# The strength of frustration and quantum fluctuations in LiVCuO

###### Abstract

We present an empirical and microscopical analysis of the main in-chain exchange constants of the edge-shared frustratrated chain cuprate LiVCuO= LiCuVO with a ferromagnetic nearest neighbour coupling which clearly exceeds the antiferromagnetic (AFM) next-nearest neighbour exchange . The measured saturation field is significantly affected by a weak 3D AFM interchain coupling leaving room for a possible Bose-Einstein condensation for several T below. The obtained exchange parameters are in agreement with the results for a realistic five-band extended Hubbard Cu O model, LSDA+ predictions as well as with inelastic neutron and magnetization data. The single chain frustration rate , including all error bars, is definitely smaller than 1.0 which correspond to strongly coupled interpenetrating AFM Heisenberg chains in contrast with opposite statements in the literature. A proper account of strong quantum fluctuations and frustration is necessary for a correct assignment of the exchange integrals. which cannot be achieved by a simple renormalization of from spin-wave theory.

###### pacs:

74.72.Jt###### pacs:

78.70.Nx###### pacs:

75.30.DsOther cuprates Neutron inelastic scattering Spin waves

## 1 1. Introduction

LiVCuO LiCuVO is one of the first [1, 2] and rather frequently studied spin-chain compounds among edge-shared cuprates

[3, 5, 4, 6, 7]. Recently it became especially interesting due to the observation of multiferroicity [8, 9, 10] and due to a possible realization of quantum spin nematics and related Bose-Einstein condensation of two-magnon bound states in high magnetic fields [11, 12, 13]. Both phenomena are still poorly understood and a precise knowledge of the main exchange interactions is of key importance to attack such complex problems in a realistic way. Unfortunately, there is no consensus about the magnitude of these couplings and in particular on the value of the in-chain frustration parameter , with as the ferromagnetic (FM) nearest neighbor (NN) and the antiferromagnetic (AFM) next-nearest neighbor (NNN)-coupling in chain direction (see Fig. 1). So far, in various studies and even above 5.5 have been predicted/reported.[14, 16, 5, 3, 15] Keeping in mind the weak interchain coupling,

the single-chain can be viewed also as two interacting and interpenetrating simple AFM Heisenberg chains (AHC) or equivalently as a single zigzag ladder. Then, one is left with a weak () or a strong coupling scenario in the opposite case because provides a direct measure of the FM ”interchain”-coupling within a zigzag-ladder or between interpenetrating AHC. Here, we give a more comprehensive presentation of arguments against weak coupling scenarios than in our short Comment [16] on Ref. [5] and rebut in more detail also arguments put forward in Ref. [6]. In addition, the striking discrepancy of a very recent parameter set obtained in Ref. [15] with available experimental data, including also Refs. [5, 6] will be shown, too. Finally, the reasons of the incorrect parameter assignment given in Refs. [3, 5, 15] are explained in terms of an inproper handling of strong quantum fluctuations (SQF). Clear evidence for SQF comes from the small values of the ordered magnetic moment of 0.31 and the low Neél-temperature K [2, 10].

## 2 2. Phenomenological Analysis

The low-energy inelastic neutron scattering (INS) data [3] have been fitted in terms of simple (bare) 3D spin wave theory (BSWT), i.e. without any renormalization or account of quantum fluctuations (see Figs. 2 (a-c,d,e)). As a result one arrives at 5.6 meV and meV despite weak interchain couplings among them meV has been claimed to be the predominant interaction responsible for the in-phase ordering of spirals in the magnetically ordered state below (see Tab. 1). Then based on these and recent high-energy data (20 meV meV data)[5] analyzed by means of a random phase approximation (RPA) approach for the account of the coupling between the AHC, an effective 1D model has been proposed in Ref. [5]:

(1) |

Thereby quantum fluctuations have been taken into account by the prefactor as for free AHC in accord with the implicite a priori assumption of a weak coupling between the AHC, in other words any renormalization related to the coupling between the AHC has been almost ignored. Such an assumption seems to be to not justified from a general many-body theory point of view. However, following our recent work [16] we will show here in more detail that the in-chain exchange integrals and , respectively, are significantly different from those suggested in Refs.[3, 5, 6, 15].

BSWT [3] | 1.6 | 5.59 | 3.49 | 0.01 | 0.4 |
---|---|---|---|---|---|

RSWT [3] | 1.6 | 3.56 | 2.23 | ||

RPA[5] | 2.4 | 3.4 | 1.42 | ||

present work | 6.95 | 5.2 | 0.75 | ||

3O2 optics | 6.31 | 5.05 | 0.8 | ||

GGA+[15] | 2.7 | 15.0 | 5.55 | -1.31 | 0.16 |

LSDA+ | 8.5 | 7.05 | 0.82 | ||

GGA+ | -6.4 | 5.45 | 0.85 |

Furthermore it has been claimed that the FM coupling can be fixed at its bare value . This would yield meV close to the result of a phemonelogical RPA-based description of the problem: 3.4 meV [5]. In passing through we note that the value predicted by Koo et al. [15] exceeds that value very much by a factor larger than three. If the interchain coupling is of less relevance for the ”high-energy” physics, the claimed 2 meV should be added to meV only for low-energy problems such as thermodynamics, i.e. relevant for the saturation field and the magnetization or the determination of the spiral’s pitch angle. With such a more convincing empirical RPA affected renormalization one would already arrive at close to the strong coupling boarder line. Up to now all considerations were based on the assumption that the FM remains fixed. However, field-theory flow-equations based approaches [17] valid at point to strong coupling renormalizations. As a consequence, might change considerably and is further scaled down. In fact, such a tendency would be compatible with our DMRG [18] results (see also below and Tab. 1) : meV, eV, and [16]. If one adopts the BSWT-parameters as a reasonable starting point, our results should be interpreted as a strong upward renormalization of both and a moderate for , too,

Turning first to the low-energy INS-data [3], we start with the two extrema of a one magnon excitation and , i.e. the peak positions near the transferred momenta and . This is the lowest two-spinon excitation (2SE) reproduced approximately by the BWST-fit taken from Ref. [3, 5]. Its dispersion is sketched by the red curves in Fig. 2 (a-c). Although the maximum corresponding to is broad, the asymmetry with respect to quantified by the dynamical asymmetry parameter is clearly visible in the INS-data in contrast with the set proposed in Ref. [15] where would occur. On absolute scale a discrepancy by factor exceeding three between the experimental and the predicted dispersion is observed (see Fig 2(c) which can be traced back to artificially large values of (see Tab. 1).

From the experiment, Fig. 2 (d), one reads off meV. Taking meV one estimates . can be obtained from fitting our dynamical DMRG [19] results for and long chains with sites

(2) | |||||

where the coupling strength has been introduced. The relation provides a convenient highly sensitive measure of the interaction regime which is heavily affected by the strong quantum fluctuations. The function is depicted in Fig. 2 (h). One realizes excellent agreement with derived in our previous paper where instead and the relative magnetization curve as a function of at low temperature have been employed [16]. Notice the large deviations if the BSWT or the RSWT would be applied to extract . Taking meV and meV from Figs. 2 (d,e) which yields almost consistent with 2.2 stated in Ref. [3]. Using the RPA-derived values one arrives at about 1.42 again in formal consistency with [5]. The strong deviations of both values from our DMRG-based value clearly show the inapplicability of simple spin-wave theory based estimates. The physical reason is the incorrect treatment of strong quantum fluctations in the title compound which manifest themselves also in a small magnetic moment as mentioned above and in relatively large pitch angles (see below).

Finally, considering briefly the calculated and the experimental INS intensities, at present only few comparisons are possible due to lacking publication of experimental spectra. Nevertheless, comparing e.g. the available data shown in Figs. 2 (d,e) one realizes that our set provides a better description of the intensity at large transferred momenta (Fig. 2(b)) as compared with that of Ref. [5]. A comparison of the theoretical shapes with more INS spectra would be helpful to reduce our error bars.

If one adopts that the experimental magnetization data up to the so-called field T ( where the peak in dM/dH occurs [12] is well-described by an effective 1D model, one arrives at the curves shown in Fig. 3. Notice the strong deviation of the weak-coupling proposal by Koo et al.[15].

Then, a dominant FM interchain coupling as proposed in Refs. [5, 3] cannot be reconcoiled with these experimental data since for such a coupling the 3D saturation field is smaller than its 1D counterpart [20]. Also the spin susceptibility is within an RPA approach for the interchain coupling best described by a total AFM interchain coupling. Anyhow, a detailed discussion of including also a consideration of the background susceptibility will be given elsewhere.

As mentioned above the presence of strong quantum fluctuations is evidenced by the small magnetic momentum of 0.31 and a low Neél-temperature K. Both values should be compared e.g. with three to four times large values for the sister compound LiCuO [21, 22] caused by a relatively strong interchain coupling [23]. In addition its small is also helpful to suppress SQF. As a consequence the spiral state is significantly driven towards almost decoupled AHC the corresponding collinear Neél state, of each AHC i.e. the experimental pitch angle analyzed within the BSWT or RSWT results in strongly overestimated -values. This is illustrated in Figs. 4 and 5, where the maximum of the static magnetic structure factor is depicted as a function of for the cases of a single frustrated - chain and a coupled pair of them, respectively. Already the latter is expected to provide a reasonable insight into the real quasi-1D situation. This point of view is supported by a detailed comparison with coupled cluster calculations to be reported elsewhere. Thus, for instance in the case of a planar arrangement of chains (i. e. a dominant 2D-interchain coupling as in the model adopted in Refs. [3, 5, 11, 12]) the effective interchain interactions and correspond approximately to

(3) |

Notice the striking failure of the classical curve especially for large (see Fig. 4). Such an effect was first addressed in Ref. [24] in the 1D-case by means of DMRG and in Ref. [25] for a plane of perpendicularly coupled chains by means of a coupled cluster approach. We stress that the experimental value of the pitch [2] is reproduced for , only, independently of the details of the weak interchain coupling. Adopting the in-chain parameters and the leading interchain coupling meV suggested in Ref. [3], one estimates from Fig. 5 a pitch angle of 89.58 () and of 89.43 in the case of the RPA derived set ( ) in contrast to 84 known experimentally [2]. Thus, the measured points clearly to a strong coupling regime in contrast to opposite statements in Refs. [3, 5, 15]. Naturally, the SWT derived ’s obey nearly the classic relation, only, (i.e. completely ignoring the strong quantum fluctuations)

(4) |

yielding 83.53 for [3] (the small deviations from 84 result from the weak interchain coupling ignored in Eq. (4) for the sake of simplicity) and 87.42 for [15]. What matters here is not the absolute value of , but the difference which differs by two orders of magnitude between the quantum and the classic case [26]. Thus, the attempt to describe the spin dynamics in a quasi-classical way is the main reason for the inproper assignment in Ref. [3].

## 3 3. Microscopic Analysis

Turning to a microscopic analysis, we compare our (DMRG) INS derived ’s with those from two independent microscopical approaches: (i) analyzing high-energy spectra from EELS, optical conductivity or RIXS data within strongly correlated extended multiband Hubbard models and a subsequent mapping of their spin-states onto the corresponding states of a spin-Hamiltonian, i.e. the 1D - model under consideration. The results are shown in Tab. 1 and Fig. 6. (ii) extracting these exchange parameters from total energy calculations of various prepared artificially magnetically ordered states (see e.g. [15]).

A mapping from a Cu3 O2 five-band Hubbard model with usual parameters which describes the -dependent dielectric response [28, 27] onto a - spin-1/2 model yields a sizeable =6.3 meV and =5.05 meV. We stress that in all closely related sister compounds [29] with a Cu-O-Cu bond angle 95 sizeable FM -values 1.6 meV have been found in fitting various data: LiCuO: 1 9.6 meV (INS [23]), CaYCuO: meV (INS [30]), LiZrCuO: meV (, [31, 14]). In particular, also for LiVCuO the -dependent optical conductivityidata [27] obtained from ellipsometry measurements can be well fitted within a five-band Cu 3 O 2 extended Hubbard model on chain-clusters with up to six CuO-plaquettes connected by edge-sharing. Thereby eV, eV meV, eV etc. has been used. As a result one arrives at in-chain ’s close to the INS-derived ones: and =5.1 meV (see Tab. 1). The value of is sensitive to the magnitude of the direct FM exchange whereas is mainly sensitive to the in-chain O-O transfer integrals. Thereby holds approximately. Notice that the contribution of is much more important for the large negative (FM) value of than that of the intra-atomic FM Hund’s rule coupling on O. In the past has been used mostly as a fitting parameter ranging from 50 to 110 meV for CuGeO [32, 33]. A reliable -value derived from an INS analysis as reported here is helpful to restrict its value and opens a door for systematic studies of this important exchange and useful comparisons with other sister compounds [29]. In fact, our empirical value corresponds to 130 meV for a -Cu-O bond to be compared with 180 meV estimated for that case and a cuprate plane in high- superconductors [34].

Considering the total energies of various magnetic states the main exchange integrals can be also extracted from LDA+ or GGA+ calculations. Thereby the results depend mainly on a single parameter , where eV denotes the Hund’s rule coupling that is rather precisely known for transition metals. For both approximations, we calculated the exchange integrals and and their ratio for the two different crystal structures refined from x-ray diffraction and neutron diffraction (labeled XRD and NRD, respectively, in Fig. 6 and below). As the key parameter, the resulting for different values of is presented in Fig. 6 and given in Table 1 ( eV). The graph indicates only small differences for the two crystal structure solutions, but essentially no difference for the two choices of the exchange-correlation potential (LDA+ vs. GGA+). For realistic parameters which describe successfully other edge-shared chain cuprates one arrives again at in sharp contrast to Koo et al. [15] who obtained unusually large - and -values not compatible with the observed pitch angle [2], the restricted two-spinon continuum, and an obviously asymmetric INS spectrum [5]. Presumably it is a consequence of the double counting procedure employed in Ref. [15] and not an artifact of the GGA as stated there because our calculations shown in Fig. 6 yield close values in the -region of interest, both for the LDA and the GGA. Also the RPA-derived value [5] could be approached for unrealistic small -values below 3 eV adopting the XRD data, only.

## 4 4. Summary

The main result of our revisited analysis of LiVCuO is the clear evidence for strong coupling of AHC as derived from four independent experimental and theoretical studies: the INS yields a dynamical asymmetry parameter and a pitch angle very sensitive to quantum fluctuations. Weak coupling would result in a nearly collinear incommensurate state and in almost vanishing dynamical anisotropy, i.e. not compatible with the diffraction and INS data.The obtained values for the main exchange integrals are supported by independent microscopic calculations based on the L(S)DA+ approach and the multiband Hubbard model.

###### Acknowledgements.

We thank the DFG [grant DR269/3-1 (S.-L.D. & J.M.) and the Emmy-Noether-program (H.R.), the programs PICS [contracts CNRS 4767, NASU 243 (R.O.K)], and ASCR(AVOZ10100520) (J.M.) for financial support.## References

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