Abstract
The conventional order parameters in quantum matters are often characterized by ‘spontaneous’ broken symmetries. However, sometimes the broken symmetries may blend with the invariant symmetries to lead to mysterious emergent phases. The heavy fermion metal URu_{2}Si_{2} is one such example, where the order parameter responsible for a secondorder phase transition at T_{h} = 17.5 K has remained a longstanding mystery. Here we propose via abinitio calculation and effective model that a novel spinorbit density wave in the fstates is responsible for the hiddenorder phase in URu_{2}Si_{2}. The staggered spinorbit order spontaneously breaks rotational and translational symmetries while timereversal symmetry remains intact. Thus it is immune to pressure, but can be destroyed by magnetic field even at T = 0 K, that means at a quantum critical point. We compute topological index of the order parameter to show that the hidden order is topologically invariant. Finally, some verifiable predictions are presented.
Introduction
Most states or phases of matter can be described by local order parameters and the associated broken symmetries in the spin, charge, orbital or momentum channel. However, recent discoveries of quantum Hall states^{1} and topological insulators^{2,3} have revamped this conventional view. It has been realized^{1,2,3,4} that systems with combined timereversal () symmetry and large spinorbit (SO) coupling can host new states of matter which are distinguished by topological quantum numbers of the bulk band structure rather than spontaneously broken symmetries. Subsequently, more such distinct phases have been proposed in the family of topological Mott insulators^{5}, topological Kondo insulators^{6}, topological antiferromagnetic insulators^{7}. In the latter cases, the combined manybody physics and symmetry governs topologically protected quantum phases. Encouraged by these breakthrough developments, we search for analogous exotic phases in the heavy fermion metal URu_{2}Si_{2}, whose lowenergy f states accommodate and strong SO coupling. This compound also naturally hosts diverse quantum mechanical phases including Kondo physics, large moment antiferromagnetism (LMAF), mysterious ‘hiddenorder’ (HO) state and superconductivity^{8}.
In URu_{2}Si_{2} the screening of felectrons due to the Kondo effect begins at relatively high temperatures, ushering the system into a heavy fermion metal at lowtemperature^{9}. Below T_{h} = 17.5 K, it enters into the HO state via a secondorder phase transition characterized by sharp discontinuities in numerous bulk properties^{10,11,12,13}. The accompanying gap is opened both in the electronic structure^{9,14,15,16} as well as in the magnetic excitation spectrum^{17}, suggesting the formation of an itinerant magnetic order at this temperature. However, the associated tiny moment (~ 0.03µ_{B}) cannot account for the large (about 24%) entropy release^{18} and other sharp thermodynamic^{10,11} and transport anomalies^{12,13} during the transition. Furthermore, very different evolutions of the HO parameter and the magnetic moment as a function of both magnetic field^{19,20} and pressure^{21,22} rule out a possible magnetic origin of the HO phase in this system. Any compelling evidence for other charge, orbital or structural ordering has also not been obtained^{23}. Existing theories include multiple spin correlator^{24}, JahnTeller distortions^{25}, unconventional spindensity wave^{26,27}, antiferromagnetic fluctuation^{28}, orbital order^{20}, helicity order^{29}, staggered quadrupole moment^{30}, octupolar moment^{31}, hexadecapolar order^{32}, linear antiferromagnetic order^{33}, incommensurate hybridization wave^{34}, spin nematic order^{35}, modulated spin liquid^{36}, jj fluctuations^{37}, unscreened Anderson lattice model^{38}, among others^{8}. However, a general consensus for the microscopic origin of the HO parameter has not yet been attained.
Formulating the correct model for the HO state requires the knowledge of the broken symmetries and the associated electronic degrees of freedom that are active during this transition. A recent torque measurement on high quality single crystal sample reveals that the fourfold rotational symmetry of the crystal becomes spontaneously broken^{23} at the onset of the HO state. Furthermore, several momentumresolved spectroscopic data unambiguously indicate the presence of a translational symmetry breaking at a longitudinal incommensurate wavevector Q_{h} = (1±0.4, 0, 0)^{14,16,18,39}. [Previous firstprinciple calculation has demonstrated that an accompanying commensurate wavevector Q_{2} = (1, 0, 0) might be responsible for the LMAF phase^{33}, which is separated from the HO state via a first order phase transition^{8,19,20,21,22}. As it is often unlikely to have two phases of same broken symmetry but separated by a phase boundary, we expect that LMAF and HO phases are different.] In general, the order parameter that emerges due to a broken symmetry relies incipiently on the good quantum number and symmetry properties of the ‘parent’ or noninteracting Hamiltonian. In case of URu_{2}Si_{2}, spin and orbital are not the good quantum numbers, rather the presence of the SO coupling renders the total angular momentum to become the good quantum number. Therefore, SU(2) symmetry can not be defined for spin or orbital alone and the ‘parent’ Hamiltonian has to be defined in representation. The ‘parent’ Hamiltonian also accommodate other symmetries coming from its crystal, wavefunction properties which we desire to incorporate to formulate the HO parameter.
Results
Abinitio band structure
In order to find out the symmetry properties of the lowlying states, we begin with investigating the abinitio ‘parent’ band dispersion and the FS of URu_{2}Si_{2}^{40,41} in Fig. 1. The electronic structure in the vicinity of the Fermi level (E_{F}) (±0.2 eV) is dominated by the 5f states of U atom in the entire Brillouin zone^{14,15,16,33,39,42}. Owing to the SO coupling and the tetragonal symmetry, the 5f states split into the octet states and the sextet states^{43}. URu_{2}Si_{2} follows a typical band progression in which the Γ_{8} bands are pushed upward to the empty states while the Γ_{6} states drop to the vicinity of E_{F}. The corresponding FS in Fig. 1d reveals that an even number of anticrossing features occurs precisely at the intersection between two oppositely dispersing conducting sheets. Unlike in topological insulators^{3,4}, the departure of the band crossing points from the invariant momenta here precludes the opening of an inverted band gap at the crossings^{2} and Diraccones crop up with Kramer's degeneracy in the bulk states. Therefore, URu_{2}Si_{2} is an intrinsically trivial topological metal above the HO transition temperature.
The SO interaction introduces two prominent FS instabilities at Q_{2} = (1, 0, 0) and at Q_{h} = (1 ± 0.4, 0, 0). The commensurate wavevector Q_{2} occurs between same orbital. Therefore, if this instability induces a gap opening, it has to be in the spinchannel, which is prohibited by symmetry and strong SO coupling. We argue (see Supplementary Information (SI) for details), in accordance with an earlier calculation^{33}, that this instability is responsible for the LMAF phase. On the other hand, the incommensurate one, Q_{h}, occurs between two different orbitals and can open a gap if a symmetry between these orbitals and spins are spontaneously broken together. In other word, since SO coupling is strong in this system, individual spin or orbitalorderings are unlikely to form unless interaction can overcome the SO coupling strength. On the other hand, a SO entangled order parameter in the twoparticle channel can collectively propagate with alternating sign in the total angular momentum at the wavelength determined by the modulation vector. This is the guiding instability that drives spontaneous rotational symmetry breaking, while the symmetry remains intact (see Fig. 2a). This is because, both SU(2) groups for spin and orbital separately are odd under , but their product becomes even. As the parent state is not a nontrivial topological phase, a gap is opened to lift the FS instability.
Lowenergy effective model
Motivated by the abovementioned experimental results and band structure symmetry properties, we formulate a simple and unified model by using the theory of invariants^{44}. We restrict our discussion to the lowlying Γ_{6} bands and neglect the unfilled Γ_{8} bands. Due to jj SO coupling and symmetry, the Γ_{6} atomic states consist of three doublets, characterized by up and down ‘pseudospins’: , , , where m_{J} is the z component of J. On entering into the HO state, the FS instability commences in between the two doubly degenerate and states only^{33,35}. If no other symmetry is broken, the degenerate state remains unaltered in the HO state^{44} and hence they are not considered in our model Hamiltonian. Throughout this paper, we consistently use two indices: orbital index and ‘pseudospin’ σ = ↑(+), ↓(−). In this notation, we consider the ‘pseudospinor’ field , where is the creation operator for an electron in the orbital with momentum k and ‘pseudospin’ σ.
The representation of the symmetry operations that belongs to the D_{4h} symmetry of the URu_{2}Si_{2} crystal structure is: symmetry, inversion symmetry , fourfold rotational symmetry and the two reflection symmetries . The SO fstate of actinides is invariant under all symmetries except the mirror reflection, which in fact allows the formation of the SO density wave into a finite gap in the HO state (see SI). On the basis of these symmetry considerations, it is possible to deduce the general form of the noninteracting Hamiltonian as:
Here, τ^{µ} (µ 0, x, y, z) depict the 2D Pauli matrices in the orbital space and τ^{0} is the identity matrix (σ^{µ} matrices will be used later to define the spin space). The invariance requires that . Under and , the symmetry of and must complement to their corresponding identity and Pauli Matrix counterparts, respectively. Hence we obtain the SlaterKoster hopping terms as: and . The obtained values of the tightbinding hopping parameters as (t, t_{1}, t_{2}, t_{z}) = −(−45,45,50,−25) in meV. The above Hamiltonian can be solved analytically which gives rise to four SOsplit energy dispersions as
Here σ = ± and τ = ± become band indices. An important difference of the present Hamiltonian with that of bulk topological insulators^{3} or quantum spinHall systems^{1} is the absence of a mass or gap parameter in the former case. The computed noninteracting bands are plotted in Fig. 2b, which exhibit several Dirac points along the highsymmetry lines. Focusing on the Dirac point close to E_{F}, we find that it occurs at the crossing between bands E^{+−} and E^{−+}, demonstrating that it hosts fourfold Kramer's degeneracy (two orbitals and two spins). Therefore, lifting this degeneracy requires the presence of a SO order parameter. However, it is important to note that the gap opening at the Dirac point is not a manifestation of the presence of degeneracy at it, but a consequence of the SO density wave caused by FS instability.
SO density wave induced HO
The ‘hotspot’ Q_{h} divides the unit cell into a reduced ‘SO Brillouin zone’ in which we can define the Nambu operator in the usual way . In this notation, the SO density wave (SODW) interaction term can be written in general as
where µ, ν ∈ {0, x, y, z}. The symbol :: represents normal ordering. Here g is the contact coupling interaction arising from screened interorbital Coulomb term embedded in Hund's coupling parameter and , τ and σ represent Pauli matrices in orbital and spin basis, respectively. Absorbing g and Γ into one term we define the meanfield order parameter
Here τ, τ′ and σ, σ′ (not in bold font) are the components of the τ^{µ} and σ^{ν} matrices, respectively. Without any loss of generality we fix the spin orientation along zdirections (ν = z). Therefore, we drop the index ν henceforth. Furthermore we define the gap vector as , where we split the interaction term g(k) into a constant onsite term and the dimensionless order parameter Δ(k). With these substitutions, we obtain the final result for the order parameter as
Eq. 7 admits a plethora of order parameters related to the SO density wave formations which break symmetry in different ways. Among them, we rule out those parameters which render gapless states by using the symmetry arguments (see SI): All four order parameters obey symmetry, while only M^{y} term is even under , because it is the product of two odd terms τ^{y} and σ (we drop the superscript ‘y’ henceforth). This is the only term which commences a finite gap opening if the translational or rotational symmetry is spontaneously broken. We have shown in SI that there exists a considerably large parameter space of coupling constant ‘g’ where this order parameter dominates.
Eq. 7 implies that spin and orbital orderings occur simultaneously along the ‘hotspot’ direction Q_{h}, as illustrated in Fig. 2a. It propagates along or directions with alternating signs (particlehole pairs) to commence a SO density wave. The resulting Hamiltonian breaks the fourfold rotational symmetry down to a twofold one and gives rise to a socalled spinorbit ‘smectic’ state which breaks both translational and C_{4} symmetry^{45}. The present invariant SO order parameter is inherently distinct from any spin or orbital or even interorbital spindensity wave order which break symmetry. This criterion also rules out any similarly between our present SO smectic state with the spinnematic phase^{35} or spinliquid state^{36}. Furthermore, the present order parameter is different from invariant ‘hybridization wave’ (between f and d orbitals of same spin), or charge density wave or others^{30,32}, as SO order involves flipping of both orbital (between split f orbitals that belong to Γ_{6} symmetry) and spin simultaneously. Taking into account the bandstructure information that Q_{h} represents the interband nesting, it is instructive to focus on only b_{12}(k) component (thus the subscript ‘12’ is eliminated hereafter). Therefore, the SO density wave does not introduce a spin or orbital moment, but a polarization in the total angular momentum δm_{J} = ±2 [for the ordering between and ].
The b vector belongs to the same irreducible point group representation, E_{g}, of the crystal with odd parity and can be defined by b(k) = 2igΔ^{x} sin k_{x}a, or 2igΔ^{y} sin k_{y}a for the wavevectors , or , respectively. The meanfield Hamiltonian for the HO state within an effective two band model reduces to the general form H_{MF} = H_{0}+H_{SODW}, where the particlehole coupling term is
In the Nambu representation, it is obvious that the HO term merely adds a mass term to the term defined above. At the bandcrossing points located where , a gap opens by the value of b(k)^{2}. Figure 2 demonstrates the development of the quasiparticle structure in the HO state. The band progression and the associated gap opening is fully consistent with the angleresolved photoemission spectroscopy (ARPES) observations^{15,16}. The scanning tunneling microscopy and spectroscopic (STM/S)^{9,14} fingerprints of the gap opening in the density of state (DOS) is also described nicely within our calculations, see Fig. 2c.
Discussion
The SO moment is , where is imaginary time. Introducing simplified indices α, β = ττ′σσ′, the correlation function of vector can be defined as , where is normal timeordering. Our numerical calculation of the within randomphase approximation (RPA) yields an inelastic neutron scattering (INS) mode with enhanced intensity at Q_{h} near ω_{Q} ~ 4.7 meV below T_{h} as shown in Fig. 1d. INS data (symbols) at a slightly large momentum agrees well with our calculation, however, a polarized INS measurement will be of considerable value to distinguish our proposed invariant mode from any spinflip and elastic background^{46}. A Tdependent study of the INS mode also reveals that this mode becomes strongly enhanced at Q_{h} rather than at the commensurate one below T_{h}^{39}.
One way to characterize the nature of a phase transition is to determine the temperature evolution of the gap value. Our computed selfconsistent values of the meanfield gap Δ(T) agree well with the extracted gap values from the STM spectra^{9} [see inset to Fig. 2c]. In general, the entropy loss at a meanfield transition is given by^{20} , where Δ is the HO gap and ξ_{F} is the Fermi energy of the gapped state. At HO the Fermi energy , where the two linearly dispersive bands near the Fermi level yields . Using the measured Sommerfield coefficient γ = 180 mJmol^{−1}K^{−2}, compared to its linear expansion of γ_{0} = 50 mJmol^{−1}K^{−2}, we obtain the mass renormalization factor Z^{−1} = γ/γ_{0} = 3.6. This gives . For the two bands that participate in the HO gap opening, we get in eV at k_{1F} = 0.5π/a and in eV at k_{2F} = 0.3π/a from Fig. 1a. Using the experimental value of Δ = 5 meV^{9,14}, we obtain ΔS~ 0.28k_{B} ln 2, which is close the experimental value of 0.3k_{B} ln 2^{10}.
We now evaluate the topological invariant index of interacting Hamiltonian in Eq. 8 to demonstrate that HO gap opening in URu_{2}Si_{2} also induces topological phase transition. To characterize the topological phenomena, we recall the FuKane classification scheme^{2} which implies that if a timereversal invariant system possess an odd value of Z_{2} invariant index, the system is guaranteed to be topologically nontrivial. Z_{2} index is evaluated by the timereversal invariant index ν_{i} = ±1, if defined, for all filled bands as Z_{2} = ν_{1}ν_{2}…ν_{n}, where n is the total number of orbitals in the Fermi sea. A more efficient method of determining the topological phase is called the adiabatic transformation scheme used earlier in realizing a large class of topological systems, especially when Z_{2} calculation is difficult^{47}. In this method, the nontrivial topological phase of a system can be realized by comparing its bandprogression with respect to an equivalent trivial topological system. URu_{2}Si_{2} is topologically trivial above the HO state, i.e. . The gap opening makes the top of the valence band (odd parity) to drop below E_{F} as shown in Fig. 2b. Thereby, an odd parity gained in the occupied level endows the system to a nontrivial topological metal. To see that we evaluate the topological index for the HO term as ν_{ho} = ∫ dkΩ(k), where the corresponding Berry curvature can be written in terms of bvector as for each spin with . Due to the odd parity symmetry of b, it is easy to show that ν_{ho} = −1 which makes the total Z_{2} value of the HO phase to be and hence we show that hidden order gapping is a topologically nontrivial phase. The consequence of a topological bulk gap is the presence of surface states^{2,47}. In our present model, we expect two surface states of opposite spin connecting different orbitals inside the HO gap. As the system is a weaktopological system, the surface states are unlikely to be topologically protected. The SO locking of these states can be probed by ARPES using circular polarized incident photon which will be a definite test of this postulate.
The HO gap is protected from any invariant perturbation such as pressure (with sufficient pressure the HO transforms into the LMAF phase), while breaking perturbation such as magnetic field will destroy the order. Remarkably, these are the hallmark features of the HO states^{23,35,48}, which find a natural explanation within our SO density wave order scenario. In what follows, the magnetic field will destroy the HO state even at T = 0 K, that means at a quantum critical point (QCP) as the HO is a spontaneously broken symmetry phase^{49}. However, due to the finite gap opening at the HO state, it requires finite field to destroy the order. The thermodynamical critical field can be obtained from^{50} , where B_{c} is the critical field and at the resonance mode that develops in the HO state. α = gµ_{B}〈Δm_{J}〉 = 2gµ_{B} and bare gfactor g = 0.8. Substituting ω_{res} = 4.7 meV, we get the location of the QCP at B ≈ 38 T, which is close the experimental value of B = 34 T^{48}.
Broken symmetry FS reconstruction leads to enhanced Nernst signal^{51}. For the case of broken symmetry SO order, we expect to generate spinresolved Nernst effect which can be measured in future experiments to verify our proposal^{52}.
In summary, we proposed a novel SO density wave order parameter for the HO state in URu_{2}Si_{2}. Such order parameter is symmetry invariant. We find no fundamental reason why such order parameter cannot develop in other systems in which both electronic correlation and SO of any kind are strong. Some of the possible materials include heavy fermion systems, Iridates^{53}, SrTiO_{3} surface states^{54}, SrTiO_{3}/LiAlO_{3} interface^{55}, HalfHeusler topological insulator^{47} and other d and felectron systems with strong SO. In particular, a Rashbatype SO appears due to relativistic effect in twodimensional electron system yielding helical FSs. In such systems, the FS instability may render similar SO density wave and the resulting quasiparticle gap opening is observed on the surface state of BiAg_{2} alloys even when the spindegeneracy remains intact^{56}. Furthermore, recent experimental findings of quasiparticle gapping in the surface state of topological insulator due to quantum phase transition even in the absence of timereversal symmetry breaking can also be interpreted as the development of some sort of spin orbit order^{57}.
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Acknowledgements
The author thanks A. V. Balatsky, M. J. Graf, A. Bansil, R. S. Markiewicz, T. Durakiewicz, J.X. Zhu, P. M. Oppeneer, J. Mydosh, P. Wölfle and J. Haraldsen for useful discussions. Work at the Los Alamos National Laboratory was supported by the U.S. DOE under contract no. DEAC5206NA25396 through the Office of Basic Energy Sciences and the UC Lab Research Program and benefited from the allocation of supercomputer time at NERSC.
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Das, T. Spinorbit density wave induced hidden topological order in URu_{2}Si_{2}. Sci Rep 2, 596 (2012). https://doi.org/10.1038/srep00596
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DOI: https://doi.org/10.1038/srep00596
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