whih llow us to study the rossover from fewE to mnyEody physis in the presene of mgneti impurityF Units. por meningful nlysisD hereD we need to hnge the size of the system long with the numer of prtiles 2 F por hrmoni trp potentilD it mkes sense to hnge the frequeny of the trp suh tht the density in the middle of the trp is (xedD ρ = k F /πD where 3 k F = (2N ↑ − 1)mω/ denotes the wve vetor ssoited with the permi energy E F = 2 k 2 F /(2m)D seeD eFgFD efF RUF uh hnge of the trp should llow for fithful omprison of physis due to the impurity in the middle of the trp @see lso efsF WD IU for relevnt disussion of moile impurities in permi gsAF sn prtieD we (x ωD nd study the fewE to mnyEody trnsition y resling the ouplings with the orresponding k . @TA por simpliityD we perform resling only for lned systemD iFeFD for N ↑ = N ↓ F por n imlned system @N ↓ = N ↑ + 1AD we useg ndG de(ned y N ↑ F pigure SX Approaching the many-body limit. oints where the groundEstte energies for lned nd imlned prtile on(gurtions re equlD iFeFD where E(N ↑ +N ↓) = E(N ↑ +(N + 1) ↓)F he dt re shown s funtion of the resled fermionEfermion intertion strengthg nd the resled mgneti impurity intertion strengthGD see iqF @TAF esults re shown for N ↑ ≤ 4D see the legendF sn ll ses the upper left @lower rightA re orresponds to the S = 1/2 @S = 0A phseF @eA hse diE grms otined from groundEstte energies extrpolted to the in(nite sis limitF rereD we lso inlude enhmrk results of the trnsorrelted method @gwA for the N ↑ = 1 nd N ↑ = 2F sn the former seD we relied on ext digonliztionF sn the ltter seD we used pgswg with 5 × 10 8 wlkersF sn oth sesD sis of 75 plne wves ws usedF @fA hse digrms t (nite uto' n b = 12 @olored symolsA ompred to the extrpolted results @gry symolsAF Numerical analysis. e extrt fewEody phseEdigrms9 from the ondition on the ground stte energiesX E(N ↑ +N ↓) = E(N ↑ +(N + 1) ↓)F pigure S presents our (ndings for systems with up to N ↑ = 4D iFeFD for systems s lrge s 4 ↑ +5 ↓F e note tht the trnsition points for N ↑ = 1 stnd out in omprison to the ones for higher prtile numers ! this is likely n rteft of our system of units whih merely (xes sle relevnt in the mnyEody limitF por lrger prtile numersD lthough not fully onvergedEtoEmnyE odyElimitD the dt otined with extrpolted groundEstte energies seem to ollpse on urveD whih is lmost independent on N ↑ D s shown in pnel e of pigF SF por exmpleD the dt for the systems with N ↑ = 2 nd N ↑ = 3 re very loseF yur interprettion is tht the impurity is sreened y only few fermionsF husD inluding more prtilesD therey inresing the size of the systemD nnot hve strong e'etF imilr ehvior ws lso oserved in other impurity systemsD see for exmple IUF pinllyD we omment on the ury of the presented dtF he mximlly ttinle uto' vlue n b is redued when we inrese the prtile numerF roweverD we n reh su0ient onvergene ross rnge of ouplings @see lso QIAF o e spei(D for N ↑ = 1 we ompute dt using up to n b = 40 sttes in the singleEprtile sis while for N ↑ = 4 we use only n b = 8 to 13 oneEody sttes 4 F yur vlues up to systems of size 3 ↑ +4 ↓ re wellEonverged nd extrpoltion is under ontrolF por lrger systemsD the ext vlues my shift slightlyF roweverD the otined ury is enough for qulittive disussionF his point is further ddressed in pnel f of pigF SD where we show the trnsition points t (nite uto' n b = 12 for ll prtile on(gurtionsF he pnel lso shows lustering of the | | = . | | = . | | = .

A B C
pigure TX Density proles. he (gure illustrtes the density pro(les for the 2 ↑ +2 ↓ system without fermionEfermion intertions @iFeFD with g = 0A for representtive vlues of GD see the legendF @eA hensity of spinEup prtilesD ρ ↑ F @fA hensity of spinEdown prtilesD ρ ↓ F @gA otl densityD ρ = ρ ↑ + ρ ↓ F sn ll pnels the lk dshed line orresponds to nonEinterting fermions without n impurityF dtD whih llows us to onlude tht more urte determintion of the groundEstte energies will not hnge the generl ehvior of the phseEdigrmF e provide further understnding of the phseEdigrm in eppF hD y ompring to the results ville in the mnyEody limitF 3.4 Other observables o this momentD we hrterized the fewEody systems only y their energiesF his lE lowed us to show similrities etween our model nd mgneti impurity in the viinity of superondutorF roweverD the energies do not provide omprehensive understnding of prtileEprtile orreltionsD whih is required to explin the physil mehnism ehind the oserved trnsitionF rereD we provide further insight into the system y onsidering exE perimentlly relevnt proes tht do not diretly onern the energies 5 F hese oservles shed dditionl light on the detils of the physis of our modelF sn the followingD we present the density pro(les of the tomi louds s well s densityE density orreltion funtions @iFeFD the shot noiseA in relE nd momentum speF hese quntities my e mesuredD for exmpleD y postEproessing timeEofE)ight @ypA imges of fewEody systems QVD RV!SHF 3.4.1 Spatial density proles en nlysis of density pro(les is n intuitive nd ommon wy of studying the ehvior of trpped permi systemsF por suh n nlysisD one lultes the spinEresolved densityX where ρ σ kl = ψ|k l|ψ denotes the groundEstte expettion vlues @|ψ is the groundE stte wve funtionA of the oneEody density mtrixD whih enodes oneEody orreltions etween singleEprtile oritlsD {φ k }D of the hrmoni osilltorF sn pigF TD we show the density for the spinEup nd spinEdown prtiles s well s the totl density for the 2 ↑ +2 ↓ systems with vrious impurity strengths GF por simpliityD 5 It is worth noting that the use of the eective interaction in exact diagonalization leads to a fast convergence not only of the energy but also of other observables, in particular, of the one-body density matrix (see Ref.
[31] for a benchmark). Therefore, we do not need to change the numerical routine in this section. the fermionEfermion intertion is omittedD iFeFD g = 0D sine we oserved tht the fetures of the density pro(les depend wekly on the onsidered vlues of gF sn pigF T eD note suppression of the ↑Edensity lose to x = 0 sine the impurity repels prtiles of this spin typeF sn pigF T fD we illustrte pek in the ↓Edensity tht grows with the vlue of GD inditing the presene of ound stte of spinEdown fermion with the impurityF ithout trpD t the singleEprtile levelD suh ound stte is desried y n exponentil wve funtion ψ(y 1 ) ∼ e −m|Gy 1 | , @VA whih lerly exhiits usp t the originF he orresponding densityD ρ(y 1 ) = |ψ(y 1 )| 2 ∼ e −2m|Gy 1 | D inherits this uspF st is impossile to hve usp in the density in numeril lultions sed upon the smooth sis set given y the hrmoni osilltor eigenfunE tionsF xeverthelessD even in the restrited sis the emergene of the ound stte eomes lerly visile in ρ ↓ F xoteD tht tehnilly the suppression of ρ ↑ round the origin should lso exhiit uspD howeverD this is smered out for the sme resonsF he emerging pek signls the lol distortion of the wve funtion round the impurity nd serves essentilly s n intuitive illustrtion of the e'et of the mgneti impurity menle for experimentl detetionF roweverD to gin more insights into the physis tht drives the trnsition presented in pigF SD one requires higherEorder orreltion funtions tht n show the e'et of the interply etween g nd GF felowD we present the densityE density orreltors whih ontin informtion out this interplyF 3.4.2 Density-density correlations rereD we nlyze densityEdensity orreltion funtions in oth rel nd momentum speF iringD whih hppens when the prtiles intert ttrtivelyD leves distint imprints on these quntities in momentum speD see for exmple efsF SI!SU where relevnt permi systems re disussedF hereforeD densityEdensity orreltion funtions n further orroorte the physil piture of the fewEody preursor of F he densityEdensity orreltion funtion in rel spe is de(ned s whereρ σ is the sptil density opertor ! its expettion vlue ws the sujet of the previous susetionF enlogouslyD in momentum speD we write G k ↑↓ (p, q) = n ↑ (p)n ↓ (q) − n ↑ (p) n ↓ (q) , @IHA wheren σ is the momentumEspe density opertor 6 F he quntities G r ↑↓ nd G k ↑↓ vnish in the se of vnishing prtileEprtile intertions @sttistilly speking the densities would e unorrelted rndom vrilesAF hereforeD the densityEdensity orreltion funtions mrk idel ndidtes for investigting e'ets indued y intertionsF e expet tht orreltions in momentum spe re in prtiulr useful for our purposesD sine the fg piring is most esily visulized in momentum speF sn pigF U eD we illustrte G r ↑↓ for g = −2.0 for the 2 ↑ +2 ↓ nd 2 ↑ +3 ↓ systems with @G = 5.0A nd without @G = 0A the mgneti impurityF e see tht the sptil orreltions in 2 ↑ +2 ↓ nd 2 ↑ +3 ↓ re similrF sn the limit of strong impurityEfermion intertions @G = 5.0AD they oth feture suppression t the enter of the trpX et the enter there is no sttistil orreltion etween the spinEup nd the spinEdown prtilesD nd the system e'etively prtitions into two opies @left nd right of the impurity respetivelyA with no rossEorreltions etween the sidesD see lso the disussion in eppF eF 6 Note that G r ↑↓ and G k ↑↓ are not simply Fourier-transforms of each other. Fourier transform of the spatial density-density correlation function would give the structure factor. pigure UX Density-density correlation functions. @eA el spe orreltion funtionD G r ↑↓ . @fA womentumEspe orreltion funtionD G k ↑↓ F ell pnels hve g = −2.0D with @G = 5.0D right olumnsA nd without @G = 0D left olumnsA the mgneti impurityF he top row is for the 2 ↑ +2 ↓ systemY the ottom row is for 2 ↑ +3 ↓F he dshed lines in pnel f show the ntiEdigonl k ↓ = −k ↑ where piring in the lned se is expetedF sn oth pnels red nd lue vlues orrespond to positive nd negtive orreltionD respetivelyF he olormps re in ritrry units nd normlized suh tht the unorrelted vlues orrespond to 0 @whiteA s well s mximl positive orreltion orresponds to 1F IS sn pigF U fD we illustrte G k ↑↓ for the sme systemF he hnge of G k ↑↓ (p, q) with G highlights the use of the fewEody preursor of in terms of piringF por the 2 ↑ +2 ↓ system with G = 0D we see strong orreltions on the ntiEdigonl line k ↓ = −k ↑ @indited s dshed line in pigF U fAF e interpret this s preursor of the fg mehnismD in whih pirs re formed etween the spinEup nd spinEdown fermions t equl ut opposite moment @iFeFD t the zero enterEofEmss momentumAF por imlned systemsD suh s the 2 ↑ +3 ↓ system shown in the ottom left frmeD piring is expeted to hppen t the respetive permi points @eyond the fewEody regimeA whih results in the two mxim t (±k F ↑ , ∓k F ↓ ) whih re now o'set with respet to the ntiEdigonlF sn the presene of strong impurityEfermion intertion @right olumn of pnel fAD the sitution is opposite to tht disussed in the previous prgrphF he lned system exhiits two mxim tht re slightly shifted wy from the k ↓ = −k ↑ lineF sn ontrstD the 2 ↑ +3 ↓ system fvors orreltions long the ntiEdigonl lineF he reson for this ehvior is the inding of spinEdown prtile to the impurityD whih turns the 2 ↑ +3 ↓ system into n e'etively lned system where fgElike pirs n e formedF 4 Conclusions sn this setionD we summrize our (ndings nd give n outlookF e lso provide rief disussion of possile experimentl systemF 4.1 Summary & Outlook sn this work we presented numeril investigtion of fewEfermion system in the presene of spinEseletive potentilD whih fetures some physis of quntum phse trnsition driven y mgneti impurity in the viinity of n sEwve superondutorF sn prtiulrD one n lower the energy of the system y introduing slight spin imlneF his ehvior is used y the ompetition of pir formtion due to the ttrtive fermionE fermion intertion nd pir reking due to the mgneti impurityF fy tuning the strength of fermionEimpurity intertionD one is le to study the rossover etween di'erent ground sttesF fesides eing of interest y itselfD the system under study is si uilding lok required to engineer more sophistited setEups inluding two or more impurities SV!TIF sn suh systemsD for smll enough sping etween the impuritiesD the ound sttes hyridize nd form suEgp hi nd TPD TQF sf in ddition spinEorit oupling is present in the host system suh setup my relize topologil superondutorF sn this seD one n potentilly oserve wjorn edge modes t the ends of the hyridized impurity hins TRD TSF puture works should fous on engineering oldEtom fewEody experiments tht n study the physis of one nd two mgneti impuritiesD nd omplement studies of mnyEody systemsD seeD eFgFD TTF e outline some relevnt ides in the next susetionF pinllyD we note tht in the present studyD the numer of prtiles ws (xed quntity preluding trnsitions etween the setors with S = 0 nd S = 1 2 F o study quntum )ututions typil for 9sD one should onsider senrio in whih these setors re oupledF por exmpleD this n e hieved if the spinEup nd spinEdown setors re onneted vi third hyper(ne stte tht is not intertingF ime dynmis in this setEup my ontin typil signtures of 9sF xote tht timeEdependent simultions of fewE ody systems eome possile TUD TVD llowing one to study time evolution in the viinity of @fewEody preursors ofA phse trnsitionsD seeD eFgFD TWF IT 4.2 Experimental considerations he rmiltonin in iqF @IA with H imp = 0 is often used to model oldEtom experiments in qusiEoneEdimensionl geometriesD see efF TDUDUH nd referenes thereinF sn prtiulrD its ury hs een tested for fewEody systems of 6 vi IUDUIF he min new element of our work from the stndpoint of those experiments is the presene of mgneti impurity9D nd we need to disuss it in some detilF he onsidered spinEseletive potentil my e nturl for systems with lrge mss imlneD eFgFD 6 viE 133 gsD lose to fvorle peshh resonneF his possiility ws very reently disussed in efF UPF felowD we shll rie)y outline two other ides for relizing H imp F yne my engineer H imp y following the ide of n tomi quntum dot UQF rereD the mgneti impurity is single tom in tight optil trpF por the ide to workD the impurity tom should e di'erent from the spinEup nd spinEdown fermionsF por instneD it ould e fermion in di'erent hyper(ne stte @not ↑ or ↓A or prtile with di'erent mss @fF URAF sn this implementtionD the fermionEimpurity intertion is of short rnge nturllyD see efF US for more informtionF purthermoreD its strength n e tuned in stndrd wy using n externl mgneti (eldF roweverD there re few disdvntges of this pprohX @iA wirotrps typilly hve µmEwidthD whih implies tht the zeroEenergy motion of the trpped tom needs to e onsideredF @iiA heterministi preprtion of suh setEups with known @smllA numer of prtiles hs not een yet demonstrtedF he item @iA does not led to muh more omplited nlysisF fy ontrstD the item @iiA my led to typil mnyEody prolem whose investigtion we leve for future studiesF elterntivelyD one my use spinEdependent potentils to mimi H imp F por exmpleD efF UT demonstrtes tht it is possile to tune g nd G independently of eh otherD nd even relize G ↑ = −G ↓ s we hve in our lultionsF he e'ets due to the (nite width of spinEdependent potentils n e esily tken into ount in our lultionsF roweverD we expet tht (nite width of the mgneti impurity does not hnge the min onlusions of this workD s long s this width is smller thn the length sle given y the permi momentumF nfortuntelyD photon sttering indues lossesD whih re not inluded in the present theoretil modelF essuming tht it will e possile to deterministilly prepre fewEody system nd spinEseletive potentilD one ould ount for losses y post seleting timeEofE)ight imges tht hve the desired numer of tomsF sn generlD in the presene of lossesD one needs to (nd suitle experimentl protoolF sn prtiulrD it seems nturl to fous on G k ↑↓ D whih should ontin tres of piring even in nonE equiliriumF e leve more elorte investigtion of this question to future studiesF Acknowledgements e knowledge fruitful disussion with ereg qhzrynD hilipp reissD elim tohim nd his group in reidelergF e thnk ereg qhzryn nd pin fruneis for omments on the mnusriptD ietro wssignn for shring with us the dt for enhmrking our numeril results without the impurityD nd éter teszenski for support with implementing the gwF pigF I ontins resoures y Pixel perfect from pltionFomF Author contributions vFF nd hFrF ontriuted eqully to this workF Funding information his work hs een supported y iuropen nion9s rorizon PHPH reserh nd innovtion progrmme under the wrie kªodowskEgurie qrnt egreeE ment xoF USRRII @eFqFFAY y the heutshe porshungsgemeinshft through rojet y PRQUGIEI @rojektnummer RIQRWSPRVA @eFqFF nd rFFrFAY y the heutshe porshungsE gemeinshft through gollortive eserh genter pf IPRS @rojektnummer PUWQVRWHUA nd y the fundesministerium für fildung und porshung under ontrt HSPIhpxf @rFFrAF vFF is supported y pUGig gonsolidtor qrnt swgyD xoF UUIVWID nd the heutshe porshungsgemeinshft @hpqD qermn eserh poundtionA under qermny9s ixellene trtegy !ig!PIII!QWHVIRVTVF he work ws prtilly supported y the Marsden Fund of xew elnd @gontrt xoF we PHHUAD from government funding mnged y the oyl oiety of xew elnd e ep© rngi @tFfFAF e lso knowledge support y the xew elnd eiene snfrstruture @xesA highEperformne omputing filities in the form of merit projet llotionF A Limiting cases: analytic insight into strong coupling A.1 Limit G → ∞ e (rst onsider the 1 ↑ +1 ↓ system with strong fermionEimpurity intertionsF sf g = 0D then the rmiltonin in iqF @IA does not ouple spinEup nd spinEdown prtilesF hereE foreD the energy of 1 ↑ +1 ↓ equls the energy of 0 ↑ +1 ↓ plus the energy of 1 ↑ +0 ↓F sn the 0 ↑ +1 ↓ systemD the fermion nd the impurity form tightlyEound stteD whose energy is G nd n e lulted nlytilly s in efsF QWD RHF xote tht for lrge vlues of GD the hrmoni trp plys minor role nd G −G 2 /2 in greement with the textook lultionsF sn the 1 ↑ +0 ↓ systemD the fermion feels n impenetrle wll in the middle of the trpF he orresponding wve funtion must vnish t x 1 = 0D whih mens tht the groundEstte energy is equl to tht of the (rst exited stte of the hrE moni osilltorD iFeFD to 3/2F hereforeD the energy of 1 ↑ +1 ↓ is G + 3/2F pinEup nd spinEdown prtiles hve no overlp if G → ∞D whih implies tht (nite vlues of g do not hnge the energy of the systemF por the 1 ↑ +2 ↓ systemD the lultions re more omplitedF por g = 0D the groundE stte energy n e lulted y onsidering the systems 1 ↑ +0 ↓ nd 0 ↑ +2 ↓ seprtelyF e otin G + 3D whih is lrger thn the energy of the 1 ↑ +1 ↓ systemF roweverD s we show elowD there is vlue of g for whih the energy of 1 ↑ +2 ↓ is equl to the energy of 1 ↑ +1 ↓F his ritil vlue of g is of our interest hereF por g = 0D the 1 ↑ +2 ↓ system n e e'etively desried using n uxiliry 1 ↑ +1 ↓ prolem in hrmoni trp with n impenetrle wll in the middle for oth spinsF he spinEup fermion feels wll due to the ondition G → ∞F he spinEdown fermion nnot go to the origin due the seond spinEdown fermion lredy ttrted y the mgneti impurityF he uxiliry prolem is desried y the rmiltonin with the oundry ondition ψ(x 1 = 0, y 1 ) = ψ(x 1 , y 1 = 0) = 0F por g > 0D the rmiltoE nin H A ws studied in efF RRF rereD we re interested in the se with g < 0F o provide some nlytil insight into the prolemD we im to (nd n pproximte vlue to the energy using vritionl nstz for the rmiltonin in the polr oordintes @x 1 = r cos φ, y 1 = r sin φAX where for the ground stte we shll only onsider 0 < φ < π/2D sine the wve funtion vnishes t the oundriesD iFeFD t φ = 0 nd φ = π/2F e suitle vritionl funtion IV reds s f = Ae −r 2 /2 r µ F (φ), @IQA where A is the normliztion onstntD µ is the vritionl prmeterD nd the funtion F hs the form sin(µ(π/2 − φ)) if φ ∈ (π/4, π/2]. @IRA st stis(es the oundry ondition F (0) = F (π/2) = 0 y onstrutionF e few omments out the vritionl funtion in iqF @IQA re in order hereF ith µ = 2D the funtion f solves the prolem t g = 0F por other vlues of µD the funtion ounts for the singulrity due the deltEfuntion intertionF yur hoie of F is motivted only for smll vlues of gF por lrger vluesD funtion tht more fithfully represents ound stte might e neededF por more detiled disussion on the physis of the employed vritionl nstzD we refer to efF UUF por the vritionl funtion from iqF IQD the expettion vlue of H A is where Γ is the qmm funtionF e minimize f |H A |f with respet to µD nd otin n pproximtion to the groundEstte energyF he orresponding pproximtion to the energy of the 1 ↑ +2 ↓ system is G + f |H A |f F por g = 0D the minimum of f |H A |f equls 3D nd it is rehed for µ = 2D s expetedF por g −1.95D the minimum of f |H A |f equls 1.5D the orresponding vlue of µ is pproximtely 1.3F o summrizeD for g −1.95D the uli pressure mkes the energy of the 1 ↑ +2 ↓ system higher thn the energy of the 1 ↑ +1 ↓ systemF por g −1.95D the fermionEfermion piring mkes the 1 ↑ +2 ↓ system energetilly more fvorle thn 1 ↑ +1 ↓F elthoughD the ritil vlue is otined here using numer of pproximtionsD we see from the results presented in the min text tht it is resonle @ompre it to the numeril results sed upon ext digonliztion demonstrted in pigF RAF A.2 Limit g → −∞ rereD we onsider the system with strong fermionEfermion intertionF he 1 ↑ +1 ↓ system with G = 0 ws investigted in efsF QWD RHF st is most esily solved y deoupling the reltive motion from the enterEofEmss oordinteD see iqF @RAF he intertion enters only in the former prtY it leds to formtion of tightly ound dimerD whose energy is g F he @totlA groundEstte energy of the 1 ↑ +1 ↓ system is g + 1/2D where 1/2 is the zeroEpoint energy of the enterEofEmss rmiltoninF por the 1 ↑ +2 ↓ systemD the ound stte eomes trnsprent to the extr fermion RTF sts energy is thus g + 1F sf we turn on GD then the net e'et of the perturing potentilD Gδ(x 1 ) − Gδ(y 1 )D on the dimer of the 1 ↑ +1 ↓ system is zeroF sndeedD the dimer is tightly oundD nd whenever the potentil Gδ(x 1 ) ts on spinEup prtileD the potentil −Gδ(y 1 ) ts on spin down prtileF his mens tht g + 1/2 is n urte pproximtion to the energy of the 1 ↑ +1 ↓ system lso for (nite vlues of GF o investigte the 1 ↑ +2 ↓ systemD we n mke use of the previously mentioned trnspreny of the stronglyEound dimer to n extr fermionF hereforeD the energies of 1 ↑ +1 ↓ nd 1 ↑ +2 ↓ re equl when the energy of the 0 ↑ +1 ↓ system vnishesD whih hppens t @see efsF QWD RHA G = 2 Γ[3/2] Γ[1/4] 0.5. @ITA ell in llD the onsidered limiting ses suggest urve in prmeter spe g − G tht seprtes the S = 0 from S = 1 2 setorsF sn the min textD we (nd this urve numerillyD see pigF RF A.3 Limiting cases for the many-body system por lrge systemsD we n nlyze the limiting se g → −∞ following the disussion oveF he limit G → ∞ requires more involved lultionsD nd we leve its investigtion to future studiesF por g → −∞D spinEup nd spinEdown fermion form dimerF he lned system is then equivlent to the onksEqirrdeu gsD seeD eFgFD efF UVF he N ↓ = N ↑ + 1 system hs n dditionl fermionD whih does not intert with the onksEqirrdeu gsF he sene of fermionEdimer intertion mens tht the ritil vlue of G is independent of the size of the systemF hereforeD we n use the result of iqF @ITAF sn the resled unitsD this eqution n e written sG 0.5 2N ↑ − 1 . @IUA xote tht this vlue vnishes for the mnyEody system @N ↑ → ∞AF B Extrapolation routine for the eective interaction sn this ppendixD we rie)y outline our routine for extrpolting the groundEstte energies tht we otin with (nite n b to the limit n b → ∞F o this endD we employ the method of lest squresX we ompute energies for severl vlues of n b nd (t them to the funtionl form where a nd E ∞ re (t prmeters nd σ is the exponent tht determines the rte of onvergene to the true9 groundEstte energy with inresing the singleEprtile uto'D n b F sn prinipleD when working with su0iently mny uto' vluesD the est proedure is to inlude σ s n open prmeterF roweverD when the numer of dt points is rther smll @s is the se for lrger systems where vlues re expensive to otinA this strtegy will led to extrpolted energies tht re fr from the ext solution due to only few vlues in the sling windowF yftenD etter hoie here is to (x σD idelly relying on some theoretil insightsF es mentioned in the min textD for the se of re intertions in Ih it ws shown thtD t leding orderD groundEstte energies onverge to the in(niteEsis limit with σ = 0.5 @seeD eFgFD RSAF he leding order exponent for the e'etive intertion is not knownD howeverD empirilly we oserved tht σ > 1 2 nd the est results hve een otined with σ 1D s found y omprison to the trnsorrelted method nd the ext solution for the 1 ↑ +1 ↓ prolemF hereforeD we employ σ = 1 unless otherwise notedF st is worth noting tht not only the vlue of σ ut lso the vlue of the preftor a is oserved to e more fvorle for the e'etive intertionD ompred to the onvergene properties for the re intertionF sn mny sesD the fvorle onvergene properties would llow one to skip extrpoltion n b → ∞ ltogether within resonle uryF he results of our (tting proedure re summrized in pigF VX sn pnel e we show omprison of the three extrpoltion routines mentioned ove for vlue of g = −4.0D where it is pprent tht the (xed vlue σ = 1.0 yields the est resultsF nel f of the sme (gure shows the reltive error of our extrpolted ground stte energy for the 1 ↑ [ , ]

A B C
pigure VX Extrapolation of eective interaction results without magnetic impurity for the 1 ↑ +1 ↓ systemF @eA ixtrpoltion of the groundEstte energy to the limit n b → ∞ for g = −4.0 for σ = 0.5 nd 1.0 @lue dshedEdotted nd red solid urvesD respetivelyA s well s for open σ @green dshed urveAF he lk dshed line re)ets the ext twoEody solutionF @fA eltive error of the extrpolted result with respet to the ext solution @olor oding s in eAF he vertil dshed line mrks zeroErossing of the groundEstte energyF @gA pitted vlues for σ when it is unonstrined s funtion of the intertion strength gF the fermionEfermion intertion gF he lrge reltive error t the gry line is used y zeroErossing of the energyF he low reltive unertinty further shows tht σ (xed to 1.0 yields the est results for ll onsidered ouplings strengthsF pinllyD in pnel g of pigF V the extrted vlues for n open σ prmeter re plotted for the sme rnge of intertion strengthsF por g −2D we used uto' vlues in the rnge [20, 30] for extrpoltionF por stronger ttrtive intertions we used [20, 40]F xote tht the prmeter σ in pigF V g is lose to IF B.1 Extrapolation for systems with a magnetic impurity sn the presene of mgneti impurityD iFeFD when G = 0D slight omplitions riseF pirst of llD n oddEeven stggering 7 s funtion of the uto' prmeter n b is oservedD whih n e mitigted y seprtely extrpolting the vlues of odd nd even uto' vluesF he resulting extrpolted groundEstte energies hve een oserved to lie within the hievle unertintiesF eondlyD n interply etween the di'erent intertions in the rmiltonin my led to distint ehviorX for smll impurity strength G the dt onverges from ove with inresing uto' wheres t lrge impurity strength onvergene from elow is foundF his ehvior is shown for 1 ↑ +1 ↓ system t g = −2.0 in pnels e nd g of pigF WF woreoverD this entils region where results re virtully independent of the uto'D s shown in pnel B of the sme (gure @note the tiny extent of the yExis in ll sesAF sn suh se the dt my not e (tted with simple powerElwF hereforeD we merely verge the ville dtpoints to otin our extrpolted resultF his ehvior is n rteft of the e'etive intertion pprohD sine the digonliztion with re intertion yields vritionl energies even with (nite vlues of G nd hene should lwys disply onvergene from oveF 7 To understand this note that the odd basis functions cannot feel the potential at the origin.

A B C
pigure WX Extrapolation of eective interaction results in the presence of a magnetic impurity for 1 ↑ +1 ↓ t g = −2.0F @eA gonvergene from ove without the impurityF @fA ek uto' dependene in the viinity of the rossover pointD symptoti vlues re indited y dotted linesF @gA gonvergene from elow when G is lrgeF pigure IHX Phase diagram for individual cuto values. hse digrm for the 1 ↑ +1 ↓ vs 1 ↑ +2 ↓ for individul uto' vlues shown y the shded reF he upper edge of the re shows energies for uto' 20D the lower edge for uto' 30 @g > −2.0A nd 40 @g < −2.0AF he extrpolted results with the (xed exponent σ = 0.5 @σ = 1A is shown y the solid lue @dshed redA urveF C Additional data sn this ppendix we present dditionl dt for the fewEody phseEdigrms t vrious uto' vlues nd prtile numersF C.1 Few-body phase diagram for individual cuto values o provide dditionl insight into (niteEuto' e'etsD we here ompre the phseEdigrm otined with (nite singleEprtile uto's to the extrpolted fewEody @iFeFD 1 ↑ +1 ↓ vs 1 ↑ +2 ↓A phseEdigrm tht ws lredy shown in pigF Q of the min textF es pprent from pigF IHD the overll form of the phseEdigrm does not hnge drstilly t (nite uto' vlues @vlues etween the lowest nd highest uto' n b re summrized y the green semiEtrnsprent ndA s ompred to the extrpolted resultF he ltter is shown for extrpoltions n b → ∞ with the oe0ients E 0 ∝ n −0.5 b @solid lue lineA nd E 0 ∝ n −1 b @dshed red lineAD respetivelyF elthough the ext vlues for the di'er slightlyD in prtiulr t intermedite prtile oupling nd impurity strengthD the pigure IIX Few-body system interacting with an impurity. he gp |∆ 32 | etween the groundEstte energies in the 2 ↑ +2 ↓ nd 2 ↑ +3 ↓ setorsF sn pnel e @no fermionE fermion intertionsD iFeFD g = 0A nd f @no impurityD iFeFD G = 0A the gp is lwys openF sf G = 0 nd g = 0 @pnel gA the ground stte of the system my trnsition from the S = 0 to S = 1 2 setorsF @hA pewEoy phse digrm in the g!G plneD dt points re otined y extrpolting to the limit n b → ∞F overll form is onsistent etween ll versionsF sn onlusionD within the e'etive intertion frmework the qulittive fetures of the phse digrm do not depend strongly on the singleEprtile uto' n b in the prmE eter rnge up to |g| ∼ 6.0 t lestF his oservtion rries over from the doumented onvergene properties of the e'etive intertion pproh for more generl fewEody systems QIF C.2 Crossover between the 2 ↑ +2 ↓ and 2 ↑ +3 ↓ sectors sn ddition to the nlysis shown in the min text s well s in the preeding susetionD we here show some omplimentry dt for the fewEody in 2 ↑ +2 ↓ nd 2 ↑ +3 ↓ setors to highlight the similrities of this system to the smllest setEup with single spinEup prtileF sn pigF II we present |∆ 32 |D iFeF the energy di'erene etween the lowest energy level of eh setorD de(ned nlogously to etF QFPFI of the min textF he limiting ses of G > 0, g = 0 nd G = 0, g < 0 re shown in pnels e nd fD respetivelyF es lredy oserved for the nlogue system of fewer prtilesD there is no groundEstte level rossing nd the 2 ↑ +2 ↓ setor remins the lowest in the energy for ll ses where one of the oupling strengths vnishesF sn pnel g of pigF IID the gp to the lowest energy level is shown for oth setors s funtion of the impurity strength G for onstnt prtile intertion strengthF sn dditionD in pnels f nd g the di'erent types of symols re)et results t vriE ous singleEprtile uto' vluesF he smll spred in energy indites tht the e'etive intertion pproh is well onverged in the proed regime nd produes quntittively meningful resultsF sn the present se t g = −3.0D only t very lrge impurity strength one is le to distinguish the vlues for di'erent uto's y eye on this sleF pinllyD in pnel h of pigF II we show the resulting fewEody phseEdigrm t (xed uto' of n b = 12F he gry symols re)et the tul dt pointsD the rrows indite the lines overed in pnels e E g of the sme (gureF D Connection to the many-body model rereD we ttempt to onnet the results presented in pigF S to the mnyEody limitF e mnyEody rmiltonin tht n e used to study mgneti impurity in the viinity of k,σ nd a k,σ re retion nd nnihiltion opertorsY ∆ M B is the energy gp in the spetrum of the mnyEody systemY J determines the strength of the impurityEfermion intertionF xote tht the numer of prtiles is not onserved hereF sn the thermodynmi limitD it is strightforwrd to onnet the prmeter J in iqF @IWA to GX G = −J/(4ρ)Y the prmeter G is de(ned in iqF @PA vi G ↑ = −G ↓ ≡ G > 0D nd ρ is the density of spinEup prtiles @ρ would e equl to N ↑ /L for ox potentil of length LAF he rmiltonin in iqF @IWA is qudrti nd hene extly solvle PID PPF st is esy to show tht for |J| < 4πρ 2 D the ground stte of H hs S = 0F por lrger vlues of |J|D the ground stte hs S = 1 2 F xote tht the trnsition point etween the sttes with di'erent fermioni prities does not depend on the vlue of ∆ M B F vet us ompre this predition to the results presented in pigF @SAF o this endD we (rst oserve tht in our lultions the di'erene etween the energies of systems with N ↓ nd N ↓ + 1 in the limit N ↓ 1 is given not only y ∆ M B ut y ∆ M B + π 2 ρ 2 /2 8 F sing this oservtionD we present our results in the ∆ M B − G plneD see pigF IPF e hve ssumed tht the urve with N ↑ = 2 pproximtes the mnyEody limit wellF o onnet our vlue of g to ∆ M B D we hve used the expression where the (rst term ws derived in the fg limit using the fethe nstz QTD nd the seond term is due to the dditionl permi energy present in our seF e simple sutrtion of these energies is not rigorousF st follows from the oservtion tht point of trnsition |J cr | = 4πρ 2 does not depend on ∆ M B F sn spite of numer of ssumptions mde long the wyD our dt grees well with wht is expeted from the mnyEody rmiltonin @IWAD supporting our interprettion of the resultsF he di'erene n e ttriuted to the fg formultion of the rmiltonin H s well s to the (niteEsize e'ets present in our dtF wore importntlyD there is di'erene due to di'erent formultions of the prolem ! in our se we dd the permi energy when we dd spinEdown prtileF 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orld ienti(D doiXIHFIIRPGSSSP @PHHRAF RU pF qleisergD F onneergerD F hlöder nd gF immermnnD Noninteracting fermions in a one-dimensional harmonic atom trap: Exact one-particle properties at zero temperatureD hysF evF e 62D HTQTHP @PHHHAD doiXIHFIIHQGhyseveFTPFHTQTHPF RV eF fergshneiderD F wF ulinkhmerD tF rF feherD F ulemtD qF ürnD F wF reiss nd F tohimD Spin-resolved single-atom imaging of 6 Li in free spaceD hysF evF e 97D HTQTIQ @PHIVAD doiXIHFIIHQGhyseveFWUFHTQTIQF RW eF fergshneiderD F wF ulinkhmerD tF rF feherD F ulemtD vF lmD qF ürnD F tohim nd F wF reissD Experimental characterization of two-particle entanglement through position and momentum correlationsD xtF hysF 15@UAD TRH @PHIWAD doiXIHFIHQVGsRISTUEHIWEHSHVETF SH wF roltenD vF fyhD uF urmninD gF reintzeD F wF reiss nd F tohimD Observation of Pauli crystalsD hysF evF vettF 126D HPHRHI @PHPIAD doiXIHFIIHQGhysevvettFIPTFHPHRHIF SI vF wtheyD iF eltmn nd eF ishwnthD Noise correlations in onedimensional systems of ultracold fermionsD hysF evF vettF 100D PRHRHI @PHHVAD doiXIHFIIHQGhysevvettFIHHFPRHRHIF SP eF vüsherD eF wF väuhli nd F wF xokD Spatial noise correlations of a chain of ultracold fermions: A numerical studyD hysF evF e 76D HRQTIR @PHHUAD doiXIHFIIHQGhyseveFUTFHRQTIRF SQ eF vüsherD F wF xok nd eF wF väuhliD Fulde-Ferrell-Larkin-Ovchinnikov state in the one-dimensional attractive hubbard model and its ngerprint in spatial noise correlationsD hysF evF e 78D HIQTQU @PHHVAD doiXIHFIIHQGhyseveFUVFHIQTQUF SR hF ¦k nd F owi«skiD Intercomponent correlations in attractive onedimensional mass-imbalanced few-body mixturesD hysF evF e 99D HRQTIP @PHIWAD doiXIHFIIHQGhyseveFWWFHRQTIPF SS hF ¦k nd F owi«skiD Signatures of unconventional pairing in spin-imbalanced one-dimensional few-fermion systemsD hysF evF eserh 2D HIPHUU @PHPHAD doiXIHFIIHQGhyseveserhFPFHIPHUUF ST vF mmelmüllerD tF iF hrut nd tF frunD Pairing patterns in onedimensional spin-and mass-imbalanced Fermi gasesD iost hysF 9D IR @PHPHAD 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