[Paper Review] Making use of self-energy functionals: The variational cluster approximation

Michael Potthoff|arXiv (Cornell University)|Jul 15, 2014

Advanced Thermodynamics and Statistical Mechanics5 citations

TL;DR

This paper presents the variational cluster approximation (VCA) as a non-perturbative, self-energy-functional approach to strongly correlated electron systems, using a cluster mean-field framework with variational Weiss fields to capture symmetry-broken phases. By minimizing the grand potential with respect to variational parameters, VCA enables accurate description of antiferromagnetic order and other broken-symmetry states beyond standard cluster perturbation theory.

ABSTRACT

A pedagogical introduction to the cluster-perturbation theory, the variational cluster approximation and to self-energy-functional theory is given. Some standard applications and the relation to dynamical mean-field theory are discussed.

Motivation & Objective

• To develop a consistent, non-perturbative framework for studying strongly correlated fermionic systems with broken symmetries.
• To overcome the limitations of cluster perturbation theory (CPT) in describing spontaneous symmetry breaking, such as antiferromagnetism in finite clusters.
• To generalize the Weiss field approach to arbitrary variational parameters, enabling optimization of cluster-level order parameters.
• To unify and re-derive existing cluster mean-field methods (e.g., DCA, cellular DMFT) within the self-energy-functional theory (SFT) framework.
• To establish a foundation for future extensions to non-equilibrium dynamics and non-local interactions in correlated lattices.

Proposed method

• The VCA constructs an approximate solution for the infinite lattice by solving the Hubbard model on isolated clusters with a variational Weiss field, which breaks symmetry and enables finite-order parameters.
• The grand potential Ω(B′) is computed as a functional of the Weiss field B′, and the optimal field B′_opt is determined by minimizing ∂Ω/∂B′ = 0.
• The method uses all-order perturbation theory to connect the non-interacting cluster solutions via inter-cluster hopping, avoiding the mean-field limitations of CPT.
• The self-energy-functional theory (SFT) provides the theoretical foundation, allowing the VCA to be derived as a variational principle over cluster-level parameters.
• The approach generalizes to arbitrary variational parameters λ′, not just staggered magnetic fields, enabling optimization over multiple order parameters.
• The framework allows re-derivation of other cluster methods such as DCA and cellular DMFT, demonstrating its unifying power and consistency.

Experimental results

Research questions

• RQ1 How can a variational cluster approximation be constructed to describe symmetry-broken phases such as antiferromagnetism in strongly correlated systems?
• RQ2 What is the role of the Weiss field in stabilizing broken-symmetry states when the original system has no external field?
• RQ3 How can the self-energy-functional theory be used to derive consistent, non-perturbative approximations beyond standard cluster perturbation theory?
• RQ4 In what way does the VCA unify or generalize existing cluster mean-field theories like DCA and cellular DMFT?
• RQ5 What are the limitations of the VCA and how can it be extended to non-local interactions and non-equilibrium dynamics?

Key findings

• The VCA successfully captures antiferromagnetic order in the Hubbard model by optimizing a variational Weiss field, with B′_opt > 0 indicating spontaneous symmetry breaking despite B = 0 in the original system.
• The grand potential Ω(B′) is minimized at B′_opt, providing a consistent variational principle that avoids the failure of CPT in describing long-range order.
• The VCA framework allows generalization beyond staggered magnetic fields to arbitrary variational parameters λ′, enabling optimization over multiple order parameters.
• The method recovers and unifies established approaches such as DCA and cellular DMFT within the SFT, demonstrating its theoretical coherence and broad applicability.
• The VCA is shown to be superior to the DIA and DMFT-ED in terms of internal consistency, while remaining numerically feasible with small cluster and bath sizes.
• The approach is extendable to non-equilibrium dynamics and non-local interactions, with first steps already taken in these directions.