Fluctuation-Driven Selection at Criticality in a Frustrated Magnetic System: the Case of Multiple-{\bm{k}} Partial Order on the Pyrochlore Lattice

Fluctuation-Driven Selection at Criticality in a Frustrated Magnetic System:
the Case of Multiple- Partial Order on the Pyrochlore Lattice

Behnam Javanparast, Zhihao Hao, Matthew Enjalran, Michel J. P. Gingras Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Physics Department, Southern Connecticut State University, 501 Crescent Street, New Haven, CT 06515-1355, USA
Canadian Institute for Advanced Research, 180 Dundas St. W., Toronto, Ontario, M5G 1Z8, Canada
Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada
July 5, 2019
Abstract

We study the problem of partially ordered phases with periodically arranged disordered (paramagnetic) sites on the pyrochlore lattice, a network of corner-sharing tetrahedra. The periodicity of these phases is characterized by one or more wave vectors . Starting from a general microscopic Hamiltonian including anisotropic nearest-neighbor exchange, long-range dipolar interactions and second- and third-nearest neighbor exchange, we identify using standard mean-field theory (s-MFT) an extended range of interaction parameters that support partially ordered phases. We demonstrate that thermal fluctuations ignored in s-MFT are responsible for the selection of one particular partially ordered phase, e.g. the “4-” phase over the “1-” phase. We suggest that the transition into the 4- phase is continuous with its critical properties controlled by the cubic fixed point of a Ginzburg-Landau theory with a 4-component vector order-parameter. By combining an extension of the Thouless-Anderson-Palmer method originally used to study fluctuations in spin glasses with parallel-tempering Monte-Carlo simulations, we establish the phase diagram for different types of partially ordered phases. Our results elucidate the long-standing puzzle concerning the origin of the 4- partially ordered phase observed in the GdTiO dipolar pyrochlore antiferromagnet below its paramagnetic phase transition temperature.

Figure 1: (Color online) The yellow (arrowless) sites are disordered (paramagnetic). a) a 1- state with along . The spins on the (blue) ordered sites form a pattern on a triangle that does not share a corner with a disordered site. These ordered sites form a kagome plane (blue-shaded plane) perpendicular to the direction. b) 4- state arising from the superposition of four 1- states.

Highly frustrated magnetism is one of the paradigms of modern condensed matter physics Frustrated_Magnetism (). In frustrated magnets, the combination of lattice geometry and competing interactions often leads to degenerate classical states. The degeneracies are generally accidental as they are not protected by the symmetries of the spin Hamiltonian. Yet, the degenerate states may be related by transformations that form an emergent symmetry group. Near a continuous phase transition, these approximate symmetries provide “organizing principles” in determining the critical properties by distinguishing relevant perturbations from irrelevant ones. In the most interesting case, the leading degeneracy-lifting perturbations, which may be relevant or irrelevant in the renormalization group sense, are thermal or quantum fluctuations – a phenomenon called order-by-disorder (ObD) Villain_ObD (); Henley_ObD (); Yildirim_ObD (); shender_ObD (). The competition among diverse degeneracy-lifting effects can result in a modulated long-range ordered state at nonzero wave vector , which may or may not be commensurate with the lattice Jensen (); Rossat (); Incomm_DyY (); Reimers (); Chatto (); Incomm_1 (); Incomm_2 (). In some cases, a number of superposed symmetry-related modes within the first Brillouin zone form a so-called multiple- order Jensen (); Rossat (); Mul-k_Nd (); Mul-k_pyro (). A particular interesting form of such modulated magnetism is a partially ordered state (POS) with periodically arranged “paramagnetic” sites POS_CsCoBr3 (); POS_CuFeO2 (); POS_UNi4B (); POS_Sr2YRuO6 (). These fluctuating magnetic moments decimate a fraction of the energy-costly frustrated bonds while retaining an extensive entropy, hence lowering the free energy.

In this Letter, we study the convergence of the aforementioned phenomena (emergent symmetry, multiple- POS and fluctuation-induced degeneracy-lifting) in an extensively studied class of frustrated magnetic materials, the insulating O pyrochlore oxides Gardner_RMP (). In these, the magnetic rare-earth ions (e.g. Gd, Tb, Er, Yb) occupy the vertices of a network of corner-sharing tetrahedra – the “pyrochlore” lattice (see Fig. 1). (=Ti, Sn, Zr, Ge) is non-magnetic. The competition between four types of nearest-neighbor anisotropic interactions and the nature of the single-ion magnetic anisotropy are largely responsible for the wealth of phenomena displayed by the O materials Gardner_RMP (). In this paper, we focus on a general description for the perplexing yet rich physics of multi- partial magnetic ordering in pyrochlore oxides and not on any material-specific issues.

A multi- POS (see Fig. 1b) is believed to exist in GdTiO stew () for temperature 0.7 K K in which the POS is a superposition of spin density waves with wave vectors stew (); Raju_Gd2Ti2O7 (); Champion_Gd2Ti2O7 (); Matt1 (); cepas1 (); cepas2 (); Wills_Gd2Sn2O7 (). While this compound has been the subject of a number of investigations Raju_Gd2Ti2O7 (); Champion_Gd2Ti2O7 (); Ramirez_Gd2Ti2O7 (); Bonville_Gd2Ti2O7 (); stew (); Petrenko_Gd2Ti2O7 (); Sosin_magnetocaloric (); Sosin_adiab (); Yaouanc_Gd2Ti2O7_DOS (); Reotier_GTOcrit (); Dunsiger_Gd2Ti2O7 (); Liao_Gd2Ti2O7 (); Petrenko_100 (), the mechanism responsible for the selection of 4- order has not been identified. Further, after fifteen years of research on the TbTiO spin liquid candidate Gardner_Tb2Ti2O7 (), evidence has recently begun accumulating that short-range magnetic correlations develop below K in the form of broad elastic neutron intensities at Fennell_Tb2Ti2O7 (); Petit_Tb2Ti2O7 (); Fritsch_Tb2Ti2O7 (); Taniguchi_Tb2Ti2O7 (); Guitteny_Tb2Ti2O7 (). No theory has explained the origin of these correlations. As TbTiO and GdTiO constitute two out of the six magnetic TiO pyrochlore compounds that exist (=Gd, Tb, Dy, Ho, Er and Yb), it may be that order is not unusual among the plethora of O materials, or even rare-earth spinels (=Cd, Mg; =S, Se and is a rare-earth) Lau_spinels (); Lago_spinel (); Wong ().

In our work, we identify an extensive range of exchange parameters able to support partial order through a standard mean-field theory (s-MFT) study. The s-MFT free energy displays, up to quartic order in the order parameters, an emergent symmetry. The transition into the POS is thus described by the Ginzburg-Landau free-energy, , of an -component vector model () cond (); Pelissetto_review (). The most relevant perturbation is a “cubic anisotropy” which breaks the symmetry. We thus identify the physical origin of cubic anisotropy as thermal fluctuations beyond s-MFT. The symmetry-breaking and selection of either 1- or 4- order in this model is an example of thermal ObD. We stress, however, that our problem conceptually departs significantly from the more common cases of ObD where “small” thermal or quantum fluctuations are typically considered. This is valid for temperatures much lower than the critical temperature where the harmonic approximation is often justified. For example, in GdTiO, where another transition occurs at K Ramirez_Gd2Ti2O7 (); Yaouanc_Gd2Ti2O7_DOS (); Reotier_GTOcrit (); Liao_Gd2Ti2O7 (); Petrenko_100 (), the low temperature fluctuations have no bearing on the state-selection at .

Given the above considerations, we develop an extension of the Thouless-Anderson-Palmer (TAP) method, E-TAP, to obtain the phase diagram of 1- and 4- orders. We use Monte Carlo simulations to confirm the E-TAP predictions that 1- and 4- POSs are selected in different portions of the phase diagram. We suggest that the phase transition at K in GdTiO belongs to the above cubic universality class. From a broader methodological perspective, our E-TAP method for spin models with anisotropic interactions could be applied to other problems in frustrated magnetism where the question of state selection at is of interest.

Model – We consider the general Hamiltonian, , for classical spins, , on the pyrochlore lattice:

(1a)
(1b)
(1c)

Here, includes all possible symmetry-allowed nearest-neighbor (n.n.) bilinear interactions: isotropic (), Dzyaloshinskii-Moriya (DM) (), Ising () and pseudo-dipolar () Thompson_Yb2Ti2O7 (); McClarty_Er2Ti2O7 (). Of exchange origin, all these couplings can be positive or negative. Unit vectors are chosen such that positive and negative correspond to direct and indirect DM interactions, respectively Elhajal2005 (). is the local cubic direction at site and . is the long-range magnetostatic dipole-dipole interaction with strength , where is the nearest-neighbor distance. is measured in units of and is the magnetic moment. Our convention for Thompson_Yb2Ti2O7 (); McClarty_Er2Ti2O7 () differs from that used by other groups Curnoe2008 (); balents () but the two are related by a linear-transformation. and are second and third n.n. exchange interactions, respectively.

We begin by studying using s-MFT Reimers (); suppl (); Enjalran_MFT (); Matt1 (). For a large region of parameter space, a degenerate line of modes with momenta first becomes critical at Raju_Gd2Ti2O7 (); Matt1 (); cepas1 (); cepas2 (). s-MFT calculations show that lifts the degeneracy by weakly selecting soft modes at the four points within the first Brillouin zone, which we label () cepas1 (); Matt1 (). Other perturbations to , such as and , can have similar effects suppl (); Matt1 (); cepas1 (); cepas2 (); Wills_Gd2Sn2O7 (). Here, we focus on since it can be of prominence in rare-earth pyrochlore oxides Gardner_RMP (). In particular, we choose the GdTiO value Raju_Gd2Ti2O7 () as an examplar. For completeness, and of possible relevance to magnetic pyrochlores with ions having a small magnetic moment, we present some results in the Supplemental Material (Sup. Mat.) suppl () that use the notation of Ref. balents () for and where the degeneracy-lifting originates from as opposed to .

The direct space spin configurations corresponding to a 1- state, with , is illustrated in Fig. 1a. Denoting the corresponding spin direction at site as , with () defined similarly suppl (), we introduce the order parameter for a particular ordering wave vector as , where the summation goes over all sites of the pyrochlore lattice and defines a 4-dimensional vector.

For a wide range of couplings, s-MFT predicts a second order transition with ordering at momenta . Interestingly, we find that up to quartic order in the order parameters, the s-MFT free-energy displays an emergent symmetry suppl (). While the symmetry is spoiled at higher order in , no selection of 1- (which has only one ) vs 4- (which has four with the same nonzero amplitude) occurs at any order suppl ().

The observation of an emergent symmetry at quartic order in s-MFT helps us to recognize that a “cubic anisotropy” in the is the most relevant symmetry-allowed perturbation. We then write the Ginzburg-Landau free energy suppl () of the system, up to quartic order in the , that includes a cubic anisotropy term ():

(2)

Eq. (2) is the celebrated -vector model () with cubic anisotropy cond () with and . This model, a cornerstone of the theory of critical phenomena cond (); Pelissetto_review (), has been extensively investigated with numerous methods Ferer1981 (); Newman1982 (); Caselle1998 (); carmona2000 (). In , for and carmona2000 (), the model undergoes a second-order transition into a phase where all have the same amplitude if . The universality class is controlled by the cubic fixed point cond () with distinct critical exponents from those of the isotropic fixed point Pelissetto_review (); carmona2000 (). For , the phase transition is a fluctuation-induced first-order transition to a state with only one nonzero . Therefore, we conclude that for the magnetic order is defined by a superposition of four spin states (4- state) while for the order is defined by a single spin structure (1- state). Both states are POSs since the spins on of the sites remain disordered (see Fig. 1). As stated above, s-MFT predicts that 1- and 4- have the same (i.e. ). We thus identify thermal fluctuations as the mechanism for generating leading to a selection of 1- vs 4-. To expose how fluctuations may lead at the microscopic level to a selection directly at , we devise and then use an E-TAP method TAPmain (); Yadidah (); Plefka ().

Extended TAP Method – A magnetic moment at a particular lattice site experiences a local field due to its neighbors. At the s-MFT level, the presence of the spin at the site of interest affects its local field indirectly. This is an artifact of s-MFT. The Onsager reaction field (ORF) introduces a term that cancels this unphysical effect. The TAP approach provides a systematic way to implement the ORF correction naiveTAP (). In this work, we devise an extension of the TAP method, E-TAP, where the ORF is the first term of a series originating from nonzero on-site fluctuations, e.g. suppl (); Plefka (); Yadidah ().

To compute the E-TAP corrections, we consider a perturbative expansion of the Gibbs free-energy, , in inverse temperature suppl (); Yadidah ().

(3)

Here is the local magnetization and the vector is a Lagrange multiplier; different from the local mean field at site by a factor of suppl (); Yadidah (). Defining , the first and second terms in the expansion, and , where the prime represents differentiation with respect to , are the s-MFT entropy and energy, respectively. The third term, , is the first correction beyond s-MFT, arising from fluctuations suppl (); Yadidah ():

(4)

Here, is the on-site susceptibility. Since is quadratic in suppl (), in Eq. (54) is therefore of quartic order in . The first order E-TAP correction, , can thus, in principle, generate a finite cubic anisotropy term in and select 1- or 4- depending on the bilinear spin-spin interaction matrix defined through in Eq. (1).

We calculate suppl () for the region in parameter space with ordering (the dark-shaded 1-/4- wedge in Fig. 2). Specifically, we compute . indicates a 1- selection and conversely for , with the contribution splitting the wedge into 1- and 4- sectors. In particular, for a dipolar Heisenberg model with , relevant to GdTiO Raju_Gd2Ti2O7 (), E-TAP calculations predict a 4- state selection at , as observed in this compound stew ().

Figure 2: (Color online) Ordering wave vectors at obtained from s-MFT. The combined area denoted 1- and 4- displays order but 1- and 4- are degenerate at the s-MFT level. The first order E-TAP correction Eq. (54) applied in this regime selects either 1- or 4-. The region encompass all states for which all sites are fully ordered and each primitive 4-site tetrahedron basis has the same local spin order. The dashed (0.15) and dotted (-0.15) contours mark the boundaries for the corresponding values.

Monte Carlo Simulations – We performed parallel tempering classical Monte Carlo simulations to check the E-TAP predictions. We pick, somewhat arbitrarily, two sets of interaction parameters corresponding to the 1- and 4- regions in Fig. 2. One set is the simplest model for a spin-only (, , ) state for Gd in GdTiO, with Raju_Gd2Ti2O7 (), . We include, as in a previous Monte Carlo work cepas2 (), weak ferromagnetic second n.n. interaction (e.g. ) to stabilize a temperature range wide enough to numerically resolve order suppl (); cepas2 (). We measure the magnitude of :

(5)

where denotes a thermal average.

In addition to its magnitude, we identify the orientation of at through two additional order parameters, () defined below. On the surface of a four-dimensional unit hyper-sphere, there are eight “1- points” corresponding to 1- states including , , etc. Similarly, there are sixteen “ 4- points” on the hyper-sphere suppl (). Given a certain spin configuration, we calculate . The () are then defined as the minimum Euclidean distance between point and all of the - points. The thermal average of () is expected to decrease if the system enters a 1- (4-) state for .

The results for two sets of interaction parameters in the 1- and 4- regions of Fig. 2 are shown in the left and right columns of Fig. 3, respectively. The growth of at and shows that the system orders with . In the left column, the 1- state is selected at , as indicated by a minimum for and a maximum for . The system orders in a 4- state in the right column. The separation of and for both cases in Fig. 3 accentuates as the linear dimension of the system increases, indicating that the selection of either 1- or 4- survives in the thermodynamical limit. These results are consistent with the predictions from the E-TAP calculations at . Unfortunately, the large computational resources required for simulations with long-range dipolar interactions prevent us from investigating the order of the phase transitions in Fig. 3.

The kinks in and merging of and indicate the system enters into a distinct phase at and in the left and right columns of Fig. 3. Since in a 2- state suppl (), the results for suggest that the low-temperature region may be a 2- state. We note that the latest single-crystal neutron diffraction results Stewart_comm () indicate that the low-temperature state ( K) of GdTiO may not be the previously suggested 4- structure stew (). The results of a more in-depth numerical investigation of the low-temperature regime will be reported elsewhere.

Figure 3: (Color online) Monte Carlo simulations results. Top row: growth of . Bottom row: value of as a function of . Left column: , , , , (direct DM) and (a point in 1- region). Right column: , and (a point in 4- region). Note: The are nonzero in the paramagnetic phase because their value is equal to the average distance of a random point on a 4-dimensional hypersphere from the 1- and 4- points.

Discussion – Considering a general symmetry-allowed anisotropic Hamiltonian, we found that partial order can occur over a wide range of anisotropic exchange and long-range dipolar interactions in pyrochlore magnets. We argued that fluctuations beyond s-MFT are responsible for the stabilization of a 1- or 4- partially ordered structure. This conclusion is based on results from E-TAP calculations where on-site fluctuations are included. We used Monte Carlo simulations to illustrate that different values of the magnetic exchange interactions can, as anticipated on the basis of the E-TAP calculations, lead to either 1- or 4- order.

From our work, we have exposed a likely mechanism for the establishment of 4- order in GdTiO below its paramagnetic transition stew (). Further quantitative progress on this problem will require a better estimate of the material exchange parameters Thompson_Yb2Ti2O7 (); balents (); Savary_Er2Ti2O7 (). From this work we conclude that the transition from the paramagnetic state to the 4- phase should be second order and belong to the cubic universality class. Experimental evidence Reotier_GTOcrit () suggests that this transition is second order in GdTiO. Determining the critical exponents for this system could confirm our prediction but will be a challenge, given that the exponents for the cubic and Ising, XY and Heisenberg universality classes are proximate to one another Pelissetto_review (); carmona2000 ().

Finally, the E-TAP method for frustrated magnets formulated here could prove useful for other systems where state selection at is an open question. This is particularly so if the low-temperature state selected via (thermal or quantum) ObD is separated by a phase transition from the state selected at . In such a case, an understanding of ObD at can not be leveraged to explain the selection at . Examples include the transition to long-range order in the pyrochlore Heisenberg antiferromagnet with indirect DM interactions Elhajal2005 (); Canals_DM (); Chern_DM (), the problem of magnetization direction selection in face-centered cubic dipolar ferromagnets Roser_Corruccini (); Bouchaud () and the topical issue of state selection in pyrochlore antiferromagnets Wong (); McClarty_Er2Ti2O7 (); Savary_Er2Ti2O7 (); Champion_Er2Ti2O7 (); Wills_Er2Ti2O7 (); Stasiak_Er2Ti2O7 (); Zhitomirsky_Er2Ti2O7 (); Oitmaa_Er2Ti2O7 ().

Acknowledgements.
We thank Steve Bramwell, Alexandre Day, Jason Gardner, Paul McClarty, Oleg Petrenko, Rajiv Singh, Ross Stewart, Ettore Vicari and Andrew Wills for useful discussions. We acknowledge Pawel Stasiak for his help with the Monte Carlo simulations and Peter Holdsworth for his comments on the manuscript. This work is supported by the NSERC of Canada, the Canada Research Chair program (M.G., Tier 1) and by the Perimeter Institute for Theoretical Physics. Research at PI is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.

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Supplemental Material

In this Supplementary Material we provide the reader with some of the technical details to assist with the reading of the main body of the paper. In Section I, we derive the Ginzburg-Landau theory presented in the main text. In Section II, we show that the two types of partially ordered states, 1- and 4- states, have the same free energy in standard mean-field theory (s-MFT) with and without the presence of crystal electric field. Then, in Section III, we show that the Ginzburg-Landau theory has an emergent symmetry at the s-MFT level. In Section IV, we present the details of the extended TAP method (E-TAP) used in deriving the on-site fluctuation corrections needed to go beyond s-MFT. In Section V, we go over the technical details of our Monte Carlo simulations. Finally, in Section VI, we consider the problem of degeneracy-lifting when the selection of is induced by second nearest-neighbor exchange, , as opposed to long-range dipolar interaction (. For completeness sake, we consider a formulation of the spin Hamiltonian with anisotropic exchange couplings as expressed in Ref. [balents, ; columbic, ; ETO, ; sungbin, ].

Appendix A Derivation of the Ginzburg-Landau free energy

To proceed, we first recall the definition of the order parameters of the theory. With the spin direction at site given by (Table 1), we introduce the order parameter for a particular ordering wavevector as:

(6)

The Ginzburg-Landau free energy is constructed in terms of the four order parameters (). Each 1- order is a spin-density wave and is the amplitude of the wave. The four order parameters form the components of a four-dimensional vector which fully describes the long-range order at the four momenta.

To construct the free energy, we study how transforms under the space group symmetry of the pyrochlore lattice. Based on the real space spin configurations illustrated in Fig. 1 of the main text, we obtain how transforms. Under three fold rotations about the local direction of, say, the sublattice (see Eq. (10)), three of ’s are permuted: , and . Under a primitive FCC lattice translation, for example, two ’s, in this case and , reverse sign. All ’s reverse sign under time-reversal transformation.

Lastly, the point inversion about a site on sublattice reverses the sign of while leaving other three order parameters intact. Since the symmetry operation is responsible for the elimination of an additional term in free energy invoked in Ref. [cepas2, ], we demonstrate its consequence in detail here. Considering the 1- structure with momentum , the point inversion about a site on sublattice exchanges the spin configurations on adjacent kagome layers, which have opposite directions. As a result, . On the other hand, the point inversion about a site on any other sublattice than leaves the 1- structure of intact. In other words, such a transformation leaves unchanged. Similar arguments can be made for with .

We now construct the Ginzburg-Landau free energy to quartic order. At the quadratic order, the only invariant term is . At quartic order, there are two terms that are invariant under the above symmetry transformations: and . We note that the quartic term of the form invoked in Ref. [cepas2, ] is odd under the aforementioned point inversion symmetry and this term should not be present in the free energy. Given these considerations, we write down the Ginzburg-Landau free energy:

(7)

In this work, we make use of the following quantities:

(8a)
(8b)
(8c)

Here, is the 4-component vector order paramete and is its magnitude. is a 4-component unit vector parallel to which is convenient to parametrize the behavior of the system below the critical temperature, , in the Monte Carlo simulations discussed in Section V below.

Appendix B Free-energy degeneracy of 1- and 4- At the s-MFT Level

In this section we show that, at the s-MFT level, the 1- and 4- states have the same free energy with and without considering the effect of crystal electric field in pyrochlore lattice.

b.1 No Crystal Electric Field

We begin by defining the orientation of the moments, , on a typical tetrahedron that makes up a 1- state. Here, corresponds to one of the four bases (sublattices) of the pyrochlore structure (see Eq. (10)) and refers to ordering wave vector of the corresponding 1- state. ’s are given in Table 1 and are expressed in global Cartesian coordinates.

sublattice 0 sublattice 1 sublattice 2 sublattice 3
Table 1: Spin configurations of a single tetrahedron in 1- states

The orientation of the rest of the moments in a 1- state can be obtained from a single tetrahedron configuration using the following equation:

(9)

where are the FCC lattice position vectors, are pyrochlore lattice position vector. The vector specifies the bases’ positions in pyrochlore lattice and are given by:

(10a)
(10b)
(10c)
(10d)

The above coordinates are expressed in units of , the nearest-neighbor distance in pyrochlore structure. The spin orientations corresponding to the 4- states can be expressed in terms of linear combinations of the four 1- states:

(11)

Here conveniently keeps normalized (except for the sites with zero moments) and generate all possible 4- states.

Having defined the 1- and 4- states, we now proceed to show that both the 1- and 4- states have the same s-MFT free energy. The general form of the s-MFT free energy for a classical 3-component spin, at site readsMatt1 ():

is the bilinear spin-spin exchange coupling matrix defined through the Hamiltonian of Eq. (1) in the main text. The magnetic moment, for both 1- and 4- states at site () is obtained from the s-MFT self-consistent (Langevin function) equation,

(13)

where

(14)

In what follows, we compare the free energy, , within standard mean-field theory (s-MFT), of a 1- state, with 4-component order parameter, , with the free energy of a 4- state, with 4-component order parameter . We thus have which is also confirmed numerically by solving the Eq. (13). We define for a 1- state:

(15)

while we have for a 4- state:

(16)

In Eq. (B.1), we are employing implicit summation convention for repeated Greek superscripts and which represent Cartesian coordinates.

We now show that both terms in Eq. (B.1) are the same for 1- and 4- states and thus, at a given temperature the free-energy of 1- and 4- is the same at the s-MFT level.

First Term of Eq. (B.1): The first term, , reads for a typical 1- state:

(17)

Using Eqs. (9, 15) we have:

(18)

where and are sublattice labels and is the Fourier transform of . Similarly, using Eqs. (16, 17), for a 4- state can be written as:

(19)

Using Eq. (9) we obtain:

(20)

Consequently,

(21)

and using Eq. (18), we find:

(22)

Considering that for all , , and all are the same, we thus have:

(23)

Second Term of Eq. (B.1): In this term, the only variable that depends on different spin configurations is the magnitude of local field, , in Eq. (14),which we focus on. Using Eq. (15), the local field, , experienced by a moment at site in a 1- state, is given by:

(24)

while for a 4- state we have

(25)

where . Within s-MFT, the local field at each site is antiparallel to the moment at that site. As a result, can be written as

(26)

Here, is the same constant for each of the 1- states with (), since all of 1- states have the same free energy by definition.

Similarly, for the 4- state, using Eqs. (25, 26), we find

(27)

Since we have the same for both 1- and 4- states by construct, Eq. (27) shows that the magnitude of the local field is the same for 1- and 4- at the lattice sites with nonzero moments at a given temperature . We note that, the theory is internally consistent, since we have taken to be the same for both 1- and 4- states which is confirmed numerically as mentioned earlier. As a result, the second term in Eq. (B.1) is the same for 1- and 4-.

Based on Eqs. (23, 27) and Eq. (B.1), we therefore find that the s-MFT free energy is the same for the 1- and 4- states.

b.2 With Crystal Electric Field

The discussion in the previous subsection, as well as in the main text, has assumed for simplicity that the spin degrees of freedom, at each site , is a classical vector of fixed length. As a result, the s-MFT treatment leads for the self-consistent equation for to the Langevin function in Eq. (13). Had we assumed a quantum , Eq. (13) would be replaced , where is the Brillouin function. In this case, one can repeat the argument of the previous subsection and, again, show that the 1- and 4- states are degenerate.

One may then ask whether the single-ion anisotropy, arising from the crystal electrical field (CEF) effect, may change the conclusion that the 1- and 4- states are degenerate. Again, such a question can be asked for either classical or quantum spins . We now proceed to briefly show that introducing a crystal field anisotropy does not lift the degeneracy between the 1- and 4- states.

At the s-MFT level, the Hamiltonian including crystal electric field or single ion anisotropy () can be written as:

(28)

where can be expressed in terms of Stevens’ operators according to the symmetry of pyrochlore structureGardner_RMP ():

(29)

We consider expressed in terms of its components in the local [111] coordinate system.

The s-MFT free energy can be written as

(30)

where

(31)

In what follows, we aim to compare the s-MFT free energy of 1- and 4- states using Eq. (30) without explicitly finding its global minimum. In Eq. (31), can be treated as a classical vector or a quantum mechanical operator. However, we show that the outcome is independent of this choice. The result of performing the in Eq. (31) can be written in general form of

(32)

Since we are comparing the free energy of the 1- and 4- states, and are known for every