# Spin-state Crossover Model for the Magnetism of Iron Pnictides

###### Abstract

We propose a minimal model describing magnetic behavior of Fe-based superconductors. The key ingredient of the model is a dynamical mixing of quasi-degenerate spin states of Fe ion by intersite electron hoppings, resulting in an effective local spin . The moments tend to form singlet pairs, and may condense into a spin nematic phase due to the emergent biquadratic exchange couplings. The long-range ordered part of varies widely, , but magnon spectra are universal and scale with , resolving the puzzle of large but fluctuating Fe-moments. Unusual temperature dependences of a local moment and spin susceptibility are also explained.

###### pacs:

75.10.Jm, 74.70.Xa, 71.27.+aSince the discovery of superconductivity (SC) in doped LaFeAsO Kam08 , a number of Fe-based SC’s have been found and studied John10 . Evidence is mounting that quantum magnetism is an essential part of the physics of Fe-based SC’s. However, the origin of magnetic moments and the mechanisms that suppress their long-range order (LRO) in favor of SC remain far from being well understood.

The magnetic behavior of Fe-based SC’s is unusual. The ordered moments range from , as in spin-density wave (SDW) metals like Cr, to typical for Mott insulators, causing debates whether the spin-Heisenberg Yil09 ; Xu08 ; Si08 ; Fan08 ; Uhr09 ; Sta11 or fermionic-SDW pictures Maz08 ; Kur08 ; Chu08 ; Gra09 ; Kan09 are more adequate. At the same time, irrespective to the strength or very presence of LRO, the Fe-ions possess the fluctuating moments Gre11 ; Vil12 , even in apparently “nonmagnetic” LiFeAs and FeSe. In fact, it was noticed early on that the Fe-moments, “formed independently on fermiology” Joha10 and “present all the time” Yil09 , are instrumental to reproduce the measured bond-lengths and phonon spectra Maz09 ; Joha10 ; Yil09 ; Rez09 . Recent experiments Liu12 ; Zho13 ; Wan13 observe intense high-energy spin-waves that are almost independent of doping, further supporting a notion of local moments induced by Hund’s coupling Yin11 and coexisting Dai09 ; Kou09 ; Lv10 with metallic bands.

While the formation of the local moments in multi-orbital systems is natural, it is puzzling that these moments (residing on a simple square lattice) may remain quantum disordered in a broad phase space despite a sizable interlayer coupling; moreover, the Fe-pnictides are semimetals with strong tendency of the electron-hole pairs to form SDW state, further supporting classical LRO of the underlying moments. A fragile nature of the magnetic-LRO in Fe-pnictides thus implies the presence of a strong quantum disorder effects, not captured by ab-initio calculations that invariably lead to magnetic order over an entire phase diagram. The ideas of domain wall motion Maz09 and local spin fluctuations Yin11 were proposed as a source of spin disorder, but no clear and tractable model of quantum magnetism in Fe-based SC’s has emerged to date. Here we propose such a model.

Since Fe-pnictides are distinct among the other (Mn, Co, Ni) families, their unique physics should be rooted in specific features of the Fe-ion itself. In fact, Fe is famous for its spin-crossover Gut04 : it may adopt either of = states depending on orbital splitting, covalency, and Hund’s coupling. As the ionic radius of Fe is sensitive to its spin, Fe- bond length ( is a ligand) is also crucial. In oxides, =2 is typical and = occur at high pressures only Sta08 . In compounds with more covalent Fe- bonds (=S, As, Se), =0 is more common while = levels are higher. Here it comes the basic idea of this Letter: when the covalency and Hund’s coupling effects compete, the many-body ground state (GS) is a coherent superposition of different spin states intermixed by electron hoppings, resulting in an average effective spin whose length depends on pressure, etc. We explore this dynamical spin-crossover idea, and find that: (i) local moment may increase with temperature explaining recent data Gre13 ; (ii) interactions between contain large biquadratic exchange, and resulting spin-nematic correlations compete with magnetic-LRO; (iii) the ordered moment varies widely, but magnon spectra are universal and scale with as observed Liu12 ; Zho13 ; Par12 ; (iv) singlet correlations among lead to the increase of the spin susceptibility with temperature Kli10 .

The Fe-ions in pnictides have a formal valence state Fe. Among its possible spin states [Fig. 1(a)], low-spin ones are expected to be favored; otherwise, the ordered moment would be too large and robust. The states, “zoomed-in” further in Fig. 1(b), are most important since they can overlap in the many-body GS by an exchange of just two electrons between ions, see Fig. 1(c). The corresponding -process converts Fe(=0)–Fe(=0) pair into Fe(=1)–Fe(=1) singlet pair and vice versa; this requires the interorbital hopping which is perfectly allowed for Fe-As-Fe bonding. Basically, is a part of usual exchange process when local Hilbert space includes different spin states =0,1; hence . Coupling between =1 triplets is contributed also by their indirect interaction via the electron-hole Stoner continuum and, as expected, it reduces with doping Yar09 as the electron-hole balance of a parent semimetal becomes no longer perfect.

The Hamiltonian describing the above physics comprises three terms: on-site energy of =1 triplet relative to =0 singlet , and the bond interactions :

(1) |

The operator creates a singlet pair of spinfull -particles on bond . For a general spin of -particles, with denoting the projections; physically, . The constraint is implied SM ; note_orb .

The above model rests on three specific features of Fe-pnictides/chalcogenides: (i) spin-state flexibility of Fe that can be tuned by pressure increasing , (ii) edge-sharing Fe tetrahedral structure allowing “spin-mixing” -term, and (iii) semimetallic nature which makes values to decrease upon doping Yar09 .

Figure 1(d–f) demonstrates the behavior of spin-1 -particles () on a single bond. The GS wavefunction is a superposition of two singlets and , with the ”spin-mixing” angle . The GS energy . At , there is a sudden jump [Fig. 1(e)] from state to once the -energy compensates the cost of having two -particles. At finite , the dynamical mixing of spin states converts this transition into a spin-crossover, where the effective spin-length increases gradually. Fig. 1(f) shows that -term strongly stabilizes the singlet pair of -particles; this leads (see later) to a large biquadratic coupling which is essential in Fe-pnictides Wys11 ; Yar09 ; Yu12 .

We are ready to show the model in action, explaining recent observation of an unusual increase of the local moment upon warming Gre13 . This fact is at odds with Heisenberg and SDW pictures but easy to understand within the spin-crossover model. Indeed, the spin-length may vary as a function of which, in turn, is sensitive to lattice expansion; in fact, Gretarsson et al. found that the moment value follows -axis thermal expansion . We add (magnetoelastic) coupling in Eq. (1), affecting value, and evaluate and self-consistently. This is done by minimizing the elastic energy is the usual thermal expansion coefficient), together with the GS energy given above. This results in a linear relation between the magnetic moment () and lattice expansion. They both strongly increase with temperature if lattice is ”soft” enough (i.e., small ), as demonstrated in Fig. 1(g,h) by employing the parameters , , , , , and , providing a good fit to the experimental data of Ref. Gre13 . (

Turning to collective behavior of the model, we notice first that for and large , the GS is dominated by tightly bound singlet dimers derived from the single-bond solution. The resonance of dimers on square-lattice plaquettes then supports a columnar state Rea89 breaking lattice symmetry without magnetic LRO note_orb . In the opposite limit of , the model shows a condensation of -bosons. We found that the model relevant here is also unstable towards a condensation of -particles with . This condensate hosts interesting properties not present in a conventional Heisenberg model. We discuss them based on the following wavefunction describing Gutzwiller-projected condensate of spin-1 -bosons:

(2) |

where is the condensate density to be understood as the effective spin-length . The complex unit vectors () determine the spin structure of the condensate in terms of the coherent states of spin-1 Iva03 ; Lau06 corresponding to , , . The GS phase diagram obtained by minimizing and cross-checked by an exact diagonalization on a small cluster is presented in Fig. 2. We have included nearest-neighbor (NN) and next-NN interactions and fixed their ratio at , reflecting large next-NN overlap via As ions. Like in model, this ratio decides between and order. Fig. 2(a,b) contains, apart from a disordered (uncondensed) phase () at small , three distinct phases depending on and values: (i) Ferroquadrupolar (FQ) phase with and . This phase has zero magnetization and is characterized by the quadrupolar order parameter

The part of the phase diagram relevant to pnictides is shown in Fig. 2(d). The decrease of is associated with doping that changes the nesting conditions Yar09 , while the increase of is related to external/chemical pressure. Fig. 2(e,f) shows that the LRO-moment quickly vanishes as () values decrease (increase); however, the spin-length remains almost constant (), corresponding to a fluctuating magnetic moment . This quantum state is driven by -process which generates the spin-1 states in a form of singlet pairs.

We consider now the excitation spectrum. It is convenient to separate fast (density) and slow (spin) fluctuations. We introduce pseudospin indicating the presence of a -particle, and a vector field defining the spin-1 operator as . The resulting Hamiltonian

(3) |

is decoupled on a mean-field level. The condensate spin dynamics is then given by -symmetric Hamiltonian

(4) |

with the renormalized and . The excitations are found by introducing , , bosons according to

Shown in Fig. 3 is the excitation spectra for several points of the phase diagram. The spin-length fluctuations are high in energy. Fig. 3(b) focuses on the magnetic excitations. In the FQ phase, quadrupole/magnetic modes are degenerate and gapless at . As the AF phase is approached, the gap at decreases, and closes upon entering the magnetic phase. However, the higher energy magnons (which scale with ) are not much affected by transition, apart from getting (softer) harder in a (dis)ordered phase; this explains the persistence of well-defined high-energy magnons into nonmagnetic phases Liu12 ; Zho13 .

The magnetic modes in Fig. 3(b) resemble excitations of bilinear-biquadratic spin model Lau06 . In fact, the dispersion in FQ phase can be exactly reproduced note_bqH from an effective spin-1 model , with and given above. A large biquadratic coupling was indeed found to account for many observations in Fe-pnictides Wys11 ; Yar09 ; Sta11 . We note however, that this model possesses FQ and AF phases only and misses the ns-AF phase, where the ordered moment is reduced already at the classical level; also, it does not contain the key notion of the original model, i.e., formation of the effective spin and its fluctuations.

Singlet correlations inherent to the model may also lead to increase of the paramagnetic susceptibility with temperature Kli10 . Considering nonmagnetic phase, we find that for the field parallel to the director , is temperature dependent,

To conclude, we proposed the model describing quantum magnetism of Fe-pnictides. Their universal magnetic spectra, wide-range variations of the LRO-moments, emergent biquadratic-spin couplings are explained. The model stands also on its own: extending the Heisenberg models to the case of “mixed-spin” ions, it represents a novel many-body problem. Of a particular interest is the effect of band fermions which should have a strong impact on low energy dynamics of the model, e.g., converting the Goldstone modes into overdamped spin-nematic fluctuations. Understanding the effects of coupling between local moments and band fermions, including implications for SC, should be the next step towards a complete theory of Fe-pnictides.

J.C. acknowledges support by the Alexander von Humboldt Foundation, ERDF under project CEITEC (CZ.1.05/1.1.00/02.0068) and EC 7 Framework Programme (286154/SYLICA).

## References

- (1) Y. Kamihara et al., J. Am. Chem. Soc. 130, 3296 (2008).
- (2) For a review of the experimental data, see, e.g., D.C. Johnston, Adv. Phys. 59, 803 (2010).
- (3) T. Yildirim, Physica C 469, 425 (2009).
- (4) C. Xu, M. Müller, and S. Sachdev, Phys. Rev. B 78, 020501(R) (2008).
- (5) Q. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401 (2008).
- (6) C. Fang et al., Phys. Rev. B 77, 224509 (2008).
- (7) G. Uhrig et al., Phys. Rev. B 79, 092416 (2009).
- (8) D. Stanek, O.P. Sushkov, and G. Uhrig, Phys. Rev. B 84, 064505 (2011).
- (9) I.I. Mazin, D.J. Singh, M.D. Johannes, and M.H. Du, Phys. Rev. Lett. 101, 057003 (2008).
- (10) K. Kuroki et al., Phys. Rev. Lett. 101, 087004 (2008).
- (11) A.V. Chubukov, D.V. Efremov, and I. Eremin, Phys. Rev. B 78, 134512 (2008).
- (12) S. Graser, T.A. Maier, P.J. Hirschfeld, and D.J. Scalapino, New J. Phys. 11, 025016 (2009).
- (13) E. Kaneshita, T. Morinari, and T. Tohyama, Phys. Rev. Lett. 103, 247202 (2009).
- (14) H. Gretarsson et al., Phys. Rev. B 84, 100509(R) (2011).
- (15) P. Vilmercati et al., Phys. Rev. B 85, 220503(R) (2012).
- (16) M.D. Johannes, I.I. Mazin, and D.S. Parker, Phys. Rev. B 82, 024527 (2010).
- (17) I.I. Mazin and M.D. Johannes, Nature Phys. 5, 141 (2009).
- (18) D. Reznik et al., Phys. Rev. B 80, 214534 (2009).
- (19) M. Liu et al., Nature Phys. 8, 376 (2012).
- (20) K.-J. Zhou et al., Nature Commun. 4, 1470 (2013).
- (21) M. Wang et al., arXiv:1303.7339.
- (22) Z.P. Yin, K. Haule, and G. Kotliar, Nature Mater. 10, 932 (2011).
- (23) J. Dai, Q. Si, J.-X. Zhu, and E. Abrahams, Proc. Natl. Acad. Sci. U.S.A. 106, 4118 (2009).
- (24) S.-P. Kou, T. Li, and Z.-Y. Weng, Europhys. Lett. 88, 17010 (2009).
- (25) W. Lv, F. Krüger, and P. Phillips, Phys. Rev. B 82, 045125 (2010).
- (26) P. Gütlich and H.A. Goodwin (Eds.), Spin Crossover in Transition Metal Compounds I (Springer, Berlin, 2004).
- (27) S. Stackhouse, Nature Geosci. 1, 648 (2008).
- (28) H. Gretarsson et al., Phys. Rev. Lett. 110, 047003 (2013).
- (29) J.T. Park et al., Phys. Rev. B 86, 024437 (2012).
- (30) R. Klingeler et al., Phys. Rev. B 81, 024506 (2010).
- (31) A.N. Yaresko, G.-Q. Liu, V.N. Antonov, and O.K. Andersen, Phys. Rev. B 79, 144421 (2009).
- (32) In the Supplemental Material, we derive the Hamiltonian (1) from a two-orbital Hubbard model.
- (33) To address a tetra/ortho structural (”orbital order”) transition, we may include also orbital degeneracy of triplets; this is left for future work.
- (34) A.L. Wysocki, K.D. Belashchenko, and V.P. Antropov, Nature Phys. 7, 485 (2011).
- (35) R. Yu et al., Phys. Rev. B 86, 085148 (2012).
- (36) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
- (37) B.A. Ivanov and A.K. Kolezhuk, Phys. Rev. B 68, 052401 (2003).
- (38) A. Läuchli, F. Mila, and K. Penc, Phys. Rev. Lett. 97, 087205 (2006).
- (39) H. Tsunetsugu and M. Arikawa, J. Phys. Soc. Jpn. 75, 083701 (2006).
- (40) K. Harada and N. Kawashima, Phys. Rev. B 65, 052403 (2002).
- (41) E. Demler and F. Zhou, Phys. Rev. Lett. 88, 163001 (2002); A. Imambekov, M. Lukin, and E. Demler, Phys. Rev. B 68, 063602 (2003).
- (42) S.K. Yip, Phys. Rev. Lett. 90, 250402 (2003).
- (43) C.M. Puetter, M.J. Lawler, and H.-Y. Kee, Phys. Rev. B 78, 165121 (2008).
- (44) M. Serbyn, T. Senthil, and P.A. Lee, Phys. Rev. B 84, 180403(R) (2011).
- (45) This can be understood using the identity . If , like in the FQ phase or close to it in the ns-AF phase, we recover the -term of Eq. (4): .

Supplemental Material for Spin-state crossover model for the magnetism of iron pnictides

Jiří Chaloupka and Giniyat Khaliullin Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

Central European Institute of Technology, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic

Here we analyze two-orbital Hubbard model in the regime of large Hund’s coupling and large interorbital hopping, and explicitly demonstrate the emergence of the effective model proposed in the main paper. We also provide estimates of the model parameters in terms of the microscopic parameters of the Hubbard model.

Based on the ”orbital-differentiation” mechanism – which is particularly pronounced in multiorbital systems with large Hund’s coupling (see Ref. Geo12 for recent discussion) – we assume a coexistence of strongly correlated orbitals (hosting magnetic moments) and more itinerant bands (responsible for the charge transport and Fermi-surface related physics). For the Fe-pnictide/chalcogenide families, a minimal model for the ”magnetic” sector is a two-orbital Hubbard model which may accommodate magnetic moments ranging from zero to per Fe-ion, depending on the parameter regime. This possible moment-window is what observed in Fe-pnictide/chalcogenides Gre11 (and also consistent with the model of the main text). We assume that these two orbitals (labeled and below) are populated by two electrons per site on average, while the remaining four electrons out of Fe- configuration form a semimetallic band structure. The itinerant bands are not a prime source of magnetic moments but, as noticed in the main text, we keep in mind that they may mediate the interactions between local moments and hence support their long-range order Joh10 ; Joh09 .

Let us focus now on the ”magnetic” sector, i.e. two-orbital Hubbard Hamiltonian. As usual, it comprises two parts, local interactions and intersite hoppings: . Its local part includes the crystal field splitting between and orbitals (their precise structure in terms of original states is not essential here) and local correlations:

(S1) |

The local pair-hopping term is neglected, and the relation between inter- and intra-orbital Coulomb interactions will be used. The kinetic term of the Hamiltonian contains the intersite hopping of both intra- and inter-orbital character

(S2) |

Similar model was recently considered in Ref. Kun11 to address the spin-transition physics in cobaltates. The key difference of our model is the presence of interorbital hopping , which converts the transitions found in Ref. Kun11 into a smooth spin-crossover such that the ground state magnetic moment length (not long-range order parameter!) may acquire any value from zero to .

Our aim is to obtain the model Hamiltonian of the main paper as an effective low-energy Hamiltonian resulting from in the appropriate regime of parameters etc. This is achieved by a standard procedure – we select the relevant bond states from the eigenbasis of and obtain effective interactions on the bonds by eliminating perturbatively, employing the low-energy and configurations as the intermediate states. To check the validity of this approach, the exact eigenstates of on a single bond are calculated and the results compared with those of the effective Hamiltonian we have derived.

In the spin-crossover regime discussed in the main paper, large Hund’s coupling nearly compensates the crystal field splitting (i.e., ) and makes the on-site singlet and the three triplet states , quasidegenerate. These states thus form the relevant low-energy sector while the other states (such as ) are much higher in energy and can be ignored. , and

To be able to extract the effective Hamiltonian on a bond, it is convenient to consider the subspaces with total spin separately. In sector, the relevant bond states are the two configurations depicted in Fig. S1(a, b): with the local energy , and with the local energy . The bond interaction originates from virtual processes employing as the intermediate states mainly the low-lying and configurations presented in Fig. S1(c, d). They are denoted as and and their local energy amounts to . The other bond states have a negligible contribution to the groundstate, due to their high energy or due to kinematic (no hopping) reasons. The lowest state in the sector is composed of a pair of on-site triplets and states analogous to and but having total spin one. Finally, the only states in the are the combinations of two on-site triplets. These states are unaffected by hopping.

The validity of the above classification of low-energy levels of Hubbard model is demonstrated in Fig. S2 showing the results of an exact diagonalization of full two-orbital model on a single bond. We consider a representative set of parameters and such that a spin-crossover regime, where the on-site singlet and triplet states are quasidegenerate, is realized: . Focusing on sector in Fig. S2(b), we can observe that with increasing , the state gets gradually involved into the groundstate, which becomes a mixture of , and the higher energy states , serving as the intermediate states for the -processes. The contribution of the other states, which are neglected in our derivation of the effective Hamiltonian below, is indeed negligible.

Having selected our basis states and evaluated their local energy, we proceed now by incorporating the intersite hopping within this basis. First, the initial Hamiltonian is projected to the selected subspace of total spin and denoted accordingly as (where ). In the next step, the intermediate states are eliminated from -matrix by perturbation theory. After these steps, we will obtain an effective Hamiltonian that operates within configuration alone, and compare it with the model Hamiltonian used in the main paper.

In the most interesting subspace, after the elimination of intermediate states, the Hamiltonian projected to the subspace spanned by , , , states

(S3) |

operating now within the and singlet states of configuration. Here, denotes the excitation energy. In the second order perturbation theory , but by diagonalizing the energy dependent self-consistently, one can exactly reproduce the groundstate energy and the ratio of and coefficients obtained by diagonalizing the original matrix . In the following, we therefore take with being the groundstate energy of .

Using the same procedure, the pairs of local triplets of total spin obtain an energy with being the excitation energy, and those of total spin remain at an energy .

The effective Hamiltonian can now be exactly mapped to the model Hamiltonian proposed in the main paper. For a single bond, using the same notations, the corresponding matrix elements of read as

(S4) |

To make the correspondence between and matrices complete, we had to include small biquadratic exchange . The term-by-term comparison of the matrix elements of and yields the following values of the model parameters

(S5) |

As evidenced by Fig. S3(a), the effective model gives an adequate description of the lowest states of the Hubbard model. The obtained model parameters entering Eqs. (S4) and (S5) are presented in Fig. S3(b) as functions of the interorbital hopping amplitude . The realistic range of and is obtained by taking . The small biquadratic exchange contained in can be neglected at this point because the much larger effective biquadratic contribution is in fact generated by the -processes dynamically (see main text).

It is worth noticing that the strength of the key process of the model is finite due to interorbital hopping . This process is thus ineffective in perovskite lattices, but it is perfectly allowed for the Fe-(As/Te)-Fe bonding geometry of Fe-pnictides/chalcogenides and leads to the spin-crossover mechanism (”soft” magnetism) in these compounds (see main text). Concerning the role of intra-orbital -hopping in the mapping, it did not enter the above formulas, since does not connect any pair of the selected low energy states. The intermediate states that can be reached by have an energy higher by than those involved by , so that the -effect on and values is relatively weak. It is only found to increase by about .

To conclude, we have shown that the model Hamiltonian proposed in the paper naturally emerges from the two-orbital Hubbard model with strong Hund’s coupling, when a regime of spin-state quasidegeneracy is realized. The model parameters that follow from this derivation are well within the ranges that we have explored in our study.

## References

- (S1) A. Georges, L. de’ Medici, and J. Mravlje, arXiv:1207.3033.
- (S2) H. Gretarsson et al., Phys. Rev. B 84, 100509(R) (2011).
- (S3) M.D. Johannes, I.I. Mazin, and D.S. Parker, Phys. Rev. B 82, 024527 (2010).
- (S4) M.D. Johannes and I.I. Mazin, Phys. Rev. B 79, 220510(R) (2009).
- (S5) J. Kuneš and V. Křápek, Phys. Rev. Lett. 106, 256401 (2011).