Spinons and helimagnons in the frustrated Heisenberg chain
We investigate the dynamical spin structure factor for the Heisenberg chain with ferromagnetic nearest () and antiferromagnetic next-nearest () neighbor exchange using bosonization and a time-dependent density-matrix renormalization group algorithm. For and low energies we analytically find and numerically confirm two spinon branches with different velocities and different spectral weights. Following the evolution of with decreasing we find that helimagnons develop at high energies just before entering the ferromagnetic phase. Furthermore, we show that a recent interpretation of neutron scattering data for LiCuVO in terms of two weakly coupled antiferromagnetic chains () is not viable. We demonstrate that the data are instead fully consistent with a dominant ferromagnetic coupling, .
Frustrated spin models often show complicated phase diagrams. Apart from phases with conventional (quasi) long-range magnetic order, also valence bond solid, chirally ordered, multipolar as well as spin liquid phases are possible. In addition to identifying and studying the different phases, also the quantum phase transitions between them have attracted considerable interest Senthil et al. (2004); Sirker et al. (2011). Numerical studies are hindered by the so-called sign problem making quantum Monte-Carlo simulations impracticable. Our knowledge about frustrated systems in dimensions is therefore still limited Yan et al. (2011); Sirker et al. (2006); Jiang et al. (2011). In one dimension, on the other hand, bosonization techniques can be used to analytically study the weakly frustrated case White and Affleck (1996); Nersesyan et al. (1998). Furthermore, with the density-matrix renormalization group (DMRG) White (1992) a powerful numerical method is available to calculate static properties at zero White and Affleck (1996) and finite temperatures Sirker (2010). Lately, it has been shown that it is a useful strategy to study frustration in two or three dimensions by a successive coupling of one-dimensional chains Kohno et al. (2007); Yan et al. (2011) thus further motivating investigations of the static and dynamic properties of frustrated spin chains.
The standard spin model to study frustration in one dimension is the Heisenberg model
with antiferromagnetic coupling . Here is a spin- operator acting at site . This model has recently been studied intensely for ferromagnetic coupling Bursill et al. (1995); Hikihara et al. (2008); Sudan et al. (2009); Vekua et al. (2007); Heidrich-Meisner et al. (2006), driven by the discovery of edge-sharing cuprate chain compounds for which Eq. (1) seems to be the minimal model. These cuprates show fascinating properties including multiferroicity Masuda et al. (2004); Park et al. (2007); Seki et al. (2008); Drechsler et al. (2007); Schrettle et al. (2008), i.e., an intricate interplay between incommensurate magnetic and ferroelectric order Katsura et al. (2005). It is, however, so far unclear if all of their magnetic properties can be reasonably well described within the simple minimal model (1). In one of the best studied edge-sharing chain cuprates LiCuVO, for example, susceptibility data seem to point to a dominant ferromagnetic coupling Büttgen et al. (2007); Sirker (2010) while neutron scattering data have been interpreted in terms of two weakly coupled antiferromagnetic chains Enderle et al. (2010), a conclusion which has later been challenged Drechsler et al. (2011); Nishimoto et al. (2011); Enderle et al. (2011).
Apart from being relevant for the edge-sharing cuprate chains, the dynamical properties of the chain are also of fundamental interest. For the model consists of two decoupled antiferromagnetic chains whose elementary gapless excitations are spinons. Introducing a small coupling between the chains leads classically to a spiral magnetic order while bosonization predicts incommensurate spin correlations and an exponentially small gap in the quantum case Nersesyan et al. (1998). At there is an unusual quantum critical point Sirker et al. (2011) separating the incommensurate from a ferromagnetic phase. Changing the frustration ratio thus turns antiferromagnetic spinons through an incommensurate phase into ferromagnetic magnons.
In this letter we present a systematic study of the dynamical spin structure factor
of the - Heisenberg model (1) from the limit of decoupled antiferromagnetic Heisenberg chains, across the quantum critical point , into the ferromagnetic phase, . Note that due to symmetry it is sufficient to consider the longitudinal correlation function in (2). For we compare our data with results obtained by bosonization while for we compare with spin wave theory. Finally, we present a comparison of our results with recent neutron scattering data for the multiferroic cuprate LiCuVO Enderle et al. (2010).
We start by considering the weak coupling limit by bosonization. On each of the two antiferromagnetically coupled sublattices we write
Here are bosonic fields obeying the standard commutation rules and are the Luttinger parameters. Let us first consider the free fermion case with . Ignoring irrelevant terms, bosonization leads to
for each of the chains . In this case and with in units of the lattice constant due to a doubling of the unit cell. Apart from irrelevant terms, the interchain coupling introduces a density-density type interaction
We can absorb this term into the Gaussian part (4) by defining the new fields and . The Hamiltonian (4) stays invariant under this transformation with being replaced by but the velocities are renormalized with .
In the isotropic Heisenberg case for the low-energy properties are still described by (4) but with and as known from Bethe ansatz. To keep the symmetry intact, we now use non-Abelian instead of the Abelian bosonization (Spinons and helimagnons in the frustrated Heisenberg chain) for the interchain interaction . This leads to various marginal terms Nersesyan et al. (1998); Allen and Sénéchal (1997) and, in particular, a term which can again be expressed as in (5) but with a different prefactor leading to a velocity renormalization
The other marginal terms induced by the coupling have been studied by renormalization group methods in Nersesyan et al. (1998) and seem to lead to an exponentially small gap. Numerically, however, no gap in the parameter regime has been confirmed yet and we will therefore neglect these terms for the moment.
At weak coupling , the structure factor will have low-energy contributions at . We now calculate these contributions in the isotropic case using the free boson Hamiltonian (4) with and velocities given by Eq. (6). Using the identity (Spinons and helimagnons in the frustrated Heisenberg chain)—rewritten in terms of —and the standard result for the boson propagator we obtain
The singularity at dominates while the singularity at is dominant for . The other contribution is in both cases surpressed by a factor . Note that because . Irrelevant band curvature and interaction terms will lead to a finite linewidth Pereira et al. (2006, 2007), an aspect which is beyond the scope of the present study. For and sufficiently different we find
Here are the Luttinger parameters for the two modes at weak coupling . Note that the divergencies in (Spinons and helimagnons in the frustrated Heisenberg chain) are weaker than the square-root divergence for the pure isotropic Heisenberg chain Schulz (1986). Taking the additional marginal current-current interactions into account perturbatively, we find furthermore an asymmetry with small.
To test the analytic predictions in the weak coupling limit and extend our study to the experimentally relevant regime of strong interchain coupling, we now turn to a numerical calculation of the dynamical spin structure factor. We use an adaptive time-dependent density-matrix renormalization group (DMRG) algorithm with a second order Trotter-Suzuki decomposition of the time evolution operator Feiguin and White (2005). As time step we choose . We present results for an open chain with sites with states kept in the adaptive Hilbert space. In order to perform the Fourier transform in time in Eq. (2) one has to deal with the problem that numerical data are only available for a finite time interval . The maximal simulation time, , up to which our numerical results are reliable has been estimated by keeping track of the discarded weight and by comparing with exact results for the model and Bethe ansatz results for the isotropic Heisenberg chain Caux et al. (2005). We calculate the spin correlations . Since two-point spin correlations are negligible for distances much larger than , we obtain results almost unaffected by the boundaries in the accessible time interval. These data are then extended in time using linear prediction G. U. Yule (1927); Barthel et al. (2009) leading to a smooth exponentially decaying extrapolation of the data. We want to stress that linear prediction does not allow to obtain reliable results for but rather represents a smooth cutoff which does not affect the data for .
In Fig. 1 we show our results for for various frustrations. A further check of the quality of the numerical data is obtained by considering the sum rules where and . For all frustrations shown we find that the sum rules are fulfilled with an absolute error of at most . As an example, we show results for at in Fig. 2(a).
For weak frustrations (Fig. 1(a,b)) two excitations with different velocities are clearly visible near . As expected from bosonization, the smaller velocity agrees with that of the dominant excitation at while the one at has velocity . The velocities extracted from the numerical data also agree fairly well with the prediction from bosonization, see Fig. 2(b).
A closer inspection of the data for also reveals the second mode with velocity at with a weight suppressed by approximately relative to the dominant mode, see inset of Fig. 1(b).
At larger frustrations the weak coupling picture from bosonization clearly breaks down. For shown in Fig. 1(c) and (d) the low energy spectral weight is concentrated at an incommensurate wave vector . Classically, frustration leads to the formation of a helical state with a pitch vector . For the quantum model, incommensurate spin correlations have been shown to occur with wave vectors which approach with increasing much faster than in the classical case Bursill et al. (1995); Sirker (2010) in full agreement with our dynamical data. For we observe—in addition to the low-energy spectral weight near —the development of three magnon-like dispersions at higher energies. For the classically expected state with long-range spiral order at wave vector , spin-waves have the dispersion
with , where Nagamiya (1967); Zhitomirsky and Zaliznyak (1996). In the quantum model, long-range order is destroyed. However, close to the quantum critical point where the model starts to develop a long-ranged ordered ferromagnetic state, the correlation length will become large so that the excitations at high energy remain helimagnon-like. As shown in Fig. 1(d) the helimagnon dispersion (9) does indeed describe the high-energy modes very well with a renormalized effective corresponding to a pitch vector . The three modes Nagamiya (1967) are then given by and where is the incommensurate wave vector of the quantum model.
For the ground state is a simple ferromagnet, , and (9) reduces to the magnon dispersion which is in excellent agreement with the numerical data as shown in Fig. 1(f). At the quantum critical point, , the magnon dispersion becomes quartic, at small . As already noticed in Sirker et al. (2011) spin-wave theory does not describe the quantum critical point correctly at low energies due to the degeneracy of the ferromagnetic with valence bond solid states Hamada et al. (1988). This is confirmed by our data (see inset of Fig. 1(e)) showing that the low-energy spectral weight does not follow the magnon dispersion while the agreement is good at higher energies.
The experimental data for LiCuVO have been interpreted in terms of the - Heisenberg model with . However, none of the features predicted by bosonization in this limit and visible in Fig. 1(a,b) have been observed. The neutron data in Fig. 2 of Enderle et al. (2010) look instead remarkably similar to our numerical results for . Comparing to the numerically calculated structure factor one has to keep in mind that the neutron scattering data are slightly supressed at higher momenta due to the -dependence of the atomic form factor for Cu ions and the whole spectrum is broadened due to a finite , resolution. However, this does not affect the qualitative features of the spectrum. In fact, we can even obtain a quantitatively satisfying description of the data, see Fig. 3. For small momenta the agreement becomes excellent when convoluting the numerical data with a Gaussian to take the finite instrumental resolution into account. A frustration implies an incommensurate wave vector , see Fig. 2(a), consistent with the neutron data and susceptibility measurements Büttgen et al. (2007); Sirker (2010). For such a strong frustration the explanation of the spectral weight at high energies in terms of multispinon excitations of the antiferromagnetic chains offered in Ref. Enderle et al. (2010) is not viable. Instead, this weight is related to the incommensurate spin correlations in this material caused by the dominant ferromagnetic coupling between the chains. Let us also briefly comment on very recent neutron scattering results on powder samples of LiCuSbO Dutton et al. (2011). The powder averaging prevents a detailed analysis, however, the concentration of spectral weight at the incommensurate wave vector points again to a strong frustration which is fully consistent with an analysis of susceptibility and specific heat data Dutton et al. (2011).
Acknowledgements.The authors thank J.-S. Caux for sending us his data for the Heisenberg chain. J.S. acknowledges support by the DFG via the SFB/TR 49 and by the graduate school of excellence MAINZ and J. R. by the National Natural Science Foundation of China (NO.11104021).
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