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arXiv:1208.3989 · Physical Review B

Spin- 12 Heisenberg antiferromagnet on an anisotropic kagome lattice

P. H. Y. Li, R. F. Bishop, C. E. Campbell, DAMIAN J. J. FARNELL, Oliver Götze, Johannes Richter

Published Dec 4, 2012• 29 citations
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Abstract

We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-$\frac{1}{2}$ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants ${J}_{1}>0$ along two of the three lattice directions and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$ along the third. In the classical limit, the ground-state (GS) phase for $\ensuremath{\kappa}<\frac{1}{2}$ has collinear ferrimagnetic (N\'eel${}^{\ensuremath{'}}$) order where the ${J}_{2}$-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for $\ensuremath{\kappa}>\frac{1}{2}$ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-$\frac{1}{2}$ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter $\ensuremath{\kappa}$, namely, for $0<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}$ for the N\'eel${}^{\ensuremath{'}}$ state and for (at least part of) the region $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}$ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$, which includes the isotropic kagome point $\ensuremath{\kappa}=1$ where the stable GS phase is now believed to be a topological (${\mathbb{Z}}_{2}$) spin liquid. Our best numerical estimates are ${\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015$ and ${\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03$.

DOI
10.1103/physrevb.86.214403

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We use the coupled-cluster method to study the zero-temperature properties of an extended two-dimensional Heisenberg antiferromagnet formed from spin-12\frac{1}{2}21​ moments on an infinite spatially anisotropic kagome lattice of corner-sharing isosceles triangles, with nearest-neighbor bonds only. The bonds have exchange constants J1>0{J}_{1}>0J1​>0 along two of the three lattice directions and J2\ensuremath≡\ensuremathκJ1>0{J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0J2​\ensuremath≡\ensuremathκJ1​>0 along the third. In the classical limit, the ground-state (GS) phase for \ensuremathκ<12\ensuremath{\kappa}<\frac{1}{2}\ensuremathκ<21​ has collinear ferrimagnetic (N'eel\ensuremath′{}^{\ensuremath{'}}\ensuremath′) order where the J2{J}_{2}J2​-coupled chain spins are ferromagnetically ordered in one direction with the remaining spins aligned in the opposite direction, while for \ensuremathκ>12\ensuremath{\kappa}>\frac{1}{2}\ensuremathκ>21​ there exists an infinite GS family of canted ferrimagnetic spin states, which are energetically degenerate. For the spin-12\frac{1}{2}21​ case, we find that quantum analogs of both these classical states continue to exist as stable GS phases in some regions of the anisotropy parameter \ensuremathκ\ensuremath{\kappa}\ensuremathκ, namely, for 0<\ensuremathκ<\ensuremathκc10<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}0<\ensuremathκ<\ensuremathκc1​​ for the N'eel\ensuremath′{}^{\ensuremath{'}}\ensuremath′ state and for (at least part of) the region \ensuremathκ>\ensuremathκc2\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}\ensuremathκ>\ensuremathκc2​​ for the canted phase. However, they are now separated by a paramagnetic phase without either sort of magnetic order in the region \ensuremathκc1<\ensuremathκ<\ensuremathκc2{\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}\ensuremathκc1​​<\ensuremathκ<\ensuremathκc2​​, which includes the isotropic kagome point \ensuremathκ=1\ensuremath{\kappa}=1\ensuremathκ=1 where the stable GS phase is now believed to be a topological (Z2{\mathbb{Z}}_{2}Z2​) spin liquid. Our best numerical estimates are \ensuremathκc1=0.515\ifmmode±\else\textpm\fi0.015{\ensuremath{\kappa}}_{{c}_{1}}=0.515\ifmmode\pm\else\textpm\fi{}0.015\ensuremathκc1​​=0.515\ifmmode±\else\textpm\fi0.015 and \ensuremathκc2=1.82\ifmmode±\else\textpm\fi0.03{\ensuremath{\kappa}}_{{c}_{2}}=1.82\ifmmode\pm\else\textpm\fi{}0.03\ensuremathκc2​​=1.82\ifmmode±\else\textpm\fi0.03.
We use the coupled cluster method (CCM) to study the zero-temperature ground-state (GS) properties of a spin-12\frac{1}{2}21​ ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg antiferromagnet on a triangular lattice with competing nearest-neighbor and next-nearest-neighbor exchange couplings J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, in the window $0\ensuremath{\le}\ensuremath{\kappa}<1$. The classical version of the model has a single GS phase transition at ${\ensuremath{\kappa}}^{\mathrm{cl}}=\frac{1}{8}$ in this window from a phase with 3-sublattice antiferromagnetic (AFM) ${120}^{\ensuremath{\circ}}$ N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}^{\mathrm{cl}}$ to an infinitely degenerate family of 4-sublattice AFM N\'eel phases for $\ensuremath{\kappa}>{\ensuremath{\kappa}}^{\mathrm{cl}}$. This classical accidental degeneracy is lifted by quantum fluctuations, which favor a 2-sublattice AFM striped phase. For the quantum model we work directly in the thermodynamic limit of an infinite number of spins, with no consequent need for any finite-size scaling analysis of our results. We perform high-order CCM calculations within a well-controlled hierarchy of approximations, which we show how to extrapolate to the exact limit. In this way we find results for the case $\ensuremath{\kappa}=0$ of the spin-12\frac{1}{2}21​ model for the GS energy per spin, $E/N=\ensuremath{-}0.5521(2){J}_{1}$, and the GS magnetic order parameter, M=0.198(5)M=0.198(5)M=0.198(5) (in units where the classical value is Mcl=12{M}^{\mathrm{cl}}=\frac{1}{2}Mcl=21​), which are among the best available. For the spin-12\frac{1}{2}21​ ${J}_{1}\ensuremath{-}{J}_{2}$ model we find that the classical transition at $\ensuremath{\kappa}={\ensuremath{\kappa}}^{\mathrm{cl}}$ is split into two quantum phase transitions at ${\ensuremath{\kappa}}_{1}^{c}=0.060(10)$ and ${\ensuremath{\kappa}}_{2}^{c}=0.165(5)$. The two quasiclassical AFM states (viz., the ${120}^{\ensuremath{\circ}}$ N\'eel state and the striped state) are found to be the stable GS phases in the regime $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{1}^{c}$ and $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{2}^{c}$, respectively, while in the intermediate regimes ${\ensuremath{\kappa}}_{1}^{c}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{2}^{c}$ the stable GS phase has no evident long-range magnetic order.
We study the properties of the Heisenberg antiferromagnet with spatially anisotropic nearest-neighbour exchange couplings on the kagome net, i.e. with coupling J in one lattice direction and couplings J' along the other two directions. For J/J' > 1, this model is believed to describe the magnetic properties of the mineral volborthite. In the classical limit, it exhibits two kinds of ground states: a ferrimagnetic state for J/J' 1/2. To include quantum effects self-consistently, we investigate the Sp(N) symmetric generalisation of the original SU(2) symmetric model in the large-N limit. In addition to the dependence on the anisotropy, the Sp(N) symmetric model depends on a parameter kappa that measures the importance of quantum effects. Our numerical calculations reveal that in the kappa-J/J' plane, the system shows a rich phase diagram containing a ferrimagnetic phase, an incommensurate phase, and a decoupled chain phase, the latter two with short- and long-range order. We corroborate these results by showing that the boundaries between the various phases and several other features of the Sp(N) phase diagram can be determined by analytical calculations. Finally, the application of a block-spin perturbation expansion to the trimerised version of the original spin-1/2 model leads us to suggest that in the limit of strong anisotropy, J/J' >> 1, the ground state of the original model is a collinearly ordered antiferromagnet, which is separated from the incommensurate state by a quantum phase transition.
Using exact numerical diagonalizations as well as the semiclassical spin-wave approach, we study the quantum phase diagram of the spin-12 Heisenberg kagome strip, which is a particular cut-out from the kagome lattice. The unit cell contains five spins placed on the central cite and on the four end sites building two spin-12 along the strip. The model can be characterized by three different constants for the nearest-neighbor exchange bonds between (i) the central and chain spins (J1), (ii) the end spins in a cluster (J2 > 0), and (ii) the spins of neighboring clusters (J′2>0) In the present work we extend the previous studies of the model, which were restricted to antiferromagnetic J1 bonds (J1 > 0), by including the region with ferromagnetic J1 couplings (J1<0). It is demonstrated that for dominating ferromagnetic J1 couplings there is another gapless phase, which can be related to the critical ground state of the spin-52 Heisenberg chain. The phase boundaries separating the discussed critical phases, as well as the behavior of their low-lying excitations close to these boundaries, are examined as well.
Using the coupled cluster method (CCM) we study the full (zero-temperature) ground-state (GS) phase diagram of a spin-half (s=12s=\frac{1}{2}s=21​) J1{J}_{1}J1​-J2{J}_{2}J2​ Heisenberg model on a cross-striped square lattice. Each site of the square lattice has four nearest-neighbor exchange bonds of strength J1{J}_{1}J1​ and two next-nearest-neighbor (diagonal) bonds of strength J2{J}_{2}J2​. The J2{J}_{2}J2​ bonds are arranged so that the basic square plaquettes in alternating columns have either both or no J2{J}_{2}J2​ bonds included. The classical ($s\ensuremath{\rightarrow}\ensuremath{\infty}$) version of the model has four collinear phases when J1{J}_{1}J1​ and J2{J}_{2}J2​ can take either sign. Three phases are antiferromagnetic (AFM), showing so-called N\'eel, double N\'eel, and double columnar striped order, respectively, while the fourth is ferromagnetic. For the quantum s=12s=\frac{1}{2}s=21​ model we use the three classical AFM phases as CCM reference states, on top of which the multispin-flip configurations arising from quantum fluctuations are incorporated in a systematic truncation hierarchy. Calculations of the corresponding GS energy, magnetic order parameter, and the susceptibilities of the states to various forms of valence-bond crystalline (VBC) order are thus carried out numerically to high orders of approximation and then extrapolated to the (exact) physical limit. We find that the s=12s=\frac{1}{2}s=21​ model has five phases, which correspond to the four classical phases plus a new quantum phase with plaquette VBC order. The positions of the five quantum critical points are determined with high accuracy. While all four phase transitions in the classical model are first order, we find strong evidence that three of the five quantum phase transitions in the s=12s=\frac{1}{2}s=21​ model are of continuous deconfined type.
We use the coupled cluster method (CCM) to study the zero-temperature phase diagram of a two-dimensional frustrated spin-half antiferromagnet, the so-called Union Jack model. It is defined on a square lattice such that all nearest-neighbor pairs are connected by bonds with a strength J1>0{J}_{1}>0J1​>0, but only half the next-nearest-neighbor pairs are connected by bonds with a strength ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$. The bonds are arranged such that on the $2\ifmmode\times\else\texttimes\fi{}2$ unit cell they form the pattern of the Union Jack flag. Alternating sites on the square lattice are thus four-connected and eight-connected. We find strong evidence for a first phase transition between a N\'eel antiferromagnetic phase and a canted ferrimagnetic phase at a critical coupling ${\ensuremath{\kappa}}_{{c}_{1}}=0.66\ifmmode\pm\else\textpm\fi{}0.02$. The transition is an interesting one, at which the energy and its first derivative seem continuous, thus providing a typical scenario of a second-order transition (just as in the classical case for the model), although a weakly first-order transition cannot be excluded. By contrast, the average on-site magnetization approaches a nonzero value ${M}_{{c}_{1}}=0.195\ifmmode\pm\else\textpm\fi{}0.005$ on both sides of the transition, which is more typical of a first-order transition. The slope, $dM/d\ensuremath{\kappa}$, of the order parameter curve as a function of the coupling strength $\ensuremath{\kappa}$, also appears to be continuous, or very nearly so, at the critical point ${\ensuremath{\kappa}}_{{c}_{1}}$, thereby providing further evidence of the subtle nature of the transition between the N\'eel and canted phases. Our CCM calculations provide strong evidence that the canted ferrimagnetic phase becomes unstable at large values of $\ensuremath{\kappa}$, and hence we have also used the CCM with a model collinear semistripe-ordered ferrimagnetic state in which alternating rows (and columns) are ferromagnetically and antiferromagnetically ordered, and in which the spins connected by J2{J}_{2}J2​ bonds are antiparallel to one another. We find tentative evidence, based on the relative energies of the two states, for a second zero-temperature phase transition between the canted and semistripe-ordered ferrimagnetic states at a large value of the coupling parameter around ${\ensuremath{\kappa}}_{{c}_{2}}\ensuremath{\approx}125\ifmmode\pm\else\textpm\fi{}5$. This prediction, however, is based on an extrapolation of the CCM results for the canted state into regimes where the solutions have already become unstable and the CCM equations based on the canted state at any level of approximation beyond the lowest have no solutions. Our prediction for ${\ensuremath{\kappa}}_{{c}_{2}}$ is hence less reliable than that for ${\ensuremath{\kappa}}_{{c}_{1}}$. Nevertheless, if this second transition at ${\ensuremath{\kappa}}_{{c}_{2}}$ does exist, our results clearly indicate it to be of first-order type.
We study the zero-temperature quantum phase diagram of a spin-1 Heisenberg antiferromagnet on the honeycomb lattice with both nearest-neighbor exchange coupling J1>0{J}_{1}>0J1​>0 and frustrating next-nearest-neighbor coupling ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, using the coupled cluster method implemented to high orders of approximation, and based on model states with different forms of classical magnetic order. For each we calculate directly in the bulk thermodynamic limit both ground-state low-energy parameters (including the energy per spin, magnetic order parameter, spin stiffness coefficient, and zero-field uniform transverse magnetic susceptibility) and their generalized susceptibilities to various forms of valence-bond crystalline (VBC) order, as well as the energy gap to the lowest-lying spin-triplet excitation. In the range $0<\ensuremath{\kappa}<1$ we find evidence for four distinct phases. Two of these are quasiclassical phases with antiferromagnetic long-range order, one with two-sublattice N\'eel order for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{1}}=0.250(5)$, and another with four-sublattice N\'eel-II order for $\ensuremath{\kappa}>{\ensuremath{\kappa}}_{{c}_{2}}=0.340(5)$. Two different paramagnetic phases are found to exist in the intermediate region. Over the range ${\ensuremath{\kappa}}_{{c}_{1}}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}^{i}=0.305(5)$ we find a gapless phase with no discernible magnetic order, which is a strong candidate for being a quantum spin liquid, while over the range ${\ensuremath{\kappa}}_{c}^{i}<\ensuremath{\kappa}<{\ensuremath{\kappa}}_{{c}_{2}}$ we find a gapped phase, which is most likely a lattice nematic with staggered dimer VBC order that breaks the lattice rotational symmetry.