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arXiv:cond-mat/9911365 · Physical review. B, Condensed matter

Phase transition in the transverse Ising model using the extended coupled-cluster method

J. Rosenfeld, N.E. Ligterink

Published Jul 1, 2000• 3 citations
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Abstract

The phase transition present in the linear-chain and square-lattice cases of the transverse Ising model is examined. The extended coupled-cluster method (ECCM) can describe both sides of the phase transition with a unified approach. The correlation length and the excitation energy are determined. The ECCM is able to use both the weak- and the strong-coupling starting states within a calculation based on the same uncorrelated model state.

DOI
10.1103/physrevb.62.308

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The phase transition present in the linear-chain and square-lattice cases of the transverse Ising model is examined. The extended coupled-cluster method (ECCM) can describe both sides of the phase transition with a unified approach. The correlation length and the excitation energy are determined. The ECCM is able to use both the weak- and the strong-coupling starting states within a calculation based on the same uncorrelated model state.
The spin-half XXZ\mathrm{XXZ}XXZ model on the linear chain and the square lattice are examined with the extended coupled-cluster method (ECCM) of quantum many-body theory. We are able to describe both the Ising-Heisenberg phase and the XY\mathrm{XY}XY-Heisenberg phase, starting from known wave functions in the Ising limit and at the phase transition point between the XY\mathrm{XY}XY-Heisenberg and ferromagnetic phases, respectively, and by systematically incorporating correlations on top of them. The ECCM yields good numerical results via a diagrammatic approach, which makes the numerical implementation of higher-order truncation schemes feasible. In particular, the best nonextrapolated coupled-cluster result for the sublattice magnetization is obtained, which indicates the employment of an improved wave function. Furthermore, the ECCM finds the expected qualitatively different behaviors of the linear-chain and square-lattice cases.
Earlier papers in this series have shown how the extended coupled-cluster method (ECCM) provides a widely applicable biorthogonal formulation of quantum many-body and quantum field theory. Rules for calculating matrix elements of arbitrary operators have been formulated, and an exact description of the excited states has been given in terms of a generalized random-phase approximation for the many-body, quasilocal c-number (classical), ECCM cluster amplitudes which completely and exactly parametrize the system. The present paper introduces a variational functional in order to treat the excited states differently within the ECCM framework, so as to allow extra flexibility in the choice of approximation schemes. The method is tested on the SU(2) quasispin model of Lipkin, Meshkov, and Glick [Nucl. Phys. 62, 188 (1965); 62, 199 (1965)], which contains the spontaneous breakdown of parity, and which models a shape transition in finite atomic nuclei. The method very well describes both the ground and excited states of the model Hamiltonian over a broad range of coupling parameters on both sides of the critical value corresponding to the transition.
A high-order coupled cluster method (CCM) for the zero-temperature ground states of quantum lattice spin systems of general spin quantum number (s> 1/2)(s>~1/2)(s> 1/2) is described and illustrated via the spin-1 anisotropic Heisenberg antiferromagnet on the square lattice. Results for the ground-state energy and sublattice magnetization are seen to be in excellent agreement with the best of other approximate methods. Furthermore, we are able to follow the solution to the CCM equations at a given level of approximation with respect to the anisotropy parameter $\ensuremath{\Delta}$ from the trivial Ising limit $(\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Delta}}\ensuremath{\infty})$ down to a critical value ${\ensuremath{\Delta}}_{c},$ at which point the physical solution terminates. The results for the ground-state energy, sublattice magnetization, and phase transition point thus support our choice of this model as a good nontrivial test case of our CCM formalism.
Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce 2L collective structures, referred to as patterns, for the TIM with L ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength J_{c}=1. We systematically explore this process for small lattice sizes of L=6,8,10,12, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to L=128, the actual QPT point located at J_{c}=1 in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.
The coupled-cluster method is used to obtain the ground-state energy of the isotropic Heisenberg-biquadratic quantum spin-one chain as a function of the ratio of the magnitudes of the two terms in the Hamiltonian. Two different model states are used which are expected to be valid in different regimes. In both cases we use simple approximation schemes to obtain numerical results for the ground-state energy which are compared with results of exact diagonalizations of short chains. For both cases we are able to incorporate some of the long-range correlations explicitly, using the so-called full-SUB2 approximation schemes, and this leads to evidence of phase changes at certain points. These are discussed in the light of known and conjectured phase transitions in this system.