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=24D=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 22 bands of spinful fermions of \mathrm{SU}(2)_\mathrm{spin} \otimes \mathrm{SU}(2)_\mathrm{orb}SU(2)spin⊗SU(2)orb symmetry, the other with 33 flavors of spinless fermions of \mathrm{SU}(3)_\mathrm{flavor}SU(3)flavor symmetry.
Highlights
We focus on the corner transfer matrix method (CTM) [57, 58], which is well suited for Infinite projected entangled pair states (iPEPS) applications on square-lattice geometries
This formulation builds on two simple “fermionization” rules discussed below, that were pioneered in the context of fermionic MERA by Refs. [85] and [46], and later adapted to the Projected entangled pair states (PEPS) and iPEPS framework [45]
Quantum states can be organized into irreducible symmetry multiplets that carry an additional label qz that specifies the internal structure of an individual multiplet, e.g. |ql〉 → |ql; qz〉
Summary
Ever since the discovery of high-Tc superconductivity, there is a great need for developing and improving numerical approaches for studying one-band and multi-band fermionic many-body systems in two spatial dimensions. PEPS has not yet reached its full potential in application to frustrated and fermionic 2D systems This is mostly due to the technical complexity of the algorithm, especially when dealing with fermionic signs [24] and when implementing symmetries explicitly [25,26,27,28,29,30,31,32,33]. A first application of our non-abelian fermionic iPEPS code, published concurrently with this tutorial review, is a study of the 2D fermionic t-J model [51] – by exploiting either U(1) or SU(2) symmetry to allow or forbid spontaneous spin symmetry breaking, we elucidate the interplay between antiferromagnetic order, stripe formation and pairing correlations. We further illustrate the power of non-abelian iPEPS by presenting some exemplary results for two 2D fermionic Hubbard models of higher complexity: a model with two degenerate bands of spinful fermions, featuring SU(2)spin ⊗ SU(2)orb symmetry, and a model with three degenerate bands of spinless fermions, featuring SU(3)flavor symmetry
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