A beginner's guide to non-abelian iPEPS for correlated fermions
Benedikt Bruognolo, Jheng-Wei Li, Jan von Delft, Andreas Weichselbaum
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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>24</mml:mn> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> bands of spinful fermions of \mathrm{SU}(2)_\mathrm{spin} \otimes \mathrm{SU}(2)_\mathrm{orb} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mstyle mathvariant="normal"> <mml:mi>s</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> <mml:mo>⊗</mml:mo> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mstyle mathvariant="normal"> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> <mml:mi>b</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> symmetry, the other with 3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>3</mml:mn> </mml:math> flavors of spinless fermions of \mathrm{SU}(3)_\mathrm{flavor} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>3</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mstyle mathvariant="normal"> <mml:mi>f</mml:mi> <mml:mi>l</mml:mi> <mml:mi>a</mml:mi> <mml:mi>v</mml:mi> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> symmetry.