inga fischer, nayana shah and achim rosch- crystalline phases in chiral ferromagnets:...

8/3/2019 Inga Fischer, Nayana Shah and Achim Rosch- Crystalline phases in chiral ferromagnets: Destabilization of helical order

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rXiv:cond-mat/070

2287v2

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Crystalline phases in chiral ferromagnets: Destabilization of helical order

Inga Fischer1, Nayana Shah2, and Achim Rosch11 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany,

2 Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St, Urbana, IL 61801, USA(Dated: April 11, 2008)

In chiral ferromagnets, weak spin-orbit interactions twists the ferromagnetic order into spiralsleading to helical order. We investigate an extended Ginzburg-Landau theory of such systemswhere the helical order is destabilized in favour of crystalline phases. These crystalline phases arebased on periodic arrangements of double-twist cylinders and are strongly reminiscent of blue phasesin liquid crystals. We discuss the relevance of such blue phases for the phase diagram of the chiralferromagnet MnSi.

PACS numbers: 75.10.-b,75.30.Kz,75.90.+w

I. INTRODUCTION

In metallic magnets without inversion symmetry so-called chiral ferromagnets spin-orbit coupling effects de-

termine the physics decisively: they can twist the magne-tization into a helix in the ordered phase of the material.Prominent examples are MnSi and FeGe, which have fora long time been known to exhibit spiral order1. Re-cently, interest in such materials has been renewed witha number of experiments that on the one hand foundnon-Fermi liquid behaviour in a large temperature andpressure region of the phase diagram2, and on the otherhand uncovered signs of a peculiar partially ordered statein neutron scattering experiments3.

In the ordered state of MnSi, helical order is observedin neutron scattering experiments in the form of Braggpeaks situated on a sphere in reciprocal space. The ra-

dius of this sphere is proportional to the inverse pitchof the helix, and the peaks are positioned in the 111-directions. This locking of the helices is due to higherorder spin-orbit coupling effects. For temperatures downto 12 K the phase diagram exhibits a second-order phasetransition out of the helical into the disordered, isotropicphase when external pressure is applied. Below 12 K,helical order is lost at external pressure of 12-14 kbar viaa first-order phase transition into a partially-orderedstate. This state seems to retain remnants of helicalorder on intermediate time and length scales: neutronscattering experiments reveal a signal on the surface ofa sphere in reciprocal space, which is resolution-limitedin the radial direction but smeared out over the surfaceof the sphere3. Maxima of the signal now point into the110-directions but are no longer sharply peaked.

Motivated by these experiments, we investigated waysof destabilizing helical order in chiral ferromagnets andthe possible new phases that can exist besides helicalorder. Such phases are well known for cholesteric liquidcrystals4, i.e. chiral nematics, where several so-called bluephases have been observed. In these phases (more pre-cisely in the phases I and II) crystalline order is formedwhich can be interpreted as a periodic network of ordered

cylinders, see below. The lattice spacing is often of theorder of a few hundred nanometers, leading to the color-ful appearance of these phases (including the color bluein some variants).

As the Ginzburg-Landau theory for the director of achiral nematic liquid is very similar to that for the vec-tor order parameter of a chiral magnet (see discussionbelow), the question arises whether similar phases arerealized in magnetic systems. This question has beenaddressed in some detail by Wright and Mermin4 manyyears ago. They showed that amplitude fluctuations,which are essential for the stability of blue phases, cost afactor 3 more energy for ferromagnets compared to liquidcrystals and concluded that such phases do not appearwithin a Ginzburg Landau theory with local interactions.

Motivated by the physics of MnSi, several groups havereexamined this issue. Roler et al.5 added a further pa-

rameter to the Ginzburg Landau theory, allowing themto reduce the energy of amplitude fluctuations arguingthat such a term might arise from higher-order fluctu-ation corrections. The term considered in Ref. [5] is,however, non-analytic in the Ginzburg-Landau order pa-rameter. In the presence of such a term, they were able toshow that a crystalline array of cylinders can have lowerenergy than a uniform helix. We will discuss a similarstructure below. An alternative route (followed also byus) to destabilize the helical solution are non-local inter-actions as suggested by Binz et al.6. They consideredcrystals formed from superpositions of helices to be re-sponsible for the partially ordered phase of MnSi.

As pointed out by Wright and Mermin4, is is useful todistinguish two rather different limits when investigatingthe physics of blue phases. Here one has to comparethe correlation length for amplitude fluctuations of theordered state (i.e. the width of a typical domain wall)with the wavelength 2/q0 of the helical state. For q0 1, the high-chirality limit, the magnetic structure canbe described by a superposition of a few helices. Thisapproach was taken by Binz et al.6.

The factor q0 is proportional to the strength of spin-orbit coupling1 and therefore expected to become large
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only very close to the second-order phase transitionwhere diverges. As the pitch of the helix3 in MnSi,2/q0 175A, is very large compared to all other mi-croscopic length scales, we think that the low-chiralitylimit, q0 1, is more appropriate for the descriptionof the high-pressure phase of this material. In this limit,amplitude fluctuations of the order parameter cost moreenergy than twists of the phase and therefore one has

to look for order-parameter configurations with as littleamplitude fluctuations as possible.In the following we will first investigate which terms in

the Ginzburg Landau theory destabilize the helical state.Then we will suggest variational solutions appropriate inthe low-chirality limit. Finally we investigate experimen-tal consequences of the resulting structures.

II. CHIRAL MAGNETS AND BLUE PHASES

Starting point of our investigation is a Ginzburg-Landau theory at finite temperature, assuming that all

modes with non-zero Matsubara frequencies are massiveand can be integrated out.Up to second order in spin-orbit coupling, the

Ginzburg-Landau theory for chiral ferromagnets is givenby1

f(r) =

2

(Mi)

2 + M (M) + fFM, (1)

where M = M(r) is the position-dependent magneti-zation and fFM =

2

(M)2 + u(M)4 is the Landau freeenergy of the underlying ferromagnet. Spin-orbit cou-pling is present in the form of the Dzyaloshinsky-Moriyainteraction . In order to facilitate the following dis-cussion, we also provide a second form for (1). It can be

rewritten by setting M = n, where is the amplitudeand n is the direction of the magnetization M:

f(r) =

22inj +

ijknk

2+

2()2

2

2

+ fFM(). (2)

with fFM() =2

2 + u4.Terms that break rotational symmetry are of higher

order in spin-orbit coupling and take the form e.g.

B1

(xMx)2 + (yMy)

2 + (zMz)2

+ B2 (2xM)2 + (2yM)2 + (2zM)2+ B3(M4x + M

4y + M

4z ). (3)

(to fourth order in spin-orbit coupling) for the pointgroup P213 of MnSi

7. These terms will be neglected forthe moment.

The helical state

nhelix(r) = x cos(q0z) + y sin(q0z), (4)

with q0 = / can be shown to be the lowest energystate of (1) if one assumes that the amplitude of the

magnetization is constant and was argued in Ref. [4] to bethe only ordered phase possible for chiral ferromagnets.Higher order terms in the Ginzburg-Landau expansioncan, however, serve to destabilize this helical order andinduce other phases. The simplest (i.e. of lowest orderin spin-orbit coupling) of these terms will be the focus ofour investigation in this paper:

i

MiMi2

= 14

(M2)2. (5)

This term acts only on the amplitude of the magnetiza-tion, not on its direction. Therefore, it gives no contribu-tion to the free energy density in the helical phase, whichhas a uniform amplitude. If (5) has a negative prefactor,then in a certain parameter regime, it can be expected todestabilize helical order in favour of an order parameterwith a fluctuating amplitude. In the following we willuse the shorthand (MM)2 for the expression (5).

In order to determine the new states that could bestabilized by (5), it is instructive to explore the close

analogies between chiral ferromagnets and chiral liquidcrystals4. The order parameter of chiral liquid crystals isa director, and as a consequence topological defects arefundamentally different in both systems. However, chiralliquid crystals can be described by a free energy densitythat is quite similar in form to (2), the only differencesbeing an additional cubic term in and an extra factor1/3 in front of the term ()2.

Locally, the blue phases are based on a configurationof the magnetization that can be shown to be even lowerin energy than the helix:

ndt(r) = z cos(qr) sin(qr), (6)

in cylinder coordinates, see Fig. 1. This can easily beseen4 by comparing the energy density of the uniform

helix, 22

2+fFM(), to the result obtained by pluggingEq. (6) into Eq. (2): For = const. and q = q0, thefirst two terms in (2) vanish for r = 0, and therefore the

energy density at r = 0 is given by 2 2 + fFM(), i.e.lowered by an extra factor 2

22.

This so-called double-twist configuration is cylindri-cally symmetric: sheets of constant magnetization arerolled up around a common cylinder axis (see Fig. 1).The configuration (6) is, however, only favoured in thevicinity of the cylinder axis the energy difference be-tween the helix and the double-twist configuration dimin-ishes and even becomes positive as the distance from thecylinder axis is increased. Isolated double-twist configu-rations therefore necessarily have to occur in the form ofcylinders with an amplitude that becomes zero at a cer-tain distance from the cylinder axis. However, amplitudefluctuations cost energy, as can be seen from the secondterm in (2). In the case of liquid crystals, the prefactorof ()2 is small enough for double-twist cylinders tobecome energetically lower in energy than the helix (seediscussion above). Crystals made of these double twist


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FIG. 1: Cut through a double-twist cylinder: Double-twist-configuration of the magnetization. Sheets of constant mag-netization are wrapped as cylinders around a common axis.

cylinders are indeed believed to lie at the heart of theblue phases I and II in liquid crystals4. In the case of theferromagnet, however, the energy cost of amplitude fluc-tuations outweighs the energy gains due to directionalfluctuations within the double-twist structure, and fora free energy density given by (1) (single) double-twistcylinders do not occur4.

Adding the term (5) to the free energy density can

invalidate this conclusion: if the prefactor of (5) is al-lowed to become negative, it can reduce the cost of am-plitude fluctuations and allow for the appearance of bluephases even in chiral ferromagnets. If (5) has a nega-tive prefactor, then the inclusion of higher order terms inthe Ginzburg-Landau theory becomes necessary in orderto obtain a stable solution and avoid e.g. an unboundedmagnetization.

In the following, we want to analyze the possible occur-rence of the analogue of blue phases in chiral ferromag-nets. By rescaling the magnetization, the free energydensity and the momenta, the free energy density can becast into the following form:

f(r) = M2 +

(Mi)2 + M (M) + M4

+ (MM)2 + hi(M), (7)

where < 0, > 0 and hi(M) is a term containing kpowers ofM and l derivatives (k > 4, l 2), which isused to stabilize solutions against an unbounded mag-netization and oscillations thereof. Possible choices arefor example h1(M) = (MM)2

(Mi)2, h2(M) =

(MM)4 and h3(M) = M2(MM)2. In the rest of

this paper, we will use h1(M) exclusively: this term isthe one with the least number of derivatives and powersofM that on the one hand stabilizes single double twist

cylinders and on the other hand is identically zero for thesingle helix.

III. FROM CYLINDERS TO CRYSTALS

As a first step, we investigated single double twistcylinders for = 1/4. At this point, the helical andisotropic phase are degenerate with fhelical = fisotropic =0. If a single cylinder can now be shown to have negative

1 2 3 4

0.5

1

1.5

2

2.5

3

r

FIG. 2: Amplitude function (r) which minimizes (7) usingthe directional dependence (6), the stabilizing term h1 (with = 6.5, = 5) and the boundary condition (r) = 0 forr > R. Here q = 0.38q0 and R = 3.98 are free variationalparameters where q0 = 1/2 is the helix wave vector. Theresulting free energy gain per length is given by fcyl = 0.21.

free energy, then the system can lower its free energy fur-

ther by creating extended networks of double-twist cylin-ders: crystalline phases, the analogy to blue phases inliquid crystals, can be expected to form in chiral ferro-magnets.

Setting M= ndt(r)(r), with ndt(r) given by (6), wecalculated numerically the amplitude function (r) thatminimizes the free energy density (7) (for certain valuesof and ), subject to the condition that the magneti-zation drops to zero at a certain distance from the cylin-der axis. For a given and sufficiently negative valuesof , single cylinders are stable configurations within aGinzburg-Landau theory of the form (7), see Fig. 2. Inorder to minimize its free energy, the system will try to

produce many such cylinders, packed as tightly as possi-ble. In these configurations, the magnetization only hasto drop to zero on lines or points, if at all. Consideringthat it is the competition between phase and amplitudefluctuations that either stabilizes double-twist structuresor not, it can be expected that crystalline arrangementsof the helices can exist even in parameter regimes wheresingle cylinders are unstable.

The possible crystalline structures are subject to thecondition that the magnetization matches where cylin-ders touch, in order to avoid discontinuities in the magne-tization. This condition is much more restrictive for fer-romagnets compared to liquid crystals4. We have found8

only two allowed structures constructed as networks ofdouble-twist cylinders: a square and a cubic lattice, seeFig. 3. The square lattice is invariant with respect to ro-tations by multiples of/2 around the z-axis, translationsalong the z-axis as well as a translation by a( 1

2, 12

, 0) com-bined with time reversal (M M). In this structure,the magnetization has to go to zero on the line ( a

2, 0, z)

and symmetry-equivalent lines. The symmetry transfor-mations that leave the second structure (cubic lattice)invariant are cyclic permutations of the axes and a trans-lation by a(1

2, 12

, 12

) combined with time reversal. In this


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(a)

y

xa

(b)

z

xy

a

FIG. 3: Crystalline structures built from double-twist cylin-ders: (a) Unit cell of the square lattice of double-twist cylin-ders in the x-y-plane. The magnetization on the cylinder axesas well as at the points where cylinders touch is shown explic-itly. (b) Unit cell of the cubic lattice of double-twist cylinders.At the points where cylinders touch, the magnetization hastwisted to an angle of 45 from the cylinder axis.

structure, the magnetization vanishes only at a 18 , 58 , 38,a38 , 38 , 38

, and symmetry-equivalent points in the unit

cell.We calculated the free energy density of these struc-

tures by means of a variational ansatz for the amplitudeof a single double-twist cylinder:

(r) = y0 (r0 r)er/r1(r0 r), (8)

where y0, r1 and the double-twist wave vector q are vari-ational parameters. This ansatz is based on the numericsfor a single cylinder. The cylinders were then arrangedin crystals as shown in Fig. 3. The cutoffr0

5/(2q)

was chosen for convenience to avoid within the calcula-

tion the tiny overlap of cylinders which are far apart.A phase diagram as a function of the remaining threefree parameters can now be computed. Here, we set to a fixed value, assuming that it is the least susceptibleto variations of external pressure, and established thephase diagram as a function of the two remaining freeparameters. The result for = 0.05 can be seen in Fig. 4.

One immediately notices that there is no parameterregime in which the square lattice shown in Fig. 3 (a)has the lowest free energy density; this is also true fordifferent values of not shown here. While the fillingfraction of double-twist regions is certainly larger in thesquare lattice than in the cubic structure, the magnetiza-tion in the cubic crystal only has to go to zero at pointsand not on lines, as it is the case for the square lat-tice. Furthermore, in the cubic lattice the magnetizationonly has to twist outwards by 45 degrees until cylinderstouch, compared to 90 degrees in the case of the squarelattice (see Fig. 3). These factors conspire to make thefree energy of the cubic structure even lower than that ofthe square lattice. Within our variational ansatz we findthat the transition from the crystalline to the isotropicphase is very weakly first order [the relevant free energydifferences are less than 0.02 in units of Eq. (7)]. Note,

1.36 1.34 1.32 1.3 1.28

2

1.5

1

0.5

0

0.5

helical order

cubic crystal

isotropic phase

FIG. 4: Phase diagram for = 0.05. In addition to thehelical and the isotropic phases there is a parameter rangewhere the cubic lattice of double-twist cylinders minimizesthe free energy.

-1 -0.5 0 0.5

0.5

1

q/q0

= 1.32 = 1.35 = 1.36 = 1.37

FIG. 5: Ratio of double twist wave vector q [see Eq. (6)]versus helix wave vector q0 [see Eq. (4)] for = 0.05: q/q0varies considerably but remains of order 1 in the parameterregion where the cubic crystal is stable.

however, that our analysis might break down close to thetransition, as one enters the high-chirality regime, seeabove.

Since the double twist wave vector q is now also a vari-ational parameter, it is no longer necessarily identical tothe helix wave vector q0. In Fig. 5 we show that q de-pends considerably on the microscopic parameters butremains of order q0.

What signal can these structures be expected to pro-duce in neutron scattering experiments? As for any crys-tal, peaks in neutron scattering originate from the lat-tice structure itself, and the configuration of the magne-tization within the Wigner-Seitz cell only enters in theshape of form factors. The lattice constants are func-tions of the double-twist wave vector and determined bythe requirement that the magnetization has to matchwhere cylinders touch. For the square lattice, one ob-tains a =

2/q, and for the cubic lattice a = 2/q.


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In neutron scattering, higher order Bragg peaks canalso be expected to be be generated by the lattice struc-tures. The elastic scattering crossection9 can be calcu-lated from a Fourier transform of the magnetization:

d

d

el

| (M() )|2. (9)

In order to compare with experimental data, the quanti-

ties of interest are the positions of the Bragg peaks andthe relative intensities of the Bragg peaks.

For the square lattice, normalizing intensities to giveunity for the first reflection with Miller indices 10, the21 and the 30-reflections have intensities 0.08 and0.01, respectively. The invariance of the square lat-tice with respect to a translation by a(1

2, 12

) combinedwith M M constrains all Bragg peaks hk withh + k = 2n to vanish. In the case of the square lattice,the lowest order reflexes already account for 84% of thetotal intensity.

For the cubic lattice, assuming that the lowest Braggpeak, i.e. the

100

-peak, has intensity 1.0, the

210

-

and 300-peaks have relative intensities 0.17 and 0.03.The 110-, and 200-peaks vanish for symmetry rea-sons, while the 111-peak vanishes as a consequence ofour ansatz based on a linear combination of cylinders inx, y and z direction, and should really be non-zero asallowed by symmetry. Our approach suggests, however,that these peaks have small weight. The 100-peaks ofthe cubic lattice only represent 46% of the total scatter-ing intensity Icubic. The 100-, 210- and 300-peaksof the cubic lattice add up to 79% of Icubic.

Anisotropic terms that orient the helix also act onthe crystalline structures and determine their orientationwith respect to the crystal lattice of the substance. For

the square lattice, where all cylinders are arranged in par-allel, we find using Eq. (3) for B1 = B2 = 0, B3 = 0 thatweak anisotropic terms align the cylinder axes parallelto the preferred direction of the helix vector, i.e. eitherin the 111 or 100 direction, depending on the sign ofB3. However, the orientation of the crystal in perpen-dicular direction is not affected to leading order by theanisotropy term, but higher-order terms would lock a per-fect crystal. Motivated by the partial order observedin the high-pressure phase of MnSi (see introduction), weinvestigate the expected signature in neutron scatteringassuming that these higher-order terms are not effective.In this case, the square lattice will produce rings in planesnormal to the

111

-direction (the orientation of the helix

in the low-pressure phase). These rings intersect to pro-duce maxima in the 110-direction on a circle with radius

2q in reciprocal space. For parameters with

2q q0,this is consistent with the observed signatures3 in neu-tron scattering. Note, however, that at least within ourmodel, the square lattice never has the lowest energy.

In Ref. [5] , it was argued that an amorphous textureof parallel cylinders (which the authors called skyrmions)aligned preferentially along the 111 direction would alsoproduces such rings. However, such a scenario would not

explain the resolution limited width in radial direction,i.e. at least on length scales of 2000A the square crystalwould have to remain intact.

In the case of the cubic lattice, the numerical calcu-lation of the free energy density shows that for B3 > 0the anisotropic terms are minimized if one of the axes ofthe cubic lattice is (1/2, 1/2, 0) and the other axesare

(

1/2, 1/2, 1/

2) and

(1/2,

1/2, 1/

2), respec-

tively. While such structures would produce peaks in the110 direction (as observed in the high-pressure phaseof MnSi), one expects also considerable intensity ratherclose to 111 (as e.g. (1/2, 1/2, 1/2) differs onlyby about 10 degrees from (111)). Experimentally, how-ever, one observes a minimum of the intensity in the 111direction. For B1 = B2 = 0, B3 < 0, the system wouldprefer to align the cubic structure with the crystal latticeof MnSi.

Binz et al.6 have argued that the Bragg peaks in neu-tron scattering should be smeared out because the mag-netization has a varying amplitude and is therefore sus-ceptible to interactions with non-magnetic impurities.

The same argument would apply for the crystalline struc-tures presented in this paper.

IV. CONCLUSIONS

In conclusion, we have constructed blue phases inchiral ferromagnets. Is it likely that these blue phasesare realized in MnSi? The signatures in neutron scat-tering seem to be consistent with the square structure(also considered in Ref. [5]), which is, however, neverthe ground state within the models which we considered.Our results suggest that the intensity in neutron scatter-

ing is located on rings intersecting in the 110-direction.It would be interesting to check experimentally whetherthe intensity distribution arising from such a picture fitsquantitatively.

In the case of the cubic lattice, the positions of theexpected maxima do not match the ones observed. Inany case, the smoking gun experiment to detect crys-talline structures, would be the observation of higher or-der Bragg peaks (this applies also for other structures,e.g. those suggested in Refs. [5,6]) and we hope that ourestimates may guide future experiments in this direction.

However, it is still unclear whether the partially or-dered state of MnSi is indeed a separate phase. Re-cent measurements of the thermal expansion coefficientgive no trace of a phase transition from the partially or-dered state to the isotropic phase10. Furthermore, SRexperiments11 give no evidence for static order in thisregime and it remains unclear on which time scales thepartial order survives3.

It therefore remains an open question whether distinctphases other than the helical one are present in chiral fer-romagnets that are currently investigated experimentally.In fact, the partially ordered phase in MnSi might bemore analogous to the blue phase III than to blue phases


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I and II. While the precise structure of the blue phaseIII is not completely clear, it seems to show some type ofcrystalline short range order but is a liquid on long lengthscales12,13. Therefore it has been conjectured14 that anamorphous arrangement of double-twist cylinders is rele-vant for this blue phase III in liquid crystals this mightalso be true3,5,15 for the partially ordered phase in MnSi.

Acknowledgments

We thank B. Binz, C. Pfleiderer, U. K. Roler, A. Vish-wanath for useful discussions and the SFB 608 of theDFG for financial support.

1 P. Bak and M. Hgh, J. Phys. C: Solid St. Phys. 13,881 (1980); O. Nakanishi, A. Yanase, A. Hasegawa, andM. Kataoka, Solid State Commun., 35, 995 (1980).

2 C. Pfleiderer, S.R. Julian, and G.G.Lonzarich, Nature 414,427 (2003).

3 C. Pfleiderer, D. Reznik, L. Pintschovius, H. v. Lohneysen,M. Garst, A. Rosch, Nature 427, 227 (2004).

4 D.C. Wright and N.D. Mermin, Rev. Mod. Phys. 61, 385(1989).

5 U.K. Roler, A.N. Bogdanov, and C. Pfleiderer, Na-ture 442, 797 (2006); see also supplementary notescond-mat/0603104.

6 B. Binz, A. Vishwanath, and V. Aji, Phys. Rev. Lett. 96,207202 (2006); B. Binz and A. Vishwanath, Phys. Rev. B74, 214408 (2006).

7 D. Belitz, T.R. Kirkpatrick, and A. Rosch, Phys. Rev. B73, 054431 (2006).

8 Although the three-dimensional structures proposed inRef. [6] cannot be realized with double-twist cylinders, theynonetheless contain large, clearly identifyable regions that

exhibit double-twist-structures.9 G.L. Squires, Introduction to the Theory of Thermal Neu-

tron Scattering (Dover Publications, New York, 1996).10 C. Pfleiderer, P. Boni, T. Keller, U. K. Roler, A. Rosch,

Science 316, 1871 (2007)11 Y. J. Uemura, T. Goko, I. M. Gat-Malureanu, J. P. Carlo,

P. L. Russo, A. T. Savici, A. Aczel, G. J. MacDougall, J.A. Rodriguez, G. M. Luke, S. R. Dunsiger, A. McCollam,J. Arai, Ch. Pfleiderer, P. Boni, K. Yoshimura, E. Baggio-Saitovitch, M. B. Fontes, J. Larrea, Y. V. Sushko, and J.Sereni, Nature Physics 3, 29 (2007).

12 H.-S. Kitzerow and C.E. Bahr, Chirality in liquid crystals

(Springer Verlag, Berlin, 2001).13 E.P. Koistinen and P.H. Keyes, Phys. Rev. Lett. 74, 4460

(1995).14 R.M. Hornreich, M. Kugler, and S. Shtrikman,

Phys. Rev. Lett. 48, 1404 (1982).15 S. Tewari, D. Belitz, and T.R. Kirkpatrick,

Phys. Rev. Lett. 96, 047207 (2006).
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inga fischer, nayana shah and achim rosch- crystalline phases in chiral ferromagnets:...

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