Spin-orbit density wave induced hidden topological order in URu2Si2

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 URu2Si2 is one such example, where the order parameter responsible for a second-order phase transition at Th = 17.5 K has remained a long-standing mystery. Here we propose via ab-initio calculation and effective model that a novel spin-orbit density wave in the f-states is responsible for the hidden-order phase in URu2Si2. The staggered spin-orbit order spontaneously breaks rotational, and translational symmetries while time-reversal 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.

In the main text, we have provided an effective two band model which is relevant for the study of hidden-order (HO) gap structure coming from Fermi surface (FS) instability. Here, we expand on how such an effective Hamiltonian is deduced. The spin-orbit splitting state is, in general, studied within either L.S coupling or j − j coupling approaches. In the former case, the total L and S are formed due to strong Hund's coupling prior to the formation of spin-orbit coupled eigenstates. Such process mainly occurs in insulating compounds with localized f states. [1] However, in actinides the spin-orbit (SO) coupling is stronger than the Hund's coupling. [2] Therefore, total angular momentum J = L + S is the good quantum number for this state. In such systems, a SO density wave order in the two-particle channel can arise at some critical value of the coupling constant g, by taking advantage of any instability, such as FS 'nesting' (shown in the main text), even when the time-reversal symmetry remains invariant. The coupling g can be related to some form of 'screened' interorbital Coulomb term. Physically, SO density wave is different from a spin density wave because here spin-flip occurs between two different orbitals without breaking time-reversal symmetry. Therefore, we desire to study a Hamiltonian: where H 0 is the non-interacting part and H SODW is the SO density wave term. In the 'pseudospin' basis introduced in the main textΨ , the representation of the symmetry operations for URu 2 Si 2 system which belong to the D 4h symmetry is: timereversal symmetry T R = K · iσ y ⊗ τ 0 , inversion symmetry I = τ z ⊗ τ 0 , four-fold rotation symmetry around the z axis C 4 = exp [i(π/4)σ z ⊗ τ 0 ] and the two reflection symmetries P x/y = iσ x/y which map x → −x (where x is in Γ-X direction) and y → −y (where y is orthogonal to Γ-X direction), respectively. Here, K is complex conjugation operator, and σ x,y,z and τ x,y,z depict the two-dimensional Pauli matrices in the 'pseudospin' and orbital space, respectively where τ 0 is the unitary matrix.
Each symmetry operation transforms the time-reversal invariant f -electron field as: These symmetry operations imply that the spin-orbit coupled actinide f -state is an odd-parity wavefunction. Using these symmetry properties, the non-interacting Hamiltonian is deduced in the main text, using standard procedure, see for example Refs. 3, 4.

I. HIDDEN ORDER PARAMETER
Considering the non-interacting FS nesting at Q h = (1 ± 0.4, 0, 0) for the hidden order state as demonstrated in the main text, we expand the above-mentioned 'pseudospinor' in the Nambu notation asΨ(k) = Ψ (k),Ψ(k + Q h ) . In this notation, the modulated SOC interaction term can be written in general as where µ, ν ∈ {0, x, y, z}. The symbol :: represents normal ordering. Here g is the coupling constant discussed latter and Γ µν = τ µ ⊗ σ ν , τ and σ represent Pauli matrices in orbital and spin basis, respectively. Absorbing g and Γ into one term we define the mean-field 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 z-directions (ν = z). Therefore, we drop the index ν henceforth. Furthermore we define the gap vector as where momentum dependence of the gap function transforms according to the same irreducible representation of the pointgroup symmetry (g is absorbed in the gap function defined below). With these substitutions, we obtain the final result for the order parameter as For the unidirectional modulation vector Q h , all components of M µ break C 4 rotational symmetry. All the symmetry properties of the order parameters are given in Table I. M 0 , M x and M z break time-reversal and thus are ruled out as the hidden-order state is arguably does not exhibit any timereversal symmetry breaking. [5][6][7]. Although some evidences 1. a, Black and blue symbols are ab-initio FSs, as shown in Fig. 1 of main text, but plotted here only two relevant bands. Two large square FSs for band 1 are shifted along the x−direction by nesting vectors Q11 = Q2 and Q12, while the two small square FSs for band 2 are shifted along −x direction. Good nestings for all three vectors are observed, but the intraband one are protected by strong SOC, where Q12 = Q h is the hidden order nesting that opens a gap. b, Phase diagram as a function of intra-orbital (g0ij = g012 = g021) and inter-orbital coupling constant (g0ii = g011 = g022) and nearest neighbor hopping (t) with respect to the Fermi energy (EF ) for all thee nesting instabilities. We find that for smaller inter-orbital coupling than the intra-orbital one, the hidden order parameter arises if EF /t ration is large. With large t (proportional to pressure), the LMAF phase at the commensurate nesting wins. This result is consistent, at least, in qualitative level, with the phase diagram of URu2Si2. [8] for the time-reversal symmetry breaking are also present, but it is not well established if the measurements are done in single crystal where the time-reversal symmetry breaking LMAF state and time-reversal symmetry invariant HO state are not mixed. In microscopic sense, LMAF and HO state are separated by phase transition, and thus both phases cannot inherit same broken symmetry. On the other hand, M 2 is even under time-reversal but odd under parity, and thus adds a mass term to the Hamiltonian which opens a gap at the nested portion of the Fermi surface. This is the term that represents modulated spin-orbit coupling. A trivial check can be performed by explicitly writing down the the spin-orbit coupling 3 2 ,σ ). With this definition, the total Hamiltonian in the reduced Brillouin zone can be written as Where h is the non-interacting part of the Hamiltonian defined in Eq. 2 of the main text. The prime over a summation implies that the summation is performed in the reduced Brillouin zone. Diagonalizing the above Hamiltonian, we deduce the quasiparticle states are: E τ σ k,ν = ξ τ σ k+ + νE τ σ k,0 , and from Eq. 3 of the main text. ν, ν = ±. The corresponding coherence factors are

II. SELF-CONSISTENT GAP EQUATION
We can easily derive the self-consistent gap equation from Eq. 9 using Bogolyubov treatment. We substitute the fermion operator f in terms of a Bogolyubov operators as From Eqs. 7, 8, and 9, we get the gap function as Here we have substituted γ † k,τ σ γ k,τ σ = n ν f,τ σ (k), where n is the Fermi function. We find that g 12 = 27 meV gives ∆ =5 meV in consistent with experiments. For this temperature independent value of g 12 , we obtain T h = 22K, which is higher than the experimental value of T h = 17.5 meV. Possible reason for overestimating the value of T h are the neglect of quantum fluctuation, disorder which can reduce its value. In this context, it can be noted that recently, a 'pseudogap' phase upto 20 K is marked from experimental features, [17] which arguably suggests that there are indeed fluctuations present above T h .

III. SPIN-ORBIT CORRELATION FUNCTION
The non-interacting single-particle Green's function for the Hamiltonian given in Eq. 1 above is defined as Here T is the imaginary time, f kτ σ (T ) is the imaginary time evolution of the fermionic operator f kτ σ , n is the fermionic Matsubara frequency, and β = 1/k B T , where k B is Boltzmann constant. T T gives normal time-ordering. The anomalous part of the Green's function is By Fourier transforming in the Matsubara frequency space, we obtain the explicit forms of two Green's functions as The orbital overlap matrix-element R ν τ τ σσ (k) = ψ ν τ σ (k)ψ ν † τ σ (k), where ψ is the eigenvector of the noninteracting Hamiltonian, projects the Green's function from band basis to the orbital one. However, to simplify our calculation, we assume that each block state E τ σ corresponds to each orbital which makes R = 1. By taking analytical continuation of the Matsubara frequency to the real frequency in the above Green's function, it is easy to show that the gap function ∆ can be obtained by averaging the anomalous Green's function.
The general form of the polarization vector is (18) To simplify the notations, we define composite indices α, β = τ τ σσ , in which the correlation function of J z α vector can now be defined as Substituting J z α and then applying standard Wick's decomposition to the electron bracket [9] yields, Here S αβ (k, q) = R α (k)R β (k + q) is the matrix-element term which projects the susceptibility from the band representation to the orbital one. χ νν are the fermionic oscillator terms in the band space ν, ν = ±: The momentum, energy, α indices on both sides of the above equation are implicit.

A. Hidden order instability
The divergence in the real-part of the χ αβ (q, ω = 0) indicates an instability due to the FS nestings between two orbitals α and β (in the explicit notation χ 11 means χ 1111 , etc). The two intraband nestings at Q 11 and Q 22 can give rise to spin-density wave or antiferromagnetism, if the time-reversal symmetry is broken. We believe that Q 11 is responsible for the LMAF phase. Our present interest is the interband one at Q 12 , shown in supplementary Fig. 1(A) which happens at (1 ± 0.4, 0, 0). Since in this case, both orbital and spin flip together, the time-reversal symmetry remains intact.
The corresponding critical value of the coupling constants g αβ at which a gap opening or an order parameter develops can be evaluated within random-phase approximation (RPA). The stoner criterion for an instable state implies that 1−χ αβ (Q αβ , 0)g αβ ≥ 0, or g αβ ≤ 1/χ αβ (Q αβ , 0). Looking at the FS areas for each band, shown in supplementary Fig. 1a, we immediately see that 1/χ 11 (Q 11 , 0) < 1/χ 22 (Q 22 , 0) < 1/χ 12 (Q 12 , 0), leading to a phase diagram shown in supplementary Fig. 1(B). The present calculation does not incorporate the possible coexistence state between different phases. The phase diagram implies that there is a considerably large parameter space, where the Q 12 = Q h nesting dominates. We have not considered all possible phases except M y for Q 12 nestings discussed in Eq. 9 above, because constrained by the symmetry arguments given in Table I, others render gapless state.

B. Neutron mode at the Hidden order state
Eqs. 22 and 23, imply that the hidden order transition accompanies an inelastic neutron scattering mode with en- FIG. 2. The real and imaginary part of the bare susceptibility calculated from Eqs. 22-23 in the hidden order state. The red line gives the INS mode obtained using RPA calculation at g12 = 28.4 meV whose inverse is shown by horizontal dashed line. The RPA value has a huge peak (resonance) at ωQ =4.7 meV when χ (Q h , ωQ) = 1/g12, and we have rescaled its intensity arbitrarily to fit it into the same figure with the bare values for comparison. The obtained resonance is in good agreement with experiment shown by symbols. [11] hanced intensity at Q h whose energy scale is approximately given by . C is the prefactor which has to be evaluated rigorously, but it does not contribute to the peak position in bare χ . Here we have substituted the condition that non-interacting bands are nested on the FS at Q h such as E ν k = |∆(k)| ≈5 meV. The RPA correction shifts the resonance energy which depends on the value of coupling constant. At g 12 = 28.4 meV, the 1/g 12 line cuts twice to χ (Q h , ω), however, the resonance is stronger at the energy where the broadening function coming from χ is weaker. Therefore, the strong intensity or a resonance occurs at ω Q ∼ 4.7 meV, see Fig. 2. INS measurements have observed this resonance at the incommensurate nesting vector Q = (1.4, 0, 0). [11][12][13] However, one have to be careful to directly compare our result with this data. Because, in the present case, we expect a mode which does not break timereversal symmetry. Therefore, in order to observe this mode, one requires to perform a polarized INS measurement.
On the same reasoning, INS should see more resonances at Q 11 and Q 22 even in the non-polarized condition. However, the energy scale and intensity of those modes will depend on the location of the phase diagram of URu 2 Si 2 where the ex-periment is performed. As we argued earlier, Q 11 is most likely responsible for the LMAF phase, therefore, we can expect a mode at twice of the corresponding gap opening. The peak in the INS spectra at the commensurate vector is also observed at a much lower in energy, however, the peak is much broader than the resonance peak observed at the incommensurate vector. [11,13] 18. A QCP at T = 0 along the field directions is expected from our theory, and also observed in experiment. [18] Deducing the phase diagram for SC and other possible phases that may arise above the QCP is beyond the scope of the present calculation.

IV. QUANTUM CRITICAL POINT
As mentioned in the main text, any T R breaking perturbation such as magnetic field will destroy the T R invariant HO phase. As the HO state incipiently is a spontaneously broken symmetry, it will exhibit a second order phase transition along the field axis. In what follows we expect to obtain a quantum critical point (QCP) at T = 0, as extensively proposed to be associated with any second order phase transition, [19] along the magnetic field axis. We find the critical field to be B ≈38 T, whereas the experimental data [18] indicates that the QCP resides around B ∼ 34 T. The present model can not deduce the phase diagram for the superconducting (SC) state, possibly intervening the HO state, or any other phases that may arise above the QCP. [18] More experimental data and theoretical modelling are required to understand the details of this regime of the phase diagram.