# Pauli paramagnetism of triplet Cooper pairs in a nematic superconductor

D. A. Khokhlov  Dukhov Research Institute of Automatics, Moscow, 127055 Russia
Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141700
Russia  National Research University Higher School of Economics, 101000
Moscow, Russia  R. S. Akzyanov  Dukhov Research Institute of Automatics,
Moscow, 127055 Russia  Moscow Institute of Physics and Technology,
Dolgoprudny, Moscow Region, 141700 Russia  Institute for Theoretical and
Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia

###### Abstract

We investigate the response of a doped topological insulator Bi2Se3 with spin-
triplet nematic superconductivity to external magnetization. We calculate the
Zeeman part of magnetic susceptibility for nematic and chiral superconducting
phases near Tc in Ginzburg-Landau formalism. Superconducting order parameter
from $E_{u}$ representation has non-trivial coupling with the transversal
Zeeman field that results in a paramagnetic response to a magnetization. The
topology of a Fermi surface has a strong influence on magnetic susceptibility.
Lifshitz transition from closed to open Fermi surface eventually leads to
phase transition from the nematic to chiral phase. At the transition point,
magnetic susceptibility diverges. Also, we study the effects of the electron-
electron interaction on the competition between nematic and chiral phases. We
found that in a real system, electron-electron interaction can drive nematic
to chiral phase only in the vicinity of the phase transition. We compare our
results with the existing experimental data.

##  I Introduction

Several years ago topological superconductivity in the bulk of doped
topological insulator (TI) AxBi2Se3 was discovered [1, 2, 3]. Dopant atom A
can be Cu [1, 2, 3, 4, 5, 6, 7], Sr [8, 9, 10, 11, 12, 13] or Nb [14, 15, 16,
17]. Superconducting phase in these materials shows C2 rotational symmetry
that breaks C3 symmetry of the normal state. Such a rotational symmetry
breaking was observed in measurements of specific heat [5], magnetic resonance
[16], in-plane upper critical field [12, 10], vortex core form [6].
Observation of nuclear magnetic resonance indicates the spin-triplet character
of this superconductivity [7].

Such C2 rotational symmetry is possible due to the realization of the
superconducting vector order parameter that belongs to Eu representation of
crystalline D3d point group [18, 19]. One of possible solutions within such a
representation is a nematic order parameter that is a two-component real
vector $\boldsymbol{\eta}=\eta(\cos\theta;\sin\theta)$. The direction of this
vector is associated with the nematicity axis. Orientation of the nematicity
axis shows the direction of two-fold symmetry breaking [20, 11]. Nematic order
parameter brings several interesting features such as vestigial nematic order
[21], surface Andreev bound states [22], unconventional Higgs mode [23], and
quasiparticle interference [24, 25].

Another possible solution for the order parameter is called chiral and
corresponds to the vector with an imaginary component
$\boldsymbol{\eta}\propto(1;\pm i)$. This order parameter spontaneously breaks
time-reversal symmetry, keeping rotational crystalline symmetry intact [26].
In NbxBi2Se3 such spontaneous breaking of time-reversal symmetry was found
[14]. Muon spin rotational experiment shows time-reversal symmetry breaking in
Sr0.1Bi2Se3 superconductor [13]. However, another research of NbxBi2Se3 shows
the presence of time-reversal symmetry in the superconducting phase [16, 15,
17]. Theoretical calculations predict the chiral phase is the ground state in
2D films of doped Bi2Se3 [26, 27] while in a 3D system with a closed Fermi
surface, the nematic phase has the lower free energy than the chiral state
[28, 29, 30]. Experimentally, the Fermi surface of doped Bi2Se3 with the
superconductivity has a topology of an open cylinder [31, 32] which is an
intermediate case between 2D and 3D Fermi surface. This transition to the open
Fermi surface leads to the phase transition from nematic to chiral phase [33].

Near the critical temperature, physical properties of the superconducting
state are described by the Ginzburg-Landau (GL) functional that for the
nematic superconductor was obtained in Refs. [19, 20]. One of the interesting
features of this functional is the presence of the linear in magnetization’s
powers, coupling between the magnetic and superconducting degrees of freedom.
Such coupling can lead to the transition from the nematic to chiral state with
the spontaneous magnetization [34]. It was predicted that transition from the
nematic to chiral state occurs upon doping by magnetic ions [28, 29].

Magnetization measurements show the presence of the diamagnetic Meissner
effect[17]. Muon spin rotational ($\mu$SR) experiment can be used to determine
local magnetic moments in superconductors [35, 36, 37, 38]. Recent $\mu$SR
experiment did not find time-reversal breaking in superconducting Nb0.25Bi2Se3
in the absence of magnetic field [17]. However, $\mu$SR shows that the
superconducting state has an additional paramagnetic magnetization compared to
the normal state in the magnetic field. In this work, we show how non-trivial
coupling between magnetization and the nematic superconductivity can explain
such a paramagnetic response.

In this paper, we calculate the magnetic susceptibility of the nematic
superconductor near the critical temperature $T_{c}$. We start with the
microscopic derivation of the GL free energy of the doped topological
insulator with the open Fermi surface and finite magnetization that is induced
by the magnetic field following the common procedure [20, 28, 29]. We solve GL
equations for the order parameter and get that the magnetic field influences
the form of the order parameter for the ground state. We calculate ground-
state free energy as a function of a magnetic field. Using this free energy,
we get that in both nematic and chiral states a paramagnetic contribution to
the magnetic susceptibility exists in the system. We interpret this phenomenon
as a Pauli paramagnetism of spin $1$ Cooper’s pairs of the nematic
superconductor. Such magnetic susceptibility diverges near the transition from
the nematic to the chiral state. In addition, we show that coupling between
magnetism and superconductivity is rather weak and electron-electron
interaction can not cause the phase transition between nematic and chiral
superconductivity.

The paper is organized as follows: in Sec. II we describe the normal and
superconducting state in the presence of an external Zeeman field and
calculate GL coefficients. Sec. III is dedicated to the Zeeman susceptibility
of the superconductor. In Sec. IV we consider the possibility of the phase
transition due to electron-electron interaction. We discuss and summarize
obtained results in Sec. V.

##  II Model

###  II.1 Normal phase

We describe bulk electrons in a doped topological insulator of Bi2Se3 family
by low-energy $\mathbf{k\cdot p}$ two-orbital Hamiltonian [39]:

$\displaystyle\hat{H}_{0}(\mathbf{k})=-\mu+m\sigma_{z}+v_{z}k_{z}\sigma_{y}+v(k_{x}s_{y}-k_{y}s_{x})\sigma_{x},$
(1)

where $\mu$ is the chemical potential, $2m$ is a single-electron gap at zero
chemical potential, Fermi velocities $v$ and $v_{z}$ describe motion in the
$(\Gamma K;\Gamma M)$ plane and along $\Gamma Z$ direction correspondingly.
Pauli matrices $s_{i}$ act in spin space while matrices $\sigma_{i}$ act in
space of Bi and Se orbitals $\mathbf{p}=(P^{1},P^{2})$, where $i=\\{x,y,z\\}$,
Planck constant $\hbar=1$. The Hamiltonian (1) obeys time-reversal symmetry
$\hat{\mathcal{T}}\hat{H}_{0}(\mathbf{k})\hat{\mathcal{T}}^{-1}=\hat{H}_{0}(-\mathbf{k})$,
where $\hat{\mathcal{T}}=is_{y}\hat{K}$, $\hat{\mathcal{T}}^{2}=-1$ is time-
reversal operator and $\hat{K}$ provides complex conjugation. Also, this
Hamiltonian has inversion symmetry
$\hat{P}\hat{H}_{0}(\mathbf{k})\hat{P}=\hat{H}_{0}(-\mathbf{k})$, where
$\hat{P}=\sigma_{z}$, $\hat{P}^{2}=1$ is the operator of the inversion [39].

Figure 1: Fermi surface before and after Lifshitz transition in dimensionless
coordinates. The dashed blue line gives boundaries of the first BZ (white
background) and second BZ (yellow background). Panel a): Gray curve gives
closed Fermi surface at $\mu=1.2m$. Panel b): Gray line gives open Fermi
surface at $\mu=1.4m$, cutoff angle $\theta_{0}$ is defined by Eq. (2). Panel
c): The same Fermi surface as in panel a) where parts of Fermi surface with
$|k_{z}|<q/2$ are highlighted by thick red lines. In these red sheets of FS,
the electron-phonon coupling is the strongest, see Ref. [40] for details.

Angle-resolved photoemission spectroscopy and measurements of Shubnikov-de
Haas oscillations show that the Fermi surface in doped Bi2Se3 is open in
$\Gamma Z$-direction [31, 32]. An increase of the chemical potential by doping
transforms the closed ellipsoid Fermi surface of undoped samples to a
corrugated cylinder Fermi surface of doped samples. For high chemical
potential, dispersion in $z$-direction disappears, and the system becomes
effectively 2D. We introduce a finite length of 001 lattices constant $c$ into
the model. Momentum is bounded by BZ size $|k_{z}|<\pi/c,$ or in elliptical
coordinates $(vk_{x},vk_{y},v_{z}k_{z})$ polar angle stays in range
$\theta\in[\theta_{0};\pi-\theta_{0}]$, where we define cutoff angle
$\theta_{0}$ as follows

$\displaystyle\cos(\theta_{0})=\text{min}\left(1,\frac{\pi
v_{z}}{c\sqrt{\mu^{2}-m^{2}}}\right).$ (2)

In this model, the Fermi surface is closed at a low chemical potential $\mu$.
When chemical potential reaches a critical value
$\mu_{\text{LT}}=\sqrt{\pi^{2}v_{z}^{2}/c^{2}-m^{2}}$, the Fermi surface
reaches the boundary of the first Brillouin zone (BZ) and changes topology
from ellipsoid to open cylinder. That is the Lifshitz transition (LT) [41]. We
show closed and open Fermi surfaces in Figs. 1a) and b) correspondingly.

Experiment with inelastic neutron scattering [40] shows linewidth of phonon
spectra has singularity for phonon momentum $\mathbf{q}$ oriented along
$\Gamma\text{Z}$ direction while $q\to 0$. The attraction between electrons
with momenta $\mathbf{k}$ and $-\mathbf{k}$ in the Cooper channel occurs only
for electrons whose $k_{z}$ components are close enough to each other. Thus,
only electrons with the momentum $|k_{z}|<q/2$ participate in superconducting
pairing. This singular coupling can be modeled by the Heaviside step function
$\theta(q-2k_{z})$ and the Fermi surface where Cooper pair’s can be formed is
effectively cut off. This new effective Fermi surface has the topology of an
open cylinder, see Fig. 1.

###  II.2 Superconducting phase

We describe superconductivity in Nambu-II basis, where wave function is

$\displaystyle\Psi_{\mathbf{k}}=(\phi_{\mathbf{k}}^{t},-i\phi_{\mathbf{k}}^{\dagger}s_{y})^{t},$
(3)

where
$\phi_{\mathbf{k}}=(\phi_{\uparrow,1,\mathbf{k}},\phi_{\downarrow,1,\mathbf{k}},\phi_{\uparrow,2,\mathbf{k}},\phi_{\downarrow,2,\mathbf{k}})^{t}$,
symbol $t$ is transpose and symbol † is Hermitian conjugate. Operator
$\phi_{\uparrow(\downarrow),\sigma,\mathbf{k}}^{(\dagger)}$ annihilates
(creates) electron with up (down) spin on the orbital $\sigma={P^{1},P^{2}}$
with momentum $\mathbf{k}$. Superconducting order parameter from Eu
representation of D3d crystalline point group has the following matrix
structure [26]:

$\displaystyle\hat{\Delta}=\eta_{x}s_{x}\sigma_{y}(\tau_{x}+i\tau_{y})+\eta_{x}^{*}s_{x}\sigma_{y}(\tau_{x}-i\tau_{y})+$
(4)
$\displaystyle\eta_{y}s_{y}\sigma_{y}(\tau_{x}+i\tau_{y})+\eta_{y}^{*}s_{y}\sigma_{y}(\tau_{x}-i\tau_{y}).$

The superconducting term depends on two components of vector order parameters
$\eta_{x}=\eta\sin(\alpha)e^{i\phi_{1}}$ and
$\eta_{y}=\eta\cos(\alpha)e^{i\phi_{2}}$, where $\phi=\phi_{1}-\phi_{2}$.
Matrices $\tau_{i}$ acts in electron-hole space. We assume that only the
electrons in the Debye window participate in the superconductivity
$-\omega_{D}<\epsilon_{\mathbf{k}}<\omega_{D}$, where $\epsilon_{\mathbf{k}}$
is the band’s dispersion of the Hamiltonian (1). This order parameter violate
inversion symmetry of the normal state.

The BdG Hamiltonian in Nambu-II basis is [18]

$\displaystyle\hat{H}_{BdG}(\mathbf{k})=\tau_{z}\hat{H}_{0}(\mathbf{k})+\hat{\Delta}.$
(5)

We find GL free energy from microscopical theory as [20, 28, 29]

$\displaystyle
F=F_{0}-T\sum_{\omega}\int\frac{d\mathbf{k}^{3}}{(2\pi)^{3}}\text{Tr}\left[\log(1-\hat{G}_{0}\hat{\Sigma})\right],$
(6)

where
$F_{0}=-T\sum_{\omega}\int\frac{d\mathbf{k}^{3}}{(2\pi)^{3}}\text{Tr}\left[\log(\hat{G}_{0}^{-1})\right]$
is a free energy of a normal state, $\hat{\Sigma}$ is a self energy. Matrix
$\hat{G}_{0}=(i\omega-\hat{H}_{0})^{-1}$ is Matsubara Green’s functions in the
Nambu II basis and fermionic Matsubara frequency $\omega=(2n+1)\pi T$, where
$n$ is integer. We take trace $\text{Tr}[..]$ over spin, orbital and electron-
hole degrees of freedom. Then we expand logarithm from Eq. (6) into Taylor
series
$\log(1-\hat{G}_{0}\hat{\Sigma})=-\sum_{n}\frac{(\hat{G}_{0}\hat{\Sigma})^{n}}{n}$
in powers of a perturbation $\hat{\Sigma}$ and combine terms of the series in
powers of order parameters. Calculated GL coefficients are given in table 1.
Similar calculations are provided in Refs. [29, 26, 28].

We start from $\hat{\Sigma}=\hat{\Delta}$ and obtain superconducting part of
GL free energy [26] up to terms $\propto\eta^{4}$

$\displaystyle
F_{sc}(\eta_{x},\eta_{y})=A(|\eta_{x}|^{2}+|\eta_{y}|^{2})\\!+\\!B_{1}(|\eta_{x}|^{2}+|\eta_{y}|^{2})^{2}\\!+$
(7) $\displaystyle B_{2}|\eta_{x}^{*}\eta_{y}-\eta_{x}\eta_{y}^{*}|^{2}.$

The coefficient $A\propto(T-T_{c})$ changes sign from positive to negative
under cooling from a temperature above $T_{c}$ to a temperature under $T_{c}$
inducing superconducting phase transition [26].

In the chiral phase expression
$|\eta_{x}^{*}\eta_{y}-\eta_{x}\eta_{y}^{*}|=\eta^{2}$, while in the nematic
phase this expression is zero. Thus, the free energy of the chiral phase has
an additional term $B_{2}\eta^{4}$. Therefore, in the system with positive
$B_{2}>0$, the nematic phase is the ground state, while negative $B_{2}<0$
promotes the chiral phase. The sum of coefficients $B_{1}+B_{2}$ is always
positive. Direct calculations in model with infinite Brillouin zone and the
Hamiltonian (1) show $B_{2}>0$ in 3D system [28] and $B_{2}<0$ in 2D case
[27].

Figure 2: GL coefficient $B_{2}(\mu)$ in units of $B_{2,LT}$ at the Lifshitz
transition point vs dimensionless chemical potential $\mu/m$. The solid red
curve corresponds to $B_{2}$ calculated in a model with finite BZ. Dash red
curve corresponds to $B_{2}$ calculated in model with infinite BZ. Dash gray
horizontal line corresponds to $B_{2}=0$. The vertical dash blue line shows
the transition between the nematic phase with $B_{2}>0$ and the chiral phase
with $B_{2}<0$.

In the absence of magnetism, nematic phase has a two-component vector order
parameter with real components
$\boldsymbol{\eta}=\eta(\cos\alpha;\sin\alpha)$. In chiral phase order
parameter is a two-component vector again with one real component and another
one pure imaginary $\boldsymbol{\eta}=\frac{\eta}{\sqrt{2}}(1;\pm i)$, which
breaks time-reversal symmetry.

We calculate coefficient $B_{2}$ for arbitrary lattice constant $c$ in a model
with finite BZ. Results are given in Table 1. The coefficient $B_{2}$ is
plotted as a function of Fermi energy $\mu$ in Fig. 2 in models with finite
and infinite BZ. Fermi surface is closed at low chemical potential even when
BZ is finite and both models are the same. In the model with an infinite BZ,
coefficient $B_{2}>0$ grows with an increase in Fermi energy. Thus nematic
order parameter has lower energy than the chiral one. Lifshitz transition
occurs at $\mu_{LT}=\sqrt{\frac{\pi^{2}v_{z}^{2}}{c^{2}}+m^{2}}$ in model with
finite BZ and leads to decline in $B_{2}$. This coefficient reaches zero in
point
$\mu^{*}=\sqrt{\frac{v_{z}^{2}\pi^{2}}{c^{2}\cos^{2}(\theta_{0}^{*})}+m^{2}}$,
where angle $\theta_{0}^{*}\approx 0.985$ does not depend on any parameters.
Critical chemical potential $\mu^{*}$ indicates phase transition between
nematic and chiral superconductivity. In all numerical calculations we set
superconducting critical temperature $T_{c}=10^{-3}m$, in-plane velocity
$v=1/3mc$ and $\Gamma Z$-velocity $v_{z}=\frac{2}{3}v$.

###  II.3 Coupling between the magnetism and superconductivity

In presence of transversal magnetic field electrons in doped Bi2Se3 experience
orbital depended Zeeman magnetization. We decompose this magnetization into
ferromagnetic (FM) and antiferromagnetic (AFM) with corresponding Lande
g-factors $\beta_{f}$ and $\beta_{a}$, see Ref. [39] and write it down as

$\displaystyle\hat{\Sigma}_{m}=\mu_{B}\beta_{f}Hs_{z}+\mu_{B}\beta_{a}Hs_{z}\sigma_{z}.$
(8)

Figure 3: GL coefficients g${}_{f}(\mu)$ and g${}_{a}(\mu)$ in units of
$\frac{m}{6\pi^{2}v^{2}v_{z}}$ versus dimensionless chemical potential
$\mu/m$. Red curves correspond to ferromagnetic coupling gf while blue curves
corresponds to antiferromagnetic coupling $g_{a}$. Solid lines correspond to
calculations in a model with finite BZ. Dash curves correspond to calculations
in a model with infinite BZ. Dash gray line indicates zero level. The green
circles mark Lifshitz transition.

We calculate free energy via Eq. (6), where perturbation is
$\hat{\Sigma}=\hat{\Sigma}_{m}+\hat{\Delta}$. We expand
$\log(1-\hat{G}_{0}\hat{\Sigma})$ in powers of $H$ and $\eta$ up to
$O(\hat{\Sigma}^{2})$. The free energy receive additional terms [29, 28]

$\displaystyle
F_{\alpha}(H,\eta_{x},\eta_{y})=-2ig_{\alpha}\mu_{B}\beta_{\alpha}H(\eta_{x}^{*}\eta_{y}-\eta_{x}\eta_{y}^{*})+$
(9) $\displaystyle a_{\alpha}(\mu_{B}\beta_{\alpha}H)^{2},$ $\displaystyle
F_{m}(H)=a_{m}\beta_{f}\beta_{a}\mu_{B}^{2}H^{2},$ (10)

where $\alpha=\\{f,a\\}$ refers to FM or AFM terms. Constants $a_{f},a_{a}$
and $a_{m}$ describes response of the normal state to a transverse
magnetization, see Table 1. We combine terms that $\propto H^{2}$ into one
effective term
$a_{eff}\mu_{B}^{2}H^{2}=(a_{f}\beta_{f}^{2}+a_{a}\beta_{a}^{2}+a_{m}\beta_{f}\beta_{a})\mu_{B}^{2}H^{2}$.

Coefficients $g_{f}$ and $g_{a}$ describe the coupling between magnetism and
superconductivity. Note, that in-plane magnetization does not produce such
terms. We calculate these coefficients in models with finite and infinite BZ,
see Fig 3. Ferromagnetic coefficient $g_{f}$ changes sign with the increase of
the chemical potential, while antiferromagnetic coefficient $g_{a}$ is always
positive. According to expression for $g_{f}$ at arbitrary geometry of Fermi
surface given in Table 1, it changes sign $g_{f}=0$ if
$\cos^{2}(\theta_{0}^{**})=\frac{3\left(\log(j\omega/T)\left(m^{2}-\mu^{2}\right)+m^{2}\right)}{-3\mu^{2}\log(j\omega/T)+2\mu^{2}+m^{2}}$.
In this point coupling between superconductivity and FM perturbation
disappears.

Figure 4: Effective coupling between superconductivity and Zeeman field
g${}_{eff}(\mu)$ in units of $\frac{m}{6\pi^{2}v^{2}v_{z}}$ versus
dimensionless chemical potential $\mu/m$. The solid red curve gives geff
calculated in a model with finite BZ, while dash red curves correspond to
calculations in a model with infinite BZ. Dash gray curve indicates zero
level. The green circle marks Lifshitz’s transition.

The total coupling between Zeeman field and superconductivity is described by
effective coefficient $g_{eff}=\beta_{f}g_{f}+\beta_{a}g_{a}$. We plot
$g_{eff}(\mu)$ in Fig. 4. Coefficient $g_{eff}$ is negative at low chemical
potential since both FM and AFM g-factors are negative. Then $g_{eff}$ change
sign from negative to positive at some value of Fermi energy. Zero-point of
$g_{eff}$ occurs at higher chemical potential than zero-point of $g_{f}$ in
model with open FS. In numerical calculations we set Lande g-factors
$\beta_{f}=-5.3$, $\beta_{a}=-7.4$.

GL coeff. |  $\theta_{0}=0$: Closed FS |  $\theta_{0}=\pi/2$: cylindrical FS |  $0<\theta_{0}<\pi/2$: Open FS
---|---|---|---
A |  μ2-m23(T-Tc)3π2μTcv2vz |  (μ2-m2)(T-Tc)4πcμTcv2 |  μ2-m23(T-Tc)cos(θ0)(7+cos(2θ0))24π2μTcv2vz
B_1 |  7ζ(3)μ2-m2530π4Tc2μ3v2vz |  21ζ(3)(μ2-m2)128πc Tc2μ3v2 |  7ζ(3)cos(θ0)μ2-m25(427+76cos(2θ0)+9cos(4θ0))15360π2Tc2μ3v2vz
B_2 |  7ζ(3)μ2-m2560π4Tc2μ3v2vz |  -7ζ(3)(μ2-m2)128πc Tc2μ3v2 |  7ζ(3)cos(θ0)μ2-m25(71+188cos(2θ0)-3cos(4θ0))15360π2Tc2μ3v2vz
g_f |  μ2-m26π2μ2v2vz× (3m^2log(jωDTc)+2(m^2-μ^2)) |  12πc v2μ2× $\displaystyle\Bigg{(}m^{2}+(m^{2}-\mu^{2})\log\left(\frac{j\omega_{D}}{T_{c}}\right)\Bigg{)}$ |  μ2-m2cos(θ0)6π2v2vzμ2× $\displaystyle\Bigg{\\{}m^{2}\left[3-\cos^{2}(\theta_{0})+3\log\left(\frac{j\omega_{D}}{T_{c}}\right)\right]-$ $\displaystyle\mu^{2}\Bigg{[}2\cos^{2}\theta_{0}+3\sin^{2}\theta_{0}\log\left(\frac{j\omega_{D}}{T_{c}}\right)\Bigg{]}\Bigg{\\}}$
g_a |  mμ2-m26π2μ3v2vz× (3μ^2log(jω_D/T_c)+μ^2-m^2) |  mμ2-m22πc v2μ |  -mμ2-m2cos(θ0)6π2v2vzμ3× $\displaystyle\Bigg{(}m^{2}\cos^{2}(\theta_{0})-\mu^{2}\Bigg{[}2-\cos(2\theta_{0})+3\cos^{2}\theta_{0}\log\left(\frac{j\omega_{D}}{T_{c}}\right)\Bigg{]}\Bigg{)}$
a_f,z |  -μ2-m2(2m2+μ2)12π2v2vzμ |  -m24πv2cμ |  -μ2-m2cos(θ0)(5m2+μ2+cos(2θ0)(μ2-m2))24π2v2vzμ
a_a,z |  -μ2-m2(m2+2μ2)12π2v2vzμ |  -μ4πv2c |  -μ2-m2cos(θ0)(m2+5μ2-cos(2θ0)(μ2-m2))24π2v2vzμ
a_a,f |  $\frac{m}{\pi^{2}\mu}$
$a_{\alpha,e}=1/V_{\alpha}+a_{\alpha,z}$
Table 1: GL coefficients $A$, $B_{1}$, $B_{2}$, $g$ and $a$ for systems with
different Fermi surface and different magnetic interaction. Left column
describes 3D system with closed FS. Coefficients in 2D system are shown in
central column. Right column gives general case of open FS. Coefficient
$j=\frac{2}{\pi}e^{c}$, where $c$ is Euler–Mascheroni constant and $\zeta$ is
Riemann zeta function.

##  III Magnetic susceptibility

Response of the superconductor to a magnetic field can be decomposed into a
response to orbital and Zeeman parts of the magnetic field. Orbital part
induces Meissner currents and dominates overall response. Zeeman part occurs
due to change of the order parameter in case of finite magnetization. In this
paper, we focus only the second term. Total GL free energy of $E_{u}$
superconductor in the Zeeman field is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!F(H,\eta_{x},\eta_{y})\\!=\\!F_{sc}(\eta_{x},\eta_{y})\\!-\\!2ig_{eff}\mu_{B}H(\eta_{x}^{*}\eta_{y}\\!-\\!\eta_{x}\eta_{y}^{*})\\!+\\!$
(11) $\displaystyle a_{eff}H^{2}.$

The free energy depends on the value of complex values components of the order
parameter $\eta_{x}$ and $\eta_{y}$ as well as it depends on external magnetic
field $H$. We minimize free energy (11) as a function of $\boldsymbol{\eta}$
and find equilibrium free energy in given Zeeman field $H$. The exact
expression for the equilibrium free energy depends on the sign of the $B_{2}$
coefficient. We start from nematic case $B_{2}>0$. Finite $H$ breaks time-
reversal symmetry and for the ground state $\sin(2\alpha)\sin(\phi)\propto H$
becomes non zero, however we still name this phase nematic [34]. In this case
free energy is written as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!F_{min}^{nem}(H)\\!=\\!\\!-\frac{A^{2}}{4B_{1}}\\!+\\!a_{eff}H^{2}\\!\\!-\\!\frac{g_{eff}^{2}H^{2}}{B_{2}}\\!\\!\left(\\!\\!1\\!\\!-\\!\\!\frac{g_{eff}^{2}}{B_{2}a_{eff}}\\!\\!\right)\\!\\!.$
(12)

The first term gives free energy of the nematic phase in a zero field. The
second term describes diamagnetism of the normal state. The third term appears
due to coupling between the nematic superconductivity and Zeeman field and
describes the superconducting Pauli paramagnetism. We set
$\left(1-\frac{g_{eff}^{2}}{B_{2}a_{eff}}\right)\sim\left(1-\frac{T_{c}^{2}}{\mu^{2}}\right)\simeq
1$ since coupling $g_{eff}$ is small enough. This approximation is valid
everywhere besides the small neighborhood of point $B_{2}=0$.

Zeeman part of magnetic susceptibility is

$\displaystyle\chi_{ns}=-\frac{\partial^{2}F_{min}^{nem}(H)}{\partial
H^{2}}=-2a_{eff}+\frac{2g_{eff}^{2}}{B_{2}}.$ (13)

Here the first term corresponds to Zeeman susceptibility of normal phase, and
the second term arises due to coupling between nematic superconductivity and
magnetism. Zeeman susceptibility has a jump under phase transition between
normal and nematic superconducting phases

$\displaystyle\chi_{ns}-\chi_{n}=\frac{g_{eff}^{2}\mu_{B}^{2}}{2B_{2}}>0.$
(14)

This jump is relatively small compared to the Pauli paramagnetism of the
normal phase $\chi_{n}$ with a small parameter
$(\chi_{ns}-\chi_{n})/\chi_{n}=g_{eff}^{2}/(4B_{2}a_{eff})\sim(T_{c}/\mu)^{2}$.
However, this additional paramagnetism is enhanced close to area $B_{2}=0$,
where phase transition between nematic and chiral phases occurs.

Now we minimize free energy (11) as a function of $\eta_{x}$ and $\eta_{y}$
for $B_{2}<0$ In this case order parameter chiral and free energy is

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!F_{min}^{ch}(H)\\!=\\!\\!\frac{-A^{2}}{\\!4(B_{1}\\!+\\!B_{2})}\\!+\\!\frac{Ag_{eff}H}{B_{1}\\!+\\!B_{2}}\\!-\\!\frac{g_{eff}^{2}H^{2}}{B_{1}+B_{2}}+\\!a_{eff}H^{2}\\!.\\!\\!$
(15)

Jump of susceptibility between the chiral superconducting and normal phase is

$\displaystyle\chi_{cs}-\chi_{n}=\frac{g_{eff}^{2}}{4(B_{1}+B_{2})}>0,$ (16)

which has no singularities in contrast with $\chi_{ns}$ since $B_{1}+B_{2}>0$.

Figure 5: Jump of Zeeman susceptibility between superconducting and normal
phases $\Delta\chi^{f}$ normalized by its value $\Delta\chi^{f}_{LT}$ in
Lifshitz transition point as a function of the dimensionless chemical
potential $\mu/m$ in model with only one Lande g-factor $\beta_{f}=-5.3$. The
red solid curve corresponds to finite BZ. The red dashed curve corresponds to
infinite BZ. Green circle indicates Lifshitz transition. The vertical blue
dashed line shows the transition between nematic and chiral phases. Figure 6:
Jump of Zeeman susceptibility between superconducting and normal phases
$\Delta\chi^{eff}$ in units of its value $\Delta\chi^{eff}_{LT}$ in Lifshitz
transition point as a function of dimensionless chemical potential $\mu/m$ in
model with both Lande g-factors $\beta_{f}=-5.3$ and $\beta_{a}=-7.4$. The red
solid curve corresponds to finite BZ. The Red dashed curve corresponds to
infinite BZ. Green circle indicates Lifshitz transition. The vertical blue
dashed line shows the transition between nematic and chiral phases.

We calculate the Zeeman susceptibility as a function of Fermi energy. We
consider two different cases when Zeeman field appears (i) only in FM channel
with $g_{f}=-5.3$ and $g_{a}=0$, see Fig. 5 and (ii) in both FM and AFM
channels with $g_{f}=-5.3$ and $g_{a}=-7.4$, see Fig. 6. We see that the
inclusion of AFM response to the magnetic field significantly changes the
magnetic susceptibility.

In the model with finite BZ, Lifshitz transition occurs at $\mu_{LT}\approx
1.25$. Both susceptibilities $\Delta\chi^{f}$ and $\Delta\chi^{eff}$ diverges
for $\mu\approx 1.6$, where $B_{2}$ changes sign and phase transition from
nematic to chiral phase occurs. Susceptibility $\Delta\chi^{f}$ vanishes in
nematic phase in point $\mu\approx 1.4$, while $\Delta\chi^{eff}$ vanishes in
chiral phase in point $\mu\approx 1.8$. This disappearance of susceptibilities
occurs in points, where $g_{f}$ and $g_{eff}$ reaches zero value, see Figs. 5,
6. The model with an infinite BZ system always stays in the nematic phase.
Thus no divergence in susceptibility occurs.

Superconductor with Eu pairing has triplet structure i.e. Cooper pairs have
spin equal to one [19]. It can be seen if we write down order parameter using
second quantization form $\eta_{x}\propto
c_{1\uparrow}c_{2\uparrow}+c_{1\downarrow}c_{2\downarrow}$ and
$\eta_{y}\propto i(c_{1\uparrow}c_{2\uparrow}-c_{1\downarrow}c_{2\downarrow})$
where $c_{is}$ is the operator of the annihilation of the electron on $i$-th
orbital with the spin projection $s$. We can see that the order parameter
represents two Cooper’s pairs with opposite spins $\pm 1$. Each Cooper’s pair
couples electrons with the same spin on different orbitals. In presence of the
magnetic field number of spin up and spin down Cooper’s pairs is not equal.
Such spin imbalance influences the form of the order parameter that defines
the response to the magnetic field.

Spin density of the system is expressed through full Green’s function as
$S_{z}=-T\sum_{\omega}\int\frac{d^{3}\mathbf{}k}{(2\pi)^{3}}\text{Tr}[\hat{G}s_{z}]$.
We expand superconducting Green’s function
$\hat{G}\approx\hat{G}_{0}+\hat{G}_{0}\hat{\Delta}\hat{G}_{0}+\hat{G}_{0}\hat{\Delta}\hat{G}_{0}\hat{\Delta}\hat{G}_{0}$
in powers of superconducting order parameter 4 that gives us
$S_{z}=-2ig_{eff}(\eta_{x}^{*}\eta_{y}-\eta_{x}\eta_{y}^{*})\propto\sin(2\alpha)\sin(\phi)\propto
n_{\uparrow 1}n_{\uparrow 2}-n_{\downarrow 1}n_{\downarrow 2}$ where
$n_{s\sigma}=\phi^{\dagger}_{s\sigma}\phi_{s\sigma}$ is density of electrons
from orbital $\sigma$ with spin $s$ [19]. We see that parameter
$\sin(2\alpha)\sin(\phi)$ is responsible for spin imbalance of Cooper pair’s.
While the order parameter is real, angle $\phi=0$, spin density is zero and
number of spin up and spin down Cooper’s pairs is equal. Turning on Zeeman
field makes product $\sin(2\alpha)\sin(\phi)\sim H$ and spin density that
arises due to imbalance between spin up and spin down Cooper’s pairs becomes
$S_{z}=\frac{g_{eff}^{2}\mu_{B}H}{2B_{2}}\sim
k_{F}^{3}\beta_{eff}^{2}(\mu_{B}H/\mu)(T_{c}/\mu)^{2}$. Such spin imbalance
decreases the energy of the system in a magnetic field similarly to the Pauli
paramagnetism of the electrons. Thus, we refer this effect to a Pauli
paramagnetism of Cooper pairs.

Average spin density of Cooper pairs in chiral phase is
$S_{z}=g_{eff}\eta\approx Ag_{eff}/(B_{1}+B_{2})\sim\delta T\cdot
T_{c}k_{F}^{3}/\mu^{2}$. Therefore, chiral phase has nonzero magnetization
even in absence of magnetic field. For comparison, spin density of (i) a
normal-phase Bi2Se3 and (ii) a fully spin-polarized metal in Zeeman field $H$
have order of (i) $k_{F}^{3}\beta_{eff}\mu_{B}H/\mu$ and (ii) $k_{F}^{3}$
correspondingly. Density of Cooper pairs in chiral phase enlarges in presence
of Zeeman field $\eta=\frac{A+2|g_{eff}H|}{2(B_{1}+B_{2})}$. Thus, spin
density receives a term $\frac{g_{eff}^{2}\mu_{B}H}{4(B_{1}+B_{2})}$ and a
sample receives extra polarization proportional to Zeeman field $H$.
Therefore, chiral phase has Zeeman susceptibility despite been fully spin-
polarized.

Note that the chiral phase degenerates regarding the sign of the spin density.
The magnetic field removes such degeneracy, and the chiral phase with the
spins of Cooper’s pairs aligned with the magnetic field becomes a ground
state. If the sign of the spin density is fixed by some other field, then the
response to the magnetic field that aligned against the spins of Cooper’s
pairs will be diamagnetic.

##  IV Phase transition under electron-electron repulsion

Coupling between magnetic and superconducting order parameters can lead to the
emergence of ferromagnetic or antiferromagnetic phase in the superconducting
state. We consider effects of point-like electron-electron repulsion. We treat
this interaction in the mean-field approach. Only FM and AFM order parameters
couples with $E_{u}$ superconductivity $M_{z}=\langle
n_{\uparrow}\rangle-\langle n_{\downarrow}\rangle$ and $L_{z}=\langle
n_{\uparrow,1}\rangle+\langle n_{\downarrow,2}\rangle-\langle
n_{\uparrow,2}\rangle-\langle n_{\downarrow,1}\rangle$ correspondingly. Here
$n_{s\sigma}$ is local density of electrons with spin $s$ from orbital
$\sigma$. Both order parameters enter in free energy in the same way as Zeeman
field (9) where factors $\mu_{B}H\beta_{f(a)}$ should be replaced by
$V_{f(a)}M_{z}(L_{z})$. Coefficients $g_{f(a)}$ and $a_{m}$ do not change
while coefficient $a_{f(a)}\to a_{f(a),e}=a_{f(a)}+1/V_{f(a)}$, see table 1.
This coupling may inspire the phase transition from nematic to chiral phase
that can occur even under $B_{2}>0$. Particularly while $g_{a}=0$ phase
transition occurs when $\frac{g_{f}^{2}}{a_{f,e}B_{2}}>1$, see Ref. [34].
Direct calculations show the coefficient order $(T/\mu)^{2}\ll 1$. While both
FM and AFM order parameters exist, we find a generalized condition on the
phase transition

$\displaystyle\lambda=\frac{a_{a,e}g_{f}^{2}-a_{m}g_{f}g_{a}+a_{f,e}g_{a}^{2}}{(4a_{f,e}a_{a,e}-a_{m}^{2})B_{2}}>1.$
(17)

Since AFM coupling $g_{a}\sim g_{f}$, generalized condition is met only in
neighbourhood of a point $B_{2}=0$ or point $4a_{f,e}a_{a,e}-a_{m}^{2}=0$. The
condition (17) means the system stays extremely close to the appearance of
magnetic order parameters even in the normal phase that seems unrealistic for
a real system. Such a degree of proximity to the phase transition imposes
strong conditions on electron-electron interaction $V_{f}$ and $V_{a}$.
Therefore, the emergence of the chiral phase from the nematic phase under
electron-electron interaction in the absence of magnetic ordering in the
normal phase seems unlikely. For example, we find $\lambda\approx 6\cdot
10^{-7}\ll 1$ when coupling constants are $V_{f}=200\text{eV}\cdot\text{\AA}$
and $V_{a}=100\text{eV}\cdot\text{\AA}$, Fermi energy $\mu/m=1.3$ and other
parameters the same as at the end of Sec. II.2. Thus, electron-electron
interaction cannot drive the system from the nematic to the chiral state for
the experimentally achievable conditions.

##  V Discussion

Recently, superconducting powder of Nb0.25Bi2Se3 was investigated in $\mu$SR
experiment [17]. The authors measured the magnetization of a sample at
different temperatures in the external magnetic field. In contrast with
previous experiment [14], here authors find the system has time-reversal
symmetry in the absence of a magnetic field. In Fig.3b,Ref. [17] authors show
temperature dependence of magnetization in superconducting phase counted from
magnetization in normal phase at $T=5K$ (above $T_{c}$). Surprisingly, the
magnetic moment of the superconducting phase is higher than in the normal one,
which indicates the appearance of an additional paramagnetism associated with
superconductivity. We claim that Pauli paramagnetism of triplet Cooper’s pairs
is observed in this experiment. Note that measurements of the magnetic
susceptibility of the large monocrystal Nb0.25Bi2Se3 using SQUID show a strong
diamagnetic response that is expected in superconductors [17].

Typically, superconductors in the low magnetic fields demonstrate diamagnetism
due to Meissner effects, which occurs due to response to an orbital part of a
magnetic field [42, 43]. In this paper, we focus on the coupling between spin
and magnetization and do not take into account Meissner currents since they
are calculated in previous works. Due to rather weak coupling between
magnetization and superconductivity, we expect that diamagnetic susceptibility
from the Meissner effect will dominate paramagnetic susceptibility for large
samples. However, the Meissner effect can be suppressed for large thin films
and near the critical temperature $T_{c}$.

Magnetism can induce phase transition between nematic and chiral
superconductivity [28, 29]. We investigate this phase transition, assuming
magnetism appears because of electron-electron repulsion with order parameters
in FM and AFM channels. We find general condition on the phase transition
(17). Since coupling between magnetism and superconductivity is small, this
condition is not met, and the phase transition can not be induced by electron-
electron repulsion.

In Ref. [32] evolution of the Fermi surface of Bi2Se3 upon doping was
investigated. Authors showed that doping increases carrier density that leads
to the transformation of closed Fermi surface to open. This transformation
coincides with the emergence of the superconductivity and occurs at carrier
density $2\cdot 10^{19}\text{cm}^{-3}\lesssim n\lesssim
10^{20}\text{cm}^{-3}$. According to Ref. [31] Lifshitz transition occurs at
carrier density is $n=5\cdot 10^{19}\text{cm}^{-3}$. In Ref. [4] authors
investigated excessive Cu doping in Bi2Se3. They obtain several samples with
Cu concentration $0.28<x<0.54$. According to Knight shift measurements,
carrier density at $x>0.37$ strongly increases, which perhaps turns the system
to the chiral phase. Thus, the realization of the chiral superconductivity in
overdoped samples is possible in such samples. Since doping is non-uniform in
general, macroscopic parts of the superconductor can be in the vicinity of the
nematic to chiral phase transitions. Since paramagnetic susceptibility
diverges near the transition, such parts can give a substantial contribution
to the susceptibility that can be used to track phase transition.

## Acknowledgment

Authors acknowledge support by the Russian Science Foundation under Grant No
20-72-00030 and partial support from the Foundation for the Advancement of
Theoretical Physics and Mathematics “BASIS”.

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