Magnetic Correlations in a Periodic Anderson Model with Non-Uniform Conduction Electron Coordination N. Hartman1, W.-T. Chiu2 and R.T. Scalettar2 1Department of Physics, Southern Methodist University, Dallas, Texas 75275-0175 and 2Department of Physics, One Shields Ave., University of California, Davis, California 95616, USA The Periodic Anderson Model (PAM) is widely studied to understand strong correlation physics and especially the competition of antiferromagnetism and singlet formation. Quantum Monte Carlo (QMC) studies have focused both on issues such as the nature of screening and locating the quantumcriticalpoint(QCP)atzerotemperatureandalsoonpossibleexperimentalconnectionsto phenomena ranging from the Cerium volume collapse to the relation of the magnetic susceptibility and Knight shift in heavy fermions. In this paper we extend QMC work to lattices in which the 6 conductionelectronsitescanhavevariablecoordination. Thissituationisrelevantbothtorecently 1 discovered magnetic quasicrystals and also to magnetism in doped heavy fermion systems. 0 2 PACSnumbers: 71.10.Fd,71.30.+h,02.70.Uu b e F INTRODUCTION modification allows for intersite, rather than on-site, hybridization between conduction and local orbitals and 1 The single band Hubbard Hamiltonian[1–4] captures hencemetallicbehaviorintheabsenceofinteractions[29]. ] severalofthemostfundamentalconsequencesofelectron- Another motivation is provided by chemical substitution l e electron interactions in solids, namely magnetic order in heavy fermion materials, either by the replacement - r and the Mott metal-insulator transition. Although of some of the local moment atoms by non-magnetic st the question is still open, it may even contain the ones, as in (Ce,La)CoIn5[30] or by changes to the t. fundamental physics of d-wave superconductivity in conduction orbitals, as in the alloying of Cd onto ma the cuprates[5, 6]. Multi-band Hamiltonians like the In sites in CeCo(In,Cd)5 [31, 32]. In the latter periodic Anderson model (PAM) [7, 8] examine what situation, Vfd is reduced locally, and AF droplets can d- happens when two species of electrons, one delocalized form around the impurity sites. A second motivation n ‘conduction’ band (often d), and another localized is the recent observation of a quantum critical state o band (often f) are present. Here a central effect in magnetic quasicrystals[33, 34]. In these Au-Al-Yb c is the competition between singlet formation[9], when alloys (Au51Al34Yb15), measurements of the magnetic [ the conduction and localized electrons are strongly susceptibility χ and specific heat C diverge as T → 2 hybridized, and ordering of the local moments mediated 0. This non-Fermi liquid (NFL) behavior is associated v indirectly through the Ruderman-Kittel-Kasuya-Yosida with strongly correlated 4f Yb electrons. These two 4 interaction[10–13]. sets of materials share a common feature which is that 1 2 In heavy fermion materials[14, 15], this competition the coordination number of the different atoms is no 7 is believed to explain different low temperature phases, longer spatially uniform. The effects of these unique 0 e.g. non-magnetic CeAl where the f moments are local environments can be probed with nuclear magnetic 3 1. screened by the conduction electrons, and CeAl2 which resonance[35]. 0 becomes antiferromagnetic (AF) at low temperatures. The NFL behavior of Au-Al-Yb alloys has recently 6 Likewise, in Ce metal itself, as well as in other rare- been studied by solving the U = ∞ Anderson Impurity 1 earths, the tuning of the f-d hybridization V through Model (AIM) for a single local moment coupled to : fd v pressure is believed to play a crucial role in the ‘volume conduction electrons in a quasicrystal approximant i collapse’ transition[16–21]. geometry[36]. The crucial result is that singular X Quantum Monte Carlo (QMC) studies of the PAM responses in χ and C occur as a consequence of a broad r a have explored some of this physics, in one[22, 23], (power law) distribution of Kondo temperatures which two[24], and three dimensions[25]. The focus has been delays screening of a large fraction of the magnetic on bipartite lattices which, at half-filling, host AF order moments until very low temperatures. without frustration and are also free of the fermion In this paper we study the PAM in two different sign problem[26, 27]. QMC in infinite dimensions[28] geometries: the Lieb lattice and a 2D “Ammann- complements work in finite d by allowing simulations Beenker”tiling[37,38]. Quasicrystalineapproximates[39] at very low temperature, at or even well below the forAu Al Yb arein3D;thequasi-periodicAmmann- 51 34 15 Kondo scale, at the expense of some of the knowledge Beenker tiling is a more tractable 2D alternative of correlations in space. for QMC, which is limited in the number of sites There is interest in understanding the magnetic which can be simulated. Our goal is to explore correlations in more general geometries. One such the nature of magnetic correlations as a function 2 of f-d hybridization, and, specifically, to understand There are six bands corresponding to the six sites (three the competition of antiferromagnetic order and singlet conductionandthreelocalized)perunitcell. Thelattice formation in geometries where the coordination number is bipartite with four of the six sites on one sublattice of different sites in the lattice is non-uniform. Our and two on the other. Hence, in accordance with Lieb’s work extends that of [36] by examining a dense array theorem[43] there are two flat bands (at E = ±1). As of local orbitals and also by including the effect of in the case of the PAM on a square lattice with on-site finite U . We begin with the Lieb lattice, because hybridization, the half-filled lattice is a band insulator f it contains two separate coordination numbers, z = in the non-interacting limit. However, by comparing 2,4 while still retaining very simple lattice periodicity. calculations for on-site and intersite V , the latter being fd We then turn to the more complicated quasicrystal metallic at half-filling, it has been shown that many approximantstructure. Wedonotatpresentaddressthe properties of the PAM when Uf/t >∼ 4 are insensitive anomalous NFL behavior of the magnetic susceptibility, to the presence of a U /t=0 band gap[29]. f sincethosephenomenaappeartobeassociatedonlywith the quasicrystal itself, and not its approximant[32]. The magnetic properties of quantum antiferromagnets in geometries which have variable coordination number (the crown, dice, and CaVO lattices) have also been studied in the context of the spin-1/2 Heisenberg Hamiltonian [40–42]. Surprisingly, unlike the case of regular geometries where the ordered moment increases fromthehoneycomb(z =3)tosquare(z =4)lattices,it isfoundthatthelocalAForderparameterdecreaseswith z. We will comment on this further in our conclusions. MODEL AND METHODS ThePAMisatightbindingHamiltonianforwhicheach spatial site contains both an extended and a localized FIG. 1: (Color online) The Lieb lattice geometry under orbital, consideration in this paper. (Cluster shown has 4x4 unit cells with 48 sites). Each site contains both a conduction (d) (cid:88) (cid:88) H=−t (d† d +d† d )−V (d† f +f† d ) orbital and a localized (f) orbital, so that there is a total iσ jσ jσ iσ fd iσ iσ iσ iσ (cid:104)ij(cid:105),σ iσ of 96 sites/orbitals. Lines correspond to the d-d hopping t, with eight possible coordinations z = 1,2,.···8. We use (cid:88) 1 1 +U (nf − )(nf − ) (1) periodic boundary conditions (pbc). There are two possible f i↑ 2 i↓ 2 conduction orbital coordinations, z = 2,4. The local f i orbitals are connected to the d orbital on the same site by Here t is the hybridization between conduction orbitals hybridization V . fd with creation(destruction) operators d† (d ) on near iσ iσ neighbor sites (cid:104)ij(cid:105). In this paper we consider the ThemagneticpropertiesofthePAMarecharacterized two conduction electron geometries shown in Figs. 1,2 by intra- and inter-orbital spin-spin correlations, corresponding to “Lieb” and “quasicrystal” lattices respectively. Each site of these lattices also contains czz(cid:48)(r)=(cid:104)f† f f†f (cid:105) ff i+r↓ i+r↑ i↑ i↓ a localized orbital, creation(destruction) operators f† (f ). U is the on-site interaction between spin up czdzd(cid:48)(r)=(cid:104)d†i+r↓di+r↑d†i↑di↓(cid:105) iσ iσ f and spin down electrons on the localized orbital, and Vfd czfdz(cid:48)(r)=(cid:104)fi†+r↓fi+r↑d†i↑di↓(cid:105) (2) is the conduction-localized orbital hybridization. Both geometries of Figs. 1,2 are bipartite. In H we have Here the superscripts z,z(cid:48) refer to the coordination writtentheinteractionterm,in‘particle-hole’symmetric number of the conduction orbital on site i and i + form, and set the site energy difference between f and d r respectively. This separation allows us to isolate orbitals to zero, so that the lattice is half-filled for all the effects of the number of neighbors on the spin temperatures T and Hamiltonian parameters t,U ,V . correlations [44]. We focus here on czz(cid:48)(r) which f fd ff Half-filling optimizes the tendency for AF correlations, measures intersite magnetic correlations between the and also allows DQMC simulations at low temperature local electrons, and czz(r = 0), the singlet correlator fd since the sign problem[26] is absent. between local and conduction electrons onthe same site. Figure3showsthedensityofstatesandbandstructure The spin-spin correlations are translationally invariant of the PAM on a Lieb lattice for t = 1 and V = 1. foruniformgeometriesandperiodicboundaryconditions, fd 3 1 Quasicrystal Lieb (flat band) 0.8 Lieb 0.6 ) E ( D 0.4 0.2 0 -4 -2 0 2 4 E 4 Student Version of MATLAB 2 ) k 0 ( E -2 FIG. 2: (Color online) Top (bottom): Approximants to the Au Al Yb crystalline lattice for N = 41(239) sites. In 51 34 15 each case, sites shown contain both a conduction (d) orbital and a localized (f) orbital. Lines correspond to the d-d -4 hoppingt. Thelocalf orbitalsareconnectedtothedorbital Γ M X Γ on the same site by hybridization V . For this geometry we fdStudent Version of MATLAB use open boundary conditions (OBC) to avoid frustration. The conduction electron sites range in coordination from z=1toz=8. Theuseoftwocolorsforthesitesemphasizes FIG. 3: (Color online) DOS (top) and band structure that, despite its complexity, the geometry is still bipartite. (bottom) of the PAM on a Lieb lattice. Here t = 1 and V = 1. The two completely flat bands at E = ±1 give fd rise to δ function spikes in D(E) which are indicated by but depend more generally on both i and r in irregular dashed vertical lines). D(E = 0) vanishes: the system is lattices. a band insulator at half-filling. The DOS for the N = 239 We also measure the structure factor, quasicrystal approximant is also shown in the top panel. As is the case for the Lieb lattice PAM, the quasicrystal PAM Sz =(cid:88)(cid:88)czz(cid:48)(r)(−1)r also has a hybridization gap at E =0. The single band case ff ff is metallic[36, 42]. r z(cid:48) Stot =(cid:88)(cid:88)gzz(cid:48)czz(cid:48)(r)(−1)r (3) ff ff r zz(cid:48) correlations decay exponentially with separation r and which sums the spin-spin correlations to all distances r Stot getscontributionsonlyfromasmallnumberr <ξof ff from sites i with a given z. The staggered phase factor local correlations. It becomes temperature independent (−1)r takes the value ±1 on the two sublattices of the below a relatively high T set by the singlet energy scale. bipartite geometry and hence measures AF order. The Inanorderedphase, ontheotherhand, Stot willdepend ff z-resolvedcontributionstothetotalstructurefactorStot ontemperaturedowntomuchlowerT asthecorrelation ff are weighted by the fractions of sites in the lattice with length ξ diverges. Thus a T dependence of Stot can be ff a given coordination gzz(cid:48). In the singlet phase, the spin used as an indicator of AF order. 4 OurcomputationalapproachisdeterminantQuantum Monte Carlo (DQMC)[45, 46]. This method allows the solution of interacting tight-binding Hamiltionians like the PAM through an exact mapping onto a problem of non-interacting particles moving in a space and 0.1 (imaginary) time dependent auxiliary field. This field is sampled stochastically to obtain the expectation ) r values of different correlation functions. The update z’( 0 z moves require the non-local computation of the fermion cff Green’s function, which also the quantity needed to measure equal time observables including the energy, -0.1 z=2 z=4 double occupation, and spin correlations. The algorithm involves matrix operations and scales as the cube of -0.2 the product of the number of spatial lattice sites and V=0.8 (AF phase) fd the number of orbitals. In certain special situations, V=1.3 (singlet phase) fd including the PAM on the geometries studied here, the sampling is free of the sign problem[26] so that the 0 1 2 3 0 1 2 3 simulation may be conducted on large lattices (here r r several hundreds of spatial sites) at low temperature (here T/t<∼1/30). FIG. 4: (Color online) Spin-spin correlation function czz(cid:48)(r) ff as a function of r for the half-filled Lieb lattice with U =4t f andβ =30. r=1istheunitcellsize: integerrcorrespondto PAM ON THE LIEB LATTICE z(cid:48) =zandhalf-integervaluestoz(cid:48) (cid:54)=z. Theleft(right)panels showV =0.8t(1.3t)respectively. ForV =0.8tthereisAF fd fd We begin with the Lieb lattice which has 2N/3 sites order to large r, while for V = 1.3, czz(cid:48)(r) decays rapidly fd ff of coordination number z =2 and N/3 sites with z =4. to zero. In the AF regime, larger z increases czz(cid:48)(r). Data Figure4showsczffz(cid:48)(r)forVfd =0.8andVfd =1.3. Inthe depictedbyx(Vfd =0.8)and∗(Vfd =1.3)arefoffrthesquare former case, the correlation function alternates between lattice. r = 0.5 is for near-neighbors, and r = 1.0 for next positive and negative values, with a correlation length near-neighbors. which exceeds the linear lattice size, as is characteristic of an AF phase. r = 1 corresponds to the separation betweenunitcells,sothatintegervaluesofrarebetween In Fig. 6 we turn to the AF structure factor, Eq. 3, sites with z(cid:48) = z (and hence the same sublattice) whichsumsthespincorrelationsonthelocalizedorbitals and half-integer values have z(cid:48) (cid:54)= z (and hence occupy over the whole lattice. In the singlet phase, czffz(cid:48)(r) is different sublattices). The AF correlations are evident shortrangedandtemperatureindependent,achievingits in both z = 2 and z = 4, although they are larger for ground state value at T ∼ Vf2d/Uf. In the AF phase, higher coordination number. This reflects the collective on the other hand, the correlation length grows as T is nature of the AF order, which is more robust as the lowered, and hence czffz(cid:48)(r) contributes to the structure numberofneighborsgrows. Actually,becausetheAand factor out to larger and larger distances. The structure B sublattices have different numbers, the ordered phase factor becomes temperature dependent at low T. These is Ferrimagnetic[43], with N↑ (cid:54)= N↓ in addition to the two regimes are evident, and are separated by Vc ∼ 1.1. staggered pattern seen in the Figure. For V = 1.3, on This is suggestive, but certainly not conclusive, evidence fd the other hand, czz(cid:48)(r) falls rapidly to zero, indicative of of the presence of a QCP. At the end of the following ff a singlet phase. section we will provide a finite size scaling analysis of The AF and singlet regimes can also be distinguished this data to ascertain whether there is true long range by czfdz(r = 0), as shown in Fig. 5. (Here since r = 0 order below Vc . Note that the reduction in Sfftot as Vfd the coordination numbers z(cid:48) = z.) czz(r = 0) vanishes is reduced below Vfd ≈0.7 is a finite temperature effect. for Vfd =0 where the localized and cofdnduction fermions The RKKY exchange scales as Vf2d and T = t/30 (βt = are decoupled, and saturates at a large value for V → 30) is no longer low enough to reach the ground state. fd ∞. For the sites with larger coordination number z(cid:48) = z = 4, singlet correlations develop at larger V than fd for sites with z(cid:48) = z = 2. As might be expected for a PAM ON A QUASICRYSTAL LATTICE local quantity, the singlet correlator for the z = 4 sites matches quite well to those on a square lattice. (The We turn now to the quasicrystal geometry. Our 4×4 square lattice is anomalous because of its unusual discussionwillparallelthatoftheprecedingsection. For additional symmetries, and is not shown.) thislattice,thechoicesforcoordinationnumberaremore 5 0 4x4 β=30 Uf=4 44xx44 zz==22 ββ==1250 3 56xx56 ββ==3300 5x5 z=2 β=15 -0.2 4x4 z=4 β=15 4x4 z=4 β=20 ) 2 =0 5x5 z=4 β=15 ot zz(r 5x5 Sqr β=15 tSff d cf -0.4 1 -0.6 0 0 0.5 1 1.5 2 0 0.5 1 1.5 V V fd fd FIG. 5: (Color online) Local singlet correlator czz(cid:48)(r = 0) FIG. 6: (Color online) Localized electron antiferromagnetic fd for the half-filled Lieb lattice with Uf = 4t. The singlet structure factor for the Lieb lattice. For Vfd >∼ 1.1, Sfftot is correlations develop more rapidly for z = 2 than for z = 4 independent of temperature and lattice size N. However, since for smaller coordination number the competition with when T is decreased for Vfd <∼ 1.1, Sfftot grows as the system AForderisreduced. Datafordifferent(β =15,20)aswellas is cooled. These distinct behaviors reflect the completely differentsystemsizes(4x4and6x6)overlap: thisshortrange localnatureofmagneticcorrelationsinthesingletphase,and correlation function converges rather quickly as T is lowered an increasing correlation length at low T in the AF phase. and N is increased. Vertical dashed lines at V = 0.8,1.3 VerticaldashedlinesatV =0.8,1.3demarkthevaluesused fd fd demark the values used for the real space spin correlation for the real space spin correlation data of Fig. 4. data of Fig. 4. Square lattice data coincide well with Lieb sites with z=4. in Fig. 8 for N = 41 and N = 239 are similar), and it also converges with β fairly quickly. (Data in Fig. 8 for numerous, z = 1,2,···8, as evident in Fig. 2. The β =15 and β =20 are similar). z =1,2 sites originate in our use of OBC, a choice made The sum of the spin-spin correlation function of to avoid frustration of AF order[47]. It is important to localized fermions in the quasi-crystal geometry yields emphasizethatthesecoordinationnumbersoccuronlyat the structure factor and is given in Fig. 9 as a function the lattice edges. Their contribution to the properties of of Vfd. For hybridizations Vfd >∼ 1.1, where results in the system will vanish in the thermodynamic limit. Fig. 8 suggest singlet formation is robust for all z, Stot czz(cid:48)(r)forthequasi-crystalgeometryisgiveninFig.7 is temperature independent. Below V ∼1.1, curves fffor ff fd and shows a differentiation between long range behavior different β break apart, suggesting that AF correlations for Vfd = 0.8 and rapid decay to zero for Vfd = 1.4. are present and increasing as T is lowered. As noted in Similar to the Lieb case, czz(cid:48)(r) is larger for z = 4 the discussion of Fig. 6, the reduction in the structure ff than z = 2. Data for other z (not shown) confirm this factor at low V is a finite temperature effect: the fd trend. The AF correlations extending outward from a effective RKKY coupling goes as V2 and hence even fd site become more and more robust as the coordination largerβ isneededforAFcorrelationstodevelopatsmall number of the conduction orbital increases. V . See also [48]. fd Figure 8 shows the singlet correlator for the N = In the presence of long range order (LRO) the 239 site quasicrystal geometry of Fig. 2(bottom). The correlationapproachesanonzeroasymptoticvaluec(r → appearance of well-formed singlets depends on the ∞) → m2, where m is the order parameter, and the coordination number z of the conduction electron site- structure factor scales as S = Nm2. Even if LRO is the point of maximum change of cz (r = 0) shifts from present only at T = 0, as is the cases in d = 2 with fd V ∼ 0.4 to V ∼ 1.1 as z increases. This reflects the continuous symmetries, this scaling is observed at T low fd fd fact that AF is favored by a larger number of neighbors, enough that the correlation length exceeds the largest so that the cross-over to singlets requires larger V as linear lattice size studied. We expect S >Nm2 on finite fd z increases. Since czz(r = 0) is a local quantity, its lattices, since c(r) > m2 at small distances, and these fd value is relatively unaffected by total lattice size (data shortrangecontributionscanbesubstantialifthelattice 6 z = 1, N = 239, β = 20 z = 2, N = 239, β = 20 ) z = 3, N = 239, β = 20 =0 0 z = 4, N = 239, β = 20 (r z = 5, N = 239, β = 20 0.1 z’ z = 6, N = 239, β = 20 z d z = 7, N = 239, β = 20 cf z = 8, N = 239, β = 20 -0.2 z = 3, N = 41, β = 20 0 z = 6, N = 41, β = 20 r) z = 4, N = 239, β = 15 ( z’ z cff-0.1 -0.4 z=2 z=4 -0.2 V = 0.8 (AF phase) -0.6 V = 1.4 (singlet phase) 0 5 0 5 0 0.4 0.8 1.2 1.6 2 r r V fd FIG. 7: (Color online) z resolved spin-spin correlation FIG. 8: (Color online) The singlet correlators (circles) for function between localized orbitals for V = 0.8 (AF phase) the quasicrystal geometry with N = 239 sites and inverse fd and V = 1.4 (singlet phase) for the N = 41 quasicrystal temperature β =20 shown as functions of V . cz (r =0) is fd fd fd lattice at β =30. In the former case, cz(r) remains non-zero largestin magnitudeforsmallest z=1. Thesingletsbecome ff outtolargeseparations, whileinthelattercaseitfallsoffto lessandlesswell-formedaszincreases. DataforN =41sites zero. Left(right) panels are z=2(4). (squares) indicate that finite size effects are relatively small. Similarly, data for β = 15 (diamonds) show that the low T limit has been reached. Vertical dashed lines show the V fd size is small. A finite size scaling plot is given in Fig. 10. values of Fig. 7. CONCLUSIONS 6 We have explored the competition between antiferromagnetic order and singlet formation in N = 41, β = 15 5 N = 41, β = 20 the periodic Anderson model in 2D geometries which N = 41, β = 25 are unfrustrated, but which have conduction electron N = 41, β = 30 coordination which varies from site to site. As is 4 N = 239, β = 5 N = 239, β = 10 intuitively reasonable, singlet formation depends on N = 239, β = 15 z, and is delayed to larger interorbital hybridization ffStot3 N = 239, β = 20 V as z increases. Our data suggest that, as in the fd case of uniform z, AF order is present in the ground 2 state at low V and absent at large V . Related fd fd issues arise in models in which site dilution provides 1 different conduction electron coordination [49, 50] or in which variation in conduction electron-local orbital hybridization is considered [51]. It is interesting to note 0 0 0.5 1 1.5 2 that the anomalous tendency for the staggered moment V fd to go down with increasing z in the spin-1/2 Heisenberg model on quasicrystal lattices [38, 40–42] would be even more evident in itinerant Hamiltonians such as that FIG. 9: (Color online) Stot as a function of V for several ff fd studied here, since the greater coordination number inverse temperatures β and quasicrystal lattice sizes N = reduces the local moment. A number of experimental 41,239. As for the Lieb lattice, curves coincide for different systems also exhibit a similar behavior in which T β in the singlet phase at large Vfd, but break apart at Vfd ≈ Neel 1.0−1.1. This signals the development of antiferromagnetic canbehigheratthe(lowerz)surfacethaninthe(higher correlations at large spatial separations at low V . Vertical z) bulk.[52] fd dashed lines show the V values of Fig. 7. fd Both geometries studied have unusual U = 0 single f 7 Although we have emphasized here the presence of sites with different conduction electron lattice coordination numbers, an alternate perspective on our work is 0.1 that of a study of a PAM in which the conduction electronsthemselveshaveseveralbands. TheLieblattice geometry, for example, has three sites per unit cell, and 0.08 hencethreeconductionbands(Fig.3),inadditiontothe localized orbitals. Our DQMC simulations indicate that thecompetitionbetweensingletformationandAForder 0.06 N isnotfundamentallyaffectedbythismorecomplexband / ot structure. tSff 0.04 ACKNOWLEDGEMENTS 0.02 The work of NH was supported by NSF-PHY-1263201 (REU program). RTS and WTC were supported by de- sc0014671. 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