Baryon squishing in synthetic dimensions by effectiveSU(M)gauge fields

Abstract

We investigate few body physics in a cold atomic system with synthetic dimensions (Celi et al., PRL 112, 043001 (2014)) which realizes a Hofstadter model with long-ranged interactions along the synthetic dimension. We show that the problem can be mapped to a system of particles (with SU (M ) symmetric interactions) which experience an SU (M ) Zeeman field at each lattice site and a non-Abelian SU (M ) gauge potential that affects their hopping from one site to another. This mapping brings out the possibility of generating non-local interactions (interaction between particles at different physical sites). It also shows that the non-Abelian gauge field, which induces a flavororbital coupling, mitigates the "baryon breaking" effects of the Zeeman field. For M particles, the SU (M ) singlet baryon which is site localized, is "deformed" to be a nonlocal object ("squished" baryon) by the combination of the Zeeman and the non-Abelian gauge potential, an effect that we conclusively demonstrate by analytical arguments and exact (numerical) diagonalization studies. These results not only promise a rich phase diagram in the many body setting, but also suggests possibility of using cold atom systems to address problems that are inconceivable in traditional condensed matter systems. As an example, we show that the system can be adapted to realize Hamiltonians akin to the SU (M ) random flux model. Celi et al. [38] proposed the concept of "synthetic dimensions" which achieves the goal of realizing finite sized "strip" of a Hofstadter model. Their idea, illustrated in Fig. 1, involves atoms with M internal states (labeled 1 . . . γ) in a 1D (this can also be in higher dimensions) optical lattice. The hopping of the atoms from a site j (with coordinate x j = jd, d is the spacing of the optical lattice) to its neighbour does not change its internal state, and the amplitude t is independent of γ. The internal states at a site j are now coherently coupled such that an atom in state γ at site j can "hop" to the state γ + 1 at j with an amplitude Ω j γ . This produces, as shown in Fig. 1, a square lattice strip of finite width with M sites along the "synthetic dimension". Since the coherent coupling is produced by a light of wavenumber k , we have Ω Another interesting aspect of the problem is that the SU (M ) symmetric interactions between the atoms at a site j manifest as "infinite-ranged" (distanceindependent) interactions along the synthetic dimension. For example, two atoms at site j (see Fig. 1) with γ = 1 and γ = 2 will interact with the same strength as γ = 1 and γ = 4(M ). It is the physics of such a system that is the subject of this paper, i. e., to understand interplay between the flux p/q and the SU (M ) interactions. It is essential to focus, as we do, on the physics of few particles since it provides crucial insights into constructing a many body phase diagram of the system. Previous studies [23,[25][26][27] of fermionic atoms with attractive SU (M )-interactions in a simple 1d lattice (no flux, i. e., p q = 0, Ω γ =...

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