Baryon squishing in synthetic dimensions by effective $\text{SU}\left(M\right)$ gauge fields

The “synthetic dimension” proposal [A. Celi et al., Phys. Rev. Lett. 112, 043001 (2014)] uses atoms with $M$ internal states (“flavors”) in a one-dimensional (1D) optical lattice, to realize a hopping Hamiltonian equivalent to the Hofstadter model (tight-binding model with a given magnetic flux per plaquette) on an $M$-sites-wide square lattice strip. We investigate the physics of $\text{SU}\left(M\right)$ symmetric interactions in the synthetic dimension system. We show that this system is equivalent to particles [with $\text{SU}\left(M\right)$ symmetric interactions] experiencing an $\text{SU}\left(M\right)$ Zeeman field at each lattice site and a non-Abelian $\text{SU}\left(M\right)$ gauge potential that affects their hopping. This equivalence brings out the possibility of generating nonlocal interactions between particles at different sites of the optical lattice. In addition, the gauge field induces a flavor-orbital coupling, which mitigates the “baryon breaking” effect of the Zeeman field. For $M$ particles, concomitantly, the $\text{SU}\left(M\right)$ singlet baryon which is site localized in the usual 1D optical lattice, is deformed to a nonlocal object (“squished baryon”). We conclusively demonstrate this effect by analytical arguments and exact (numerical) diagonalization studies. Our study promises a rich many-body phase diagram for this system. It also uncovers the possibility of using the synthetic dimension system to laboratory realize condensed-matter models such as the $\text{SU}\left(M\right)$ random flux model, inconceivable in conventional experimental systems.

Figure 1

Nonlocal interaction: The top panel shows the state of two fermions when $t=0$ with $M=2$, which has ${\omega }_{\zeta =1}=-\mathrm{\Omega }$ and ${\omega }_{\zeta =2}=\mathrm{\Omega }$. The arrows in the left bottom panel show the hopping pattern when $t\ne 0$ in the presence of a $\frac{1}{2}$ flux. If the two particles are in the neighboring sites as shown, then this baryon can effectively hop on a dual lattice shown by crosses (bottom right) by hybridizing with the degenerate baryon (vertical shaded bond), gaining kinetic energy. This produces a net attractive interaction between particles at neighboring sites with $\zeta =1$.

Figure 2

$M=2$. (a) and (b): “Phase diagram” of two particles showing dependence of (a) $I_{xx}$ and (b) $\langle \zeta \rangle$ on flux $p/q$ and $\mathrm{\Omega }/U$ for $t/U=0.1$. The insets show the type of bound state stabilized. (c)–(i), $p/q=\frac{1}{2}$: $E_b,I_{xx}$, and $\langle \zeta \rangle$ are respectively shown in (c)–(e). (f) shows the dependence of $E_b$ on $\mathrm{\Omega }/U$ for $t/U=0.1$, while (g) shows $I_{xx}$ and $\langle \zeta \rangle$ for the same case. The vertical black dotted lines show $\mathrm{\Omega }={\mathrm{\Omega }}_c$. The dependence of $E_b$ in (h) and $I_{xx},\langle \zeta \rangle$ in (i) on $t/U$ at $\mathrm{\Omega }={\mathrm{\Omega }}_c=U/2$ are shown. The dashed lines in (h) are results of analytical considerations at small and large $t/U$.

Figure 3

$M=3$. Phase diagram, (a) and (b): Three-particle phase diagram showing the dependence of (a) $I_{xx}$ and (b) $\langle \zeta \rangle$ on flux $p/q$ and $\mathrm{\Omega }/U$ for $t/U=0.1$ $\left(N_q=18\right)$. Insets: Type of bound state. (c) and (d), $p/q=\frac{1}{2}$: (c) and (d) show the binding energy, and $I_{xx}$ and $\langle \zeta \rangle$ vs $\mathrm{\Omega }/U$ with $t/U=0.1$. (e) and (f), $p/q=\frac{1}{3}$: (e) and (f) show same quantities as (c) and (d) for $\frac{1}{3}$ flux. The vertical black dotted lines show $\mathrm{\Omega }={\mathrm{\Omega }}_c$.

Figure 4

$M=4$. Phase diagram, (a) and (b): Four-particle phase diagram showing the dependence of (a) $I_{xx}$ and (b) $\langle \zeta \rangle$ on flux $p/q$ and $\mathrm{\Omega }/U$ for $t/U=0.1$ obtained with $N_q=8$. The insets show the type of bound state stabilized. (c) and (d), $p/q=\frac{1}{2}$: (c) and (d) show the dependence of the binding energy, and $I_{xx}$ and $\langle \zeta \rangle$ on $\mathrm{\Omega }/U$ with $t/U=0.1$. (e) and (f), $p/q=\frac{1}{4}$: (e) and (f) show same quantities as (c) and (d) for the $\frac{1}{4}$-flux case. The vertical black dotted lines show $\mathrm{\Omega }={\mathrm{\Omega }}_{c1}$ and $\mathrm{\Omega }={\mathrm{\Omega }}_{c2}$ $\left({\mathrm{\Omega }}_{c1}<{\mathrm{\Omega }}_{c2}\right)$.