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arXiv:1103.1247

CCM Calculations For The Ground-State Properties Of The One-Dimensional Spin-Half $J_1$--$J_2$ Model: Possible Evidence of Collinear Ordering for $J_2/J_1 > \frac 12$?

DAMIAN J. J. FARNELL

Published Mar 7, 2011• 0 citations
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Abstract

In this article we investigate the linear-chain spin-half $J_1$--$J_2$ model by using high-order coupled cluster method (CCM) calculations. We employ three model states, namely, a nearest-neighbour (n.n.) Néel model state in which neighbouring spins are antiparallel, a next-nearest-neighbour (n.n.n.) Néel model state in which next-neighbouring spins are antiparallel, and finally a type of "double spiral" model state with {\it two} pitch angles. For $J_2/J_1 \le \frac 12$, we find that the n.n. Néel model state produces the lowest energies. For $J_2/J_1 > \frac 12$, we find that the stable states for the quantum system are those for the "traditional" spiral state in which the two pitch angles are identical and the collinear n.n.n. Néel model state. We show that ground-state energies for the collinear n.n.n. model state are lower than those of the spiral state in a finite region for $J_2/J_1>\frac 12$. We produce results for the dimer order parameter by using the n.n.n. Néel model state and these results are in reasonable agreement with previous results of DMRG. Again for the n.n.n. Néel model state, it is shown that the correlation length increases with increasing $J_2/J_1$ above $J_2/J_1=\frac 12$ using this model state and that shift in the peak of the structure function occurs from $q=π$ to $q=π/2$ as one increases $J_2/J_1$. Although these results are not conclusive, they are intriguing because they suggest that a ground state exhibiting collinear order might occur for some region for $J_2/J_1 \ge \frac 12$.

DOI
10.48550/arxiv.1103.1247

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We study the ground-state (GS) properties of the frustrated spin-$\frac{1}{2}$ ${J}_{1}$--${J}_{2}$--${J}_{3}$ Heisenberg model on the two-dimensional honeycomb lattice with ferromagnetic nearest-neighbor (${J}_{1}=\ensuremath{-}1$) exchange and frustrating antiferromagnetic next-nearest-neighbor (${J}_{2}>0$) and next-next-nearest-neighbor (${J}_{3}>0$) exchanges, for the case ${J}_{3}={J}_{2}$. 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 ${J}_{2}^{\mathrm{cl}}=\frac{1}{7}$ ($\ensuremath{\approx}0.143$) for the classical version of the model, in which we momentarily ignore the intervening noncollinear spiral phase in the region $\frac{1}{10}<{J}_{2}<\frac{1}{5}$. 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 ${J}_{1}=+1$). The magnetic order parameter is found to behave similarly for the ferromagnetic and the antiferromagnetic models for large values of the frustration parameter ${J}_{2}$. 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 ${J}_{1}=+1$) 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 ${J}_{1}=+1$), where a plaquette valence-bond crystal phase intrudes between the two corresponding quasiclassical antiferromagnetic phases (with N\'eel and striped order) for ${J}_{2}^{{c}_{1}}<{J}_{2}<{J}_{2}^{{c}_{2}}$, 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 $\frac{1}{10}<{J}_{2}<\frac{1}{5}$.

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CCM Calculations For The Ground-State Properties Of The One-Dimensional Spin-Half J1J_1J1​--J2J_2J2​ Model: Possible Evidence of Collinear Ordering for J2/J1>12J_2/J_1 > \frac 12J2​/J1​>21​?
In this article we investigate the linear-chain spin-half J1J_1J1​--J2J_2J2​ model by using high-order coupled cluster method (CCM) calculations. We employ three model states, namely, a nearest-neighbour (n.n.) Néel model state in which neighbouring spins are antiparallel, a next-nearest-neighbour (n.n.n.) Néel model state in which next-neighbouring spins are antiparallel, and finally a type of "double spiral" model state with {\it two} pitch angles. For J2/J1≤12J_2/J_1 \le \frac 12J2​/J1​≤21​, we find that the n.n. Néel model state produces the lowest energies. For J2/J1>12J_2/J_1 > \frac 12J2​/J1​>21​, we find that the stable states for the quantum system are those for the "traditional" spiral state in which the two pitch angles are identical and the collinear n.n.n. Néel model state. We show that ground-state energies for the collinear n.n.n. model state are lower than those of the spiral state in a finite region for J2/J1>12J_2/J_1>\frac 12J2​/J1​>21​. We produce results for the dimer order parameter by using the n.n.n. Néel model state and these results are in reasonable agreement with previous results of DMRG. Again for the n.n.n. Néel model state, it is shown that the correlation length increases with increasing J2/J1J_2/J_1J2​/J1​ above J2/J1=12J_2/J_1=\frac 12J2​/J1​=21​ using this model state and that shift in the peak of the structure function occurs from q=πq=πq=π to q=π/2q=π/2q=π/2 as one increases J2/J1J_2/J_1J2​/J1​. Although these results are not conclusive, they are intriguing because they suggest that a ground state exhibiting collinear order might occur for some region for J2/J1≥12J_2/J_1 \ge \frac 12J2​/J1​≥21​.
In this article we present new formalism for high-order coupled cluster method (CCM) calculations for "generalized" ground-state expectation values and the excited states of quantum magnetic systems with spin quantum number s ≥ 1/2. We use high-order CCM to demonstrate spontaneous symmetry breaking in the spin-half J1-J2 model for the linear chain using the coupled cluster method (CCM). We show that we are able to reproduce exactly the dimerized ground (ket) state at the Majumdar-Ghosh point (J2/J1= 1/2) using a Néel model state. We show that the onset of dimerized phase is indicated by a bifurcation of the nearest-neighbour ket- and bra-state correlation coefficients for the nearest-neighbour Néel model state. We show that ground-state energies are in good agreement with the results of exact diagonalizations of finite-length chains across this entire regime (i. e., J1>0 and J2≤1/2). The effects of the bifurcation point are also observed for the sublattice magnetization for the nearest-neighbour model state. Finally, we use the new formalism for the excited state in order to obtain the excitation energy as a function of J2/J1 for the nearest-neighbour model state by solving up to the LSUB14 level of approximation. We obtain an extrapolated value for the excited-state energy gap of -0.0036 at J2/J1=0.0 and of 0.2310 at J2/J1=0.5. We show that an excitation energy gap opens up at J2/J1 ≈ 0.24, although the gap only becomes large at J2/J1≈ 0.4.
Recent developments of high-order CCM have been to extend existing formalism and codes to s≥12s \ge \frac 12s≥21​ for both the ground and excited states, and independently to "generalised" expectation values for a wide range of one- and two-body spin operators. An advantage of the CCM is that the Goldstone linked-cluster theorem is obeyed at all levels of approximation and so it provides results in the infinite lattice limit N→∞N \to \inftyN→∞ from the outset. However, recent results have also shown that the CCM can provide exact (symmetry-breaking) results for the spin-half linear-chain J1J_1J1​--J2J_2J2​ at the Majumdar-Ghosh point J2/J1=0.5J_2/J_1=0.5J2​/J1​=0.5 by identifying special solutions of the CCM equations for the usual Néel model state. Interestingly, the CCM provides exact (non-symmetry-breaking) results for systems in which small magnetic clusters become de-coupled from each other when the bonds connecting them tend to zero. These exact results involve the identification of "special solutions" of the CCM equations for the Néel state. An example of this is given by a spin-half system with nearest-neighbour bonds for an underlying lattice corresponding to the magnetic material CaV4_44​O9_99​ (CAVO) in which one of the two different types of bonds on the lattice tend to zero. Larger finite-sized systems may be considered by appropriate choice of the unit cell and the bonds on it. We show here that exact diagonalisation results for ground-state energy and excitation energy gap for the spin-half and spin-one linear Heisenberg model on chains of length up to N=12 sites for s=1/2 and N=6 sites for s=1 with periodic boundary conditions are reproduced exactly using high-order CCM via this "brute-force" approach; i.e., one in which none of the translational or point-group symmetries of the finite lattice are used.
The ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model is a ``canonical'' model in the field of quantum magnetism in order to study the interplay between frustration and quantum fluctuations as well as quantum phase transitions driven by frustration. Here we apply the coupled cluster method (CCM) to study the spin-half ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model with antiferromagnetic nearest-neighbor bonds J1>0{J}_{1}>0J1​>0 and next-nearest-neighbor bonds J2>0{J}_{2}>0J2​>0 for the simple cubic (sc) and body-centered cubic (bcc) lattices. In particular, we wish to study the ground-state ordering of these systems as a function of the frustration parameter p=z2J2/z1J1p={z}_{2}{J}_{2}/{z}_{1}{J}_{1}p=z2​J2​/z1​J1​, where z1(z2){z}_{1} ({z}_{2})z1​(z2​) is the number of nearest (next-nearest) neighbors. We wish to determine the positions of the phase transitions using the CCM and we aim to resolve the nature of the phase transition points. We consider the ground-state energy, order parameters, spin-spin correlation functions, as well as the spin stiffness in order to determine the ground-state phase diagrams of these models. We find a direct first-order phase transition at a value of p=0.528p=0.528p=0.528 from a state of nearest-neighbor N\'eel order to next-nearest-neighbor N\'eel order for the bcc lattice. For the sc lattice the situation is more subtle. CCM results for the energy, the order parameter, the spin-spin correlation functions, and the spin stiffness indicate that there is no direct first-order transition between ground-state phases with magnetic long-range order, rather it is more likely that two phases with antiferromagnetic long range are separated by a narrow region of a spin-liquid-like quantum phase around p=0.55p=0.55p=0.55. Thus the strong frustration present in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the sc lattice may open a window for an unconventional quantum ground state in this three-dimensional spin model.
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 investigate the influence of an interchain coupling on the spiral ground-state correlations of a spin-1/2 Heisenberg model consisting of a two-dimensional array of coupled chains with nearest- (J1)({J}_{1})(J1​) and frustrating next-nearest-neighbor (J2)({J}_{2})(J2​) in-chain exchange couplings. Using the coupled cluster method we calculate the transition point between the commensurate and the incommensurate (spiral) ground states as well as the quantum pitch angle of the spiral ground state. In addition, we provide a simple empirical formula which describes the relation between the quantum pitch angle and the frustration parameter J2/J1{J}_{2}/{J}_{1}J2​/J1​.
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​.