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arXiv:hep-ph/0008040 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Quantum phase transitions and the extended coupled cluster method

N.E. Ligterink, Niels R. Walet, R. F. Bishop

Published Feb 27, 2001• 1 citation
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Abstract

We discuss the application of an extended version of the coupled cluster method to systems exhibiting a quantum phase transition. We use the lattice O(4) nonlinear sigma model in (1+1) and (3+1) dimensions as an example. We show how simple predictions get modified, leading to the absence of a phase transition in (1+1) dimensions, and strong indications for a phase transition in (3+1) dimensions.

DOI
10.1103/physreve.63.037103

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We discuss the application of an extended version of the coupled cluster method to systems exhibiting a quantum phase transition. We use the lattice O(4) nonlinear sigma model in (1+1) and (3+1) dimensions as an example. We show how simple predictions get modified, leading to the absence of a phase transition in (1+1) dimensions, and strong indications for a phase transition in (3+1) dimensions.
The coupled cluster method (CCM) has become one of the most pervasive of all ab initio formulations of quantum many-body theory.It has yielded numerical results which are among the most accurate available for a wide range of both nite and extended physical systems de ned on a spatial continuum.This widespread success has spurred many recent applications to quantum systems de ned on a lattice.We discuss here a typical example of a two-dimensional spin-half Heisenberg magnet with two kinds of nearest-neighbour bonds.We show how the CCM can successfully describe the in uence of strong quantum uctuations on the zero-temperature phases and their quantum phase transitions.The model shows how the CCM can successfully describe the eects of competition between magnetic bonds with and without the presence of frustration.The frustrated case is particulary important since many other methods, including quantum Monte Carlo simulations, typical fail in this regime.
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.
We present a mixed-spin cluster expansion method to discuss the quantum phase transitions for the Haldane system in two and three dimensions. By mapping the s=1s=1s=1 antiferromagnetic spin model on square and cubic lattices to the equivalent mixed-spin model, we study the competition among the Haldane, the dimer, and the magnetically ordered phases. The mixed-spin cluster expansion proposed here realizes the notion of the valence bond solid in a perturbation theory. This method allows us to directly deal with the Haldane phase, which may not be reached by standard series expansion methods. The zero-temperature phase diagram is thus determined rather precisely for the two- and three-dimensional Haldane systems.
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.
The coupled cluster method is widely recognized nowadays as providing an ab initio method of great versatility, power, and accuracy for handling in a fully microscopic and systematic way the correlations between particles in quantum many-body systems. The number of successful applications made to date within both chemistry and physics is impressive. In this article we review our recent extensions of the method which now provide a unifying framework for also dealing with strongly interacting infinite quantum lattice systems described by a Hamiltonian. Such systems include both spin-lattice models (such as the anisotropic Heisenberg or XXZ model) exhibiting interesting magnetic properties, and electron lattice models (such as the tJ and Hubbard models), where the spins or fermions are localized on the sites of a regular lattice; as well as lattice gauge theories [such as the Abelian U(1) model of quantum electrodynamics and non-Abelian SU(n) models]. Illustrative results are given for both the XXZ spin lattice model and U(1) lattice gauge theory. © 1994 John Wiley & Sons, Inc.