Dynamical symmetry between spin and charge excitations studied by a plaquette mean-field approach in two dimensions

The real-time dynamics of local occupation numbers in a Hubbard model on a $6\times 6$ square lattice is studied by means of the nonequilibrium generalization of the cluster-perturbation theory. The cluster approach is adapted to studies of two-dimensional lattice systems by using concepts of multiple-scattering theory and a component decomposition of the nonequilibrium Green's function on the Keldysh-Matsubara contour. We consider “classical” initial states formed as tensor products of states on $2\times 2$ plaquettes and trace the effects of the interplaquette hopping in the final-state dynamics. Two different initially excited states are considered on an individual plaquette, a fully polarized staggered spin state (Néel) and a fully polarized charge-density wave (CDW). The final-state dynamics is constrained by a dynamical symmetry; i.e., the time-evolution operator and certain observables are invariant under an antiunitary transformation composed of time reversal, an asymmetric particle-hole, and a staggered sign transformation. We find an interesting interrelation between this dynamical symmetry and the separation of energy and time scales: In the case of a global excitation with all plaquettes excited, the initial Néel and the initial CDW states are linked by the transformation. This prevents an efficient relaxation of the CDW state on the short time scale governing the dynamics of charge degrees of freedom. Contrarily, the CDW state is found to relax much faster than the Néel state in the case of a local excitation on a single plaquette where the symmetry relation between the two states is broken by the coupling to the environment.

Figure 1

Contour $\mathcal{C}$ in the complex time plane. $t_0$ is the time at which the system is prepared in the initial state. $t_{\text{max}}$ is the maximum time up to which observables are traced. $\beta$ is the inverse temperature. For a pure initial state $\left|\Psi \right(t_0)\rangle$ we have $\beta \to \infty$.

Figure 2

Decomposition of the Hubbard model on the two-dimensional square lattice into disconnected plaquettes. $T$ denotes the intra- and $V$ the interplaquette hopping. $U$ is the on-site Hubbard interaction. See text for discussion.

Figure 3

Pictorial representation of the Hamiltonians $B$ and $H$. The system is prepared as the ground state of $\mathcal{B}$ in the presence of a staggered field $h$ coupling to the $z$ component of the local spin or to the charge density. The evolution of the respective final states is given by the Hamiltonian $H$ with interplaquette hopping $V=T$ (nearest-neighbor hopping $T=-1$). Here, the decomposition into plaquettes is artificial and defines the NE-CPT scheme. Lattice sites as well as plaquettes are numbered. Interactions and hoppings are indicated exemplarily only.

Figure 4

Time evolution of the local spin-$\uparrow$ occupation numbers at the sites $j=1,...$ , $j=4$ for the system displayed in Fig. 3 (with the staggered field applied to every plaquette). Intercluster hopping $V=T$, Hubbard interaction $U=8\left|T\right|$. Energy and time scales are set by $T=-1$. Lines: Initial “spin-excitation state” $|\text{Néel}\rangle$ obtained with $h=100$ in Eq. (16). Dots: Initial “charge-excitation state” $|\mathrm{CDW}\rangle$, $h=100$; see Eq. (17).

Figure 5

Time evolution of the local spin-$\uparrow$ occupation numbers at the sites $j=1$,..., $j=4$ as in Fig. 4 ($U=8\left|T\right|$, $T=-1$) but with the initial field (strength $h=100$) applied to the sites in plaquette $p=1$ only. The initial states of plaquettes $p=2$,..., $p=9$ are the respective ground states $|\text{GS},p\rangle$. Top [(a), (b)]: Intraplaquette dynamics only ($V=0$). Bottom [(c), (d)]: Plaquettes coupled via NE-CPT ($V=T$). Left [(a), (c)]: Initial “spin-excitation state” on $p=1$, i.e., $|\text{Néel},1\rangle$. Right [(b), (d)]: Initial “charge-excitation state” on $p=1$, i.e., $|\mathrm{CDW},1\rangle$.