The many faces of quantum kagome materials:
Interplay of further-neighbour exchange and Dzyaloshinskii-Moriya interaction
The field of frustrated magnetism has been enriched significantly by the discovery of various kagome lattice compounds. These materials exhibit a great variety of macroscopic behaviours ranging from magnetic orders to quantum spin liquids. Using large-scale exact diagonalization, we construct the phase diagram of the - kagome Heisenberg model with -axis Dzyaloshinskii-Moriya interaction . We show that this model can systematically account for many of the experimentally observed phases. Small and can stabilize respectively a gapped and a gapless spin liquid. When or is substantial, the ground state develops a , 120°antiferromagnetic order. The critical strengths for inducing magnetic transition are at , and at . The previously reported values of and for herbertsmithite [ZnCu(OH)Cl] place the compound in close proximity to a quantum critical point.
pacs:75.10.Jm, 75.10.Kt, 75.30.Kz
In frustrated magnetism Ramirez (1994); Diep (2005); Lacroix et al. (2011), the kagome lattice has become the paradigmatic system of choice for studying novel spin-liquid phases that result from geometric frustration and quantum fluctuation Balents (2010). For example, the antiferromagnetic (AFM) kagome Heisenberg model with only the nearest-neighbour (NN) exchange is magnetically disordered with a spin correlation length less than one lattice spacing Zeng and Elser (1990); Sachdev (1992); Leung and Elser (1993); Nakamura and Miyashita (1995); Lecheminant et al. (1997); Jiang et al. (2008); Yan et al. (2011); Depenbrock et al. (2012). The ground state is characterized as a gapped topological spin liquid with a finite triplet gap Waldmann et al. (1998); Li et al. (2007); Singh and Huse (2008); Jiang et al. (2008); Läuchli et al. (2011); Yan et al. (2011); Depenbrock et al. (2012); Nishimoto et al. (2013) and a nonzero topological entanglement entropy Jiang et al. (2012); Depenbrock et al. (2012). Advances in computational methods and theoretical ideas have led to a deeper understanding and classification of these exotic states of matter, which do not break any symmetry and sustain emergent fractional excitations. However, a wide gulf still separates theory and experiments.
The celebrated herbertsmithite, ZnCu(OH)Cl Braithwaite et al. (2004); Shores et al. (2005), has nearly perfect kagome planes consisting of =1/2, Cu atoms. Despite a predominant K, this compound does not develop any long-range magnetic order down to mK Ofer et al. (); Mendels et al. (2007); Helton et al. (2007); Lee et al. (2007); Han et al. (2012a), agreeing with the NN AFM kagome Heisenberg model. But there are disagreements: Without finding any apparent gap down to meV, neutron scattering suggests gapless excitations Helton et al. (2007); Han et al. (2012a). Magnetic susceptibility shows an upturn at low temperature Ofer et al. (); Helton et al. (2007), which is also unexpected; a gapped spin liquid would otherwise show a vanishingly small close to . To account for these discrepancies, Dzyaloshinskii-Moriya (DM) interaction Rigol and Singh (2007a, b); Zorko et al. (2008); Cépas et al. (2008); Rousochatzakis et al. (2009), exchange anisotropy Ofer and Keren (2009); Han et al. (2012b), or quenched site dilution Dommange et al. (2003); Rigol and Singh (2007b); Rozenberg and Chitra (2008); Rousochatzakis et al. (2009) have been investigated for herbertsmithite. On the other hand, not all kagome compounds are spin liquids. In materials such as Cu(1,3-bdc) [Cu-benzenedicarboxylate] Nytko et al. (2008); Marcipar et al. (2009), vesignieite [BaCu(VO)(OH)] Colman et al. (2011); Quilliam et al. (2011); Yoshida et al. (2012, 2013), and CsCuSnF Ono et al. (2009); Matan et al. (2010); Katayama et al. (), the ground states is a , 120°AFM order, which highlights potential interactions beyond the NN exchange. Determining the importance of additional couplings is thereby key to a comprehensive understanding of the diversified properties in kagome materials Iqbal et al. ().
In this Letter, we study the effects of the -axis DM interaction and second NN exchange coupling , which are arguably the two most relevant perturbations for isotropic kagome compounds. For the first time, we construct the phase diagram of the -- model using large-scale exact diagonalization (ED) with cluster sizes up to sites. By also examining the excitation gaps and static structure factors, we show that the model can sustain various phases including long-range AFM order and quantum spin liquids with or without a finite spin gap. The distinct ground states among different kagome compounds can be systematically accounted for with varying interaction strengths of the systems. In particular, the reported values of and for herbertsmithite indicate a ground state closely proximal to a magnetic quantum critical point, where small extra perturbations suffice to suppress its long-range magnetism. Based on a numerically unbiased method, our study provides a direct road map for gauging and in materials with nearly prefect isotropic kagome structures.
Model and Method – We consider the - kagome Heisenberg model with DM interactions:
where the first two terms are respectively superexchange interactions between NN and second NN sites. The third DM-interaction term originates from relativistic spin-orbit coupling and is nonzero when lattice inversion symmetry is absent Dzyaloshinsky (1958); Moriya (1960). Here we focus on the -axis component of the DM interaction , using the convention that when all links are oriented clockwise [see inset of Fig. 1(a)] Nakano and Sakai (2011). We neglect the in-plane component , as it is reported to be smaller than in materials of interests and also reducible to second order in with a spin basis rotation Shekhtman et al. (1992); Cheng et al. (2007); Cépas et al. (2008). Throughout the paper we consider antiferromagnetic couplings ( ) and set .
We solve Eq. (The many faces of quantum kagome materials: Interplay of further-neighbour exchange and Dzyaloshinskii-Moriya interaction) systematically by numerical diagonalization on clusters of size up to . The Hamiltonian matrix is constructed utilizing translational symmetry and conservation Sandvik (2010). The resulting sparse matrix eigenvalue problem is solved by the Krylov-Schur algorithm as implemented in SLEPc Hernandez et al. (2005) and PETSc Balay et al. (1997) libraries. The cluster choices and calculation details are given in the Supplemental Material.
Phase Diagram – We first establish the phase diagram of the -- model. Without and , the system is magnetically disordered. A finite or could support a , 120°AFM ground state with spins lying in the -plane [see inset of Fig. 1(a)]. Therefore, we proceed to map out the phase boundary between the quantum AFM state and the magnetically disordered region by studying the transverse spin-spin correlation function Cépas et al. (2008):
Here, are unit-cell positions and are intra-unit-cell site indices. represents the elements of a matrix and peaks at . The largest matrix eigenvalue corresponds to the 120°AFM spin arrangements, and its classical value is equal to 1 with the pre-factor choice of Eq. (2) Cépas et al. (2008). Spontaneous spin symmetry breaking can be identified on finite-size clusters by a linear extrapolation Huse (1988); Neuberger and Ziman (1989); Sandvik (2010). When the extrapolated , long-range magnetic order develops.
Figure 1(a) shows the phase diagram obtained by a grid interpolation of points on the plane. The generic features are computed with cluster sizes , and further calculations are performed to more precisely locate the phase boundary. The blue region of Fig. 1(a) represents a magnetically disordered ground state (), and the red represents the , 120°AFM phase (). Figures 1(b) and 1(c) show linear extrapolations of close to the critical transition points along the - and -axes, respectively. The critical strengths are found to be at , and at . We note that Fig. 1(c) shows a more apparent finite-size effect with an even-/odd- alternation. With additional calculations in the zero momentum sector, the critical strengths when extrapolated independently are and at . is further reduced when is finite, and vice versa.
The - and - models have been separately investigated before. In particular, at was also reported by previous ED studies Cépas et al. (2008); Rousochatzakis et al. (2009). at has been computed by a number of methods Tay and Motrunich (2011); Suttner et al. (2014); Iqbal et al. (2015); Gong et al. (2015); Kolley et al. (2015), with the reported critical strength ranging from 0.2 to 0.7. Classically, a positive infinitesimal would favour a , 120°AFM long-range order () Harris et al. (1992); Spenke and Guertler (2012). Our finding of highlights the role of quantum fluctuation in destabilizing magnetism. Nonetheless, when becomes substantial, the quantum , 120°AFM ground state can be stabilized.
Materials Relevance – We next discuss the relevance of our phase diagram to different , Cu-based materials with (nearly) perfect isotropic kagome structures, as denoted by the geometric symbols in Fig. 1(a). In Cu(1,3-bdc), the material develops the , 120°AFM order with a critical transition temperature K Nytko et al. (2008); Marcipar et al. (2009). Its interaction strengths estimated by first-principles calculations Liu et al. () indeed correspond to a positive in our phase diagram. The compound vesignieite also develops the magnetic order at K Colman et al. (2011); Quilliam et al. (2011); Yoshida et al. (2012, 2013). Its experimentally estimated DM interaction renders a positive . Similarly, CsCuSnF is ordered at K Ono et al. (2009); Matan et al. (2010); Katayama et al. (), where the reported could be as large as , leading to . Interestingly, a higher in experiments seems to be correlated with a larger positive in our phase diagram.
Herbertsmithite, however, is not magnetically ordered. This could mean that its interactions are below the critical strengths . On the other hand, although is likely small compared to , is reported to be comparable to in this compound. For example, electron spin resonance experiment estimates a DM-interaction strength Zorko et al. (2008). Theoretical fit of indicates a Rigol and Singh (2007b). In fact, first-principles calculations find for both Cu(1,3-bdc) and herbertsmithite Liu et al. (), whereas the former is magnetically ordered but the latter is not. When the system resides in close proximity to the boundary of quantum phase transition, additional perturbations such as spin-space exchange anisotropy Ofer and Keren (2009); Han et al. (2012b); Kuroda and Miyashita (1995); Bekhechi and Southern (2003); Chernyshev and Zhitomirsky (2014); Götze and Richter (2015) or quenched dilution of Cu sites Dommange et al. (2003); Rigol and Singh (2007b); Rozenberg and Chitra (2008); Rousochatzakis et al. (2009) in herbertsmithite could more easily suppress its long-range magnetism.
More recently, a structurally perfect kagome compound —the barlowite [Cu(OH)FBr]— has been synthesized Han et al. (2014). This material develops long-range magnetic order at K with a weak ferromagnetic moment. In terms of our phase diagram, such a relatively high would imply a strong , which in conjunction with a small can lead to a , canted AFM state with a net ferromagnetic moment pointing outward from the kagome plane Elhajal et al. (2002); Jeschke et al. (). Future single-crystal measurements can further clarify the nature of barlowite’s low-temperature magnetic structure.
Gapped versus Gapless Spin Liquids – We next address the issue of spin gap by focusing on the excitation: . Here, is the lowest energy in a given sector, and corresponds to the triplet excitation when spin symmetry is present. We will consider only even- clusters based on the gap definition.
We first note that extrapolating the gap on finite-size clusters is more difficult. In the disordered region, the scaling form is a priori unknown. The result can depend largely on the extrapolation function Waldmann et al. (1998); Läuchli et al. (2011); Nakano and Sakai (2011), as well as the cluster size and shape Nishimoto et al. (2013). In the ordered phase, the energy splitting between quasi-degenerate ground states in quantum antiferromagnets would scale to zero as Anderson (1952); Bernu et al. (1992), and the linear-dispersing Goldstone modes for spontaneous broken symmetries would scale as Yildirim and Harris (2006); Messio et al. (2010); Chernyshev and Zhitomirsky (2014). But there may be no clear separation between these states in relatively small systems. Despite these difficulties, however, the variation of the gap with parameters would be robust and distinguishable in our numerically exact data. Therefore, instead of making precise quantitative statements, we would mainly focus on the trends.
Figure 2 shows the spin gap and the ground state energy per site on different size clusters. As shown in Fig. 2(a), is quickly enhanced by Sindzingre and Lhuillier (2009), but the rate of increase becomes smaller above and tends to level off with increasing . If a simple scaling is employed in the disordered region as in previous ED study Waldmann et al. (1998), would reach its maximum at and then decrease monotonically above it. This extrapolated gap behaviour agrees well with recent density matrix renormalization group calculations Kolley et al. (2015), implying that the gapped spin liquid phase is most stable around . At large , although the raw data of appear to grow with across the magnetic phase boundary , we note that the absolute value of the ground state energy is also increasing [Fig. 2(c)]. The ratio becomes nearly flat and decreases systematically with increasing in the ordered regime. The gap would eventually scale to zero in the thermodynamic limit, corresponding to spontaneous spin symmetry breaking of the ordered ground state.
The effect of on the spin gap is quite different. As shown in Fig. 2(b), is rapidly reduced in the presence of a small Mohakud et al. (2012). In fact, a simple extrapolation would indicate that is already zero in the disordered region before reaching the critical point . This shows the possibility of a gapless spin liquid ground state induced by spin exchange anisotropy Ryu et al. (2007); Hu et al. (). At 0.1, the ratio stays flat and again decreases systematically with increasing . The thermodynamic-limit remains gapless in the magnetic phase.
The above results show that small and could stabilize respectively a gapped and a gapless spin liquid. Based on the magnetically disordered ground state and gapless excitations found in herbertsmithite, our study suggests in this material a DM-interaction strength , closely proximal to the quantum critical point . With the prevalent observation of gapless excitations in putative spin-liquid phases, our study also implies that the DM interaction is in general non-negligible in isotropic kagome systems.
Static Structure Factor – To connect with neutron scattering experiments, we study the static structure factor:
where , . Neutron scattering also provides energy-resolved spectra by measuring the dynamic structure factor , where . A spin liquid phase would produce continuous or diffusive scattering spectra, whereas an ordered magnet would generate sharp, discrete Bragg peaks.
Figure 3 shows the transverse component computed on the cluster. When and are both zero [Fig. 3(a)], the spectrum is close to being uniformly distributed along the extended Brillouin zone (BZ); the first BZ contains little spectral weight Läuchli and Lhuillier (). This suggests that spin correlations are predominantly antiferromagnetic, while correlation lengths are on the order of lattice spacing. Due to spin symmetry at , the longitudinal component is identical to , and both components are zero at the point.
When , the , 120°AFM ground state manifests a structure factor that peaks at the midpoints of the extended BZ edges [Fig. 3(b)]. In this case, is much weaker than at , and spins mainly lie in the -plane. When the system is ordered, at large and at large are in general similar, except that (i) is no longer zero in the former, and (ii) is further suppressed in the latter. In addition, the overall spectra do not undergo a sharp transition across the critical point or . These features can be seen in Fig. 3(c) that shows high-symmetry line cuts of at various with .
In herbertsmithite, neutron scattering signals are diffused for all the measured energies between 0.25 to 11 meV (where meV) Han et al. (2012a). The spectral weight is concentrated in the extend BZ but does not peak at any specific point, although at 0.75 meV additional peak appears at the midpoints of the extended BZ edges. These results agree with our calculation for the magnetically disordered state [Fig. 3(a)]. The experimental spectra also contain a small but finite weight at the point, which could result from the DM interaction. We note, however, that the experimental intensity integrated up to 11 meV contains only of the total spectral weight. A more detailed theory-experiment comparison would require direct calculations of .
In conclusion, we have studied the interplay between further-neighbour exchange and Dzyaloshinskii-Moriya interaction on the kagome lattice. The phase diagram of the -- model is shown to contain various novel states of matter, including a = 0, 120°antiferromagnetic long-range order, as well as gapped and gapless quantum spin liquids. A small variation of the parameters near the phase transition boundary could potentially account for the distinct properties observed in different kagome materials. The phase diagram thereby serves as a benchmark for determining the importance of these additional perturbations. Studying the dynamical properties in different parts of the phase diagram and making further connection to inelastic neutron or x-ray scattering measurements would be important for future research.
Acknowledgements.The authors acknowledge discussions with Keun Hyuk Ahn, Hong-Chen Jiang, and Zhenyue Zhu. C.C.C. is supported by the Aneesur Rahman Postdoctoral Fellowship at Argonne National Laboratory, operated by the U.S. Department of Energy (DOE) Contract No. DE-AC02-06CH11357. R.R.P.S. is supported by the National Science Foundation Grant No. DMR-1306048. T.F.S. and M.v.V are supported by the U.S. DOE, Office of Basic Energy Sciences, under Award No. DE-FG02-03ER46097, and by the Institute for Nanoscience, Engineering and Technology at Northern Illinois University. This research used resources of the National Energy Research Scientific Computing Center, supported by the U.S. DOE under Contract No. DE-AC02-05CH11231.
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