Partially disordered state near ferromagnetic transition in MnSi
Graph1
Partially disordered state near ferromagnetic transition in MnSi
S.V.Maleyev∗ and S. V. Grigoriev
Petersburg Nuclear Physics Institute, Gatchina, Leningrad District 188300, Russia
(Dated: November 4, 2018)
The polarized neutron scattering in helimagnetic MnSi at low T reveals existence of a partially
disordered chiral state at ambient pressure in the magnetic field applied along 〈111〉 axis below
the first order transition to the non-chiral ferromagnetic state. This unexpected phenomenon is
explained by the analysis of the spin-wave spectrum. We demonstrate that the square of the spin-
wave gap becomes negative under magnetic field applied along 〈111〉 and 〈110〉 but not along the
〈100〉 direction. It is a result of competition between the spin-wave interaction and cubic anisotropy.
This negative sign means an instability of the spin wave spectrum for the helix and leads to a
destruction of the helical order, giving rise to the partially disordered state below the first order
ferromagnetic transition.
PACS numbers: 75.25.+z,61.12.Ex
Non-centrosymmetric cubic helimagnets such as MnSi, FeGe, FeCoSi are the subject of the intensive experi-
mental and theoretical studies for the last several decades. Their single-handed helical structure was explained by
Dzyaloshinskii1. The full set of interactions responsible for observed helical structure (Bak-Jensen model) was es-
tablished later in2,3 in agreement with existing experimental data (see for example4 and references therein). The
renascence in this field began with a discovery of the quantum phase transition to a disordered (partially ordered)
state in MnSi at high pressure5 and6. The following properties of this state attract the main attention: i) non-Fermi-
liquid conductivity, ii) spherical neutron scattering surface with the weak maxima along the 〈110〉 axes7,8, whereas
at ambient pressure Bragg reflections are observed along 〈111〉4. These features and the structure of the partially
ordered state were discussed in several theoretical papers (see9,10,11,12 and references therein). It should be noted also
that the spherical scattering surface with maxima along 〈111〉 was observed at ambient pressure just above critical
temperature Tc ≃ 29K. This experiment was explained using the Bak-Jensen model13.
These studies shadowed an important problem of the helix structure evolution in the external magnetic field H . In
particular, simple phenomenological14 and microscopical15 theories predict the smooth second order transition from
the conical to the ferromagnetic state. The spin component of the cone parallel to the applied field is proportional
to the magnetization and it increases as (H/HC) up to its saturated value. This prediction is in agreement with
experiment6. The perpendicular, rotating spin components fade away with the field and according to the plain theory
the helical Bragg reflections must decrease as (H2C − H2)/H2C where HC is the critical field for the ferromagnetic
transition. This, however, contradicts to the experimental facts if H ‖ [111] (See16 and Fig.1). The experiment shows
that the transition is of the first order and the Bragg intensity does not follow the law given above.
In this paper we demonstrate that although from a general expression for the ground state energy one can expect
the second order transition at the critical field HC , this is true for H ‖ [100] only but the situation changes if H ‖ [111]
or H ‖ [110]. In the last cases the spin-wave spectrum is unstable in the field range H1 < H < HC due to the cubic
anisotropy as the square of the spin-wave gap becomes negative and the helical long-range order decays in this field
interval. The ferromagnetic state occurs above HC . Hence we have a region below HC where the partially disordered
state coexists with the almost saturated magnetization. It is a reason why this state has not been noticed in the
earlier macroscopic measurements5,6.
Let us consider the conical helix with the lattice spin SR = S
ζ̂R + S
η̂R + S
ξ̂R where
ζ̂R = ĉ sinα+ (Ae
ik·R + c.h.) cosα;
η̂R = i(Ae
ik·R − c.h.);
ξ̂R = ĉ cosα− (Aeik·R + c.h.) sinα,
whereA = (â−ib̂)/2 and the unit vectors ζ̂, η̂ and ξ̂ form the right-handed frame. If α = 0 we have a plain helix15. The
spin operators are given by the well known expressions: S
= S−(a+a)R, SηR = −i(S/2)1/2[aR−a
−(a+a2)R/(2S)]
and S
= (S/2)1/2[aR + a
− (a+a2)R/(2S)].
http://arxiv.org/abs/0704.0683v1
Similar to in15, we use the following Hamiltonian
{−JqSq · S−q + 2iDq[Sq × S−q]
l=x,y,z
Sq,lS−q,lq
l }+K
l=x,y,z
Sq,lSp,lSh,lSf ,l
+N1/2H · S0,
where the first term is the ferromagnetic exchange interaction, the second term is the Dzyaloshinskii interaction (DI)
responsible for the helix structure1. The following terms are the anisotropic exchange, the cubic anisotropy and the
Zeeman energy. They determine the orientation of the helix vector k in the magnetic field (see15,16 and17). The
following hierarchy of interactions holds2,3: J0 >> D0/a >> F0/a
2 ∼ K where a is the lattice constant. Replacing
SR → Sζ̂R one gets the classical energy15
Ecl = S
2L(k)
−SD0(k · [â× b̂])
cos2α+ SH‖ sinα+ Ecub
where A = S(J0 − Jk)/k2 is the spin-wave stiffness at q >> k, L(k) =
k̂2l (â
l + b̂
l ) and H‖ is the field component
parallel to k.
Using (3) and (4) and taking into account that the single ion contribution Ecub does not depend on k we get
kl = k(D0/|D0|)[ĉl − SF0(â2l + b̂2l )/(2A)]
Ecl = −(SAk2/2)[1− SF0L(ĉ)/(2A)] cos2 α+H‖ sinα+ Ecub,
where k = S|D0/A| and L(ĉ) is a cubic invariant. For D0 > 0 or D0 < 0 we have the right or the left helix,
respectively3. For F0 < (>)0 the helix vector k is oriented along the 〈111〉 (〈100〉) axes as the invariant L has two
extrema 2/3(0)3.
Neglecting Ecub in Eq.(4) one obtains the conical state with sinα = −H‖/HC if H‖ < HC where HC = Ak2(1 −
SF0L/(2A)] ≃ Ak2 and the ferromagnetic state for H‖ > HC15,19. It can be shown that Ecub gives a negligible
contribution to HC and to sinα. The principal parameters of the magnetic structure for MnSi are: A ≃ 52meV Å
k ≃ 0.038Å and HC ≃ 0.6T ≃ Ak2 in agreement with the theory (see15). Another energy F0k2 ∼ 0.01meV ≃ 0.1T
was estimated from the anisotropy of the critical neutron scattering13 and from the reorientation of the helix axis in
the magnetic field16. The value of K will be estimated below.
The classical energy depends on H‖ only. The general expression for the ground state energy, derived in
15. It
contains yet another term of purely quantum nature, which depends on the spin-wave gap ∆ as a parameter. As shown
in15 if H⊥ > ∆
2 the helix wave vector k is directed along the magnetic field. For MnSi ∆ ≃ 12µeV ≃ 0.1T ≪ HC16.
In a linear spin-wave theory the gap appears due to the cubic anisotropy but it equals to zero at H‖ = 0
20. There
is yet another contribution to the gap, which is a result of the spin-wave interaction. We begin with the former. For
evaluation of ∆2 one has to consider the uniform part of the bilinear Hamiltonian, which is given by
H0 = E0a
0 a0 +B0(a
0 + a
0 )/2, (5)
and ∆20 = E
0 −B20 . If one neglects the cubic anisotropy at H‖ < HC then one has E0 = B0 = (HC/2) cos2 α and the
gap is zero15,18. Taking into account Hcub, one obtains after simple but rather tedious calculations
E0 = (HC cos
2 α)/2 + E1 + E2;B0 = (HC cos
2 α)/2− E2
E1 = −4Λ[(1− L) sin4 α+ 3L sin2 α cos2 α
+ (3/8)(2− L) cos4 α];
E2 = (3Λ/4)[4L sin
2 α+ (2− L) cos2 α];
∆2Cub = (HC cos
2 α+ E1)(E1 + 2E2),
where Λ = S3K. These expressions hold at H‖ < HC and, indeed, ∆
Cub = 0 in zero field only
For H ≥ HC we have cosα = 0 and if without cubic anisotropy E0 = H−HC , B0 = 0 and the gap ∆ = H−HC15,
while if taking it into account the cubic anisotropy we get
∆2Cub = [H −HC − 4Λ(1− L)][H −HC + Λ(10L− 4)]. (7)
For the two principal directions L111 = 2/3 and L001 = 0 and we obtain:
∆2Cub,[1,1,1] = (H −HC − 4Λ/3)(H −HC + 8Λ/3);
∆2Cub,[1,0,0] = (H −HC − 4Λ)
Thus, one comes to the important conclusion: ∆2
Cub,[1,1,1]
is negative at H = HC for both signs of K
21. This
circumstance is decisive for the stability of the system if one takes into account that the contribution to ∆2, stem
from the spin-wave interaction, is proportional to cos4 α and disappears at H close to HC .
Let’s consider the spin wave interaction to the gap ∆2Int. As the DI breaks the total spin conservation law it must
lead to the spin-wave gap. However, in cubic crystals this interaction is very soft [see Eq.(2)] and the gap appears as
a result of the spin-wave interaction only similar to the case of pseudo-dipolar interaction in antiferromagnets22,23.
In the one-loop approximation it consists of both the Hartree-Fock (HF) part evaluated in15 at H = 0 and the
second order contribution from three-point spin-wave interaction, which appears due to the helical structure. It was
ignored in15. The diagrams for both contributions are shown in Fig.2 where lines correspond to Green functions
Gq = − < Taq, a+q > and Fq = F+q = − < Taq, a−q >, which in ω-representation are given by15:
Gq(iω) =
Eq + iω
(iω)2 − ǫ2q
; Fq(iω) = −
(iω)2 − ǫ2q
, (9)
where Eq = S(Mo −Mq) +Bq;Bq = (S/2)(Mq − Jq) cos2 α;Mq = J+q + 2Dq(k · ĉ), J±q = (Jq+k ± Jq−k)/2 and the
spin-wave energy ǫq = (E
q − B2q)1/2. Although at q ≪ 1/a these expressions give the same result as obtained in15:
Eq = Aq
2 + Bq; Bq = (Ak
2/2) cos2 α and ǫq = Aq(k
2 cos2 α + q2)1/2. However for the present consideration we
need them for all q as the formulae for ∆2Int contains the sums, which saturate at q ∼ 1/a.
The contribution of the forth-point interaction to ∆2 was analyzed in15 at H = 0 and T = 0. At an arbitrary H
the gap consists of two terms
V1 = (1/4N)
(M1 +M2 −M1−3 −M2−3)a+1 a
2 a3a4;
V2 = (1/4N)
(M1 − J1)[2(a+a)1(a+a)−1 sin2 α
− (a+a2)1(a1 + a+−1) cos2 α],
where 1 = q1 etc. At small momenta we have in parentheses −2A(3 · 4)/S and Ak2/S for V1 and V2, respectively.
Now one has to consider the V1 interaction and the second part of V2 interaction together. They give the principal
contribution to the gap ∆2Int
∆2Int =
SAk2 cos4 α
(Mq − Jq) =
(Ak2)2 cos4 α
, (11)
where in r.h.s we take into account that
J+q =
Jq = 0 and according to Eq.(4) (k · ĉ) = Ak2/(SD0). This
contribution is T -independent.
The contribution of the first term in V2 is more complicated one. Its T -independent part is proportional to (ka)
and may be neglected [for MnSi (ka)2 ≈ 0.03]. The T -dependent contribution consists of two parts. The excitations
with q ≫ k have a quadratic dispersion ǫq ≈ Aq215 and at T ≫ Ak2 they are responsible for the first part, which
has the form ∆22,1 = (1/2)(Ak
2)2ζ(3/2)(ka)3[T/(2πAk2)]3/2 sin2 α cos2 α. In spite of small factor (ka)3 it may be
important at sufficiently high T .
The second part is not so trivial. According to15 (cf. also11) at q . k and H < HC the spin-wave spectrum
becomes strongly anisotropic due to umklapp processes connecting excitations with q and q± k. In zero field taking
into account the gap we have
ǫq = [A
2(k2q2‖ + 3q
⊥/8) + ∆
2]1/2. (12)
As the field increases this anisotropy becomes weaker since the term in the expression for ǫq appears to be proportional
to q2⊥. In the ferromagnetic state (H > HC) this anisotropy vanishes. In a very weak field we can use Eq.(12) and
then the second part is given by
∆22,2 = (1/π)Ak
3/2 ln(Ak2/∆) sin2 α cos4 α. (13)
This expression diverges if ∆ → 0. Similar divergences were discussed in25. However, we can neglect this contribution
due to the small factor (ka)3 and decreasing of logarithm when H increases.
The three-point interaction is given by V3 = V− cosα+V+ sinα cosα where V± = (2S/N)
+a)−q(a−q±
a+q ) and
C(−)q = [J
q +Dq(q · ĉ)] ∼ (A/S)(q · k)(qa)2
C(+)q = [(Jq − J (+)q )/2−Dq(k · ĉ)] ≃ −Ak2/(2S),
where the r.h.s. expressions are derived from Eq.(4) at q ≪ 1/a. The corresponding contributions to ∆2 were
evaluated as in22. The T -independent term is proportional to (ka)2 and the second one is equal to −2∆2,2/3. Both
of them are small and can be neglected. Finally we have
∆2 = ∆2Int +∆
Cub. (15)
The field dependence of the ratio ∆2(H)/∆2(0) for the three directions 〈111〉, 〈100〉 and 〈110〉 are shown in Fig.3
with HC = 0.565T, ∆(0) = 0.1 T in agreement with the experiment. The cubic anisotropy Λ was chosen to be
equal to −0.05 T. In case of H ‖ [100] ∆2 remains positive and the spin-wave spectrum is stable at all H . In this
case at H > HC along with ferromagnetic spin configuration the spin-wave components of the lattice spins has to
remain rotating [see Eq.(1)] as was discussed in15. For 〈111〉 direction ∆2 is negative at H > H1 ≃ 0.72HC and the
spin-wave spectrum becomes unstable. Hence the long-range helical order demolishes and the corresponding Bragg
peaks disappear and the scattering has to be spread around them. Such decrease of the intensity along [111] is shown
in Fig.1 in qualitative agreement with the theory. However, more detailed measurements are needed. A similar
instability has to be along [110] also contrary to the expectations for H ‖ [100].
The width of this scattering may be estimated from the condition ǫ2q = 0, as for larger q the spin-wave spectrum
can not feel the disorder. Near H1 we have ǫ
q ∼ (Aqk)2 + ∆2 and the inverse correlation length of the disorder
κ = k(|∆|/Ak2) ≪ k. Close to HC ǫq ∼ Aq215 and κ = k(|∆|/Ak2)1/2.
This disordered state have a strong chirality, which is demonstrated by a constant polarization of the scattered
neutrons in the whole field range below HC (see inset in Fig.1). This polarization is determined as P = (I− −
I+)/(I− + I+) and according to the general theory
26 the ratio P/P0, where P0 is the initial neutron polarization, is
the chirality at given q. A strong drop of the polarization at HC is a signature of the first order transition to the
uniform ferromagnetic state with weak chiral fluctuations.
In conclusion, we analyzed thoroughly the field behavior of the spin wave gap in the spin-wave spectrum of cubic
helimagnets. It is shown that if the field is applied parallel to both [111] and [110] directions a partially disordered
state has to take place at H1 < HC . We demonstrated that this state appears when the square of the spin-wave gap
becomes negative and the spin-wave spectrum unstable. We presented the first observation in MnSi of this partially
disordered chiral state in the magnetic field.
The work is supported in part by the RFBR (projects No 05-02-19889, 06-02-16702 and 07-02-01318) and the
Russian State Programs ”Quantum Macrophysics” and ”Strongly correlated electrons in Semiconductors, Metals,
Superconductors and magnetic Materials” and Russian State Program ”Neutron research of solids”.
∗ Electronic address: maleyev@sm8283.spb.edu
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12 B.Binz, A.Vishwanath, Phys.Rev. B 74, 214408 (2006).
13 S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, R.Georgii, P.Böni, D.Lamago, H.Eckerlebe, K.Pranzas,
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15 S.V.Maleyev, Phys.Rev. B 73,174402 (2006).
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mailto:maleyev@sm8283.spb.edu
350 400 450 500 550 600 650
350 400 450 500 550 600 650
H (mT)
H (mT)
FIG. 1: The intensity of the Bragg reflection in MnSi at T = 15 K as function of the field at H ‖ [111]. The full line is the
theoretical prediction (see the text). Inset: the spin chirality as a function of the field measured by polarized neutrons(see
text).
(a) (b)
FIG. 2: Hartree-Fock (a) and three-point diagrams for the spin-wave gap (b).
17 S.V.Grigoriev, S.V.Maleyev, A.I.Okorokov, Yu.O.Chetverikov, H.Eckerlebe, J. Phys.: Condens. Matter 19 145286 (2007).
18 There are misprints in Eqs. (13) and (21,22) in Ref.15: the F terms has to have negative sign.
19 Due to demagnetization HC has additional term depending on the sample form
20 Eq.(53) in15 is erroneous.
21 The same trues in [1, 1, 1] direction where L = 1/2
22 D.Petitgrand, S.V.Maleyev, Ph.Bourges, A.S.Ivanov, Phys.Rev B 59, 1079 (1999).
23 Results presented below were obtained as in22 using modification of the Belyaev’s technique24 adjusted to non-Hermitian
spin-wave interaction in Dayson-Maleyev representation.
24 A.A.Abrikosov, L.P.Gor’kov, I.E.Dzyaloshinskii, Quantum field theoretical Methods in Statistical Physics, (Pergamon, New
York, 1965).
25 T.R.Kirpatrick D.Belitz, Phys.Rev.Lett. 97, 267205 (2006).
26 S.V.Maleyev, Phys. Usp.,45, 569 (2002); Physica B 350, 26 (2004).
0,0 0,2 0,4 0,6 0,8 1,0 1,2
 H || (100)
 H || (111)
 H || (110)
 = - 0.5
FIG. 3: The magnetic field dependence of the ratio ∆2(H)/∆2(0). Parameters are given in the text.
	References
