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Iron pnictides as a new setting for quantum criticality

Contributed by Elihu Abrahams, January 26, 2009 (received for review December 26, 2008)
This article has a correction. Please see:
Abstract
Two major themes in the physics of condensed matter are quantum critical phenomena and unconventional superconductivity. These usually occur in the context of competing interactions in systems of strongly correlated electrons. All this interesting physics comes together in the behavior of the recently discovered iron pnictide compounds that have generated enormous interest because of their moderately hightemperature superconductivity. The ubiquity of antiferromagnetic ordering in their phase diagrams naturally raises the question of the relevance of magnetic quantum criticality, but the answer remains uncertain both theoretically and experimentally. Here, we show that the undoped iron pnictides feature a unique type of magnetic quantum critical point, which results from a competition between electronic localization and itinerancy. Our theory provides a mechanism to understand the experimentally observed variation of the ordered moment among the undoped iron pnictides. We suggest P substitution for As in the undoped iron pnictides as a means to access this example of magnetic quantum criticality in an unmasked fashion. Our findings point to the iron pnictides as a muchneeded setting for quantum criticality, one that offers a unique set of control parameters.
The recent discovery of copperfree highT_{c} superconductors has triggered intense interest in the homologous iron pnictides. The parent compound of the lanthanumiron oxyarsenide, LaOFeAs (1), exhibits a tetragonalorthorhombic structural transition and longrange antiferromagnetic order (2). Electron doping, via fluorine substitution for oxygen, suppresses both and induces superconductivity. Other families of the arsenide compounds show a similar interplay among structure, antiferromagnetism, and superconductivity. These include the oxyarsenide systems obtained through replacing lanthanum by other rareearth elements such as Ce, Pr, Nd, Sm, and Gd (3–6), as well as oxygenfree arsenides, such as BaFe_{2}As_{2} (7) and SrFe_{2}As_{2} (8).
Quantum Criticality in the Pnictides
The existence of the antiferromagnetic state naturally raises the possibility of carrierdopinginduced quantum phase transitions in the iron pnictides (9–11), but the situation is not yet certain. Theoretically, the evolution of the Fermi surface as a function of carrier doping is not yet well understood, and this limits the study of quantum criticality. Experimentally, earlier measurements in LaO_{1−x}F_{x}FeAs (1) and SmO_{1−x}F_{x}FeAs (12) show a moderate suppression of the magnetic/structural transition temperature(s) as x is increased; beyond x of about ∼7%, the transitions are interrupted by superconductivity. Further experiments have led to conflicting reports for the firstorder or secondorder nature of the carrierinduced zerotemperature magnetic and structural phase transitions (13–15).
We propose that an alternative to a possible dopinginduced quantum phase transition is one that is accessed by changing the relative strength of electron–electron correlations. Thus, we suggest that the iron pnictides may exhibit an example and setting for quantum criticality. Our approach is motivated by the phenomenological and theoretical evidence that the parent iron pnictide is a “bad metal“ (9, 16, 17). Accordingly, we formulate our considerations in terms of an incipient Mott insulator: the electron–electron interactions lie close to, but do not exceed the critical value for the insulating state. Within this picture, the electronic excitations comprise an incoherent part away from the Fermi energy, and a coherent part in its vicinity. The incoherent electronic excitations are described in terms of localized Fe magnetic moments, with frustrating superexchange interactions. The latter have been discussed earlier by two of us (9) and others (18). This division of the electron spectrum is a simple and convenient way of analyzing the complex behavior of a bad metal close to the Mott transition, whose spectrum exhibits incipient upper and lower Hubbard bands and a coherent quasiparticle peak at the Fermi energy (19).
The coupling of the local moments to the coherent electronic excitations competes against the magnetic ordering. A magnetic quantum critical point arises when the spectral weight of the coherent electronic excitations is increased to some threshold value.
The Electron Spectrum
The incoherent and coherent parts of the singleelectron spectral function are illustrated schematically in Fig. 1 The central peak describes the coherent itinerant carriers; these are the electronic excitations that are responsible for a Drude optical response and that are adiabatically connected to their noninteracting counterparts. The side peaks describe the incoherent excitations, vestiges of the lower and upper Hubbard bands associated with a Mott insulator that would arise if the electron–electron interactions were larger than the Mott localization threshold. Each of the three peaks may in general have a complex structure due to the multiorbital nature of the iron pnictides. The decomposition of the electronic spectral weight into coherent and incoherent parts is natural for a metal near a Mott transition (19, 20).
We use w to denote the percentage of the spectral weight lying in the coherent part of the spectrum. A relatively small w may be inferred for the iron pnictides, because the Drude weight seen in the optical conductivity (21–23) is very small (on the order of 5% of the total spectral weight integrated to ≈2 eV). A small w corresponds to an interaction strength sufficiently large that the system is close to the Mott transition, albeit on the metallic side; this implies a large electron–electron scattering rate, consistent with the observed large electrical resistivity (on the order of 0.5 mΩ · cm for single crystals and 5 mΩ · cm for polycrystals) at room temperature. In terms of electrical conduction, the iron pnictides are similar to, e.g., V_{2}O_{3}, a bad metal (with a room temperature resistivity (24) of about 0.5 mΩ · cm) that is known to be on the verge of a Mott transition, and is very different from, e.g., Cr, a simple metal [with a room temperature resistivity (25) of ≈0.01 mΩ · cm] which orders into a spindensitywave ground state.
Effective Hamiltonian
To study the magnetism, the incoherent spectrum is naturally described in terms of localized magnetic moments, leading to a matrix J_{1}–J_{2} model (9): Here, J_{1} and J_{2} label the superexchange interactions between two nearestneighbor (n.n., 〈ij〉) and nextnearestneighbor (n.n.n., 〈〈ij〉〉) Fe sites, respectively. Both are matrices in the orbital basis, α, β with these indices summed when repeated. J_{H} is the Hund's coupling.
Eq. 1 reflects the projection of the full interacting problem to the lowenergy subspace when the system is a Mott insulator (w = 0) and the singleelectron excitations have only the incoherent part. When the singleelectron excitations also contain the coherent part (w being nonzero but small, see Fig. 1), these coherent electronic excitations must be included in the lowenergy subspace as well.
We will use the projection procedure of ref. 26 to construct the effective lowenergy Hamiltonian. We denote by d_{kασ}^{coh} the delectron operator projected to the coherent part of the electronic states near the Fermi energy, and define the incoherent part through d_{kασ} ≡ d_{kασ}^{coh} + d_{kασ}^{incoh}. Therefore, unlike the full delectron operator, d_{kασ}^{coh} does not satisfy the fermion anticommutation rule. Indeed, its spectral function integrated over frequency defines w. We therefore introduce c_{kασ} = (1/
We then have the effective lowenergy Hamiltonian terms for the coherent itinerant carriers (H_{c}) and for their mixing with the local moments (H_{m}):
Here, τ labels the three Pauli matrices. In the projection procedure leading to Eq. 2, we keep d_{kασ}^{coh} as part of the lowenergy degrees of freedom; the prefactor w in the first equation comes from the rescaling c_{kασ} = (1/
J_{1}–J_{2} Competition
The superexchange interactions in the iron pnictides contain n.n. and n.n.n. terms because of the specific relative locations of the ligand As atoms and Fe atoms (9, 18, 27). To assess the tunability of J_{1} and J_{2}, we consider an oversimplified case, illustrated in Fig. 2. Here, only one Fe 3d orbital is considered. We assume that the 3d orbital on each of the 4 corners of a square plaquette has an identical hybridization matrix element, V, with one As 4p orbital located above the center of the plaquette. The superexchange interaction is found to be h_{J} ∝ ∑_{r} [∑_{□}s(r)]^{2}, where r labels a plaquette in the 2D square lattice and the summation ∑_{□} is over the 4 Fe sites of a plaquette. For classical spins, this is the canonical case of magnetic frustration: all states with ∑_{□}s(r) = 0 are degenerate. Written in the form of Eq. 1, this corresponds to J_{2} = J_{1}/2. This discussion is instructive for the understanding of the realistic exchange interactions in the iron pnictides. Several aspects are neglected in the simplified analysis given above. First, multiple 3d orbitals are important, and the hybridization is orbitalsensitive. Both the J_{1} and J_{2} interactions are therefore matrices. Second, the real band structures must be described by more complex dp, pp, and dd tightbinding parameters. Both features spoil the elementary J_{2} = J_{1}/2 relationship. Still, the simple considerations given above suggest that the overall strength of J_{2} and J_{1}, i.e., the largest eigenvalues of the J_{2} and J_{1} matrices, are comparable with each other. Detailed analysis of the matrix elements indicates that there are more entries in the J_{2} matrix than in the J_{1} matrix that correspond to the dominating antiferromagnetic component, and that the overall magnitude (the largest eigenvalue) of the J_{2} matrix will be somewhat larger than half of that of the J_{1} matrix. This conclusion is supported by the fitting of the ab initio results of the groundstate energies for various magnetic configurations in terms of J_{1} and J_{2} parameters in the (nonmatrix) Heisenberg form (18, 27). This range of J_{2}/J_{1} leads to a twosublattice collinear antiferromagnetic ground state, consistent with the results of the neutron scattering experiment (2).
On the one hand, the above argument implies that the magnetic frustration effect is strong, and can provide significant quantum fluctuations leading to a reduced ordered moment. On the other hand, it suggests that the degree to which J_{2}/J_{1} can be tuned in practice could be limited.
Magnetic Quantum Critical Point
The order parameter for the twosublattice antiferromagnet appropriate for J_{2}/J_{1} > 1/2 is the staggered magnetization, m, at wave vector Q = (π,0). The effective theory for the H_{J} term alone corresponds to a φ^{4} theory whose action is of the form S ∼ rϕ^{2} + uϕ^{4}. The coupling to the coherent quasiparticles is given by the H_{m} term of Eq. 2; it causes a shift of the tuning parameter r and also introduces a damping term. These contributions to the r coefficient are given by: Here, f(ɛ) is the Fermi–Dirac distribution function and a_{γ} is an orbitaldependent coefficient: ∑_{γ} a_{γ} s_{γ} appears in the order parameter for the (π,0) antiferromagnet. Note that both g_{k,qαβγ} and ɛ_{k,β}, ɛ_{k+q,α} are linear order in w. We can infer from Eq. 4 that the damping term is of the order w^{0} at low energies: for ω_{n} ≪ wW (W is the bandwidth), Γ = γω_{n}, where γ is, to leading order in w, the constant value associated with the couplings and density of states of the w = 1 case. Note that γ is nonzero because, for the parent compounds, Q connects the hole pockets near the Γ point of the Brillouin zone (BZ) and the electron pockets near the M points (in the unfolded BZ notation). At the same time, γ does not diverge since the nesting is not perfect. The existence of the linear in ω damping term is in contrast to the doped case, where Q no longer connects the hole and electronFermi surfaces (11). Importantly, we can also infer from Eq. 4 that the leading frequency and temperatureindependent term Δr = wA_{Q} is linear in w, with A_{Q} = ∑_{k,α,β,γ} G_{k,qαβγ}^{2} a_{γ}^{2} [Θ(E_{F} − E_{k + Q}) − Θ(E_{F}−E_{k})]/(E_{k,β} − E_{k+Q,α}) (where Θ is the Heaviside function) is independent of w, and positive.
The lowenergy Ginzburg–Landau theory then takes the form, where r(w) = r(w = 0) + wA_{Q}. r(w = 0) is negative, placing the system at w = 0 to be antiferromagnetically ordered. The linear in w shift, wA_{Q}, causes r(w) to vanish at a w = w_{c}, leading to a quantum critical point. In terms of the external control parameter δ, shown in Fig. 3, w = w_{c} defines δ = δ_{c}. The ϕ^{4} theory describes a z = 2 (where z is the dynamical exponent) antiferromagnetic quantum phase transition, which is generically second order.
The O(3) vector m, corresponding to the (π,0) order, is accompanied by another O(3) vector, m′ that describes the (0,π) order. These two vector order parameters accommodate a composite scalar, m · m′, the order parameter for an Ising transition (10, 11, 28). In turn, the Ginzburg–Landau action, Eq. 5, contains a quartic coupling
The magnetic quantum criticality will strongly contribute to the electronic and magnetic properties in the quantum critical regime. We note that since d = z = 2, there are (marginal) logarithmic corrections to simple Gaussian critical behavior (29). Following discussions in, e.g., ref. 29, we expect that the specificheat coefficient will be C/T ∼ ln (1/T), the NMR relaxation rate 1/T_{1} ∝ const, and (in the presence of disorder scattering that smears the Fermi surface) the resistivity ρ ∝ T.
Tuning Parameter and Variation of Magnetic Order
The parent materials of the different iron arsenides will have different internal pressures and “c/a“ ratios, and will correspondingly have different ratios of the electron–electron interaction to the effective bandwidth. According to our theory, the resulting variation of the coherent spectral weight w will, in turn, tune the control parameter r in Eq. 5, and the ordered moment will change accordingly across the different compounds.
Neutron scattering experiments have indeed found that the ordered moment does vary across the parent arsenides. The moment associated with Feordering at low temperatures is ≈0.20.3 μ_{B}/Fe in NdOFeAs (30, 31), 0.4 μ_{B}/Fe in LaOFeAs (2), 0.5 μ_{B}/Fe in PrOFeAs (32, 33), and 0.81.0 μ_{B}/Fe in CeOFeAs (15), BaFe_{2}As_{2} (34), and SrFe_{2}As_{2} (35).
As_{1−δ}P_{δ} Series of the Parent Iron Pnictides
Because the clattice constant in LaOFeP is smaller than that in LaOFeAs, these considerations suggest that the coherentelectron spectral weight of the iron phosphides is larger than that of the iron arsenides. A consequence is that, in contrast to the arsenide, the phosphide does not have a magnetic transition (36). We then propose that a parent iron pnictide series created by P doping of As presents a means to unmask a magnetic quantum critical point. Our purpose is better served the weaker the superconductivity is in the P end material. LaOFeAs_{1−δ}P_{δ} is promising, since LaOFeP is a weak superconductor whose T_{c} is only a few Kelvin or may even vanish (37–39). CeOFeAs_{1−δ}P_{δ} may also be of interest in this context. While CeOFeAs (15) is antiferromagnetic, CeOFeP is a paramagnetic metal (40). We remark in passing that Pdoping for As is more advantageous than external pressure, because the latter is known to cause a volume collapse (41). It would be interesting to search for a substitution for As such that w could be reduced, leading toward to the Mott insulating state.
To understand the tuning of the microscopic electronic parameters, we have carried out densityfunctionaltheory (DFT) calculations on both CeOFeAs and CeOFeP for comparison. We find that the dp hybridization matrix is larger in CeOFeP than in CeOFeAs. This is consistent with the qualitative consideration that, compared with CeOFeAs, CeOFeP has a higher internal pressure and, hence, a higher kinetic energy and smaller ratio of the interaction to the bandwidth, thus a larger coherent weight w.
Comparison with DFT Studies
We have considered the mechanism for quantum fluctuations having in mind the proximity to the Mott limit, where the instantaneous atomic moment is large (a few μ_{B}/Fe) to begin with. Most DFT calculations have shown that the ordered moment in the antiferromagnetic ground state is large, of the order 2 μ_{B}/Fe. Moreover, such a large ordered moment was found within DFT not only for the parent iron pnictides, but also for their doped counterparts.
Since DFT calculations neglect quantum fluctuations, we are tempted to interpret the large DFTcalculated moment as essentially the instantaneous atomic moment. Quantum fluctuations will then lead to a reduced ordered moment in the true ground state. The J_{1}–J_{2} competition together with the coupling of the local moments to the coherent itinerant electronic excitations arising naturally in the Mottproximity picture we have described is just such a mechanism for quantum fluctuations.
Discussion
We have developed a framework to describe the quantum magnetism of the iron pnictides, appropriate for electron–electron interactions that are of an intermediate strength to place the materials at the delicate boundary between itinerancy and localization. Our description takes into account the interplay between the itinerant and localmoment aspects, which are naturally associated with the interactioninduced coherent and incoherent parts of the electronic excitations. Enhancement of the spectral weight associated with the coherent electronic excitations weakens the magnetic order, and induces a magnetic quantum critical point. Our characterization of the magnetic excitations is important not only for the understanding of the existing and future experiments in the normal state, but also for the microscopic understanding of hightemperature superconductivity in the iron pnictides and related metallic systems close to a Mott transition. In addition, realization of a magnetic quantum critical point in the iron pnictides provides a new setting to explore some of the rich complexities (42, 43) of quantum criticality; this is much needed since quantum critical points have so far been explicitly observed only in a very small number of materials.
Acknowledgments
We thank G. Cao, P. Coleman, C. Geibel, A. Jesche, C. Krellner, Z.Y. Lu, E. Morosan, D. Natelson, C. Xu, and Z.A. Xu for useful discussions. This work was supported by the National Science Foundation of China, the 973 Program, and the Program for Changjian Scholars and Innovative Research Team in University (RT0754) of the Education Ministry of China (J.D.), the Robert A. Welch Foundation (Q.S.), and the Department of Energy (J.X.Z.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: abrahams{at}physics.rutgers.edu

Author contributions: J.D., Q.S., J.X.Z., and E.A. designed research; J.D., Q.S., J.X.Z., and E.A. performed research; and Q.S. and E.A. wrote the paper.

The authors declare no conflict of interest.
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 Article
 Abstract
 Quantum Criticality in the Pnictides
 The Electron Spectrum
 Effective Hamiltonian
 J_{1}–J_{2} Competition
 Magnetic Quantum Critical Point
 Tuning Parameter and Variation of Magnetic Order
 As_{1−δ}P_{δ} Series of the Parent Iron Pnictides
 Comparison with DFT Studies
 Discussion
 Acknowledgments
 Footnotes
 References
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