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arXiv:1407.1074 · AIP conference proceedings

Highly frustrated spin-lattice models of magnetism and their quantum phase transitions: A microscopic treatment via the coupled cluster method

R. F. Bishop, P. H. Y. Li, C. E. Campbell

Published Jan 1, 2014• 8 citations
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Abstract

We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spinlattice models of interest in quantum magnetism, including their quantum phase transitions.The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (socalled LSUBm) approximations, by the use of computer-algebraic techniques.The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the only approximation made.All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half J XXZ 1 -J XXZ 2 model on a honeycomb lattice with nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 1 > 0, respectively, where both interactions are of the same anisotropic XXZ type.We show how the method can be used to determine the entire zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ Δ ≤ 1 of the spin-space anisotropy parameter.In particular, we identify a candidate quantum spin-liquid region in the phase space.

DOI
10.1063/1.4899216

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We outline how the coupled cluster method of microscopic quantum many-body theory can be utilized in practice to give highly accurate results for the ground-state properties of a wide variety of highly frustrated and strongly correlated spinlattice models of interest in quantum magnetism, including their quantum phase transitions.The method itself is described, and it is shown how it may be implemented in practice to high orders in a systematically improvable hierarchy of (socalled LSUBm) approximations, by the use of computer-algebraic techniques.The method works from the outset in the thermodynamic limit of an infinite lattice at all levels of approximation, and it is shown both how the "raw" LSUBm results are themselves generally excellent in the sense that they converge rapidly, and how they may accurately be extrapolated to the exact limit, m → ∞, of the truncation index m, which denotes the only approximation made.All of this is illustrated via a specific application to a two-dimensional, frustrated, spin-half J XXZ 1 -J XXZ 2 model on a honeycomb lattice with nearestneighbor and next-nearest-neighbor interactions with exchange couplings J 1 > 0 and J 2 ≡ κJ 1 > 0, respectively, where both interactions are of the same anisotropic XXZ type.We show how the method can be used to determine the entire zerotemperature ground-state phase diagram of the model in the range 0 ≤ κ ≤ 1 of the frustration parameter and 0 ≤ Δ ≤ 1 of the spin-space anisotropy parameter.In particular, we identify a candidate quantum spin-liquid region in the phase space.
Quantum spins arranged on regular geometrical lattices, and interacting through a wide diversity of specified model Hamiltonians, provide a rich variety of magnetically ordered phases with many subtle properties. The zero-temperature quantum phase transitions between states of different order can be explored by the variation of internal control parameters within the interaction Hamiltonian. We illustrate how the systematic inclusion of multi-spin correlations for such lattice systems may be efficiently and effectively implemented via the coupled cluster method (CCM), which has been widely demonstrated on many previous occasions to be one of the most versatile methods of microscopic quantum many-body theory. Its power is illustrated by application to the spin-½ anisotropic Heisenberg (or XXZ) model on an infinite square lattice. The results obtained for such properties as the ground-state energy and sublattice magnetization are found to be as accurate as those obtained by any available alternative method. We show also how the CCM can now provide accurate ab initio estimates for the critical values of the parameters which mark the positions of phase boundaries between states of different quantum order, as well as for the critical indices associated with the singular behaviour near these transition points of various physical quantities.
Strongly interacting quantum spin–lattice models exhibit a wide variety of phases with diverse and subtle magnetic ordering properties. Their detailed description within a unified microscopic framework poses a real challenge for the many-body theorist. By specific application to the spin-½ anisotropic Heisenberg model on a square lattice, we show how the ab initio coupled cluster method, which has already been very successfully applied to a wide variety of quantum many-body and field-theoretic systems, may be very efficiently and systematically implemented for spin–lattice models. Results for such local properties as the ground-state energy and sublattice magnetization are thereby obtained which are on a par with those from the best of the available alternative methods. Furthermore, we demonstrate explicitly how the coupled cluster method now also provides an effective and fully microscopic tool to yield systematic and accurate estimates of the zero-temperature quantum phase transition boundaries between states of different quantum order, as well as of the critical behaviour of the system in the vicinity of the transition points.
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We study the zero-temperature (T=0T=0T=0) ground-state (GS) properties of a frustrated spin-half J1XXZ{J}_{1}^{\mathit{X}XZ}J1XXZ​--J2XXZ{J}_{2}^{\mathit{X}XZ}J2XXZ​ model on the honeycomb lattice with nearest-neighbor and next-nearest-neighbor interactions with exchange couplings J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively, using the coupled cluster method. Both interactions are of the anisotropic XXZ type. We present the T=0T=0T=0 GS phase diagram of the model in the ranges $0\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}1$ of the spin-space anisotropy parameter and $0\ensuremath{\le}\ensuremath{\kappa}\ensuremath{\le}1$ of the frustration parameter. A possible quantum spin-liquid region is identified.
We use the coupled cluster method to study the zero-temperature quantum phase diagram of the spin-$\frac{1}{2}{J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{1}^{\ensuremath{\perp}}$ model on the honeycomb bilayer lattice. In each layer, we include both nearest-neighbor and frustrating next-nearest-neighbor antiferromagnetic exchange couplings, of strength J1>0{J}_{1}>0J1​>0 and ${J}_{2}\ensuremath{\equiv}\ensuremath{\kappa}{J}_{1}>0$, respectively. The two layers are coupled by an interlayer nearest-neighbor exchange, with coupling constant ${J}_{1}^{\ensuremath{\perp}}\ensuremath{\equiv}\ensuremath{\delta}{J}_{1}>0$. We calculate directly in the infinite-lattice limit both the ground-state energy per spin and the N\'eel magnetic order parameter, as well as the triplet spin gap. By implementing the method to very high orders of approximation we obtain an accurate estimate for the full boundary of the N\'eel phase in the $\ensuremath{\kappa}\ensuremath{\delta}$ plane. For each value $\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(0)\ensuremath{\approx}1.70(5)$, we find an upper critical value ${\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$, such that N\'eel order is present for $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(\ensuremath{\delta})$. Conversely, for each value $\ensuremath{\kappa}<{\ensuremath{\kappa}}_{c}(0)\ensuremath{\approx}0.19(1)$, we find an upper critical value ${\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, such that N\'eel order persists for $0<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$. Most interestingly, for values of $\ensuremath{\kappa}$ in the range ${\ensuremath{\kappa}}_{c}(0)<\ensuremath{\kappa}<{\ensuremath{\kappa}}^{>}\ensuremath{\approx}0.215(2)$, we find a reentrant behavior such that N\'eel order exists only in the range ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})<\ensuremath{\delta}<{\ensuremath{\delta}}_{c}^{>}(\ensuremath{\kappa})$, with ${\ensuremath{\delta}}_{c}^{<}(\ensuremath{\kappa})>0$. These latter upper and lower critical values coalesce when $\ensuremath{\kappa}={\ensuremath{\kappa}}^{>}$, such that ${\ensuremath{\delta}}_{c}^{<}({\ensuremath{\kappa}}^{>})={\ensuremath{\delta}}_{c}^{>}({\ensuremath{\kappa}}^{>})\ensuremath{\approx}0.25(5)$.
In this article we study the ground-state properties of two square-lattice Heisenberg quantum spin models with competing bonds using a high-order coupled cluster treatment. The first model is a spin-half model with competing nearest-neighbour bonds with and without frustration. We discuss the influence of quantum fluctuations on the ground-state phase diagram and in particular on the nature of the zero-temperature phase transitions from phases with collinear magnetic order at small frustration to phases with noncollinear spiral order at large frustration. The second model is a highly frustrated ferrimagnet, which contains one sublattice (A) entirely populated with spin-one spins and an other sublattice (B) entirely populated with spin-half spins. Sublattice A sites are nearest-neighbours to sublattice B sites and vice versa and frustration is introduced by next-nearest-neighbour bonds. The model shows two collinear ordered phases and a noncollinear phase in which (classically) the spin-one spins are allowed to cant at an angle. Both examples show that the coupled-cluster method is able to describe the zero-temperature transitions well and provides a consistent description of collinear, noncollinear, and disordered phases, for cases in which other standard techniques (e.g. the quantum Monte Carlo technique for spin systems which are frustrated) are not applicable.