Reactive Hall and Edelstein effects in a tight-binding model with spin-orbit coupling

The reactive Hall constant R_H, described by reactive (nondissipative) conductivities, is analyzed within linear response theory in the presence of spin-orbit interaction. Within a two dimensional tight-binding model the effect of van Hove singularities is studied. Along the same line a formulation of the Edelstein constant is proposed and studied as a function of coupling parameters and fermion filling.


INTRODUCTION
The spin-orbit interaction (SOI) [1,2] plays a prominent role in the field of spintronics.It has been extensively studied as a key mechanism in the anomalous Hall effect [3], the spin Hall effect [4][5][6][7], the induced magnetoelectric torque and edge currents [8,9], the Edelstein effect [10,11] to name just a very few.
From a different perspective, in a seminal work [12] W. Kohn brought attention to the reactive response of an electronic system, as criterion of the Mott metal-insulator transition, given by the imaginary part of the conductivity, σ ′′ (ω → 0) = 2D/ω.The prefactor D, referred to as "Drude" weight in numerous recent theoretical studies and also equal to the weight of the zero-frequency δ-function in the regular part of the conductivity, is finite in a non-interacting gapless system without disorder at all temperatures or a gapless strongly correlated integrable one [13].In the presence of scattering (e.g.phonons, disorder) the δ-function broadens to a peak of width 1/τ (where τ is a characteristic scattering time) and integrated weight D which is experimentally studied.
Along the same line, a formulation of the reactive Hall response was proposed [14] in order to address the problem of Hall constant sign change as a function of doping observed in experiments in high T c superconductors.In this formulation the Hall constant is given by the logarithmic derivative of the Drude weight with respect to the particle density.This reactive Hall constant approach has recently been extensively used in theoretical [15] and experimental [16] studies of interacting (and synthetic) quantum systems.
In this work, we will first study the effect of the spin-orbit interaction on the reactive Hall effect of a two-dimensional tight binding model of non-interacting fermions, showing ballistic transport, as a function of spin-orbit coupling.We study its behavior as the Fermi surface evolves as a function of filling.We obtain the signatures on R H of Fermi surface topological transitions, with the associated van Hove singularities in a quantum mechanical description.Earlier work, using the Boltzmann equation studied these effects at a semiclassical level [38] and a recent one, on the spin and orbital Edelstein effect in a bilayer system with Rashba interaction [39].Next, we develop a novel formula, in analogy to the reactive Hall constant, for the Edelstein effect which describes the appearance of a transverse magnetization due to a charge current in a two-dimensional system with spin-orbit interaction.It would be fascinating to experimentally study the reactive Edelstein effect, for instance in synthetic systems [16], as the reactive Hall effect.

MODEL
We consider a generic Hamiltonian for fermions on a lattice, where for simplicity we describe the kinetic energy term by a one band tight binding model; it is straightforward to extend this formulation to a manyband or continuum system.The sites are labeled l(m) along the x(ŷ)-direction with periodic boundary conditions in both directions: arXiv:2310.11253v2[cond-mat.str-el]4 Mar 2024 where t x,y are the hopping parameters, λ x,y the Rashba spin-orbit couplings, σ α , α = x, y, z spin-1/2 Pauli matrices and the fermion creation (annihilation) operators are denoted as, c † l,m = (c † ↑l,m c † ↓l,m ).We take a unit lattice constant, electric charge e = 1 and h = 1.We add a magnetic field along the ẑ-direction, modulated by a one component wave-vector-q along the ŷ-direction, generated by the vector potential A m ; this allows to take the zero magnetic field limit smoothly: (for convenience, we will present the long wavelength limit, substituting 2 sin(q/2) → q).Electric fields along the x, ŷ directions are generated by time dependent vector potentials: Currents are defined through derivatives of the Hamiltonian in ϕ x,y : with the paramagnetic parts:

REACTIVE HALL RESPONSE
We will analyze the reactive Hall response within standard linear response theory, closely following the development in [14].
Reactive -nondissipative -response occurs in a generic uniform (without disorder) interacting system at zero temperature, in which case the brackets ⟨...⟩ denote ground state average ( [14]) or at finite temperatures in a uniform noninteracting system in which case the brackets ⟨...⟩ denote thermal average.Here we study the reactive Hall constant for the Hamiltonian (1) at finite temperatures in the presence of a magnetic field, with the conductivities given by, In contrast to the usual derivation of the Hall constant expression, we keep the q−dependence explicit by converting the current-current to current-density correlations using the continuity equation: . At this point we consider the "screening" (or slow) response in the ŷ−direction, by taking the (q, ω) limits in the order ω → 0 first and q → 0 last.For H(λ, µ), using the following identity, ρ the electron density.The Drude weight D can be evaluated by Kohn's expression [12] and its finite temperatures extension [40] as the second derivative of energy eigenvalues E n with respect to a uniform fictitious flux ϕ x , where p n = e −βEn / n e −βEn , β = 1/k B T the inverse temperature, k B = 1 the Boltzmann constant and ρ the fermion density.
To evaluate the eigenvalues/eigenstates we diagonalize the Hamiltonian (1) in the presence of the fictitious flux We obtain H = k± ϵ k± c † k± c k± with the two spin states turning by the spin-orbit interaction to two chirality eigenstates c k± = 1 √ 2 (c k↑ ± e −iϕ k c k↓ ) and eigenvalues, Finally the reactive Hall constant is given by, In Fig. 1, without any spin-orbit coupling, we observe a clear indication in the Hall constant of the van Hove singularities.When there is a Fermi surface topological transition, there is a discontinuity in the derivative of R H with respect to ρ.In addition, there is a change of sign when there is a transition from an electron-like to a hole-like dispersion.At finite temperature the behavior is continuous and the signature of the transition from an electron-like to a hole-like dispersion is reflected on the monotonic behavior of R H .This goes away at very high temperature where R H has a monotonic behavior independent of the parameters.

EDELSTEIN COEFFICIENT
Along the same line, we can also study the Edelstein effect in a two dimensional system with spin-orbit coupling where a charge current in the x-direction induces a bulk magnetization density ⟨s y ⟩ = ⟨ 1 pointing in the ŷ-direction.The corresponding response is given by, Here we consider the system (1) in zero magnetic field A m = 0 and zero electric field ϕ y m+1/2 (t) = 0 along the ŷ-direction.Now we can define a "reactive Edelstein constant" as the prefactor of the 1/ω imaginary part of α yx , which using the identity (2) can be written similarly to the Kohn formula, Here the energy derivatives are over a uniform fictitious flux ϕ x in the x-direction and a Zeeman field h y along the ŷ-direction.
For the noninteracting fermion Hamiltonian (1), the imaginary part of the response function becomes, As discussed above, this expression for the imaginary part of the response function, can be recast in the form, To evaluate the derivative ∂ 2 ϵ k± ∂h y ∂ϕx | h y ,ϕx→0 we add in the Hamiltonian (1) a magnetic field in the ŷ-direction (+h y s y ), modifying the eigenvalues as, and finally obtaining, Note from ( 5) and ( 6) that inverting λ x → −λ x , D E → −D E , while D E remains invariant changing the sign of λ y .
As we are interested in probing Fermi surface topological transitions, the derivatives of D E with respect to chemical potential, trace FS topological transitions.At the points where the curvature of D E as a function of fermion density ρ changes sign, we've got the relation: The ratio of the derivatives of D E on the left-hand side, probe Fermi surface topological transitions because at T=0 the integrations are along Fermi surface paths within the first BZ: ) Fig. 3 shows D E at zero temperature as a function of fermion density for t x = t y = 1 and λ x = λ y = 0.1.We see that D E changes sign as we go from a closed to an open Fermi surface.As we noted before, this behavior captures the transition from electron-like to hole-like pocket.Similarly, Fig. 4 illustrates the same physics for the case of anisotropic spin-orbit coupling, while in Fig. 5 the chosen parameters correspond to anisotropic strength of hopping elements.In Fig. 6 we show separately the opposite contributions of the two chiralities to D E in (5).We find that D E is the result of a large cancellation between the two components each of which however keeps track of the FS topological transitions.
The low density limit k → 0 can be treated analytically for the isotropic parameters t At T = 0 the upper cutoff of the integrals becomes

CONCLUSIONS
In this work we study the effect of spin-orbit interaction on the reactive Hall constant R H within a noninteracting fermion Hamiltonian model.In particular we show the role of van Hove singularities in the density of states as cusps in the density dependence of R H . Furthermore, we introduce the concept of reactive Edelstein constant, an effect in spin-orbit coupled systems and derive an expression analogous to the Kohn formula for the reactive response.
The main open question is the effect of interactions, introduced e.g. by a Hubbard U term in the Hamiltonian.An emerging Mott gap, typically at half filling, leads to a vanishing D as Kohn pointed out, the criterion for a Mott metal-insulator transition.Thus the interaction effects in combination with the presence of (higher-order) van Hove singularities can have drastic consequence on phase formation [41].At first, this can be studied by a mean field approach [42,43] which essentially leads to effective single particle states allowing a direct computation of R H and D E .Next, the interaction effects can be studied by either numerical simulations or many-body theory techniques [41].An interesting question is the temperature dependence of the Edelstein constant.While it is known that the Drude weight vanishes at any finite temperature in an interacting system evidence of dissipation due to interactions, the temperature dependence of D E is not clear.
Figs.1,2 the Hall constant for different values of the hopping and spin-orbit parameters as a function of fermion density ρ in parallel to the density of states, ρ) − ϵ k± ).

FIG. 3 .FIG. 4 .FIG. 5 .
FIG. 3. Left-hand side: the Edelstein constant DE as a function of fermion density, for tx = ty= 1 and λx = λy = 0.1, right-hand side: the chemical potential as a function of fermion density for the same parameters.