Diagrammatic approach in the variational coupled-cluster method

TL;DR

This paper introduces a diagrammatic technique for the variational coupled-cluster method, enabling systematic resummation of diagrams and improving calculations of quantum spin models beyond traditional spin-wave theory.

Contribution

It presents a novel diagrammatic approach to the variational CCM, including resummation techniques that enhance accuracy over spin-wave theory and connect with correlated-basis functionals.

Findings

Reproduces spin-wave theory through ring diagram resummation.

Achieves convergent, precise order-parameter calculations for 1D models.

Extends beyond SWT with super-ring diagram resummation.

Abstract

Recently, as demonstrated by an antiferromagnetic spin-lattice application, we have successfully extended the coupled-cluster method (CCM) to a variational formalism in which two sets of distribution functions are introduced to evaluate Hamiltonian expectation. We calculated these distribution functions by employing an algebraic scheme. Here we present an alternative calculation based on a diagrammatic technique. Similar to the method of correlated-basis functionals (CBF), a generating functional is introduced and calculated by a linked-cluster expansion in terms of diagrams which are categorized and constructed according to a few simple rules and using correlation coefficients and Pauli exclusion principle (or Pauli line) as basic elements. Infinite resummations of diagrams can then be done in a straightforward manner. One such resummation, which includes all so-called ring diagrams and ignores Pauli exclusion principle, reproduces spin-wave theory (SWT). Approximations beyond SWT are also given. Interestingly, one such approximation including all so-called super-ring diagrams by a resummation of infinite Pauli lines in additional to resummations of ring diagrams produces a convergent, precise number for the order-parameter of the one-dimensional isotropic model, contrast to the well-known divergence of SWT. We also discuss the direct relation between our variational CCM and CBF and discuss a possible unification of the two theories.