Symmetry Protected Dynamical Symmetry in the Generalized Hubbard Models

In this Letter we present a theorem on the dynamics of the generalized Hubbard models. This theorem shows that the symmetry of the single-particle Hamiltonian can protect a kind of dynamical symmetry driven by the interactions. Here the dynamical symmetry refers to the phenomenon that time evolution of certain observables are symmetric between the repulsive and attractive Hubbard models. We demonstrate our theorem with three different examples in which the symmetry involves bipartite lattice symmetry, reflection symmetry, and translation symmetry, respectively. Each of these examples relates to one recent cold atom experiment on the dynamics in the optical lattices where such a dynamical symmetry is manifested. These experiments include expansion dynamics of cold atoms, chirality of atomic motion within a synthetic magnetic field, and melting of charge-density-wave order. Therefore, our theorem provides a unified view of these seemingly disparate phenomena.

Figure 1

The presence (absence) of dynamical symmetry in the square (triangular) ladder. Panels (a) and (b) show the ladder configuration. (c) The time evolution of the local densities of the square ladder with attractive (left half) and repulsive (right half) interactions. The initial state is a two-boson state located at the central rung: $|{\mathrm{\Psi }}_0⟩={\widehat{c}}_{0,0}^{†}{\widehat{c}}_{0,1}^{†}|0⟩$. Here we take $(J_x,J_y,U)=2\pi \times (11,34,131)\mathrm{Hz}$. (d) The same as (c), except that we consider the triangular ladder case.

Figure 2

Dynamical symmetry of the Harper-Hofstadter model. (a) A schematic illustration of the Harper-Hofstadter model. An orbital magnetic field is applied to the square lattice, giving rise to flux $\varphi$ per plaquette. (b) The time-dependent shearing for the attractive (left half) and repulsive (right half) interaction cases. The initial state is a two-body state located at the central rung [Eq. (15)], as shown schematically by the inset. The parameters are the same as the ones in the experiment [16]: $\varphi =0.55\pi$, $(J_x,J_y,U)=2\pi \times (11,34,131)\mathrm{Hz}$. (c) The same as (b), except that we take a different initial state [Eq. (21)] as shown schematically by the inset.

Figure 3

Dynamical symmetry of the fermionic Aubry-André model. (a) The dashed line shows the superlattice potential $V(x)=\mathrm{\Delta }\cos (2\pi \beta x+\theta )$ with $\beta =1/4$ and $\theta =1.0$. The open circles indicate the equilibrium positions of the particles in the absence of $\mathrm{\Delta }$. When $\mathrm{\Delta }\ne 0$, the equilibrium positions are shifted, as shown by the filled circles. A unit cell for this case consists of four lattice sites, which are denoted as $A$, $B$, $C$, and $D$, respectively. (b)–(d) The time-dependent imbalance for different lattice periodicities ($\beta =1/4$ and $1/6$) and initial states [Eqs. (25) and ((26))]. The other parameters in (b)–(d) are the same: $\theta =1.0$, and $(J,\mathrm{\Delta },U)=2\pi \times (100,200,500)\mathrm{Hz}$.