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arXiv:1202.3214 · Physical Review B

Degeneracy and ordering of the noncoplanar phase of the classical bilinear-biquadratic Heisenberg model on the triangular lattice

S. E. Korshunov, Frédéric Mila, Karlo Penc

Published May 14, 2012• 5 citations
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Abstract

We investigate the zero-temperature behavior of the classical Heisenberg model on the triangular lattice in which the competition between exchange interactions of different orders favors a relative angle between neighboring spins $\ensuremath{\Phi}\ensuremath{\in}(0,2\ensuremath{\pi}/3)$. In this situation, the ground states are noncoplanar and have an infinite discrete degeneracy. In the generic case, i.e., when $\ensuremath{\Phi}\ensuremath{\ne}\ensuremath{\pi}/2,\mathrm{arccos}(\ensuremath{-}1/3)$, the ground-state manifold is in one-to-one correspondence (up to a global rotation) with the set of noncrossing loop coverings of the three equivalent honeycomb sublattices into which the bonds of the triangular lattice can be partitioned. This allows one to identify the order parameter space as an infinite Cayley tree with coordination number 3. Building on the duality between a similar loop model and the ferromagnetic $O(3)$ model on the honeycomb lattice, we argue that a typical ground state should have long-range order in terms of spin orientation. This conclusion is further supported by the comparison with the four-state antiferromagnetic Potts model [describing the $\ensuremath{\Phi}=\mathrm{arccos}(\ensuremath{-}1/3)$ case], which at zero temperature is critical and in terms of the solid-on-solid representation is located exactly at the point of roughening transition. At $\ensuremath{\Phi}\ensuremath{\ne}\mathrm{arccos}(\ensuremath{-}1/3)$, an additional constraint appears, whose presence drives the system into an ordered phase (unless $\ensuremath{\Phi}=\ensuremath{\pi}/2$, when another constraint is removed and the model becomes trivially exactly solvable).

DOI
10.1103/physrevb.85.174420

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We investigate the zero-temperature behavior of the classical Heisenberg model on the triangular lattice in which the competition between exchange interactions of different orders favors a relative angle between neighboring spins \ensuremathΦ\ensuremath∈(0,2\ensuremathπ/3)\ensuremath{\Phi}\ensuremath{\in}(0,2\ensuremath{\pi}/3)\ensuremathΦ\ensuremath∈(0,2\ensuremathπ/3). In this situation, the ground states are noncoplanar and have an infinite discrete degeneracy. In the generic case, i.e., when \ensuremathΦ\ensuremath≠\ensuremathπ/2,arccos(\ensuremath−1/3)\ensuremath{\Phi}\ensuremath{\ne}\ensuremath{\pi}/2,\mathrm{arccos}(\ensuremath{-}1/3)\ensuremathΦ\ensuremath=\ensuremathπ/2,arccos(\ensuremath−1/3), the ground-state manifold is in one-to-one correspondence (up to a global rotation) with the set of noncrossing loop coverings of the three equivalent honeycomb sublattices into which the bonds of the triangular lattice can be partitioned. This allows one to identify the order parameter space as an infinite Cayley tree with coordination number 3. Building on the duality between a similar loop model and the ferromagnetic O(3)O(3)O(3) model on the honeycomb lattice, we argue that a typical ground state should have long-range order in terms of spin orientation. This conclusion is further supported by the comparison with the four-state antiferromagnetic Potts model [describing the \ensuremathΦ=arccos(\ensuremath−1/3)\ensuremath{\Phi}=\mathrm{arccos}(\ensuremath{-}1/3)\ensuremathΦ=arccos(\ensuremath−1/3) case], which at zero temperature is critical and in terms of the solid-on-solid representation is located exactly at the point of roughening transition. At \ensuremathΦ\ensuremath≠arccos(\ensuremath−1/3)\ensuremath{\Phi}\ensuremath{\ne}\mathrm{arccos}(\ensuremath{-}1/3)\ensuremathΦ\ensuremath=arccos(\ensuremath−1/3), an additional constraint appears, whose presence drives the system into an ordered phase (unless \ensuremathΦ=\ensuremathπ/2\ensuremath{\Phi}=\ensuremath{\pi}/2\ensuremathΦ=\ensuremathπ/2, when another constraint is removed and the model becomes trivially exactly solvable).
Making use of infinite projected entangled pair states, we investigate the ground-state phase diagram of the nearest-neighbor spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. In agreement with previous studies, we find the ferromagnetic, ${120}^{\ensuremath{\circ}}$ magnetically ordered, ferroquadrupolar and antiferroquadrupolar phases, and confirm that all corresponding phase transitions are first order. Moreover, we provide an accurate estimate of the location of the ferroquadrupolar to ${120}^{\ensuremath{\circ}}$ magnetically ordered phase transition, thereby fully establishing the phase diagram. Also, we do not encounter any signs of the existence of a quantum paramagnetic phase. In particular, contrary to the equivalent square lattice model, we demonstrate that on the triangular lattice the one-dimensional Haldane phase does not reach all the way up to the two-dimensional limit. Finally, we investigate the possibility of an intermediate partially magnetic partially quadrupolar phase close to $\ensuremath{\theta}=\ensuremath{\pi}/2$, and we show that, also contrary to the square lattice case, this phase is not present on the triangular lattice.
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.
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.
Abstract Motivated by the fact that the study of disordered phases at zero temperature is of great interest, I study the spin-one quantum antiferromagnet with a next-nearest neighbor interaction on a triangular lattice with bilinear and biquadratic exchange interactions and a single ion anisotropy, using a SU(3) Schwinger boson mean-field theory. I calculate the critical properties, at zero temperature, for values of the single ion anisotropy parameter D above a critical value D C , where a quantum phase transition takes place from a higher D disordered phase to a lower D ordered phase.
We propose a non-disordered antiferromagnetic Heisenberg spin model on a triangular lattice, which exhibits a critical state and a glassy state at low temperatures. The system has three phases, the paramagnetic, the intermediate critical, and the glassy phase. The intermediate phase is characterized by a quasi-long-range order, whereas only short-range order exists in the glassy phase. Using Monte Carlo simulation, the phase transitions and properties of the two phases at low temperatures are examined. Our spin model shows that the variable-length spin and the competition between bilinear and biquadratic interactions are essential for forming the critical and glassy phases.
We study the ground-state (GS) properties of the frustrated spin-12\frac{1}{2}21​ J1{J}_{1}J1​--J2{J}_{2}J2​--J3{J}_{3}J3​ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (J2>0{J}_{2}>0J2​>0) and next-next-nearest-neighbor (J3>0{J}_{3}>0J3​>0) exchanges, for the case J3=J2{J}_{3}={J}_{2}J3​=J2​. We use the coupled-cluster method implemented to high orders of approximation, complemented by the Lanczos exact diagonalization of a large finite lattice with 32 sites, in order to calculate the GS energy, magnetic order parameter, and spin-spin correlation functions. In one scenario we find a quantum phase transition point between regions characterized by ferromagnetic order and a form of antiferromagnetic (``striped'') collinear order at ${J}_{2}^{c}\ensuremath{\approx}0.1095\ifmmode\pm\else\textpm\fi{}0.0005$, which is below the corresponding hypothetical transition point at J2cl=17{J}_{2}^{\mathrm{cl}}=\frac{1}{7}J2cl​=71​ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​. Hence we see that quantum fluctuations appear to stabilize somewhat the collinear antiferromagnetic order in preference to the ferromagnetic order in this model. We compare results for the present ferromagnetic case (with ${J}_{1}=\ensuremath{-}1$) to previous results for the corresponding antiferromagnetic case (with J1=+1{J}_{1}=+1J1​=+1). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter J2{J}_{2}J2​. However, there are considerable differences in the behavior of the order parameters for the two models for ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}0.6$. For example, the quasiclassical collinear magnetic long-range order for the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1) breaks down at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.60$, whereas the ``equivalent'' point for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) occurs at ${J}_{2}^{c}\ensuremath{\approx}0.11$. Unlike in the antiferromagnetic model (with J1=+1{J}_{1}=+1J1​=+1), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for J2c1<J2<J2c2{J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}J2c1​​<J2​<J2c2​​, with ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.47$, we find no clear indications at all in the ferromagnetic model for an intermediate magnetically disordered phase between the corresponding phases exhibiting ferromagnetic and striped order. Instead the evidence for the ferromagnetic model (with ${J}_{1}=\ensuremath{-}1$) points to one of two scenarios: either there is a direct first-order transition between the two magnetically ordered phases, as mentioned above; or there exists an intervening phase between them in the very narrow range $0.10\ensuremath{\lesssim}{J}_{2}\ensuremath{\lesssim}0.12$, which is probably a remnant of the spiral phase that exists in the classical counterpart of the model over the larger range 110<J2<15\frac{1}{10}<{J}_{2}<\frac{1}{5}101​<J2​<51​.