Orbital Edelstein effect from densitywave order
Abstract
Coupling between charge and spin, and magnetoelectric effects more generally, have been an area of great interest for several years, with the soughtafter ability to control magnetic degrees of freedom via charge currents serving as an impetus. The orbital Edelstein effect (OEE) is a kinetic magnetoelectric effect consisting of a bulk orbital magnetization induced by a charge current. It is the orbital analogue of the spin Edelstein effect in spinorbit coupled materials, in which a charge current drives nonzero electron spin magnetization. The OEE has recently been investigated in the context of Weyl semimetals and Weyl metals. Motivated by these developments, we study a model of electrons without spinorbit coupling which exhibits line nodes that get gapped out by via symmetry breaking due to an interactioninduced charge density wave order. This model is shown to exhibit a temperature dependent OEE, which appears due to symmetry reduction into a gyrotropic crystal class.
I Introduction
The field of magnetoelectric effects has seen a revival in interest in the past decades on several fronts. The discovery and study of multiferroicity in correlated materials has uncovered unconventional mechanisms which can give rise to a large effective magnetoelectric coupling Spaldin and Fiebig (2005); Eerenstein et al. (2006); Ramesh and Spaldin (2007); Cheong and Mostovoy (2007); Spaldin et al. (2010); Lawes and Srinivasan (2011); FuentesCobas et al. (2015); Chu et al. (2018). Similarly, the discovery of threedimensional (3D) topological insulators has led to an exploration of novel magnetoelectric effects due to emergent axion electrodynamics in such topological phases Hasan and Kane (2010); Qi and Zhang (2011); Grushin and de Juan (2012); Pesin and MacDonald (2013); Schmeltzer and Saxena (2013); Mal’Shukov et al. (2013); Baasanjav et al. (2014); Morimoto et al. (2015); Xiao et al. (2018); Tokura et al. (2019). A part of the reason for the wide interest in such magnetoelectric effects partly stems from the technological potential of controlling charge degrees of freedom via magnetic fields or, conversely, tuning magnetic degrees of freedom via an applied electric field.
A prominent example of such a magnetoelectric effect is the nonequilibrium phenomenon of currentinduced magnetization, which is also termed as kinetic magnetoelectric effect (KME)Levitov et al. (1985); Şahin et al. (2018). The intrinsicspin variant of this effect, wherein a charge current in a spinorbitcoupled conductor gives rise to bulk spin polarization and, hence, a net magnetization, is referred to as the Edelstein effect or the inverse spingalvanic effectSinova et al. (2015); Manchon et al. (2015) and has been under study for several decadesGanichev et al. (2016). A great deal of experimental work has focused on the Edelstein effect in 2D systems, notably in thinfilm semiconductorsKato et al. (2004); Silov et al. (2004); Kato et al. (2005); Sih et al. (2005); Stern et al. (2006); Yang et al. (2006) and at metal surfacesZhang et al. (2014). However, experiments on 3D materials have been scant, although some recent studies have reported its observation in trigonal tellurium Shalygin et al. (2012); Furukawa et al. (2017).
In recent years, it has come to light that 3D systems can have an intrinsic orbital contribution to the KME, analogous to the spin part and arising as a consequence of the orbital magnetic moment of Bloch bandsYoda et al. (2015); Zhong et al. (2016); Rou et al. (2017); Yoda et al. (2018); Şahin et al. (2018); Tsirkin et al. (2018); Flicker et al. (2018); Niu et al. (2018); Shi and Song (2019), notably in trigonal selenium and tellurium. Whereas the ordinary Edelstein effect (hereafter referred to as the spin Edelstein effect, SEE) relies on crystalline spinorbit coupling (SOC) to give Bloch states a spin texture and, hence, is limited by the size of the SOC, the orbital Edelstein effect (OEE), also referred to as the inverse gyrotropic magnetic effectZhong et al. (2016), is determined solely by the geometry of the crystalYoda et al. (2015, 2018).
Chiral crystals are a subset of those that can exhibit the KME. Previous studies have considered trigonal selenium and tellurium and viewed their chiral nature as descending from a chargedensitywave (CDW) instability of a hypothetical parent phaseSilva et al. (2018), and others have studied optical gyrotropy as a probe for symmetry breaking in the chiral CDW phase of  Gradhand and van Wezel (2015) and in stripeordered cupratesOrenstein and Moore (2013). It has also been shown that Weyl nodes at the Fermi level can yield a large intrinsic contribution to the KME Yoda et al. (2015, 2018).
Our work builds on this theme, and explores the KME induced by symmetry breaking in a system with line nodes in the electronic band structure. The resulting phase is a nonchiral but gyrotropic crystal, and we study the concomitant temperaturedependent KME as a probe of the densitywave order. Below, we briefly review the OEE, before introducing our model Hamiltonian and presenting its theoretical study.
Ii Orbital Edelstein effect
Electrons in crystalline solids form Bloch bands with an intrinsic spin magnetic moment
(1) 
where is the spin operator for the electrons. The modern theory of magnetization in solidsResta (2010); Thonhauser (2011); Vanderbilt (2018); Resta (2018) has discovered that such Bloch bands also host an intrinsic orbital magnetic moment given by
(2) 
where is the Bloch Hamiltonian, is the band index, and . This orbital magnetization has been shown to arise from the selfrotation of wavepackets in the semiclassical theory of electron dynamicsXiao et al. (2010); Thonhauser (2011). Since under time reversal and under spatial inversion, it is clear that at least one of these symmetries must be broken in order for to not be identically zero.
From the viewpoint of semiclassical dynamics, given an electron distribution function , the instrinsic contribution to the net electronic magnetization is given byXiao et al. (2010); Resta (2010); Şahin et al. (2018)
(3) 
where is the crystal volume. In thermodynamic equilibrium for a timereversal symmetric system, the FermiDirac distribution, forces zero a net magnetization because of cancellation between contributions from opposite crystal momentaYoda et al. (2018). However, an asymmetric distribution function, such as that arising from an applied electric field, can generally give rise to nonzero net bulk magnetization.
Explicitly, to lowest order in an applied uniform DC electric field, the distribution function is modified asAshcroft and Mermin (1976)
(4) 
where is the electronic group velocity, is the impurityscattering relaxation time in relaxationtime approximation, and is the elementary charge. Hence, the magnetization arises as a linear response to an applied electric field,
(5) 
with the linear response tensor
(6a)  
(6b) 
where and are Cartesian indices.
The form of the tensor is significantly constrained by crystal symmetryYoda et al. (2018); Ganichev et al. (2016); Nye (1957); Landau and Lifshitz (1960). is an axial ranktwo tensor since it relates a polar vector, , to a axial vector, . Crystal classes whose pointgroup symmetries allow for nonzero axial ranktwo response tensors are known as gyrotropic. The reason for this name is that the tensor governing natural optical activity, or gyrotropy, transforms in the same way as ; thus KME and optical gyrotropy go hand in hand.
We note that the same symmetry constraints govern the appearance of nonzero spin and orbital contributions to , so both are expected to arise together, and there is no clear route to disentangling them in a 3D systemYoda et al. (2018); Furukawa et al. (2017). Indeed, the authors of Ref. Furukawa et al., 2017 conclude by speculating that the currentinduced magnetization they observe in trigonal tellurium may be due not only to the wellknown SEE, but also to the OEE.
Iii Model
As an example of the OEE brought about by symmetry breaking, we consider a tightbinding toy model of spinless fermions moving in a tetragonal crystal as shown in Fig. 1, consisting of identical atoms are arranged in layered square lattices. We assume a single isotropic orbital, and ignore the electron spin below; there are many cases where this is a useful starting point. In crystals with densitywave order driven by nearestneighbor repulsion, such as we will consider below, both spin components behave in the same manner. Including spin then only leads to an extra factoroftwo in certain equations below. The other case where spin may be ignored is in spinpolarized systems which might be a useful description of states in a large energy interval around the Fermi energy in strong ferromagnets (i.e., in halfmetals).
We define the nearestneighbor (NN) lattice constant , and lattice constant along the stacking axis to be . In the  planes, we include NN hopping and nextnearestneighbor (NNN) hopping . In the  and  planes, we include NN hopping and the peculiar NNN hoppings depicted in Fig. 1, with . This last choice differentiates staggered and sublattices, and endows the crystal with primitive translations , , and . The crystal space group is , and its associated crystal class is . Importantly, this crystal has centers of inversion at the middle point of every NN bond.
The inequivalence of the hopping amplitudes and for symmetry may be rationalized as depicted in Fig. 1: if a secondary set of atoms is present only along the red bonds—which would be consistent with the spacegroup symmetry of the model—and their energy levels are far from the Fermi energy, their dynamics could be integrated out, with the end result of renormalizing the hopping amplitude between the primary atoms (shown in gray), effectively leading to . The same reasoning holds if there are additional secondary atoms on the green bonds provided they are of a species different from those on the red bonds.
We use the following values for the hopping parameters: , , , and .
As is true for any crystal class with inversion symmetry, is nongyrotropicNye (1957); Landau and Lifshitz (1960). However, if the symmetry of the crystal were to be reduced, for instance by the onset of CDW order, the inversion symmetry could be broken and, indeed, the ordered structure could fall into a gyrotropic class. Below, we first study the band structure of this model in the absence of interactions, before turning to the impact of nearestneighbor repulsion.
iii.1 Noninteracting band structure
The noninteracting Hamiltonian is , where creates an electron on sublattice of unit cell (that is, on the atom at position ). In momentum space
(7) 
, is the number of unit cells, is the vector of Pauli matrices (acting in sublattice space), and . Because of timereversal symmetry, the hopping amplitudes are necessarily real valued. We will measure momenta in units of inverse lattice spacing, and henceforth set . With this, we arrive at
(8a)  
(8b)  
(8c)  
(8d) 
The inversion symmetry mentioned above combined with the Hamiltonian’s timereversal symmetry constrain identically; for this reason, band touchings for this crystal will generically arise as line nodes, as seen in the noninteracting band structure (top panel of Fig. 6) Figure 2 shows the location of the line nodes in the Brillouin zone.
Spontaneous symmetry breaking (SSB), however, could change the crystal class to a less symmetric one. A simple scenario is a CDW phase in which the densities on the and atoms are inequal, in which case the space group becomes , whose crystal class——is gyrotropic.
iii.2 Repulsive interactions
Next, we include repulsive interactions between NNs and nextnearest neighbors (NNNs); that is,
(9) 
where is the number operator for the atom and we take . We take nonzero repulsions on the same bonds as the hopping amplitudes and use a similar naming scheme, as shown in Fig. 1. Note that, unlike for the hopping amplitudes , we take the repulsion strengths to be on both the red and green bonds — we make this choice since it serves as a representative slice through the full parameter space and serves to illustrate some of the key ideas. Hence, the full Hamiltonian is given by .
In the absence of spin order, a Hubbard term would modify the MF Hamiltonian by a mere uniform offset in the chemical potential . Hence, as we will restrict ourselves to ansätze without spin order, we do not include the Hubbard interaction.
We performed a MF calculation of the CDW order in the above system, allowing for a finite set of commensurate wavevectors. We identified the ordering wavevectors favoured by the interaction by considering a simple model of classical charges resting at each atomic site of the crystal of Fig. 1—see Appendix A. Based on this, we included in our ansatz the four ordering wavevectors
The order parameters for the meanfield theory are the Fourier amplitudes for the listed above, where . However, we find it convenient to express the amplitudes on the and sublattices in terms of a symmetric and an antisymmetric part, respectively defined as
(10a)  
(10b) 
Since , the electron filling, we are left with 7 independent MFs.
Iv Results
iv.1 Meanfield theory of chargedensitywave order
We focus on a cut of parameter space parametrized by such that
(11) 
According to the model of electrostatic charges, at these relative repulsion values (marked by a white star in Fig. 10), the interaction most favours an ordering with ; however, the region of parameter space with most favoured is nearby (see Appendix A).
The MF calculation, for which the zerotemperature phase diagram is shown in Fig. 3, does indeed identify a swath of pure order for sufficiently large and for approximately —call this order phase I. In addition to this phase, we discover mixed phases at high and low filling in which both and , where is either or —call the order in these regions phase II.
For , we observe that the phase diagram is approximately symmetric under ; this is expected given the particlehole symmetry of , which becomes an approximate symmetry of when is nonzero but small compared with .
The phase diagram at as a function of and is shown in Fig. 4 and reveals that raising the system temperature progressively supresses phaseI order.
iv.2 Magnetoelectric response in phase I
We studied the temperaturedependent SSBinduced OEE in brokensymmetry phase I, whose sole order parameter is . Figure 5 shows the evolution of and the components of the response tensor at half filling () and fixed interaction strength (). The phase transition, which is seen to be continuous, takes place at .
As alluded to previously, the pointgroup symmetry in phase I is the , and a symmetry analysis reveals that symmetry constrains the response tensor to the formNye (1957)
(12) 
As seen in the lower panel of Fig. 5, the calculated is indeed of this form. The nonmonotonicity of is understood via the MF band structure, seen in Fig. 6. The top panel shows the hightemperature () band structure. Although the system is in a metallic phase with many states within (gray shading) from the chemical potential (dotted line), in the highsymmetry phase, identically, so . The middle panel shows the band structure at an intermediate temperature (near the peak in the ); the phaseI charge order has broken inversion symmetry and split the bands throughout the Brillouin zone, and the reduced symmetry of phase allows gyrotropic response, so . As the temperature is further lowered, the size of the band splitting increases and a full gap develops, with within this gap (lower panel of Fig. 5). Hence, it is clear that in the lowtemperature limit, as the sum in Eq. 6 approaches a Fermisurface integral, should again vanish.
iv.3 Role of line nodes
In this subsection, we inspect the contribution of the gappedout line nodes to the magnetoelectric response tensor in phase I. As depicted in Fig. 2, in the highsymmetry phase, three independent line nodes exist in the first Brillouin zone: 1. the  segment, 2. the  segment, and 3. the  segment—all other line nodes are related by symmetry.
In phase I, the charge order breaks the inversion symmetry ; accordingly, in the MF Hamiltonian for phase I, is no longer constrained to be zero, so the band touchings disappear. Rather, is set by the phaseI order parameter . As a shorthand, define
(13) 
where , defined in Appendix B, is proportional to the phaseI order parameter .
We expand about the (gapped) line nodes, denoting small deviations in momenum by . In the neighborhood of the (gapped) nodes, we have

, ,
(14a) (14b) (14c) (14d) 
(15a) (15b) (15c) (15d) 
(16a) (16b) (16c) (16d)
We can compute the orbital magnetic moment about such a (gapped) line node: for a twoband model with Bloch Hamiltonian , can be written in the closed formZhong et al. (2016); Yoda et al. (2018)
(17) 
where , , and are Cartesian indices.
In the neighborhood of a (gapped) line node in the direction,
(18) 
where , , , and are functions of , and, hence, the components of are given by
(19a)  
(19b)  
(19c) 
where primes denote differentiation of the singlevariable functions. To describe a line node in another direction, the indices in Eqs. 18 and 19 must be interchanged appropriately.
Figure 7 shows that the main contributions to originate from the the line nodes. The figure, for which we take as in the middle panel of Fig. 6, shows the orbital magnetic moment (left panel) as well as the product for the bottom (, middle panel) and top (, top panel) bands, which enter into the calculation of . [The component is related by symmetry (Eq. 12), while all other components vanish.] These quantities are shown along slices , , , , and .
The form of in momentum space reflects all the symmetries of the point group of phase I. The large values of are attributable to the presence of line nodes and are as expected fom Eqs. 19: for the slices , we see the signature of the line node transverse to , for which in proximity to the node, and for the slices and , we also recognize contributions from the line nodes longitudinal to , which are largest at the locus of the line nodes. The product still retains the general form of , though with a modified symmetry that allows a nonzero integral over the Fermi surface.
The difference in the magnitude of the longitudinal contributions at and can be retraced to the different for the line nodes in those planes: while for the former, for the latter, and with our parameter values.
As shown by Eq. 6, up to constant factors, the integrand for is multiplied by , a weighting factor concentrated within around the chemical potential . Hence, the equalenergy surfaces at (solid black lines) and (dashed black lines) reveal the main contributions to the integral. At this temperature, for our choice of parameters, the largest contributions to are from the (gapped) line nodes at .
V Discussion and conclusion
It is sometimes overlooked that gyrotropy can arise without chiral or timereversal symmetry breakingOrenstein and Moore (2013); Ganichev et al. (2016). Since the point group of the phase I order contains mirror symmetries (in our coordinates, one perpendicular to and another perpendicular to ), the structure of this brokensymmetry phase is an example of a nonchiral gyrotropic structure. This is unlike many of the examples of the OEE studied so farYoda et al. (2015, 2018), such as trigonal selenium and telluriumRou et al. (2017); Şahin et al. (2018). Furthermore, it constitutes a concrete example of longitudinal magnetization induced by a current in a mirrorsymmetric structure, while it has been implied that this is not allowed by symmetry Yoda et al. (2015). While longitudinal magnetization is forbidden by mirrors perpendicular or parallel to the current, it is not forbidden by mirrors at angles from the current.
Similarly to a previously studied modelYoda et al. (2018), a picture of current flowing through solenoids provides a qualitative understanding of the OEE in phase I. Furthermore, this picture makes physically clearer why the OEE response vanishes (i) when — hence clarifying physically the necessity of assuming from the start —and (ii) in the absence of charge order, that is, when the atoms are indistinguishable. We imagine a current driven in the direction: the simplest solenoidlike paths that result in net displacement purely in the direction are shown in Fig. 8 and 9.
(i) Each row of atoms, whether on sublattice or , is surrounded by a solenoid traced out of (green) hoppings and another traced out of (red) hoppings (Fig. 8). These two solenoids are of opposite helicites, and if , their induced magnetic fields cancel identically, precluding any currentinduced magnetizationwhether in the symmetric phase or in a chargeordered phase. Indeed, if , there exist additional mirror planes (perpendicular to and to ) that interchange the two solenoids.
(ii) If , let us presume without loss of generality that (in red) is dominant and ignore the solenoids. Figure 9 shows that a solenoid whose central axis is a row of sites is of opposite helicity as one whose central axis is a row of sites. If the atoms on the two sublattices are indistinguishable, the magnetic fields induced by these two solenoids are equal and opposite and cancel when averaged over several lattice spacings. If the and atoms are distinguishable, however, the two sets of solenoids are distinct and can give rise to net magnetization. Indeed, while in the symmetric phase, there exist glide planes perpendicular to and along that interchange the dottedline and solidline solenoids.
It is surprising a priori that such a solenoid picture can exist in a nonchiral crystal (previous examples have focused on chiral structuresYoda et al. (2015, 2018); Şahin et al. (2018)); however, the two mirror symmetries of point group —one perpendicular to and the other perpendicular to —do not bring the solenoids into themselves, but rather exchange the axis solenoids with the axis solenoids. The mirrors instead explain why the OEE coefficients along the and axis are opposite in sign, since the current is a polar vector and the magnetization is an axial vector.
In summary, we have discussed a simple model of a crystal with lines nodes, and shown how charge order can lead to a nonzero OEE. Such an OEE could be probed by nuclear magnetic resonance experiments. We have also discussed how the magnetoelectric response has a large contribution arising from the vicinity of line nodes. Our work suggests that lightly doped linenode semimetals in materials with weak spinorbit coupling might be a promising place to search for large OEE.
Note
While completing this manuscript, we came across a recent theory preprint Ishitobi and Hattori (2019) which discusses the OEE induced by quadrupolar symmetry breaking in certain diamond lattice materials.
Acknowledgements
We thank D. A. Pesin for fruitful discussions. This research was funded by the Natural Sciences and Engineering Research Council of Canada and the Canadian Institute for Advanced Research. G. M. is supported by the Fonds de recherche du Québec  Nature et technologies. This research was enabled in part by support provided by Compute Ontario, Westgrid and Compute Canada. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation; the Government of Ontario; Ontario Research Fund  Research Excellence; and the University of Toronto.
Appendix A Classical phase diagram
The ordering wavevectors favoured by the interaction of Eq. 9 were identified by determining the energy of CDW modes according to in a classical picture of electrostatic charges. In Fig. 10, we show which mode has the lowest energy (according to ) as a function of the (relative) sizes of the repulsion strengths , , , and . Note that, unlike in the main text, the modes are given in  coordinates as (where we have set ), since it is not necessary to work with a multiatom basis when considering only the repulsion term . Furthermore, note that the yellow phase implicitly stands for itself as well as its symmetryrelated counterpart ; the two are degenerate as expected by symmetry.
The ansatz for the meanfield theory described in the main text was chosen to potentially allow all the ground states that occur in this simplified model.
Appendix B Details of the meanfield calculation
Here, we provide additional information regarding the selfconsistent MF calculation of the CDW order. The interaction of Eq. 9 was decomposed in the density channel, giving rise to the following MF interaction term:
(20a)  
(20b) 
Here, is the number of unit cells in the crystal, (and likewise for ), and is the Fourier transorm of , which is invariant under simultaneous translation of and .
Choosing a closed set of commensurate wavevectors defines a reduced Brillouin zone (RBZ), which is mapped to the full Brillouin zone under addition of the wavevectors . This property allows us to rewrite the integral of reciprocalspaceperiodic functions as
(21) 
where the domain of the momentum sums is indicated above the summation symbol. Hence, we diagonalized the Hamiltonian in the RBZ by writing it in the form
(22)  
(23) 
where and is the Bloch Hamiltonian, in our case given in Eq. 7: . Starting from a series of randomized values for the MFs , we iterated until the computed expectation values agree with the input MFs to within . We compared the Helmholtz free energy of the different ground states thusly obtained and selected the one with minimal at every point in parameter space.
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