Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2405.01001 · Proceedings of the American Mathematical Society

Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems

Hajime Moriya, Heide Narnhofer

Published Jul 19, 2024• 0 citations
arXivHTMLPDF

Abstract

We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.

DOI
10.1090/proc/16919

Related papers

On implementation of discrete and continuous shift translations on quantum lattice systems (Mathematical Aspects of Quantum Fields and Related Topics)

Hajime Moriya

We discuss the continuous extension of the discrete shift translations on the one-dimensional fermion lattice system to C*-flows.

数理解析研究所講究録•May 1, 2025•0 citations

Jordan-Wigner transformation in a higher-dimensional lattice

Wang Shaofeng

A type of Jordan-Wigner transformation is constructed for a spin-one-half quantum system on a higher-dimensional lattice. The properties of the transformation are discussed briefly. No flux is attached to spinless fermions in the present case.

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Feb 1, 1995•11 citations

Does causal dynamics imply local interactions?

Zoltán Zimborás, Terry Farrelly, Szilárd Farkas +1 more

We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian. On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.

arXiv:2006.10707•Quantum•Jun 29, 2022•11 citations

Symmetric Jordan-Wigner transformation in higher dimensions

Hoi Chun Po

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.

arXiv:2107.10842•Jul 22, 2021•8 citations

Quantum transport in d -dimensional lattices

Daniel Manzano, Chern Chuang, Jianshu Cao

We show that both fermionic and bosonic uniform d -dimensional lattices can be reduced to a set of independent one-dimensional chains. This reduction leads to the expression for ballistic energy fluxes in uniform fermionic and bosonic lattices. By the use of the Jordan–Wigner transformation we can extend our analysis to spin lattices, proving the coexistence of both ballistic and non-ballistic subspaces in any dimension and for any system size. We then relate the nature of transport to the number of excitations in the homogeneous spin lattice, indicating that a single excitation always propagates ballistically and that the non-ballistic behaviour of uniform spin lattices is a consequence of the interaction between different excitations.

arXiv:1507.05705•New Journal of Physics•Apr 28, 2016•36 citations

Bosonization in three spatial dimensions and a 2-form gauge theory

Yu-An Chen, Anton Kapustin

The well-known Jordan-Wigner transformation maps an arbitrary system of fermions on a one-dimensional lattice to a system of spins. An essential property of this transformation is that it preserves the locality of observables. This transformation has been recently extended to two-dimensional lattices, where it maps an arbitrary system of fermions to a gauge theory, with locality preserved. Here, the authors construct a bosonization map for arbitrary systems of fermions in three dimensions. As an application, the authors design a model of interacting spins on a cubic lattice, which is exactly soluble and is equivalent to a tight-binding model of free fermions with Dirac points.

arXiv:1807.07081•Physical review. B./Physical review. B•Dec 16, 2019•67 citations
We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.
We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian. On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.
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-1/21/21/2 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 ttt-JJJ models.
We show that both fermionic and bosonic uniform d -dimensional lattices can be reduced to a set of independent one-dimensional chains. This reduction leads to the expression for ballistic energy fluxes in uniform fermionic and bosonic lattices. By the use of the Jordan–Wigner transformation we can extend our analysis to spin lattices, proving the coexistence of both ballistic and non-ballistic subspaces in any dimension and for any system size. We then relate the nature of transport to the number of excitations in the homogeneous spin lattice, indicating that a single excitation always propagates ballistically and that the non-ballistic behaviour of uniform spin lattices is a consequence of the interaction between different excitations.
The well-known Jordan-Wigner transformation maps an arbitrary system of fermions on a one-dimensional lattice to a system of spins. An essential property of this transformation is that it preserves the locality of observables. This transformation has been recently extended to two-dimensional lattices, where it maps an arbitrary system of fermions to a gauge theory, with locality preserved. Here, the authors construct a bosonization map for arbitrary systems of fermions in three dimensions. As an application, the authors design a model of interacting spins on a cubic lattice, which is exactly soluble and is equivalent to a tight-binding model of free fermions with Dirac points.