Identifying materials with charge–spin physics using charge–spin susceptibility computed from first principles

The conventional paradigm of condensed matter physics involves the partitioning of electronic ground states into descriptions of spin and electronic degrees of freedom, potentially with small coupling terms between them. Many materials fall under this paradigm, such as antiferromagnetic insulators, ferromagnetic metals, and non-magnetic insulators. However, this weak coupling paradigm is insufficient to describe some materials, which often show unconventional ground states and excitations. Examples of unconventional behavior include the unconventional superconductors,^1,2 unconventional magnetic states,^3–5 spin liquids,⁶ and strong magnetodielectric effects.⁷ It is still an open question how to predict a priori whether a material breaks the weakly coupled spin/electron paradigm, even if such coupling has been previously studied, mostly in the context of model Hamiltonians.^1,8–12

One of the most prominent examples of charge–spin interactions are the high-temperature unconventional superconductors such as copper oxides and iron-based pnictides and chalcogenides. As pointed out by Scalapino,¹ there is vast experimental evidence such as the proximity between magnetic ordered phases and superconductivity, suggesting that in these materials, the coupling between orbital (charge) and magnetic (spins) degrees of freedom plays a crucial role in determining the physics.

Despite recent progress in the study of model Hamiltonians commonly associated with materials showing unconventional properties,^13–18 the study of real materials with such properties remains challenging. There have been several attempts to computationally predict whether specific groups of materials show unconventional electronic phases, for example, unconventional superconductivity.^19–21 However, to our knowledge, most of these similarity-based searches have attained limited success. In this manuscript, we adopt a somewhat different approach: we concentrate on exploring a new computational probe of charge–spin coupling in materials. We present and test the charge–spin susceptibility, which is computed from first principles in a simple way. This quantity measures the response of a material’s charge density to changes in its spin density.

We use density functional theory (DFT) and quantum Monte Carlo (QMC) calculations to estimate the charge–spin coupling in a set of 28 layered strongly correlated transition metal compounds including unconventional behavior such as unconventional superconductivity in the cuprates and iron-based superconductors,^2,22 disordered magnetic ground states, and bad metallic behavior in materials such as BaCo₂As₂ and Sr₂VO₄.^4,5,23 In order to assess the quality of the DFT results, for a small set of materials, we compare the DFT-derived charge–spin susceptibility predictions with those obtained from fixed-node diffusion Monte Carlo. We find that while DFT+U predicts a different charge–spin response from the QMC results, it is sufficient to distinguish large responses from small responses. Using DFT+U calculations on the entire set of materials, we find that materials with a large charge–spin susceptibility often present unconventional phases, while materials in the same class with smaller charge–spin susceptibility do not.

In this work, we concentrate on investigating a set of 28 magnetic layered materials containing transition metal atoms with magnetic moments arranged in diverse two dimensional structural motifs. Some of these materials are well known to present unconventional phases of matter as described in Sec. I. In Table I, we list the materials in our test set along with the low-temperature magnetic and electronic phases they show upon chemical doping or pressure.

We shall start by clarifying the relation between the charge–spin susceptibility and the coupling between charge and spin degrees of freedom in model effective Hamiltonians. While our methodology does not depend on any particular effective Hamiltonian being applicable, we will illustrate this relation using a toy model. Consider the effective Hamiltonian,

where H_o describes the orbital degrees of freedom, while H_S describes spin states. The term H_oS accounts for interactions between the latter two sets of degrees of freedom, which are controlled by the coupling λ.

In order to compute the relation between the charge–spin coupling λ and the charge–spin susceptibility in a system governed by Eq. (1), consider a small deformation of the electronic wave function away from the ground state. Assume that this deformation amounts to a change in the ground state’s magnetic order, which results in a change in the system’s spin density, Δs_i(r) ≡ s_i(r) − s₀(r), where s₀(r) stands for the ground state spin density, while s_i(r) stands for the new/deformed state’s spin density. In such a case, one can show (see the  Appendix) that to first order in the deformation, the resulting change in the charge density, Δρ_i(r) ≡ ρ_i(r) − ρ₀(r) (with ρ₀ and ρ_i standing for the ground state and deformed state charge density), is proportional to the change in the spin density,

In the above expression, λ is the coupling constant connecting the orbital and the spin levels, while w is the energy scale associated with the orbital degrees of freedom, and X_i(r) is a numerical factor related to the type of spin deformation. Thus, the ratio Δρ_i(r)/Δs_i(r) gives direct access to the magnitude of λ/w.

A simple way of estimating the magnitude of the coupling λ/w is to compute the average charge–spin susceptibility, χ_cs, defined as⁸³ 

where N stands for the number of different magnetic orders considered for each material (see online data⁸⁴). χ_i stands for the pairwise charge–spin susceptibility of each magnetic order with respect to the ground state. This is defined as

where Δρ_i (Δs_i) stands for the spatial fluctuations in charge (spin) density relative to the lowest-energy magnetic state. The former are given by

where ρ₀(r) and s₀(r) are the charge and spin distributions of the lowest-energy magnetic state.

We calculate χ_cs as defined in Eqs. (3)–(5), by generating several low-energy magnetic textures for each material. As represented in Fig. 1, in order to obtain a new magnetic order, we optimize initial magnetic textures that differ from the ground state’s one by a few flipped transition metal atoms’ magnetic moments. Then, we compute the charge density and spin density differences between the ground and the new state obtaining Δρ_i and Δs_i from Eq. (5). With several different magnetic orders, we compute the average charge–spin susceptibility χ_cs from Eqs. (3) and (4).

Since our objective is to screen a large set of materials against the charge–spin susceptibility, we decided to base our search protocol on a low-cost but sufficiently accurate computational method. With that in mind, we chose Kohn–Sham density functional theory (KS–DFT).⁸⁵ Most of the calculations presented in this work were performed using the KS–DFT approach,⁸⁵ as implemented in the QUANTUM ESPRESSO code.⁸⁶ The exchange–correlation energy was approximated by the generalized gradient approximation (GGA) using the Perdew–Burke–Ernzerhof (PBE) functional.⁸⁷ To improve the description of the d orbitals, we used the DFT+U scheme of Cococcioni and de Gironcoli.⁸⁸ Interactions between valence and core electrons were described by pseudopotentials in the accurate set of the Standard Solid-State Pseudopotentials library.^89,90 The Kohn–Sham orbitals were expanded in a plane-wave basis with a cutoff energy E_c (Ry), while a cutoff of 4E_c was used for the charge density (see online data for E_c of each material⁸⁴). The E_c of a given compound was chosen to be the largest E_c among those of its constituent chemical elements. The E_c of an atomic species was estimated from checking for convergence of the single-atom’s total energy against E_c. Convergence was assumed when total energy changed less than 0.01 Ry upon an increase of E_c by 10 Ry. The Brillouin zone (BZ) was sampled using a Γ-centered 6 × 6 × 6 grid following the scheme proposed by Monkhorst–Pack.⁹¹ Total energy convergence against the BZ grid density was tested by doing 7 × 7 × 7 grid calculations for unpolarized and ferromagnetic textures. The crystal structure for each material was set up with the information available on the ICSD database92—see online data⁸⁴ for the CIF(s) used in the calculations of each material. A supercell was used whenever the material unit cell had less than four transition metal atoms per unit cell. This ensures that we can generate sufficient magnetic textures to properly estimate the charge–spin susceptibility.

For each material, we performed multiple DFT+U calculations (with U = 0 eV, 5 eV, 10 eV) in order to assess the uncertainty in the charge–spin susceptibility estimate. With the aim of converging different magnetic orders, we performed calculations in which the self-consistent cycle started from different magnetic states (see Fig. 1), i.e., different orderings and magnitudes for the magnetic moments on the transition metal atoms. In Ref. 84, the reader can find data specifying all the DFT+U calculations that were performed, including material name, crystallographic identifier (CIF), Hubbard U, E_c cutoff, supercell size, k-point mesh, starting magnetic state, final magnetic state, band gap estimate, and total energy.

To check the accuracy of the results obtained from the DFT+U calculations, in Subsection III B, we compare charge–spin susceptibilities obtained from DFT+U with those obtained from the PBE and PBE0 hybrid functionals⁹³ (as implemented in the CRYSTAL17 code⁹⁴), as well as those obtained from the highly accurate fixed-node diffusion Monte Carlo (DMC)⁹⁵ (as implemented on the quantum Monte Carlo package QWALK⁹⁶).

Fixed-node diffusion Monte Carlo (DMC) is a fully first-principles stochastic framework to solve the Schrödinger equation, which yields a variational upper bound to the ground state.⁹⁵ We employed a Slater–Jastrow trial wavefunction, as implemented in the QWalk package.⁹⁶ We constructed the Slater determinant with orbitals from DFT calculations using the Crystal code⁹⁴ employing the PBE0 functional.⁹³ Previous studies have shown that per comparison to other commonly used DFT functionals, PBE0 typically gives the best wave function nodes.^97–99 The Brillouin zone (BZ) was also sampled using a Gamma-centered 6 × 6 × 6 Monkhorst–Pack grid.⁹¹ We used Dirac–Fock pseudopotentials and ECPs specially constructed for quantum Monte Carlo computations.^100,101 We controlled finite-size errors by using 2 × 1 × 1 supercells and averaging over the sampled k-points. We used a time step of 0.005 Ha⁻¹. This setup has been shown to give a good description of challenging materials such as the cuprates¹⁰² and FeSe.⁹⁹ 

For each material in Subsection III B, we considered several magnetic textures. Some of these have zero total spin projection S_z = 0 (i.e., antiferromagnetic-like orders), while others have S_z ≠ 0 (ferromagnetic or flip orders). The datasets provided online⁸⁴ identify the magnetic textures considered for each material. There we have also included figures representing those magnetic textures.⁸⁴ 

In Fig. 2, we plot the change in charge and spin density between the lowest and second-lowest energy magnetic textures for K₂NiF₄, SrCuO₂, FeSe, and BaCo₂As₂. Stoichiometric K₂NiF₄ is a typical Mott insulator showing Néel order.⁸¹ SrCuO₂ is a Néel ordered magnetic insulator, while FeSe is a Hund’s metal without long-range magnetic order at atmospheric pressure. SrCuO₂ and FeSe are representative of the families of cuprate and iron-based high-temperature unconventional superconductors, well known to support a wide range of uncommon phases, ranging from strange metallic behavior to non-trivial magnetic states and high-temperature unconventional superconductivity.^2,44,45 BaCo₂As₂ is a disordered magnetic metal.⁴ In the bottom row of Fig. 2, we show the pairwise charge–spin susceptibility χ_i [see Eqs. (4) and (5)] resulting from those magnetic textures.

In the leftmost column of Fig. 2, we see that the charge density of K₂NiF₄ is just slightly rearranged when the magnetic texture is modified. This weak charge density response to changes in magnetic order persists for other magnetic textures of K₂NiF₄, which indicates that charge and spin degrees of freedom are weakly coupled in this material.

In the remaining columns of Fig. 2, we can see that the charge density response in SrCuO₂, FeSe, and BaCo₂As₂ is much stronger than that in K₂NiF₄ whose iso-charge density surfaces were fivefold magnified with respect to those of the other materials. However, the charge–spin response in SrCuO₂, FeSe, and BaCo₂As₂ is rather different. Both the way in which charge rearranges and the magnitude of that rearrangement vary across these materials, as can be inferred from the shape and size of the isodensity-difference surfaces in Fig. 2. For instance, changing the magnetic order from Néel (ground state) to bicolinear order in SrCuO₂ mostly results in electrons moving from oxygen 2p_x (2p_y) orbitals into copper $3dx2−y2$ ones. In FeSe, changing from the stripe (ground state) to the Néel order seems to largely transfer electrons from iron’s 3d_xz, 3d_yz, and 3d_xy orbitals into its $3dz2−r2$⁠. Similarly, the change from bicollinear to collinear magnetic order in BaCo₂As₂ mostly redistributes electrons among the 3d orbitals of cobalt.

The spin density differences for these four materials (see second row of Fig. 2) have similar magnitudes even if they are qualitatively different. This thus suggests that the charge–spin response in K₂NiF₄ should be much weaker than that in SrCuO₂ and FeSe, which, in turn, seem to have a somewhat weaker response than BaCo₂As₂. Those observations are corroborated by the values of the pairwise charge–spin susceptibility χ_i calculated for these magnetic textures and shown in the third row of Fig. 2.

We now compare the charge–spin susceptibility computed from a few different methods: PBE+U^87,88 with U = 0 eV, 5 eV, 10 eV (with the plane-wave code quantum espresso⁸⁶); PBE0⁸⁷ (using the localized basis code CRYSTAL17⁹⁴); and fixed-node diffusion Monte Carlo⁹⁵ (using the quantum Monte Carlo package QWalk⁹⁶). Due to the high computational cost of the DMC calculations, we performed this comparison for a set of four barium arsenides with a ThCr₂Si₂-like structure: BaM₂As₂ with M = Cr, Mn, Fe, and Mn. In Fig. 3, we compare the pairwise charge–spin susceptibility χ_i obtained with diffusion Monte Carlo (x-axis) with the χ_i obtained from density functional theory (y axis). Each panel in Fig. 3 makes this comparison for DFT calculations done with each of the four functionals mentioned above: PBE+U = 0, 5, 10, and PBE0.

In this figure, the PBE+U derived pairwise charge–spin susceptibilities generally follow the trends of the more expensive PBE0 and DMC results. The ordering of these four materials according to their values of pairwise charge–spin susceptibility χ_i resulting from DFT calculations with the PBE0 and DFT+U = 5 eV, 10 eV functionals is the same as that resulting from DMC: χ_Mn ≲ χ_Cr ≲ χ_Fe ≲ χ_Co.

The deviation between the DFT-derived pairwise charge–spin susceptibilities and those calculated from DMC is shown in the table of Fig. 3. PBE0 deviates the least from DMC, with a root mean square deviation of RMSD_PBE0 = 0.04 considerably smaller than those resulting from the DFT+U calculations. Part of the discrepancy between the PBE+U and PBE0/DMC χ_i’s arises from the fact that some magnetic textures obtained with PBE+U are often quantitatively different from those obtained with PBE0/DMC. Among the PBE+U functionals, U = 5 eV is the one that better captures the response of charge to changes in the spin texture of these materials. The PBE+U = 5 eV χ_i root mean square deviation from DMC is RMSD_U=5 = 0.11, while that of U = 0 eV and U = 10 eV is slightly larger: RMSD_U=0 = 0.17 and RMSD_U=10 = 0.14.

Even if the quantitative agreement between the DFT+U methods and DMC is not perfect, these still capture the qualitative trends in charge–spin susceptibility, enabling their use in charge–spin susceptibility screenings of large sets of materials. In what follows, we will show the results for DFT+U with U = 5 eV since this functional minimizes deviations from the DMC results—see table of Fig. 3.

In some materials, the charge–spin response is strongly dependent on the type of change in the magnetic texture. That can be seen in Fig. 3 where the pairwise charge–spin susceptibilities show a large spread for BaFe₂As₂ but a small one for BaCr₂As₂. This is reminiscent of electron–phonon coupling physics, in which some phonons are more strongly coupled to the material’s electronic degrees of freedom than others. For simplicity, we take the average [see Eq. (3)] but have checked that different strategies do not affect the results in Sec. III C.

In Fig. 4, we show the charge–spin susceptibility for the materials in the entire test set from Table I obtained using DFT+U with U = 5 eV. Each material is colored according to its family: copper oxides, barium arsenides, MPX₃’s, iron chalcogenides, transition metal dichalcogenides, and 214 materials. Overall, we find that materials showing large charge–spin susceptibility in Fig. 4, typically present unconventional properties either in their stoichiometric form, or when put under pressure or chemically doped. This is shown in Fig. 5 where we color materials according to whether their pressure vs doping phase diagram shows unconventional ground states. Both these families are well separated: χ_conv = 0.08 ± 0.10 for conventional materials, while χ_unc = 0.28 ± 0.12 for unconventional ones. Using a two-sided t-test, the populations have different means with a p-value of 3 × 10⁻⁵. In this section, we discuss in some detail the materials in our test set showing large charge–spin susceptibility.

The results in Fig. 4 show four copper oxides with sizable values of charge–spin susceptibility: T-La₂CuO₄, CaCuO₂, and SrCuO₂ all have χ_cs ≈ 0.15, while T′-La₂CuO₄ shows χ_cs ≈ 0.40. All these four materials are Néel ordered^2,27,50,52 insulators and well known to become unconventional superconductors under chemical doping.^2,27,50,53 Several other uncommon phases have been observed in these materials, ranging from a pseudogap phase, to strange metallicity, short-range magnetic and charge order.^2,27,50,52

Two other members of the copper oxide family, TeCuO₃ and SeCuO₃, show very small charge–spin susceptibility. TeCuO₃ is a Néel ordered insulator, while SeCuO₃ is a ferromagnetic insulator,⁷ both well known to show magnetodielectric properties.⁷ As opposed to the four cuprates with copper-oxide planes discussed above, unconventional phases have not been observed in either TeCuO₃ or SeCuO₃, consistent with the small value of susceptibility computed here.

The iron pnictides and chalcogenides o-FeSe, t-FeSe, FeS, FeTe, t-BaFe₂As₂, and o-BaFe₂As₂ all show χ_cs ≈ 0.30–0.37. These materials are all metallic magnets, some showing stripe magnetic textures (BaFe₂As₂ and FeTe^34,39) ,while others only present short-range order (FeSe and FeS^42,44). All these materials have unconventional superconducting phases induced by pressure or doping,^36,40,41,45 as well as short-range magnetic order and bad metal phases.^{33,35,39,40,44}

Sr₂FeO₄ has χ_cs = 0.22 and is an antiferromagnetic semiconductor. Chemically doping this compound weakens both its antiferromagnetic ordering and semiconducting character without completely suppressing the electronic gap.^103–105 It has been shown that pressure induces a semiconductor-to-metal transition at P ≈ 18 GPa,⁴⁷ but so far no unconventional phases have been observed on its phase diagram down to 5 K for pressures up to 30 GPa.

Sr₂CoO₄ is ferromagnetic and metallic at low temperatures.²⁸ Upon chemical doping with Y,²⁸ La,²⁹ and Nd,³⁰ the ferromagnetism weakens and semiconducting behavior arises. This material becomes a superconductor at around 5 K when doped with H₂O,³¹ with very similar properties to the cuprate superconductors. We were not aware of this result prior to the study; the material was identified purely due to the charge–spin descriptor, χ_cs = 0.39.

Sr₂VO₄ is a multi-orbital Mott insulator with no long-range magnetic order⁵ that can be driven into a metallic state by hydrostatic pressure (≈20 GPa–24 GPa).²³ An unconventional metal emerges at low temperatures in the vicinity of the pressure-driven transition.²³ In contrast to Sr₂VO₄ thin films,¹⁰⁶ attempts to chemically dope the bulk crystal did not succeed in making it metallic.¹⁰⁷ Since our calculations suggest it has a strong charge–spin coupling, χ_cs = 0.42, a more comprehensive exploration of different ways of chemically doping this material may reveal novel phases.

BaCo₂As₂ has a rather large charge–spin susceptibility, χ_cs = 0.45. It is a disordered magnetic metal⁴ that seems to remain so upon both chemical doping with K⁴ and hydrostatic pressure (up to 8 GPa).¹⁰⁸ It thus seems similar to Sr₂VO₄ in the sense that there is a disordered magnetic state. We note that these two systems are not ordinary magnetic materials, and both exhibit nonstandard ground states. Thus, the charge–spin descriptor succeeded in identifying unusual physics in these materials.

K₂CoF₄ has been classified as a 2D Ising magnet⁵⁵ owing to its strongly anisotropic magnetic interactions. To our knowledge, this material’s behavior under pressure or chemical doping has been very sparsely studied,¹⁰⁹ potentially due to the presence of fluorine. Its charge–spin susceptibility, χ_cs = 0.10, is slightly below that of some superconducting cuprates.

TaS₂ is a Mott insulator associated with a charge-density wave (CDW) phase.⁵⁹ Pressure induces a metal-to-insulator transition and a superconducting state below 5 K.⁵⁹ Recent experiments⁵⁸ suggest that the low-temperature CDW phase has short-range magnetic order, which supported proposals that this material might realize a quantum spin liquid state.¹¹⁰ Our calculations, done with the undistorted crystal structure, give χ_cs = 0.09 just below what we get for some cuprates.

The compounds with lower charge–spin susceptibility in Fig. 5 (listed on the bottom of Table I) comprise materials that show conventional phases, mostly insulators with Néel antiferromagnetic order and ferromagnetic order. Some of these materials are known to become metallic (e.g., BaMn₂As₂) or to acquire spin and/or charge stripe order (e.g., La₂NiO₄ and La₂CoO₄) upon doping or pressure, but none of them presents unconventional electronic phases. This indicates that, as suggested by our calculations, their charge and spin degrees of freedom are weakly coupled.

We have presented a new way of probing charge–spin coupling in materials. It is based on the charge–spin susceptibility, a quantity that estimates the magnitude of the coupling between charge and spin degrees of freedom in a material. This quantity is straightforward to compute in a high-throughput workflow, and when applied to a collection of layered materials containing transition metal atoms suggests that materials with high charge–spin coupling exhibit unconventional phenomena.

All the materials in our test set known to present unconventional phases (green colored in Fig. 5) show average charge–spin susceptibilities χ_cs ≥ 0.09. Among the 16 materials with charge–spin susceptibility above this value, only three have not been observed to show unconventional physics, Sr₂CoO₄, Sr₂FeO₄, and K₂CoF₄, which is a rather small rate of false positive identifications. Perhaps more importantly, the false negative rate was zero within our test set. This rate is certainly good enough to motivate experimental investigation into materials.

The computation of the charge–spin susceptibility for the materials in our test set was performed using DFT+U, a low-cost method which is sufficiently accurate to highlight the same qualitative trends found using the highly accurate many-body method fixed-node diffusion Monte Carlo. These calculations are inexpensive enough that they can be used in large-scale probes of the strength of the coupling between electronic and magnetic degrees of freedom in materials. The trends found here suggest that the charge–spin susceptibility is a valuable quantity for computational searches for new unconventional ground states, including unconventional superconductivity similar to iron-based and cuprate superconductors.

The data that support the findings of this study are openly available in Materials Data Facility^111,112 at http://doi.org/10.18126/6oby-l2lp, Ref. 84.

This work was supported by the Center for Emergent Superconductivity, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DEAC0298CH1088. The authors thank Daniel Shoemaker for many illuminating discussions. The computational resources used in this work were provided by the University of Illinois Campus Cluster and the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (Award Nos. OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Superconducting Applications. L.K.W. was supported by a grant from the Simons Foundation as part of the Simons Collaboration on the many-electron problem.

In the context of a system governed by the Hamiltonian in Eq. (1), assume that we fix the portion of the wave function associated with the spin degrees of freedom, ϕ(r), to a particular magnetic order. Then, the orbital degrees of freedom will be described by φ(r; ϕ). The charge and the spin density of such configuration will be given by

where σ = ± identifies the spin projection, while φ_σ is a short-hand for φ_σ(r; ϕ).

Let us apply an infinitesimal deformation away from the ground state on the portion of the wave function describing the spin degrees of freedom, δϕ_σ(r) = ϕ_σ(r) − ϕ_0σ(r) (where ϕ_σ is the deformed wave function and ϕ_0σ is the ground state one). We can then write the differential of each component of the charge and spin densities as follows:

where we used $δ|φσ|=∑αδ|φσ|δ|ϕα|δ|ϕα|$ under the assumption that the wave function’s orbital degrees of freedom component only depend on the absolute value of the spin states component, φ_σ(r; ϕ_σ) ≃ φ_σ(r; |ϕ_σ|).

We can write the differential of the charge and spin densities as

where δ_σα is the Kronecker delta.

Consider now that the small deformation is such that it only changes the spin states’ magnetic order, preserving their contribution to the charge density, i.e., δ∑_σ|ϕ_σ|² ≃ 0. This implies that ∑_σ2|ϕ_σ|δ|ϕ_σ| ≃ 0, which allows us to write

Under this approximation, we can write δρ and δs as

where Υ_α ≡ ∑_σ|φ_σ|f_σα and $Ξα≡λwϒα+|ϕα|$⁠. In these expressions, we used the definition $δ|φσ|δ|ϕα|≡λwfσα$⁠, where according to the main text’s notation, λ stands for the coupling between the spin states and the orbital degrees of freedom, while w corresponds to a material-specific energy scale.

Using the above expressions, we can write δρ in terms of δs as

which using Eq. (A4) can be simplified into

where X(r) = Υ₊(r)/|ϕ₊(r)|.

Equation (A7) has the same form of main text’s Eq. (2), only that in the latter we use a perturbation theory notation: Δs_i(r) = s_i(r) − s₀(r) [instead of δs(r)] and Δρ_i(r) = ρ_i(r) − ρ₀(r) [instead of δρ(r)], where ρ_i and s_i stand for the charge and spin densities of the i deformation away from the ground state’s charge and spin densities, ρ₀(r) and s₀(r).