Toolbox for Abelian lattice gauge theories with synthetic matter

Omjyoti Dutta, Luca Tagliacozzo, Maciej Lewenstein, and Jakub Zakrzewski

Phys. Rev. A 95, 053608 – Published 4 May 2017

Abstract

Fundamental forces of nature are described by field theories, also known as gauge theories, based on a local gauge invariance. The simplest of them is quantum electrodynamics (QED), which is an example of an Abelian gauge theory. Such theories describe the dynamics of massless photons and their coupling to matter. However, in two spatial dimensions (2D), they are known to exhibit gapped phases at low temperature. In the realm of quantum spin systems, it remains a subject of considerable debate if their low-energy physics can be described by emergent gauge degrees of freedom. Here we present a class of simple two-dimensional models that admit a low-energy description in terms of an Abelian gauge theory. We find rich phase diagrams for these models comprising exotic deconfined phases and gapless phases—a rare example for 2D Abelian gauge theories. The counterintuitive presence of gapless phases in 2D results from the emergence of additional symmetry in the models. Moreover, we propose schemes to realize our model with current experiments using ultracold bosonic atoms in optical lattices.

Received 10 October 2016
Revised 10 March 2017

DOI:https://doi.org/10.1103/PhysRevA.95.053608

Physics Subject Headings (PhySH)

Cold atoms & matter waves Cold gases in optical lattices Quantum spin liquid

Condensed Matter, Materials & Applied PhysicsInterdisciplinary PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Omjyoti Dutta^1,2,4,*, Luca Tagliacozzo^3,4, Maciej Lewenstein^4,5, and Jakub Zakrzewski^1,6

¹Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagielloński, ulica Łojasiewicza 11, PL-30-348 Kraków, Poland
²Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain
³Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
⁴ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
⁵ICREA, Passeig Lluis Companys 23, 08010 Barcelona, Spain
⁶Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Kraków, Poland

^*omjyoti@gmail.com

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Issue

Vol. 95, Iss. 5 — May 2017

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Images

Figure 1
Key ingredients: (a) Left panel: The auxiliary particles (black dots) are trapped in superlattice geometry following the pattern in the figure. The tunneling rate is large (small) along dark (dashed) bonds. There is one particle for every dark bond. Middle panel: We need a large occupation of $b$ bosons on every site. This is achieved by trapping them along one-dimensional tubes (blue ovals) arranged in a square lattice geometry. Right panel: Upon appropriately shaking the setup (see text for details), the effective tunneling of the $a$ bosons, at low frequencies, is modulated in phase by the presence of the $b$ bosons. (b) Left panel: By further increasing the tunneling along dark bonds, the $a$ bosons delocalize on that bond. Middle panel: At second order in perturbation theory with respect to the weak tunneling, the virtual processes depicted create an effective plaquette interaction for the $b$ bosons (yellow sphere). Right panel: We can thus change variable to plaquette variables that can be thought as belonging to a coarse-grained lattice (shown by the wiggly blue lines) where the electric fields and the vector potentials live.
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Figure 2
Low-energy excitations and phase diagram: (a) At weak coupling, the low-energy excitations are plaquette excitations, magnetic fluxes of strength $\pm 2 / N$ . Excitations can only be created in pairs inside a column and are free to move along that column. Alternatively, dipoles of magnetic fluxes involving the excitations of two adjacent plaquettes are free to move along both lattice directions. (b) Qualitative phase diagram of the Hamiltonian $H_{plaq}$ in the $g - N$ plane. The upper shaded region denotes a gapped phase in the strong coupling limit. In the weak coupling limit, for low $N$ , the system is gapped but deconfined (region shaded in light blue). In the $U (1)$ limit, the system becomes gapless and an exotic dipole liquid phase emerges (region shaded in dark blue). In the intermediate region, the system is effectively in a one-dimensional gapless Bose liquid phase.
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Figure 3
A cartoon showing the Wilson loop in the coarse-grained lattice. The red line depicts the loop along which the vector potential is calculated. All the plaquettes inside the red line contribute to Wilson loop. Moreover, we have shown three possible positions for a single dipole. The top left and the bottom right dipole positions contributes to the Wilson loop a factor of $exp (\pm i 2 π / N)$ . The dipole placed in the top right position is totally inside the loop area and has a unity contribution.
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