RVB signatures in the spin dynamics of the square-lattice Heisenberg antiferromagnet

0.35 MB·

by  E. A. Ghioldi

epl draft
RVB signatures in the spin dynamics of the square-lattice Heisen-
berg antiferromagnet
6 E. A. Ghioldi, M. G. Gonzalez, L. O. Manuel and A. E. Trumper
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0
Instituto de F´ısica Rosario (CONICET) and Universidad Nacional de Rosario, Boulevard 27 de Febrero 210 bis,
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(2000) Rosario, Argentina

r
a
M
PACS 75.10.Jm–Quantized spin models
1
3 Abstract – We investigate the spin dynamics of the square-lattice spin-1 Heisenberg antiferro-
2
magnet by means of an improved mean field Schwinger boson calculation. By identifying both,
]
l the long range N´eel and the RVB-like components of the ground state, we propose an educated
e
guessfor themeanfield magneticexcitation consistingon alinearcombination oflocal and bond
-
r spinflipstocomputethedynamicalstructurefactor. Ourmainresult isthatwhen thismagnetic
t
s excitation is optimized in such a way that the corresponding sum rule is fulfilled, we recover the
. low and high energy spectral weight features of the experimental spectrum. In particular, the
t
a anomalous spectral weight depletion at (π,0) found in recent inelastic neutron scattering experi-
m
mentscanbeattributedtotheinterferenceofthetripletbondexcitationsoftheRVBcomponent
- of theground state. Weconcludethat theSchwingerboson theory seems tobea good candidate
d
to adequately interpret thedynamic properties of thesquare-lattice Heisenberg antiferromagnet.
n
o
c
[

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v Introduction. – The nature of the spin excitations Signals of these features were also found in the undoped
0 in two dimensional (2D) quantum antiferromagnets (AF) cuprates,beingtheiroriginattributedtothepossiblepres-
0
represents one of the major challenges in strongly cor- enceofextraringexchangeinteractionsduetochargefluc-
2
related electron systems. Originally motivated by the tuations effects [9]. In CFDT, however, the electrons are
1
0 cuprates superconductors [1], the square-lattice Heisen- much more localized implying a negligible role of charge
. bergantiferromagnethaslongbeentheprototypicalmodel fluctuations[14]. Infact,thedispersionrelationmeasured
1
0 toinvestigatethevalidityofthespinwave[2]andthe res- in CFDT has a deviation with respect to linear spin wave
6 onant valence bond (RVB) descriptions [3]. Here, in con- theoryat(π,0) butagreesverywellwithseriesexpansion
1 trast to the one dimensional case [4–7], the ground state [10]andquantumMonteCarlo[11]calculationsperformed
:
v has long range N´eel order, so it is expected that spin-1 in the pure Heisenberg model. Furthermore, spin wave
i magnon excitations take correctly into account the low calculations reproduce the correct spectral weight of the
X
energy part of the spectrum [2]. However, due to the spectrumaround(π,π) while the anomalyat (π,0) seems
r
presence of strong quantum fluctuations, there is an in- tobe reproducedby aGutzwiller projectedwavefunction
a
creasing belief that the high energy part of the spectrum calculation where the isotropic continuum is interpreted
can be described by pairs of spin-1 spinons, representing as spatially extended pairs of fermionic spinons [8]. This
2
the excitations of the isotropic component of the ground calculation, however, fails to consistently describe at the
state [3]. sametimethelowenergypartofthespectrumnear(π,π).
Recent high resolution inelastic neutron scattering ex-
On the other hand, recently, a whole description of the
periments performed in the metal organic compound
dispersion relation in terms of magnons was performed
Cu(DCOO) 4D O (CFDT) –a knownrealizationof the
2 2 using a continuous similarity transformation [12]. In this
·
square-latticeHeisenbergAFmodel–seemtosupportthis
context, the downward renormalization at (π,0) was at-
argument[8]. Specifically,thespectrumhasi)welldefined
tributed to a significant spectral weight transfer from the
low energy magnon peaks around (π,π); ii) a wipe out of
singlemagnonstatestothethreemagnoncontinuum. Un-
the intensity along with a downward renormalization of
fortunately, the corresponding dynamical structure factor
the dispersion near (π,0) and iii) the continuum excita-
has not been computed; so a close comparison with the
tion at (π,0) is isotropic, in contrast to the (π,π) one.
2 2 measured spectral weight has not been carried out yet.
p-1
E. A. Ghioldi et al.
Alternatively, two decades ago, Arovas and Auerbach
J J
developed a bosonic spinon-based theory [13]. Using a Sˆ Sˆ = S2 2Aˆ† Aˆ , (1)
mean field Schwinger boson (MFSB) theory they com- 2 X i· j 2 X h − ij iji
<i,j> <i,j>
puted the dynamical structure factor S(k,ω) in the
square-lattice antiferromagnet (see fig. 1(a)). Even if the where J > 0 is the exchange interaction between near-
spectrum shows a low energy dominant spectral weight est neighbors <ij > and Aˆ†ij = 21 σσˆb†iσˆb†jσ¯ is a sin-
around(π,π) with a magnon dispersion that matches the glet bond operator [15]. IntroduciPng a Lagrange mul-
spin wave result, the spectral weight at (π,0) and (π,π) tiplier λ to impose the local constraint on average and
2 2
are practically the same, namely, at odds with the exper- performing a mean field decoupling of eq. (1), such that
imental spectrum (see fig. 1(b)). In view of the recent Aij = hAˆiji = hAˆ†iji, the diagonalized mean field Hamil-
neutron scattering experiments on CFDT, and the theo- tonian yields [13,16,18]
retical difficulties mentioned above, the search for a con-
sistent theoretical description that takes into account the HˆMF =Egs+ ωk αˆ†k↑αˆk↑+αˆ†−k↓αˆ−k↓ ,
main features of the spectrum has become an important Xk h i
goal.
where
In this paper. we perform an improved mean field
Schwinger boson calculation of the dynamical structure N
factor for the square-lattice Heisenberg antiferromagnet. Egs = ωk− 2 J(S2−A2δ)−λN(2S+1)
X X
k δ
Using the fact that the mean field ground state can be
described by a N´eel and an averaged RVB component, we is the groundstate energy,with A chosenrealandδ con-
δ
investigatethe correspondingspectralpropertiesandpro- necting all the first neighbors of a square lattice, while
pose an educated guess for the magnetic excitations con-
sistingofalinearcombinationoflocalandbondspinflips. ωk↑ =ωk↓ =ωk =[λ2−(γkA)2]21,
Notably, when this magnetic excitation is optimized in
suchawaythatthesumrule dω S(k,ω)=NS(S+1) is the spinon dispersion relation with γA =
k k
is fulfilled we find that the mRainPspectral weight features 1J A sin(k.δ), The ground state wave function
2 δ δ
of the experimental spectrum are reproduced quite well gsPissuchthatαˆ gs =αˆ gs =0andhasaJastrow
k↑ k↓
| i | i | i
(see fig. 3(a)). In particular, our results support the idea form [19]
thattheanomalousspectralweightdepletionat(π,0)can
be attributed to the interference of the triplet bond exci- gs =e kfkb†k↑b†−k↓ 0 b, (2)
tations corresponding to the averaged RVB component of | i P | i
where 0 is the vacuum of Schwinger bosons b and
the ground state [14]. b
| i
fk = −vk/uk, with uk = [21(1 + ωλk)]12 and vk =
N´eel and RVB components of the mean field ısgn(γkA)[12(−1+ωλk)]21 theBogoliubovcoefficientsusedto
Schwinger boson ground state. – It is firmly es- diagonalizeHˆ . Theselfconsistentmeanfieldequations
MF
tablished that the ground state of the spin-12 Heisenberg for Aδand λ are,
modelonthe square-latticeis anSU(2)brokensymmetry
quantum N´eel state [2]. Nonetheless, the nature of the
zero point quantum fluctuations and the spin excitations 1 γA
A = k sin(k.δ) (3)
above the ground state are still controversial. It has been δ 2N ω
X k
k
proposed that the zero point quantum fluctuations have
1 1 λ
both local and RVB character, the latter being related S+ = . (4)
2 2N ω
to the anomaly found at (π,0) in the neutron scattering Xk k
experiments of CFDT [14]. In this section we show that
The rotational invariance preserved by the MFSB allows
the groundstate providedby the MFSB canbe relatedto
us to study finite size systems directly, avoiding the sin-
the above proposal.
gularities that arise when a broken symmetry state is as-
sumed, likein spin wavetheory [20]. Numericalcomputa-
Here, we present the main steps of the mean field
tionoftheaboveselfconsistentequationsshowstwogauge
Schwinger boson theory, originally developed by Arovas
related singlet solutions, the s wave (A =A ) [13] and
andAuerbach[13]. WithintheSchwingerbosonrepresen- δx δy
the d wave (A = A ) [21] solutions. In particular, as
tation the spin operators are expressed as Sˆ = 1b†~σ b , δx − δy
i 2 i i soonasthesystemsizeN increases,bothsolutionsdevelop
withthespinorb†i=(ˆb†i↑;ˆb†i↓)composedbythebosonicop- 180◦N´eelcorrelationssignaledbytheminimumgapofthe
eratorsˆb† andˆb† ,and~σ=(σx,σy,σz) the Paulimatrices. spinondispersionat (π,π) that vanishesin the thermo-
i↑ i↓ ± 2 2
To fulfill the spin algebra the constraint of 2S bosons per dynamic limit. This closing of the gap is related to the
site, ˆb† ˆb = 2S, must be imposed. Using this repre- spontaneous SU(2) broken symmetry N´eel state [19,22].
σ iσ iσ
sentaPtion the AF Heisenberg Hamiltonian results [13], By the way, it is instructive to rearrange eq. (4) as
p-2
RVB signatures in the spin dynamics of the square-lattice Heisenberg antiferromagnet
with
S = 1 |fk|2 . (5) D† =b† b† +b† b†
N [1 f 2] iδ i↑ i+δ↓ i↓ i+δ↑
X k
k −| |
that creates a triplet of z component equal to zero;
In the thermodynamic limit the self consistent solutions while operators like T† = −b† b† and T† = b† b†
yield |f±(π2,π2)| → 1. Then, the quantum corrected mag- create triplets of z coimδ↑poneni↑t ei+quδ↑al to 1i,δ↓respeci↓tivi+elδy↓.
netization m can be obtained as the singular part of eq. − ±
Actually, other kind of triplet bond excitations such as
(5) [19], C† = b† b b† b , can be constructed. However,
m= 2 |f(π2,π2)|2 . (6) afitδer a siy↑sti+emδ↑a−ticis↓tuid+yδ↓of all possible triplet bond exci-
N [1 f(π,π) 2] tations we have found that the correctspectralproperties
−| 2 2 |
are recovered once the excitations corresponding to D† ,
From eq. (2), it is clear that the singular behavior is due iδ
to the condensation of spin up and spin down bosons at T† ,andT† areincorporatedinthedynamicalstructure
iδ↑ iδ↓
k = (π,π). Consequently, in the thermodynamic limit, factor calculation (see below).
± 2 2
the ground state can be splitted as
Dynamical structure factor study. – In this sec-
gs = c n ,
| i | i| i tion we show the difficulty of the original MFSB the-
where ory [13] to recover the anomaly of the spectrum at (π,0)
andwe propose animprovedcalculationof the dynamical
structurefactorbyconsidering,explicitly,thetripletbond
c =e√N2m(cid:16)b†(π2,π2)↑+b†−(π2,π2)↑+ib†(π2,π2)↓−ib†−(π2,π2)↓(cid:17) 0 b excitationsDˆq† mentionedinthe previoussection. Atzero
| i | i temperature the dynamical structure factor is defined as
(7)
is the condensate part which represents the quantum cor-
rected N´eel state, CSz C =( 1)rx+rym, and S(k,ω)= gsSˆk n 2δ(ω (ǫn Egs)), (9)
h | r| i − X|h | | i| − −
n
|ni =ePk6=±(π2,π2)fkb†k↑b†−k↓|0ib where |ni are the spin-1 excited states. Plugging the cor-
responding mean field states in eq. (9) results in
is the isotropic normal fluid part which represents the
zero point quantum fluctuations of the ground state [19].
1
A practical advantage of the MFSB theory is that, even S(k,ω)= u v u v 2δ(ω (ω +ω )).
k+q q q k+q −q k+q
4N | − | −
workingonfinitesystems,itispossibletokeeptrackofthe Xq
putativemagneticorder. Forinstance,afinitesizescaling (10)
ofeq. (6)givesm=0.3034,inagreementwithArovasand
Auerbachresult[13]. Goingbacktorealspacethe normal By exploiting the fact that the MFSB theory fulfills the
fluid part is, approximately, Mermin Wagner theorem [23], Arovas and Auerbach ac-
cessed to the above zero temperature dynamical struc-
n e ijfijAˆ†ij 0 b, (8) turefactorcomingfromthefinitetemperatureregime[13].
| i ≈ P | i
Here, instead, we work at zero temperature and use finite
where f is the Fourier transform of f . Equation
ij k sizesystemstocomputeeq. (10)wherethecontributionof
(8) shows explicitly the singlet bond structure of the
the x,y and z components are identical due to the SU(2)
normal fluid part of the ground state. Although it is
symmetry of the ground state. However, due to the de-
not a true RVB state, because the constraint is only
velopment of long range N´eel order, the spectrum shows
satisfied on average, one can still interpret the normal
BraggpeaksandlowenergyGoldstonemodesatk=(0,0)
fluid component of the ground state as an averaged RVB
and (π,π). This is shown in fig. 1(a) where eq. (10) is
component.
displayed in an intensity curve. Even if the spectrum is
expressed in terms of free pairs of spinons it is expected
Keeping in mind this picture for the ground state, two
that gauge fluctuations, dynamically generated, will con-
kind of magnetic excitations can be envisaged within the
finethemintomagnonicexcitation[24,25]. Infact,asim-
MFSB: local spin flips and bond spin flips acting on the
ple firstordercalculationinperturbationtheory supports
condensateandnormalfluidcomponents,respectively. On
the picture of tightly bond pair of spinons in the neigh-
one hand, magnonic-like excitations are created by Sˆz ,
q borhood of the Goldstone modes [26]. Furthermore, the
which is a linear combination of the local operator Sˆz .
i spectrum shows a low energy dominant spectral weight
Ontheotherhand,tripletbondexcitationsoftheaveraged
around(π,π) with a magnon dispersion that matches the
RVB component can be created by D†, which is a linear
q spin wave result. However the spectral weight at (π,0)
combination of the bond operator and(π,π)arepracticallythesame(seefig. 1(b)),thatis,
2 2
at odds with the anomaly observed at (π,0) in the neu-
1
Dˆi† = 2(Di†δx +Di†δy), tron scattering experiments of CFDT [8]. This anomaly
p-3
E. A. Ghioldi et al.
Fig. 1: Dynamical structure factor S(k,ω) within the MFSB Fig. 2: Dynamical structure factor D(k,ω) within the MFSB
(eq. (10)). (a) intensity curve along a path of the Brillouin (eq. (12)). (a) intensity curve along a path of the Brillouin
zone. (b) comparison between k = (π,π) (solid line) and zone. (b) comparison between k = (π,π) (solid line) and
2 2 2 2
k=(π,0) (dashed line). k=(π,0) (dashed line).
was previously interpreted as a quantum mechanical in- energies. Notice the high energy spectral weight trans-
terference due to the entanglement of the RVB compo- fer with respect to S(k,ω) (see fig. 1(a)). In particular,
nent of the ground state [14]. This motivated us to focus at (π,0) the spectral weight transfer is complete (see fig.
onthe spectralpropertiesof the tripletbond operatorD† 2(b)). It is important to note that among all possible
i
which,withinthecontextofourapproximation,represents tripletbondexcitationsmentionedinthe previoussection
apropermagneticexcitationofthe averagedRVBcompo- only the spectrum of D(q,ω) shows this interference at
nent of the mean field ground state. The corresponding (π,0).
dynamical structure factor is In principle, such anomaly should appear in a rigorous
calculationofS(k,ω). Butitisknownthat,duetothere-
laxationofthe localconstraint,the excitationscreatedby
D(k,ω)= gsDˆk n 2δ(ω (ǫn Egs)), (11) Sˆq in the MFSB are not completely physical,producing a
|h | | i| − −
Xn factor 3 in the sum rule, dω S(k,ω)= 3NS(S+1)
2 k 2
[13]. Therefore, to improRve thPe S(k,ω) calculation one
with Dˆk the Fourier transform of Di†. Within the MFSB should project the mean field ground state and the spin-
eq. (11) results 1 states onto the physical Hilbert space gsP−1S P n ,
k
h | | i
an operation that is very difficult to implement not only
analytically [27,28] but numerically [29–31]. In our case,
1
D(k,ω)= [(γ +γ )u u ]2δ(ω ω ω ) the proper way to correct the mean field, or saddle point
k k+q k k+q k k+q
N − −
Xq approximation, is to include Gaussian fluctuations of the
(12) mean field parameters [32]; although its concrete compu-
with γ = 1 cosk.δ. In fig. 2(a) is plotted eq. (12) tation for the dynamic structure factor may turn out a
k 2 δ
in an intensitPy curve. As expected, the spectral weight of quite long task that is out of the scope of the present
these triplet bond fluctuations is mostly located at high work[33]. Alternatively, welook foraneffective magnetic
p-4
RVB signatures in the spin dynamics of the square-lattice Heisenberg antiferromagnet
excitation that somehow mimics the effect of the above
projection. To do that, based on the nature of MFSB
groundstatediscussedintheprevioussection,wepropose
an educated guess for the magnetic excitation consisting
onalinearcombination(1 β)Sz+βD† insuchawaythat
− k k
the free parameter β can be adjusted to enforce the cor-
rectsumrule. Hereitisimportanttonotethatboth, Sˆz
q
and Dˆ†, produces the same change in the z-component
q
of the total spin, ∆Sz = 0, when they are applied to
total
thegroundstate. Then,themodifieddynamicalstructure
factor yields
S(β,k,ω)=3 |hgs|(1−β)Szk+βDk†|ni|2δ(ω−(ǫn−Egs)),
X
n
(13)
where the factor 3 is due to rotational invariance. Notice
that β = 0 is equivalent to eq. (10); while β = 1 corre-
sponds to eq. (12) , up to a factor 3 . After a little of
algebra,
3(1 β)2
S(β,q,ω)= 4−N Ω2k,q δ(ω−ωk−ωk+q)+
X
k
3β2
+ Γ2 δ(ω ω ω )+ (14)
N k,q − k− k+q
X
k
+3(1−β)β Γk,q Ωk,q δ(ω ωk ωk+q) Fig. 3: Modified dynamical structure factor S(β,k,ω) within
N X − − the MFSB (eq. (14)). For β∗ =0.315 the sum rule is fulfilled
k
(eq. (15)). (a) intensity curve along a path of the Brillouin
with zone. The white box shows theanomalous behavior at (π,0) .
Γk,q = uk+qvq uqvk+q (b) comparison between k=(π2,π2) (solid line) and k=(π,0)
| − | (dashed line).
and
Ω =[(γ +γ )u u ].
k,q k k+q k k+q
We have computed eq. (14) finding that the correct sum that if the d wave solution (A = A ) is used the re-
δx − δy
rule sults are the same if the triplet bond operator is changed
to Dˆ† = 1(D† D† ). This shows the sensitivity of
i 2 iδx − iδy
S(β,k,ω)dω =NS(S+1) (15) the MFSB to capture the intimate connection between
Z
X the structure of the ground state and the corresponding
k
triplet bond excitation.
is fulfilled for β∗ = 0.315. Notably, S(β∗,q,ω) repro-
Finally, it is important to point out that the anomaly at
duces qualitativelyquite wellthe lowand the high energy
(π,0)isonlynoticeableforthequantumspincaseS =1/2.
features of the expected spectrum. This is shown in fig.
In fact, we have found that, as soon as S is increased,the
3(a) where it is clear that the partial depletion of spec-
contributionof the bond spin flips to S(β∗,k,ω) becomes
tral weight at (π,0) (white box of fig. 3 (a)) is directly
negligible with respect to the local spin flip one, thus re-
related to the presence of the triplet bond spin flips (see
covering the expected large S spin wave result.
S(k,ω) of fig. 1(a))). Therefore, we can conclude that
the modified dynamicalstructurefactor S(β∗,k,ω)corre- Summary and concluding remarks. – Motivated
sponding to the optimized operator (1 β∗)Sz +β∗D† bytherecentinelasticneutronscatteringexperimentsper-
− k k
is mimicking the aimed effect of the projection operation; formed in CFDT [8], the experimental realization of the
although, actually, we are not strictly imposing the local square-lattice quantum antiferromagnet, we have investi-
constraint. However,thefactthattheexpectedfeaturesof gated the anomaly found in the spectrum at (π,0) using
thespectrumarerecoveredoncethesumruleisfulfilledis mean field Schwinger bosons. Based on the proposal that
encouraging. Sofarthedynamicalstructurefactorresults suchanomalycouldbeduetothequantummechanicalde-
for D(k,ω) and S(β∗,k,ω) correspond to the s wave so- structiveinterferenceoftheRVBcomponentoftheground
lution (A =A ) of the ground state. We have checked state[14], wehavestudied the spectrumby exploitingthe
δx δy
p-5
E. A. Ghioldi et al.
ability of the MFSB to properly describe the magnetic [9] Coldea R,, Hayden S. M., Aeppli, Perring T. G.,
excitations above the N´eel and the averaged RVB com- Frost C. D., Mason E., Cheong S.-W. and Z. Fisk,
ponents of the ground state. In particular,we have found Phys. Rev. Lett., 86 (2001) 5377. Peres N. M. R. and
thatthetripletD† bondspinexcitations,thenaturalexci- AraujoM.A.N.,Phys.Rev.B,65(2002)132404.Capri-
k otti L., L˚auchli A. and Paramekanti A., Phys. Rev.
tationsabovetheaveragedRVB,haveananomalousspec-
B, 72 (2005) 214433. Headings N. S., Hayden S. M.,
tralpropertyat(π,0) thatcanbe associatedto the above
Coldea R. and Perring T. G., Phys. Rev. Lett., 105
mentioned interference. Then, in order to improve the
(2010) 247001. Dean M. P. M., Springell R. S., Mon-
original dynamical structure factor S(k,ω) [13] we have
ney C., Zhou K. J., Pereiro J., Boˇzovic´ I., Dalla
proposedacombinedmagneticexcitation(1−β)Szk+βDk† Piazza B., Rønnow H. M., Morenzoni E., van den
that gives rise to a modified dynamical structure factor Brink J., Schmitt T. and Hill J. P., Nat. Mat., 11
S(β,k,ω). Remarkably,onceitisoptimizedtoenforcethe (2012) 850.
sum rule, the main features of the spectrum at low and [10] Zheng, W., Oitmaa, J. and Hamer, C. J., Phys. Rev.
highenergiesarereproducedquitewell. Unfortunatelythe B, 71 (2005) 184440.
optimized S(β∗,k,ω) does not recoverthe rotonic feature [11] Sylju˚asen, O. F. and Rønnow, H. M., J. Phys. Con-
dens. Matter, 12, (2000) L405.
at(π,0) but this is because the localconstraintis not im-
[12] Powalski M., Uhrig G.S. and Schmidt K.P., Phys.
posedexactly. If itis imposed carefully[29], asit has also
Rev. Lett., 115 (2015) 207202.
beendoneinthefermionicspinoncase[8],therotonicfea-
[13] Auerbach A. and Arovas D. P., Phys. Rev. Lett., 61
tureswillberecovered. Regardingthepossibilitythatfree
(1988) 617;Arovas D. P. and Auerbach A.,Phys. Rev.
bosonic spinons can survive, or not, at (π,0) is an issue B, 38 (1988) 316.
that is beyond the scope of the present work. However, [14] Christensen, N. B., Rønnow H. M., McMorrow D.
we think that the present results are calling for a more F.,HarrisonA.,PerringT.G.,EnderleM.,Coldea
sophisticated calculation of the dynamical structure fac- R.,RegnaultL.P.andAeppliG.,Proc.NatlAcad.Sci.
tor within the context of the Schwinger boson formalism, USA, 104 (2007) 15264.
suchasvariationalMonte Carlo[29–31]or 1/Ncorrection [15] For frustrated AF it has been discussed in the literature
[13,27,32,33]. Work in the latter direction is in progress. the necessity of using a second singlet bond operator [16–
18] but for the unfrustrated case it is enough the original
formulation with only thesinglet bond operator Aˆ† .
ij
∗∗∗ [16] Ceccatto H. A., Gazza C. J. and Trumper A. E.,
Phys. Rev. B, 47 (1993) 12329.
We thank A. Lobos for the careful reading of the
[17] Flint R. and Coleman P., Phys. Rev. B, 79 (2009)
manuscriptandH.M.Rønnowforusefuldiscussions. This
014424.
workwassupportedbyCONICET(PIP2012)undergrant
[18] Mezio A., Sposetti C. N., Manuel L. O. and
Nro 1060. Trumper A. E., Europhys. Lett., 94 (2011) 47001.
[19] Chandra P., Coleman P. and Larkin A. I., J. Phys.
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