U(1)$\times$SU(2) Gauge Invariance Made Simple for Density Functional Approximations

TL;DR

This paper introduces a method to generate gauge-invariant density functional approximations for spin-orbit coupled systems by applying a minimal substitution to existing functionals, enabling more accurate and versatile computational modeling.

Contribution

It proposes a simple minimal substitution technique to create gauge-invariant functionals from existing density functionals, facilitating practical applications in complex magnetic and topological systems.

Findings

Gauge-invariant functionals can be generated from existing ones using minimal substitution.

The approach simplifies the development of non-empirical functionals for inhomogeneous systems.

Potential for wide applicability in atomic, molecular, and condensed matter physics.

Abstract

A semi-relativistic density-functional theory that includes spin-orbit couplings and Zeeman fields on equal footing with the electromagnetic potentials, is an appealing framework to develop a unified first-principles computational approach for non-collinear magnetism, spintronics, orbitronics, and topological states. The basic variables of this theory include the paramagnetic current and the spin-current density, besides the particle and the spin density, and the corresponding exchange-correlation (xc) energy functional is invariant under local U(1)$\times$SU(2) gauge transformations. The xc-energy functional must be approximated to enable practical applications, but, contrary to the case of the standard density functional theory, finding simple approximations suited to deal with realistic atomistic inhomogeneities has been a long-standing challenge. Here, we propose a way out of this impasse by showing that approximate gauge-invariant functionals can be easily generated from existing approximate functionals of ordinary density-functional theory by applying a simple {\it minimal substitution} on the kinetic energy density, which controls the short-range behavior of the exchange hole. Our proposal opens the way to the construction of approximate, yet non-empirical functionals, which do not assume weak inhomogeneity and should therefore have a wide range of applicability in atomic, molecular and condensed matter physics.

Equations

$$\tau \to \tilde{\tau }=\tau -\frac{\mathbf{j}\cdot \mathbf{j}}{2n},$$

$$\hat{H}$$

$$\hat{W}=\frac{1}{2}\iint {\mathrm{d}}^3{r}_1{\mathrm{d}}^3{r}_2\frac{:{\hat{\Psi }}^{†}({\mathbf{r}}_1)\hat{\Psi }({\mathbf{r}}_1){\hat{\Psi }}^{†}({\mathbf{r}}_2)\hat{\Psi }({\mathbf{r}}_2):}{|{\mathbf{r}}_1-{\mathbf{r}}_2|},$$

$$\hat{n}(\mathbf{r})={\hat{\Psi }}^{†}(\mathbf{r})\hat{\Psi }(\mathbf{r})$$

$$\hat{s}(\mathbf{r})={\hat{\Psi }}^{†}(\mathbf{r}){\sigma }^a\hat{\Psi }(\mathbf{r})$$

$$\hat{H}$$

$$\hat{\tau }=\frac{1}{2}\left(\nabla {\hat{\Psi }}^{†}\right)\cdot \left(\nabla \hat{\Psi }\right)$$

$$\hat{\mathbf{j}}=\frac{1}{2i}\left[{\hat{\Psi }}^{†}\nabla \hat{\Psi }-\left(\nabla {\hat{\Psi }}^{†}\right)\hat{\Psi }\right]$$

$${\hat{\mathbf{J}}}^a=\frac{1}{2i}\left[{\hat{\Psi }}^{†}{\sigma }^a\nabla \hat{\Psi }-\left(\nabla {\hat{\Psi }}^{†}\right){\sigma }^a\hat{\Psi }\right]$$

$$\tilde{v}=v+\frac{1}{2c^2}\left[\mathbf{A}\cdot \mathbf{A}+\frac{{\mu }_B^2}{4}{\mathbf{A}}^a\cdot {\mathbf{A}}^a\right],$$

$${\tilde{B}}^a=B^a+\frac{1}{2c^2}\mathbf{A}\cdot {\mathbf{A}}^a.$$

$$E$$

$$+\frac{1}{c}\int d^{3}r\mathbf{j}(\mathbf{r})\cdot \mathbf{A}(\mathbf{r})+\frac{{\mu }_B}{2c}\int d^{3}r{\mathbf{J}}^a(\mathbf{r})\cdot {\mathbf{A}}^a(\mathbf{r})$$

$$\displaystyle+\int d^{3}r~{}{n}({\bf r})\tilde{v}({\bf r})+\mu_{B}\int d^{3}r~{}{s}^{a}({\bf r})\tilde{B}^{a}(\bf r)\Big{\}}$$

$$F[n,s^a,\mathbf{j},{\mathbf{J}}^a]=\underset{|{\Psi }^{\prime }⟩\to (n,s^a,\mathbf{j},{\mathbf{J}}^a)}{\min }⟨\Psi |\hat{T}+\hat{W}|\Psi ⟩$$

$$F[n,s^a,\mathbf{j},{\mathbf{J}}^a]$$

$$\left[\frac{1}{2}{\left(-i\nabla +\frac{1}{c}{\text{mathbfcal}A}_{\mathrm{KS}}\right)}^2+{\mathcal{V}}_{\mathrm{KS}}\right]{\Phi }_{\mu }={\epsilon }_k{\Phi }_k$$

$${\text{mathbfcal}A}_{\mathrm{KS}}=\left(\mathbf{A}+{\mathbf{A}}_{\mathrm{xc}}\right)+{\sigma }^a\left({\mathbf{A}}^a+{\mathbf{A}}_{\mathrm{xc}}^a\right),$$

$${\mathcal{V}}_{\mathrm{KS}}$$

$$+\frac{1}{2c^2}\left[{\left(\mathbf{A}+{\sigma }^a{\mathbf{A}}^a\right)}^2-{\text{mathbfcal}A}_{\mathrm{KS}}^2\right],$$

$$\hat{\Psi }(\mathbf{r})\to {\hat{\Psi }}^{\prime }(\mathbf{r})=\exp \left[\frac{i}{c}\chi (\mathbf{r})\right]\hat{\Psi }(\mathbf{r})$$

$$\hat{\Psi }(\mathbf{r})\to {\hat{\Psi }}^{\prime }(\mathbf{r})$$

$$E_{\mathrm{xc}}[n^{\prime },{s^{\prime }}^a,{\mathbf{j}}^{\prime },{{\mathbf{j}}^{\prime }}^a]=E_{\mathrm{xc}}[n,s^a,\mathbf{j},{\mathbf{J}}^a],$$

$$n\to n^{\prime }=n$$

$$s^a\to {s^{\prime }}^a=R^{ab}{s}^b$$

$$\mathbf{j}\to {\mathbf{j}}^{\prime }=\mathbf{j}+\frac{1}{c}n\nabla \chi -\frac{i}{2}{s}^a\mathrm{Tr}\left({\sigma }^a{U}_{\mathrm{S}}^{†}\nabla U_{\mathrm{S}}\right)$$

$${\mathbf{J}}^a\to {{\mathbf{J}}^{\prime }}^a=R^{ab}\left[{\mathbf{J}}^b+\frac{1}{c}{s}^b\nabla \chi -\frac{i}{2}n\mathrm{Tr}\left({\sigma }^b{U}_{\mathrm{S}}^{†}\nabla U_{\mathrm{S}}\right)\right].$$

$${\mathbf{A}}_{\mathrm{xc}}^{\prime }={\mathbf{A}}_{\mathrm{xc}}$$

$${{\mathbf{A}}^{\prime }}_{\mathrm{xc}}^a=R^{ab}{\mathbf{A}}_{\mathrm{xc}}^b$$

$${B^{\prime }}_{\mathrm{xc}}^a=R^{ab}\left[B_{\mathrm{xc}}^b+\frac{i}{2}{\mathbf{A}}_{\mathrm{xc}}\cdot \mathrm{Tr}\left({\sigma }_b{U}_{\mathrm{S}}^{†}\nabla U_{\mathrm{S}}\right)-{\mathbf{A}}_{\mathrm{xc}}^b\cdot \nabla \chi \right]$$

$$v_{\mathrm{xc}}^{\prime }=v_{\mathrm{xc}}-\frac{1}{c}{\mathbf{A}}_{\mathrm{xc}}\cdot \nabla \chi +\frac{i}{2}{\mathbf{A}}_{\mathrm{xc}}^a\cdot \mathrm{Tr}\left({\sigma }^a{U}_{\mathrm{S}}^{†}\nabla U_{\mathrm{S}}\right).$$

$${\partial }_{\mu }\left[n{A}_{\mathrm{xc},\mu }+\vec{s}\cdot {\vec{A}}_{\mathrm{xc},\mu }\right]=0$$

$$\frac{1}{2c}{\partial }_{\mu }\cdot \left[A_{\mathrm{xc},\mu }\vec{s}+n{\vec{A}}_{\mathrm{xc},\mu }\right]=\frac{1}{c}{\vec{A}}_{\mathrm{xc},\mu }\times {\vec{j}}_{\mu }+{\vec{B}}_{\mathrm{xc}}\times \vec{s}.$$

$$E_{\mathrm{x}}=-\frac{1}{2}\int d^{3}r\int d^3{r}^{\prime }\frac{\mathrm{Tr}\left\{\Gamma (\mathbf{r},{\mathbf{r}}^{\prime })\Gamma ({\mathbf{r}}^{\prime },\mathbf{r})\right\}}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}$$

$$\Gamma (\mathbf{r},{\mathbf{r}}^{\prime })=\sum _{k=1}^N{\Phi }_k(\mathbf{r}){\Phi }_k^{†}({\mathbf{r}}^{\prime })$$

$$h_{\mathrm{x}}(\mathbf{r},{\mathbf{r}}^{\prime }):=-\frac{\mathrm{Tr}\left\{\Gamma (\mathbf{r},{\mathbf{r}}^{\prime })\Gamma ({\mathbf{r}}^{\prime },\mathbf{r})\right\}}{n(\mathbf{r})}.$$

$$h_{\mathrm{x}}(\mathbf{r},u):=\frac{1}{4\pi }\int d{\Omega }_{\mathbf{u}}{h}_{\mathrm{x}}(\mathbf{r},\mathbf{r}+\mathbf{u})$$

Full text

I Introduction

Density-functional methods are the most widely used approach to efficiently compute the electronic structure of atoms, molecules and solids. Based on the Hohenberg-Kohn theorem Hohenberg and Kohn (1964), the electronic ground-state energy of interacting electrons is computed via the solution of the Schrödinger equation for fictitious noninteracting electrons – the so-called Kohn-Sham (KS) electrons Kohn and Sham (1965). At the heart of Density-Functional Theory (DFT) lies the idea that the exchange-correlation (xc) energy, i.e., the energy due to the electron-electron interactions beyond the classical Hartree energy, can be approximated by a universal functional of appropriate local densities. In its original incarnation, DFT considered only an external scalar potential coupled to the charge density, which characterizes the interacting system. This means that the universal functional for the xc energy could be written as a functional of the charge density alone. However, as soon as additional couplings, e.g., the Zeeman term or the coupling to an external vector potential, are present, the universality of the xc-energy functional is lost, unless additional densities are included as fundamental variables. This observation led over the years to the creation of multivariate DFTs, such as Spin-DFT (SDFT) von Barth and Hedin (1972) for including the Zeeman coupling and Current-DFT (CDFT) Vignale and Rasolt (1987) for orbital magnetism. In these theories, the spin density and the paramagnetic current density are included as basic variables, respectively. For the description of two dimensional heterostructures Winkler (2003); Dresselhaus (1955); Rashba and Sheka (1959) or topological insulators Kane and Mele (2005); Qi and Zhang (2011); Ando (2013), Spin-Orbit Coupling (SOC) plays a crucial role Hasan and Kane (2010). Since SOC is naturally described in terms of spin-dependent vector potentials, its density-functional treatment requires the additional inclusion of spin current densities as basic variables for a universal xc-energy functional. The corresponding extension of DFT has been dubbed Spin-Current-DFT (SCDFT) Vignale and Rasolt (1988); Bencheikh (2003).

Upgrading DFT to include additional variables not only leads to universal functionals for systems with SOCs and in strong magnetic fields, but also brings forth an important physical concept, namely, gauge symmetry Fröhlich and Studer (1993), which places strong constraints on the admissible dependence of the xc-energy functional on the basic variables and offers guidance in the construction of approximate functionals. In CDFT, for example, the xc-energy functional is invariant under local U(1) gauge transformations. Similarly, for spin-orbit coupled systems, the xc-energy functional is also invariant under local SU(2) gauge transformations (we will come back to this point in the next section extensively). In order to guarantee the U(1) invariance of the theory, Vignale and Rasolt argued that the xc energy should not be expressed in terms of the paramagnetic current, which is gauge dependent, but in terms of the gauge invariant vorticity Vignale and Rasolt (1987): thus, they arrived at the first universal local density approximation for electrons in a magnetic field. Subsequent experimentation showed that the U(1)-invariant vorticity is not well suited to deal with strongly inhomogeneous systems, such as atoms Zhu and Trickey (2006). Later, Abedinpour, Vignale and Tokatly proposed the generalized U(1)$\times$SU(2) covariant vorticity as the fundamental variable for constructing gauge-invariant approximations to the xc-energy functional in SCDFT Abedinpour et al. (2010). Its definition depends on an arbitrary choice of a “linking path” in physical space (cf. Ref. Abedinpour et al., 2010 for details) – an arbitrariness that, however, disappears in the limit of slowly varying densities. To the best of our knowledge, no experience has yet been gained on the practical use of the SU(2)-covariant vorticity. In any case, this quantity only emerges naturally when analyzing systems with minor inhomogeneities, such as an almost-uniform electron gas, and it may therefore suffer from the same shortcomings as its U(1) counterpart, when applied to atomistic systems.

One lesson we learn from these examples is that the choice of the gauge-invariant “building blocks” of the DFT is a nontrivial and important task, which requires much ingenuity as well as extensive experimentation on realistic systems. For example, the development of CDFT and, to a lesser extent, of SDFT, has been hampered for many years by the lack of functionals suitable to work with strong inhomogeneities. In the case of CDFT, the difficulty arose from the vorticity being too closely tied to the uniform free electron gas model. In the case of SDFT, as applied to magnetic systems with non-collinear spins, it is well known that a straightforward generalization of functionals derived in the collinear SDFT framework Kübler et al. (1988) only accounts for longitudinal fluctuations of the spin magnetization. Various attempts have been made recently Scalmani and Frisch (2012); Eich and Gross (2013); Eich et al. (2013) to include a dependence on transverse fluctuation of the spin magnetization within SDFT, but none of them, for different reasons, has been proven fully satisfactory.

In this paper, we propose a way out of the impasse, following a suggestion by Tao and Perdew Tao and Perdew (2005) who noticed that functionals of the Meta-Generalized-Gradient-Approximation (MGGA) family, such as the TPSS and similar Tao et al. (2003), which use the kinetic energy density $\tau({\bf r})$ as a basic variable, could be made current-dependent by enforcing U(1) gauge invariance through the minimal substitution

[TABLE]

where ${\bf j}$ is the paramagnetic current density. Earlier evidence of the relevance of their suggestion can be found in ideas by Dobson Dobson (1993) and, particularly, Becke Becke (1996), whose current-dependent functional, based on a careful study of the short-range behavior of the exchange hole (x-hole), greatly improved the description of degenerate ground states in open shell atoms Becke (2002). Follow-up works extended these ideas and demonstrated their usefulness in applications Maximoff et al. (2004); Pittalis et al. (2007); Zhu et al. (2016).

Along similar lines, we show that DFT-MGGA forms can be readily upgraded to SCDFT-MGGA forms (i.e., they can be made spin-dependent and spin-current dependent) by enforcing the U(1)$\times$SU(2) gauge invariance through minimal substitutions to be performed on the curvature of the exchange hole. These substitutions implicate new quantities such as the “spin-kinetic energy” (defined below), the spin-currents and the spin density combined with its gradients. Additionally, a trivial modification has to account for an extra dependence of the extended on-top exchange-hole on the squared modulus of the magnetization (and no other extra combinations). In this manner, successful approximations of ordinary DFT can be readily turned into approximations for SCDFT and non-collinear SDFT. Our proposal opens the way to the construction of approximate, yet non-empirical functionals, which do not assume weak inhomogeneity and should therefore have a wide range of applicability in atomic, molecular, and condensed matter physics.

This paper is organized as follows. In Sec. II, we review the theoretical background of U(1)$\times$ SU(2) gauge invariance in SCDFT and take the opportunity to remark that the xc-fields – generated, as usual, through functional derivatives of the xc-energy functional with respect to the basic densities – can exert non-trivial torques on the spin density and paramagnetic spin currents. In Sec III, we derive the generalized short-range behavior of exchange-only quantities, which allows us to extract very useful U(1) $\times$ SU(2) gauge-invariant quantities for the construction of functional approximations in SCDFT and, therefore, we point out the aforementioned minimal substitutions. In Sec. IV, we give examples of new approximate functionals of the MGGA form. Technical details concerning the gauge transformations of the various fundamental quantities are reported in Appendix A.

II Theoretical Background: Gauge Invariance and XC-torques

SCDFT is concerned with the calculation of the ground-state energy and densities of the semi-relativistic many-electron Pauli Hamiltonian Landau and Lifshitz (1965). In comparison to the SDFT Hamiltonian, which contains only a scalar potential, $v({\bf r})$, and a Zeeman magnetic field, $B^{a}({\bf r})$, the SCDFT also includes an Abelian vector potential, ${\bf A}({\bf r})$, and a non-Abelian vector potential ${\bf A}^{a}({\bf r})$: not (a)

[TABLE]

In Eq. (2) $\mu_{B}=1/2c$ is the Bohr magneton, we employ Einstein’s convention to sum over repeated indices ($a=x,y,z$), and a multiplication by a $2\times 2$ identity matrix is implied for the terms which are diagonal in spin space, i.e., $-i\nabla+\frac{1}{c}{\bf A}({\bf r})$ and $v({\bf r})$. The electron-electron interaction is given by

[TABLE]

where $:\cdots:$ denotes the normal ordering of the two-component Pauli field operators $\hat{\Psi}^{\dagger}=(\hat{\psi}^{\dagger}_{\uparrow}\;\hat{\psi}^{\dagger}_{\downarrow})$; finally,

[TABLE]

is the $a$-th component of the spin-density operator. Expanding the square and making use of partial integration, one obtains

[TABLE]

where

[TABLE]

We have also defined

[TABLE]

[TABLE]

Given the external fields ${\bf A}$, ${\bf A}^{a}$, $v$, and ${B}^{a}$, the ground-state energy is the expectation value of $\hat{H}$ in the corresponding ground state $|\Psi\rangle$. The ground-state energy can be determined by means of a constrained-search minimization Levy (1982); Lieb (1983):

[TABLE]

with

[TABLE]

where the inner minimization is carried out over all the many-body wave functions yielding the prescribed set of densities and the outer minimization is carried out with respect to all $N$-representable densities. Eq. (10) defines a universal density functionals, which is the direct generalization of the universal functional in standard DFT. Assuming that the same set of densities is both interacting and non-interacting $v$-representable, one can further decompose $F$

[TABLE]

in terms of the Kohn-Sham (KS) kinetic energy (see below) $T_{\rm KS}[n,{s}^{a},{\bf j},{\bf J}^{a}]$, the Hartree energy $E_{\rm H}[n]=\frac{1}{2}\int d^{3}r\int d^{3}r^{\prime}~{}\frac{{n}({\bf r}){n}({\bf r}^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}$ and a remainder, $E_{\rm xc}[n,{s}^{a},{\bf j},{\bf J}^{a}]$, which is the xc-energy functional. In this way, the problem of determining the ground-state energies is reformulated into devising practical and sufficiently accurate approximations for $E_{\rm xc}$. In order to simplify the notation in the following we redefine the external potentials $\mu_{B}{\bf B}\to{\bf B}$ and $\frac{\mu_{B}}{2}{\bf A}^{a}\to{\bf A}^{a}$.

The KS equations in SCDT have the form of single-particle Pauli equations

[TABLE]

where

[TABLE]

[TABLE]

in which ${\frac{1}{c}}{\bf A}_{\rm xc}=\frac{\delta E_{\rm xc}}{\delta{\bf j}({\bf r})}$ is the Abelian xc-vector potential, ${\frac{1}{c}}{\bf A}^{a}_{\rm xc}=\frac{\delta E_{\rm xc}}{\delta{\bf J}^{a}({\bf r})}$ is the $a$-th component of the non-Abelian xc-vector potential, ${B}^{a}_{\rm xc}=\frac{\delta E_{\rm xc}}{\delta{s^{a}}({\bf r})}$ is the $a$-th component of the xc-magnetic potential (Zeeman field), $v_{\rm xc}=\frac{\delta E_{\rm xc}}{\delta n({\bf r})}$ is the xc-scalar potential, and $v_{H}({\bf r})=\int d{\bf r}\frac{n({\bf r}^{\prime})}{|{\bf r}-{\bf r}^{\prime}|}$ is the usual Hartree potential.

A fundamental properties of $E_{\rm xc}$ is its invariance under general U(1)$\times$SU(2) gauge transformations. We recall that a local U(1) transformation, $U({\bf r})$, is defined by

[TABLE]

where $\chi(\bf r)$ is a scalar function of the position, and a local SU(2) transformation is defined by

[TABLE]

where $\lambda^{a}(\bf r)$ are the components of a vector function of the position. A detailed analysis of gauge transformations in SCDFT framework is presented in Ref.s [Vignale and Rasolt, 1988; Bencheikh, 2003]. Note that neither $F[n,{s}^{a},{\bf j},{\bf J}^{a}]$ nor $T_{\rm KS}[n,{s}^{a},{\bf j},{\bf J}^{a}]$ are invariant, but they transform in the same way because the KS system has the same densities $n$, ${s}^{a}$, ${\bf j}$, and ${\bf J}^{a}$ as the interacting system. As a result

[TABLE]

where

[TABLE]

where the matrix $R^{ab}$ is a $3\times 3$ matrix describing a rotation in $\mathbb{R}^{3}$ around $\hat{\lambda}$ – the unit vector in the direction of $\lambda^{a}$ – by an angle $\varphi=-2|\lambda|/c$ and Tr is the trace taken with respect to spin indices.

The transformation of the xc-fields can be readily deduced by combining the invariance of $E_{\rm xc}$ – as expressed in Eq. (17) – with the transformations of the densities, as given in Eq. (18). For completeness, we have

[TABLE]

It is apparent that while ${\bf A}_{{\rm xc},{\mu}}$ is invariant, $B^{a}_{\rm xc}$, ${{\bf A}^{\prime}}^{a}_{\rm xc}$, and $v_{\rm xc}$ are not invariant, but covariant not (b). Note that even in the case of a restricted U(1) transformation, $B^{a}_{\rm xc}$, ${\bf A}_{{\rm xc},\mu}$, and $v_{\rm xc}$ do not behave as standard Maxwellian fields and, in general, ${B}^{a}_{\rm xc}\neq\left[\nabla\times{\bf A}_{{\rm xc}}\right]_{a}$. The xc-fields in SCDFT should be regarded as some effective Yang-Mills fields.

Although the xc-fields are not generated by any physical field equations, they are bound to satisfy compatibility relations – having the form of conservation laws Bencheikh (2003) – due to the invariance of $E_{\rm xc}$, i.e.,

[TABLE]

In writing these expressions, we have denoted vectors in spin space (i.e., vectors with a non-Abelian index) with an arrow to highlight the torques due to $\vec{A}_{{\rm xc},\mu}$ and $\vec{B}_{\rm xc}$ on the right hand side of Eq. (20b). The KS system of SCDFT reproduces the interacting paramagnetic currents but may not reproduce the diamagnetic currents. Whatever the difference between the KS and interacting diamagnetic currents is, Eq. (20a) and Eq. (20b) ensure that the stationarity conditions for the particle and spin densities are not violated: $\vec{A}_{{\rm xc},\mu}$ and $\vec{B}_{\rm xc}$ can balance any non-vanishing xc-divergence-like contribution.

There is one non-trivial case in which we can see that the solution of the KS equations in SCDFT reduces to the solution of the analogous equations in SDFT. First, note that, to have ${A}_{{\rm xc},\mu}=0$ and $\vec{A}_{\rm xc,\mu}=0$, the torque of $\vec{B}_{\rm xc}$ must vanish as well [see Eq. (20b)]. Yet $\vec{B}_{\rm xc}$ can be non-vanishing, if it is parallel to the spin density at every point in space. If the external non-Abelian vector potential is also vanishing, we are then in a situation in which SDFT applies rigorously. Thus, in this case, we can conclude that ${v}_{\rm xc}={v}^{\rm SDFT}_{\rm xc}$ and $\vec{B}_{\rm xc}=\vec{B}^{\rm SDFT}_{\rm xc}$.

III Short-range behavior of exchange-only pair-correlation functions

Importing the standard Local-Spin-Density Approximation (LSDA) in SCDFT does not allow us to fully exploit the power of SCDFT, as the LSDA only depends on the magnitude of the spin density and the particle density. Moreover, the LSDA is insensitive to strong inhomogeneities and long-range interactions. Exact exchange would be an obvious, more sophisticated choice Rohra and Görling (2006), but its combination with suitable correlation functionals may require more involved computational approaches. Generalized-Gradient Approximations (GGAs) and, more recently, Meta-GGAs Sun et al. (2016) (MGGA) – either stand-alone or combined with the Hartree-Fock method into hybrids – are the gold standard in modern DFT calculations. Here, we report an analysis that points to the fact that MGGAs are ideal forms to satisfy U(1)$\times$SU(2) gauge-invariance while fulfilling other exact properties of the underlying pair-correlation functions.

We begin by reviewing known definitions about the so-called exchange hole. Assuming, as it is normally done, that the KS states are in the form of single Slater determinants, the exchange energy can be expressed as

[TABLE]

where

[TABLE]

is the one-body-reduced-spin density matrix obtained from the occupied spinors, which are solutions of Eq. (12). $\Gamma({\bf r},{\bf r}^{\prime})$ is a 2$\times$2 matrix in spin space. $E_{\rm x}$ is evidently invariant under general U(1) $\times$ SU(2) gauge transformations.

$E_{\rm x}$ can be usefully expressed in terms of the x-hole function, for which a convenient definition, applicable to non-collinear spin states is

[TABLE]

In practice, the spherical average

[TABLE]

is what really matters to the end of the calculation of the exchange energies.

In Eq. (24), the integration is carried out with respect to the solid angle $\Omega_{\bf u}$ formed by ${\bf r}$ and ${\bf u}$; ${\bf r}$ is the so-called reference position. Thus, we rewrite

[TABLE]

Taylor-expanding $h_{\rm x}({\bf r},u)$ for small inter-particle separations $u$, we find

[TABLE]

where

[TABLE]

is the curvature of the x-hole: the superscript “nc” emphasizes that this expression differs from the analogous quantity derived earlier for spin-unpolarized or spin-polarized but globally collinear states. By admitting complex-valued single-particle spinors, which is the natural state of affairs in non-collinear spin systems, we find terms that depend not only on the usual kinetic energy density,

[TABLE]

but also on the spin-kinetic energy density defined as

[TABLE]

Eq. (26) and Eq. (27) provide the exact short-range behavior of the x-hole function in presence of particle- and spin-currents for non-relativistic non-collinear states. The gauge-invariance of the expansion coefficients in Eq. (26) is obvious: a scalar function has been expanded with respect to a scalar variable. But one may also verify this property by direct inspection.

The on-top x-hole (i.e., the first term on the right hand side of Eq. (26)) already provides us with an explicit indication on how the on-top x-hole of an existing DFT functional should be modified to admit an extra non-empirical dependence on U(1) $\times$ SU(2) gauge-invariant quantities; simply, $n\to n\left(1+\frac{{s}^{a}{s}^{a}}{n^{2}}\right)$.

At the level of the curvature of the x-hole [Eq. (27)], notice that the combination $\left[\left(\tau-\frac{{\bf j}\cdot{\bf j}}{2n}\right)-\left(\frac{\nabla^{2}n}{4}+\frac{\nabla n\cdot\nabla n}{8n}\right)\right]$ is already known to be a U(1) gauge-invariant quantity: the difference, here, is that all the quantities are evaluated on fully non-collinear two component spinors. Thus, this expression is not SU(2) gauge invariant. The contribution $\left[\left(\frac{{s}^{a}{\tau}^{a}}{n}-\frac{{\bf J}^{a}\cdot{\bf J}^{a}}{2n}\right)-\left(\frac{{s}^{a}\cdot\nabla^{2}{s}^{a}}{4n}+\frac{(\nabla{s}^{a})\cdot(\nabla{s}^{a})}{8n}\right)\right]$ is essential to get the full invariance. A detailed discussion of the transformation of each term is presented in Appendix A.

Therefore, the minimal substitution

[TABLE]

can be used to transform a DFT-MGGA form into a SCDFT-MGGA form.

In practical applications it is often desirable to eliminate the laplacian terms, which may be difficult to evaluate numerically. To accomplish this, one substitutes the x-hole, expressed in terms of ${C}^{\rm nc}_{h_{\rm x}}$, into the expression for the exchange energy density and performs an integration by parts to show that ${C}^{\rm nc}_{h_{\rm x}}$ is actually equivalent (as far as the calculation of the exchange energy is concerned) to

[TABLE]

which no longer contains the laplacian operator. Obviously, the form of $\bar{C}^{\rm nc}_{h_{\rm x}}$ implies the minimal substitution

[TABLE]

Alternatively, functional approximations are also constructed working directly at the level of the one-body density matrix

[TABLE]

in terms of which the exchange energy can be expressed as

[TABLE]

where the vector positions are expressed with respect to the coordinates of the center of mass. Taylor-expanding $Q^{\rm nc}_{\rm x}({\bf r},u)$ with respect to $u$, we obtain

[TABLE]

where

[TABLE]

$C^{\rm nc}_{Q_{\rm x}}$ generalizes the known DFT expression to SCDFT.

Therefore, in extending a DFT-MGGA form based on the short-range of $Q_{\rm x}$, we can proceed by performing two minimal substitutions: the first one, $n^{2}\to\left(n^{2}+{s}^{a}{s}^{a}\right)$, has to be carried out only at the level of the on-top quantities; the second one is performed at the level of $\tau$

[TABLE]

Again, an intermediate integration by parts for the calculation of the exchange energies yields an alternative form of the curvature factor:

[TABLE]

which, modulo an overall multiplication by the particle density, implies the same minimal substitution as in Eq. (32).

IV Construction of functionals

We conclude our analysis by constructing two new exchange-energy functionals based on existing forms and proposing them for immediate use.

First, let us consider the construction of the BR89Becke and Roussel (1989): this was derived allowing only globally collinear spin polarization. In this approach, the x-hole of the Hydrogen atom is turned into a general model by introducing two position-dependent “parameters” $p_{1}({{\bf r}})$ and $p_{2}({{\bf r}})$

[TABLE]

to be chosen in such a way to reproduce the short-range behavior of the x-hole of an $N$-electron system.

Eq. (26) and Eq. (27) allow us to readily generalize this procedure to the non-collinear current-carrying states. As a result, the $p_{1}({{\bf r}})$ and $p_{2}({{\bf r}})$ must be determined by solving the equations

[TABLE]

As a second example, let us consider the approximations based on a Gaussian re-summation of the short-range behavior of $Q_{\rm x}({\bf r},u)$. For closed-shell systems (i.e., vanishing spin polarization), Lee and Parr Lee and Parr (1987) find

[TABLE]

For the sake of simplicity, here, we are not considering a more sophisticated form which would satisfy particle-number normalization for any system. In view of Eq. (34) and Eq. (35), the extension to non-collinear spin-polarized current-caring states is readily obtained upon the substitutions

[TABLE]

Finally, we note that ${C}^{\rm nc}_{Q_{\rm x}}$ is positive for single-particle states, for states with vanishing spin-currents, for the spin-spirals of the uniform gas, but otherwise the question remains open.

V Conclusions

This work opens the way to the extension of time-proven semi-local exchange-correlation energy functionals as well as to the derivation of novel approximations designed to deal with non-collinear spin structures which are subject of great and large ongoing interest. Specifically, we have introduced non-empirical U(1)$\times$SU(2) gauge-invariant building blocks which are, in principle, ideally suited for dealing with (static) spin-fluctuation of strongly inhomogeneous states. Our results show how the gradients of the spin-density should be combined with the spin-kinetic-energy density – an information which should be relevant even in devising semi-local forms within standard Spin-DFT – and the Kohn-Sham paramagnetic (spin-)currents appear as explicit ingredients as well. Thus, we have provided examples of extension of existing exchange-only functional forms. On passing, we have also illustrated some of the exact fundamental features of the exact exchange-correlation fields.

Acknowledgements.

S.P. was supported by the European Community through the FP7’s Marie-Curie Incoming-International Fellowship, Grant agreement No. 623413. G. V. acknowledges support from DOE Grant DE-FG02-05ER46203. F. G. E. has received funding from the European Union’s Framework Programme for Research and Innovation Horizon 2020 (2014–2020) under the Marie Skłodowska-Curie Grant Agreement No. 701796.

Appendix A Derivation of the transformation laws

The expansions worked out in the main text have the advantage to spare us from the burden to explicitly deal with the transformations of the densities under general gauge transformations – this is because we could consistently deal only with scalars. Nevertheless, in this appendix, we report the derivation of the transformation laws of the considered densities and their salient combinations. This should offer thorough clarifications and further insights.

We will use the following notational convention: Spatial indices are denoted by subscripts, spin indices by superscripts, and repeated indices are summed. Furthermore, the dependence on the position ${\bf r}$ is implied. We are interested in obtaining the transformation laws for the following densities:

[TABLE]

The combined U(1)$\times$SU(2) transformation is given by

[TABLE]

where we use that the U(1) and SU(2) transformation commute. This means that we can investigate the U(1) and SU(2) transformation laws separately.

U(1) transformation laws** **– The density, $n$, and the spin magnetization, $s^{a}$, are trivially invariant under local U(1) transformations. It is straightforward to obtain the transformation laws for the remaining densities, i.e.,

[TABLE]

It follows directly that the combinations

[TABLE]

are invariant under local U(1) transformations.

Infinitesimal SU(2) transformation laws** **– The density is trivially invariant under local SU(2) transformations. The spin magnetization, however, is not invariant. Using that

[TABLE]

where $\lambda$ is the magnitude and $\hat{\lambda}^{a}$ is the unit vector in the direction of the vector $\lambda^{a}$. We recall

[TABLE]

The matrix $R^{ab}$ is a $3\times 3$ matrix describing a rotation in $\mathbb{R}^{3}$ around the direction $\hat{\lambda}$ by an angle $\varphi=-2\lambda/c$. It follows that the spin magnetization transforms as

[TABLE]

Before embarking on the derivation of the transformation laws for the other densities, we consider the case of infinitesimal transformations. This means that we can approximate

[TABLE]

Keeping terms up to the first order in $\lambda$, we arrive at

[TABLE]

It is straightforward to verify that neither the quantity in Eq. (46a) nor the combination in Eq. (46b) are invariant under these transformations. Yet, the overall invariance of the x-only curvatures given in the main text can be now explicitly verified for arbitrary infinitesimal U(1) and SU(2) transformations in a direct calculations. This task may be further simplified by considering the identities: $\nabla s^{a}\cdot\nabla s^{a}=\frac{\nabla^{2}}{2}s^{a}s^{a}-s^{a}\nabla^{2}s^{a}$ and $\nabla n\cdot\nabla n=\frac{\nabla^{2}}{2}n^{2}-n\nabla^{2}n$.

Furthermore, the behavior of the basic densities under the same transformations suffices to establish the compatibility conditions (20a) and (20b) presented in the main text. This is achieved by using the fact that the xc energy is invariant under the corresponding U(1) $\times$ SU(2) transformations.

Finite SU(2) transformation laws** **– In principle, it is sufficient to establish invariance under infinitesimal SU(2) transformation, as an arbitrary finite SU(2) transformation can be represented as a sequence of infinitesimal transformations. However, the derivation of the transformation laws of the xc potentials requires knowledge of the explicit transformations of the basic densities under arbitrary finite SU(2) transformations. Moreover, we here intend to spell out the finite transformation of the spin-kinetic-energy density as well.

The difficulty in obtaining finite transformation laws is due to the fact that the generators of the SU(2) group, i.e., the Pauli matrices $\sigma^{a}$, do not commute. Accordingly, we have to keep in mind that

[TABLE]

Instead, using the very definition of the directional derivative, we have

[TABLE]

Now, we can use the identity

[TABLE]

– which is readily verified by noting that both sides fulfill the same first-order differential equation in $\gamma$ and vanish for $\gamma=0$ – and write

[TABLE]

where we have introduced

[TABLE]

The $3\times 3$ matrix $\Lambda^{ab}$ is the uniform average over the rotation matrices around axis $\hat{\lambda}^{a}$ with rotation angles $\varphi\in[0,-2\lambda/c]$. Note that $\Lambda^{ab}$ is not a rotation matrix itself. Using the group properties of the rotation matrices, one can easily verify the relation

[TABLE]

which is an extremely useful identity as it shows how the rotation matrix $R^{ab}$ acts on the matrix $\Lambda^{ab}$. Straightforward but tedious algebra allows us to write down the transformation laws for finite local SU(2) transformations in a concise form

[TABLE]

where we introduced $\tilde{A}_{\mu}^{a}=(\partial_{\mu}\lambda^{b})\Lambda^{ba}$, which is the non-Abelian gauge vector potential induced by the local SU(2) transformation. Note that in the main text, we wrote $\tilde{A}_{\mu}^{a}=-\frac{ic}{2}{\rm Tr}\left(\sigma^{a}U^{\dagger}_{\rm S}\nabla_{\mu}U_{\rm S}\right)$, which seems another widely adopted choice in the literature.

Finally, combining the U(1) and SU(2) transformation laws for the density, spin magnetization, and the paramagnetic current and spin current – together with the invariance of $E_{\rm xc}$ – leads to the transformation properties of the xc potential given in the main text [Eq. (19)].