# Influence of an inter-chain coupling on spiral ground-state correlations in frustrated spin-1/2 - Heisenberg chains

###### Abstract

We investigate the influence of an inter-chain coupling on the spiral ground state correlations of a spin-1/2 Heisenberg model consisting of a two-dimensional array of coupled chains with nearest () and frustrating next-nearest neighbor () in-chain exchange couplings. Using the coupled-cluster method we calculate the transition point between the commensurate and the incommensurate (spiral) ground state as well as the quantum pitch angle of the spiral ground state. In addition, we provide a simple empirical formula which describes the relation between the quantum pitch angle and the frustration parameter .

###### pacs:

75.10.Jm, 75.10.Pq, 75.40.Mg, 75.50.Ee## I Introduction

In recent years strongly frustrated quantum magnets exhibiting exotic ground state (GS) phases have been intensively investigated both theoretically and experimentally, see e.g. Refs. science, ; diep04, ; lnp04, . At zero temperature all transitions between GS phases are driven purely by the interplay of quantum fluctuations and the competition between interactions (e.g., frustration), see e.g. Refs. sachdev99, ; sachdev04, ; misg04, ; Mikeska04, ; Richter04, . Particular attention has been paid to one-dimensional (1D) - quantum Heisenberg models, which have been studied theoretically with much success over the last two decades, see Ref. Mikeska04, and references therein. From the experimental side recent investigations have shown that edge-shared chain cuprates build a special family of frustrated quantum magnets which can be described by a quasi-1D - Heisenberg model. Among others we mention here LiVCuO, LiCuO, NaCuO, LiZrCuO, and LiCuO gibson ; matsuda ; gippius ; ender ; drechs1 ; drechs3 ; drechs4 ; park ; drechsQneu ; malek ; tarui , which were identified as quasi-1D frustrated spin-1/2 magnets with ferromagnetic nearest-neighbor (NN) in-chain and antiferromagnetic next-nearest-neighbor (NNN) in-chain interactions . These cuprates have attracted much attention due to strong quantum effects and the observation of incommensurate spiral (helical) spin-spin correlations at low temperatures. Among these materials LiZrCuO and LiCuO are of particular interest, since these compounds are found to be near a quantum critical point drechsQneu ; malek .

The 1D frustrated spin-1/2 - Heisenberg model may serve as the simplest model to describe some important features of such materials. The GS properties of the model in the classical limit are well known, i.e. when the spin quantum number . In this case the GS does exhibit a second-order transition from a collinear phase (ferro or antiferro) to a non-collinear phase with spiral correlations along the chains at . For the classical spiral (pitch) angle is given by

(1) |

Similar expressions can be derived in the presence of further couplings , , etc., however, this more general case will not be considered here for the sake of simplicity. Note that in the classical limit neither the pitch angle nor the transition point depend on the inter-chain coupling . Such a classical relation as Eq. (1) between , , and the pitch angle has been used to justify 1D parameter sets obtained from fitting data capogna2005 or by LDA mapping procedures mazurenko2007 . However, ignoring this way sizable quantum effects, such an approach is not very convincing. Furthermore, in real materials additional terms in the Hamiltonian such as anisotropy or exchange coupling between the chains might be of relevance to yield a quantitative theoretical description of the experimental results. The existence of helical long-range order at low temperatures makes the importance of the inter-chain coupling evident. However, from the theoretical side such extended models so far are much less studied than the pure 1D-”parent” models.

Therefore, in the present paper we focus on the discussion of the effect of the inter-chain coupling on the GS spin-spin correlations in frustrated spin-1/2 Heisenberg model consisting of a two-dimensional (2D) array of frustrated chains coupled by . In particular, we will discuss GS’s with incommensurate spiral (i.e. noncollinear) correlations. Though, meanwhile many papers exist dealing with spiral correlations in the strictly 1D system () Hamada ; bursill ; krivnov96 ; white96 ; aligia00 ; dima06 ; dima07 , the influence of the inter-chain coupling on the pitch angle to the best of our knowledge has not been discussed so far.

The theoretical treatment of frustrated quantum antiferromagnets is far from being trivial. Although one can find exact GS’s in some exceptional cases, see e.g. Refs. Shastry, ; maju, ; Hamada, ; dim_pla, ; prl02, , standard many-body methods may fail or become computational infeasible. For instance, the quantum Monte Carlo techniques suffer from the minus-sign problem in frustrated systems. The density-matrix renormalization group (DMRG) successfully used to discuss spiral correlations in 1D magnets bursill ; white96 is essentially restricted to 1D systems, at least in the present state of the art. Also the exact diagonalization technique used in Ref. aligia00, to find the pitch angle for the 1D problem would be limited to extremely small chain lengths when a finite inter-chain coupling should be considered. A method fulfilling the requirement to be able to deal with frustrated spin systems at any dimension, including magnetic systems with incommensurate spiral GS’s, is the coupled cluster method (CCM). This method was already used for the strictly 1D - Heisenberg model and it was shown that the CCM results are in good agreement with the DMRG data bursill . Hence, in the present paper we use the CCM following the lines of Ref. bursill, but extend the CCM calculations by including the inter-chain coupling .

The frustrated spatially anisotropic 2D -- spin-1/2-Heisenberg model considered here reads

(2) | |||||

where the index labels the chains and the lattice sites within a chain . The NN in-chain coupling is fixed to either (antiferromagnetic) or (ferromagnetic). The inter-chain coupling and the frustrating NNN in-chain coupling are considered as varying parameters of the model. Note that in the case considered here the inter-chain coupling does not lead to frustration and it is practically of arbitrary strength, i.e. including also the region beyond the quasi-1D limit . Anyhow, the opposite limit will not be considered for reasons of lacking physical relevance (to the best of our knowledge). We focus on the extreme quantum case, i.e. with . The main point which will be considered here is the influence of the inter-chain coupling on the transition point between the collinear phase and the non-collinear spiral phase and on the pitch angle characterizing the spiral correlations in the quantum model with .

## Ii The coupled Cluster Method (CCM)

In this section we outline only some main features of the CCM which are relevant for the model under consideration. Again we mention that we follow the lines described in Ref. bursill, where the CCM was applied to the strictly 1D problem. For more details of the method the interested reader is referred to Refs. bursill, ; bishop91, ; bishop91a, ; zeng98, ; bishop00, ; krueger00, ; krueger01, ; ivanov02, ; farnell04, ; rachid05, ; rachid06, ; Bi:2008_j1j2j3_spinHalf, ; Bi:2008_j1j2j3_spinOne, . Special attention to the CCM treatment of non-collinear GS’s was paid in Refs. bursill, ; krueger00, ; krueger01, ; ivanov02, ; rachid05, ; rachid06, .

The starting point for the CCM calculation is the choice of a normalized reference or model state , together with a complete set of (mutually commuting) multi-configurational creation operators and the corresponding set of their Hermitian adjoints ,

(3) | |||

(4) |

With the set the CCM parametrization of the ket GS eigenvector of the considered many-body system is then given by

(5) |

The CCM correlation operator contains the correlation coefficients which can be determined by the so-called set of ket equations

(6) |

For the considered frustrated spin system we choose a reference state corresponding to the classical state of the model, i.e. the ferromagnetic state along a chain for and small and the Néel state for and small , whereas for larger frustration we have to choose a non-collinear reference state with spiral correlations along the chains (i.e., pictorially, ) characterized by a pitch angle , i.e. . Such states include the ferromagnetic state () as well as the Néel state (). In the quantum model the pitch angle may be different from the corresponding classical value . Hence, we do not choose the classical result for the pitch angle but, rather, we consider as a free parameter in the CCM calculation, which has to be determined by minimization of the GS energy given in the CCM formalism by , i.e. from , see also Appendix, Eqs. (A1)-(A6).

In order to find an appropriate set of creation operators it is convenient to perform a rotation of the local axes of each of the spins, such that all spins in the reference state align in the negative -direction. This rotation by an appropriate local angle of the spin on lattice site is equivalent to the spin-operator transformation

(7) |

The local rotation angle can be easily expressed by the pitch angle of the spiral reference state, where the relation depends on the signs of and the lattice vector .

In this new set of local spin coordinates the reference state and the corresponding creation operators are given by

(8) |

where the indices denote arbitrary lattice sites. In the new rotated coordinate frame the Hamiltonian (2) becomes dependent on the pitch angle . It reads

(9) | |||||

where , and () is the pitch angle between the two neighboring spins in a chain interacting via the NN bond , which has to be determined for the quantum model. For ferromagnetically coupled spin chains () one has to set in Eq. (9) and vice versa. Therefore, from Eq. (9) it is obvious that in the quantum case considered here the role of the (unfrustrated) inter-chain coupling introduced in Eq. (2) is different for ferromagnetic and antiferromagnetic . By contrast, in the classical case a corresponding inter-chain coupling does not affect the pitch angle at all.

The CCM formalism would be exact if we could take into account all possible multispin configurations in the correlation operator which is impossible to do in practice for a quantum many-body system. Hence, it is necessary to truncate the expansions of . In bursill it was demonstrated, that the so-called SUB2-3 approximation for the strictly 1D system leads to results of comparable accuracy to those obtained using the DMRG method. In the SUB2-3 approximation all configurations are included which span a range of no more than contiguous sites and contain only or fewer spins. A particular advantage of the SUB2-3 approximation consists of the possibility to find the relevant CCM equations (6) in closed analytical form, see Appendix A. These explicit equations provided here can be used to find the quantum pitch angle for an arbitrary set of parameters , , by simple numerical solution of them.

## Iii Results

For the model under consideration we have calculated that point where the GS state spin-spin correlations change from collinear to non-collinear spiral correlations as well as the quantum pitch angle . In what follows we call the point the ’transition point’. We mention, however, that the question for magnetic GS long-range order goes beyond the scope of the present paper. Generally one can argue that for the strictly 1D problem the GS (except the simple ferromagnetic state) does not exhibit magnetic long-range order, whereas for finite GS long-range order can exist, cf. e.g. Refs. affleck, ; wang, ; sandvik, ; kim, ; zinke08, . In particular, the GS phase for larger and small is magnetically disordered and may have a weak spontaneous dimerization along with finite-range incommensurate magnetic correlations affleck . Hence the ’transition point’ for small may locate that narrow parameter region where finite-range magnetic correlations are changing from commensurate to incommensurate ones, but does not indicate a true quantum phase transition.

We present data for and a fine net of values. For the sake of clarity in the following we asort the results into four cases, depending on the signs of and , namely (i) , ; (ii) , ; (iii) , ; (iv) , .

Since for cases (i) and (ii) the behavior is quite similar, we can discuss both cases together. The pitch angle in dependence on is shown in Figs. 1 (case (i)) and 2 (case (ii)). For comparison we also draw the corresponding DMRG data of White and Affleck white96 for the strictly 1D problem in Fig. 1. These data agree quite well with the CCM data in particular for larger . It is obvious that quantum fluctuations change the classical correlations drastically. In particular, by contrast to the classical case, the collinear quantum state can survive into the region , where classically it is already unstable. This effect is known as order from disorder villain ; shender and is widely observed in quantum spin systems henley89 ; Kubo_JPSJ ; krueger00 ; rachid05 . We find, e.g. for the quantum model with , which is in good agreement with known results, see, e.g., Fig. 3 in Ref. aligia00, .

Switching on , i.e. increasing the dimension of the spin system, the effect of quantum fluctuations should become weaker and, as a result, the collinear quantum state gives way for the state with incommensurate correlations at smaller values of , i.e. the transition point moves towards the classical value, see Fig. 3. However, even in nearly isotropic 2D systems () the quantum fluctuations are still important and one has () for case (i) (for case (ii)) which is significantly above the classical value.

The shift of the transition point leads to an interesting behaviour of the difference . It is positive for but negative for . Surprisingly we find as almost independent of , i.e. all curves vs. meet approximately in one point, see Figs. 1 and 2. Note that for large the model with strongest quantum fluctuations, i.e. , approaches the limit most rapidly.

Next we discuss the behavior of the pitch angle at the transition point. For smaller values of the inter-chain coupling ( for case (i) and for case (ii)) we find a discontinuous behavior of the quantum pitch angle. A similar jumpwise change of the pitch angle has been found for 2D frustrated quantum spin models krueger00 ; rachid05 . Note that in Ref. bursill, the CCM curve was terminated before the jump occurs so that the jump has not been observed there damian . The discontinuous change of is related to the existence of two minima in the versus curve and takes place for parameter values where both minima have equal depth, cf. Fig. 4. For larger (weaker quantum fluctuations) exhibits only one minimum and the quantum pitch angle is changing continuously from the collinear to the spiral GS. We remind the reader that for the classical case the transition from the commensurate to the incommensurate GS takes place at and that it is continuous for any value of .

Finally, in Fig. 5 we have drawn the pitch angle in dependence on the inter-chain coupling for case (i). As discussed above, with increasing the quantum pitch angle moves towards the corresponding classical value. However, in accordance with the above discussion of the change of the sign of we find two different regimes: For the increases with growing while for the decreases with .

Let us now pass to cases (iii) and (iv), i.e. when is ferromagnetic. By contrast to cases (i) and (ii), we find that here the transition from the collinear state to the spiral state is always continuous. The reason for that can be again attributed to the strength of quantum fluctuations. For ferromagnetic the relevant collinear state at consists of ferromagnetic chains, having a classical (i.e. ’non-fluctuating’) GS, coupled by . While for case (iv) the ferromagnetic inter-chain coupling does not change this ’non-fluctuating’ GS at all, for case (iii) due to the presence of antiferromagnetic couplings quantum fluctuations become relevant and the GS becomes a true quantum state, but the change of magnetic correlations within the chains for remains weak. Hence, virtually no (case (iv)) or only weak (case (iii)) quantum fluctuations occur at the transition from the collinear to the noncollinear GS. As a result the transition takes place precisely at for case (iv) while for case (iii) the transition point is above the classical value indicating again an order from disorder effect. However, the shift of is small because the quantum fluctuations are weak, see Figs. 6 and 8. Nevertheless, such a small shift of due to a finite inter-chain coupling might be important for systems such as LiZrCuO and LiCuO drechsQneu ; malek being near the transition point. Note that Bader and Schilling bader first found that the transition point is fixed at , if the collinear state is the classical ferromagnetic one. Note further that a similar change from a discontinuous to a continuous transition was discussed in Ref. krueger01, .

The quantum pitch angle for cases (iii) and (iv) is shown in Figs. 6 and 7. Obviously, there is also a significant difference between the quantum and the classical pitch angle, however, it is smaller than for cases (i) and (ii). For the largest difference of about is found at . For case (iv) for all values of and the qantum pitch angle is larger than the classical one. On the other hand, for case (iii) the shift of the transition point leads to a change of the sign of , i.e. the antiferromagnetic inter-chain coupling yields are more subtil change of the GS correlations by quantum fluctuations. Moreover, due to the shift of we find a quite large difference for and .

Similarly as for cases (i) and (ii) we observe that for large the model with strongest quantum fluctuations, i.e. , approaches the limit most rapidly. We mention here that a large value of is realized e.g. in LiCuVO gibson ; ender ; drechs4 , for which has been estimated ender ; drechs4 . The variation of with for various is shown in Fig. 9. It can be seen that the variation of with is largest for small . Furthermore, in difference to cases (i) and (ii) decreases monotonously with for all values of .

(i): | (ii): | |||

0.70 | 0.426 | 1.364 | - | - |

0.80 | 0.409 | 1.317 | 0.433 | 1.341 |

0.90 | 0.397 | 1.283 | 0.418 | 1.294 |

1.00 | 0.386 | 1.256 | 0.405 | 1.258 |

(iii): | (iv): | |||

0.00 | 0.250 | 1.972 | 0.250 | 1.972 |

0.10 | 0.256 | 1.761 | 0.250 | 1.742 |

0.20 | 0.264 | 1.662 | 0.250 | 1.630 |

0.30 | 0.270 | 1.588 | 0.250 | 1.552 |

0.40 | 0.275 | 1.530 | 0.250 | 1.494 |

0.50 | 0.278 | 1.483 | 0.250 | 1.448 |

0.60 | 0.280 | 1.443 | 0.250 | 1.411 |

0.70 | 0.282 | 1.410 | 0.250 | 1.380 |

0.80 | 0.283 | 1.381 | 0.250 | 1.354 |

0.90 | 0.284 | 1.357 | 0.250 | 1.331 |

1.00 | 0.285 | 1.336 | 0.250 | 1.312 |

As mentioned in Sect. I, a classical relation like Eq. (1) between the exchange couplings and the pitch angle has been used to discuss the ratio in Refs. capogna2005, ; mazurenko2007, this way ignoring quantum effects. To overcome this problem we will provide an empirical formula that fits the continuous part of the given in Figs. 1, 2, 6, 7 very well. Having in mind that the shape of the continuous part of the resembles the classical behavior we find that is well approximated by (written now in dimensional exchange units for the convenience of application in real experimental situations):

(10) |

with the exponent as fitting parameter. Obviously, Eqs. (10) and (1) coincide for and . In Fig. 10 we show in dependence on for the four cases (i)-(iv). We find that is always larger than the classical value . In accordance with the above discussion, decreases with increasing , i.e. it goes towards the classical exponent . From the experimental point of view the edge-shared chain cuprates are of particular interest. The parameter situation of these compounds corresponds to case (iii). Hence, we give here simple fit formulas for that case which describe the behavior of as shown in Fig. 8 and the behavior of as shown in Fig. 9 by the red line. We find that

(11) |

with , , , , and provides a reasonable fit of our data for case (iii). In addition, we present numbers for and for various values of for all cases in table 1. The fitting formulas (11) as well as the data in table 1 can be used in combination with Eq. (10) to fix the ratio using the pitch angle as an input, e.g. from neutron scattering gibson ; matsuda ; ender ; capogna2005 .

Finally, we note that recently the pitch angle of LiZrCuO has been determined from Li-NMR-data to amount tarui . This value corresponds to a predicted ratio within the framework of the classical Eq. (1). This value is surprisingly very close to the ratio = 0.3 estimated from thermodynamic properties within the 1D-quantum spin-1/2 --model drechsQneu . However, from Fig. 9 for a realistic weak effective antiferromagnetic inter-chain coupling a somewhat larger pitch angle would be predicted, namely for = 0.3. Hence, other factors such as a sizable exchange anisotropy are expected to be relevant in this material. Such an anisotropy may lead to a modification of the classical spiral as well as to a reduction of quantum fluctuation compared with the isotropic spin model considered here. The corresponding effects are outside the scope of the present paper. They will be considered elsewhere.

## Iv Summary

Based on the coupled cluster method (CCM) we have studied the GS correlations of
a 2D array of frustrated spin- - chains coupled by an
inter-chain exchange interaction .
We have discussed
the influence of quantum fluctuations, frustration and inter-chain
coupling on the pitch angle
and the transition point between a GS with collinear commensurate
correlations and a GS with incommensurate spiral correlations.
Using the CCM within the so-called SUB2-3 approximation we obtain a closed
set of analytical equations which can be used to calculate the pitch angle
for an arbitrary set of exchange parameters.
We have found that for the pitch angle of the quantum
model significantly deviates
from the classical value and can be strongly influenced by the inter-chain
coupling
. Furthermore, we have observed that the quantum pitch angle
approaches its limiting value for increasing values of
much faster than
for the classical model. For several combinations
of the sign of and we have found that the
change of the pitch angle within the quantum model could be discontinuous while
the change of the pitch angle within the classical model is always
continuous.

Acknowledgment:
The authors thank J. Schulenburg and R. Kuzian
for helpful discussions. This work was supported by the DFG (project
RI 615/16-1, DR 269/3-1).

## Appendix A The CCM SUB2-3 approximation

Within the SUB2-3 approximation scheme the CCM correlation operator , see Eq. (5), contains five non-equivalent correlation coefficients , , corresponding to the lattice configurations shown in Fig. 11. These configurations (or lattice animals) represent the arrangement of spin operators acting on the lattice spins. To each configuration belongs a corresponding ket equation, see Eq. (6). The set of these ket equations can be found by a bit tedious but straightforward calculation. It reads

(12) | |||

(13) | |||

(14) | |||

(15) | |||

(16) |

The GS energy is a function of (some of) these correlation coefficients and of the pitch angle . It reads

(17) |

Note that in the above equations one has to set () and to replace by ( by ) for ferromagnetic (antiferromagnetic) . To determine the quantum pitch angle as a function of the parameters , and one has to solve the equation , cf. Sect. II, together with the set of ket equations (A) - (A) selfconsistently by standard numerics.

Finally, we will illustrate some limiting cases contained in Eqs. (A) - (17). For one has and the remaining two non-trivial Eqs. (A) and (A) then coincide with the corresponding equations given in Ref. bursill, . In case of , , but (or alternatively, , , but ) one finds the two ket equations for the simple unfrustrated linear chain.

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