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Physical Review B

Quantum phase transition in the three-dimensional anisotropic frustrated Heisenberg antiferromagnetic model

J. Roberto Viana, J. Ricardo de Sousa, M. A. Contínentino

Published May 27, 2008• 24 citations
Publisher

Abstract

The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest $({J}_{1})$- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state $(T=0)$ and the finite-$T$ phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.

DOI
10.1103/physrevb.77.172412

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The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest (J2=\ensuremathαJ1)({J}_{2}=\ensuremath{\alpha}{J}_{1})(J2​=\ensuremathαJ1​)-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point \ensuremathαc(\ensuremathΔ){\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})\ensuremathαc​(\ensuremathΔ) between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ (\ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=0 and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of \ensuremathαc(\ensuremathΔ\ensuremath≠1)=\ensuremathαcq(\ensuremathΔ){\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})\ensuremathαc​(\ensuremathΔ\ensuremath=1)=\ensuremathαcq​(\ensuremathΔ) are smaller than those of the classical model \ensuremathαc(\ensuremathΔ=1)=\ensuremathαcc=1/4{\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4\ensuremathαc​(\ensuremathΔ=1)=\ensuremathαcc​=1/4 (for example, for \ensuremathΔ=0\ensuremath{\Delta}=0\ensuremathΔ=0, we find \ensuremathαcq(0)=0.21{\ensuremath{\alpha}}_{c}^{q}(0)=0.21\ensuremathαcq​(0)=0.21). The former monotonically increases with the anisotropy parameter \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ. The phase diagram in T\ensuremath-\ensuremathαT\text{\ensuremath{-}}\ensuremath{\alpha}T\ensuremath-\ensuremathα plane presents a second-order transition at T2(\ensuremathα){T}_{2}(\ensuremath{\alpha})T2​(\ensuremathα) between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order T1(\ensuremathα){T}_{1}(\ensuremath{\alpha})T1​(\ensuremathα) for all values of \ensuremathΔ∊[0,1]\ensuremath{\Delta}∊[0,1]\ensuremathΔ∊[0,1]. The phase transition lines T1(\ensuremathα){T}_{1}(\ensuremath{\alpha})T1​(\ensuremathα) and T2(\ensuremathα){T}_{2}(\ensuremath{\alpha})T2​(\ensuremathα) merge in a critical end point (CEP) that is dependent on \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ [i.e., TCEP(\ensuremathΔ){T}_{\text{CEP}}(\ensuremath{\Delta})TCEP​(\ensuremathΔ) and \ensuremathαCEP(\ensuremathΔ){\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})\ensuremathαCEP​(\ensuremathΔ) are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
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​.
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We present the zero-temperature phase diagram of a Heisenberg antiferromagnet on a frustrated triangular lattice with nearest-neighbor (J1)({J}_{1})(J1​) and next-nearest-neighbor (J2)({J}_{2})(J2​) interactions, in a magnetic field. We show that the classical model has an accidental degeneracy for all J2/J1{J}_{2}/{J}_{1}J2​/J1​ and all fields, but the degeneracy is lifted by quantum fluctuations. We show that at large spin SSS, for J2/J1<1/8{J}_{2}/{J}_{1}<1/8J2​/J1​<1/8, quantum fluctuations select the same sequence of three sublattice co-planar states in a field as for J2=0{J}_{2}=0J2​=0, and for 1/8<J2/J1<11/8<{J}_{2}/{J}_{1}<11/8<J2​/J1​<1, they select the canted stripe state for all nonzero fields. The transition between the two states is first order in all fields, with the hysteresis width set by quantum fluctuations. We study the model with arbitrary SSS, including S=1/2S=1/2S=1/2, near the saturation field by exploring the fact that near saturation the density of bosons is small for all SSS. We show that for S>1S>1S>1, the transition remains first order, with a finite hysteresis width, but for S=1/2S=1/2S=1/2 and, possibly, S=1S=1S=1, there appears a new intermediate phase without a quasiclassical long-range order.
We study a frustrated spin-$\frac{1}{2}\phantom{\rule{0.16em}{0ex}}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ Heisenberg antiferromagnet on an AAAAAA-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings J1,J2{J}_{1},\phantom{\rule{0.16em}{0ex}}{J}_{2}J1​,J2​, and J3{J}_{3}J3​, respectively. The two layers are coupled with an AFM NN exchange coupling ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}$. The model is studied for arbitrary values of $\ensuremath{\delta}$ along the line ${J}_{3}={J}_{2}\ensuremath{\equiv}\ensuremath{\alpha}{J}_{1}$ that includes the most highly frustrated point at $\ensuremath{\alpha}=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\ensuremath{\alpha}\ensuremath{\delta}$ plane in the window $0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1,\phantom{\rule{0.16em}{0ex}}0\ensuremath{\le}\ensuremath{\delta}\ensuremath{\le}1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\ensuremath{\delta}={\ensuremath{\delta}}_{{c}_{i}}(\ensuremath{\alpha})$, or equivalently, $\ensuremath{\alpha}={\ensuremath{\alpha}}_{{c}_{i}}(\ensuremath{\delta})$, with i=1i=1i=1 (N\'eel) and 2 (striped). The two boundaries exhibit an ``avoided crossing'' behavior with both curves being re-entrant. Thus, in this $\ensuremath{\alpha}\ensuremath{\delta}$ window, N\'eel order exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{1}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{{c}_{1}}(0)\ensuremath{\approx}0.46(2)$ and ${\ensuremath{\delta}}_{{c}_{1}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{1}}(0)<\ensuremath{\alpha}<{\ensuremath{\alpha}}_{1}^{>}\ensuremath{\approx}0.49(1)$, and striped order similarly exists only for values of $\ensuremath{\delta}$ in the range ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})<\ensuremath{\delta}<{\ensuremath{\delta}}_{{c}_{2}}^{>}(\ensuremath{\alpha})$, with ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})=0$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{{c}_{2}}(0)\ensuremath{\approx}0.600(5)$ and ${\ensuremath{\delta}}_{{c}_{2}}^{<}(\ensuremath{\alpha})>0$ for ${\ensuremath{\alpha}}_{{c}_{2}}(0)>\ensuremath{\alpha}>{\ensuremath{\alpha}}_{2}^{<}\ensuremath{\approx}0.56(1)$.
In this work the ordering processes in classical frustrated Ising models and the ground-state behavior of frustrated quantum spin models are investigated. In the first part the phase transition from a high-temperature paramagnetic phase to a antiferromagnetic ground state is analyzed for the square-lattice Ising model with competing antiferromagnetic interactions between nearest- and next-nearest-neighbor spins. In the case of spatially isotropic interactions the phase transition to a collinear state is of special interest. Using Monte-Carlo simulations on finite lattices as well as field-theoretical methods for the continuous model it is shown that depending on the frustration three different sectors can be distinguished that show (i) Ising-universal behavior (ii) Ashkin-Teller-like non-universal behavior (varying critical exponents) and (iii) non-critical first-order behavior. For spatially anisotropic interactions the emergence of an intermediate incommensurate phase is verified by means of Monte-Carlo simulations which has been predicted by earlier mean-field calculations. The nature of this phase is studied analyzing spin-spin correlation functions and its ordering wave-vector reveals a strong temperature and size dependence. In the second part the influence of quantum fluctuations on classical antiferromagnetic ground states are investigated in anisotropic quantum spin-1/2 Heisenberg models with competing interactions. Using quantum Monte-Carlo simulations, exact diagonalization and perturbation theory the ground-state phase diagrams are calculated for a square lattice with nearest- and next-nearest-neighbor interactions and a honeycomb lattice with up to third-nearest-neighbor interactions whereby the fluctuations are chosen to be non-frustrating. For both cases the stability regions of classical magnetic states are identified and an intermediate phase is detected that shows no sign of long-range magnetic order. Further calculations also rule out valence-bond-crystal or topological order for this state, thus a quantum disordered ground state is stabilized in these models.