[Paper Review] Local gauge symmetry on optical lattices?

Yuzhi Liu, Yannick Meurice+1 more|arXiv (Cornell University)|Nov 17, 2012

Cold Atom Physics and Bose-Einstein Condensates5 references3 citations

TL;DR

This paper investigates the feasibility of realizing local SU(2) lattice gauge theory using ultracold fermionic atoms in optical lattices by extending the strong coupling equivalence between the Fermi Hubbard model and SU(2) pure gauge theory in 2+1 dimensions. It proposes using dipolar molecules and external fields to generate higher-order plaquette interactions beyond leading-order strong coupling expansion, enabling experimental emulation of non-Abelian gauge dynamics in quantum simulators.

ABSTRACT

The versatile technology of cold atoms confined in optical lattices allows the creation of a vast number of lattice geometries and interactions, providing a promising platform for emulating various lattice models. This opens the possibility of letting nature take care of sign problems and real time evolution in carefully prepared situations. Up to now, experimentalists have succeeded to implement several types of Hubbard models considered by condensed matter theorists. In this proceeding, we discuss the possibility of extending this effort to lattice gauge theory. We report recent efforts to establish the strong coupling equivalence between the Fermi Hubbard model and SU(2) pure gauge theory in 2+1 dimensions by standard determinantal methods developed by Robert Sugar and collaborators. We discuss the possibility of using dipolar molecules and external fields to build models where the equivalence holds beyond the leading order in the strong coupling expansion.

Motivation & Objective

• To explore the possibility of simulating local SU(2) gauge symmetry in ultracold atomic systems on optical lattices.
• To extend the known strong coupling equivalence between the Fermi Hubbard model and SU(2) pure gauge theory beyond leading order.
• To identify experimental setups—using dipolar molecules and external fields—that can generate effective plaquette interactions necessary for non-Abelian gauge dynamics.
• To develop computational tools, including determinantal methods and HMC algorithms, to test the validity of the proposed quantum simulation framework.
• To bridge the physical parameters of ultracold atom experiments (e.g., temperature, coupling) with those of lattice gauge field theories.

Proposed method

• Utilize the strong coupling expansion of the Fermi Hubbard model to establish approximate equivalence with SU(2) lattice gauge theory in 2+1 dimensions.
• Apply standard determinantal methods developed by Robert Sugar and collaborators to compute and compare partition functions and correlation functions.
• Engineer long-range dipole-dipole interactions in spinless or spin-1/2 fermionic atoms via external fields to tune interaction anisotropy and induce bond-order or p-wave pairing phases.
• Implement a bi-partite lattice structure with S- and P-orbitals arranged on a checkerboard pattern to generate ring current-like terms resembling plaquette interactions.
• Use Fermi-Bose mixtures (e.g., 6Li and 133Cs) to couple bosonic gauge fields to fermionic matter, enabling U(1) and potentially non-Abelian gauge field realization.
• Perform higher-order strong coupling expansions of the Hubbard Hamiltonian to derive effective plaquette interactions that emerge from fermion hopping and on-site repulsion.

Experimental results

Research questions

• RQ1 Can the strong coupling equivalence between the Fermi Hubbard model and SU(2) lattice gauge theory be extended beyond second order in the coupling expansion?
• RQ2 What experimental configurations—specifically involving dipolar molecules and external fields—can generate effective plaquette interactions necessary for non-Abelian gauge symmetry?
• RQ3 How can the mapping between physical parameters in ultracold atom experiments (e.g., temperature, tunneling, interaction strength) and lattice gauge theory couplings (e.g., β or g) be established?
• RQ4 What role do higher-order terms in the strong coupling expansion play in generating emergent gauge-invariant dynamics in optical lattices?
• RQ5 Can hybrid systems of fermions and bosons in optical lattices realize U(1) or non-Abelian gauge theories via quantum link model constructions?

Key findings

• The strong coupling equivalence between the Fermi Hubbard model and SU(2) pure gauge theory holds up to second order in the expansion, but higher-order terms are required to generate plaquette interactions.
• Dipolar fermions in external fields with tunable orientation (via polar angle θ) can realize distinct phases including charge-density wave (CDW), bond-order solid (BOS), and p-wave BCS pairing, indicating controllable interaction anisotropy.
• A phase transition from CDW to BOS occurs at θ ≈ θ₁, and further increase in θ leads to a p-wave superconducting phase, suggesting tunable non-trivial order parameters.
• The use of bi-partite lattices with S- and P-orbitals enables the generation of ring currents and effective plaquette interactions resembling those in lattice gauge theories.
• Fermi-Bose mixtures, such as 6Li and 133Cs, provide a viable route to couple bosonic gauge fields to fermionic matter, supporting the construction of U(1) and potentially non-Abelian gauge theories.
• Determinantal calculations and Hybrid Monte Carlo (HMC) simulations are being developed to validate the proposed mapping between the Hubbard model and lattice gauge theory at higher orders.