The Coherence Temperature in the Diluted Periodic Anderson Model
The Kondo and Periodic Anderson Model (PAM) are known to provide a microscopic picture of many of the fundamental properties of heavy fermion materials and, more generally, a variety of strong correlation phenomena in and systems. In this paper, we apply the Determinant Quantum Monte Carlo (DQMC) method to include disorder in the PAM, specifically the removal of a fraction of the localized orbitals. We determine the evolution of the coherence temperature , where the local moments and conduction electrons become entwined in a heavy fermion fluid, with and with the hybridization between localized and conduction orbitals. We recover several of the principal observed trends in of doped heavy fermions, and also show that, within this theoretical framework, the calculated Nuclear Magnetic Resonance (NMR) relaxation rate tracks the experimentally measured behavior in pure and doped CeCoIn. Our results contribute to important issues in the interpretation of local probes of disordered, strongly correlated systems.
Materials poised at the cusp of magnetic to non-magnetic phase boundaries exhibit a myriad of complex properties. As systems ranging from the cuprate superconductorsScalapino (1995); Keimer et al. (2015) to the heavy fermionsTou et al. (1995); Coleman (2007); Gegenwart et al. (2008) and iron pnictidesChubukov et al. (2008); Si et al. (2016) are moved with pressure, chemical doping, or temperature away from a regime where magnetic order is dominant, an incredible variety of alternate patterns of spin, charge and pairing emerges. A description of the resulting competition has been an on-going challenge to the condensed matter communityDagotto (2005).
Nuclear Magnetic Resonance (NMR) has been extensively used to explore microscopic properties of correlated materials, providing great insight into their nature Kitaoka et al. (1995); Curro (2009); Walstedt (2018). In particular, NMR experiments have determined the energy scale at which a heavy fermion state emerges, i.e. when -electrons become delocalized. This scale has been associated Curro et al. (2004); Yang and Pines (2008, 2012, 2014); Yang et al. (2015, 2008, 2017) with a coherence temperature, , whose signature appears, e.g. as an anomaly in Knight Shift (KS) measurements: While for normal metals the KS tracks the magnetic susceptibility, for most heavy fermion materials this tracking breaks down below a certain temperature, which is identified with Curro et al. (2004); Yang and Pines (2008). The coherence temperature is evident in multiple experimental probes, including transport, thermodynamic, and tunnelling measurements, but the microscopic origin, and its relation to the Kondo screening temperature remain open questions Yang et al. (2008); Wirth and Steglich (2016); Yang et al. (2017).
Additional complexity is introduced by added chemical impurities Miranda and DobrosavljeviÄ (2005); Kawasaki et al. (2006); Seo et al. (2014); Wirth and Steglich (2016); Chen et al. (2018), so that treating the effects of disorder is essential to understand many of the properties of correlated electron materials. Randomness is central to the emergent physics since it acts to limit the growth of charge ordered regions Nie et al. (2014). Likewise, dopant disorder can stabilize localized antiferromagnetic (AF) regions, explaining the persistence of AF even deep in the -wave phase Andersen et al. (2007). A similar phenomenon occurs in heavy fermion materials where AF long range order is induced via Cd doping of CeCoIn Seo et al. (2014); Pham et al. (2006). Of particular interest is the crossover between Kondo screening in the single-impurity limit, and collective screening with intersite interactions among multiple sites in a lattice.
A powerful approach to investigate these crossover regimes is to systematically replace the -sites with non-magnetic atoms. This leads to inhomogeneities in the magnetic response, with some spatial regions favoring strong spin correlations, while in others a paramagnetic behavior is preferred. Thus, instead of having a single external parameter that globally tunes a system through a magnetic/non-magnetic boundary, one should also investigate how the physical quantities behave in the presence of internal, and highly inhomogeneous degrees of freedom. One expects that NMR quantities like and the spin-lattice relaxation rate to have a strong dependence with impurity doping (e.g. La substitution on Ce-based compounds) and even acquire a distribution of values depending on the local environment Nakatsuji et al. (2002, 2004); Ohishi et al. (2009); Ragel et al. (2009); Lawson et al. (2018); MacLaughlin et al. (2001, 2004).
The nature of these emergent phenomena may be described by simplified models such as the Periodic Anderson Model (PAM) Gebhard (1997); Fazekas (1999); Vekić et al. (1995); Hu et al. (2017); Schäfer et al. (2018), and the related Kondo Lattice model Doniach (1977); Lacroix and Cyrot (1979); Fazekas and Müller-Hartmann (1991); Assaad (1999); Costa et al. (2017), which consider weakly correlated ‘conduction’ electrons hybridized with strongly correlated ‘localized’ ones. Tuning the strength of the hybridization in these models leads to a quantum phase transition (QPT), in which the ground state evolves from an antiferromagnetic (AF) ordering to a spin liquid state. Recent numerical work on the homogeneous PAM has captured the KS anomaly and provided by quantitatively characterizing the different orbital contributions to the global susceptibility Jiang et al. (2014); Jiang and Yang (2017). In the context of impurity doping 111Similar results were also found in a closely related spin system, as reported in Ref. Mendes-Santos et al., 2017., the PAM successfully describes the enhancement of AF correlations around doped impurities in CeCo(InCd) Benali et al. (2016); Wei and Yang (2017); Costa et al. (2018).
Here we study the combination of randomness and strong interactions with an exact numerical approach which allows for ‘real space imaging’ of spin correlations. We investigate the behavior of the coherence temperature and NMR quantities in chemically doped heavy fermion materials, such as in CeLaCoIn, with quantum simulations which accurately incorporate sites with missing magnetic moments. Our focus is on whether the calculated trends of these quantities resemble those from experimental NMR measurements Lawson et al. (2018), on the evolution with impurity doping and external parameters. To this end, we extend previousJiang et al. (2014) Determinant Quantum Monte Carlo (DQMC) simulations to treat the randomly diluted PAM (dPAM). Our key conclusions are as follows: (1) the KS anomaly exhibits a universal scaling behavior below , even in presence of disorder; (2) increases with hybridization (or pressure) and (3) linearly decreases with impurity concentration . Finally, (4) the NMR relaxation rate, , exhibits a strongly inhomogeneous pattern throughout the lattice. These demonstrate that several of the most fundamental conclusions of NMR experiments can be predicted, including the scaling behavior of .
The Hamiltonian for the dPAM reads 222An early attempt to investigate the dPAM Hamiltonian was reported in Ref. Costi et al., 1988.
where the sums run over a two-dimensional square lattice, with denoting nearest-neighbor sites, and or ; the notation for the operators is standard. The first term corresponds to the hopping of conduction electrons (the hopping integral, , sets the energy scale), while the last describes the Coulomb repulsion on localized -orbitals. The hybridization between these two orbitals is modelled by a site-dependent hopping
Here we consider full orbital dilution, in which we randomly set on a fraction of the sites. Physically, this is equivalent to completely removing -orbitals, similarly to the replacement of a magnetic Ce atom by a La one in CeCoIn, which locally suppresses both the moment on the -orbital and the possibility of - mixing (due to the distance of the La level from the Fermi energy).
Our DQMC approach Blankenbecler et al. (1981); Hirsch (1985); White et al. (1989); dos Santos (2003); Loh et al. (1990); Troyer and Wiese (2005) (see Supplemental Materials) focuses on half-filing (). To connect with NMR measurements, the central quantities of interest are magnetic susceptibilities, from which the Knight Shift and spin-lattice relaxation rate are obtained; see below. Due to the presence of two orbitals, the total spin on a given site is , with at sites containing -orbitals, and 0 otherwise. Thus, the total magnetic susceptibility is given by
where with ; the number of lattice sites is . Similarly, the Knight shift is
where () corresponds to the hyperfine coupling between the nuclear spin of In(1) atoms and conduction (localized) electrons. is a temperature-independent term arising from orbital and diamagnetic contributions to , which we set to zero. Recall that the Knight Shift is a local quantity, which depends on the distribution of nearest-neighbor site (Ce or La) moments to the central In(1) atomLawson et al. (2018). Thus, our data correspond to the average of representative sites that couple to both and spins. Since the hyperfine couplings are generally different Curro et al. (2001, 2004), and strongly material dependent, we follow a previous study Jiang et al. (2014) and take ; general trends are not sensitive to this choiceJiang et al. (2014), Our simulations capture qualitative features of , and , but not material-specific details. In what follows, our DQMC data are averages over 20-30 different disorder configurations on a square lattice, and . Most of our results were obtained for , corresponding to the singlet region for the clean PAMHu et al. (2017); Schäfer et al. (2018).
At high temperatures, localized electrons are weakly coupled to conduction bands, so the contribution of and may be disregarded. As a result, the Knight Shift [Eq.(3)] tracks the localized electron susceptibility and, under the same assumptions, the total susceptibility as well. Following the procedure adopted in analyses of the experiments, we perform a linear fit to our DQMC Knight Shift data as a function of the susceptibility in the high temperature region, i.e. [see Fig. 1 (a)]. Next, we define the renormalized KS, which is equal to at high temperatures. This equality holds as long as the - singlet channel is small. However, since , fails to track when becomes relevant: the associated energy scale is ; see Fig. 1 (b). In this way the Knight Shift, which detects the contribution of a - channel (hence the presence of delocalized 4-electrons), provides an important tool to investigate the emergence of a heavy fermion state and its temperature scale. The KS anomaly persists even in strongly diluted materials, as displayed in Fig. 1 (c), for CeLaCoIn.
It is worth emphasizing that the continued appearance of the coherence temperature in the presence of disorder, at a value similar to that of the pure system Jiang et al. (2014), is a non-trivial observation. Indeed, electrons in unpaired orbitals (which survive dilution) are known to give large contributions to the susceptibility, regardless of whether they are conducting or localized Charlebois et al. (2015); Mendes-Santos et al. (2017); Costa et al. (2018). This could, in principle, significantly affect the assumptions under which would track , hence also the value of . As we shall see, these effects are more relevant at very low temperatures, due to the possibility of long-range order setting in the ground state. The relatively weak dependence of the NMR quantities with dilution, as presented in Figs. 1 (a)-(c), is an important step towards a global understanding of in diluted systems.
Within a two-fluid model Nakatsuji et al. (2004); Curro et al. (2004); Yang and Pines (2008); Shirer et al. (2012); feng Yang (2016), one singles out the ‘heavy fermion fluid’ contribution to the KS by subtracting its ‘normal’ (high temperature) contribution, i.e. . Remarkably, experimental results suggest a universal behavior of for many different heavy fermion materials,
where and depend on the specific material, and on external parameters. We use this phenomenological scaling form for a more accurate estimate of through the collapse of our DQMC data, as shown in Fig. 1(d).
The behavior of the KS in Fig. 1, in particular its scaling behavior [Fig. 1(d)], provides a robust evidence that DQMC simulations qualitatively reproduce trends observed experimentally, even in the presence of disorder. We now turn our attention to the dependence of with external parameters, such as the hybridization, , which is tuned in experiments by applying pressure. Figure 2 (a) displays the behavior of as a function of for different impurity concentrations, . Regardless of the level of disorder, the coherence temperature increases monotonically with . This reproduces fundamental features of NMR measurements (e.g., for CeRhInLin et al. (2015)): larger hybridization increases the probability of a hopping from -orbitals to conduction ones, which in turn increases the energy scale ().
The effect of dilution on is already apparent in Fig. 2 (a): although the clean and disordered cases share the same qualitative trend, the value of decreases with . This reflects a crossover between dense and diluted Kondo regimes. To further emphasize this crossover, Fig. 2 (b) displays as function of dilution, for different values of hybridization. Notice that has a (roughly) linear dependence with , with even at high dilution. Our DQMC predictions are in good agreement with recent NMR results for CeLaCoIn, as shown in Fig. 2 (b); see Ref. [Lawson et al., 2018]. Data from early attempts to measure in CeLaCoIn (see, e.g., Ref. [Nakatsuji et al., 2002]) are also included in Fig. 2 (b): they also display a monotonic decrease of the coherence temperature with La doping.
We now turn to the spin-lattice relaxation rate, , as obtained through our DQMC simulations. quantifies a characteristic time in which a component of the nuclear spin (of a given site) reaches equilibrium after an external perturbation (magnetic field pulse); see Supp. Materials. It is a dynamical (real frequency) quantity whose numerical evaluation usually requires an analytic continuation of the imaginary-time DQMC data. Instead, we use an approximation to this expressionRanderia et al. (1992),
Notice that, due to the lack of translational symmetry in disordered systems, we analyze local contributions to , considering two species of sites: (i) Ce sites, those with an active -orbital, and (ii) La sites, those which had their -orbitals removed. Accordingly, we define for Ce and La as the average over their individual contributions, i.e. we average over the available sites of each type, and subsequently we average over disorder configurations. Figure 3 displays the behavior of the local for fixed and for different concentrations.
For reasons apparent below, we separate the discussion of Fig. 3 to two regimes: intermediate temperatures, , and low temperatures, , when properties reflect the dominant correlations in the ground state. In the intermediate temperature range, we note that data for the spin relaxation rate on Ce sites for the clean case and for both dilution cases ( and 0.80) are almost indistinguishable; for the La sites, data for these same concentrations are also nearly identical, though much smaller than those for the Ce sites. When compared with the experimental results in Fig. 10 of Ref. [Lawson et al., 2018], we see that the same data grouping occurs, and that the decrease of as the temperature decreases (below the broad maxima) is also present; the difference in magnitude between data for Ce and La sites is also noticeable. These features therefore provide evidence that the distribution is quite inhomogeneous throughout the lattice, with Ce sites behaving as in the clean case even for strong dilution. A possible explanation for this inhomogeneity may be a local nature of singlet formation, i.e. singlets have a short correlation length.
In the low-temperature regime, the strong attenuation observed in our DQMC results for the pure case is due to the spin gapped ground state (see the Supp. Materials). For the diluted systems, however, our data on Ce sites seem to converge to finite values as decreases, consistent with gapless behavior due to either enhanced magnetic correlations or metallic (Pauli-like) behavior, depending on the degree of dilution. The data for La sites when show that increases with decreasing temperatures for , corresponding to an enhancement of magnetic correlations on these sites, a behavior also found for the regularly depleted PAM Costa et al. (2018). We note that our model for dilution may impose a bias towards an AFM ground state, since conduction sites with removed partners are unable to form singletsMendes-Santos et al. (2017).
In summary, we have presented results for the magnetic susceptibility, Knight Shift, and NMR relaxation rate computed using DQMC simulations for the diluted Periodic Anderson Model. We showed that even in the presence of disorder, the Knight Shift anomaly displays a universal behavior with a phenomenological “universal” function shared with the clean case. We have also obtained the coherence temperature, , and its dependence on - hybridization, , and with the dilution fraction . We have found that is a linearly decreasing function of , reproducing a crucial feature of the experimental results for La-doped CeCoIn. Finally, we also discussed the spin-lattice relaxation rate, which is distributed quite inhomogeneously throughout the lattice. The qualitative agreement of our results with experimental NMR measurements for CeLaCoIn suggests DQMC is a powerful theoretical tool to model accurately the nature of spin correlations in disordered heavy fermion materials.
Although we have emphasized the use of DQMC within the context of condensed matter materials, our work also has important implications for “quantum gas microscopes” and their use to explore ultracold trapped atomsCheuk et al. (2015); Kuhr (2016); Ott (2016). Like the NMR measurements described here, quantum gas microscopy allows the resolution of single atoms, doubly occupied sites, and (local) magnetic order. A central focus is on nonequilibrium properties directly connected to the relaxation times studied here.
Acknowledgements.Acknowledgments: NCC, TP, and RRdS were supported by the Brazilian Agencies CAPES, CNPq, and FAPERJ. NJC was supported by NSF, grant number DMR-1005393 and DMR-1807889, and RTS by Department of Energy, grant number DE-SC0014671.
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Supplemental Material for: The Coherence Temperature in the Diluted Periodic Anderson Model
Determinant Quantum Monte Carlo (DQMC):
The DQMC method Blankenbecler et al. (1981); Hirsch (1985); White et al. (1989); dos Santos (2003) is an unbiased technique commonly used to investigate Hubbard-like Hamiltonians: it maps a -dimensional quantum system in a classical (+1)-dimensional one, due to the inclusion of an imaginary-time coordinate. Within this approach, one separates the one-body () and two-body () pieces in the partition function by using the Trotter-Suzuki decomposition, i.e. by defining , with being the number of imaginary-time slices, and the discretization grid. Then , with an error proportional to . This is exact in the limit . The resulting partition function is rewritten in quadratic (single-body) form through a discrete Hubbard-Stratonovich transformation (HST) on the two-body terms, . This HST introduces discrete auxiliary fields with components on each of the space and imaginary-time lattice coordinates, which are sampled by Monte Carlo techniques. In this work we choose , so that the error from the Trotter-Suzuki decomposition is less than, or comparable to, that from the Monte Carlo sampling. DQMC is able to measure a general set of single- and two-particle response functions, such as susceptibilities, which can be directly compared with experimental results.
Although numerically exact, DQMC is constrained by the infamous minus-sign problem Loh et al. (1990); Troyer and Wiese (2005), which restricts our analyses to the half-filling case, i.e. when both - and -orbitals have . Determinant Quantum Monte Carlo is especially well matched to analyze the problem of disorder and the local structures which form around an impurity, since it is formulated in real space. Furthermore, many types of randomnesses such as local variations in hybridization, on-site repulsion, and site removal, do not affect particle-hole symmetry. Therefore there is no sign problem at half-filling, regardless the presence of disorder (dilution) on the lattice. This allows us to investigate the behavior of correlations in all temperature scales.
Spin-lattice relaxation rate: The relaxation rate is defined as Curro (2009)
where is the square of the Fourier transform of the hyperfine interaction, and is the gyromagnetic ratio. The latter is related to the nuclear magnetic moment by , with being the nuclear magneton, the nuclear -factor, and the nuclear spin. In order to avoid analytic continuation procedures, our analyses are performed through a commonly used approximation, presented in Eq. (5).
As a benchmark for our results for the diluted case, here we examine the behavior of for the clean () system. Previous DQMC studies Vekić et al. (1995); Hu et al. (2017) of the PAM have provided evidence of a QPT from an antiferromagnetically (AFM) ordered ground state to a spin liquid phase at . Then, one expects that the behavior of for decreasing temperatures should reflect the properties of these different ground states Mendes-Santos et al. (2017). Figure S1 displays the behavior of the relaxation rate as a function of temperature for different values of . Here we show the results from extrapolating data for lattice sizes , 10, and 12 to . Within the AFM phase, or , approaches a finite nonzero value as , consistently with the absence of a spin gap, i.e. the presence of spin-wave excitations. On the other hand, for larger , decreases monotonically when is lowered, reflecting a spin gapped (spin liquid) ground state. Notice that the change in behavior of occurs around , in line with the reported in Ref. [Hu et al., 2017].