[Paper Review] Coupling ultracold matter to dynamical gauge fields in optical lattices: From flux-attachment to Z2 lattice gauge theories

Luca Barbiero, C. Schweizer+4 more|arXiv (Cornell University)|Oct 5, 2018

Cold Atom Physics and Bose-Einstein Condensates42 references18 citations

TL;DR

This paper proposes a protocol to realize dynamical $ζ_{2}$ lattice gauge theories in ultracold atomic systems by coupling two species of atoms in optical lattices, where one species induces flux-attachment for the other. Using coherent lattice modulation and strong Hubbard interactions, the system realizes a tunable $ζ_{2}$ gauge structure, with phase transitions between electric and magnetic order regimes in a two-leg ladder model, demonstrating a fully quantum-coupled matter-gauge system.

ABSTRACT

Artificial magnetic fields and spin-orbit couplings have been recently generated in ultracold gases in view of realizing topological states of matter and frustrated magnetism in a highly-controllable environment. Despite being dynamically tunable, such artificial gauge fields are genuinely classical and exhibit no back-action from the neutral particles. Here we go beyond this paradigm, and demonstrate how quantized dynamical gauge fields can be created in mixtures of ultracold atoms in optical lattices. Specifically, we propose a protocol by which atoms of one species carry a magnetic flux felt by another species, hence realizing an instance of flux-attachment. This is obtained by combining coherent lattice modulation techniques with strong Hubbard interactions. We demonstrate how this setting can be arranged so as to implement lattice models displaying a local Z2 gauge symmetry, both in one and two dimensions. We also provide a detailed analysis of a ladder toy model, which features a global Z2 symmetry, and reveal the phase transitions that occur both in the matter and gauge sectors. Mastering flux-attachment in optical lattices envisages a new route towards the realization of strongly-correlated systems with properties dictated by an interplay of dynamical matter and gauge fields.

Motivation & Objective

• To realize dynamical gauge fields in ultracold atomic systems, moving beyond classical synthetic gauge fields.
• To implement a quantum-coupled matter-gauge system where matter and gauge fields mutually influence each other.
• To demonstrate a realization of $ζ_{2}$ lattice gauge theories in optical lattices using ultracold atoms.
• To explore phase transitions in both matter and gauge sectors via a tunable two-leg ladder model.
• To establish a platform for simulating strongly correlated quantum field theories with ultracold atoms.

Proposed method

• Use of coherent lattice modulation to engineer density-dependent tunneling and synthetic gauge fields.
• Employment of strong Hubbard interactions to induce flux-attachment between two atomic species.
• Realization of a $ζ_{2}$ gauge structure via the number imbalance of one atomic species on lattice links.
• Mapping the system to a $ζ_{2}$ lattice gauge theory Hamiltonian with local $ζ_{2}$ symmetry.
• Application of second-order perturbation theory to derive an effective spin Hamiltonian in the Mott insulating regime.
• Analysis of phase transitions via the ratio of tunneling amplitudes $\tilde{t}^{a}_{x}/\tilde{t}^{f}_{y}$, identifying BKT-type transitions.

Experimental results

Research questions

• RQ1 Can dynamical gauge fields be realized in ultracold atomic systems through flux-attachment between two atomic species?
• RQ2 How does the interplay between matter and gauge fields lead to emergent $ζ_{2}$ gauge symmetry in optical lattices?
• RQ3 What phase transitions occur in the matter and gauge sectors when the tunneling ratio $\tilde{t}^{a}_{x}/\tilde{t}^{f}_{y}$ is tuned?
• RQ4 Under what conditions does the system realize a $ζ_{2}$ gauge theory with ordered electric or magnetic phases?
• RQ5 How does the Mott gap close and the system transition to a Luttinger liquid regime as $\tilde{t}^{f}_{y}/\tilde{t}^{a}_{y} \to \infty$?

Key findings

• The system realizes a $ζ_{2}$ lattice gauge theory with local $ζ_{2}$ symmetry when the flux-attachment protocol is implemented via coherent lattice modulation and strong interactions.
• A phase transition between electric and magnetic order phases occurs at a critical ratio $\tilde{t}^{f}_{y}/\tilde{t}^{a}_{y} \to \infty$, identified as a Berezinskii-Kosterlitz-Thouless (BKT) transition.
• In the limit $\tilde{t}^{a}_{y} \ll \tilde{t}^{f}_{y}$, the Mott gap scales as $\Delta \approx 0.5(\tilde{t}^{a}_{y})^{2}/\tilde{t}^{f}_{y}$, vanishing when $\tilde{t}^{f}_{y}/\tilde{t}^{a}_{y} \to \infty$.
• The effective low-energy Hamiltonian maps to an anti-ferromagnetic XXZ spin model with $J_{x} \geq J_{y} = J_{z}$, indicating spontaneous $ζ_{2}$ symmetry breaking.
• The order parameter $\langle \hat{\tau}^{z}_{\text{rung}} \rangle \neq 0$ confirms the presence of a $ζ_{2}$ gauge field in the ordered phase.
• When the Mott gap closes, the system transitions to two decoupled Luttinger liquids, signaling the breakdown of the $ζ_{2}$ gauge structure.