Abstract
Broken symmetries in solids involving higher order multipolar degrees of freedom are historically referred to as “hidden orders” due to the formidable task of detecting them with conventional probes. In this work, we theoretically propose that magnetostriction provides a powerful and novel tool to directly detect higherorder multipolar symmetry breaking—such as the elusive octupolar order—by examining scaling behaviour of length change with respect to an applied magnetic field h. Employing a symmetrybased Landau theory, we focus on the family of Prbased cage compounds with strongly correlated felectrons, Pr(Ti,V,Ir)_{2}(Al,Zn)_{20}, whose low energy degrees of freedom are purely higherorder multipoles: quadrupoles \({\cal{O}}_{20,22}\) and octupole \({\cal{T}}_{xyz}\). We demonstrate that a magnetic field along the [111] direction induces a distinct linearinh length change below the octupolar ordering temperature. The resulting “magnetostriction coefficient” is directly proportional to the octupolar order parameter, thus providing clear access to such subtle order parameters.
Introduction
In crystalline solids, the combination of spin–orbit coupling and crystal electric fields places strong constraints on the shape of localized electronic wavefunctions^{1}. The quantum mechanically defined multipole moments provide a useful measure of the resulting complex angular distribution of the magnetization and charge densities^{2,3}. Most conventional broken symmetry phases in solids involve the magnetic dipole moment of the electron. Remarkably, a large class of strongly correlated electron materials display nontrivial higher order multipolar moments, e.g., quadrupolar or octupolar moments, whose fluctuations and ordering leads to a rich variety of phases, such as quadrupolar heavy Fermi liquids^{4,5,6}, superconductivity^{7,8,9}, and unusual multipolar symmetrybroken phases^{2,3,10,11,12,13}. While multipolar ordered phases fall under the purview of the celebrated Landau paradigm of symmetrybroken phases, they have been termed as socalled ‘hidden orders’: mysterious phases of matter whose orderings are invisible to conventional local probes (such as neutron scattering or magnetic resonance), but are remarkably still known to exist as their onset triggers nonanalytic signatures in thermodynamic measurements^{4,14,15,16,17}. Studying the mysterious ordering patterns of higher order multipoles is also often rendered challenging since they typically coexist with conventional dipolar moments. Examples of such symmetry breaking which are of great interest include spinnematic order^{18} in spin S ≥ 1 quantum magnets, quadrupolar charge order in transition metal oxides, and higher multipolar order in actinide dioxides, such as NpO_{2}^{19}, and felectron heavy fermion materials^{20}, such as URu_{2}Si_{2}^{21,22,23,24,25,26,27,28,29} and UBe_{13}^{30,31,32}. The quest to probe such orders has led to novel experimental techniques, e.g., elastoresistivity^{33,34,35} to elucidate the quadrupolar order associated with orbital nematicity in the iron pnictides. A broad understanding of the nature of these symmetrybroken phases, and means to definitively demonstrate their existence, has proven to be a challenging, yet stimulating, endeavour for both theory and experiment.
Our work is motivated by a recent series of experiments on the Prbased cage compounds Pr(Ti,V,Ir)_{2}(Al,Zn)_{20} which form an ideal setting to study multipolar moments and associated hidden orders^{8,17,36,37,38}. In these systems, the 4f^{2} electrons of Pr^{3+} ions subject to CEFs host a ground nonKramers doublet with solely higherorder moments: quadrupoles (\({\cal{O}}_{20}\) and \({\cal{O}}_{22}\)) and octupole (\({\cal{T}}_{xyz}\))^{14,39}. Uncovering and understanding the pattern of multipolar ordering across this family of materials remains an important open problem.
The nature of the quadrupolar ordering in these cage compounds has been indirectly examined with a few techniques^{40,41} such as ultrasound experiments^{42,43,44,45} (indicating softening of elastic modulus at quadrupolar ordering temperature, \(T_{\cal{Q}}\)), as well as NMR measurements (where the magnetic fieldinduced dipole moment is strongly dependent on the underlying quadrupolar phase^{46}). More recently, magnetostriction and thermal expansion strain experiments^{47} have also lent themselves as possible probes to study the transitions and the underlying quadrupolar phase. By contrast, the octupolar ordered state has continued to remain an elusive phase of matter, with only indirect hints of its existence from NMR^{48} and μSR^{49} measurements, but as yet no direct probe to reveal its existence^{50}. More recently, some of us (A.S. and S.N., unpublished) have begun experiments to study angledependent magnetostriction, the change in sample length induced by a magnetic field which can point along various crystalline directions, in a wide class of materials with multipolar degrees of freedom.
In this work, motivated by these experiments, we theoretically discuss how magnetostriction provides a novel means to directly probe multipolar order parameters. The central observation of this paper is that an applied magnetic field allows for a linear coupling between lattice strain fields and a uniform octupole moment which depends strongly on the applied field direction. In the absence of a dipolar moment, this enables measurements of the magnetostriction to directly reveal the hidden octupolar order parameter. We investigate such fieldscaling behaviour of the magnetostriction for various magnetic field directions by employing a symmetrybased Landau theory, which allows us to highlight the universal aspects of the physics and to reveal its applicability to a broader class of materials. Specifically, our Landau theory permits both antiferroquadrupolar ordering (AF\({\cal{Q}}\)) and ferrooctupolar ordering (F\({\cal{O}}\)), and we examine our theory along different field directions in three temperature regimes. Denoting the quadrupolar and octupolar transition temperatures as \(T_{\cal{Q}}\) and \(T_{\cal{O}}\), respectively, we consider the regimes (i) the paramagnetic phase above both transition temperatures (\(T \,> \, T_{\cal{Q}},T_{\cal{O}}\)), (ii) intermediate temperatures (\(T_{\cal{O}} \,< \,T \,< \,T_{\cal{Q}}\)) where the system exhibits pure quadrupolar order, and (iii) below both ordering temperatures (\(T \,< \,T_{\cal{Q}},T_{\cal{O}}\)) where the system features coexisting quadrupolar and octupolar orders. We make definite predictions for all possible combinations of length change and magnetic field directions, which can be tested in future experiments.
Results
Magnetostriction as a probe of multipolar ordering
Our studies predict a linearinh scaling behaviour for particular length changes, for \(T \,<\, T_{\cal{O}}\). The coefficient of the linearinh term, i.e. the “magnetostriction coefficient”, is directly proportional to the ordered ferrooctupolar moment, thus providing a clear and distinct means to directly probe this order parameter. This linearinh behaviour appears for magnetic fields applied along the [111] and [100] directions. For other magnetic field (and length change) directions, we predict the signature of quadrupolar ordering as a constant plus quadraticinh scaling behaviour in the length change; although the scaling behaviour explicitly involves the F\({\cal{Q}}\) order, the AF\({\cal{Q}}\) order parameter can be inferred from the F\({\cal{Q}}\), as it scales (to leading order) as the square root of the F\({\cal{Q}}\) order parameter. We present our theoretical predictions for the scaling behaviours in Table 1 for a variety of magnetic field and length change directions. A quick way to see this linearinh result is to note that the elastic energy of a strained cubic crystal is given by^{51,52}
where the crystal’s deformation is described by the components of the strain tensor \(\epsilon _{ik}\), and c_{ij} is the elastic modulus tensor describing the stiffness of the crystal. Note that we use the normal modes of the cubic lattice to write the elastic energy in this elegant form. Here c_{B} is the bulk modulus, \(\epsilon _{\mathrm{B}} \equiv \epsilon _{xx} + \epsilon _{yy} + \epsilon _{zz}\) is the volume expansion of the crystal, \(\epsilon _\nu \equiv (2\epsilon _{zz}  \epsilon _{xx}  \epsilon _{yy})/\sqrt 3\) and \(\epsilon _\mu \equiv (\epsilon _{xx}  \epsilon _{yy})\) are lattice strains that transform as the Γ_{3g} irreducible representation (irrep.) of the O_{h} group, and the offdiagonal strain components transform as the Γ_{5g} irrep. of O_{h} group. The subscript g indicates even under timereversal and spatial inversion (parity). Knowing \(\epsilon _{ij}\) determines the fractional length change along the \(\hat {\bf{\ell }}\)axis via \((\Delta L/L)_{ {\bf{\ell }}} = \mathop {\sum}\limits_{ij} {\epsilon _{ij}} \hat \ell _i\hat \ell _j\), where \(\hat \ell _i\) is the i^{th} component of unit vector \(\hat {\bf{\ell }}\); Supplementary Note 1 provides a more detailed discussion on this expression. As discussed below, an applied magnetic field enables a linear coupling between the strain field and the timereversalodd ferrooctupolar moment, m, via a term in the free energy \(\Delta F =  g_{\cal{O}}m(\epsilon _{yz}h_x + \epsilon _{xz}h_y + \epsilon _{xy}h_z)\), with a coupling constant \(g_{\cal{O}}\). Minimizing F_{lattice} + ΔF with respect to the strain, we find \(\epsilon _{xy} \propto (g_{\cal{O}}/c_{44})mh_z\), and cyclically for \(\epsilon _{yz},\epsilon _{xz}\), while diagonal components of the strain tensor vanish. As a representative example, take a [111] field, where \(h_i = h/\sqrt 3\), this leads to \((\Delta L/L)_{(1,1,1)} = (\epsilon _{xy} + \epsilon _{yz} + \epsilon _{xz})/3\) and so (ΔL/L)_{(1,1,1)} ∝ (\(g_{\cal{O}}\)/c_{44})mh. This direct relation between the linearinh magnetostriction coefficient and the ferrooctupolar order parameter for a magnetic field along the [111] direction is one of the central results of our paper. Furthermore, we predict a characteristic hysteresis in the octupolar moment and the associated parallel length change as a function of magnetic field, arising from the symmetryallowed cubicinh coupling of the magnetic field to the octupolar moment. Very recent (unpublished) experiments on PrV_{2}Al_{20} indeed appear to find a hysteretic linearinfield magnetostriction, for a [111] magnetic field, below a transition at T* ≈ 0.65 K. Our theoretical results for magnetostriction in the presence of octupolar order thus lend strong support to the idea that this approach, pursued in recent experiments performed by some of us (A.S. and S.N., unpublished), herald the unambiguous discovery of octupolar order.
The theoretical roadmap which gives rise to this striking result requires (i) the Landau free energy of the multipolar moments, and (ii) coupling between the multipolar moments and lattice strain. We present these key ingredients below.
Landau theory of multipolar order
We present in this section, for the sake of selfcontainedness and to specify our notation, the Landau theory of multipolar order first introduced in ref. ^{51}. We focus on key aspects of the model here, and relegate the symmetrybased derivation as well as the complete form of the free energy to Methods; the symmetry transformations of the multipolar moments are given in Supplementary Note 2.
The 4f^{2} electrons of Pr^{3+} ions in the family of rareearth metallic compounds Pr(Ti,V,Ir)_{2}(Al,Zn)_{20} reside on a diamond lattice of cubic space group Fd\(\bar 3\)m. Surrounding each Pr^{3+} ion is a FrankKasper (FK) cage (16 Al atom polyhedra). The crystalline electric field (CEF) of this FK cage, with T_{d} point group symmetry, splits the J = 4 multiplet of the 4f^{2} electrons. The ground states are experimentally found to be a nonKramers doublet, and they transform as the basis states of the Γ_{3g} irrep. of T_{d}; here the subscript g(erade) and u(ngerade) denote even and odd under timereversal, respectively. Moreover, this doublet is energetically well separated from the excited states, and so for energies much lower than this gap (≳50 K^{4}), the Γ_{3g} doublets form an ideal basis to describe the low energy degrees of freedom. The Γ_{3g} doublets can give rise to timereversal even quadrupolar moments \({\cal{O}}_{22} = \frac{{\sqrt 3 }}{2}(J_x^2  J_y^2)\) and \({\cal{O}}_{20} = \frac{1}{2}(2J_z^2  J_x^2  J_y^2)\) which transform as Γ_{3g}, as well as a timereversal odd octupolar moment \({\cal{T}}_{xyz} = \frac{{\sqrt {15} }}{6}\overline {J_xJ_yJ_z}\) which transforms as Γ_{2u} (where the overline represents the fully symmetrized product). Using the constructed pseudospin basis ({↑〉, ↓〉}) from the Γ_{3g} doublets, allows the multipolar moments to be neatly denoted by an effective pseudospin1/2 operator τ = (τ^{x}, τ^{y}, τ^{z})
The perpendicular component of the pseudospin vector τ^{⊥} ≡ (τ^{x}, τ^{y}) denotes the quadrupole moments, while τ^{z} denotes the octupolar moment. We also define the raising/lowering pseudospin operators τ^{±} = τ^{x} ± iτ^{y}.
The ordering of these multipolar degrees of freedom acts as a mean field on the pseudospins, and breaks the degeneracy of the nonKramers doublet. In order to describe these pseudospinsymmetrybroken phases, we resort to a Landau theory approach, focussing on the following order parameters,
Here, angular brackets 〈...〉 denote thermal averages, while the A, B subscripts denote the two sublattices of the diamond lattice. The complex scalars ϕ and \(\tilde \phi\) describe ferroquadrupolar (F\({\cal{Q}}\)) and antiferroquadrupolar (AF\({\cal{Q}}\)) orders, respectively, while the real scalars m and \(\tilde m\) denote the ferrooctupolar (F\({\cal{O}}\)) and antiferrooctupolar (AF\({\cal{O}}\)) order parameters. We henceforth use the convention of \(\tilde \phi = \tilde \phi {\mathrm{e}}^{{\mathrm{i}}\tilde \alpha }\) and ϕ = ϕe^{iα} for the complex order parameters.
In this work, we focus on a system where the primary order parameters are AF\({\cal{Q}}\) and F\({\cal{O}}\). As discussed in previous works^{53,54}, the Landau theory of a system with AF\({\cal{Q}}\) order necessarily admits a ‘parasitic’ secondary order parameter F\({\cal{Q}}\). Such mixing does not occur for the octupolar order parameter; motivated by explaining the experiments performed by some of us (A.S. and S.N., unpublished) on PrV_{2}Al_{20}, we choose to work with only F\({\cal{O}}\) order and ignore the AF\({\cal{O}}\) order parameter. We can thus construct our Landau theory using the order parameters ϕ, \(\tilde \phi\), and m, based on the local T_{d} symmetry instilled by the FK cage, \(F_{{\cal{Q}},{\cal{O}}}[\phi ,\tilde \phi ,m] = F_{\tilde \phi } + F_m + F_\phi + F_{\tilde \phi ,\phi ,m}\). Here, the free energies \(F_{\tilde \phi }\), F_{m}, and F_{ϕ} denote the independent free energies of the AF\({\cal{Q}}\), F\({\cal{O}}\), and F\({\cal{Q}}\) orders, and \(F_{\tilde \phi ,\phi ,m}\) describes the interactions between the different multipolar order parameters. Figure 1 shows the zero magnetic field phase diagram, depicting both quadrupolar and octupolar transitions; with two primary order parameters AF\({\cal{Q}}\) (and its accompanying parasitic F\({\cal{Q}}\) moment) and F\({\cal{O}}\) ordering at critical temperatures of \(T_{\cal{Q}}\) and \(T_{\cal{O}}\), respectively. The ‘kink’ in the AF\({\cal{Q}}\) (as well as F\({\cal{Q}}\)) at the octupolar ordering temperature reflects the nonanalytic behaviour of the octupolar moment at its critical temperature. We present in Supplementary Note 3 the values of the Landau parameters used for Fig. 1.
In order to study magnetostriction, it is important to understand how the magnetic field couples to the multipole moments. Due to the lack of magnetic dipole moment supported by the Γ_{3g} doublet, the magnetic field does not couple linearly to the states. One can derive the low energy magnetic field Hamiltonian by performing secondorder perturbation theory in h · J, where the low energy subspace is spanned by the Γ_{3g} doublet, and the high energy subspace is spanned by the excited triplets Γ_{4,5}. This leads to \(F_{{\mathrm{mag}}}[\phi ,\tilde \phi ]\), which involves the quadrupolar moments coupling quadratically to the magnetic field, ~h^{2}τ^{x,y}. The coupling of the magnetic field to the octupole moment (after performing thirdorder perturbation theory) is of the form ~h_{x}h_{y}h_{z}τ^{z}. The \({\cal{O}}(h^3)\) term is neglected at this stage, and its role is revived in the discussion of hysteresis.
Symmetry allowed coupling of multipoles to lattice modes
We now turn our attention to the problem of coupling the lattice normal modes of the cubic crystal to the multipolar moments. We recall that the cubic crystal structure supports macroscopic normal modes that transform as irreps. of O_{h}, while the Landau free energy of the multipolar moments (F) is constructed subject to symmetries of the local T_{d} environment. The symmetry constraints on F ensure that in principle only select normal modes of the crystal that transform as the irreps. of T_{d} are permitted to couple to the multipolar moments. In the present case, all the cubic normal modes presented in Eq. (1) also transform as irreps. under T_{d} (as can be explicitly verified), and so all of the aforementioned strain modes can participate in the coupling.
Coupling of quadrupolar moment to lattice strain
Coupling between the quadrupolar moments and the lattice normal modes appears as a natural choice, as the quadrupolar moments and the lattice strains are both even under timereversal. Moreover, both the normal modes \(\{ \epsilon _\mu ,\epsilon _\nu \}\) and the quadrupolar moments \(\{ {\cal{O}}_{22},{\cal{O}}_{20}\}\) transform as Γ_{3g} irreps. of T_{d} (the aforementioned lattice normal modes also transform as Γ_{3g} in O_{h}, as T_{d} is a subgroup of O_{h}). This similarity in how they transform under T_{d} allows a linear coupling between the aforesaid lattice normal modes and quadrupolar moments. Thus, the Landau free energy of the multipolar moments gets augmented by the following lattice elastic energy and coupling terms to quadrupolar moments,
where \(g_{\cal{Q}}\) is the coefficient of coupling between the quadrupolar moments and lattice strain tensors. Note that we include the coupling of the lattice strain to the quadrupole moment on each sublattice. Using the definition of the order parameter ϕ from Eq. (3), and minimizing \(F_{{\mathrm{strain}},{\cal{Q}}}\) with respect to \(\epsilon _\mu ,\epsilon _\nu\) yields the total strain for each normal mode
Substituting these expressions back into Eq. (4), we find that the strainoptimized \(F_{{\mathrm{strain}},{\cal{Q}}}[\phi ] =  \frac{{g_{\cal{Q}}^2}}{{2(c_{11}  c_{12})}}\phi ^2\) renormalizes the mass term of the F\({\cal{Q}}\) order.
Coupling of octupolar moment to lattice strain
A direct linear coupling between the octupolar moment \({\cal{T}}_{xyz}\) and the lattice normal modes is not permitted, as the octupolar moment is odd under timereversal. However, this potential difficulty can be alleviated by the introduction of the timereversal odd magnetic field h which assists in the coupling between the lattice degrees of freedom and octupolar moment. Thus, the Landau free energy of the multipolar moments gets augmented by the following lattice elastic energy and the coupling terms to the octupolar moments,
where we use the definition of m from Eq. (3), and \(g_{\cal{O}}\) is the coefficient of coupling between the octupolar moment and lattice strain. We also include another symmetryallowed direct coupling between the magnetic field and the same lattice normal modes (with proportionality constant γ_{c}, equivalent on both sublattices). Physically, this kind of term could arise from the independent coupling of the magnetic field and lattice strain to the conduction electrons (and after integrating out the conduction electrons). We discuss in Supplementary Note 5 how the numerical values of these coupling constants can be obtained from experimental observations in conjunction with our theoretical predictions.
Minimizing with respect to the lattice degrees of freedom yields the following expressions for the (total) lattice strains
Substituting the expression for the minimized lattice strains from Eq. (7) into Eq. (6) yields \(F_{{\mathrm{strain}},{\cal{O}}}[m]\), where the mass term of the octupolar moment is renormalized by a term quadratic in h; it also introduces an \({\cal{O}}(h^3)\) coupling term between the octupolar moment and the magnetic field, which renormalizes the coefficient of the already present h_{x}h_{y}h_{z}m from thirdorder in perturbation theory in h ⋅ J.
Length change under magnetic field along various directions
Equipped with the necessary ingredients in the previous subsections, we can now examine the relative length change, ΔL/L, for magnetic fields applied along [100], [110], [111] directions and examine the scaling in magnetic field strength, h. For the sake of clarity, we stress that we consider here the complete Landau theory of multipolar moments coupled to lattice strain fields (after having integrated out the lattice degrees of freedom): \(F[\phi ,\tilde \phi ,m] = F_{{\cal{Q}},{\cal{O}}}[\phi ,\tilde \phi ,m] + F_{{\mathrm{mag}}}[\phi ,\tilde \phi ] + F_{{\mathrm{strain}},{\cal{Q}}}[\phi ] + F_{{\mathrm{strain}}, {\cal{O}}}[m]\). The scaling relations can be inferred by substituting the expressions for the (extremized) strain in Eqs. (5) and (7) into \((\Delta L/L)_{ {\bf{\ell }}} = \mathop {\sum}\limits_{ij} {\epsilon _{ij}} \hat \ell _i\hat \ell _j\). We stress that from Eq. (7), the offdiagonal strain components involve the octupolar moment; thus to have any possibility of observing m, it requires length change expressions that are not along purely the crystal axes [100], [010], [001]. We summarize the key results in Table 1. Taking the example of length changes along the (1, ±1, 1) direction we have
This equation has a striking conclusion as it pertains to observing hidden order. The mysterious octupolar moment can now be determined (up to a proportionality constant) by measuring the slope of the linearinh behaviour of the length change both parallel and perpendicular to magnetic fields applied along the [111] direction. This provides a clear signature for the onset of the octupolar ordering as well as a means to study the general behaviour of the octupolar moment (up to a proportionality constant) with respect to other external variables such as temperature, T. Moreover, we discover that length change parallel to the magnetic field along [111] has (negative) twice the slope of the linearinh term and (negative) twice the quadratic background as the length changes perpendicular \({\bf{\ell }} = (1,  1,0),(1,1,  2)\) to the field [111]. Furthermore, from Table 1, the octupolar moment analogously appears in the length change perpendicular to the magnetic field along the [100] direction. Indeed, the sign of linearinh coefficient flips for the two presented orthogonal directions. All of these provide distinct means to validate the theory.
Next, for magnetic fields along the [110] direction, we find that the length changes parallel, \({\bf{\ell }} = (1,1,0)\), and perpendicular, \({\bf{\ell }} = (1,  1,1),(  1,1,2)\), to the field are purely quadraticinh and do not possess a linearinh scaling behaviour. Thus, these length changes (for this choice of magnetic field) do not provide information about the octupolar moment; the quadratic in h behaviour arises from the conduction electrons and/or the quadrupolar moment. We provide in Supplementary Note 4 a justification of the scaling behaviours of the multipolar moments, and in Supplementary Note 6 the corresponding general length change expressions. We note that the scaling behaviours presented here and in Supplementary Note 6 neglect the cubicinh coupling, which breaks the ℤ_{2} symmetry (m ↔ −m) of the octupolar moment. This introduces a ‘flip’ in the octupolar moment at h = 0 (and at \(T < T_{\cal{O}}\) where m has spontaneously ordered, i.e. m ≠ 0): for h = 0^{+}, the +m solution is ‘chosen’, and as we crossover to h = 0^{−}, the now physically distinct −m solution is ‘chosen’ (this is seen in Fig. 2). A similar phenomena is observed in usual ferromagnetism, below the ordering temperature. The neglect of this term is due to the consideration of weak, perturbative magnetic fields in this study. It is likely that this term could become more important (with regard to the scaling behaviour) for larger magnetic fields, but this is beyond the field ranges considered in this work.
Hysteretic behaviour of octupolar ordering
We are motivated in this section by preliminary experimental observations found by some of us (A.S. and S.N., unpublished) of hysteretic behaviour in the length change along the [111] direction below the supposedoctupolar temperature. Hysteresis arises from the existence of domains and the motion of domain walls in the presence of obstructing ‘pinning sites’, which have not been taken into account in the Landau theory we have studied. In order to incorporate such effects, we adapt the phenomenological approach due to Jiles and Atherton^{55,56} which has been used to study hysteresis loops in ferromagnetic and ferroelastic materials. This approach identifies the order parameter (obtained by minimizing the Landau free energy) as its ideal bulk value, where the Landau theory includes a direct coupling u_{f}mh^{3} of the ferrooctupolar moment m and the external [111] magnetic field. The total macroscopic octupolar moment (m_{exp}) is obtained by solving the constructed Jiles and Atherton model, which is heuristically derived in Supplementary Note 7. The key point to note is that the hysteresis loop arises from having a timereversal odd moment (and domains) coupling to the magnetic field.
We present the calculated hysteresis for \(T \,<\, T_{\cal{O}}\) in Fig. 2a. The initial condition chosen to obtain the hysteresis loop is such that at h = 0, the ideal configuration is not being met (i.e. m_{exp} ≠ m); this depicts the realistic scenario of having not all domains aligned in the same direction at h = 0. The depicted hysteresis is reminiscent of the hysteresis in ferromagnets. We obtain the corresponding length change along the (1, 1, 1) direction as shown in Fig. 2b, which for small magnetic fields displays the linearinh scaling behaviour. To better observe this linearinh scaling in the length change, we present the derivative of the length change with respect to the magnetic field in the inset of Fig. 2b. The linearinh scaling behaviour of the length change is more clearly apparent as a constant yintercept in the inset; the further linear scaling in the inset is due to the background quadraticinh scaling behaviour of ΔL/L from the conduction electrons (~γ term).
Furthermore, we note that the field strength h* corresponding to the minimum of the length change [i.e. d(ΔL/L)/dh = 0] provides a threshold above which the conduction electron background dominates over the linearinh scaling behaviour. For this particular length change direction, \(h^ \ast = \frac{{\sqrt 3 }}{2}\frac{{g_{\cal{O}}m}}{{{\gamma} _{\mathrm{c}}}}\). We note that dimensionally h has units of energy (as we have set the Bohr magneton, μ_{B} = 1 here) and the strain tensor \(\epsilon\) is dimensionless; this implies that the composite quantity \(g_{\cal{O}}m\) is dimensionless, while the conduction electron term γ_{c} (which scales like an offdiagonal magnetic susceptibility, from Eq. (6)) has units of (energy)^{−1}. Dimensional analysis thus suggests γ_{c} ~ DOS, where (DOS) is the conduction electron density of states at the Fermi level. To proceed further with the other quantities, we note that m itself is a dimensionless quantity; subsequently \(g_{\cal{O}}\) is also dimensionless. This follows from the convention used in Landau theory where \(m\sim \left( {\frac{{T_{\cal{O}}  T}}{{T_{\cal{O}}}}} \right)^\beta\), and as such for low temperatures (T → 0) we can take m as an O(1) number. If we also take \(g_{\cal{O}}\sim O(1)\), then from above h* ~ DOS^{−1}. Thus, the location of the minimum field is inversely dependent on the conduction electron DOS at the Fermi level: when the conduction electron DOS is small, the minimum field h* is correspondingly large.
Discussion
In this work, motivated by recent and ongoing experiments on Pr(Ti,V,Ir)_{2}(Al,Zn)_{20}, we have used Landau theory of multipolar orders coupled to lattice strain fields to study magnetostriction in systems with quadrupolar and octupolar orders. Our theoretical results for magnetostriction in the presence of octupolar order appear consistent with recent magnetostriction experiments performed by some of us (A.S. and S.N., unpublished) on PrV_{2}Al_{20} where the onset of unusual linearinfield and hysteretic magnetostriction is observed for fields along the [111] direction for T < 0.65 K; in particular, we predict linearinh scaling of the length change for length changes (both parallel and perpendicular) to magnetic fields applied along the [111] direction, and also for length changes perpendicular to [100], below \(T_{\cal{O}}\). Moreover, the coefficient of the linearinh term is directly proportional to the octupolar moment, thus giving a distinct signature for the onset of octupolar ordering as well as a means to detect/measure the octupolar moment. In addition, we can qualitatively understand the quadraticinfield background magnetostriction observed in these experiments; we predict that this scaling arises from the quadrupolar moments and/or direct coupling of the conduction electrons to the external magnetic field and the appropriate lattice normal modes. The summary of all the scaling behaviours is presented succinctly in Table 1.
Our results are broadly applicable to a variety of multipolar orders in cubic systems. For instance, the conclusions here are extendable to the cluster octupolar moments suggested in antiferromagnetically interacting magnetic moments in pyrochlore iridates^{57}. Furthermore, the results presented here can be extended to other more ‘typical’ probes of multipolar ordering/fluctuations. For instance, due to the permitted octupolarstrain coupling, we expect to observe elastic constant softening in the elastic constant c_{44} at the ordering temperature, \(T_{\cal{O}}\), under the application of a magnetic field. We note that it is the c_{44} elastic constant that softens, as it is the associated elastic constant with the offdiagonal strain normal modes. Similarly, we expect elastoresistivity experiments^{58} to be a probe for octupolar susceptibility. The application of an elastic strain with T_{2g} symmetry (such as xy, xz, or yz) in the presence of a magnetic field would result in an associated anisotropic resistivity (ρ_{xy}, ρ_{xz}, ρ_{yz}), which will be proportional to the octupolar susceptibility. Finally, we expect that Pr(Ti,V,Ir)_{2}(Al,Zn)_{20} compounds will possess the characteristic low frequency Raman quasielastic peak^{59}, associated with octupolar fluctuations; specifically, under the application of a magnetic field along the [001] direction, we expect the quasielastic peak to appear in the xy symmetry Raman spectra.
In terms of future work, an interesting avenue to explore is that of the coupling of the conduction electrons to the multipolar moments, as well as to the lattice strain and magnetic field. In particular, the origin of the conduction electron term in Eq. (6), introduced in our phenomenological model from symmetry arguments, is a fascinating direction to explore (as well as potential other terms arising from conduction electrons). We discuss in Supplementary Note 8 a possible origin of the conduction electron term of Eq. (6). Understanding the nature and role of the conduction electrons will also help shed light on the quantum critical behaviour and superconductivity in such multipolar Kondo lattice systems^{4,7,8,32,60,61,62,63,64}.
Methods
NonKramers ground states of Pr^{3+} ions
The ground states are experimentally found to form a nonKramers doublet written in J_{z}〉 basis as
Constructing a pseudospin basis ({↑〉, ↓〉}) from the Γ_{3g} doublets as
allows the multipolar moments to be succinctly written as the effective pseudospin1/2 operator τ = (τ^{x}, τ^{y}, τ^{z}) in Eq. (2) in the main text. The local T_{d} symmetry instilled by the FK cage provides a constraint on the possible terms permitted in the Landau theory. The generating elements of T_{d} are \({\cal{S}}_{4z}\) (improper rotation of π/2 about the \(\widehat {\mathbf{z}}\)axis) and \({\cal{C}}_{31}\) (rotation of 2π/3 about the body diagonal [111] axis). In addition to these point group symmetries, we also require that the terms in the Landau theory be invariant under spatial inversion about the diamond bond centre \({\cal{I}}\) (which swaps the A and B sublattices), as well as timereversal Θ. The behaviour of the multipolar moments under these symmetry constraints is detailed in Supplementary Table 1. As described in the main text, we construct our Landau theory using the order parameters ϕ, \(\tilde \phi\), and m.
Interacting multipolar orders
Equipped with the symmetry knowledge from Supplementary Table 1 we can now write down the Landau free energy for this particular multipolar ordered system as
Here, the free energies \(F_{\tilde \phi }\), F_{m}, and F_{ϕ} denote the independent free energies of the AF\({\cal{Q}}\), F\({\cal{O}}\), and F\({\cal{Q}}\) orders. Setting \(\tilde \phi = \tilde \phi {\mathrm{e}}^{{\mathrm{i}}\tilde \alpha }\) and ϕ = ϕe^{iα}, we get
The first two terms in Eqs. (12)–(14), in square brackets, are the usual mass and quartic interaction terms for AF\({\cal{Q}}\), F\({\cal{O}}\) and F\({\cal{Q}}\) order parameters. We choose \(t_{\tilde \phi } = (T  T_{\cal{Q}})/T_{\cal{Q}}\), and \(t_m = (T  T_{\cal{O}}^{(0)})/T_{\cal{O}}^{(0)}\) with \(T_{\cal{O}}^{(0)} \,<\, T_{\cal{Q}}\), where T denotes the temperature. Focussing on the mass term alone, decreasing T will thus lead to an antiferroquadrupolar order for T < \(T_{\cal{Q}}\), and a lower temperature transition into a state with coexisting ferrooctupolar order when \(T \,<\, T_{\cal{O}}^{(0)}\). These (bare) transition temperatures will be affected by the interplay of the two order parameters; in particular, the true octupolar transition \(T_{\cal{O}}\) will be renormalized from its bare value \(T_{\cal{O}}^{(0)}\) due to the onset of quadrupolar order (besides fluctuation effects which we do not consider here). A measure of how close the two transition temperatures are to each other is provided by the ratio (\(T_{\cal{Q}}\) − \(T_{\cal{O}}\))/(\(T_{\cal{Q}}\) + \(T_{\cal{O}}\)). Finally, since F\({\cal{Q}}\) is not considered to be a primary order parameter, we choose a large positive mass term, t_{ϕ}. The remaining nontrivial terms in Eqs. (12) and (14) are the unusual sixth order and cubic “clock” terms, with respective coefficients \(w_{\tilde \phi }\) and v_{ϕ}, which fix the phases of the AF\({\cal{Q}}\) and F\({\cal{Q}}\) order parameters. We set \(l_{\tilde \phi } \,> \, w_{\tilde \phi }\) to ensure that the free energy is bounded from below.
The couplings between the different multipolar order parameters are encapsulated in \(F_{\tilde \phi ,\phi ,m}\), namely between AF\({\cal{Q}}\) and F\({\cal{Q}}\) moments (g_{1}, g_{2}), and between the quadrupolar and the octupolar moments \((u_{\phi m},u_{\tilde \phi ,m})\)
where the term g_{1} is a symmetryallowed cubic term.
Coupling of magnetic field to multipolar moments
Due to the lack of magnetic dipole moment supported by the Γ_{3g} doublet, the magnetic field does not couple linearly to the states. One can derive the low energy magnetic field Hamiltonian by performing secondorder perturbation theory in h ⋅ J, where the low energy subspace is spanned by the Γ_{3g} doublet, and the high energy subspace is spanned by the excited triplets Γ_{4,5}; here h has units of energy as we have set the Bohr magneton, μ_{B} = 1. This leads to
In the above Eq. (16), h = (h_{x}, h_{y}, h_{z}) with h = h, and \({\gamma} _0 \equiv \frac{{  14}}{{3\Delta ({\Gamma} _4)}} + \frac{2}{{\Delta ({\Gamma} _5)}}\), where Δ(Γ_{4}), Δ(Γ_{5}) are the gaps between the low energy doublets and the corresponding triplet states at zero magnetic field. The effective coupling to the ferroquadrupolar order is via \(\psi _H \equiv \frac{{{\gamma} _0\sqrt 3 }}{4}(h_x^2  h_y^2) + {\mathrm{i}}\frac{{{\gamma} _0}}{4}(3h_z^2  h^2)\). Based on the form of the coupling in Eq. (16), we infer that ψ_{H} transforms identically to ϕ under the relevant symmetries. Going to thirdorder in perturbation theory leads to a further \({\cal{O}}\)(h^{3}) coupling of the magnetic field to octupole moment of the form ~h_{x}h_{y}h_{z}τ^{z}.
Thus, the symmetryallowed effective magnetic field coupling to the quadrupolar moments is
where \(\psi _H = \frac{{{\gamma} _0}}{4}\sqrt {3(h_x^2  h_y^2)^2 + (3h_z^2  h^2)^2}\), and \(\tan (\theta _H) = \frac{1}{{\sqrt 3 }}\frac{{3h_z^2  h^2}}{{(h_x^2  h_y^2)}}\). The first (second) line in Eq. (17) is the symmetryallowed coupling to the AF\({\cal{Q}}\) (F\({\cal{Q}}\)). The third line involves couplings permitted due to pure symmetry reasons that renormalize the mass terms of the AF\({\cal{Q}}\) and F\({\cal{Q}}\). Physically they arise from conduction electron mediated magnetic couplings (having integrated out the conduction electrons); similar coupling to the octupolar moment is also permitted [~h^{2}m^{2}], which is formally introduced via the magnetic field assisted coupling of the octupolar moment to the lattice strain. In the main text, we discuss magnetic fields applied along the [100], [110] and [111] directions. For clarity, we present the value for ψ_{H} and θ_{H} for the magnetic field directions discussed in subsequent sections in Table 2. In the presence of the magnetic field, it is possible for additional couplings between the quadrupolar and octupolar moments to be induced, such as
These terms are merely the usual quadraticinfield coupling to the quadrupolar moment (Eqs. (16) and (17)) with m^{2} multiplied into it. Due to symmetry constraints, we cannot have terms which are linear in the octupolar, quadrupolar and magnetic field (breaks \({\cal{C}}_{31}\) symmetry). These above terms do not affect the leading scaling behaviour of the magnetostriction, as they have the same order of h as previous terms in the free energy. Specifically, the terms are quadraticinh and can be thought of as renormalizing the mass term of the octupolar moment. We recall that the octupolar mass term already contains a quadraticinh term, which arose from integrating out the elastic strain in Eq. (6), and so these new terms merely modify the coefficient of the previous quadraticinh expressions/terms.
The Landau theory is numerically minimized using standard minimization/optimization schemes. The hysteresis differential equation is numerically solved using RungeKutta 4th order methods. We use the initial condition of m_{ir} = 0 for h = 0 to obtain the depicted solution, with k = 100, α = 10^{−3}, c = 0.01.
Data availability
All relevant data are available upon reasonable request to the corresponding author.
Code availability
All relevant codes are available upon reasonable request to the corresponding author.
References
Fazekas, P. Lecture Notes on Electron Correlation and Magnetism (World Scientific, 1999).
Kuramoto, Y., Kusunose, H. & Kiss, A. Multipole orders and fluctuations in strongly correlated electron systems. J. Phys. Soc. Jpn. 78, 072001 (2009).
Kusunose, H. Description of multipole in felectron systems. J. Phys. Soc. Jpn. 77, 064710 (2008).
Sakai, A. & Nakatsuji, S. Kondo effects and multipolar order in the cubic PrTr_{2}Al_{20} (Tr=Ti, V). J. Phys. Soc. Jpn. 80, 063701 (2011).
Onimaru, T. & Kusunose, H. Exotic quadrupolar phenomena in nonKramers doublet systems ‘the cases of PrT_{2}Zn_{20} (T=Ir, Rh) and PrT_{2}Al_{20} (T=V, Ti)’. J. Phys. Soc. Jpn. 85, 082002 (2016).
Flouquet, J. in Progress in Low Temperature Physics, Vol. 15, 139–281 (Elsevier, 2005).
Sakai, A., Kuga, K. & Nakatsuji, S. Superconductivity in the Ferroquadrupolar State in the Quadrupolar Kondo lattice PrTi_{2}Al_{20}. J. Phys. Soc. Jpn. 81, 083702 (2012).
Tsujimoto, M., Matsumoto, Y., Tomita, T., Sakai, A. & Nakatsuji, S. Heavyfermion superconductivity in the quadrupole ordered state of PrV_{2}Al_{20}. Phys. Rev. Lett. 113, 267001 (2014).
Kotegawa, H. et al. Evidence for unconventional strongcoupling superconductivity in PrOs_{4}Sb_{12}: an Sb nuclear Quadrupole Resonance Study. Phys. Rev. Lett. 90, 027001 (2003).
Santini, P. et al. Multipolar interactions in felectron systems: the paradigm of actinide dioxides. Rev. Mod. Phys. 81, 807–863 (2009).
Shiina, R., Shiba, H. & Thalmeier, P. Magneticfield effects on quadrupolar ordering in a Γ_{8}quartet system CeB_{6}. J. Phys. Soc. Jpn. 66, 1741–1755 (1997).
Kiss, A. Multipolar Ordering in fElectron Systems, Ph.D. thesis (Budapest University of Technology and Economics, 2004).
Kiss, A. & Fazekas, P. Group theory and octupolar order in URu_{2}Si_{2}. Phys. Rev. B 71, 054415 (2005).
Onimaru, T. et al. Antiferroquadrupolar ordering in a Prbased superconductor PrIr_{2}Zn_{20}. Phys. Rev. Lett. 106, 177001 (2011).
Iwasa, K. et al. Welldefined crystal field splitting schemes and nonKramers doublet ground states of f electrons in PrT_{2}Zn_{20} (T = Ir, Rh, and Ru). J. Phys. Soc. Jpn. 82, 043707 (2013).
Araki, K. et al. Magnetization and specific heat of the cage compound PrV_{2}Al_{20}. JPS Conf. Proc. 3, 011093 (2014).
Onimaru, T. et al. Simultaneous superconducting and antiferroquadrupolar transitions in PrRh_{2}Zn_{20}. Phys. Rev. B 86, 184426 (2012).
Podolsky, D. & Demler, E. Properties and detection of spin nematic order in strongly correlated electron systems. New J. Phys. 7, 59 (2005).
Tokunaga, Y. et al. NMR evidence for higherorder multipole order parameters in NpO_{2}. Phys. Rev. Lett. 97, 257601 (2006).
Lee, S., Paramekanti, A. & Kim, Y. B. Optical gyrotropy in quadrupolar Kondo systems. Phys. Rev. B 91, 041104 (2015).
Chandra, P., Coleman, P., Mydosh, J. A. & Tripathi, V. Hidden orbital order in the heavy fermion metal URu_{2}Si_{2}. Nature 417, 831–834 (2002).
Chandra, P., Coleman, P., Mydosh, J. A. & Tripathi, V. The case for phase separation in URu_{2}Si_{2}. J. Phys.: Condens. Matter 15, S1965–S1971 (2003).
Tripathi, V., Chandra, P. & Coleman, P. Sleuthing hidden order. Nat. Phys. 3, 78–80 (2007).
SantanderSyro, A. F. et al. Fermisurface instability at the ‘hiddenorder’transition of URu_{2}Si_{2}. Nat. Phys. 5, 637–641 (2009).
Haule, K. & Kotliar, G. Arrested Kondo effect and hidden order in URu_{2}Si_{2}. Nat. Phys. 5, 796–799 (2009).
Haule, K. & Kotliar, G. Complex LandauGinzburg theory of the hidden order in URu_{2}Si_{2}. EPL 89, 57006 (2010).
Pezzoli, M. E., Graf, M. J., Haule, K., Kotliar, G. & Balatsky, A. V. Local suppression of the hiddenorder phase by impurities in URu_{2}Si_{2}. Phys. Rev. B 83, 235106 (2011).
Okazaki, R. et al. Rotational symmetry breaking in the hiddenorder phase of URu_{2}Si_{2}. Science 331, 439–442 (2011).
Rau, J. G. & Kee, H.Y. Hidden and antiferromagnetic order as a rank5 superspin in URu_{2}Si_{2}. Phys. Rev. B 85, 245112 (2012).
Stewart, G. R. Heavyfermion systems. Rev. Mod. Phys. 56, 755–787 (1984).
Cox, D. L. Quadrupolar Kondo effect in uranium heavyelectron materials? Phys. Rev. Lett. 59, 1240–1243 (1987).
Cox, D. L. & Zawadowski, A. Exotic Kondo effects in metals: magnetic ions in a crystalline electric field and tunnelling centres. Adv. Phys. 47, 599–942 (1998).
Kuo, H.H., Shapiro, M. C., Riggs, S. C. & Fisher, I. R. Measurement of the elastoresistivity coefficients of the underdoped iron arsenide Ba(Fe_{0.975}Co_{0.025})_{2}As_{2}. Phys. Rev. B 88, 085113 (2013).
Riggs, S. C. et al. Evidence for a nematic component to the hiddenorder parameter in URu_{2}Si_{2} from differential elastoresistance measurements. Nat. Commun. 6, 6425 (2015).
Palmstrom, J. C., Hristov, A. T., Kivelson, S. A., Chu, J.H. & Fisher, I. R. Critical divergence of the symmetric (A_{1g}) nonlinear elastoresistance near the nematic transition in an ironbased superconductor. Phys. Rev. B 96, 205133 (2017).
Tokunaga, Y. et al. Magnetic excitations and cf hybridization effect in PrTi_{2}Al_{20} and PrV_{2}Al_{20}. Phys. Rev. B 88, 085124 (2013).
Matsumoto, K. T., Onimaru, T., Wakiya, K., Umeo, K. & Takabatake, T. Effect of La substitution in PrIr_{2}Zn_{20} on the superconductivity and antiferroquadrupole order. J. Phys. Soc. Jpn. 84, 063703 (2015).
Freyer, F. et al. Twostage multipolar ordering in PrT_{2}Al_{20} Kondo materials. Phys. Rev. B 97, 115111 (2018).
Sato, T. J. et al. Ferroquadrupolar ordering in PrTi_{2}Al_{20}. Phys. Rev. B 86, 184419 (2012).
Shimura, Y. et al. Fieldinduced quadrupolar quantum criticality in PrV_{2}Al_{20}. Phys. Rev. B 91, 241102 (2015).
Shimura, Y. et al. Giant anisotropic magnetoresistance due to purely orbital rearrangement in the quadrupolar heavy fermion superconductor PrV_{2}Al_{20}. Phys. Rev. Lett. 122, 256601 (2019).
Nakanishi, Y. et al. Elastic anomalies associated with two successive transitions of PrV_{2}Al_{20} probed by ultrasound measurements. Phys. B: Condens. Matter 536, 125–127 (2018).
Ishii, I. et al. Antiferroquadrupolar ordering at the lowest temperature and anisotropic magnetic fieldtemperature phase diagram in the cage compound PrIr_{2}Zn_{20}. J. Phys. Soc. Jpn. 80, 093601 (2011).
Ishii, I. et al. Antiferroquadrupolar ordering and magneticfieldinduced phase transition in the cage compound PrRh_{2}Zn_{20}. Phys. Rev. B 87, 205106 (2013).
Koseki, M. et al. Ultrasonic Investigation on a Cage structure compound PrTi_{2}Al_{2}O. J. Phys. Soc. Jpn. 80, SA049 (2011).
Taniguchi, T. et al. NMR observation of ferroquadrupole order in PrTi_{2}Al_{20}. J. Phys. Soc. Jpn. 85, 113703 (2016).
Wörl, A. et al. Highly anisotropic strain dependencies in PrIr_{2}Zn_{20}. Phys. Rev. B 99, 081117 (2019).
Santini, P. & Amoretti, G. Magneticoctupole order in neptunium dioxide? Phys. Rev. Lett. 85, 2188–2191 (2000).
Kopmann, W. et al. Magnetic order in NpO_{2} and UO_{2} studied by muon spin rotation. J. Alloy. Compd. 271273, 463–466 (1998).
Walstedt, R. E. The NMR Probe of HighTc Materials and Correlated Electron Systems, 2nd edn, 257–268 (Springer, 2018).
Landau, L. D., Lifshitz, E. M., Sykes, J. B. & Reid, W. H. Theory of Elasticity (Pergamon Press, 1986).
Lüuthi, B. Physical Acoustics in the Solid State, 1st edn (Springer, 2006).
Lee, S., Trebst, S., Kim, Y. B. & Paramekanti, A. Landau theory of multipolar orders in Pr(Y)_{2}X_{20} Kondo materials (Y=Ti, V, Rh, Ir; X=Al, Zn). Phys. Rev. B 98, 134447 (2018).
Hattori, K. & Tsunetsugu, H. Antiferro quadrupole orders in nonKramers doublet systems. J. Phys. Soc. Jpn. 83, 034709 (2014).
Jiles, D. & Atherton, D. Theory of ferromagnetic hysteresis. J. Magn. Magn. Mater. 61, 48–60 (1986).
Massad, J. E. & Smith, R. C. A domain wall model for hysteresis in ferroelastic materials. J. Intel. Mat. Syst. Str. 14, 455–471 (2003).
Arima, T.h. Timereversal symmetry breaking and consequent physical responses induced by allinallout type magnetic order on the pyrochlore lattice. J. Phys. Soc. Jpn. 82, 013705 (2013).
Rosenberg, E. W., Chu, J.H., Ruff, J. P. C., Hristov, A. T. & Fisher, I. R. Divergence of the quadrupolestrain susceptibility of the electronic nematic system YbRu_{2}Ge_{2}. Proc. Natl. Acad. Sci. 116, 7232–7237 (2019).
Gallais, Y. et al. Observation of incipient charge nematicity in Ba(Fe_{1−x}Co_{x})_{2}As_{2}. Phys. Rev. Lett. 111, 267001 (2013).
Emery, V. J. & Kivelson, S. Mapping of the twochannel Kondo problem to a resonantlevel model. Phys. Rev. B 46, 10812–10817 (1992).
Cox, D. L. & Ruckenstein, A. E. Spinflavor separation and nonFermiliquid behavior in the multichannel Kondo problem: a largeN approach. Phys. Rev. Lett. 71, 1613–1616 (1993).
Ramirez, A. P. et al. Nonlinear susceptibility: a direct test of the quadrupolar Kondo effect in UBe_{13}. Phys. Rev. Lett. 73, 3018–3021 (1994).
Tsuruta, A. & Miyake, K. NonFermi liquid and Fermi liquid in twochannel Anderson lattice model: theory for PrA_{2}Al_{20} (A = V, Ti) and PrIr_{2}Zn_{20}. J. Phys. Soc. Jpn. 84, 114714 (2015).
Onimaru, T. et al. Quadrupoledriven nonFermiliquid and magneticfieldinduced heavy fermion states in a nonKramers doublet system. Phys. Rev. B 94, 075134 (2016).
Acknowledgements
We acknowledge Premala Chandra and Piers Coleman for discussions and correspondence on the Landau theory of magnetostriction in an octupolar phase, and for letting us know about their parallel work. We thank Wonjune Choi and Li Ern Chern for helpful comments regarding the paper. Y.B.K. is supported by the Killam Research Fellowship of the Canada Council for the Arts. This work was supported by NSERC of Canada, and Canadian Institute for Advanced Research. S.B.L. is supported by the KAIST startup and National Research Foundation Grant (NRF2017R1A2B4008097). This work was partially supported by GrantsinAids for Scientific Research on Innovative Areas (15H05882 and 15H05883) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan, by CREST(JPMJCR18T3), Japan Science and Technology Agency, by GrantsinAid for Scientific Research (19H00650) from the Japanese Society for the Promotion of Science (JSPS), and by CIFAR programme “Quantum Materials” (FS201512965).
Author information
Authors and Affiliations
Contributions
Y.B.K. and S.N. conceived the research. S.L., Y.B.K. and A.P. developed the underlying Landau theory of multipolar ordering. A.S.P. and A.S. performed the respective theoretical calculations, and experimental work. All authors participated in writing the paper.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information: Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work.
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Patri, A.S., Sakai, A., Lee, S. et al. Unveiling hidden multipolar orders with magnetostriction. Nat Commun 10, 4092 (2019). https://doi.org/10.1038/s41467019119133
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467019119133
This article is cited by

Quadrupolar magnetic excitations in an isotropic spin1 antiferromagnet
Nature Communications (2022)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.