Magnetic excitations in the helical Rashba superconductor

Alireza Akbari, Peter Thalmeier

Max Planck Institute for the Chemical Physics of Solids, D-01187 Dresden,
Germany

## Abstract

We investigate the magnetic excitation spectrum in the helical state of a
noncentrosymmetric superconductor with inversion symmetry breaking and strong
Rashba spin-orbit coupling. For this purpose we derive the general expressions
of the dynamical spin response functions under the presence of strong Rashba
splitting of conduction bands, superconducting gap and external field which
lead to stabilization of Cooper pairs with finite overall momentum in a
helical state. The latter is characterized by momentum space regions of paired
and unpaired states with different quasiparticle dispersions. The magnetic
response is determined by i) excitations within and between both paired and
unpaired regions ii) anomalous coherence factors and iii) additional spin
matrix elements due to helical Rashba spin texture of bands. We show that as a
consequence typical correlated real space and spin space anisotropies appear
in the dynamical susceptibility which would be observable as a characteristic
fingerprint for a helical superconducting state in inelastic neutron
scattering investigations.

###### Contents

  1. 1 Introduction
  2. 2 Normal state Rashba bands and states
  3. 3 Helical superconductor: Segmentation into paired and unpaired regions
  4. 4 The dynamical magnetic response function for the helical Rashba superconducting state
  5. 5 Discussion of numerical results: the magnetic spectral functions
  6. 6 Summary and Conclusion
  7. A Spin operator matrix elements for the nondiagonal cartesian cases
  8. B Magnetic response function in the BCS limit

##  1 Introduction

In this work we investigate the signatures of the helical state of
noncentrosymmetric superconductors (NCS) in the magnetic excitation spectrum
obtained from inelastic neutron scattering (INS). NCS compounds have broken
inversion symmetry and consequently the gap functions in principle are
mixtures of singlet and triplet components [1, 2]. This is also possible in 2D
layered superconductors (SC) where inversion symmetry is broken in each layer
although it is preserved overall in the 3D crystal [3, 4].

In the normal state the inversion symmetry breaking leads to an odd Rashba
spin orbit coupling that results in two nondegenerate split bands
$(\lambda=\pm 1)$ with a momentum-locked spin texture characterized by
opposite helicities. The corresponding Fermi wave numbers differ by an amount
proportional to the size of Rashba spin-orbit coupling. The application of a
field shifts the two Fermi spheres perpendicular to field direction and
proportional to field strength.

In the superconducting state in an applied field this means that not only are
singlet and triplet components mixed but also the Cooper pairs $({\bf k}+{\bf
q}~{}\sigma,-{\bf k}+{\bf q}~{}\sigma^{\prime})$ will acquire a common pair
momentum $2{\bf q}$ proportional to the shift vector of Fermi surfaces and
characterized by a gap function $\Delta^{\bf k}_{{\bf q}\lambda}\exp(i{\bf
q}\cdot{\bf r})$. In this picture the orbital pair breaking is assumed to be
small. This commonly called ‘helical’ state [5] is therefore of the Fulde-
Ferrell (FF) [6] type but has a different composition of the condensation
energy due to the effect of Rashba coupling than the original Zeeman-energy
dominated FF case. The advantage of the helical state is that the finite pair
momentum appears already at moderate fields due to the shifting of Rashba
Fermi surface spheres. The modulus of its gap amplitude is, as in the FF
state, constant in real space unlike the Larkin-Ovchinnikov (LO) state [7]
which has nodal planes.

Some aspects of the helical state including Rashba coupling and Zeeman term
have been studied before, concerning mostly critical field curves [8, 9, 10].
Microscopically this state is characterized by a segmentation of Fermi surface
sheets into paired and unpaired regions determined by the balance of kinetic,
Rashba and Zeeman energies. The observation of this central aspect in the
helical (and also in the original FF) state requires spectroscopic means. It
has been proposed that STM-based quasiparticle interference (QPI) spectroscopy
may be used for FF[11] and helical Rashba NCS [12] cases. The complementary
method of magnetic inelastic neutron spectroscopy (INS) has also been
suggested for the FF state [13]. It allows to investigate the signatures of
finite momentum Cooper pairs paired/unpaired segmentation of Fermi surface
sheets on the magnetic excitation spectrum, in particular on the collective
spin resonance formation within the superconducting gap and regarding the
point group symmetry breaking anisotropy due to finite pair momentum.

In this work we generalize the analysis of magnetic excitations for the
helical Rashba SC case. We first give a brief account of the periodic Rashba
band model in the normal state. Then we discuss the simplest helical
superconducting gap model introduced by Kaur et al [5] and its condensation
energy which determines the pair momentum $2{\bf q}$ and gap size $\Delta^{\bf
k}_{\bf q}$ as function of field strength. In the main part of this work we
derive the expressions for the dynamical magnetic susecptibility tensor which
contains the static response as well as the finite frequency magnetic
spectrum. We investigate the field and momentum and polarization dependence of
the latter which are in principle accessible by INS experiments. Finally we
discuss the temperature dependence of the static homogeneous susceptibility in
view of the NMR Knight shift in the helical SC phase.

Figure 1: Field dependence of pair momentum $q$ and gap amplitude $\Delta_{q}$
for small fields $(b\ll\alpha)$ (a,d). Spectral functions i.e. quasipartilce
sheets $|E_{{\bf k}{\bf q}\lambda}^{\tau}|=\omega$ in BCS case $(q=0)$ (b,e)
for $\omega=0.06t<\Delta_{0}=0.2t$. In (b) quasiparticle sheets are absent
while they appear around nodal directions in (e). In the helical case (c,f)
$(b=0.9\Delta_{0},q>0)$ extended sheets appear for both gap symmetries due to
depairing centered around $k_{y}$ direction perpendicular to ${\bf
b}=b\hat{{\bf x}}$. The blue/red sheets correspond to $\tau=+/-$ and
outer/inner sheets to $\lambda=+/-$. In all figures the designations “s,d-
wave” refer to the gap form factors $f_{\Gamma}({\bf k})$ of the inversion
symmetric limit where $\alpha=0$ (Sec. 3).

##  2 Normal state Rashba bands and states

Here we introduce the widely used bandstructure model including the Rashba
coupling originating from inversion-symmetry breaking [1, 12]. We use the
periodic form in view of the later calculations of magnetic response functions
but occasionally discuss the features of Rashba bands in the the convenient
parabolic approximation. In the spin representation, the 2D Rashba Hamiltonian
in an external field is characterized by the following [5]

$\displaystyle H_{0}=\sum_{\bf k}\Psi^{\dagger}_{\bf k}h_{0{\bf k}}\Psi_{\bf
k};\;\;\;h_{0{\bf k}}=\xi_{\bf k}\sigma_{0}+(\alpha{\bf g}_{\bf k}+{\bf
b})\cdot{\bm{\sigma}}.$ (1)

Here $\Psi^{\dagger}_{\bf k}=(a^{\dagger}_{{\bf k}\uparrow},a^{\dagger}_{{\bf
k}\downarrow})$ represent conduction electrons with the tight binding (TB)
dispersion $\varepsilon_{\bf k}=-2t(\cos k_{x}+\cos k_{y})$ with $-\pi\leq
k_{x},k_{y}\leq\pi$. Furthermore $t>0$ is the hopping element corresponding to
a conduction band half-width $D_{c}=4t$ and $\xi_{\bf k}\equiv\varepsilon_{\bf
k}-\mu_{\rm TB}$ . The chemical potential $\mu_{\rm TB}$ falls in the interval
$-D_{c}\leq\mu_{\rm TB}\leq D_{c}$ and is counted from the band center
$\varepsilon_{\bf k}=0$. It is useful to connect this to the 2D parabolic band
model for $\mu_{\rm TB}\leq 0$ with $\varepsilon_{\bf k}=\varepsilon_{0}+{\bf
k}^{2}/2m$. With $\varepsilon_{0}=-D_{c}$ denoting the band bottom and
$m=2/D_{c}$ the effective mass. The chemical potential counted from the band
bottom is then obtained by $\mu_{P}=\mu_{\rm TB}-\varepsilon_{0}\geq 0$.
Furthermore ${\bf b}=\mu_{B}{\bf B}$ is the energy scale of the applied
magnetic field B.

The Rashba spin-orbit coupling is odd under inversion with ${\bf g}_{-{\bf
k}}=-{\bf g}_{{\bf k}}$, explicitly ${\bf g}^{P}_{\bf
k}=(k_{y},-k_{x},0)/k_{F}=(\sin\theta_{\bf k},-\cos\theta_{\bf k},0)$ in the
parabolic model. Here $\theta_{\bf k}$ is the azimuthal angle of ${\bf k}$
measured in relation to the $k_{x}$\- axis. Moreover,
$k_{F}=(2m\mu)^{\frac{1}{2}}$ is the Fermi wave number and $v_{F}=k_{F}/m$ is
the corresponding velocity. To retain consistency with the TB dispersion we
will employ the periodic form ${\bf g}^{\rm TB}_{\bf k}=(\sin k_{y},-\sin
k_{x},0)$. Both forms are normalized according to $|{\bf g}^{P}_{\bf k}|=1$
and $|{\bf g}^{\rm TB}_{\bf k}|_{\rm max}=\sqrt{2}$. Equivalency for
$k_{x},k_{y}\ll\pi$ requests that $\alpha_{P}=k_{F}\alpha_{\rm TB}$. We will
discard the indices TB, P from now on and mostly rely on the context. The
diagonalised Hamiltonian of Eq. (1) reads

$\displaystyle H_{0}=\sum_{{\bf k}\lambda}\varepsilon_{{\bf
k}\lambda}c^{\dagger}_{{\bf k}\lambda}c_{{\bf
k}\lambda};\;\;\;\varepsilon_{{\bf k}\lambda}({\bf b})=\xi_{{\bf
k}}+\lambda|\alpha{\bf g}_{{\bf k}}+{\bf b}|,$ (2)

Here $\varepsilon_{{\bf k}\lambda}({\bf b})$ are Rashba- split and Zeeman-
shifted bands (energies counted from $\mu$) with band states corresponding to
helicities $\lambda=\pm 1$. For vanishing field the two Rashba bands may be
written as

$\varepsilon^{0}_{{\bf k}\lambda}=\xi_{{\bf k}}+\lambda|\alpha{\bf g}_{{\bf
k}}|=\frac{1}{2m}(k+\lambda k_{0})^{2}-\tilde{\mu},$ (3)

with $k_{0}=\frac{1}{2}\frac{|\alpha|}{\mu}k_{F};\;{\rm
and}\;\tilde{\mu}=\mu(1+\frac{1}{4}\frac{\alpha^{2}}{\mu^{2}})$. These two
parabolic dispersions are shifted by an amount $k_{0}$. The two resulting
Fermi spheres have approximate radii $k_{F}^{\lambda}=k_{F}-\lambda
k_{0}=k_{F}(1-\frac{\lambda}{2}\frac{|\alpha|}{\mu})$ for moderately small
Rashba coupling strength $|\alpha|\ll\mu$. Then their relative deviation
$(k_{F}^{-}-k_{F}^{+})/k_{F}=|\alpha|/\mu$ is a direct measure for the size of
$\alpha$. We assumed a physically appropriate hierarchy of energy scales
described by $(b<|\alpha|<\mu<D_{c})$.

Figure 2: Brillouin-zone cuts of spectral functions (a,e,i) and corresponding
dynamical magnetic structure function for $x,y,z$ polarisation in the BCS
d-wave (b-d), helical d-wave (f-h) and helical s-wave (j-l) cases for
$\omega=0.12t<\Delta_{0}=0.2t$. In the BCS cases (b-d) the $xx$ and $yy$
response is related by a rotation $R_{\frac{\pi}{2}}$ around the $z$ axis
while the $zz$ response is by itself fourfold symmetric (Eq. (34)). These
symmetries are lost in the hellical phase for both gap models due to the
distinguished common pair momentum oriented along $q_{y}$.

For finite but small field ${\bf b}$ parallel to the plane the Rashba Fermi
surfaces resulting from Eq. (2) are shifted perpendicular to the field
orientation in opposite directions [1, 12] by a shift vector

$\displaystyle
q_{s}=\frac{b}{2\mu}k_{F}=\frac{m\mu_{B}B}{k_{F}}=\frac{b}{v_{F}}.$ (4)

In brief, the splitting of Rashba sheets is a direct measure for the Rashba
coupling strength $|\alpha|$ while their shifting perpendicular to B is a
determined only by field strength. We will use ${\bf b}=b\hat{{\bf x}}$ and
${\bf q}_{s}=q_{s}\hat{{\bf y}}$ for the geometric configuration. Finally we
give the unitary transformation from spin states to the helical Rashba
eigenstates $|{\bf k}\lambda\rangle=c^{\dagger}_{{\bf k}\lambda}|0\rangle$
which are created by operator pairs $\Phi_{\bf k}^{\dagger}=(c_{{\bf
k}+}^{\dagger}c^{\dagger}_{{\bf k}-})$ $(\lambda=\pm)$. It is defined by [14]

$\displaystyle\Phi^{\dagger}_{\bf k}=\Psi^{\dagger}_{\bf k}S_{\bf
k};\;\;\;S_{\bf k}=\frac{1}{\sqrt{2}}\left[\begin{matrix}1&ie^{-i\theta_{\bf
k}}\cr ie^{i\theta_{\bf k}}&1\end{matrix}\right],$ (5)

where the angle is obtained from

$\theta_{\bf k}=-\tan^{-1}(g_{{\bf k}x}/g_{{\bf k}y})=\tan^{-1}(\sin
k_{y}/\sin k_{x})\rightarrow\tan^{-1}(k_{y}/k_{x}).$

Here the second and last expressions correspond to TB and parabolic bands,
respectively. In the latter $\theta_{\bf k}$ is simply the azimuthal angle of
k.

##  3 Helical superconductor: Segmentation into paired and unpaired regions

In the present work we do not investigate possible mechanisms of
superconductivity in NCS compounds, for an extended review see Ref. [1]. In
the compounds with Rashba spin-orbit coupling phonons [15] and alternatively
spin-fluctuations [16, 2, 17] may provide a mechanism for Cooper pair
formation.

Here we investigate the possibility of a superconducting state with common
momentum $2{\bf q}$ of Cooper pairs due to the pairbreaking effect of the
external field in combination with Rashba spin-orbit coupling and how this
will influence the dynamical magnetic response. The actual ${\bf q}=q\hat{{\bf
y}}$ has to be evaluated by the minimization of the condensation energy in the
helical SC phase given below, one may expect it to be correlated with the
Rashba shift vector ${\bf q}_{s}=q_{s}\hat{{\bf y}}$ in Eq. (4)[18, 10]. More
general inhomogeneous SC phases with several ${\bf q}_{i}$, e.g. the ‘stripe
phase’ [5] with the pair $({\bf q},-{\bf q})$ will not be discussed here.
Furthermore we restrict the choice of the helical SC gap functions
$\Delta_{{\bf q}\lambda}({\bf k})$ to the minimal model proposed by Kaur et al
[5] whose essentials we briefly mention here: It assumes that in the limit of
$\alpha\rightarrow 0$ the SC gap is of the singlet type (e.g. s-,d-wave)
characterized by an orbital basis function $f_{\Gamma}({\bf k})$ belonging to
the irreducible representation of the tetragonal symmetry group $D_{4h}$.
Turning on a finite $\alpha$ leads to an additional k-dependence of effective
interaction and gap function due to the admixture of triplet components
enforced by the helical spin structure. For small fields $b\ll|\alpha|$ this
k-dependence may be eliminated by a phase transformation $\Delta_{{\bf
q}\pm}^{\Gamma{\bf k}}\rightarrow\pm\exp(\mp i\theta_{\bf k})\Delta_{{\bf
q}\pm}^{\Gamma{\bf k}}$ (for simplicity we keep the same symbol for the
transformed gap function). It is associated with a correspondingly transformed
effective interaction in helicity representation given by[5]

$\displaystyle\hat{V}=-\frac{V_{\Gamma}({\bf k}{\bf
k}^{\prime})}{2}(\sigma_{0}-\sigma_{x});\quad V_{\Gamma}({\bf k}{\bf
k}^{\prime})=V_{0}^{\Gamma}f_{\Gamma}({\bf k})f_{\Gamma}({\bf k}^{\prime}).$
(6)

Inserting this two-band pairing interaction into the gap equation leads to the
condition $\Delta^{\bf k}_{{\bf q}-}=-\Delta^{\bf k}_{{\bf q}+}$ [5, 12]. The
opposite sign of the two gaps is enforced by the opposite spin texture on the
two Rashba bands. For the later numerical discussion we will consider the
isotropic s-wave case $f_{\Gamma}({\bf k})=1$ and the d-wave case
$f_{\Gamma}({\bf k})=(\cos k_{x}-\cos k_{y})$ correspoinding to $\Delta^{\bf
k}_{{\bf q}\lambda}=\Delta_{{\bf q}\lambda}f_{\Gamma}({\bf k})$.

Introducing the Nambu spinors $\psi^{\dagger}_{{\bf k}{\bf
q}\lambda}=(c^{\dagger}_{{\bf k}+{\bf q}\lambda},c_{-{\bf k}+{\bf q}\lambda})$
the total BCS Hamiltonian is given by

$\displaystyle\hat{H}_{\rm BCS}=\frac{1}{2}\sum_{{\bf
k}\lambda}\psi^{\dagger}_{{\bf k}{\bf q}\lambda}\hat{h}_{{\bf k}{\bf
q}\lambda}\psi_{{\bf k}{\bf q}\lambda}+\frac{1}{2}\sum_{{\bf
k}\lambda}\varepsilon_{{\bf k}+{\bf q}\lambda}+\frac{1}{2}\sum_{{\bf
k}\lambda}\frac{\Delta^{{\bf k}2}_{{\bf q}\lambda}}{V_{0}}$ (7)

with the Hamilton matrix represented by

$\displaystyle\hat{h}_{{\bf k}{\bf q}\lambda}=\varepsilon^{a}_{{\bf k}{\bf
q}\lambda}\tau_{0}+\left[\begin{array}[]{cc}\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}&-\Delta^{\bf k}_{{\bf q}\lambda}\\\ -\Delta_{{\bf q}\lambda}^{{\bf
k}*}&-\varepsilon^{s}_{{\bf k}{\bf q}\lambda}\end{array}\right].$ (8)

Considering the symmetries $\xi_{\bf k}=\xi_{-{\bf k}}$ and ${\bf g}_{\bf
k}=-{\bf g}_{-{\bf k}}$ the diagonal matrix elements may be written as

$\displaystyle\varepsilon_{{\bf k}+{\bf q}\lambda}({\bf b})$
$\displaystyle=\xi_{{\bf k}+{\bf q}}+\lambda|\alpha{\bf g}_{{\bf k}+{\bf
q}}+{\bf b}|,$ (9) $\displaystyle\varepsilon_{-{\bf k}+{\bf q}\lambda}({\bf
b})$ $\displaystyle=\xi_{{\bf k}-{\bf q}}+\lambda|\alpha{\bf g}_{{\bf k}-{\bf
q}}-{\bf b}|.$

It is convenient to introduce symmetric (s) and antisymmetric (a) expressions
according to

$\displaystyle\varepsilon^{s,a}_{{\bf k}{\bf q}\lambda}$
$\displaystyle=\frac{1}{2}(\varepsilon_{{\bf k}+{\bf
q}\lambda}\pm\varepsilon_{-{\bf k}+{\bf q}\lambda}).$ (10)

They have even/odd symmetry $\varepsilon^{s}_{-{\bf k}{\bf
q}\lambda}=\varepsilon^{s}_{{\bf k}{\bf q}\lambda}$ and
$\varepsilon^{a}_{-{\bf k}{\bf q}\lambda}=-\varepsilon^{a}_{{\bf k}{\bf
q}\lambda}$ with respect to inversion. The latter enforces the property
$\sum_{{\bf k}\lambda}\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=0$ where the sum
over k covers both paired and unpaired regions explained below.
The second term in Eq. (8) can be diagonalized by a Bogoliubov transformation
[19, 12] leading to quasiparticle states created by $\alpha_{{\bf
k}\lambda},\beta_{{\bf k}\lambda}$ and a corresponding quasiparticle
Hamiltionian

$\displaystyle\begin{aligned} H_{\rm BCS}=&E_{G}+\frac{1}{2}\sum_{{\bf
k}\lambda}(|E^{+}_{{\bf k}{\bf q}\lambda}|\alpha^{\dagger}_{\bf k}\alpha_{\bf
k}+|E^{-}_{{\bf k}{\bf q}\lambda}|\beta^{\dagger}_{\bf k}\beta_{\bf
k}).\end{aligned}$ (11)

To visualize the quasiparticle sheets we will use the spectral function

$\displaystyle\bar{A}^{\lambda}_{{\bf k}{\bf
q}}(\omega>0)=\delta(\omega-|E^{+}_{{\bf k}{\bf
q}\lambda}|)+\delta(\omega-|E^{-}_{{\bf k}{\bf q}\lambda}|)$ (12)

in subsequent figures. Here the (positive) quasiparticle energies
$|E^{\tau}_{{\bf k}{\bf q}\lambda}|$ are given as $(\tau=\pm,\bar{\tau}=\mp)$:

$\displaystyle\begin{aligned} E^{\tau}_{{\bf k}{\bf q}\lambda}&=E_{{\bf k}{\bf
q}\lambda}+\tau\varepsilon^{a}_{{\bf k}{\bf q}\lambda}=E^{\bar{\tau}}_{-{\bf
k}{\bf q}\lambda},\\\ E_{{\bf k}{\bf q}\lambda}&=[\varepsilon^{s2}_{{\bf
k}{\bf q}\lambda}+\Delta_{{\bf q}\lambda}^{{\bf k}2}]^{\frac{1}{2}}=E_{-{\bf
k}{\bf q}\lambda}.\end{aligned}$ (13)

If both $E^{\tau}_{{\bf k}{\bf q}\lambda}>0$ $(\tau=\pm)$ for a specific k and
$\lambda$ the Cooper pair state with pair momentum $2{\bf q}$ is stable.
However, if $E^{+}_{{\bf k}{\bf q}\lambda}<0$ or $E^{-}_{{\bf k}{\bf
q}\lambda}<0$ the pair state is instable and only unpaired quasiparticle
states exist for the wave vectors ${\bf k}+{\bf q}$, $-{\bf k}+{\bf q}$.
Although for such wave vectors $|E^{\pm}_{{\bf k}{\bf q}\lambda}|$ are normal
quasiparticle excitations their energy nevertheless depends on the gap size
$\Delta_{{\bf q}\lambda}$ determined only by the paired FS sections. This is
due to the fact that in the coherent helical SC ground state the unpaired
electrons and holes also experience the pairing potential supported by the
paired electrons, although they don’t contribute to it.

The constant $E_{G}=\langle H_{\rm BCS}\rangle$ appearing in Eq. (11) is equal
the total ground state energy $E_{G}({\bf q},\Delta_{{\bf q}\pm})$ of the
helical state. Subtracting the normal state ground state energy
$E^{0}_{G}=(1/2)\sum_{{\bf k}\lambda}(\varepsilon^{0}_{{\bf
k}\lambda}-|\varepsilon^{0}_{{\bf k}\lambda}|)$ we obtain the superconducting
condensation energy $E_{c}=E_{G}-E^{0}_{G}$ as [12]

$\displaystyle\begin{aligned} E_{c}({\bf q},\Delta_{{\bf
q}\pm})=\frac{1}{2}\sum_{\lambda}\Bigg{[}&\sum_{\bf
k}N\bigl{(}\frac{|\Delta^{\bf k}_{{\bf
q}\lambda}|^{2}}{V_{0}}\bigr{)}-\sum_{\bf k}\Bigl{\\{}(E_{{\bf k}{\bf
q}\lambda}-|\varepsilon^{0}_{{\bf k}\lambda}|)+(\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}\\!-\\!\varepsilon^{0}_{{\bf k}\lambda})\\\ &\\!+[E^{+}_{{\bf k}{\bf
q}\lambda}\Theta(-E^{+}_{{\bf k}{\bf q}\lambda})+E^{-}_{{\bf k}{\bf
q}\lambda}\Theta(-E^{-}_{{\bf k}{\bf
q}\lambda})]\Bigr{\\}}\Bigg{]},\end{aligned}$ (14)

where $\Delta^{{\bf k}}_{{\bf q}\lambda}=\Delta_{{\bf
q}\lambda}f_{\Gamma}({\bf k})$. In both s,d-wave cases the normalization is
$(1/N)\sum_{{\bf k}}f_{\Gamma}({\bf k})^{2}=1$ in the first ${\bf k}$-sum of
the above equation so that this term is equal to $N|\Delta_{{\bf
q}\lambda}|^{2}/V_{0}$. Because of the separation of Eq. (10) the odd
$\varepsilon^{a}_{{\bf k}{\bf q}\lambda}$ Rashba energies enter only in the
last term of Eq. (13) but not under the square root. The above energy
functional must be minimized with respect to q and $\Delta_{{\bf q}\pm}$ for a
given size of the Rashba coupling $\alpha$ and as function of field $b$.
Possible ground states are the helical SC state $({\bf q}\neq 0,|\Delta_{{\bf
q}\lambda}|>0)$, the BCS state $({\bf q}=0,|\Delta_{0\lambda}|>0)$ or the
unpolarized normal $(b=0,{\bf q}=0,\Delta_{{\bf q}\lambda}=0)$ states. We note
again that we are restricted to the small field range $b\ll|\alpha|$ due to
the assumption of a field-independent spin texture only determined by the
Rashba term.

The minimization problem is much simplified by the equal gap magnitude
$|\Delta_{{\bf q}\pm}|=\Delta_{\bf q}$ in the model of Eq. (6). Strictly
speaking this holds only when q=0 but this minimization constraint will also
be kept for the helical SC case. The effective interaction strength $V_{0}$ in
Eqs. (6,14) is connected to the BCS gap amplitude $\Delta_{0}$ by the gap
equation $(b=0)$:

$\displaystyle\frac{1}{V_{0}}=\frac{1}{2N}\sum_{{\bf
k}\lambda}\frac{f_{\Gamma}({\bf k})^{2}}{2E_{{\bf k}\lambda}}.$ (15)

Here the BCS zero-field quasiparticle energy is simply $E_{{\bf
k}\lambda}=[\varepsilon^{02}_{{\bf k}\lambda}+\Delta_{0}^{2}]^{\frac{1}{2}}$.
Minimization of $E_{c}({\bf q},\Delta_{{\bf q}})$ with respect to
$\Delta_{{\bf q}}$ and q determines the true gap $\Delta_{\bf q}(b,\alpha)$
and wave vector ${\bf q}(b,\alpha)$ that characterise the helical SC state.
One must keep in mind, however, that the model defined in Eq. (6) is only
valid in the low field limit $b/|\alpha|<1$. An example of the resulting
$\Delta_{q}(b),q(b)$ dependence for small fields and fixed $\alpha$ is shown
in Fig. 2(a,d) for the two gap symmetries. These curves depend considerably on
the chemical potential $\mu_{P}$ and Rashba coupling $\alpha$ because for
numerical reasons the gap value $\Delta_{0}$ is not negligible compared to
them.

Figure 3: Axis cuts along $(10)$ and $(01)$ of $xx,yy$ dynamical magnetic
structure functions for s-wave case at $\omega=0.12t$. In the BCS case $(q=0)$
the $xx(10)$ and $yy(01)$ pairs (a) and likewise $yy(10)$ and $xx(01)$ pairs
(d) are equivalent due to $R_{\frac{\pi}{2}}$ rotational symmetry. In the
helical phase this symmetry is lost due to distinguished $q_{y}$ direction of
common pair momentum which introduces combined momentum space and spin space
anisotropy. This is obvious from comparison of (a) with (b,c) and (d) with
(e,f) (see also Fig. 2).

##  4 The dynamical magnetic response function for the helical Rashba
superconducting state

Now we come to the main objective of this work. The calculation of the
dynamical magnetic response function of a helical Rashba superconductor is
considerably more involved than in the simple BCS state [20, 21] or even the
centrosymmetric FF superconductor [22, 13] due to the complicated spin
textures of Rashba bands which is encoded in the unitary transformation matrix
to helical states given in Eq.(5). Here we give the details of its derivation.
For the noninteracting helical Rashba quasiparticles the dynamical magnetic
susceptibility is obtained from the bubble diagram without vertex
corrrections. Because of the helical spin structure in principle all cartesian
elements $(\alpha,\beta=x,y,z)$ of the susceptibilities may be nonzero and
different. They are defined by

$\displaystyle\chi_{\alpha\beta}(\tilde{{\bf
q}},i\nu_{m})=\frac{1}{N}\int_{0}^{\tilde{\beta}}d\tilde{\tau}e^{i\nu_{m}\tilde{\tau}}\langle
T_{\tilde{\tau}}S_{\alpha}(\tilde{{\bf
q}},\tilde{\tau})S_{\beta}^{\dagger}(\tilde{{\bf q}},0)\rangle,$ (16)

where $\tilde{\beta}=1/kT$ and the spin operators are given in spin $(a_{{\bf
k}\sigma}$) and $(c_{{\bf k}\lambda}$) helical bases as

$\displaystyle S_{\alpha}(\tilde{{\bf q}})=\frac{1}{2}\sum_{{\bf
k}\sigma\sigma^{\prime}}a^{\dagger}_{{\bf k}+\tilde{{\bf
q}}\sigma^{\prime}}\sigma^{\alpha}_{\sigma^{\prime}\sigma}a_{{\bf
k}\sigma}=\frac{1}{2}\sum_{{\bf k}\lambda\lambda^{\prime}}c^{\dagger}_{{\bf
k}+\tilde{{\bf
q}}\lambda^{\prime}}\tilde{\sigma}^{\alpha}_{\lambda^{\prime}\lambda}({\bf
k}+\tilde{{\bf q}},{\bf k})c_{{\bf k}\lambda}.$ (17)

Here the latter presentation has to be chosen as it forms the eigenbasis of
the Rashba Hamiltonian. Therefore the effective spin operators
$\frac{1}{2}\tilde{\sigma}_{\lambda^{\prime}\lambda}^{\alpha}({\bf
k}+\tilde{{\bf q}},{\bf k})$ in this basis are now momentum dependent and they
are obtained by the transformation of Eq.(5) according to

$\displaystyle\tilde{\sigma}_{\lambda^{\prime}\lambda}^{\alpha}({\bf
k}^{\prime},{\bf
k})=\sum_{\sigma,\sigma^{\prime}}S^{*}_{\sigma^{\prime}\lambda^{\prime}}({\bf
k}^{\prime})\sigma^{\alpha}_{\sigma^{\prime}\sigma}S_{\sigma\lambda}({\bf
k}).$ (18)

The matrices (with $\lambda^{\prime}\lambda$ indices) have the conjugation
property $\tilde{\sigma}^{\alpha}({\bf k}^{\prime},{\bf
k})^{\dagger}=\tilde{\sigma}^{\alpha}({\bf k},{\bf k}^{\prime})$ and obey the
commutation rules $[\tilde{\sigma}^{\alpha}({\bf k}^{\prime},{\bf
k}),{\tilde{\sigma}^{\beta}({\bf k}^{\prime},{\bf
k})}^{\dagger}]=i\epsilon_{\alpha\beta\gamma}(\tilde{\sigma}^{\gamma}({\bf
k}^{\prime})+\tilde{\sigma}^{\gamma}({\bf k}))$ where
$\epsilon_{\alpha\beta\gamma}$ is the fully antisymmetric tensor and
$\tilde{\sigma}^{\gamma}({\bf k})\equiv\tilde{\sigma}^{\gamma}({\bf k},{\bf
k})$ etc. The cartesian spin expectation values in each Rashba state $|{\bf
k}\lambda\rangle$ are given by $\langle\tilde{{\bm{\sigma}}}^{\lambda}({\bf
k})\rangle=(-\lambda\sin\theta_{\bf k},\lambda\cos\theta_{\bf k},0)$.
Therefore the spins of Rashba states are perpedicular to the momentum
direction $\hat{{\bf k}}=(\cos\theta_{\bf k},\sin\theta_{\bf k},0)$, i.e.
$\langle\tilde{{\bm{\sigma}}}^{\lambda}({\bf k})\rangle\cdot\hat{{\bf k}}=0$
and $\langle\tilde{{\bm{\sigma}}}^{\lambda}({\bf k})\rangle\times\hat{{\bf
k}}=\lambda\hat{{\bf z}}$. Furthermore they are opposite on the two Rashba
bands $\lambda=\pm$.

Using the helical eigenbase and its corresponding quasiparticle Green’s
functions and effective spin operators the dynamical susceptibility may now be
written as

$\displaystyle\chi^{\alpha\beta}_{0{\bf q}}(\tilde{{\bf
q}},i\nu_{m})=-\frac{T}{4}\frac{1}{N}\sum_{{\bf
k}n\lambda\lambda^{\prime}}\tilde{\sigma}^{\alpha}_{\lambda\lambda^{\prime}}({\bf
k}^{\prime},{\bf k})\tilde{\sigma}^{\beta}_{\lambda^{\prime}\lambda}({\bf
k},{\bf k}^{\prime})\sum_{n}Tr_{\tau}[\hat{G}_{\bf q}({\bf
k},i\omega_{n})\hat{G}_{\bf q}({\bf k}^{\prime},i\omega_{n}^{\prime})],$ (19)

where $i\omega_{n^{\prime}}=i\omega_{n}+i\nu_{m}$ with $n^{\prime}=n+m$. Here
$\tau$ is the Nambu index in particle-hole space and $Tr_{\tau}$ denotes the
corresponding trace of the product of Nambu Green’s function matrices given in
Eq. (20). The latter are obtained from Eq. (8) as

$\displaystyle\hat{G}_{{\bf q}\lambda}({\bf k},i\omega_{n})=$
$\displaystyle(i\omega_{n}-\hat{h}_{{\bf k}{\bf q}\lambda})^{-1}$ (20)
$\displaystyle=$ $\displaystyle\frac{1}{(i\omega_{n}-E^{+}_{{\bf q}{\bf
k}\lambda})(i\omega_{n}+E^{-}_{{\bf q}{\bf
k}\lambda})}\left[\begin{array}[]{cc}i\omega_{n}+\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda}&-\Delta^{\bf k}_{{\bf
q}\lambda}\\\\[5.69046pt] -\Delta^{{\bf k}*}_{{\bf
q}\lambda}&i\omega_{n}-\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}-\varepsilon^{a}_{{\bf k}{\bf q}\lambda}\end{array}\right].$

Using the definitions

$\displaystyle\begin{aligned} &M^{\alpha\beta}_{\lambda^{\prime}\lambda}({\bf
k}^{\prime},{\bf k})=\tilde{\sigma}^{\alpha}_{\lambda\lambda^{\prime}}({\bf
k}^{\prime},{\bf k})\tilde{\sigma}^{\beta}_{\lambda^{\prime}\lambda}({\bf
k},{\bf k}^{\prime});\;\;\;\\\ &\hat{\chi}_{\lambda\lambda^{\prime}}({\bf
k}{\bf k}^{\prime},i\nu_{m})=-\frac{T}{4}\sum_{n}Tr_{\tau}[\hat{G}_{\bf
q}({\bf k},i\omega_{n})\hat{G}_{\bf q}({\bf
k}^{\prime},i\omega_{n^{\prime}})],\end{aligned}$ (21)

the response function may be written as a product of helical state spin matrix
elements and a dynamical kernel, respectively, according to
$(i\nu_{m}\rightarrow\omega+i\eta)$

$\displaystyle\chi^{\alpha\beta}_{0{\bf q}}(\tilde{{\bf
q}},\omega)=\frac{1}{N}\sum_{\lambda\lambda^{\prime}}\sum_{{\bf
k}}M^{\alpha\beta}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf
k})\hat{\chi}_{\lambda\lambda^{\prime}}({\bf k}{\bf k}^{\prime},\omega).$ (22)

First we calculate the kernel where the trace may be evaluated as

$\frac{1}{2}Tr_{\tau}[\hat{G}_{\bf q}({\bf k},i\omega_{n})\hat{G}_{\bf q}({\bf
k}^{\prime},i\omega_{n^{\prime}})]=\frac{(i\omega_{n}-\varepsilon^{a}_{{\bf
k}{\bf q}\lambda})(i\omega_{n^{\prime}}-\varepsilon^{a}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})+\varepsilon^{s}_{{\bf k}{\bf
q}\lambda}\varepsilon^{s}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}+\Delta_{{\bf q}\lambda}^{\bf k}\Delta_{{\bf
q}\lambda^{\prime}}^{\bf k}}{(i\omega_{n}-E^{+}_{{\bf k}{\bf
q}})(i\omega_{n}+E^{-}_{{\bf k}{\bf q}})(i\omega_{n^{\prime}}-E^{+}_{{\bf
k}^{\prime}{\bf q}})(i\omega_{n^{\prime}}+E^{+}_{{\bf k}^{\prime}{\bf q}})}.$
(23)

Performing the sum over the Matsubara frequencies $\omega_{n}$ and
analytically continuing to the real axis according to
$i\nu_{m}\rightarrow\omega+i\eta$ a lengthy calculation leads to the final
result for the kernel in the dynamical susceptibility of Eq. (22)

$\displaystyle\hat{\chi}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime},\omega)=$ $\displaystyle\frac{1}{2}\tilde{C}^{\bf q}_{+}({\bf
k}\lambda,{\bf k}^{\prime}\lambda^{\prime})\bigl{[}\frac{f(E^{+}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}})-f(E^{+}_{{\bf k}{\bf
q}\lambda})}{\omega-(E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}-E^{+}_{{\bf k}{\bf q}\lambda})+i\eta}-\frac{f(E^{-}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}})-f(E^{-}_{{\bf k}{\bf
q}\lambda})}{\omega+(E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}-E^{-}_{{\bf k}{\bf q}\lambda})+i\eta}\bigr{]}+$ (24)
$\displaystyle\frac{1}{2}\tilde{C}^{\bf q}_{-}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\bigl{[}\frac{1-f(E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})-f(E^{+}_{{\bf k}{\bf q}\lambda})}{\omega+(E^{-}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}}+E^{+}_{{\bf k}{\bf
q}\lambda})+i\eta}+\frac{f(E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})+f(E^{-}_{{\bf k}{\bf q}\lambda})-1}{\omega-(E^{+}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}}+E^{-}_{{\bf k}{\bf
q}\lambda})+i\eta}\bigr{]},$

where $f(E)=(\exp(E/T)+1)^{-1}$ is the Fermi function. We may also obtain a
different presentation of the last two terms by using $1-f(E)=f(-E)$. The
anomalous superconducting coherence factors of magnetic response for the
helical phase are given by

$\displaystyle\begin{aligned} \tilde{C}^{\bf q}_{\pm}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})=\frac{1}{2}\bigl{[}1\pm\frac{\varepsilon^{s}_{{\bf
k}{\bf q}\lambda}\varepsilon^{s}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}+\Delta_{{\bf q}\lambda}^{\bf k}\Delta_{{\bf
q}\lambda^{\prime}}^{{\bf k}^{\prime}}}{E_{{\bf k}{\bf q}\lambda}E_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}}}\bigr{]}.\end{aligned}$ (25)

Furthermore we have to compute the matrix elements in Eq. (21) for the
evaluation of the susceptibility of Eq. (22). It is useful to note that they
satisfy Hermitean symmetry which derive from the fact that the effective spin
operators in helical representation are also Hermitean, i.e. fulfil
$\tilde{\sigma}^{\alpha}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf
k})=\tilde{\sigma}^{\alpha*}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})$. Then it follows that

$\displaystyle\begin{aligned} M^{\alpha\beta}_{\lambda^{\prime}\lambda}({\bf
k},{\bf k}^{\prime})=\;&M^{\beta\alpha}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})^{*},\\\ M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k},{\bf
k}^{\prime})=\;&M^{\alpha\alpha}_{\lambda\lambda^{\prime}}({\bf k}{\bf
k}^{\prime})^{*}=|\tilde{\sigma}^{\alpha}_{\lambda\lambda^{\prime}}({\bf
k}{\bf k}^{\prime})|^{2},\end{aligned}$ (26)

so that cartesian diagonal elements are real symmetric. Using the explicit
form given by Eqs. (18,5) we derive the latter $(\lambda^{\prime}\lambda)$
matrices as

$\displaystyle\begin{aligned} \left\\{\begin{array}[]{r}M^{xx}({\bf
k}^{\prime}{\bf k})\\\ M^{yy}({\bf k}^{\prime}{\bf
k})\end{array}\right\\}&=\frac{1}{2}\left(\begin{matrix}1\mp\cos(\theta_{\bf
k}+\theta_{{\bf k}^{\prime}})&1\pm\cos(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})\cr 1\pm\cos(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})&1\mp\cos(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})\end{matrix}\right),\\\ M^{zz}({\bf k}^{\prime}{\bf
k})&=\frac{1}{2}\left(\begin{matrix}1-\cos(\theta_{\bf k}-\theta_{{\bf
k}^{\prime}})&1+\cos(\theta_{\bf k}-\theta_{{\bf k}^{\prime}})\cr
1+\cos(\theta_{\bf k}-\theta_{{\bf k}^{\prime}})&1-\cos(\theta_{\bf
k}-\theta_{{\bf k}^{\prime}})\end{matrix}\right),\end{aligned}$ (27)

where $xx,yy$ elements correspond to upper or lower sign, respectively.
Furthermore the diagonal elements of
$M_{\lambda^{\prime}\lambda}^{\alpha\alpha}$ matrices describe intraband
$(\lambda^{\prime}=\lambda)$ and the nondiagonal ones interband
$(\lambda^{\prime}\neq\lambda)$ dipolar transitions between the Rashba bands.
The matrix elements for the nondiagonal cartesian indices are given in
Appendix A. The complete dynamical response functions may now be obtained from
Eq. (22) using Eqs. (24,25,27) and for the nondiagonal case Eq.(35) as input.

Figure 4: Dispersive plot of magnetic excitations using spin
polarisation/momentum configurations $xx(10)$ and $yy(01)$ of dynamical
magnetic structure functions for s-wave case. In the BCS limit $({\bf q}=0)$
(a) the response for the two geometries is equal and vanishing due to
$\omega<2\Delta_{0}$ (cf. Appendix B). In the helical phase (b,c) the low
energy quasiparticle sheets (Fig.2(i)) lead to non-vanishing dispersive
excitations with strong momentum and spin-space anisotropy complementary to
Figs.2,3.

The above generalized helical expressions reduce to the wellknown BCS results
for the magnetic response [20, 21, 23, 24] in the BCS limit $(b,q=0)$ which
are given in Appendix B for comparison. If we restrict to the case where ${\bf
k},{\bf k}^{\prime}$ lie both in the segment with paired states (e.g. $E_{{\bf
k}{\bf q}\lambda}>0,E_{{\bf k}^{\prime}{\bf q}\lambda^{\prime}}>0$) then the
terms in Eq. (24) may be consecutively interpreted as: quasiparticle
scattering (first two terms), pair annihilation (third) and pair creation
(fourth) terms. For general ${\bf k},{\bf k}^{\prime}$ one has to consider
processes involving quasiparticles from the paired (p) as well as the unpaired
(u) Fermi surface segments. To simplify matters in this general case we
consider the zero temperature limit when the Fermi function may be expressed
by the step function according to $f(E)=1-\Theta(E)=\Theta(-E)$. Then we
obtain

$\displaystyle\chi^{\alpha\beta}_{0{\bf q}}(\tilde{{\bf q}},\omega)=$
$\displaystyle\frac{1}{N}\sum_{{\bf
k}\lambda,\lambda^{\prime}}M^{\alpha\beta}_{\lambda^{\prime}\lambda}({\bf
k}^{\prime},{\bf k})\times$ (28) $\displaystyle\Bigl{\\{}$
$\displaystyle\frac{1}{2}\tilde{C}^{\bf q}_{+}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\bigl{[}\frac{\Theta(E^{+}_{{\bf k}{\bf
q}\lambda})-\Theta(E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})}{\omega-(E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}-E^{+}_{{\bf k}{\bf
q}\lambda})+i\eta}-\frac{\Theta(E^{-}_{{\bf k}{\bf
q}\lambda})-\Theta(E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})}{\omega+(E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}-E^{-}_{{\bf k}{\bf q}\lambda})+i\eta}\bigr{]}+$
$\displaystyle\frac{1}{2}\tilde{C}^{\bf q}_{-}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\bigl{[}\frac{\Theta(E^{+}_{{\bf k}{\bf
q}\lambda})-\Theta(-E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})}{\omega+(E^{-}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}+E^{+}_{{\bf k}{\bf
q}\lambda})+i\eta}+\frac{\Theta(-E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}})-\Theta(E^{-}_{{\bf k}{\bf
q}\lambda})}{\omega-(E^{+}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}+E^{-}_{{\bf k}{\bf q}\lambda})+i\eta}\bigr{]}\Bigr{\\}}.$

If we look at the numerators of the four terms in this equations we realize
that the first two correspond to quasiparticle scattering processes ${\bf
k}\leftrightarrow{\bf k}^{\prime}$ from paired (p) to unpaired (u) FS segments
and vice versa (p-u,u-p). whereas the third and fourth term are quasiparticle
annihilation and creation respectively, containing only processes between the
paired (p-p) or unpaired (u-u) segments. The dynamical structure function for
low temperature (without the Bose factor) which is proportional to the INS
cross section [25] is then obtained as

$S(\tilde{{\bf
Q}},\omega)=\sum_{\alpha\beta}(\delta_{\alpha\beta}-\hat{Q}_{\alpha}\hat{Q}_{\beta})\frac{1}{\pi}{\rm
Im}\chi^{\alpha\beta}_{0{\bf q}}(\tilde{{\bf Q}},\omega),$ (29)

where $\tilde{{\bf Q}}={\bf k}^{\prime}-{\bf k}=\tilde{{\bf q}}+{\bf K}$ is
the total momentum transfer with K denoting a reciprocal lattice vector and
$\hat{{\bf Q}}=\tilde{{\bf Q}}/|\tilde{{\bf Q}}|$ denoting the unit vector or
direction of total momentum transfer. The prefactor projects out only
scattering processes where the magnetic moment is perpendicular to
$\tilde{{\bf Q}}$. By chosing various appropriate values of the latter the
individual susceptibility components, in particicualr the diagonal ones
$(\alpha=\beta)$ on which we will focus in the discussion below can be
accessed by INS.

Finallly we may also consider the special case of the static staggered
susceptibility components by setting $\omega=0$ in Eq. (24). After some
rearrangements we obtain $(\bar{\tau}=-\tau)$:

$\displaystyle\chi^{\alpha\beta}_{0{\bf q}}(\tilde{{\bf
q}})=\frac{1}{4N}\sum_{{\bf
k}\lambda\lambda^{\prime}\tau}M^{\alpha\beta}_{\lambda^{\prime}\lambda}({\bf
k}^{\prime},{\bf k})\Bigl{\\{}$ $\displaystyle\tilde{C}^{\bf q}_{+}({\bf
k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\frac{\tanh\frac{\beta}{2}E^{\tau}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}}-\tanh\frac{\beta}{2}E^{\tau}_{{\bf k}{\bf
q}\lambda}}{E^{\tau}_{{\bf k}^{\prime}{\bf q}\lambda^{\prime}}-E^{\tau}_{{\bf
k}{\bf q}\lambda}}+$ (30) $\displaystyle\tilde{C}^{\bf q}_{-}({\bf
k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\frac{\tanh\frac{\beta}{2}E^{\bar{\tau}}_{{\bf
k}^{\prime}{\bf q}\lambda^{\prime}}+\tanh\frac{\beta}{2}E^{\tau}_{{\bf k}{\bf
q}\lambda}}{E^{\bar{\tau}}_{{\bf k}^{\prime}{\bf
q}\lambda^{\prime}}+E^{\tau}_{{\bf k}{\bf q}\lambda}}\Bigr{\\}}.$

We may further specify to $\tilde{{\bf q}}=0$ which is the homogeneous spin
susceptibility. In this case the nondiagonal $(\lambda\neq\lambda^{\prime})$
contributions are interband vanVleck terms with a large energy denominator
whose modulus is $2|\alpha|k_{F}\gg 2\Delta_{0}$. Therefore they may be
neglected compared to intraband terms $(\lambda^{\prime}=\lambda)$. Then using
Using $\tilde{C}_{+}^{\bf q}({\bf k}\lambda{\bf k}\lambda)=1$ and
$\tilde{C}_{-}^{\bf q}({\bf k}\lambda{\bf k}\lambda)=0$ we arrive at

$\chi^{\alpha\beta}_{0{\bf q}}(0)\simeq\frac{1}{2N}\sum_{{\bf
k}\lambda\tau}M^{\alpha\beta}({\bf k}\lambda)\Bigl{(}-\frac{\partial
f}{\partial E^{\tau}_{{\bf k}{\bf
q}\lambda}}\Bigr{)}=\frac{\beta}{4}\frac{1}{2N}\sum_{{\bf
k}\lambda\tau}M^{\alpha\beta}({\bf
k}\lambda)\Bigl{(}\frac{1}{\cosh^{2}\frac{\beta}{2}E^{\tau}_{{\bf k}{\bf
q}\lambda}}\Bigr{)}.$ (31)

Where the diagonal helical matrix elements now simplify to

$\displaystyle M^{xx}({\bf k}\lambda)=\frac{1}{2}(1-\cos\theta_{\bf k});\quad
M^{yy}({\bf k}\lambda)=\frac{1}{2}(1+\cos\theta_{\bf k});\quad M^{zz}({\bf
k}\lambda)=0.$ (32)

In the parabolic band approximation (for $\mu\ll D_{c}$) with a 2D DOS
$N^{0}_{\lambda}=m^{*}k^{\lambda}_{F}/2\pi$ and effective mass $m^{*}=2/D_{c}$
and Fermi vector $k_{F}^{\lambda}=k_{F}-\lambda k_{0}$,
$k^{0}_{F}=(2m^{*}\mu)^{\frac{1}{2}}$ the homogeneous susceptibility may
approximately be written as

$\displaystyle\chi^{\alpha\alpha}_{0{\bf
q}}(0,T)=\sum_{\lambda\tau}N^{0}_{\lambda}\int\frac{d\theta_{\bf
k}}{2\pi}M^{\alpha\alpha}(\theta_{\bf
k}\lambda)Y_{q\lambda}^{\tau}(\theta_{\bf k},T);\;\;\;Y^{\tau}_{{\bf
q}\lambda}(\theta_{\bf
k},T)=\frac{1}{4\pi}\int\frac{d\varepsilon_{\lambda}}{\cosh^{2}\frac{\beta}{2}E^{\tau}_{{\bf
k}{\bf q}\lambda}},$ (33)

where $Y^{\sigma}_{q\lambda}(T,\theta_{\bf k})$ is a generalized angular
resolved Yosida function [26]. The static homogeneous susceptibility
$\chi^{\alpha\alpha}_{0{\bf q}}(\tilde{{\bf q}})$ describes the temperature
dependence of the NMR Knight shift [27, 13] of the singlet superconductor (for
$\alpha=0$) in the helical phase. Because of Eq. (32) $xx$ and $yy$ components
are equivalent and the $zz$ component vanishes. For plotting the temperature
dependence of the intra-band contribution we use a phenomenological
temperature dependence of the helical gap $\Delta_{q}$ given by
$\Delta_{q}(t)=\Delta_{q}\tanh[1.74\sqrt{\frac{1-t}{t}}]$ where $t=T/T_{c}(H)$
is the reduced temperature. The comparison of $\chi^{\alpha\alpha}_{0{\bf
q}}(0,T)$ in the BCS $({\bf q}=0)$ and helical $({\bf q}\neq 0)$ case in the
interval $t\in[0,1]$ is shown in Fig. 5.

Figure 5: Temperature dependence of static homogeneous susceptibility
contributing to NMR Knight shift for BCS and helical phase. The large nonzero
value for $T\rightarrow 0$ in (b) is due to normal quasiparticles in depaired
momentum space region.

##  5 Discussion of numerical results: the magnetic spectral functions

Here we discuss typical numerical results for the magnetic spectrum that may
be obtained from the theory developed in the previous sections. In particular
we focus on the resulting various momentum-space and cartesian spin space
anisotropies of the response function with respect to the direction of the
field ${\bf b}=b\hat{{\bf x}}$ and the corresponding overall pair momentum
direction ${\bf q}=q\hat{{\bf y}}$. In this discussion we have to restrict to
results for the small field range $b\ll\alpha$ where the assumption of field-
independent spin texture of Rashba states is still acceptable. This is the
basis for the simplified gap models [5] employed here. For the feasibility of
numerical computations we had to use a sizable gap amplitude of
$\Delta_{0}/t=0.2$. We use the s- and d- wave gap models. We stress that this
designation refers to the limiting case of $\alpha=0$ as mentioned in Sec. 3.
In the majority of results presented here we use a Rashba parameter value
$\alpha=0.6t$ and field strength $b=0.9\Delta_{0}$ (i.e. $b/\alpha=0.3$).
Furthermore the chemical potential is set to $\mu=\mu_{TB}=-2.8t$ (or
$\mu_{P}=1.2t$) to achieve quasi- circular Rashba Fermi surfaces. For
$b=0.9\Delta_{0}$, we found ($q/\pi=0.011$; $\Delta_{q}=0.75\Delta_{0}$) and
($q/\pi=0.015$; $\Delta_{q}=0.85\Delta_{0}$) for s-wave and d-wave,
respectively.

In Fig. 1 we show the field dependence of $q,\Delta_{q}$ in the small field
region (a,b) and the associated change in the quasiparticle sheets represented
by the spectral function for both gap symmetries. In the BCS case the fully
gapped s-wave state has none while in the d-wave case the sheets for both
Rashba bands (inner and outer circle) appear first around the nodal
directions. In the helical phase with finite overall pair momentum $q$ large
quasiparticle sheets exist around the depaired momentum space regions
perpendicular to the field direction. It is obvious that the fourfold symmetry
of the BCS state is broken by the distinguished q-direction and this should be
visible in the dynamical magnetic structure function as a momentum-space
anisotropy of the response. Because in the Rashba state with dominating
$\alpha$ spin and momentum directions are locked this should also be
transfered to a spin-space anisotropy, expressed by non-equivalence of
$\chi_{0}^{\alpha\alpha}({\bf q},\omega)$ for the cartesian directions
$\alpha=x,y,z$. While the former effect is already observed in the
centrosymmetric FF state [13], the latter is characteristic for the helical
Rashba SC due to the spin-momentum locking.

This is nicely illustrated by the constant-$\omega$ $(=0.12t=0.6\Delta_{0})$
cut of the spectrum (imaginary part of the dynamical susceptibility) in the
panels of Fig 2. It shows the BCS and helical d-wave case in the top and
center row and the helical s-wave case in the bottom row (in-the BCS s-wave
case there is no intensity at finite $\omega<2\Delta_{0}$ due to the full
gapping (Appendix B)). The spectral functions are depicted on the left column
and the $xx,yy,zz$ dynamical magnetic response in the three consecutive
columns. We notice that in the BCS case the combined spin/momentum space
rotational symmetries

$\displaystyle\chi_{00}^{xx}(R_{\frac{\pi}{2}}\tilde{{\bf
q}}.\omega)=\chi_{00}^{yy}(\tilde{{\bf
q}},\omega);\quad\chi_{00}^{yy}(R_{\frac{\pi}{2}}\tilde{{\bf
q}},\omega)=\chi_{00}^{xx}(\tilde{{\bf
q}},\omega);\quad\chi_{00}^{zz}(R_{\frac{\pi}{2}}\tilde{{\bf
q}},\omega)=\chi_{00}^{zz}(\tilde{{\bf q}},\omega),$ (34)

hold for any $\tilde{{\bf q}}$ in the BZ where $R_{\frac{\pi}{2}}$ denotes the
rotation by $\frac{\pi}{2}$ around $\tilde{q}_{z}$\- axis. These symmetries
are all violated in the helical phases where the common pair momentum ${\bf
q}=q\hat{{\bf y}}$ leads to the anisotropic response depicted in (f-h) and
(j-l). This spin/momentum space anisotropy in the $xy$ and $q_{x}q_{y}$ planes
is a fingerprint of the helical SC phase. It should be accessible
experimentally by constant-$\omega$ scans of the INS intensity in the BZ. The
various cartesian spin channels for $\alpha=x,y,z$ may be selected by using a
suitable total momentum transfer according to Eq. (29).

There are complementary ways of presenting these anisotropies
characteristically appearing in the helical phase. One possibility is to make
cuts through the BZ and plot the dynamical response along that direction. This
is shown in Fig. 3 for $(10)$ and $(01)$ directions and $xx,yy$ comonents in
the s-wave case. For BCS limit we have again pairwise equivalence of response
according to Eq.(34) as evident from (a,d). In the helical phase these
symmetries are destroyed as is shown by the further cuts (b,c) and (e,f).
Another complementary presentation of the helical anisotropies is shown in the
dispersive plots of Fig. 4, again for the s-wave case. In the BCS limit of (a)
the dynamical response in $xx(10)$ $yy(01)$ is equivalent and vanishing for
$\omega<2\Delta_{0}$. For the helical case in the same configurations the
dispersive magnetic excitations are clearly present due to low energy
quasiparticles in the depaired momentum space sectors but strongly anisotropic
for the two cases, in agreement with previous presentations.

Another interesting result of this investigation is the temperature dependence
of the static spin susceptibility which in principle determines the NMR Knight
shift. It is shown in Fig. 5 in comparison for BCS and helical states. Note
that according to Sec.4 $xx$ and $yy$ susceptibility are equivalent for
$\tilde{{\bf q}}=0$ for both BCS and helical state, therefore we plot only the
$xx$ component. In the BCS case the well known exponential and power law
behaviour of the susceptibility for the s- and d-wave cases are observed. The
appearance of large sheets of low energy quasiparticles in the helical phase
leads to large residual low temperature susceptibility which should be
observable. We note that this happens although the helical order parameter
does not have nodes in real space as it is true, e.g. for the LO phase in the
case without inversion symmetry breaking [28].

##  6 Summary and Conclusion

This work completes our previous investigations of spectroscopic (QPI, INS)
properties of superconducting states with finite momentum pairing [11, 12,
13]. We have derived the general expressions of dynamical magnetic response in
noncentrosymmetric superconductors with Rashba spin orbit coupling that may
stabilize the helical phase in an applied field. We have shown which type of
quasiparticle excitations appear in the response functions and derived the
corresponding anomalous coherence factors and matrix elements due to helical
spin textures of the Rashba bands. The obtained expressions of the response
function generalize the known results for the centrosymmetric BCS case to the
two-band Rashba case with spin-momentum locking under the presence of applied
fields which are small compared to the Rashba coupling energy.

As a major result we demonstrated that the combined spin-/momentum space
symmetries of the BCS phase response is broken in the helical phase. It leads
to characteristic spin/momentum asymmetries and anisotropies which can in
principle be detected by inelastic neutron scattering experiments. In
particular the constant-$\omega$ cuts of the magnetic spectrum by scanning
through the BZ should give a fingerprint of the helical phase. From the type
of observed anisotropies it should be possible to conclude about the direction
of the q\- pair momentum vector and observe its dependence on applied field
direction. A further possibility is to look for the frequency dependence at
certain $\tilde{{\bf q}}$ momentum transfer and whether spin resonance
excitations may form as a consequence of quasiparticle interactions similar as
has been predicted for the centrosymmetric case [13].

Finally we investigated the generic temperature dependence of the static spin
susceptibillity important to know for NMR experiments. We demonstrated that in
the helical phase a large residual low temperature spin susceptibility remains
which signifies the appearance of unpaired states in this phase, although
these are not due to real space nodal planes of the order parameter itself. In
summary our investigation gives a solid theoretical foundation for magnetic
spectroscopy of the helical Rashba superconducting state.

##  Appendix A Spin operator matrix elements for the nondiagonal cartesian
cases

In a similar manner as in Sec. 4 the matrix elements
$M^{\alpha\beta}_{\lambda\lambda^{\prime}}$ for $\alpha\neq\beta$ may be
derived as

$\displaystyle M^{xy}({\bf k}^{\prime}{\bf k})=$
$\displaystyle\frac{1}{2}\left(\begin{matrix}-\sin(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})&\sin(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})\cr\sin(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})&-\sin(\theta_{\bf k}+\theta_{{\bf
k}^{\prime}})\end{matrix}\right),$ (35) $\displaystyle M^{xz}({\bf
k}^{\prime}{\bf k})=$
$\displaystyle\frac{1}{2}\left(\begin{matrix}i(\cos\theta_{\bf
k}-\cos\theta_{{\bf k}^{\prime}})&-i(\cos\theta_{\bf k}+\cos\theta_{{\bf
k}^{\prime}})\cr i(\cos\theta_{\bf k}+\cos\theta_{{\bf
k}^{\prime}})&-i(\cos\theta_{\bf k}-\cos\theta_{{\bf
k}^{\prime}})\end{matrix}\right),$ $\displaystyle M^{yz}({\bf k}^{\prime}{\bf
k})=$ $\displaystyle\frac{1}{2}\left(\begin{matrix}i(\sin\theta_{\bf
k}-\sin\theta_{{\bf k}^{\prime}})&-i(\sin\theta_{\bf k}+\sin\theta_{{\bf
k}^{\prime}})\cr i(\sin\theta_{\bf k}+\sin\theta_{{\bf
k}^{\prime}})&-i(\sin\theta_{\bf k}-\sin\theta_{{\bf
k}^{\prime}})\end{matrix}\right).$

They appear in the expressions for the nondiagonal susceptibility components
$\chi_{0}^{\alpha\beta}(\tilde{{\bf q}},\omega)$.

##  Appendix B Magnetic response function in the BCS limit

The diagonal cartesian susceptibilities $(\alpha=\beta)$ of Eqs. (22,24) may
be rewritten by using the symmetry of matrix elements
$M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf k})$ and
coherence factors $\tilde{C}^{\bf q}_{\pm}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})$ against interchange of primed and unprimed
arguments. Furthermore, since the summation over both is identical, the first
two terms of Eqs. (22,24) may be contracted into one term. Then in the BCS
case $(b,q=0)$ when quasiparticle bands are given by $E^{\tau}_{{\bf k}{\bf
q}\lambda}=E_{{\bf k}\lambda}=(\varepsilon_{{\bf k}\lambda}^{02}+\Delta_{\bf
k}^{2})^{\frac{1}{2}}$ with $\varepsilon_{{\bf k}\lambda}^{0}$ defined by Eq.
(3) we obtain the simplified result

$\displaystyle\chi^{\alpha\alpha}_{0}(\tilde{{\bf q}},\omega)=$
$\displaystyle\frac{1}{N}\sum_{\lambda\lambda^{\prime}}\sum_{{\bf
k}}M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf
k})\Bigl{\\{}\tilde{C}_{+}({\bf k}\lambda,{\bf
k}^{\prime}\lambda^{\prime})\frac{f(E_{{\bf
k}^{\prime}\lambda^{\prime}})-f(E_{{\bf k}\lambda})}{\omega-(E_{{\bf
k}^{\prime}\lambda^{\prime}}-E_{{\bf k}\lambda})+i\eta}$ (36)
$\displaystyle+\frac{1}{2}\tilde{C}_{-}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\bigl{[}\frac{1-f(E_{{\bf
k}^{\prime}\lambda^{\prime}})-f(E_{{\bf k}\lambda})}{\omega+(E_{{\bf
k}^{\prime}\lambda^{\prime}}+E_{{\bf k}\lambda})+i\eta}+\frac{f(E_{{\bf
k}^{\prime}\lambda^{\prime}})+f(E_{{\bf k}\lambda})-1}{\omega-(E_{{\bf
k}^{\prime}\lambda^{\prime}}+E_{{\bf k}\lambda})+i\eta}\bigr{]}\Bigr{\\}}.$

This is a generalisation of BCS expressions given in Refs. [20, 21, 23, 24] to
the magnetic response for the two-band Rashba-BCS superconductor. The
anomalous coherence factors of Eq. (25) now simplify to

$\tilde{C}_{\pm}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})=\frac{1}{2}\bigl{[}1\pm\frac{\varepsilon^{0}_{{\bf
k}\lambda}\varepsilon^{0}_{{\bf
k}^{\prime}\lambda^{\prime}}+\Delta_{\lambda}^{\bf
k}\Delta_{\lambda^{\prime}}^{{\bf k}^{\prime}}}{E_{{\bf k}\lambda}E_{{\bf
k}^{\prime}\lambda^{\prime}}}\bigr{]}.$ (37)

In the low temperature limit and for positive frequencies only the last term
survives leading to

$\displaystyle\chi^{\alpha\alpha}_{0}(\tilde{{\bf
q}},\omega)=-\frac{1}{N}\sum_{\lambda\lambda^{\prime}}\sum_{{\bf
k}}M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf
k})\frac{\frac{1}{2}\tilde{C}_{-}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})}{\omega-(E_{{\bf
k}^{\prime}\lambda^{\prime}}+E_{{\bf k}\lambda})+i\eta}.$ (38)

The spectrum of the $T=0$ diagonal BCS response functions is then given by
($\tilde{{\bf q}}\in 1^{st}$ BZ)

$\displaystyle\hat{S}_{\alpha\alpha}(\tilde{{\bf q}},\omega)=\frac{1}{\pi}{\rm
Im}\chi^{\alpha\alpha}_{0}(\tilde{{\bf
q}},\omega)=\frac{1}{N}\sum_{\lambda\lambda^{\prime}}\sum_{{\bf
k}}M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k}^{\prime},{\bf
k})\frac{1}{2}\tilde{C}_{-}({\bf k}\lambda{\bf
k}^{\prime}\lambda^{\prime})\delta[\omega-(E_{{\bf
k}^{\prime}\lambda^{\prime}}+E_{{\bf k}\lambda})],$ (39)

which contains both intra- and interband transitions between the Rashba-split
quasiparticle bands. For $\tilde{{\bf q}}=0$ i.e. ${\bf k}={\bf k}^{\prime}$
we obtain

$\displaystyle\hat{S}_{\alpha\alpha}(\omega)=\frac{1}{N}\sum_{\lambda\lambda^{\prime}}\sum_{{\bf
k}}M^{\alpha\alpha}_{\lambda^{\prime}\lambda}({\bf k},{\bf
k})\frac{1}{2}\tilde{C}_{-}({\bf k}\lambda{\bf
k}\lambda^{\prime})\delta[\omega-(E_{{\bf k}\lambda^{\prime}}+E_{{\bf
k}\lambda})].$ (40)

The threshold values for this spectrum are different for intra- and interband
transitions. For the former $(\lambda^{\prime}=\lambda)$ it is given by
$2\Delta_{0}$ for the latter $(\lambda^{\prime}\neq\lambda)$ by
$\omega^{\pm}_{0}=|\alpha|k^{\pm}_{F}\gg 2\Delta_{0}$. Therefore the threshold
of the total spectrum is the intraband threshold $2\Delta_{0}$.

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