Symmetric Jordan-Wigner transformation in higher dimensions

Abstract

The Jordan-Wigner transformation is traditionally applied to one dimensional systems, but recent works have generalized the transformation to fermionic lattice systems in higher dimensions while keeping locality manifest. These developments could aid the theoretical or even experimental studies of strongly correlated electronic problems through their bosonic counterparts. In this work, we develop a scheme for higher-dimensional Jordan-Wigner transformation which keeps all relevant symmetries manifest on the bosonic side. Our approach connects the discussion of exact lattice bosonization to the familiar notions of fractionalized partons like spinons and chargeons, and works for spin-1/2 fermions-like the physical electrons-on four-coordinated lattices. The construction is applied to fermions defined on the square, kagome and diamond lattices, and we provide explicit expressions for the bosonized versions of well-known models of strongly correlated electrons, like the Hubbard and t-J models.

Long-range entanglement from measuring symmetry-protected topological phases

“…Third, this allows for the preparation of states that are not realized by stabilizer codes, such as topological order described by twisted gauge theories or non-Abelian fracton orders [60][61][62][63][64][65][66][67][68][69]. Fourth, we achieve a new perspective on Kramers-Wannier (KW) [18,[70][71][72][73][74][75][76][77] and Jordan-Wigner (JW) [78][79][80][81][82][83][84][85][86] transformations. Indeed, we show how these nonlocal transformations can be efficiently implemented in a finite time by adding SPT entanglers to arbitrary initial states 3 and subsequently performing single-site measurements.…”

Higher-dimensional Jordan-Wigner Transformation and Auxiliary Majorana Fermions

“…In a recent work [27], we propose an approach for performing higher-dimensional Jordan-Wigner transformation while keeping all symmetries manifest. The symmetry transformation on the bosonic side can all be traced down to that of a collection of operators denoted by Λ αβ , which could be viewed as the bosonic analog (up to a Jordan-Wigner string) of the Majorana fermions defining the physical fermionic Hilbert space.…”

“…The approach in Ref. 27, however, is limited to four-coordinated lattices like the 2D square and 3D diamond lattices with a single orbital per site. In this work, we generalize our construction to lattices with even coordination number and an arbitrary number of orbitals.…”

“…IV we provide a parallel discussion for the triangular lattice, which is six-coordinated and goes beyond the approach developed in Ref. 27. For a single-orbital triangular lattice model with spin-1/2 fermions, Λ ij will be a 4 × 6 operator-valued matrix, i.e., we have four η's and six χ's.…”

“…The operator constraints defining the fermionic subspace is more complicated in three dimensions, and we provide a strategy for finding independent constraints. We also analyze in details how symmetries are implemented on the bosonized model, and discuss how fermion-odd operators can also be identified on the bosonic side [27]. We conclude in Sec.…”