Exotic phases of frustrated antiferromagnet LiCu2O2
A. A. Bush N. Büttgen A. A. Gippius M. Horvatić M. Jeong W. Kraetschmer V. I. Marchenko Yu. A. Sakhratov L. E. Svistov

APS/123-QED




Moscow Institute of Radioengineering, Electronics and Automation, 117454 Moscow, Russia


Center for Electronic Correlations and Magnetism EKM,
Experimentalphysik V, Universität Augsburg, D–86135 Augsburg, Germany


Faculty of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia
P. N. Lebedev Physical Institute RAS, 119991 Moscow, Russia


Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS (UPR3228), 
 EMFL, UGA, UPS, and INSA, Boîte Postale 166, 38042, Grenoble Cedex 9, France


[Present address: ]School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT,
United Kingdom
Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS (UPR3228), 
 EMFL, UGA, UPS, and INSA, Boîte Postale 166, 38042, Grenoble Cedex 9, France


Center for Electronic Correlations and Magnetism EKM,
Experimentalphysik V, Universität Augsburg, D–86135 Augsburg, Germany


P. L. Kapitza Institute for Physical Problems RAS, 119334 Moscow, Russia


Kazan State Power Engineering University, 420066 Kazan, Russia


svistov@kapitza.ras.ru
P. L. Kapitza Institute for Physical Problems RAS, 119334 Moscow, Russia




^7Li NMR spectra were measured in a magnetic field up to 17 T at temperatures 5-30 K on single crystalline LiCu_2O_2. Earlier reported anomalies on magnetization curves correspond to magnetic field values where we observe changes of the NMR spectral shape. For the interpretation of the field and temperature evolutions of our NMR spectra, the magnetic structures were analyzed in the frame of the phenomenological theoretical approach of the Dzyaloshinskii-Landau theory. A set of possible planar and collinear structures was obtained. Most of these structures have an unusual configuration; they are characterized by a two-component order parameter and their magnetic moments vary harmonically not only in direction, but also in size. From the modeling of the observed spectra, a possible scenario of magnetic structure transformations is obtained.


75.50.Ee, 76.60.-k, 75.10.Jm, 75.10.Pq

Exotic phases of frustrated antiferromagnet LiCu_2O_2
    L. E. Svistov
    December 30, 2023
=====================================================





§ INTRODUCTION


Unconventional magnetic orders and phases in frustrated quantum-spin chains appear under a fine balance of the exchange interactions and are sometimes caused by much weaker interactions or fluctuations.<cit.> A kind of frustration in quasi-one-dimensional (1D) chain magnets is provided by competing interactions, when the intrachain nearest neighbor (NN) exchange is ferromagnetic (J_NN < 0) and the next-nearest neighbor (NNN) exchange is antiferromagnetic (J_NNN > 0). Numerical investigations of frustrated chain magnets within different models<cit.> have predicted a number of exotic magnetic phases in the magnetization process, such as planar, spiral and different multipolar phases. The theoretical study of the magnetic phase diagram shows that real magnetic phases are very sensitive to interchain interactions and anisotropic interactions.

LiCu_2O_2 is an example of Cu^2+ (S = 1/2) magnet with frustrated
exchange interactions akin to the quantum spin-chain compound LiCuVO_4.<cit.> However, the magnetic structure of LiCu_2O_2 appears to be complicated by interchain interactions between coupled
chains of magnetic Cu^2+ ions. Superexchange interactions via oxygen ions of edge-shared CuO_4 squares (see Fig. 1) provide frustration of the intrachain exchange interactions (J_NN < 0, J_NNN > 0).<cit.>
According to Ref. [Hikihara_2008], for the 1D model with the intrachain exchange constants of LiCu_2O_2, a chiral long-range order in the low-field range and a quasi-long-range ordered spin-density-wave phase in higher applied magnetic fields H are expected similar to the above mentioned LiCuVO_4. Experimentally an incommensurate spin structure was observed at T < T_N  in the low-field range, which was ascribed to a planar helical spin structure.<cit.>
Despite the fact that the magnetic properties of LiCu_2O_2 and structurally isomorphic NaCu_2O_2 have been intensively studied for more than ten years, the magnetic structures of these compounds have not been unambiguously determined. The lack of a reliable interpretation of the magnetic structure makes it impossible to unequivocally explain the nature of the multiferroic properties of LiCu_2O_2 and their absence in NaCu_2O_2.<cit.>

In the present work,  ^7Li NMR spectra of untwinned single crystals of LiCu_2O_2 were studied in a magnetic field up to 17 T at temperatures 5-30 K. Field and temperature dependencies of the spectra are discussed in Sec. IV. The evolving magnetic structures were analyzed in the frame of the Dzyaloshinskii-Landau  theory of magnetic phase transitions. This analysis was performed in the exchange approximation, i.e., under the assumption that the exchange interactions predominate the interactions of relativistic nature (Sec. V).
A theoretical analysis of the relativistic effects, such as anisotropy and electric polarization, is given in Appendix B. From the simulations of the NMR line shape of our observed spectra we elaborate
the underlying magnetic structures theoretically and provide a scenario for the case of LiCu_2O_2. The structures exhibit an extraordinary configuration; a two-component order parameter characterizes the ordering of the magnetic moments which appear to be not only rotated  but also harmonically modulated in size from site to site (Sec. VI).
Note that elliptical structures in magnets with a two-component order parameter arise quite often.<cit.> However, it is commonly accepted that the ellipticity of the spiral structure is due to relativistic interactions, while the proposed structures for LiCu_2O_2 are elliptical already in the exchange approximation.



§ CRYSTAL AND MAGNETIC STRUCTURE


LiCu_2O_2 crystallizes in an orthorhombic lattice (space group Pnma)
with the unit cell parameters a=5.73 Å, b=2.86 Å  and c=12.42
Å.<cit.> The unit cell parameter a is approximately twice the
unit cell parameter b. Consequently, LiCu_2O_2 single crystals, as a rule, exhibit considerable
twinning due to the formation of crystallographic domains rotated by 90^∘ around their crystallographically common c axis.

The unit cell of the LiCu_2O_2 crystal contains
four monovalent nonmagnetic cations Cu^+ and four divalent cations Cu^2+
with S=1/2. The positions of all ions in the crystal lattice are schematically shown in Fig. 1.
The unit cell is selected with a dashed line. There are four crystallographically equivalent positions of
the magnetic Cu^2+ ions in the crystal unit cell of LiCu_2O_2,
denoted as I, II, III, and IV.




The two-stage transition into a magnetically ordered state occurs at
T_c1=24.6 K and T_c2=23.2 K.<cit.> Neutron scattering and NMR experiments
revealed an incommensurate magnetic structure in the magnetically ordered state
(T<T_c1).<cit.> The wave vector
of the incommensurate magnetic structure coincides with the chain direction
(b-axis). The magnitude of the propagation vector at T < 17 K is almost
temperature independent and is equal to 0.827×2π/b. The neutron
scattering experiments have shown that that adjacent magnetic moments along the a-direction are oriented antiparallel, whereas those along the c-direction are mutually coaligned.
The intra-chain and inter-chain exchange constants were determined from the analysis of the spin wave
spectra.<cit.> The large number of exchange bonds stipulates the ambiguity of the main exchange parameters obtained from Ref. [Masuda_2005]. Theoretical analysis based on local density approach
(LDA) and cluster calculations as well as a phenomenological analysis of the temperature dependence of the magnetic susceptibility allow to map the most significant exchange paths. Using the result of these investigations<cit.> we choose the most suitable set of parameters proposed from neutron experiments: the nearest neighbor exchange interaction is ferromagnetic J_1=-7.00 meV, while the next nearest neighbor exchange interaction is antiferromagnetic J_2=3.75 meV.
The competition between these intra-chain interactions leads to an incommensurate magnetic structure.
The antiparallel orientation of magnetic moments of Cu^2+ between nearby chains is
caused by strong antiferromagnetic exchange interaction J_3=3.4 meV.
These main exchange paths are shown in Fig. 1.
The coupling of the Cu^2+ moments along the c direction and the couplings
between the magnetic ions in other crystallographic positions are much
weaker.<cit.> Thus, LiCu_2O_2 can be considered
as a quasi-two dimensional system. The quasi-two-dimensional character of magnetic
interactions in LiCu_2O_2 compound was also proved by resonant soft
x-ray magnetic scattering experiments.<cit.>

The magnetic structure of LiCu_2O_2 at zero magnetic field was studied by
several groups by means of neutron diffraction experiments.<cit.> The authors of Ref. [Masuda_2004] have
proposed the planar spiral spin structure with the spins confined to the
ab plane. Polarized neutron scattering measurements<cit.> have detected the spin component along the
c direction, indicating the spiral magnetic structure in the bc plane.
The authors of Ref. [Kobayashi_2009], alternatively, have proposed a
spiral spin structure confined to the (1,1,0) plane. It was attempted to
extract information about the magnetic structure of LiCu_2O_2 from the studies
of electric polarization, which accompanies magnetic
ordering.<cit.> Unfortunately, at the moment, the nature
of this polarization  is not clear<cit.> and, hence,
does not allow to draw an unambiguous conclusion about the zero-field
magnetic structure from this type of experiment.

The magnetic structure of LiCu_2O_2 was also studied by ^63,65Cu and ^7Li NMR in Ref. [Sadykov_2012]. The authors of this work describe their results in the frame of planar spin structure and come to the conclusion that the spiral planes do not coincide with any of the crystallographic planes ab, ac, or bc, respectively.



§ SAMPLE PREPARATION AND EXPERIMENTAL DETAILS


Untwinned single crystals of LiCu_2O_2 with the size of several cubic
millimeters were prepared by the solution in the melt method as described in Ref. [Svistov_2009].
The quality of the crystals under investigation was studied in Refs. [Kamentsev_2013, Bush_2004] and the magnetic properties were found to be identical for all the samples from different batches.

^7Li nuclei (nuclear spin I=3/2, gyromagnetic ratio γ/2π=16.5471 MHz/T) were
probed using pulsed NMR technique. The spectra were obtained by summing fast Fourier transforms (FFT)
or integrating the averaged spin-echo signals as the field was swept through the resonance line.
NMR spin echoes were obtained using τ_p - τ_D - τ_p pulse sequences, where the pulse
lengths τ_p were 1.5 μs, the delay times between the pulses τ_D were 28 μs.
Measurements were carried out in the temperature range 5≤ T ≤ 30 K stabilized with a
precision better than 0.1 K.



§ EXPERIMENTAL RESULTS


^7Li NMR spectra were studied for four orientations of the static magnetic field:
H∥a (Figs. 2, 3, 4), H∥b (Figs. 5 and 10),
H∥c (Figs. 6, 11, and 12), and H∥ (c+15^∘) (H in bc plane at an angle of 15^∘ with respect to c axis, Fig. 7).



Figures 2, 5, 6, and 7 show field evolutions of the spectra. The measurements were performed at the lowest temperature 5 K (4.5 K for H∥a and ν = 10, 90, 127 MHz), i.e., well below the magnetic ordering temperature (T_N ≈ 24 K). The spectra lines are shifted along the vertical axis for clarity. The resonance field is defined by the vector sum of the external field H and the effective field H_eff from neighboring magnetic environment. For the case H ≫ H_eff the resonance field H_res≈ H + H_eff, where H_eff is the projection of the effective field on the direction of the external field. In figs. 2-7, the field scale for each NMR spectrum is shifted by the value of the undisturbed Larmor field 2πν / γ at ^7Li nuclei (γ / 2π = 16.5471 MHz/T). In such a representation, the horizontal axes show H_eff at the nuclei of nonmagnetic ions.



The magnetic field H_eff is defined by the magnetic environment.
If the effective field H_eff varies harmonically in space and the wave length of this variation is incommensurate with the crystal lattice, we expect a broad NMR spectral line where its shape is dominated by two characteristic maxima at the edges. These maxima occur for the case H_eff∥H (H ≫ H_eff) and we designate this particular shape as a double-horn pattern.



There are eight chains of Li^+ ions along the b direction in the magnetic structure of LiCu_2O_2 with the vector
q = (1/2, q, 0). <cit.> All other lithium chains can be obtained by translations. For each of the eight chains we expect a harmonically oscillating effective field generated by the magnetic surrounding.
Therefore, in the low-temperature magnetic phase the NMR spectra comprise a superposition of eight double-horn patterns, where some of them may coincide due to symmetry restrictions.
All the observed spectra are well described by a sum of not more than four double-horn spectra with nearly the same integral intensity. This fact shows that lithium chains, distanced by the lattice constant a, yield identical NMR spectra.



Figure 2 shows ^7Li NMR field-swept spectra for H∥a. For the fields below
μ_0H_c1≈ 15 T all four lithium chains demonstrate identical double-horn shaped lines. This means that the a-projections of effective fields on ^7Li nuclei oscillate along the lithium chains with the same amplitudes. At higher fields the spectrum transforms into a sum of four double-horn shaped spectra. The value of H_c1 agrees with the field of an anomaly in the magnetization curves. <cit.> The step-like increase of the magnetic susceptibility at this field was previously associated with a spin-flop transition. The value of H_c1 is in satisfactory agreement with the value evaluated from ESR in an investigation of the spin-flop transition in LiCu_2O_2. <cit.> At fields nearby the phase transition the spectra exhibit hysteretic feature: The spectra recorded upon field increase differ from the spectra recorded upon field decrease (Fig. 2, 2πν / γ = 15 T) which indicates that the phase transition observed at this field is of first order typical for spin-flop transitions.





Figures 3 and 4 show the temperature evolutions of NMR spectra measured around (μ_0H≈ 7.68 T) and (μ_0H≈ 16.87 T), i.e., below and above spin-flop field H_c1, respectively. The single line of the paramagnetic phase is significantly broadened in the magnetically ordered phase at T_c1 = (24.8 ± 0.2) and (24.3 ± 0.2) K for 7.68 and 16.87 T, respectively. Within a range of 1 K below T_c1, the NMR spectrum can be considered as a superposition of a double-horn spectrum and a paramagnetic solitary line. These spectra are blue colored and observed between the temperatures T_c1 and T_c2 of the two step transition into the magnetically ordered phase. The intensity of the paramagnetic line rapidly decreases with decreasing temperature. At temperatures below T_c2, the double-horn spectrum transforms into two (four) double-horn spectra. These spectra are red colored in Figs. 3 and 4. We suggest that they are obtained within spin-flopped phase. The boundary between phases with blue and red colored spectra for the fields up to 9 T was presumably observed in Ref. [Svistov_2012] by dielectric constant measurements.





Figures 5, 6, and 7 show the field evolutions of Li NMR spectra measured for the field directions H∥b, H∥c, and H∥ (c+15^∘), respectively.
All spectra were measured at high enough fields, where the normal vector of the magnetically ordered spin plane appears to follow the field direction. <cit.> For H∥b a continuous transformation from two double-horn shaped lines to one double-horn shaped line is observed upon field increase. For H∥c the spectra do not change in the full range of applied magnetic fields as it is documented in Fig. 6.
An applied field direction with small deviation by 15^∘ off the crystallographic c-axis yields a duplication of the number of double-horn pattern towards lower applied magnetic fields (see Fig. 7).
In other words, for this field orientation all four lithium ions show the double-horn spectrum with individual amplitude of oscillating projection H_eff. The field splitting decreases with increasing field and disappears at elevated fields higher than (12.5 ± 0.5) T.

Figures 8 and 9 show the field dependencies of the resonance field values of all edge singularities within the multi-horn spectral pattern. The fields where the number of edge singularities is halved are 17 and 12.5 T for H∥b and H∥ (c+15^∘), respectively, corresponding to the anomalies observed in the magnetization measurements. <cit.> The fields values of these anomalies were designated as H_c2. Such an anomaly was also observed in the magnetization measurements for H∥a at μ_0H_c2≈ 20 T, that is out of range of the present experiments.
It appears that we also observed this transition for H∥a within the temperature scan at μ_0H≈ 16.87 T (see Fig. 4). There, also the number of edge singularities halves for temperatures 15-20 K and the multi-horn shape of the spectra pattern in this temperature range strongly resembles to the shapes observed at lowest temperatures for H > H_c2.







The temperature evolutions  of  the lithium spectra for H∥b and H∥c are given in Figs. 10-12. For fields lower than the critical field H_c2, the transition from the low-temperature magnetic phase to the paramagnetic phase occurs through an intermediate phase, with one solitary double-horn spectral line (see Fig. 11). This intermediate phase manifests itself within the temperature range between T_c1 and T_c2. For elevated fields higher than H_c2 (Fig. 12), the presence of the intermediate phase is not established.
In this case, the single line spectrum of the paramagnetic phase transforms immediately into the characteristic spectral pattern of the low-temperature phase, even within the temperature steps of our measurements. Any intermediate magnetically ordered phase is skipped.



§ THEORY. EXCHANGE APPROXIMATION
 


The crystal cell of LiCu_2O_2 contains four magnetic ions Cu^2+ (S = 1/2). At the magnetic transition a doubling of the period along the twofold C_2x axis occurs, and as a result eight spiral spin chains appear. Their mutual orientation, amplitudes, phases and possible ellipticity are unknown.

As a first step to interpret the spin structure of LiCu_2O_2, it is useful to find out
the types of the structures with the wave vector (1/2,q,0) occurring in the Dzyaloshinskii-Landau theory.
As it is usually done applying the Landau theory of second-order phase transitions, we assume the region of critical fluctuations to be small, which in our case fits with the experimental observations. In this case, the small value of the spin (S = 1/2) is not essential at all, since the quantum fluctuations near the transition are always small in comparison with the thermodynamic ones.

In this section we define a list of such structures – candidates for description of the phase realized in LiCu_2O_2
at temperatures below T_c2, as well as within the intermediate phase (T_c2<T<T_c1).
It was found that the splitting of the transition is most likely due to small relativistic effects, and that even an additional transition is possible between the temperatures T_ c1 and T_ c2.

The crystal symmetry group of LiCu_2O_2 – Pnma  (D_2h^16) – is defined by the three translations
τ_a: x→ x+1,
τ_b: y→ y+1, τ_c: z→ z+1, the inversion I: (x,y,z)→(-x,-y,-z) and
the two screw rotations 
    C_2x:  (x,y,z)→(x+1/2,-y+1/2,-z+1/2),


    C_2y: (x,y,z)→(-x,y+1/2,-z).


Two-dimensional complex representation corresponding to the wave vector (1/2,q,0) (compare with the last case considered in Sec. 134 of the book Ref. [LL5]) is implemented by the functions sinπ xe^± iqy, cosπ xe^± iqy (the coordinates x,y,z are measured in the unit cell parameters a,b,c, respectively). The functions that transform according to other possible representations associated with this wavevector differ by the factors which are insignificant for constructing the invariants even in the order parameter. For example, there is a representation with a pseudo-scalar
factor sin2π xsin2π ysin2π z.

The spin density arising at a second-order transition over the considered representation is

    s( r)=η{sinπ
    x(μe^iqy+μ^*e^-iqy)+
     +cosπ
    x(νe^iqy+ν^*e^-iqy)}f(
    r),
 where η is the magnitude of the magnetic order parameter; μ, ν are the complex vectors in the spin space, normalized by the condition μμ^*+
νν^*=1; f( r) is a scalar function of coordinates which is invariant with
respect to the crystal symmetry group in paramagnetic phase.

To select the exchange effects, we suppose<cit.> that the effect of crystalline transformations on the function s( r) is reduced to a corresponding change of coordinates (x, y, z) at a fixed orientation of the spin space. In the Landau theory of second-order phase transitions, it is convenient to transfer
the laws of crystalline transformations from the coordinate functions to the coefficients

    I:    μ→-μ^*, ν→ν^*;
    
    C_2x:    μ→
    -e^-iq/2ν^*, ν→
    e^-iq/2μ^*;
    
     C_2y:    μ→-e^iq/2μ, ν→ e^iq/2ν;
    τ_a:    μ→-μ,
     ν→-ν;
    τ_b:    μ→ e^iqμ,
     ν→ e^iqν.


For the considered representation, there is the Lifshitz exchange invariant

    μ∂_xν^*- ν^*∂_xμ+
    μ^*∂_xν- ν∂_xμ^*,

leading to the instability of the phase transition.
Therefore observing a continuous transition over the representation with the wavevector (1/2,q,0) in LiCu_2O_2 implies that the impact of the invariant (<ref>) in this antiferromagnet is small compared to the anisotropy effects, and we will not take it into account when considering the phases structure at T<T_c2.

In the exchange approximation, the Dzyaloshinskii-Landau expansion of free energy up to the fourth-order terms has the form   
    F=τη^2 +β_0+B/2η^4,


    B=β_1(μν^*+
    μ^*ν)^2
    +β_2(μ^*μ-
    ν^*ν)^2-
    
    -β_3(μν^*-
    μ^*ν)^2+
    β_4(μ^2+ν^2)
    (μ^*2+ν^*2)+
    β_5(μ^2-ν^2)
    (μ^*2-ν^*2)+
    4β_6(μν)
    (μ^*ν^*).


Note that in the case of spin-1/2 the fourth order terms of the Landau expansion should be treated
as the result of thermodynamic averaging of the microscopic exchange Hamiltonian, which takes into
account simultaneous pair permutations of spins belonging to four or more atoms, for example,
a biquadratic term of the form (σ_1 σ_2) (σ_3 σ_4), where σ_i is the spin operator of the ith atom.
This peculiarity of the spin-1/2 case was established earlier in the study of the antiferromagnetic
phase of crystalline ^3He in Ref. Roger_1983.


The real form
p=μν^*+μ^*ν
is transformed as the z-component of the vector, and the real forms
r=μμ^*-νν^*,  s=i{μν^*-νμ^*}
are transformed as components of deformation tensors u_xz and
u_xy. For the rest spin convolutions we introduce the following notation:
ζ=μ^2+ν^2, ω=μ^2-ν^2, ξ=μν.

The free energy F does not change under the calibration transformation

    μ→ e^iκμ,  ν→ e^iκν,

which corresponds to the incommensurability of arising spin structure to the crystal spacing along the y axis.
Using this invariance, we assume ω is real-valued.

The free energy F is invariant under the replacement
μ⇆ν,
which is connected with the solution transformation under the reflection σ_x=IC_2x.
Taking this into account, we will consider only the solutions with μ≠0.

The free energy (<ref>) does not change under the transformation

    ν→ iν,  β_1⇆β_3,  β_4⇆β_5.

Hence extrema that either remain unchanged or transform into each other under this transformation are possible.
Note that the Lifshitz invariant (<ref>) breaks this random symmetry.

Performing the transformation (<ref>) two times we obtain invariance F under the replacement
μ→μ,  ν→-ν.
This invariance is associated with the solution transformation under the crystal rotation τ_aC_2y.

The extrema conditions of the free energy (<ref>) are
 
    -(B-β_2r)μ+(β_1p
    -iβ_3s)ν+
    +(β_4ζ+
    β_5ω)μ^*+2β_6ξν^*=0,


    -(B+β_2r)ν+(β_1p
    +iβ_3s)μ+
    +(β_4ζ-
    β_5ω)ν^*+2β_6ξμ^*=0.


Performing scalar multiplication of the equation (<ref>) by μ^*
and separating the imaginary part, we find

    (β_1-β_3)ps+(β_4+
    β_5)ζ”ω=0,
 where ζ”=Im(ζ).

Multiplying the equation (<ref>) by ν^*, for the imaginary part of the product we get

    (β_1-β_3)ps+(β_4-β_5)ζ”ω=0.


From the equations (<ref>,<ref>) it follows that
ps=ζ”ω=0. Hence in view of invariance of (<ref>), with s→ p→-s, we can assume that, for example, p=0.
Thus, we have two cases

    1)   p=ζ”=0,   2)   p=ω=0.


At ω=0 we can again use the calibration symmetry and choose ζ of real-valued. Thus, in all cases
it will be ζ”=0.

Performing scalar multiplication of the equation (<ref>) by μ and separating the imaginary part
at p=ζ”=0, we find

    sξ'=0.


Substraction of the equation (<ref>) multiplied by ν, and the equation (<ref>) multiplied
by μ, at p=0 gives

    2(β_2-β_6)rξ+i(-β_3+β_4)sζ=0.

Hence at ζ”=0 we find rξ'=0. Combining this result with the conditions (<ref>,<ref>) we obtain
the following possibilities

    1) p=ζ”=r=s=0,
       2) p=ζ”=ξ'=0.


Let us introduce four real-valued vectors a, b, c, d, such that μ= a+i b, ν= c+i d. For all possible cases from the general conditions of reality of ω,
ζ it follows, that μ^2, ν^2 are real, which leads to the orthogonality condition
ab= cd=0.

According to the Appendix A the magnetic structure should be planar and can therefore be written as:

    μ=c_α(
    lc_γ+i ks_γ),
       ν=s_α{c_ϵ( lc_φ+
    ks_φ) -is_ϵ( ls_φ-
    kc_φ)},
 where l,  k
are mutually orthogonal unit vectors (we use short notation c_α=cosα,   s_α=sinα).

In such parametrization, we have p=s_2αc_φc_-,
r=c_2α, s=-s_2αs_φs_+,
ξ'=s_2αc_φc_+, and

    B=β_1s^2_2αc^2_φc^2_-+
    β_2c^2_2α+β_3s^2_2αs^2_φs^2_+
    
    +β_4(c_+c_--c_2αs_+s_-)^2
    
    +β_5(c_+c_-c_2α-s_+s_-)^2
    
    +β_6s^2_2α(c^2_φc^2_++ s^2_φs^2_-)

wheres_±=s_γ±ϵ, c_±=c_γ±ϵ.
At the phase transition a spin structure corresponding to the minimum of B arises,
in this case the magnitude of the order parameter takes the maximum value η=√(-τ/(β_0+B)).
According to (<ref>), we get five scenarios for solving simultaneous equations (<ref>,<ref>)

    A)     c_2α=0, c_φ=s_+=0;
    
    B)     c_2α=0, s_φ=c_-=0;
    
    C)     s_2α=0;
    
    D)     s_2α≠0, c_+=c_-=0;
    
    E)     s_2α≠0, c_φ=0.


In the framework of these scenarios solving is reduced to elementary minimization of the function (<ref>)
over the remaining free angular parameters in each of them. <cit.>
We find 8 solutions, whose form is independent of the values of β_i:

A_1:μ   =l+ik/2, 
ν=±k+il/2,   B=β_6;

A_2:μ   =l/√(2), 
ν=k/√(2),   B=β_4;

B_1:μ   =l+ik/2, 
ν=±l-ik/2,   B=β_6;

B_2:μ   =l/√(2), 
ν=ik/√(2),    B=β_5;

C_1:μ   =l, ν=0,   
B=β_2+β_4+β_5;

C_2:μ   =l+ik/√(2), 
ν=0,    B=β_2;

D:μ   =l/√(2), 
ν=±il/√(2),   
B=β_3+β_5+β_6;

E: μ   =l+ik/2, 
ν=±k-il/2,    B=β_3.


    


The degeneracy of energy for the solutions A_1 and B_1 will be removed if we take into account the next terms of
the Dzyaloshinskii-Landau expansion. The solutions A_1 and B_1, as well as A_2 and B_2 are transformed into
each other under the transformation (<ref>). To the list of solutions we obviously need to add two more associated solutions:

D^*:μ   =±ν=l/√(2),    B=β_1+β_4+β_6;

E^*: μ   =±ν=l+ik/2,    B=β_1.

    


All the phases found can be presented on the phase diagram. Indeed, the value of B for each solution is obviously an absolute minimum of the function B, if the parameters β_i belonging to it are negative, and the rest of the parameters β_i are positive.

Besides the magnetic structures (<ref>,<ref>) with fixed values of the phase parameters α,γ,ϵ, the symmetry allows the existence of magnetic structures, for which these parameters are functions of the coefficients of energy expansion β_i. These solutions may occur in a certain range of values of the parameters β_i only in the scenario E, when φ=π/2, see equation (<ref>),

    μ=c_α(
    lc_γ+i ks_γ),  ν=s_α(c_ϵ
    k -is_ϵ l),

where the parameters α,γ,ϵ are functions of β_i.
In this case rather complicated expressions occur, whose form is not necessary to determine the structure realized in
LiCu_2O_2 – it is enough to have the formulas (<ref>) with three free angles.

As an illustration, let us get the solution in the limit of large values of magnitude of the parameter β_2:

    c_+^2=β_3+β_5/β_4+β_5<1, c_-^2=β_6+β_5/β_4+β_5<1,
     c_2α=β_4+β_5/β_2c_+c_-s_+s_-,
    B≈β_3s_+^2+β_4c_+^2c_-^2+β_5s_+^2s_-^2+β_6s_-^2.

There is also an associate of this solution, corresponding to the transformation (<ref>).

It should be noted that the Lifshitz exchange invariants (<ref>) occur in the phases A_2, B_2, E, E^*,
as well as in the set of phases described by (<ref>).

In the presence of magnetic field there is an invariant

    H_iH_k(μ_iμ_k^*+μ_kμ_i^*+ν_iν_k^*+ν_kν^*_i),

giving the anisotropy of the magnetic susceptibility tensor of exchange approximation.
For all the structures (<ref>,<ref>) this tensor has obvious axial symmetry. For the structures described by (<ref>) the two main axes of the tensor belong to spin plane.

A spontaneous electric polarization of exchange nature as a quadratic effect in the order parameter (the effect predicted by Indenbom <cit.>) arises in the phases D^* and E^*, where the quadratic form p (transforming as z component of vector) is not equal to zero.

Exchange striction corresponding to the symmetry breaking D_2h arises in the phases C_1,C_2 (invariant ru_xz) and in the phases D,E (invariant su_xy).

The  relativistic  effects  are  discussed  in  Appendix B.




§ DISCUSSION




We discuss the observed NMR spectra using the magnetic phases obtained in Sec. <ref>.
The theoretical analysis assumes that in the entire range of fields and temperatures the crystal structure of LiCu_2O_2 is described by the symmetry group Pnma, and the low-temperature magnetic structure is described by the wave vector (1/2, q, 0). The proposed magnetic phases are obtained under the assumption that the magnetic structure is defined by the dominating exchange interactions, whereas its orientation with respect to the crystal axes is determined by the relativistic interactions with the crystal environment and the applied magnetic field. It was also demonstrated (see Appendix A) that the exchange interactions in LiCu_2O_2 can lead to planar or collinear magnetic structures only. Therefore, non-coplanar magnetic structures will be excluded from further consideration.

In the following we compare the experimental NMR spectra with the simulated ones.
We assume that the magnetic moments are localized at the Cu^2+ ions. This assumption accounts for the scalar function f(r) = δ (r - R_i) in Eq. (<ref>), where R_i is the radius-vector of ith Cu^2+ ion. Using Eqs. (<ref>) and (<ref>) the magnetic moment of Cu^2+ ion with coordinates (x, y, z) for the coplanar magnetic structure with a wave vector (1/2, q, 0) can be written as follows:


    M(x,y,z)= η{l[ η_l1sin(π x)cos(qy+φ_lc) + 
    
    + η_l2cos(π x)sin(qy+φ_ls)] + 
    
    + k[ η_k1sin(π x)cos(qy+φ_kc) + 
    
    + η_k2cos(π x)sin(qy+φ_ks)]
    }.

Here, l and k are two mutually perpendicular unit vectors, defining the spin plane, and η is the magnitude of the two component magnetic order parameter. The parameters η_l1, η_l2, η_k1 and η_k2 take the values 0 and 1. The angles φ_lc, φ_ls, φ_kc and φ_ks denote harmonic phase angles. According to Sec. <ref> there are ten possible magnetic phases A_1, A_2, B_1, B_2, C_1, C_2, D, E, D^*, and E^* with fixed phase parameters. An overview of all values of these parameters is given in Table I.

The letters “C” and “P” in the table mark collinear and planar phases, respectively. The value of the magnetic moment for the collinear phases C_1, D, and D^* oscillates harmonically along the chains. The phases C_2, E, and E^* are circular. For them, the Cu^2+ magnetic moment with constant absolute value rotates by an angle defined by q.
Note, the absolute values of magnetic moments differ from chain to neighboring chains within the phases C_2 and E^*. The structures A_1, B_1, and B_2 can be considered as two embedded elliptical phases with large elliptical axes which are oriented perpendicular with respect to each other between neighboring axes. The planar structure A_2 consists of two embedded collinear structures in which the value of the magnetic moment varies along the chains. In this structure the magnetic moments of neighboring chains are oriented perpendicular with respect to each other. To visualize the structures, we included their sketches in the Table I (the spin plane coincides with the easel plane, l is horizontal, k is vertical). The sketches show the ends of the magnetic moment vector for four chains I, II, III, and IV, the arrows indicate the moment rotation direction for the translation along the chain for one of magnetic domains. The magnetic phases which allow magnetically induced electrical polarization (multiferroicity) are marked with the letter “p”.

Besides the structures given by Eqs. (<ref>) and (<ref>), there are the set of structures given by Eq. (<ref>). These structures are described by Eq. (<ref>) with the following values of parameters: φ_lc = φ_ls = 0 and φ_kc = φ_ks = π/2. Other parameters are defined by:
η_l1 = 2 c_α c_γ, η_l2 = 2 s_α s_ϵ,
η_k1 = 2 c_α s_γ, and η_k2 = 2 s_α c_ϵ,
where α, γ, and ϵ are arbitrary. The set of structures Eq. (<ref>) contains the phases A_1, A_2, D, and E and adjoins to the phases C_1 and C_2.

Concluding the phase description, we note that all suggested magnetic phases are unusual. The values of the magnetic moments of all Cu^2+ ions are identical only for the E phase, for other phases this value oscillates.

Space orientation of the spin structure is defined by the vectors l and k. For further discussion we suppose that initially the vectors l and k are directed along a and b crystal axes, respectively. An arbitrary orientation of the spin plane can be obtained by successive rotation of the structure with
l∥a and k∥b by an angle Θ_a around a axis, Θ_b around b axis and Θ_c around c axis. The number 1 or 2 or 3 in the brackets after the angle Θ specifies the order of the corresponding rotation where it matters. Thus the distribution of the magnetic moments is defined by the four parameters η, Θ_a, Θ_b, and Θ_c for the structures Eq. (<ref>,<ref>) and by the seven parameters η, α, γ, ϵ, Θ_a, Θ_b, and Θ_c for the structures Eq. (<ref>). For large enough static fields (H > H_c1), the spin plane orientation is defined by the field direction, whereas in small fields the orientation is defined by relativistic interactions with the crystallographic environment.

In our spectra simulations we suppose that the effective field on lithium nuclei is defined by the dipolar fields and the contact Fermi fields. We took into account the dipolar fields from the neighboring moments in the sphere of radius 20 Å and the hyperfine contact fields from the four nearest moments belonging to neighboring chains (Fig. 1). The constants defining the contact fields for three field orientations H∥a, b, c were obtained as follows. First, the shift of ^7Li NMR line at a certain temperature in the paramagnetic state was determined. Then, using the magnetization data from Ref. [Svistov_2012], the magnetic moment of individual Cu^2+ ion at the same temperature and field was calculated. The difference between the computed dipole field at the lithium nuclei and the effective field observed in the experiment was ascribed to the contact field. Thus obtained contact field is in good agreement with the results of Ref. [Sadykov_2017]. This value was ascribed to two pairs of the nearest magnetic moments. The first pair is located along a axis, the second pair is located along b axis (see Fig. 1). According to the first-principles calculations, <cit.> the contact field from the first pair is expected to be much larger than the contribution from the second one. It is important that in the magnetically ordered phase the magnetic moments from the first pair are antiparallel; as a result, we can exclude the contact part of effective field in the magnetically ordered phase from further consideration. <cit.>

We have simulated the spectra for all magnetic structures from Table I and orientations of the applied magnetic field
H∥a, b, c, and (c+15^∘), respectively. For H∥a, the spectral shape does not change with applied magnetic fields up to ≈ 15 T. For higher fields the susceptibility sharply increases, which was interpreted as a spin flop transition. <cit.> For all other field directions the shape changes with field monotonically. For H∥b and (c+15^∘) some pairs of maxima continuously approach each other and merge into one single spectral line at elevated applied fields. This disappearance of the line splitting is accomplished at a field value where anomalies of the magnetic susceptibility were already observed in Ref. [Svistov_2012]. We tried to describe the field evolution of the spectra by changing the spin plane orientation of one individual magnetic phase. A special program was written for spectra simulation. <cit.> It calculates NMR spectra for the suggested magnetic structures (<ref>,<ref>,<ref>) for given orientations of the static magnetic field with respect to the spin plane. The number of maxima in the spectrum was chosen as the main criterion in order to assess the matching between the observed NMR spectrum and the simulated one.
The calculations for all the structures Eqs. (<ref>) and (<ref>) performed with an angular step of 10^∘ showed that the observed field evolution of the spectra for H∥a, b, c, and (c+15^∘), respectively, can not be fitted assuming one single structure.



For H∥a at the spin-flop field H_c1≈ 15 T the spin plane turns abruptly from the state with n∥b (Θ_a = 90^∘, Θ_b = 0^∘, and Θ_c = 0^∘) to the state with n∥a (Θ_a = 0^∘, Θ_b(2) = 90^∘, and Θ_c(1) = 90^∘). Figures 13(a) and 13(b) show simulated spectra for H < H_c1 and H > H_c1, respectively.
The best coincidence between experiment and simulation above the spin-flop transition is achieved if one assumes that a phase from the set Eq. (<ref>) is realized. Note here that the inflated value of the magnetic moment used for modeling at H∥a, in comparison with the expected value of 1 μ_B, is most probably explained by the contact fields. This additional contribution to the local magnetic field at the ^7Li nuclear site can be significant for the magnetic field applied in the spin plane.



For H∥c and (c+15^∘), respectively, the spectra evolution can be modelled by rotation of the spin plane around the c axis within the structure A_1. When H > H_c2 the spin plane rotation stops. For H∥c, the spectrum exhibits four maxima throughout the entire field range under investigation in this work. Figures 13(e) and 13(f) show simulated spectra for H < H_c2 and H > H_c2, respectively. For H∥ (c+15^∘) in the low field range H < H_c2 the number of maxima doubles. When the field is increased to the value μ_0 H_c2≈ 12.5 T the number of maxima becomes equal to four. Figures 13(g) and 13(h) show the simulated spectra for H < H_c2 and H > H_c2, respectively.

The orientation of the spin plane with respect to the crystallographic axes for different directions of the applied magnetic field for the phase A_1 can be explained by assuming the hierarchy of the anisotropy constants
λ_2 > 0, λ_1 < 0, |λ_4| ≪ |λ_2| < |λ_1|.

For H∥b we were unable to describe the field evolution of the spectra in the frame of our established structures which we introduce in Eqs. (<ref>) and (<ref>).
Figures 13(c) and 13(d) show a possible scenario in the frame of the phases Eq. (<ref>).
The low field phase exhibits a similar structure like the phase A_1.
As the field increases and gets closer to H_c2, the phase Eq. (<ref>) gradually transforms into the phase C_1. Both phases A_1 and C_1 are located on the boundary of the phases set described by Eq. (<ref>).

Fig. 14 shows the sketches of the structures used in the proposed scenario.

As a conclusion note, we cannot judge the uniqueness of the proposed scenario of the spectral field evolution within the magnetic structures set by Eq. (<ref>), since the number of free parameters defining these structures is too large.

As the temperature increases, the spectra shape does not change within a scale factor. This indicates that the magnetic structure is not changed until the temperatures approach the transition temperature, and the width of the spectra is determined by the order parameter η, see Eq. (<ref>). The transition occurs through an intermediate phase observed in a narrow temperature range T_c2 < T < T_c1. In this temperature range for all field orientations spectra with two characteristic maxima at the edges and a narrow central line ascribed to the paramagnetic state are observed. Relative intensities of these lines change with temperature. As the temperature increases, the intensity of the paramagnetic line increases, while the intensity of the double-horn pattern with the significant maxima at its edges decreases. A phase transition that takes place in two stages by passing through nearby transition temperatures is a characteristic fingerprint of a transition into a planar low-temperature phase in magnetic compounds
having easy-axis anisotropy. The observed spectra with such two characteristic maxima are well described by the collinear phase D.



§ CONCLUSIONS


1. ^7Li NMR spectra were measured in applied magnetic fields μ_0 H up to 17 T with the temperature range 5-30 K. Four different orientations of the field H∥a, b, c, and (c+15^∘) with respect to the crystallographic axes where employed. The field dependent evolution of the spectra was studied in full detail. Anomalies in bulk measurements of the magnetization curves at fields H_c1 and H_c2, recently reported in the literature, were found to correspond to the fields values where the shape of our NMR spectra changes.
At H_c1,  a first-order phase transition is observed for H∥a,
which is accompanied by the increase of magnetic susceptibility and most likely indicates a spin-reorientation transition. The observation of the spin-reorientation transition for H∥a supports the model of a planar two-component magnetic structure with the magnetic susceptibility in the field perpendicular to the spin plane larger than that in the field directed within the spin plane. This result is also in agreement with the biaxial character of the magnetic anisotropy suggested in Refs. [Svistov_2009, Svistov_2012].
For other directions of the magnetic field at H < H_c2, the shape of the spectra monotonically evolves to a state with less number of maxima upon field increase. At the point where the evolution stops the magnetic susceptibility decreases. Such monotonous field evolution could be explained by the rotation of the spin plane or by monotonous change of magnetic structure. The modeling of our experimental NMR spectra shows that  the second scenario is realized.

2. We analyzed the underlying magnetic structures in the frame of Dzyaloshinskii-Landau theory of magnetic phase transitions. A set of possible planar and collinear structures for low temperatures and close to the transition temperature was obtained. Most of these structures have an unusual configuration. They are characterized by a two-component magnetic order parameter and their magnetic moments vary harmonically not only in direction, but also in size. A theoretical analysis made within the exchange approximation, revealed a set of magnetic structures given by Eqs. (<ref>) and (<ref>), which are determined by fixed spin configurations within each individual chain and fixed inter-chain phase relations. Besides these solutions a possibility that the minimum of exchange energy is achieved for a set of magnetic structures described by Eq. (<ref>) was found. For such a case, the configuration of the magnetic structure depends on the values of the coefficients of the energy expansion written in Eq. (<ref>), β_i. For these structures, monotonous temperature and field evolutions of the spectra are expected and degeneracy removing occurs due to weak interactions. For such structures, the application of a magnetic field can lead not only to a rotation of the spin plane, but also yields a change into a structure out of the set which we present in Eq. (<ref>).

3. Our simulation of the experimental NMR spectra was performed ab initio from the structures
in Eqs. (<ref>), (<ref>), and (<ref>). It has been shown
that the experimentally observed spectra for H∥a, b, c, and (c+15^∘),
respectively, and for temperatures T ≪ T_N, and applied fields H ≪ H_sat can only be described in the frame of the structures given in Eq. (<ref>). This result allowed us to propose a possible scenario of the magnetic structure evolution which should easily be confirmed by elastic neutron scattering experiments.



We gratefully acknowledge support by the German Research Society (DFG) via TRR80 (Augsburg, Munich).
The work was supported by the Program of the Presidium of Russian Academy of Sciences and by Russian Foundation for Basic Research (Grant No. 16-02-00688). A. A. Gippius acknowledge support from the RFBR BRICS Grant No. 17-52-80036.



§ APPENDIX A

Let us write the real and imaginary parts of the equation obtained by
scalar multiplication of the equations (<ref>,<ref>) by e=[ab], g=[cd]:

    (-B+β_2r+β_4ζ+
    β_5ω) ag=0,
    
    (-B+β_2r-β_4ζ-
    β_5ω) bg=0,
    
    (β_1p+2β_6ξ') ag
    +(2β_6ξ”-β_3s) bg=0,
    
    (2β_6ξ”+β_3s) ag+(β_1p-2β_6ξ') bg=0,
    
    (-B-β_2r+β_4ζ-β_5ω) ce=0,
    
    (-B-β_2r-β_4ζ+β_5ω) de=0,
    
    (β_1p+2β_6ξ') ce+(2β_6ξ”+β_3s) de=0,
    
    (2β_6ξ”-β_3s) ce+(β_1p-2β_6ξ') de=0.

These simultaneous equations lead to four types of solutions
(accurate within the above mentioned transformations
μ⇆ν,
μ→iμ):

I. If ag= bg=0, then the vectors a, b,
c, d are coplanar and therefor ce=
de=0. These structures were considered above.

II. If ag= ce=0,  bg, de≠0,
i.e. when simultaneously three vectors a, c, d are coplanar
and three vectors a, b, c are coplanar,
this is possible only at a|| c. Then p=s=ξ=0, β_2r=-β_5ω, and
B=-β_4ζ.

III. If ce=0,  ag, bg, de≠0, then
β_1^2p^2+β_3^2s^2=4β_6^2ξξ^* and
B=β_2r=-β_4ζ=β_5ω.

IV. If ag, bg, ce, de≠0, then
r=ζ=ω=B=0,
β_1^2p^2+β_3^2s^2=4β_6^2ξξ^*.

Thus, the possibility of existence analysis for non-coplanar structures
is reduced to the analysis of 2 * 3 = 6 cases. Here the same solution can be obtained repeatedly, but no one solution will be missed.

Let us introduce two orthonormal bases in the spin space
l, k, n and l̃,k̃,ñ,
and the parameters α γ, ϵ, such that

    a= lc_αc_γ,  b= kc_αs_γ,


    c=l̃s_αc_ϵ,  d=k̃sαs_ϵ.


Let us introduce Euler angles θ, φ, ψ,
to specify the mutual orientation of the spin bases:

    l̃= l(c_ψ c_φ-
    s_ψ c_θ s_φ)+  k(c_ψ s_φ+s_ψ
    c_θ c_φ)+  ns_ψ s_θ,
       k̃=- l(s_ψ c_φ+c_ψ c_θ s_φ)
     + k(c_ψ c_θ c_φ-s_ψ s_φ)+
      nc_ψ s_θ,
       ñ= ls_θ s_φ- ks_θ c_φ+
     nc_θ.


II. From the condition a|| c it follows, that

    μ=c_α( lc_γ+i ks_γ), ν=s_α{c_ϵ l
    +is_ϵ( kc_θ+ ns_θ)}.

In this case, the condition s=0 is performed automatically.
The condition p=ξ=0 for the considered non-coplanar structures (c_α s_α≠0)
is reduced to the relation c_γ c_ϵ=s_γ s_ϵ c_θ=0.
Hence for non-coplanar structures we have either c_γ=c_θ=0, or
c_ϵ=c_θ=0. In the first case

    μ=ic_α k, ν=s_α(c_ϵ l +is_ϵ
    n),
 
    B=β_2c^2_2α+β_4(s^2_α
    c_2ϵ-c^2_α)^2+
    β_5(s^2_αc_2ϵ+c^2_α)^2.

It is easy to verify that there are no non-coplanar solutions here.
The same result is obtained in the case of c_ϵ=c_θ=0.

III. From the condition ce=0 for non-coplanar structures it follows that s_ψc_ϵ=0.

IIIA. If s_ψ=0, then

    μ=c_α( lc_γ+i ks_γ),


    ν=s_α{c_ϵ( lc_φ+ 
    ks_φ)+is_ϵ(- lc_θs_φ+
    kc_θc_φ+ ns_θ)}.
 From the general for 1a,b
condition p=0 we found, that either c_φ=0, and then a|| c, i.e.
we return to the already considered type of solution II, or

    c_γ c_ϵ=-s_γ
    s_ϵ c_θ.


In the case 1a) we have  r=0, whence it follows, that c_2α=0, i.e.

    μ= lc_γ+i ks_γ/√(2),


    ν=c_ϵ( lc_φ+ 
    ks_φ)+is_ϵ(- lc_θs_φ+
    kc_θc_φ+ ns_θ)/√(2),
 and, according to III, ζ=ω=0.
Hence we found
c_2γ=c_2ϵ=0, i.e., according to (<ref>),
c_θ=γϵ=± 1, i.e. we return to coplanar structure again.

In the case 1b) from the condition ξ'=0 we find

    c_γ c_ϵ=s_γ
    s_ϵ c_θ.

For non-coplanar structure s_ϵ≠0,
and from the equations (<ref>,<ref>) it follows,
that c_γc_ϵ=s_γc_θ=0. Hence
there are three options of solution:

1. c_γ=c_θ=0: μ=ic_α k,

    ν=s_α{c_ϵ( lc_φ+ 
    ks_φ)-is_ϵ n},


    B=β_2c^2_2α+
    (β_3+β_6)s^2_2αc^2_ϵs^2_φ+


    β_4(s^2_αc_2ϵ-c^2_α)+
    β_5(s^2_αc_2ϵ+c^2_α);


2. c_ϵ=s_γ=0: μ=c_α l,

    ν=is_α{c_θ(- ls_φ+
    kc_φ)+ ns_θ},


    B=(β_2+β_4)c^2_2α+
    (β_3+β_6)s^2_2αc^2_θs^2_φ+β_5;


3. c_ϵ=c_θ=0: ν=is_α
k, μ=c_α( lc_γ+i
ks_γ),

    B=β_2c^2_2α+β_4(c^2_αc_2γ+s^2_α)^2
    +β_5(c^2_αc_2γ-s^2_α)^2.

It is easy to verify that in these three options only coplanar structures appear.

IIIB. If c_ϵ=0, then the vector c=0. Let us direct
ñ along the vector d, then

    μ=c_α(c_γ l+is_γ k)


    ν=is_α( ls_θ s_φ-
     ks_θ c_φ+  nc_θ),


    B=(β_1+β_3)c^2_γs^2_2αs^2_θc^2_φ+
    (β_3+β_6)c^2_γs^2_2αs^2_θs^2_φ+


    β_2c^2_2α+
    β_4(c^2_αc_2γ-s^2_α)^2+
    β_5(c^2_αc_2γ+s^2_α)^2.

Here also only coplanar structures appear.

IV. From the condition r=ζ=ω=p=0 it follows, that
α=γ=ϵ=π/4; ψ+φ=π/2. Thus,
2μ= l+i k and

    2ν=
     ls_φc_φ(1-c_θ)+  k(s^2_φ+c_θc^2_φ)+  nc_φs_θ,


    +i{- l(c^2_φ+c_θs^2_φ)
     + ks_φc_φ(c_θ-1)+
      ns_φs_θ,}


    B=β_3/4(1+c_θ)^2+
    β_6/4(1-c_θ)^2.
 Minimizing the function B, we found

    c_θ=β_6-β_3/β_6+β_3, B=β_3β_6/β_3+β_6.

We see, that the condition B=0, as it should be
for the solutions type IV, is only performed when β_3 or β_6 equals to zero.

Thus, there are no extrema of the function B, corresponding to
non-coplanar structures.



§ APPENDIX B


Relativistic effects. To obtain relativistic invariants the action of rotational elements, in contrast to the exchange symmetry transformations (<ref>), has to be extended to the spin indices

    C_2x:    μ_x→-e^-iq/2ν_x^*, μ_y,z→ e^-iq/2ν_y,z^*,
       ν_x→ e^-iq/2μ_x^*, ν_y,z→-e^-iq/2μ_y,z^*;
     C_2y:    μ_y→-e^iq/2μ_y, μ_x,z→ e^iq/2μ_x,z,
       ν_y→ e^iq/2ν_y, ν_x,z→-e^iq/2ν_x,z.


Anisotropy energy in general case has the form

    λ_1(μ_xμ^*_x+ν_xν^*_x)+ λ_2(μ_yμ^*_y+ν_yν^*_y)+
    
    +λ_3(μ_xμ^*_z+μ^*_xμ_z-ν^*_xν_z-ν_xν^*_z)+
    
    +iλ_4(μ_xν^*_y-μ^*_xν_y+ν^*_xμ_y-ν_xμ^*_y),

here the common factor η^2 has been omitted.

Let us give the expressions for the anisotropy energy of the phases:

    A_1:   -λ_1/2n^2_x-λ_2/2n^2_y±λ_4(l_xl_y-k_xk_y);
       A_2:   -λ_1/2n^2_x-λ_2/2n^2_y +λ_3(l_xl_z-k_xk_z);
       B_1:   -λ_1/2n^2_x-λ_2/2n^2_y ±λ_4(l_xk_y+k_xl_y);
       B_2:   -λ_1/2n^2_x-λ_2/2n^2_y+ λ_3(l_xl_z-k_xk_z)+
        ±λ_4(l_xk_y+k_xl_y);
       C_1:  λ_1/2l^2_x+λ_2/2l^2_y +2λ_3l_xl_z;
       C_2:   -λ_1/2n^2_x-λ_2/2n^2_y -λ_3n_xn_z;
       D:  λ_1/2l^2_x+λ_2/2l^2_y∓2λ_4l_xl_y;
       E:   -λ_1/2n^2_x-λ_2/2n^2_y ±λ_4n_xn_y;
       D^*:  λ_1/2l^2_x+λ_2/2l^2_y;
       E^*:   -λ_1/2n^2_x-λ_2/2n^2_y.

Here for the planar structures we introduced the normal vector to the spin plane n=[ l k]
and used the relation

    l_αl_β+k_αk_β+n_αn_β= δ_αβ.


Electric polarization of relativistic nature is due to the following invariants:

    -d_1E_x(μ_xν^*_z+μ^*_xν_z+μ_zν^*_x+μ^*_zν_x)
       -d_2E_y(μ_yν^*_z+μ^*_yν_z+μ_zν^*_y+μ^*_zν_y)
       -d_3E_z(μ_zν^*_z+μ^*_zν_z).

Accordingly, in five phases of twelve components of polarization P_x,P_y,P_z occur:

    A_1)  d_1(l_xk_z+k_xl_z),d_2(l_yk_z+k_yl_z),d_3l_zk_z;
       A_2)  d_1(l_xk_z+k_zl_x),d_2(l_yk_z+k_yl_z),d_3l_zk_z;
       B_1)  d_1(l_xl_z-k_xk_z),d_2(l_yl_z-k_yk_z),d_3(l^2_z-k^2_z);
       D^*)  2d_1l_xl_z,2d_2l_yl_z,P_0+d_3l^2_z;
       E^*)  d_1(l_xl_z+k_xk_z),d_2(l_yl_z+k_yk_z),P_0-d_3n^2_z.

Here for the phases D^* и E^* we added the already mentioned polarization of exchange nature P_0.
Note that P_0 and the relativistic contribution to the polarization, as well as the "anomalous"  terms of anisotropy (λ_3,λ_4) are given for the one of four possible domains, differed by the sign of ν and the replacement μ⇆ν.

Thus, if we exclude the collinear structure D^*, there are four candidates for the phase, observed in LiCu_2O_2 at the temperature T<23,2 K.

Due to the low symmetry of the crystal, there are too many invariants caused by the relativistic effects of magnetostriction:
η_xxu_xx, η_xxu_yy, η_xxu_zz, η_yyu_xx, η_yyu_yy, η_yyu_zz;
η_zzu_xx, η_zzu_yy, η_zzu_zz;
η_xyu_xy, η_yzu_yz, η_zxu_zx;
r_xxu_xz, r_yyu_xz, r_zzu_xz, r_xzu_xx, r_xzu_yy, r_xzu_zz, r_xyu_yz, r_yzu_xy; s_xyu_xx, s_xyu_yy, s_xyu_zz,
s_xxu_xy, s_yyu_xy, s_zzu_xy, s_yzu_xz, s_xzu_zy;
here, the following forms are introduced for brevity

    η_ik=μ_iμ^*_k+μ_kμ^*_i+ν_iν^*_k+ν_kν^*_i,


    r_ik=μ_iμ^*_k+μ_kμ^*_i-ν_iν^*_k-ν_kν^*_i,


    s_ik=μ_iν^*_k+μ_kν^*_i-μ^*_iν_k-μ^*_kν_i,

therefore the use of strain measurements for the task of structure selection is difficult.

Spin-orbit splitting of phase transition.
The above regular accounting for relativistic effects under the perturbation theory may become inapplicable in the vicinity of the transition, where their contribution necessarily becomes comparable to the exchange quadratic term of the Dzyaloshinskii-Landau expansion τη^2. In this case, an intermediate magnetic phase may appear in a small vicinity of the transition in the cases, when active representation of the exchange approximation is not one-dimensional. Let us find out the features of this phase, suggesting the Lifshitz instability is weak.

Let us select the real and imaginary parts in the order parameter ημ= a+i b,
ην= c+i d, without imposing any restrictions on the real-valued vectors a, b, c, d. In these variables the quadratic part of the Dzyaloshinskii-Landau expansion has the form

    (τ+λ_1)(a_x^2+b_x^2+c_x^2+d_x^2)+
       +(τ+λ_2)(a_y^2+b_y^2+c_y^2+d_y^2)+
       +τ(a_z^2+b_z^2+c_z^2+d_z^2)+
       +2λ_3(a_xa_z+b_xb_z-c_xc_z-d_xd_z)+
       +2λ_4(a_xd_y-b_xc_y+d_xa_y-c_xb_y).


In the immediate vicinity of the transition we can not neglect the Lifshitz invariant (<ref>), and should also
consider suppressive instability quadratic in the gradients exchange invariant

    L(ac'-ca'+
    bd'-db')+
    +
    g(a'^2+b'^2+c'^2+d'^2),

where prime means differentiation with respect to coordinate x.

In the general case this is a rather complicated task. However, in LiCu_2O_2, according to the experimental data,<cit.> if the parameter λ_1 is negative and substantially exceeds the value of the remaining anisotropy constants, then the analysis of the transition is much easier. The transition then should occur at τ≈-λ_1
and we may neglect y and z components of the vectors in the vicinity of T_c1. As a result, the quadratic term in the Dzyaloshinskii-Landau expansion in the order parameter has the form

    F_2=τ̃η^2+
    L(ac'-ca'+bd'-db')+
    +
    g(a'^2+b'^2+c'^2+d'^2),

where η^2=a^2+b^2+c^2+d^2, the same for all vectors index x has been omitted.

Let us turn to the Fourier components a=a_ke^ikx+a^*_ke^-ikx, ...,  k>0.
For the contribution of single harmonic we have

    F_2=(τ̃+gk^2)(a_k^*a_k+b_k^*b_k+c_k^*c_k+d_k^*d_k)+
    + ikL(a_k^*c_k-c_k^*a_k+b_k^*d_k-d_kb_k^*).

For given magnitude a_k^*a_k+b_k^*b_k+c_k^*c_k+d_k^*d_k and wavevector this contribution is minimal if

    c_k=ia_k,   d_k=ib_k,

then

    F_2=2(τ̃-Lk+gk^2)(a_k^*a_k+b_k^*b_k).


The phase transition occurs when the temperature decreases and the minimal value of the coefficient in (<ref>)
changes sign at k=k_0=L/2g. The relative phase and amplitude of the components a_k and b_k
are determined by the minimum condition of the function

    B=4β_1(ac+bd)^2+
       β_2(a^2+b^2-c^2-d^2)^2+4β_3(ad-bc)^2+
       β_4[(a^2-b^2+c^2-d^2)^2+4(ab+cd)^2]+
       β_5[(a^2-b^2-c^2+d^2)^2+4(ab-cd)^2]+
       4β_6(a^2c^2+b^2d^2+a^2d^2+b^2c^2)

at the fixed magnitude η^2=2(a_k^*a_k+b_k^*b_k) and fulfillment the condition (<ref>).

Turning to the Fourier components in the expression (<ref>) and integrating over the volume we get

    8(β_1+β_2)(a^2+b^2)(a^*2+b^*2)-16β_3(a^*b-ab^*)^2
       +16β_4(a^2a^*2+b^2b^*2+a^2b^*2+a^*2b^2)
       +8(β_5+β_6)[(a^2-b^2)(a^*2-b^*2)+4aa^*bb^*],

here the indices k have been omitted.

Introducing parametrization

    a_k=η/2e^-iφs_ϕ,  b_k=η/2c_ϕe^i(δ-φ),

we get

    B=B_0+As^2_2ϕs^2_δ,

where

    B_0=1/2(β_1+β_2+2β_4+β_5+β_6),
       A=-1/2(β_1+β_2+2β_3+2β_4+β_5+β_6).


If A<0 then the minimum B occurs at ϕ=π/4, δ=π/2. If A>0 then
there are two solutions 1) ϕ=0 (or ϕ=π/2), and the value of δ is not relevant, and 2) δ=0,π. Thus, the following solutions are the candidates for the intermediate phase

I_1:μ_x   =±iν_x=e^-i(kx-φ)/√(2)

I_2:μ_x   =cos(kx-φ),   ν_x=±sin(kx-φ);

I_3:μ_x   =cos(kx-φ),   ν_x=±isin(kx-φ).


Let us note that there are no associated solutions transforming into each other under the transformation (<ref>) here, since, as noted above, the Lifshitz invariant (<ref>) breaks this random symmetry.
One of these phases has to be realized at the transition point T_ c1. Note that only in the phase I_2 electric polarization of exchange nature occurs.

There are two possible scenarios with decreasing temperature: either a first-order transition into one of the phases considered above at the temperature T_c2 or, if the low-temperature phase is noncollinear, then a first-order transition into another intermediate collinear phase C_1,D or D^* is possible.



37

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Contact_field_note We tried to fit the experimental spectra in the model where the second pair of copper ions induces the dominant contact field. This assumption does not change the conclusions of our paper.
Program The program can be found at http://www.kapitza.ras.ru/rgroups/esrgroup/numa.html.




