# Emergent magnetic degeneracy in iron pnictides due to the interplay between spin-orbit coupling and quantum fluctuations

## Abstract

Recent experiments in iron pnictide superconductors reveal that, as the putative magnetic quantum critical point is approached, different types of magnetic order coexist over a narrow region of the phase diagram. Although these magnetic configurations share the same wave-vectors, they break distinct symmetries of the lattice. Importantly, the highest superconducting transition temperature takes place close to this proliferation of near-degenerate magnetic states. In this paper, we employ a renormalization group calculation to show that such a behavior naturally arises due to the effects of spin-orbit coupling on the quantum magnetic fluctuations. In particular, the spin anisotropies introduced by spin-orbit coupling profoundly affect the renormalization group flow, leading to the appearance of a stable Gaussian fixed point with a large basin of attraction, resulting in the emergence of an enhanced magnetic degeneracy. We also discuss the implications of our findings to the superconductivity of the iron pnictides.

*Introduction.–* Magnetism in the iron pnictide superconductors remains an intensely studied subject, not least due to its impact on unconventional superconductivity (1); (3); (2), but also as a playground for exploring unusual types of magnetic orders (4); (5); (6).
While early experiments reported the prevalence of a stripe spin-density
wave (SSDW) in the phase diagrams of these systems (7); (8),
a series of recent experiments in several different compounds found
a richer behavior (9); (10); (12); (16); (14); (15); (20); (17); (18); (19); (13); (21); (24); (22); (23); (11). As optimal doping is approached and the SSDW transition temperature
is suppressed to zero, signaling a putative magnetic quantum critical
point (QCP), other types of magnetic order proliferate and coexist
with SSDW order. Although they are characterized by the same wave-vectors as the SSDW phase, , they do not break the tetragonal symmetry of the lattice – hence being dubbed magnetic phases (12). The proximity of the superconducting dome to this peculiar regime of intertwined magnetic phases with comparable transition temperatures raises important questions about the interplay between magnetism, quantum fluctuations, and superconductivity. Phenomenologically, these phases can be understood as double-Q configurations corresponding to a collinear or coplanar equal-weight superposition
of and orders –
in contrast to the single-Q SSDW phase (6).
Several microscopic mechanisms have been proposed to explain their
origin (4); (5); (25); (26); (32); (27); (28); (29); (30); (35); (33); (34); (31),
however, most of these works are mean-field calculations, which do
not account for the strong fluctuations characteristic of a QCP.

In this paper, we go beyond mean-field theory and employ a renormalization group (RG) approach to investigate the universal properties of the magnetic phase diagram of the iron pnictides near the putative QCP. For isotropic spins, we find that fluctuations drive the system deep into either the SSDW or one of the phases, completely lifting any near-degeneracy present at the mean-field level. We also recover previous results that the phase transition is driven first-order by the fluctuations (36); (37); (38); (39), signaled by the fact that the RG flow has fixed runaway trajectories instead of fixed points. We argue, however, that spin anisotropies must be taken into account due to the spin-orbit coupling (SOC) present in these systems. The typical energy scale of the SOC, as measured by ARPES experiments, is meV (40). Although near a finite temperature magnetic phase transition the effects of the SOC may be limited to a small temperature window (41), near the putative QCP the SOC is always a relevant perturbation.

Indeed, we find that the spin anisotropies induced by the SOC completely change the RG flow at . In particular, we find a stable Gaussian fixed point, absent in the spin isotropic case, with a large basin of attraction in parameter space. Importantly, at this Gaussian fixed point the SSDW and phases become degenerate, and quantum critical behavior is restored, as the first-order transition of the isotropic case is avoided. Higher order terms that are RG-irrelevant distinguish between the different orders. Our main point is that this Gaussian fixed point can explain the proliferation of near-degenerate magnetic phases observed in different pnictide materials as optimal doping is approached. This is shown schematically in Fig. 1. We discuss the broad implications of our results for the understanding of magnetic quantum criticality in general, and for the elucidation of the superconducting state of the pnictides.

*Renormalization group flow of the isotropic case.–*
Magnetic order in the iron pnictides is characterized by two distinct
ordering wave-vectors and
(using single iron Brillouin zone notation). The magnetic moment on
iron site is given by .
The relative orientations and amplitudes of the magnetic component
vector order parameters distinguishes three types of
order (6), as illustrated in Fig. 1:
(i) A single-Q SSDW, which takes place when only one of the
is non-zero; (ii) A collinear double-Q order
dubbed charge-spin density-wave (CSDW), corresponding to ;
(iii) A coplanar double-Q order dubbed spin-vortex crystal
(SVC), characterized by and .
Although the three types of order share the same magnetic wave-vectors,
they break distinct symmetries of the lattice: the SSDW phase is orthorhombic
(i.e. symmetric) whereas the CSDW and SVC phases are tetragonal
(i.e. symmetric) (4); (29); (6). Experimentally, nearly all parent iron
pnictide compounds display SSDW order (42); (43); (44).
However, upon approaching optimal doping, magnetic phases
have been observed to coexist with SSDW order in hole-doped BaFeAs compounds (12); (14); (13); (19) and electron-doped CaKFeAs compounds (11). At least in the case of Na-doped BaFeAs and Ni-doped CaKFeAs, it has been shown that the phases realized are the CSDW and SVC, respectively (13); (11).
Note that certain iron chalcogenides (FeTe and *A*FeSe)
display magnetic orders with other ordering wave-vectors (45); (46); (47). We will not discuss these particular cases here.

To discuss the universal properties of the magnetic phase diagram, we introduce the magnetic action (in the case of isotropic spins) (4); (5); (25); (28):

(1) |

Here, is the inverse bare susceptibility, with the bare mass parameter , tuning the distance to the mean-field QCP, momentum , bosonic Matsubara frequency with temperature , and Landau damping coefficient . The latter arises from the decay of magnetic excitations into particle-hole pairs. Note that and with and . The mean-field phase diagram as a function of the quartic coefficients has been obtained from straightforward minimization (4); (32) and is shown here in Fig. 2. The type of magnetic order is determined entirely by the coefficients and , whereas the coefficient ensures stability of the functional. The values of these coefficients are determined by the underlying microscopic model.

Previous works have found different scenarios in which doping leads to a change in the quartic coefficients that move the ground state from the SSDW phase to either the CSDW or the SVC phase (25); (39); (28); (29); (30); (31). However, the values obtained in these works are the mean-field (i.e. bare) values of the coefficients. A crucial question, particularly near the putative QCP, is how fluctuations renormalize the bare coefficients and affect the mean-field transition lines in the phase diagram of Fig. 2. At the QCP, and the two-dimensional system is at its upper critical dimension, , with dynamic critical exponent . In this case, we can employ an RG procedure to first order in the logarithms (48), and systematically integrate out high-energy degrees of freedom from the cutoff scale to a lower energy . The result is that the coefficients of the action (1) become functions of the energy , which satisfy three coupled non-linear differential equations (36); (37); (38); (39):

(2) | ||||

(3) | ||||

(4) |

with . Importantly, the RG equations exhibit no fixed points, but three fixed trajectories, , , and , which together characterize the runaway flow of the coefficients. Hence, even though some coefficients diverge at , the ratios between the coefficients remain fixed. Previous works have focused on the first of these fixed trajectories, relevant for the SSDW phase (36); (37); (38); (39). Here, we discuss their implications for the mean-field phase diagram. Our stability analysis reveals that these trajectories serve as separatrices in the phase diagram of Fig. 2: while their repulsive parts act as phase boundaries, their attractive parts are buried deep inside each phase. The key point is that the existence of these fixed trajectories ensures that the mean-field phases are stable against fluctuations. We further confirm this conclusion by a numerical solution of the flow equations for different initial conditions, which we show as lines projected onto the - plane in Fig. 2.

The flow of is not shown as it does not distinguish between the different magnetic orders. However, it does determine the order of the magnetic transition. We find that in all cases , which, when combined with the flow of and , violates the stability criterion of the free energy, indicating that quantum fluctuations render all transitions first-order, regardless of the magnetic ground state. Importantly, the fluctuations move the system deep into the ordered state selected at the mean-field level. As a result, any near degeneracy between different magnetic phases obtained in mean-field is strongly lifted, which makes it hard to reconcile with the experimental observations described above.

*Action in the presence of spin anisotropies.–* A
crucial ingredient missing in the analysis above is the spin anisotropy
that is generated by the spin-orbit coupling present in these systems (40); (49); (50).
Indeed, experimentally, the magnetic moments of each configuration
are found to point to well-defined directions: in the SSDW phase,
the moments point in-plane, parallel to the wave-vector direction (8);
in the SVC phase, the moments also point in-plane, making
with respect to the wave-vector directions (11); in the
CSDW phase, the moments point out-of-plane (15); (13)
(see Fig. 1). At the quadratic level,
the spin-orbit coupling gives rise to three different spin-anisotropic
terms (51); (30):

(5) |

The physical interpretation of each term is apparent: a small favors in-plane moments parallel to the wave-vector directions; a small favors in-plane moments perpendicular to the wave-vector directions; and a small favors out-of-plane moments. While the SSDW supports any of these three magnetization directions, SVC is only compatible with the and terms, and CSDW only with the term. Thus, the presence of spin anisotropies makes it impossible for the three magnetic ground states to be nearly degenerate, but they do allow, in principle, for SSDW to be near-degenerate with either CSDW or SVC.

Near a finite-temperature phase transition , the impact of the spin-anisotropy coefficients on the magnetic configuration is primarily restricted to a narrow temperature range immediately below (a detailed analysis is presented in the accompanying paper (41)). However, at the putative QCP, where , spin anisotropy is always a relevant perturbation. Moreover, the RG flows of the coefficients naturally generate anisotropies in the quartic coefficients of the action, which can be recast in the form:

(6) |

Here the indices correspond to the spin directions , and denotes one of the two magnetic order paramaters. In the isotropic limit, and , , , recovering the action (1) with and . Deriving the corresponding RG equations of the action (5) and (6) is tedious but straightforward; the explicit expressions are presented in the accompanying paper (41).

*Renormalization group flow of the anisotropic case.–*
To understand the solutions of the RG flow equations in the general
case, it is sufficient to note that the quadratic coefficients ,
due to their scaling dimension, can display two possible asymptotic
behaviors as . Either ,
in which case the associated spin components are quenched and do not
contribute to the action, or ,
signaling a transition and the condensation of the spin components related
to (in practice, the flow terminates prior to the divergence
due to a singularity in the flow equations at the cutoff scale, i.e.
). Importantly, only the smallest
of the quadratic coefficients will flow towards the
ordered phase, while the other two flow to the disordered state. Hence,
the with the smallest bare value selects which components
will condense. Thus, to understand the general RG flow structure of
the anisotropic problem, it is enough to consider three different
cases, corresponding to whether , , or
has the smallest bare value.

Let us first consider the case in which initially . The possible ground states are the SSDW phase with moments pointing parallel to the ordering vectors and the (hedgehog)-SVC phase (11), see Fig. 1. According to the discussion above, the components associated with ( and ) will condense while , , , and can be neglected, as they acquire an asymptotically infinite mass. Hence, the universal properties of the action are the same as those of the action restricted only to the and degrees of freedom:

(7) |

with effective coefficients and , where we used . Note that the coefficients drop out of the action. The RG flow of this action has two fixed runaway trajectories, and , with , indicative of first-order quantum phase transitions. However, we also find a Gaussian fixed point , which signals a second-order quantum phase transition and a near-degeneracy between the SSDW and SVC states.

In Fig. 3(a), we show the basins of attraction of the fixed point and fixed trajectories. Interestingly, we find a wide range of parameters for which the Gaussian fixed point is attractive, in contrast to the isotropic case. This implies that, even if the bare (mean-field) values of and are not near the phase boundary between SVC and SSDW, quantum fluctuations will bring the system to this special point of the phase diagram. We note that this Gaussian fixed point is reminiscent of that found previously in the isotropic case in the particular case (39).

A similar result is found for the case , whose action is analogous to Eq. (7) with . As for the case of dominant out-of-plane anisotropy, , the effective low-energy action depends on and . Although all quartic coefficients remain present, they only appear in two different combinations. In this case, the action has the same form as Eq. (7), albeit with replacing and replacing . In terms of the original coefficients, , , and . The possible ground states in this case are the SSDW and CSDW with out-of-plane moments. The fixed point and fixed trajectories are identical to those of Eq. (7), with the change of coefficients given above, see Fig. 3(b). In this case the Gaussian fixed point is associated with nearly-degenerate SSDW and CSDW phases.

Our main result is the emergence of a stable Gaussian fixed point when spin anisotropies due to SOC are included in the model. Although the analysis presented here relied on the asymptotic behavior of the quadratic coefficient, which allowed us to analyze the simpler action (7), numerical solutions of the coupled RG equations of the full action (5)-(6) yield identical results (see our accompanying paper (41)).

*Discussion.–* Our results provide a compelling scenario
to explain the observed proliferation of phases in close proximity to the symmetric SSDW phase as optimal doping is approached in different hole-doped,
electron-doped, and undoped iron pnictides (9); (10); (12); (16); (14); (15); (20); (17); (18); (19); (13); (21); (24); (22); (23); (11).
The variety of compounds displaying such similar behavior suggests
that it may be a universal property of iron-based materials tuned
to their putative magnetic QCP. Our analysis reveals that, near the
phase transition, the spin
anisotropies caused by the spin-orbit coupling are always relevant,
and profoundly modify the critical properties of the isotropic case.
In the latter case the RG flow is governed exclusively by fixed
trajectories that drive the quantum phase transition first-order and
lift any trace of degeneracy obtained at the mean-field level. However, in the anisotropic case there appears a Gaussian fixed point with a large basin of attraction. Because it is precisely at this Gaussian fixed point that the phase boundaries of the SSDW and the magnetic phase meet, this result implies that the quantum fluctuations, in the presence of spin anisotropy, not only restore the second-order
character of the transition, but also drive the system closer to a
near-degeneracy between and SSDW phases. We argue here that
this degeneracy is responsible for the observed emergence of
phases near optimal doping.

An important question concerns the impact of these results to superconductivity, since the maximum takes place around the putative magnetic QCP. While a detailed analysis is beyond the scope of this work, we note that the near degeneracy between and SSDW phases effectively enhances the symmetry of the magnetic order parameter from Ising-like (due to the spin anisotropy) to XY-like at the Gaussian fixed point, since two of the magnetic order parameter components only appear together as a quadratic form, e.g. in Eq. (7). This suggests an enhanced phase space for magnetic fluctuations. If superconductivity is indeed mediated by such fluctuations (1); (3); (2), this enhancement is likely to help increase .

###### Acknowledgements.

The authors are grateful to W. R. Meier, A. E. Böhmer, J. Kang, A. Kreisel, D. D. Scherer, and M. Schütt for valuable discussions. M.H.C. and R.M.F. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award number DE-SC0012336. B.M.A. acknowledges financial support from a Lundbeckfond fellowship (Grant No. A9318). P.P.O. acknowledges support from Iowa State University Startup Funds.### References

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