Continuity equation and local gauge invariance for the N3LO nuclear Energy Density Functionals

Abstract

Background: The next-to-next-to-next-to-leading order (N3LO) nuclear energy
density functional extends the standard Skyrme functional with new terms
depending on higher-order derivatives of densities, introduced to gain better
precision in the nuclear many-body calculations. A thorough study of the
transformation properties of the functional with respect to different
symmetries is required, as a step preliminary to the adjustment of the coupling
constants. Purpose: Determine to which extent the presence of higher-order
derivatives in the functional can be compatible with the continuity equation.
In particular, to study the relations between the validity of the continuity
equation and invariance of the functional under gauge transformations. Methods:
Derive conditions for the validity of the continuity equation in the framework
of time-dependent density functional theory. The conditions apply separately to
the four spin-isospin channels of the one-body density matrix. Results: We
obtained four sets of constraints on the coupling constants of the N3LO energy
density functional that guarantee the validity of the continuity equation in
all spin-isospin channels. In particular, for the scalar-isoscalar channel, the
constraints are the same as those resulting from imposing the standard U(1)
local-gauge-invariance conditions. Conclusions: Validity of the continuity
equation in the four spin-isospin channels is equivalent to the local-gauge
invariance of the energy density functional. For vector and isovector channels,
such validity requires the invariance of the functional under local rotations
in the spin and isospin spaces.

Full text

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arXiv:1110.3027v1 [nucl-th] 13 Oct 2011
Continuity equation and local gauge invariance for the N3LO nuclear Energy Density
Functionals
F. Raimondi,1, ∗B. G. Carlsson,2J. Dobaczewski,1,3and J. Toivanen1
1Department of Physics, P.O. Box 35 (YFL) FI-40014 University of Jyv¨ askyl¨ a, Finland.
2Department of Physics, Lund University, P.O. Box 118 Lund 22100, Sweden.
3Institute of Theoretical Physics, Faculty of Physics,
University of Warsaw, ul. Ho˙ za 69, PL-00-681 Warsaw, Poland.
(Dated: October 14, 2011)
Background: The next-to-next-to-next-to-leading order (N3LO) nuclear energy density functional extends the standard
Skyrme functional with new terms depending on higher-order derivatives of densities, introduced to gain better pre-
cision in the nuclear many-body calculations. A thorough study of the transformation properties of the functional with
respect to different symmetries is required, as a step preliminary to the adjustment of the coupling constants.
Purpose: Determine to which extent the presence of higher-order derivatives in the functional can be compatible with the
continuity equation. In particular, to study the relations between the validity of the continuity equation and invariance
of the functional under gauge transformations.
Methods: Derive conditions for the validity of the continuity equation in the framework of time-dependent density functional
theory. The conditions apply separately to the four spin-isospin channels of the one-body density matrix.
Results: We obtained four sets of constraints on the coupling constants of the N3LO energy density functional that guarantee
the validity of the continuity equation in all spin-isospin channels. In particular, for the scalar-isoscalar channel, the
constraints are the same as those resulting from imposing the standard U(1) local-gauge-invariance conditions.
Conclusions: Validity of the continuity equation in the four spin-isospin channels is equivalent to the local-gauge invariance
of the energy density functional. For vector and isovector channels, such validity requires the invariance of the functional
under local rotations in the spin and isospin spaces.
PACS numbers: 21.60.Jz, 11.30.-j
I.INTRODUCTION
In recent years, methods using energy density func-
tionals (EDFs) [1] to describe nuclear properties are be-
ing developed in three complementary directions. First,
the ideas of effective theories [2, 3] are employed in deter-
mining the EDFs from first principles [4–7]. These devel-
opments are supplemented by a renewed interest [8–10]
in the density-matrix expansion (DME) methods [11, 12],
which allow for treating exchange correlations in terms of
(quasi)local functionals. Second, the coupling constants
of the well-known EDFs undergo a thorough scrutiny,
including an advanced work on the readjustment of pa-
rameters [13, 14] and study of inter-parameter correla-
tions [15]. Finally, the standard functionals are extended
by adding new terms [16–19], so as to gain increased pre-
cision of description and predictability, in quest for the
spectroscopic-quality [20] and universal [21] EDFs.
In the present work we study properties of EDFs [16]
and pseudopotentials [18] extended by adding terms that
depend on higher-order derivatives up to sixth, next-to-
next-to-next-to-leading order (N3LO). Such extensions
lead to self-consistent mean-field Hamiltonians that are
sixth-order differential operators [22], that is, they de-
pend on up to sixth power of the momentum operator.
This makes them unusual objects, in the sense that stan-
∗Electronic address: francesco.raimondi75@gmail.com
dard second-order one-body Hamiltonians contain only
the Laplace operator in the kinetic-energy term and pos-
sibly the angular-momentum operator in the spin-orbit
term. The main question we address here is whether the
presence of higher powers of momenta is compatible with
the continuity equation (CE).
The CE is a differential equation that describes a con-
servative transport of some physical quantity [23]. In
quantum mechanics, it relates the time variation of the
probability density to the probability current [24]. In our
case, it appears when the N3LO EDFs or pseudopoten-
tials are employed within a time-dependent theory. For
the standard Skyrme (NLO) functional, the validity of
the CE has been checked explicitly [25]. Our goal here
is to derive constraints on the coupling constant of the
N3LO EDF or parameters of the pseudopotential that
would guarantee the validity of the CE. Apart from link-
ing the CE to the local gauge symmetry [26], we also
analyze the CEs in vector and isovector channels and
link them to the local non-abelian gauge symmetries.
The paper is organized as follows.
present the standard quantal CE for a single particle and
introduce the vector CE. Then, in Sec. IIB we discuss
the CEs within the time-dependent density functional
theory and in Sec. IIC we specify the case to the N3LO
quasilocal functional. The main body of results obtained
for the CEs in the four spin-isospin channels is presented
in Sec. III and Appendices A–C. Finally in Sec. IV we
formulate the conclusions of the present study.
In Sec. IIA we

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II. CONTINUITY EQUATION IN THE EDF
APPROACH
A.Time evolution of a spin-1
2particle
We begin by recalling the well-known [24] CE for a
single particle. The time evolution of a non-relativistic
spin-1
2particle moving in a local potential is given by the
Schr¨ odinger equation,
i?∂
∂tψ(rσ,t) = −?2
+
?
where V0(r,t) and V1µ(r,t) are scalar and vector real
time-dependent potentials, respectively, and ?σ|σµ|σ′?
are the standard Pauli matrices. By multiplying Eq. (1)
with ψ∗(rσ,t), summing up over σ, and taking the imag-
inary part, we obtain the standard CE for the probability
density ρ(r,t) in terms of the current j(r,t),
2m∆ψ(rσ,t) + V0(r,t)ψ(rσ,t)
V1µ(r,t)
?
µ=x,y,z
σ′
?σ|σµ|σ′?ψ(rσ′,t),(1)
∂
∂tρ(r,t) = −?
m∇ · j(r,t),(2)
where
ρ(r,t) =
?
?
σ
|ψ(rσ,t)|2, (3a)
j(r,t) =
σ
Im
?
ψ∗(rσ,t)∇ψ(rσ,t)
?
.(3b)
We see that the hermiticity of the local potential guar-
antees that the potential energy does not contribute to
the CE of Eq. (2).
Similarly,bymultiplying
ψ∗(rσ′′,t)?σ′′|σν|σ?,
and taking the imaginary part, we obtain the CE for the
spin density sν(r,t) in terms of the spin current Jν(r,t),
Eq.(1)with
summing up over σ′′
and σ,
∂
∂tsν(r,t) = −?
m∇ · Jν(r,t) +1
?
?
V1(r,t)×s(r,t)
?
ν,(4)
where
sν(r,t) =
?
?
σ′σ
ψ∗(rσ′,t)?σ′|σν|σ?ψ(rσ,t),(5a)
Jν(r,t) =
σ′σ
Im
?
ψ∗(rσ′,t)?σ′|σν|σ?∇ψ(rσ,t)
?
.(5b)
We see that the spin CE does depend on the vector po-
tential, and the second term in Eq. (4) is responsible,
e.g., for the spin precession in magnetic field.
It is interesting to note that when potential V1(r,t) is
parallel to the spin density s(r,t) (non-linear Schr¨ odinger
equation), all components of the spin density fulfill the
CEs. In fact, this is exactly the case for the TDHF equa-
tion induced by a zero-range two-body interaction, see
below. Another interesting case corresponds to the vec-
tor potential aligned along a fixed direction in space,
say, along the z axis, that is V1(r,t) = V1(r,t)ez. In
this case, the time evolutions of the spin-up and spin-
down components decouple from one another, that is,
s(r,0) = s(r,0)ez implies s(r,t) = s(r,t)ez, and the
spin-up and spin-down components individually obey the
corresponding CEs.
We also note here that for a nonlocal potential-energy
term,
(ˆV ψ)(rσ,t) =
?
d3r′?
σ′
V (rσ,r′σ′,t)ψ(r′σ′,t),(6)
the time evolution does not, in general, lead to a CE.
B.Time-dependent density functional theory
In the framework of the time-dependent Hartree-Fock
(TDHF) approximation or time-dependent density func-
tional theory (TDDFT), the so-called memory effects are
often neglected and it is assumed that the potential at
time t is just the static potential evaluated at the in-
stantaneous density [27]. For these two time-dependent
approaches, the starting point is the equation of motion
for the one-body density matrix ραβ[26, 28],
i?d
dtρ = [h,ρ],(7)
where the mean-field Hamiltonian hαβ is defined as the
derivative of the total energy E{ρ} with respect to the
density matrix,
hαβ=∂E{ρ}
∂ρβα
. (8)
In the present study we are concerned with the Kohn-
Sham approach [29], whereby the total energy is the sum
of the kinetic and potential-energy terms,
E{ρ} = Ek{ρ} + Ep{ρ},(9)
where
Ek{ρ} =
?2
2m
?
d3rτ0
0(r,t)(10)
and τ0
0(r,t) =
??
στ∇ · ∇′ρ(rστ,r′στ,t)
?
|r=r′ is the
scalar-isoscalar kinetic density, see, e.g., Ref. [30] for def-
initions. The nonlocal density, can be defined in terms
of either the fixed-basis orbitals, ψα(rστ),
ρ(rστ,r′σ′τ′,t) =
?
βα
ψβ(rστ)ρβα(t)ψ∗
α(r′σ′τ′), (11)
or instantaneous Kohn-Sham orbitals, φi(rστ,t),
ρ(rστ,r′σ′τ′,t) =
A
?
i=1
φi(rστ,t)φ∗
i(r′σ′τ′,t).(12)

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The mean-field Hamiltonian is the sum of kinetic and
potential-energy terms, hαβ= Tαβ+ Γαβ, where
Tαβ=
?
d3r
?
στ
ψ∗
α(rστ)−?2
2m∆ψβ(rστ) (13)
and
Γαβ=∂Ep{ρ}
∂ρβα
.(14)
Let us now assume that the potential energy is invari-
ant with respect to a unitary transformation of the den-
sity matrix [26, 28], U = exp(iηG), that is, for all param-
eters η we have,
Ep{ρ} = Ep{UρU+},(15)
where Gαβ is the hermitian matrix of a one-body sym-
metry generator. Then, the first-order expansion in η,
Ep{UρU+} ≃ Ep{ρ} + η
?
βα
?∂Ep{ρ}
∂ρβα
∂(UρU+)βα
∂η
?
η=0
(16)
,
gives a condition for the energy to be invariant with re-
spect to this unitary transformation, that is
TrΓ[G,ρ] ≡ TrG[Γ,ρ] = 0,(17)
which allows us to derive the equation of motion for the
average value of ?G? = TrGρ. Indeed, from the TDDFT
equation (7) we then have:
i?d
dt?G? = i?TrGd
dtρ = TrG[h,ρ] = TrG[T,ρ],(18)
that is, the time evolution of ?G? is governed solely by
the kinetic term of the mean-field Hamiltonian.
1.Continuity equation for the scalar-isoscalar density
The CE now results from specifying ηG to the local
gauge transformation [26, 31] that is defined as
ψ′
α(rστ) ≡ (Uψα)(rστ) = eiγ(r)ψα(rστ).
Then, Eq. (11) gives:
(19)
ρ′(rστ,r′σ′τ′,t) = ei(γ(r)−γ(r′))ρ(rστ,r′σ′τ′,t).(20)
Matrix elements of the local-gauge angle γ(r) are given
by local integrals,
γαβ=
?
d3r
?
στ
ψ∗
α(rστ)γ(r)ψβ(rστ);(21)
therefore, from Eq. (11) again, the average value of the
gauge angle, ?γ? = Trγρ, depends on the scalar-isoscalar
local density ρ0
0(r,t) =?
?γ? =
στρ(rστ,rστ,t), that is,
?
d3rγ(r)ρ0
0(r,t). (22)
Now, the assumed local-gauge invariance of the poten-
tial energy implies the equation of motion for the average
value ?γ?, which from Eq. (18) reads
d
dt?γ? = −?
m
?
d3rγ(r)∇ · j0
0(r,t), (23)
where the standard scalar-isoscalar current is defined as
[30] j0
0(r,t) =?
corresponds to a specific dependence of the gauge angle
on position, γ(r) = P0· r, represents the Galilean in-
variance of the potential energy for the system boosted
to momentum P0. Then, equation of motion (23) simply
represents the classical equation for the center-of-mass
velocity,
στ
1
2i[(∇ − ∇′)ρ(rστ,r′στ,t)]r=r′.
We note here [26, 31], that the gauge invariance that
d
dt
?r?
A
≡d
dtRCM=?P?
mA≡?−i?∇?
mA
.(24)
In the general case, that is, when the potential energy
is gauge-invariant and the gauge angle γ(r) is an arbi-
trary function of r, Eq. (23) gives the CE that reads
d
dtρ0
0(r,t) = −?
m∇ · j0
0(r,t). (25)
Thus for a gauge-invariant potential energy density, the
TDHF or TDDFT equation of motion implies the CE,
that is, the gauge invariance is a sufficient condition for
the validity of the CE. By proceeding in the opposite di-
rection, we can prove that it is also a necessary condition.
Indeed, the CE of Eq. (25) implies the first-order condi-
tion (17), and then the full gauge invariance stems from
the fact that the gauge transformations form local U(1)
groups.
2.Continuity equation for densities in spin-isospin
channels
We can now repeat derivations presented in Eqs. (19)-
(23) by considering the spin-isospin local-gauge groups,
and derive CEs in other spin-isospin channels. To this
end, we first express the nuclear one-body density matrix
(11)–(12) as a linear combination of nonlocal spin-isospin
densities ρt
v(r,r′) [9],
ρ(rστ,r′σ′τ′) =
1
4
?
t=0,1
v=0,1
?√3
?v+t?
σσσ′
v
?τt
ττ′ρt
v(r,r′)?0?
0, (26)
where the sums run over the spin (v = 0,1) and isospin
(t = 0,1) indices denoted by subscripts and superscripts,
respectively, coupled to total scalar and isoscalar. Here
and below we use the coupling of spherical tensors both
for angular momentum and isospin tensors; therefore, in
Eq. (26) the factor of?√3?v+twas included so as to can-
efficients, and to maintain the standard normalization
cel the corresponding values of the Clebsch-Gordan co-

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of the spin-isospin densities. The spin-isospin densities
can be conversely expressed as the following traces of the
density matrix,
ρt
v(r,r′) =
?
στ,σ′τ′
σσ′σ
v
τt
τ′τρ(rστ,r′σ′τ′).(27)
The CEs for densities in the scalar-isoscalar (v = 0,
t = 0), scalar-isovector (v = 0, t = 1), vector-isoscalar
(v = 1, t = 0), and vector-isovector (v = 1, t = 1)
channels,
d
dtρt
v(r) = −?
m∇ · Jt
v(r),(28)
where Jt
ρt
v(r,r), are now equivalent to the local gauge invari-
ances, respectively, with respect to the four local spin-
isospin groups:
v(r) =
1
2i(∇ − ∇′)ρt
v(r,r′)|r′=r and ρt
v(r) =
Ut
v(r) = exp
?
i??γt
v(r)σv
?
0τt?0?
.(29)
Of course, the standard CE derived in Sec. IIB1 cor-
responds to γ(r) ≡ γ0
groups are different: U0
0(r) gives the standard abelian
gauge group U(1), U0
gauge groups SU(2), whereas U1
non-abelian gauge group SU(2)×SU(2).
0(r).Note that the four gauge
1(r) and U1
0(r) form the non-abelian
1(r) corresponds to the
C.The N3LO quasilocal functional
We are now in a position to discuss the CE for the
N3LO quasilocal functional introduced by Carlsson et
al. [16].By imposing on the functional the gauge-
invariance conditions, we can then confirm and explicitly
rederive the results of Sec. IIB. The explicit derivation
will also allow us to discuss the CEs for densities in other
spin-isospin channels analyzed in Sec. IIB2.
Below we consider the EDF given in terms of a local
integral of the energy density HE(r),
?
which is represented as a sum of the kinetic and potential
energies conforming to Eq. (9),
E{ρ} =d3r HE(r),(30)
HE(r) =?2
2mτ0
0(r) +
?
t=0,1
Ht(r).(31)
To lighten the notation and avoid confusion with the
isospin index t = 0,1, in this section we do not explicitly
show the time argument of densities, which within the
TDDFT all depend on time.
The quasilocal N3LO EDF was constructed [16] by
building the t = 0 and t = 1 potential-energy densities
Ht(r) from isoscalar and isovector densities, respectively,
and their derivatives up to sixth order. For clarity, we
give here a brief summary of definitions and notations
used in this construction.
The local higher-order primary densities are defined
by the coupling of relative-momentum tensors KnL[16]
with nonlocal densities (27) to total angular momentum
J, that is,
ρt
nLvJ(r) =??KnLρt
v(r,r′)?
J
?
r′=r. (32)
Then, a general term of the N3LO functional can be writ-
ten, in the language of the spherical tensors, as
Tn′L′v′J′,t
mI,nLvJ(r)=
??ρt
n′L′v′J′(r)?DmIρt
nLvJ(r)?
J′
?0?
0, (33)
where the local secondary densities, [DmIρt
obtained by acting with derivatives DmIon primary den-
sities and coupling them to total angular momentum J′.
Each term (33) is multiplied by the corresponding cou-
pling constant Cn′L′v′J′,t
mI,nLvJthat is denoted by the same set
of indices as those in the term itself.
We note here that the definition of the isovector terms
depends on whether one uses Cartesian or spherical rep-
resentation of tensors in isospace. On the one hand, the
use of the standard Cartesian representation, see, e.g.,
Refs. [16, 30], implies that the isovector terms depend on
products of differences of neutron and proton densities.
On the other hand, the use of the spherical representa-
tion, which was assumed in Ref. [18] and is also used in
the present study, involves the coupling of two isovectors
to a scalar, whereby there appears a Clebsch-Gordan co-
efficient of?√3?−1. Therefore, for the isospace spherical
the factor of
larger than those for the Cartesian
representation.
In the remaining part of this section, we employ
the compact notation introduced in Ref. [22], whereby
the grouped indices, such as the Greek indices α =
{nαLαvαJα} and the Roman indices a = {maIa}, denote
all the quantum numbers of the local densities ρα(r) and
derivative operators Da, respectively. In this notation,
the N3LO potential-energy density of Eq. (31) reads
nLvJ(r)]J′, are
representation, the isovector coupling constants are by
?√3?
Ht(r) =
?
aαβ
Cβ,t
a,αTβ,t
a,α(r).(34)
Our following discussion of the CE is mainly focused on
the one-body potential-energy term, defined in Eq. (14)
as the variation of the potential energy with respect to
the density matrix. For the N3LO functional, this term
was derived in Ref. [22], where it was shown that in space
coordinates it has the form of a one-body pseudopoten-
tial,
ˆΓσσ′
ττ′(r) =
?
γ,t
??
Ut
γ(r)
?
DnγLγσσσ′
vγ
?
Jγ
?
0
τt
ττ′
?0
.(35)
An equivalent form of the one-body pseudopotential,
which can be obtained by recoupling spherical tensors

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within a scalar, and which separates out the spin Pauli
matrices, reads
ˆΓσσ′
ττ′(r) =
?
γ,t
???Ut
γ(r)DnγLγ
?
vγσσσ′
vγ
?
0τt
ττ′
?0
.(36)
In turn, potentials Ut
binations of the secondary densities,
γ(r) were derived as linear com-
Ut
γ(r) =
?
aαβ;dδ
Cβ,t
a,αχβ;dδ
a,α;γ
?Ddρt
δ(r)?
Jγ, (37)
where χβ;dδ
body operatorˆΓσσ′
fined in terms of potentials Ut
tors DnγLγacting on single-particle wave functions. We
note here, that in Eqs. (35) and (36) potentials always
appear to the left of all derivatives; nonetheless, the one-
body pseudopotential is a hermitian operator, which is
guaranteed by specific conditions obeyed by potentials
Ut
γ(r), which were derived in Ref. [22].
Fortheone-body pseudopotential
Schr¨ odinger equation that gives the time evolution
of single-particle Kohn-Sham wave functions in space
coordinates reads,
a,α;γare numerical coefficients. We call the one-
ττ′(r) pseudopotential, because it is de-
γ(r) and differential opera-
(35),the
i?∂
∂tφi(rστ,t) = −?2
2m∆φi(rστ,t)
?
+
σ′τ′
ˆΓσσ′
ττ′(r)φi(rσ′τ′,t).(38)
By multiplying the Schr¨ odinger equation with the
complex-conjugated wave function, φ∗
summing over the single-particle index i we obtain the
time-evolution equation of the density matrix (12), that
is,
i(r′σ′τ′,t) and
i?∂
∂tρ(rστ,r′σ′τ′,t) =
−?2
?
−ˆΓσ′σ′′∗
2m(∆ − ∆′)ρ(rστ,r′σ′τ′,t)
?ˆΓσσ′′
τ′τ′′ (r′)ρ(rστ,r′σ′′τ′′,t)
+
σ′′τ′′
ττ′′(r)ρ(rσ′′τ′′,r′σ′τ′,t)
?
. (39)
Before we proceed, we must first consider the complex-
conjugated pseudopotentialˆΓσ′σ′′∗
use the property of the Biedenharn-Rose phase conven-
tion employed in Refs. [16, 22], by which all scalars are
always real. Note that for the spherical representation of
Pauli matrices, the Biedenharn-Rose phase convention
implies the transposition of spin indices, that is,
τ′τ′′ (r′). To this end, we
?
σσσ′
vµ
?∗
= (−1)v−µσσ′σ
v,−µ, (40)
where µ = 0 for v = 0 and µ = −1,0,1 for v = 1 denote
tensor components of scalar and vector Pauli matrices,
respectively.
Finally,inEqs.(36)
conjugationonlyaffects
(−1)nγ+ma+mdχβ;dδ
and
coefficients
(37), thecomplex
χ∗β;dδ
a,α;γ
=
a,α;γ[22], which gives,
ˆΓσ′σ′′∗
τ′τ′′ (r′) =ˆΓ
′σ′′σ′
τ′′τ′ (r′), (41)
for
ˆΓ
′σσ′
ττ′ (r′) =
?
γ,t
???
U
′t
γ(r′)D′
nγLγ
?
vγσσσ′
vγ
?
0
τt
ττ′
?0
(42)
and
U
′t
γ(r′) =
?
aαβ;dδ
(−1)nγ+ma+mdCβ,t
a,αχβ;dδ
a,α;γ
?D′
dρt
δ(r′)?
Jγ.
(43)
It means that in all further derivations we must use the
second set of potentials U
γ(r′) with signs of terms mod-
ified according to the phase (−1)nγ+ma+md. It is now
obvious that the CEs will hold independently of the spin-
isospin coordinates if, and only if, the pseudopotentials
fulfill the condition
′t
?
σ′′τ′′
?
−Γ
Γσσ′′
ττ′′(r)ρ(rσ′′τ′′,r′σ′τ′)
′σ′′σ′
τ′′τ′ (r′)ρ(rστ,r′σ′′τ′′)
?
r′=r= 0. (44)
We are now in a position to separate the four spin-
isospin channels in Eq. (39). We do so by multiplying
both sides of the equation with σσ′σ
over στ,σ′τ′.From Eq. (27) it is then obvious that,
in close analogy to Sec. IIA, after setting r′= r, we
obtain the CEs (28) in the four spin-isospin channels,
provided terms coming from one-body pseudopotentials
do not contribute, as in Eq. (44). When evaluating this
condition for the four spin-isospin channels, we use the
expression for the trace of three Paul matrices in spheri-
cal representation, which reads [32],
v
τt
τ′τand summing
Tr
?
σvµσv′µ′σv′′µ′′
?
= A(v + v′+ v′′)(−1)v−µCv,−µ
v′µ′v′′µ′′,
(45)
where we introduced A(v+v′+v′′) as a shorthand symbol
for numerical coefficients coming from the computation
of the trace. In the calculation, we only need values of
A(0) = 2, A(2) = 2√3, A(3) = 2√2i. After a trivial but
lengthy calculation, we obtain the final result: