$\mathrm{SU}(3)$ Fermi-Hubbard gas with three-body losses: symmetries and dark states

Abstract

We study an $\mathrm{SU}(3)$ invariant Fermi-Hubbard gas undergoing on-site three-body losses. The model presents eight independent strong symmetries preventing the complete depletion of the gas. By making use of a basis of semi-standard Young tableaux states, we reveal the presence of a rich phenomenology of stationary states. We classify the latter according to the irreducible representation of $\mathrm{SU}(3)$ to which they belong. We finally discuss the presence of three-particle stationary states that are not protected by the $\mathrm{SU}(3)$ symmetry.

Figures (11)

• Figure 1: (a) From left to right: a Young diagram of shape $\alpha = [3,3,3,2,1,1]$, a semi-standard Young tableau of the same shape, a Young diagram of shape $\overline{\alpha}=[6,4,3]$. (b) Example of a state written in terms of semi-standard Young tableaux for $N_{\rm f} = 7$ particles on $L=4$ sites, with $N_A = 2$, $N_B=3$ and $N_C=2$. From the spin part, we see that $p=2$, $q=1$, $r=1$ and $I = 1/2$. The total number of boxes in one tableau is $N_{\rm f} = p+2q+3r$. (c) Basis of the Hilbert subspace for $N_{\rm f}=3$ particles on $L=2$ lattice sites in the sector with one fermion $A$, one $B$, and one $C$. In this $8$-dimensional subspace, $4$ basis states belong to the $(p=0,q=0)$-irrep of $\mathrm{SU}(3)$ (in red) and $4$ basis states belong to the $(p=1,q=1)$-irrep (in blue).
• Figure 2: Possible states added at step $k$ corresponding to $n_k$ fermions located at the lattice site $k$ and belonging to the $(p_k,q_k)$-irrep of $\mathrm{SU}(3)$. Here, $i_k(i_k+1)$, $\lambda_k^3$ and $\lambda_8^k$ are the eigenvalues of the operators $I^2$, $\Lambda_3$ and $\Lambda_8$ respectively.
• Figure 3: Modification rules for a ssYT state $\ket{\Psi_{\rm ini}}$, having $n_j=3$, which undergoes a loss at site $j$.
• Figure 4: Mean number of fermions remaining on the lattice as a function of time for the parameters $(L,J, U_2, U_3,\gamma)=(3,1,1,0,1)$ and periodic boundary conditions. (a) The initial state is $\ket{\Psi_0} = c_{1A}^\dag c_{2B}^\dag c_{3C}^\dag \ket{\rm vac}$. (b) The initial state is $\ket{\Psi_0} = \left( c_{1A}^\dag c_{2B}^\dag c_{3C}^\dag + c_{2A}^\dag c_{3B}^\dag c_{1C}^\dag + c_{3A}^\dag c_{1B}^\dag c_{2C}^\dag \right) \ket{\rm vac}/\sqrt{3}$. For the two panels, the blue curve is computed via numerical simulations using stochastic quantum trajectories approach with $50000$ trajectories. The black dotted line is the lower bound in Eq. \ref{['eq:LowerBound_SU3']} (see Appendix \ref{['ap:L3N3']} for detailed calculation). The numerical calculation is performed with the open-source python-framework QuTiP JOHANSSON20121760JOHANSSON20131234.
• Figure 5: (a) Weight diagram for $(p=1,q=4)$. A weight space with degeneracy one is represented by a dot, while a weight space of degeneracy two is represented by a circled dot. The purple dots are weight spaces that maximize the imbalance between the three spin flavors. (b) Relation between the weight diagram and the spin ssYTs when $N_{\rm f}=p+2q$ for $(p=1,q=4)$. (c) Young diagram for the orbital part in the case $(p=1,q=4,r=0)$.