A beginner's guide to non-abelian iPEPS for correlated fermions

Abstract

Infinite projected entangled pair states (iPEPS) have emerged as a powerful tool for studying interacting two-dimensional fermionic systems. In this review, we discuss the iPEPS construction and some basic properties of this tensor network (TN) ansatz. Special focus is put on (i) a gentle introduction of the diagrammatic TN representations forming the basis for deriving the complex numerical algorithm, and (ii) the technical advance of fully exploiting non-abelian symmetries for fermionic iPEPS treatments of multi-band lattice models. The exploitation of non-abelian symmetries substantially increases the performance of the algorithm, enabling the treatment of fermionic systems up to a bond dimension $D=24$ on a square lattice. A variety of complex two-dimensional (2D) models thus become numerically accessible. Here, we present first promising results for two types of multi-band Hubbard models, one with $2$ bands of spinful fermions of $\mathrm{SU}(2{)}_{\mathrm{spin}}\otimes \mathrm{SU}(2{)}_{\mathrm{orb}}$ symmetry, the other with $3$ flavors of spinless fermions of $\mathrm{SU}(3{)}_{\mathrm{flavor}}$ symmetry.

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• Fermionic tensor network contraction for arbitrary geometries
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• Magnetic phases of the periodic Anderson model in two dimensions
• Density Matrix Renormalization Group simulations of the SU(N) Fermi-Hubbard chain implementing the full SU(N) symmetry via Semi-Standard Young Tableaux and Unitary Group Subduction Coefficients
• Non-local order parameters for fermion chains via the partial transpose
• Grassmann tensor networks
• Dispersion Relations in Two- and Three-Dimensional Quantum Systems