{'fraction_non_alphanumeric': 0.03554514183159679, 'fraction_numerical': 0.013460292138192333, 'mean_word_length': 4.462962962962963, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Magnetic Correlations in a Periodic Anderson Model with Non-Uniform Conduction Electron Coordination

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 quantum critical point (QCP) at zero temperature and also on possible experimental connections to 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 conduction electron sites can have variable coordination. This situation is relevant both to recently discovered magnetic quasicrystals and also to magnetism in doped heavy fermion systems.


INTRODUCTION
The single band Hubbard Hamiltonian [1][2][3][4] captures several of the most fundamental consequences of electronelectron interactions in solids, namely magnetic order and the Mott metal-insulator transition. Although the question is still open, it may even contain the fundamental physics of d-wave superconductivity in the cuprates [5,6]. Multi-band Hamiltonians like the periodic Anderson model (PAM) [7,8] examine what happens when two species of electrons, one delocalized 'conduction' band (often d), and another localized band (often f ) are present. Here a central effect is the competition between singlet formation [9], when the conduction and localized electrons are strongly hybridized, and ordering of the local moments mediated indirectly through the Ruderman-Kittel-Kasuya-Yosida interaction [10][11][12][13].
In heavy fermion materials [14,15], this competition is believed to explain different low temperature phases, e.g. non-magnetic CeAl 3 where the f moments are screened by the conduction electrons, and CeAl 2 which becomes antiferromagnetic (AF) at low temperatures. Likewise, in Ce metal itself, as well as in other rareearths, the tuning of the f -d hybridization V fd through pressure is believed to play a crucial role in the 'volume collapse' transition [16][17][18][19][20][21].
Quantum Monte Carlo (QMC) studies of the PAM have explored some of this physics, in one [22,23], two [24], and three dimensions [25]. The focus has been on bipartite lattices which, at half-filling, host AF order without frustration and are also free of the fermion sign problem [26,27]. QMC in infinite dimensions [28] complements work in finite d by allowing simulations at very low temperature, at or even well below the Kondo scale, at the expense of some of the knowledge of correlations in space.
There is interest in understanding the magnetic correlations in more general geometries. One such modification allows for intersite, rather than on-site, hybridization between conduction and local orbitals and hence metallic behavior in the absence of interactions [29]. Another motivation is provided by chemical substitution in heavy fermion materials, either by the replacement of some of the local moment atoms by non-magnetic ones, as in (Ce,La)CoIn 5 [30] or by changes to the conduction orbitals, as in the alloying of Cd onto In sites in CeCo(In,Cd) 5 [31,32].
In the latter situation, V fd is reduced locally, and AF droplets can form around the impurity sites. A second motivation is the recent observation of a quantum critical state in magnetic quasicrystals [33,34]. In these Au-Al-Yb alloys (Au 51 Al 34 Yb 15 ), measurements of the magnetic susceptibility χ and specific heat C diverge as T → 0. This non-Fermi liquid (NFL) behavior is associated with strongly correlated 4f Yb electrons. These two sets of materials share a common feature which is that the coordination number of the different atoms is no longer spatially uniform. The effects of these unique local environments can be probed with nuclear magnetic resonance [35].
The NFL behavior of Au-Al-Yb alloys has recently been studied by solving the U = ∞ Anderson Impurity Model (AIM) for a single local moment coupled to conduction electrons in a quasicrystal approximant geometry [36].
The crucial result is that singular responses in χ and C occur as a consequence of a broad (power law) distribution of Kondo temperatures which delays screening of a large fraction of the magnetic moments until very low temperatures.
In this paper we study the PAM in two different geometries: the Lieb lattice and a 2D "Ammann-Beenker" tiling [37,38]. Quasicrystaline approximates [39] for Au 51 Al 34 Yb 15 are in 3D; the quasi-periodic Ammann-Beenker tiling is a more tractable 2D alternative for QMC, which is limited in the number of sites which can be simulated.
Our goal is to explore the nature of magnetic correlations as a function of f -d hybridization, and, specifically, to understand the competition of antiferromagnetic order and singlet formation in geometries where the coordination number of different sites in the lattice is non-uniform. Our work extends that of [36] by examining a dense array of local orbitals and also by including the effect of finite U f . We begin with the Lieb lattice, because it contains two separate coordination numbers, z = 2, 4 while still retaining very simple lattice periodicity. We then turn to the more complicated quasicrystal approximant structure. We do not at present address the anomalous NFL behavior of the magnetic susceptibility, since those phenomena appear to be associated only with 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][41][42]. Surprisingly, unlike the case of regular geometries where the ordered moment increases from the honeycomb (z = 3) to square (z = 4) lattices, it is found that the local AF order parameter decreases with z. We will comment on this further in our conclusions.

MODEL AND METHODS
The PAM is a tight binding Hamiltonian for which each spatial site contains both an extended and a localized orbital, Here t is the hybridization between conduction orbitals with creation(destruction) operators d † iσ (d iσ ) on near neighbor sites ij . In this paper we consider the two conduction electron geometries shown in Figs. 1,2 corresponding to "Lieb" and "quasicrystal" lattices respectively. Each site of these lattices also contains a localized orbital, creation(destruction) operators f † iσ (f iσ ). U f is the on-site interaction between spin up and spin down electrons on the localized orbital, and V fd is the conduction-localized orbital hybridization. Both geometries of Figs. 1,2 are bipartite. In H we have written the interaction term, in 'particle-hole' symmetric form, and set the site energy difference between f and d orbitals to zero, so that the lattice is half-filled for all temperatures T and Hamiltonian parameters t, U f , V fd . Half-filling optimizes the tendency for AF correlations, and also allows DQMC simulations at low temperature since the sign problem [26] is absent. Figure 3 shows the density of states and band structure of the PAM on a Lieb lattice for t = 1 and V fd = 1.
There are six bands corresponding to the six sites (three conduction and three localized) per unit cell. The lattice is bipartite with four of the six sites on one sublattice and two on the other. Hence, in accordance with Lieb's theorem [43] there are two flat bands (at E = ±1). As in the case of the PAM on a square lattice with on-site hybridization, the half-filled lattice is a band insulator in the non-interacting limit. However, by comparing calculations for on-site and intersite V fd , the latter being metallic at half-filling, it has been shown that many properties of the PAM when U f /t > ∼ 4 are insensitive to the presence of a U f /t = 0 band gap [29]. The magnetic properties of the PAM are characterized by intra-and inter-orbital spin-spin correlations, Here the superscripts z, z refer to the coordination number of the conduction orbital on site i and i + r respectively. This separation allows us to isolate the effects of the number of neighbors on the spin correlations [44]. We focus here on c zz ff (r) which measures intersite magnetic correlations between the local electrons, and c zz fd (r = 0), the singlet correlator between local and conduction electrons on the same site. The conduction electron sites range in coordination from z = 1 to z = 8. The use of two colors for the sites emphasizes that, despite its complexity, the geometry is still bipartite.
but depend more generally on both i and r in irregular lattices.
We also measure the structure factor, which sums the spin-spin correlations to all distances r from sites i with a given z. The staggered phase factor (−1) r takes the value ±1 on the two sublattices of the bipartite geometry and hence measures AF order. The z-resolved contributions to the total structure factor S tot ff are weighted by the fractions of sites in the lattice with a given coordination g zz . In the singlet phase, the spin . D(E = 0) vanishes: the system is a band insulator at half-filling. The DOS for the N = 239 quasicrystal approximant is also shown in the top panel. As is the case for the Lieb lattice PAM, the quasicrystal PAM also has a hybridization gap at E = 0. The single band case is metallic [36,42].
correlations decay exponentially with separation r and S tot ff gets contributions only from a small number r < ξ of local correlations. It becomes temperature independent below a relatively high T set by the singlet energy scale.
In an ordered phase, on the other hand, S tot ff will depend on temperature down to much lower T as the correlation length ξ diverges. Thus a T dependence of S tot ff can be used as an indicator of AF order.
Our computational approach is determinant Quantum 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 (imaginary) time dependent auxiliary field. This field is sampled stochastically to obtain the expectation values of different correlation functions. The update moves require the non-local computation of the fermion Green's function, which also the quantity needed to measure equal time observables including the energy, double occupation, and spin correlations. The algorithm involves matrix operations and scales as the cube of the product of the number of spatial lattice sites and the number of orbitals. In certain special situations, including the PAM on the geometries studied here, the sampling is free of the sign problem [26] so that the simulation may be conducted on large lattices (here several hundreds of spatial sites) at low temperature (here T /t < ∼ 1/30).

PAM ON THE LIEB LATTICE
We begin with the Lieb lattice which has 2N/3 sites of coordination number z = 2 and N/3 sites with z = 4. Figure 4 shows c zz ff (r) for V fd = 0.8 and V fd = 1.3. In the former case, the correlation function alternates between positive and negative values, with a correlation length which exceeds the linear lattice size, as is characteristic of an AF phase. r = 1 corresponds to the separation between unit cells, so that integer values of r are between sites with z = z (and hence the same sublattice) and half-integer values have z = z (and hence occupy different sublattices). The AF correlations are evident in both z = 2 and z = 4, although they are larger for higher coordination number. This reflects the collective nature of the AF order, which is more robust as the number of neighbors grows. Actually, because the A and B sublattices have different numbers, the ordered phase is Ferrimagnetic [43], with N ↑ = N ↓ in addition to the staggered pattern seen in the Figure. For V fd = 1.3, on the other hand, c zz ff (r) falls rapidly to zero, indicative of a singlet phase.
The AF and singlet regimes can also be distinguished by c zz fd (r = 0), as shown in Fig. 5. (Here since r = 0 the coordination numbers z = z.) c zz fd (r = 0) vanishes for V fd = 0 where the localized and conduction fermions are decoupled, and saturates at a large value for V fd → ∞. For the sites with larger coordination number z = z = 4, singlet correlations develop at larger V f d than for sites with z = z = 2. As might be expected for a local quantity, the singlet correlator for the z = 4 sites matches quite well to those on a square lattice. (The 4 × 4 square lattice is anomalous because of its unusual additional symmetries, and is not shown.) In Fig. 6 we turn to the AF structure factor, Eq. 3, which sums the spin correlations on the localized orbitals over the whole lattice. In the singlet phase, c zz ff (r) is short ranged and temperature independent, achieving its ground state value at T ∼ V 2 fd /U f . In the AF phase, on the other hand, the correlation length grows as T is lowered, and hence c zz ff (r) contributes to the structure factor out to larger and larger distances. The structure factor becomes temperature dependent at low T . These two regimes are evident, and are separated by V c ∼ 1.1. This is suggestive, but certainly not conclusive, evidence of the presence of a QCP. At the end of the following section we will provide a finite size scaling analysis of this data to ascertain whether there is true long range order below V c . Note that the reduction in S tot ff as V f d is reduced below V f d ≈ 0.7 is a finite temperature effect. The RKKY exchange scales as V 2 f d and T = t/30 (βt = 30) is no longer low enough to reach the ground state.

PAM ON A QUASICRYSTAL LATTICE
We turn now to the quasicrystal geometry. Our discussion will parallel that of the preceding section. For this lattice, the choices for coordination number are more numerous, z = 1, 2, · · · 8, as evident in Fig. 2. The z = 1, 2 sites originate in our use of OBC, a choice made to avoid frustration of AF order [47]. It is important to emphasize that these coordination numbers occur only at the lattice edges. Their contribution to the properties of the system will vanish in the thermodynamic limit.
c zz ff (r) for the quasi-crystal geometry is given in Fig. 7 and shows a differentiation between long range behavior for V fd = 0.8 and rapid decay to zero for V fd = 1.4. Similar to the Lieb case, c zz ff (r) is larger for z = 4 than z = 2. Data for other z (not shown) confirm this trend. The AF correlations extending outward from a site become more and more robust as the coordination number of the conduction orbital increases. Figure 8 shows the singlet correlator for the N = 239 site quasicrystal geometry of Fig. 2(bottom). The appearance of well-formed singlets depends on the coordination number z of the conduction electron sitethe point of maximum change of c z fd (r = 0) shifts from V fd ∼ 0.4 to V fd ∼ 1.1 as z increases. This reflects the fact that AF is favored by a larger number of neighbors, so that the cross-over to singlets requires larger V fd as z increases. Since c zz fd (r = 0) is a local quantity, its value is relatively unaffected by total lattice size (data The sum of the spin-spin correlation function of localized fermions in the quasi-crystal geometry yields the structure factor and is given in Fig. 9 as a function of V fd . For hybridizations V fd > ∼ 1.1, where results in Fig. 8 suggest singlet formation is robust for all z, S tot ff is temperature independent. Below V fd ∼ 1.1, curves for different β break apart, suggesting that AF correlations are present and increasing as T is lowered. As noted in the discussion of Fig. 6, the reduction in the structure factor at low V fd is a finite temperature effect: the effective RKKY coupling goes as V 2 fd and hence even larger β is needed for AF correlations to develop at small V fd . See also [48]. In the presence of long range order (LRO) the correlation approaches a nonzero asymptotic value c(r → ∞) → m 2 , where m is the order parameter, and the structure factor scales as S = N m 2 . Even if LRO is present only at T = 0, as is the cases in d = 2 with continuous symmetries, this scaling is observed at T low enough that the correlation length exceeds the largest linear lattice size studied. We expect S > N m 2 on finite lattices, since c(r) > m 2 at small distances, and these short range contributions can be substantial if the lattice size is small. A finite size scaling plot is given in Fig. 10.

CONCLUSIONS
We have explored the competition between antiferromagnetic order and singlet formation in the periodic Anderson model in 2D geometries which are unfrustrated, but which have conduction electron coordination which varies from site to site. As is intuitively reasonable, singlet formation depends on z, and is delayed to larger interorbital hybridization V fd as z increases. Our data suggest that, as in the case of uniform z, AF order is present in the ground state at low V fd and absent at large V fd . Related issues arise in models in which site dilution provides different conduction electron coordination [49,50] or in which variation in conduction electron-local orbital hybridization is considered [51]. It is interesting to note that the anomalous tendency for the staggered moment to go down with increasing z in the spin-1/2 Heisenberg model on quasicrystal lattices [38,[40][41][42] would be even more evident in itinerant Hamiltonians such as that studied here, since the greater coordination number reduces the local moment. A number of experimental systems also exhibit a similar behavior in which T N eel can be higher at the (lower z) surface than in the (higher z) bulk. [52] Both geometries studied have unusual U f = 0 single  Using 4x4, 6x6, and 8x8 unit cells, the normalized structure factor S tot ff /N scales to a nonzero value for V fd = 0.8. Here the inverse temperature β = 30 which is large enough that ground state properties have been reached for lattices of the sizes shown. particle eigenstates. In the case of the Lieb lattice, the inequivalence of the number of sites in the A and B sublattices leads to the presence of flat bands. In the true quasicrystal geometry (of which we explore only an approximant) the eigenstates exhibit an intermediate scaling with system size, between the limits where the participation ratio P grows with the number of sites, as occurs for Bloch states, and P ∼ o(1) for conventional localization [32,36]. Thus our work also explores the effects of those features of the noninteracting spectrum on magnetic correlations in the presence of interactions. Limitations on accessible system sizes and temperatures prevent us from addressing high precision questions such as whether the distribution of z destroys a sharp quantum critical point (QCP) between the AF and singlet phases and replaces it with a more gradual cross-over. Related DMFT work [36] indicates that there is a range of Kondo temperatures. It is interesting to note that, to within the accuracy DQMC simulations provide, there is remarkably little variation between the critical value of V fd for different 2D conduction electron geometry. The QCP appears to be quite close to the square lattice value V fd /t ≈ 1 for the Lieb and quasicrystal lattices considered here.
The usual view of the PAM is of a two band (conduction and localized) tight binding Hamiltonian.
Although we have emphasized here the presence of sites with different conduction electron lattice coordination numbers, an alternate perspective on our work is that of a study of a PAM in which the conduction electrons themselves have several bands. The Lieb lattice geometry, for example, has three sites per unit cell, and hence three conduction bands (Fig. 3), in addition to the localized orbitals. Our DQMC simulations indicate that the competition between singlet formation and AF order is not fundamentally affected by this more complex band structure.