Doping control of realization of an extended Nagaoka ferromagnetic state from the Mott state Hiroaki Onishi∗ Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan Seiji Miyashita† Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan and CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan 5 (Dated: January 13, 2015) 1 0 Inspired by the Nagaoka ferromagnetism, we propose an itinerant model to study the transition 2 between the Mott singlet state and a ferromagnetic state by emulating a doping process in finite lattices. In the Nagaoka ferromagnetism, the total spin of the system takes the maximum value n whenanelectronisremovedfrom thehalf-filled system. Toincorporate aprocedureoftheelectron a removal, our model contains extra sites as a reservoir of electrons, and the chemical potential of J the reservoir controls the distribution of electrons. As a function of the chemical potential, the 3 system exhibits ground-state phase transitions among various values of the total spin, including a 1 saturated ferromagnetic stateduetotheNagaoka mechanism at finiteholedensity. Wediscuss the nature of the ferromagnetism by measuring various physical quantities, such as the distribution of ] l electrons, the spin correlation functions, the magnetization process in the magnetic field, and also e theentanglement entropy. - r st PACSnumbers: 75.10.-b,75.45.+j,71.10.Fd . t a m I. INTRODUCTION the ferromagnetism at finite hole densities. In a two-leg ladder, numerical results have shown that the saturated - d ferromagnetic ground state keeps stable up to a critical The itinerant ferromagnetismis inherently a quantum n hole density.17–19 The critical hole density is insensitive o phenomenoninwhichtheelectroncorrelationisessential. totheladderwidthforwiderfour-legandsix-legladders, c Microscopic origin for the itinerant ferromagnetism has suggestingthatthe ferromagnetisminthetwo-legladder [ beenstudiedinthe frameworkoftheHubbardmodel.1–3 smoothly connects with that in two dimensions as the A conventionalmean-field treatment leads to the Stoner 1 ladder width increases.17,18 Regarding the experimental v criterion for the occurrence of band ferromagnetism.4 It observation of the Nagaoka ferromagnetism, it has been 9 tells us that the ferromagnetism occurs if the Coulomb proposed to use cold-atom optical lattice systems, since 9 repulsion and/or the density of states at the Fermi level the Hubbard model is realizable in a clean environment 8 are large enough. However, such a mean-field treatment 2 with high tunability and controllability.23,24 overestimates the stability of the ferromagnetism and it 0 . is not adequate for the effect of the electron correlation. In sharp contrast to the Nagaoka ferromagnetism, the 1 In fact, according to the multiple scattering theory,2 the groundstateathalf-fillingisaMottstatewithzerototal 0 Coulombrepulsionisrevisedbyarenormalizedoneinthe spininabipartitelatticewithanequalnumberofsitesin 5 Stoner criterion, which improves the stability condition. eachsublattice.25Namely,withthechangeofthenumber 1 : Ontheotherhand,theoccurrenceoftheferromagnetism of electrons by one, we can see a drastic change between v intheHubbardmodelhasbeenprovenrigorouslyinsome theMottstatewithzerototalspinandtheferromagnetic Xi limiting conditions.5–7 The Nagaokaferromagnetismis a state with the maximumtotalspin. Since the number of r well-knownrigorousresult,5 indicatingthatsystemshave electrons is a conserved quantity in the Hubbard model, a asaturatedferromagneticgroundstatewhenthereisone we usually study these states independently by changing hole added to the half-filling and the Coulomb repulsion the numberofelectronsonebyone. Toshedlightonthe is infinitely large on appropriate lattices that satisfy the quantum mechanical transition between the two states, so-called connectivity condition. we have introduced a quantum mechanical procedure to Note that the introduction of one hole corresponds to removeanelectronfromthesystem,consideringamodel aninfinitesimalhole dopinginthe thermodynamic limit. with a four-site plaquette and an extra site, as depicted In order to explore the ferromagnetism in more realistic in Fig. 1.26 There the electron occupation is controlled conditions, much effort has been made to know how the throughachemicalpotentialattheextrasite. Theextra ferromagneticphase extends in the case with finite holes site can be regarded as a particle reservoir for a part of and finite Coulomb repulsion.8–22 In a square lattice, for thesystemwithouttheextrasite. Wehavealsodiscussed instance, many authors have tried to clarify whether the types of itinerant ferromagnetism for particles with S > saturatedferromagneticstatesurvivesoverafiniterange 1/2,whichcouldberealizedinopticallatticeswithlaser- of hole density,12–17 and most results are supportive for cooled atoms.27 2 where S = S and h···i denotes the expectation tot Pi i value in the ground state. Note that spin-1/2 operators μ are described by the electron creation and annihilation operators and the Pauli matrices σ as FIG. 1: (Color online) A lattice with fivesites, which is used 1 as a unit structure to construct an extended lattice. Here Si = 2Xc†iσσσσ′ciσ′. (3) we put four electrons. Solid circles are occupied by an elec- σ,σ′ tron, while open circles are vacant. The electron occupation is controlled bythe chemical potential at the centersite µ. The total magnetization, defined by M =XhSizi, (4) In this paper we study a mechanism for the control of i themagneticpropertythroughalocalchemicalpotential in systems which are made by a five-site unit structure isalsoa conservedquantity. Inthe following,weanalyze in Fig. 1. We investigate the ground-state properties by the ground state with M =0 unless otherwise specified. numerical methods such as Lanczos diagonalization and The ground state at half-filling, i.e., one electron per density-matrix renormalization group (DMRG).28,29 As site (Ne = N, where N is the number of sites and Ne is the chemicalpotentialis varied,weobserveground-state the number of electrons), is a Mott state with zero total transitions among various values of the total spin due to spin in a bipartite lattice with equal number of sites in the changeofthe distributionofelectrons. Inparticular, each sublattice. In contrast, if we remove one electron wefindasaturatedferromagneticstateinabroadregion from the half-filling (Ne = N −1), the ground state is of the chemical potential irrespective of the system size, a saturatedferromagneticstate with the maximumtotal suggestingthattheferromagnetismisrealizedduetothe spin,assuming thatU issufficiently largeandthe lattice present mechanism in the thermodynamic limit. satisfies the connectivity condition, which is well known The organization of the paper is as follows. In Sec. II as the Nagaoka ferromagnetism.5 Here we note that the we explain our basic idea of a mechanism to switch the hoppingcoefficient−tisnegative,andinordertoensure ground state between the Mott state and the Nagaoka the symmetric ground state, the lattice should have a ferromagneticstate,basedonthe caseofasmallfive-site bipartite structure, in which we can change the sign of lattice.26 In Sec. III we introduce an extended model of the hopping amplitude by a gauge transformationciσ → largersites. Wepresentnumericalresultstodemonstrate −ciσ. that a saturated ferromagnetic state occurs. In Sec. IV In the present study, as we will explain in the next we discussthe characteristicsofphasesfromaviewpoint subsection (Sec. II B), instead of removing one electron of correlations. Section V is devoted to summary and fromthehalf-filledsystem,weconsiderakindofparticle discussion. bath to control the number of electrons in a part of the system effectively. That is, the system is composed of a subsystem and the particle bath. We expect a kind of II. ROUTE FROM MOTT STATE TO Nagaoka ferromagnetism when the number of electrons FERROMAGNETIC STATE in the subsystem is reduced from the half-filled case of thesubsystem. Wecallthepresentmechanism“extended A. Ferromagnetism in itinerant electron system: Nagaoka ferromagnetism.” Nagaoka ferromagnetism The Hubbard model is one of the simplest models for B. Control of total spin by mechanism of Nagaoka ferromagnetism itinerantelectronsystems. Itiscomposedoftheelectron hopping term and the on-site Coulomb repulsion term, described by We have proposeda possible mechanismto switch the ground state between the Mott state and the Nagaoka H=−t X (c†iσcjσ +h.c.)+UXni↑ni↓, (1) ferromagnetic state.26 Here we explain its basic idea for the completeness of the paper, and make a few remarks hiji,σ i relevant for the present study. where c is an annihilation operatorof an electron with Since the number of electrons is a conserved quantity iσ spin σ (=↑,↓) at site i, n = c† c , t is the hopping intheHubbardmodel,itisdifficulttodescribeaprocess iσ iσ iσ amplitude, and U is the on-site Coulomb repulsion. We toremoveanelectronfromthe systeminaHamiltonian. set t = 1 and take it as the energy unit. Because of the Inordertodescribeaprocedurefortheelectronremoval, SU(2)symmetry,thegroundstateischaracterizedbythe weprepareanextrasitetowhichanelectroncanescape. total spin S , given by InFig.1wepresentanexampleofsuchalatticewithfive tot siteswhereweputfourelectrons. Thelatticeiscomposed S (S +1)=hS2 i, (2) of a four-site ring anda center site. We call the four-site tot tot tot 3 (a) 5 (b) N=8 4 b 3.98 u s, Note 23 subNe N=11 St 3.96 1 S tot N es ub N=14 0 -20 -10 0 10 20 7 8 9 μ μ 13 17 center sites 7 8 9 10 11 12 upper leg FIG.2: (Coloronline)(a)Theµdependenceofthetotalspin N=17 subsystem Stot and the number of electrons in the subsystem Nesub for thesystemwithfivesitesandfourelectronsatU =1000. (b) 1 2 3 4 5 6 lower leg Nesub aroundµ=8in amagnified scale, whereweclearly see a jump. FIG. 3: Extended lattices with 8, 11, 14, and 17 sites. The site numbering is presented for N = 17 as an example. The ring“subsystem.” TheHamiltonianisexplicitlygivenby lattice sites are labeled in the order of the sites in the lower leginthesubsystem,thoseintheupperleginthesubsystem, H = −t X(c†iσcjσ +h.c.)+UXni↑ni↓ and those in thecentersites. hiji,σ i +µ(n5↑+n5↓), (5) ferromagnetisminthesubsystemleadstothetotalspinof the subsystem of 3/2. Note that in the present case, the wherehijidenotespairsofsitesconnectedbyasolidline total spin of the subsystem S and that of the center sub inFig.1,thecentersiteisnamed“5,”andµistheon-site site S are not conserved quantities, but they are still c energy,whichwecall“chemicalpotential.” Notethatthe approximately represented by S = 3/2 and S = 1/2, sub c Coulomb repulsion is active in all the sites including the where we define the total spin of a part of the system as subsystemandthe centersite. S andM areconserved tot quantities in the same way as the Hubbard model (1). S (S +1)=hS2 i, (7) sub sub sub We note that in the previous paper26 we defined the chemical potential with the opposite minus sign. In this paper we will introduce an extended lattice composedof Sc(Sc+1)=hS2ci, (8) a subsystem and center sites in the next section. We with S = S and S = S . We regardthecentersitesasareservoirofelectrons,andthe sub Pi∈subsystem i c Pi∈center i observe that the total spin of the whole system is given presentsignis more appropriatein its physicalmeaning. by the combination of those of the subsystem and the That is, if the chemical potential is large, electrons tend center site as S =3/2+1/2=2. Although this could tomoveoutfromthecentersites. Thechemicalpotential tot be, in principle, S = 3/2−1/2 = 1, the symmetrized represents the electron affinity at the center site. tot state is realized in the present case. In this way, we can InFig.2weplotthe totalspinS andthe numberof tot controlthe totalspinbythe chemicalpotentialinalocal electrons in the subsystem Nsub, given by e site. Nesub = X hniσi, (6) areIftwraepfpuertdheartdtehcerecaesneteµrdsoitwen, stuogµge.sti−nUg,thtwatoweleecctorounlds i∈subsystem,σ possiblychangeNsub inawiderrange,includingcasesof e where the summation of i is taken over the sites of the thedoublyoccupiedcentersite. Infact,forhugenegative subsystem, as a function of µ at U = 1000. We clearly µaroundµ=−U,itisobservedthatthecentersitetraps seeaground-statetransitionataroundµ=8,whereStot two electrons. However, it turns out that Stot is simply exhibits a jump. We also find a small jump of Nsub, as zerowithoutexhibiting anymagneticstates. Thisfactis e depictedinthe magnifiedfigure. Atthe transitionpoint, similarly found in an extended lattice introduced in the Nsubisabout3.97. Weenvisagethatthissituationwould following section. In the present paper, we focus on the e correspondto a hole doping into the subsystem to result case of |µ| ≪ U, since we find a complete ferromagnetic in a change of the magnetic property. state only for |µ|≪U in an extended lattice. For large µ, electrons are repelled from the center site and stay in the subsystem. The subsystem is half-filled andthegroundstateisessentiallyintheMottstatewith III. EXTENDED LATTICE antiferromagnetic correlations. For negative µ, electrons are attracted onto the center site, so that an electron is Here we extend the five-site lattice depicted in Fig. 1 removed from the subsystem. Then the total spin takes to larger system sizes by simply repeating it as a unit in the maximum value S = Smax = 2, indicating a fully onedirection,asshowninFig.3. Theextendedlatticeis tot tot symmetrized state. We should remark that the Nagaoka composedofasubsysteminladdershapeandcentersites. 4 Forinstance,thelatticeof11sitesconsistsofthreeunits, (a) 4 and the system has a subsystem of an eight-site ladder cNe 3 SN teoc t abnydththerseuemceonftetrhasitteisn. tThehesunbusmysbteermofNssiutbesaNnditshgaitveinn S, tot 12 N=11 the center sites Nc as N =Nsub+Nc. We consider the 0 -20 -10 0 10 20 same type of Hamiltonian as Eq. (5), μ (b) 5 H = −t X(c†iσcjσ +h.c.)+UXni↑ni↓ cNe 34 SN teoc t hiji,σ i , ot 2 N=14 +µ X (ni↑+ni↓), (9) St 1 0 i∈center -20 -10 μ0 10 20 where the chemical potential µ is effective for the center (c) 6 ssiutbLesse.ytsWutesemcuo,snie.sei.od,peNreneth=beoNsuynssudtbea.mryFwocriotnihndseittlaeiocnntcrseo.,ninswthheiclhatfitilclethoef cS, Netot 12345 N=SN1 t7eoc t 11 sites, we put eight electrons to fill the outside eight- 0 -20 -10 0 10 20 site ladder. This is a natural extension of the five-site μ case in Sec. II, since the center sites are empty and the subsystem is half-filled for large µ, while electrons turn to occupy the center sites as µ decreases. FIG.4: (Coloronline) ThetotalspinStot andthenumberof We mention that the present system is also regarded electrons in the center sites Nec for several system sizes: (a) as a periodic Anderson lattice, which is a typical model (N,Ne) = (11,8), (b) (14,10), and (c) (17,12). Here we set U =1000. forheavy-electronsystems,assumingthatthesubsystem represents conduction-electron sites and the center sites correspond to f-electron sites. Here µ plays a role of a this isahardtaskinthe presentmodel. Since the model local f-electron level. Regarding the hybridization, due involveschargeandspindegreesoffreedom,manynearly to the lattice structure in Fig. 3, each f-electron site is degenerate low-energy states appear because of a subtle connected to two conduction-electron sites, indicating a balanceofmultiple degreesoffreedom. Insuchacase,it multiband system. Moreover, when the f-electron sites is difficult to determine the total spin by comparing the are singly occupied and they are considered as localized ground-stateenergiesnumerically. Instead,weobtainthe spins, the system is equivalent to a Kondo lattice. The ground-statewavefunctionwithM =0andevaluateS tot ferromagnetism in periodic Anderson and Kondo lattice byusingEq.(2). Inaddition,weneedtoperformalarge models has been studied extensively.30–35 number of Lanczos iteration steps to obtain an accurate We investigate the magnetic properties by making use ground-state wave function which gives an integer value of numerical techniques. We examine the µ dependence ofS . Typically,severalthousandstepsarerequiredfor tot of various physical quantities with U = 1000 fixed. The the good convergence near transition points and in the U dependence is also discussed in Sec. III B. We use the negative µ region. Lanczos diagonalization method for small clusters up to In Fig. 4 we show the µ dependence of the total spin N =17 to obtain numerical results with relatively small S and the number of electrons in the center sites Nc, computationalcosts. For largerlattices, we also perform tot e given by extensive DMRG calculations to grasp the ground-state padrooppetrotipeesninbothuendtahreyrmcoonddyintaiomniscfolirmLita.nWczeosnaontedtDhMatRwGe Nec = X hniσi, (10) i∈center,σ calculations for consistency. for N = 11, 14, and 17. Note that N = Nsub+Nc by e e e definition. For large µ, the center sites are vacant and A. Extended Nagaoka ferromagnetic state the subsystem is half-filled, so that the Mott state with S = 0 is realized. With decreasing µ, Nc gradually tot e 1. Lanczos results increases, since electrons come to the center sites. The same amount of holes is introduced into the subsystem. Before going into the discussionof Lanczos results, let Insuchasituationweexpectaground-statechangefrom usmakeafewcommentsontechnicaldetails. Becauseof the Mott state to a ferromagnetic state in a similar way theSU(2)symmetry,weusuallydeterminethetotalspin tothefive-sitemodel. Indeed,weobservethatS jumps tot fromtheground-statedegeneracy. Thatis,bycomparing from zero to N /2−2 and stays there in a short period e the ground-state energies of different values of the total around µ=8, and then it increases up to the maximum magnetization M, if the ground states with |M|≤S are value N /2. With further decreasing µ, S is reduced e tot degenerate,thetotalspinisestimatedtobeS. However, from N /2. Sudden jumps of S signal transitions of e tot 5 first order. We note that the complete ferromagnetic (a) 1 state is found in 2 . µ . 8 similarly for N = 11, 14, N=11 and 17, implying that the complete ferromagnetic state N=14 isrealizedwithoutsignificantfinite-sizeeffects. Notealso cN N=17 thatthecompleteferromagneticstateappearsinaregion c / e 0.5 N where the amount of holes doped into the subsystem is moderately small. Here let us discuss the underlying mechanism of the 0-20 -10 0 10 20 μ complete ferromagnetic state from the viewpoint of hole doping into the subsystem. We note that the subsystem (b) 1 is equivalent to a two-leg ladder, eliminating the center N=11 0.9 sitesfromthewholesystem. Thuswerefertotheground ub N=14 state of the two-leg ladder Hubbard model as a function sN 0.8 N=17 of the hole doping rate.17–19 When we have one electron ub / 0.7 less than half-filling, the Nagaoka ferromagnetism takes sNe place. Evenwhenweaddholes,thegroundstateremains 0.6 a ferromagnetic state due to the Nagaoka mechanism in 0.5 -20 -10 0 10 20 some range of the hole density. When the hole density μ exceeds a critical point, the ground state changes to a partially spin-polarized state, and eventually becomes a spin-singlet state. In the present model we expect the FIG. 5: (Color online) (a) The electron density in the center samebehaviorforthe subsystemwhen µ varies. Thatis, sites Nec/Nc, and (b) that in the subsystem Nesub/Nsub, for the subsystem exhibits a ferromagnetic state in a range several system sizes at U =1000. of µ, and it is destabilized with the decrease of µ due to holedopingintothesubsystem. Asaresult,thecomplete ferromagnetic state in the whole system is broken down. 2. DMRG results Thus, the complete ferromagnetic state is attributed to the Nagaokaferromagnetismat finite hole density in the AsfortheefficiencyofthepresentDMRGcalculations, subsystem. we mention that the computational cost highly depends on the value of µ. At large µ, the center sites are vacant We mention that in the region near µ =0, S shows tot andthesubsystemishalf-filled,sothatchargedegreesof acomplicateddependence. Whenµ is decreasedfurther, freedom are frozen out and only spin degrees of freedom all the center sites are singly occupied, corresponding to arerelevantinthestrong-couplingregime. Insuchacase the Kondo lattice regime. In that region we do not find we can easily obtain the ground state in high precision. the complete ferromagnetic state with S = N /2, but tot e However, with decreasing µ, charge degrees of freedom observe that S =3 for N =11, 14, and 17. We would tot shouldalsobecomerelevant,sinceelectronsturntomove expect that S = 3 even for larger system sizes, but tot around the whole system. This indicates that we need actually this is not the case, as we will discuss based on to keep a large number of DMRG states to describe the DMRG results later. groundstate. Forinstance,intheregionnearµ=0,even In Fig. 5(a) we show the µ dependence of the electron if we keep 1000 states, in which the truncation error is density inthe center sites Nec/Nc for N =11,14,and17 estimatedto be around10−6, itis still difficult to obtain inthesameplot. Weclearlyobservethatwithdecreasing thetrue groundstate,andwehaveincorrectresultssuch µ, Nec/Nc graduallyincreases from zero to unity, and its as a non-integer value of Stot. µ dependence is independent of the system size. On the Figure 6(a) presents DMRG results of S for various tot other hand, as shown in Fig. 5(b), the electron density values of the system size N. We find that with decreas- in the subsystem Nsub/Nsub decreases from unity to a e ing µ, Stot changes from zero to Ne/2−2 and then it constant value that depends on N, takes the complete ferromagnetic value N /2 in a broad e region 2 . µ . 8 in the same way as Lanczos results in Fig. 4. This tendency is found for all the system sizes Nsub Nsub−Nc e = , (11) we have analyzed. In Fig. 6(b) we show the size depen- Nsub Nsub dence of critical points of µ where S changes: µ for tot c1 the change between S = 0 and N /2−2; µ for the tot e c2 and it becomes 0.5 in the large N limit. Here electrons change between S = N /2−2 and N /2; and µ for tot e e c3 escape from the subsystem to the center sites. In other the change between S = N /2 and a lower value. We tot e words, holes are doped into the subsystem. The doping clearly find that they saturate well at large values of N rate is controlled by µ. In this situation we can regard withoutanysignificantsizedependence, andthus webe- the center sites as a reservoir of electrons, i.e., a kind of lievethatthesecriticalvaluesexistinthethermodynamic particle bath. limit. Moreover,theµdependenciesofNc ofthesystems e 6 (a) 1 (a) 4 0.8 NN==2203 μc 3 μc 2 cNe 3 SN teoc t maxot 0.6 NNN===223692 S, tot 12 St N=35 0 / ot 0.4 -20 -10 μ0 10 20 St 0.2 (b) 135 μc 1 0-20 -10 0 10 20 Uc 130 μ (b) 10 125 0 0.02 0.04 0.06 0.08 0.1 μc 1 1/N μc 2 μ 5 FIG.7: (Coloronline)(a)ThetotalspinStot andthenumber of electrons in the center sites Nec at U = 100 for N = 11. (b) The size dependence of the critical value of U for the μc 3 appearance of the ferromagnetism Uc with N = 11, 14, 17, and 20. 0 0 0.05 0.1 1 / N 20 μμcc 12 S tot =N e /2−2 15 μc 3 Mott state FIG.6: (Coloronline) DMRGresultswith largesystemsizes S tot =0 uptoN =35atU =1000. (a)ThetotalspinStotnormalized bythemaximum valueStmotax =Ne/2, i.e., 0≤Stot/Stmotax ≤1 μ 10 regardlessofthesystemsize. Intheshadedregionnearµ=0 complete ferromagnetic state we cannot obtain well-converged data byDMRGsimulations 5 S tot =N e /2 even if we keep 1000 states. (b) The size dependence of the transition points µc’s denoted byarrows in (a). Herewe also 0 plot Lanczos results for N =11, 14, and 17 together. 0 1000 2000 3000 4000 5000 U with17<N ≤35collapseaswesawforN ≤17inFig.5 FIG. 8: (Color online) The ground-state phase diagram in (not shown). the coordinate (U,µ) for N =11. The transition points µc’s On the other hand, we do not see saturated behavior are defined in the same way as those in Fig. 6. At several ofS asafunctionofN fornegativeµ. Infact,S =3 tot tot points denoted by crosses, we have confirmed the realization forN ≤17,butS =0forN =20and26,andS =1 tot tot of the complete ferromagnetic state by DMRG calculations forN =23and29. Thiscomplicationwouldbeprobably up to N =110. duetothefactthattheelectrondensityinthesubsystem varies with the system size as Eq. (11). Moreover, we point out that even if we keep 1000 states, the DMRG U = 100 in Fig. 7(a). There S does not reach the tot doesnotgivefairlywell-convergeddataintheregionnear maximum value Smax = 4 for any values of µ, although tot µ = 0, denoted by the shaded region in Fig. 6(a). Note Nc varies with µ in the same way as compared with the e that the Lanczos results for small clusters also show the case of U = 1000 in Fig. 4(a). The critical value of U complicated dependence near µ = 0. These regions are abovewhichthe complete ferromagneticstate appearsis interestingtostudyforapossiblerealizationofmagnetic estimatedatU =127.3for N =11,andit further shifts c states other than the complete ferromagnetic state such toU =130.4forN =14,indicatingfinite-size effectson c as a partially spin-polarized state, but we leave it for a the locationofthephaseboundaryinthe regionofsmall future issue. U. In Fig. 7(b) we plot the size dependence of U . The c curve is bent such that the slope becomes gentle as the system size increases, which suggests a tendency toward B. U dependence convergence. Note, however, that the systems studied are still small and we need calculations with larger sizes ItshouldbenotedthatthevalueofU whichcancause fortheextrapolation. Incontrast,forlargeU,wefindno thecomopleteferromagneticstatedependsonthesystem significant size dependence at U = 1000 up to N = 35, size. In fact, U = 100 is not large enough to realize the asshowninFig.6(b). Inthe presentpaperwehaveused complete ferromagnetic state for N ≥11, while U =100 U =1000whichissufficientlylargeenoughtorealizethe was enough to generate it in the system of N = 5.26 As complete ferromagnetic state. a typical example we depict S and Nc for N = 11 at In Fig. 8 we present the ground-state phase diagram tot e 7 in the coordinate (U,µ) for N =11. The regionbetween subsystem center sites µ and µ is of the complete ferromagnetic state with lower leg upper leg c2 c3 (a) 0.25 cSotomtp=letNeef/e2rr.omWaegcnleetaicrlystasetee sthhraitnkthsewritahngdeecorfeaµsionfgtUhe, 6,j) μ=10 anditeventuallydisappearsatsmallU,aswementioned (xy 0 C above. Here we note again that finite-size effects should -0.25 be carefully considered to discuss the phase diagram in 5 10 15site 2j0 25 30 35 the thermodynamic limit. In the region of small U, the (b) 0.25 phaseboundaryissupposedtobedeformedtoreducethe ) μ=8 ferromagneticregion,since the criticalpointofU forthe 6,j appearanceofthe ferromagnetismshifts towardlargerU (y 0 x C with increasing the system size. For large U, numerical -0.25 results of 11 ≤ N ≤ 35 at U = 1000 are indicative that 5 10 15 20 25 30 35 site j the finite-size correction is small. In addition, at several points denoted by crosses in Fig. 8, we have performed (c) 0.25 DMRG calculations up to N = 110 and confirmed that 6,j) μ=5 the complete ferromagneticstate isrealized. Thus,these (y 0 x points are expected to be included in the ferromagnetic C phaseinthe thermodynamic limit, althoughitishardto -0.25 5 10 15 20 25 30 35 site j determine the entire phase boundary on the basis of the present numerical results for small systems. FIG. 9: The transverse spin correlation function Cxy(i,j) C. Spin correlation function for typical values of µ at U = 1000: (a) µ = 10 for the Mottantiferromagnetic statewith Stot =0;(b)µ=8forthe partially spin-polarized state with Stot = Ne/2−2; and (c) Inordertoclarifythecharacteristicsofmagneticstates µ=5forthecompleteferromagnetic statewithStot =Ne/2, fromamicroscopicviewpoint,itisusefultomeasurespin obtained by DMRG with N = 35. The correlation function correlationfunctions. Here we study the ground state in is measured with the middle site of the subsystem (site 6 thesubspaceofM =0. Insuchacase,whenweconsider denoted by an open circle) as starting point. a spin-polarized ground state, the spin moment lies in the xy plane and the ferromagnetic correlation develops in the xy plane. Thus we investigate the transverse spin the subsystem, as shown in Fig. 9(c). We note that the correlation function, spin correlation between the subsystem and the center sitesisstillveryweak,sincewehaveonlyasmallnumber 1 1 C (i,j)= h(SxSx+SySy)i= h(S+S−+S−S+)i. of electrons in the center sites at µ=5. xy 2 i j i j 4 i j i j Here we investigate the longitudinal spin correlation (12) function, In Fig. 9 we show C (i,j) measured from the middle xy of the subsystem (site 6 denoted by an open circle), for C (i,j)=hSzSzi. (13) z i j typicalvaluesofµ. Theplotsintheleftsideofthedouble line are for the correlation within the subsystem, and In Fig. 10(a) we present C (i,j) for µ = 10. Since the z thoseintherightsidedenotethecorrelationbetweensite maximum value of M is zero in the singlet groundstate, 6inthesubsystemandthecentersites. Inthesubsystem, the spin correlationis isotropic, i.e., C (i,j)=C (i,j). z xy which has a ladder shape, the left part ofthe dotted line In contrast, the ground state is spin-polarized for µ = 8 represent the lower leg, while the right part is the upper and 5, so that we can see anisotropic behavior between leg. The site numbering is givenin the bottom of Fig. 3. C (i,j)andC (i,j),ifwecomparetheminthesubspace z xy As shown in Fig. 9(a), for µ=10, we find a Ne´el-type of M = 0. In Figs. 10(b) and 10(c) we observe that the antiferromagnetic correlation corresponding to the Mott longitudinalspincorrelationtakesasmallnegativevalue state in the subsystem. On the other hand, we observe in the subsystem. This reflects the sum rule that the spin correlationbetween the subsystem and the N centersitesisalmostzeroduetotheabsenceofelectrons hM2i= e +XCz(i,j)=0, (14) in the center sites. At µ=8, where S =N /2−2, the 4 tot e i6=j system is nearly ordered ferromagnetically. As for the microscopic spin configuration, we find a ferromagnetic and we expect that C (i,j)∼−1/N . z e correlation in the subsystem except for two corner sites, We notice that the maximum value of M is N /2−2 e while thetwocornerspinsaligninthe oppositedirection for µ = 8, suggesting that the spin configuration can be to the others, as shown in Fig. 9(b). At µ = 5, where deduced from C (i,j) in the subspace of M = N /2−2 z e S =N /2,thesystemisinthecompleteferromagnetic insteadofM =0. Indeed,asshowninFig.11,weclearly tot e state, and we find a simple ferromagnetic correlation in see that the spins at the two corners are antiparallel to 8 subsystem center sites (a) 4 lower leg upper leg (a) 0.25 3 ) μ=10 (6,j 0 M2 Cz B S=1/2 B S=1/2 1 μ=10 μ=10 μ=10 -0.25 5 10 15 20 25 30 35 T=0 T=0.001 T=0.01 site j 0 0 0.005 0.010 0.005 0.010 0.05 0.1 (b) 0.25 H H H ) μ=8 6,j (b) 4 ( 0 z C 3 -0.25 5 10 15 20 25 30 35 site j M2 B S=1/2 B S=1/2 B S=2 B S=2 (c) 0.25 1 μ=8 μ=8 μ=8 ) μ=5 T=0 T=0.001 T=0.01 6,j ( 0 0 z 0 0.005 0.010 0.005 0.01 0 0.05 0.1 C H H H -0.25 5 10 15site 2j0 25 30 35 (c) 4 3 FIG. 10: The longitudinal spin correlation function Cz(i,j) M2 B S=1/2 B S=1/2 for typical values of µ at U = 1000: (a) µ = 10 for the B S=4 B S=4 Mott antiferromagnetic statewith Stot =0;(b)µ=8forthe 1 μ=5 μ=5 μ=5 T=0 T=0.001 T=0.01 partially spin-polarized state with Stot = Ne/2−2; and (c) µ=5forthecompleteferromagnetic statewithStot =Ne/2, 00 0.005 0.01 0 0.005 0.010 0.05 0.1 H H H obtained by DMRG with N = 35. The correlation function is measured with the middle site of the subsystem (site 6 denoted by an open circle) as starting point. FIG. 12: (Color online) The magnetization curve for typical valuesofµatU =1000andseveraltemperatures: (a)µ=10 subsystem center sites lower leg upper leg fortheMottantiferromagnetic statewithStot =0; (b)µ=8 0.25 forthepartiallyspin-polarizedstatewithStot =Ne/2−2;and C(6,j)z 0 μ=8 (tNcee)m/µp2.e=rFato5urrfeeoasrcThthve=acluo0em, o0pf.l0eµ0te1w,feearnsrhdomow0a.0gm1n.eatgiOncepstetiznaatteciiorwnciltechsurdSveteonstoat=et numerical results with N = 11, and the Brillouin functions -0.25 5 10 15site 2j0 25 30 35 BS for the corresponding temperatures are plotted by solid or dashed curves. FIG. 11: The longitudinal spin correlation function Cz(i,j) for the partially spin-polarized state with Stot = Ne/2−2 where M(H) is the magnetization of model (9) with an in the subspace of M = Ne/2−2 at µ = 8 and U = 1000, additionalZeemanterm−HPiSiz,andh···iT istheex- obtained by DMRG with N = 35. The correlation function pectation value at temperature T. Here, for the thermal is measured with the middle site of the subsystem (site 6 average,we need all the eigenvalues and wave functions, denoted by an open circle) as starting point. computedby fulldiagonalization(Householdermethod). In Fig. 12 we show the magnetization curve for N = 11 attypicalvaluesofµandT. Forµ=10,asshowninthe theothers,inasimilarwaytothecaseofCxy(i,j)inthe left panel of Fig. 12(a), the magnetizationcurve shows a subspace of M =0 shown in Fig. 9(b). stepwise increase starting from zero up to the maximum value 4 at T = 0, because S = 0 in the ground state tot and there is an energy gap between states of different values of S due to the finite system size. The stepwise D. Magnetization tot structure is rapidly smeared out at finite temperatures, since the energy gap is rather small. In the right panel We investigate the magnetization curve as a function ofFig.12(a)wefindthatthemagnetizationcurveagrees of an applied magnetic field H, well with the Brillouin function of S =1/2 at T =0.01, indicatingthateachspinfluctuatesindependentlydueto M(H,T)=hM(H)i , (15) thermal fluctuations. T 9 (a) 4 (a) 4 Ssub 23 c, Neot 23 t'=SN0. te5oc t 1 St 1 0 0 -20 -10 0 10 20 -20 -10 0 10 20 μ μ (b) 1.5 (b) 4 Sc 0.51 cS, Ntote 123 t'=SN0. te1oc t 0 0 -20 -10 0 10 20 -20 -10 0 10 20 μ μ FIG. 13: (a) The total spin of the subsystem Ssub and (b) FIG. 14: (Color online) The total spin Stot and the number thatofthecentersitesSc atU =1000forN =11. Notethat of electrons in the center sites Nec for (a) t′ = 0.5 and (b) thetotalspinofthewholesystemStot wasgiveninFig.4(a). t′ = 0.1 at U = 1000 for N = 11. Note that the plot for t′=1.0 was given in Fig. 4(a). Figure12(b) showsthe magnetizationcurveforµ=8, 10 wherethe groundstate is apartiallyspin-polarizedstate μc 1 S tot =N e /2−2 μc 2 Mott state with Stot =2. According to the total spin in the ground μc 3 S tot =0 state, the Brillouin function of S = 2 is realized at low temperatureT =0.001,anditchangestothatofS =1/2 μ 5 at high temperature T =0.01. In Fig. 12(c) we can also complete ferromagnetic state see a similar behavior for µ=5, where the ground state S tot =N e /2 is a complete ferromagnetic state with S =4. Indeed, tot the magnetization curve is represented by the Brillouin function of S = 4 at T = 0.001, and it approaches that 00 0.2 0.4 0.6 0.8 1 t' of S =1/2 as the temperature increases. FIG. 15: (Color online) The ground-state phase diagram in IV. CORRELATION BETWEEN SUBSYSTEM thecoordinate(t′,µ)atU =1000forN =11. Thetransition AND CENTER SITES points µc’s are definedin thesame way as those in Fig. 6. A. Total spin Fig. 14 we show the µ dependence of S and Nc for tot e InFig.13(a)and13(b)wepresenttheµdependenceof t′ =0.5and0.1. The plot fort′ =1 wasalreadygivenin thetotalspinofthesubsystemSsubandthatofthecenter Fig. 4(a). As t′ decreases,the regionwhere Nec increases sites Sc, respectively, at U = 1000. Comparing with the from zero to three (= Nc) becomes narrower, while Nec totalspinofthewholesystem,giveninFig.4(a),wefind exhibits a sharp change with a plateaulike structure at thatthe totalspinsofthe subsystemandthe centersites integers. There appears a compelete ferromagnetic state are correlated positively, i.e., Stot = Ssub+Sc, for most in a region where Nec varies from zero to one. values of µ except for the region near µ = 0. We note In Fig. 15 we depict the ground-state phase diagram that they couple negatively, i.e., Stot = |Ssub −Sc|, for in the coordinate (t′,µ) for N =11, where µc1, µc2, and 0.4 . µ . 2.1, and the total spin takes an intermediate µ are defined in the same way as those in Fig. 8. We c3 value for −2.2.µ.0.4. observe that µ does not depend on t′ so much, while c3 µ is reducedast′ decreases. As a result,the rangeofµ c2 ofthecompleteferromagneticstatebecomesnarrowwith B. Hopping amplitude between subsystem and decreasing t′. center sites In the limit of t′ = 0, the subsystem and the cen- ter sites are independent by definition. We find that in Now, let us introduce a different value of the hopping the period µ < µ < µ , the subsystem contains just c2 c3 amplitude between the subsystem and the center sites seven electrons and the remaining one electron stays at t′ from that within the subsystem t, in order to control a center site. In this situation, the Nagaoka ferromag- the degree of correlationbetween the subsystem and the netism is realized in the subsystem, i.e., S = 7/2, sub center sites in a direct way. That is, the center sites are while S = 1/2. Here the states of S = S ± S c tot sub c separated from the subsystem in the limit of t′ = 0. In are degenerate, and thus we obtain the complete ferro- 10 magnetic state. Note that in this limit we do not have (a) 3 (b) 6 t'=1 N=8 N=8 t'=1 the phase of Stot = Ne/2−2. We find that µc1 appears t'=0.5 5 N=11 t'=0.1 N=14 above t′ ≃0.19. 2 4 Esub Esub 3 C. Entanglement entropy 1 2 In Figs. 9 and 10 we found that the spin correlation 1 between spins in the subsystem and those in the center 0 0 -20 -10 0 10 20 -20 -10 0 10 20 sitesareveryweakregardlessofthesituationofthetotal μ μ spin. Note that the spin correlation is still weak even in the case of µ=0 (not shown). However, we expect that FIG. 16: (Color online) (a) The entanglement entropy of the strong quantum correlations occur in the ferromagnetic state where all the spins contribute to form the totally subsystemEsub forseveralvaluesoft′ forN =8. (b)Esub at t′=1 for several system sizes. HereU =1000. symmetric wave function. Now we study the nature of correlation in the system by making use of the entanglement entropy36 instead of relevantspinstatesasfollows. Forlargenegativeµ,each the spin correlationfunctions. In particular,we measure centersitetrapsoneelectron. Notethatthecontribution theentanglemententropyofthesubsystembytracingout of the double occupied states is negligibly small because the degrees of freedom in the center sites. The reduced of the Coulomb repulsion unless we do not decrease µ density matrix of the subsystem is given by down to −U. If t′ is small, the center sites are isolated fromthe system, so that we canconsider subspaces with ρ =Tr |GihG|, (16) sub c different spin states in the center sites independently. In the case with N = 8 and N = 6, we have two electrons where |Gi is the ground state of the whole system, and e inthetwocentersites,andthereappearfourspinstates, the degrees of freedom of the center sites are traced out. i.e., (↑,↑), (↓,↓), (↑,↓), and (↓,↑). These four states Theentanglemententropyofthesubsystemisdefinedby correspond to the subspaces with (Nsub,Nsub) = (1,3), ↑ ↓ E =−Tr ρ lnρ , (17) (3,1),(2,2), and(2,2),respectively, whereNsub denotes sub sub sub sub σ the number of spin-σ electrons in the subsystem. The wherethetraceistakenforthesubsystem. Wenotethat electronhoppingt′ mixesthestatesofthelowestenergies the entanglement entropy of the center sites can also be of the four cases, which gives the entanglements. That definedasE =−Tr ρ lnρ withρ =Tr |GihG|. The is, the exchange ofthe spin state between the subsystem c c c c c sub relation E =E holds for the ground state, although, and the center sites brings the entanglements. As the sub c in general, E 6=E for a mixed state. first approximation, let us assume that the four states sub c In Fig. 16(a) we present the µ dependence of E for have an equal weight in the ground state, i.e., the four sub several values of t′. In the Mott-state regime at large µ, statesaremaximallyentangled. Then,the entanglement since the center sites are empty, the subsystem and the entropy is given by center sites are practically separated. Therefore, E is sub 1 1 suppressed,anditapproacheszerointhelimitofµ→∞. E˜sub =− X ln =1.386, (18) As t′ decreases,E decaysto zero at lower µ, since the 4 4 sub (↑,↑),(↓,↓),(↑,↓),(↓,↑) subsystem and the center sites are disconnected. Withdecreasingµ,electronsturntocometothecenter where the summation is taken over allowed patterns of sites as well as the subsystem, and thus we expect that the spins trapped in the center sites. The thus obtained the correlation between the subsystem and the center valueisclosetotheresultinFig.16(a),i.e.,Esub ≃1.328 sites becomes significant. In the region of the complete at µ=−20 for t′ =0.1. Here we find a slight difference, ferromagneticstate,thefullysymmetrizedstateofallthe because relative weights of the four states are different spinstakesplace,andE isenhanced,asexpected. We in reality. If we take account of the relative weights, we sub find a peak near µ=0 where electrons canmove around have a more precise estimation of Esub. In fact, the four thewholesystemwithoutthedisturbanceofthechemical largest eigenvalues of ρsub are λ(↑,↑) = λ(↓,↓) ≃ 0.167, potential. This enhancement of the entanglement is due λ((↑,↓)+(↓,↑)) ≃ 0.310, and λ((↑,↓)−(↓,↑)) ≃ 0.357, to the electron motion, i.e., the subsystem is entangled and these eigenvalues gives the entanglement entropy by the quantum motion among the subsystem and the 4 cinenttheerlsiimteist.oWf te′n=ot0e. that the entanglement occurs even E˜sub = −Xλilnλi i=1 Here let us discuss the behavior for large negative µ, = −0.167×ln0.167−0.167×ln0.167 where wefind thatE convergesto a constantvalue as sub µ decreases independent of t′. We can understand this −0.310×ln0.310−0.357×ln0.357 convergent behavior by considering the contribution of = 1.328, (19)