Quench dynamics of Anderson impurity model using configuration interaction method

We study the quench dynamics of an Anderson impurity model using the configuration interaction (CI) method. In particular, we focus on the relaxation behavior of the impurity occupation. The system is found to behave very differently in the weak-coupling and strong-coupling regimes. In the weak-coupling regime, the impurity occupation relaxes to a time-independent constant quickly after only a few oscillations. In the strong-coupling regime, the impurity occupation develops a fast oscillation, with a much slower relaxation. We show that it is the multipeak structure in the many-body energy spectrum that separates these two regimes. The characteristic behavior, including the power-law decay and the period of oscillation, can also be related to certain features in the many-body energy spectrum. The respective advantages of several impurity solvers are discussed, and the convergence of different CI truncation schemes is provided.

Figure 1

Illustration of CI steps to systematically construct a small subspace. In a CI scheme, a state is classified by the number of particle-hole pairs with respect to a reference Fock state and the orbitals by its occupation. (a) Based on the former, $\text{CI-}{n}_{p-h}$ is to denote a subspace containing states of $0,1...,n_{p-h}$ p-h pairs. CI-3 is illustrated. (b) Based on the latter, each kept Fock state has mostly empty secondary orbitals and mostly occupied inactive orbitals but has no restriction on the active orbitals. (c) Out of the full CI space, the truncated subspace is chosen by some CI scheme.

Figure 2

Time dependence of impurity occupation for $N_o=N_e=8, U=2$, and $\mu =-U/2=-1$. $N_o=8$ is chosen so that the exact evaluation is available. The exact result (black curve) is obtained by diagonalizing the full Hamiltonian without truncating the Hilbert space. The CAS(4,8) result (red curve) is also given, and a good agreement is seen.

Figure 3

(a) Time dependence of impurity occupation for $N_o=N_e=20, U=2$, and $\mu =-U/2=-1$. Five initial states are chosen as the noninteracting ground state of different impurity potential $\mu =-2,-1,0,1,2$. They all converge to $n_{\mathrm{imp}}\sim 1$. (b) Time dependence of impurity occupation for $N_o=20, N_e=16, U=2$, and $\mu =-U/2=-1$. Three initial states are chosen as the noninteracting ground states of different impurity potential $\mu =0,1$, and 2. They all converge to the values close to $n_{\mathrm{imp}}\sim 0.815$ (blue dashed line).

Figure 4

Time dependence of impurity occupation for $U=2, \mu =-U/2=-1$, and $N_o=N_e=10,20$, and 30. The initial state is chosen as the ground state of impurity potential $\mu =-2$ and $U=0$. Including more bath orbitals leads to a long-time $n_{\mathrm{imp}}$ value closer to one.

Figure 5

Time dependence of the impurity double occupancy $D_{\mathrm{imp}}\left(t\right)\equiv \langle \mathrm{\Psi }\left(t\right)|n_{\mathrm{imp},\uparrow }{n}_{\mathrm{imp},\downarrow }|\mathrm{\Psi }\left(t\right)\rangle$. (a) $D_{\mathrm{imp}}\left(t\right)$ under the Hamiltonian with $N_o=20, N_e=20, U=2, \mu =-1$, with two initial conditions chosen as the ground states of the Hamiltonian with $U=0, \mu =-2$, and 2. (b) $D_{\mathrm{imp}}\left(t\right)$ under the Hamiltonian with $N_o=20, N_e=20, U=1, \mu =-0.5$, with two initial conditions chosen as the ground states of the Hamiltonian with $U=0, \mu =-1$, and 1. The blue dashed line indicates the $\langle n_{\mathrm{imp},\uparrow }{n}_{\mathrm{imp},\downarrow }\rangle$ computed from the interacting ground state.

Figure 6

(Left) Evolution of the impurity occupation under Hamiltonians of $U=2$ (weak coupling), 6 (intermediate coupling), and 20 (strong coupling), with $\mu =-U/2$. The initial states are chosen as the noninteracting ground state of $\mu =0.5,2,-1$, respectively. $N_e=N_o=20$ is used. The dotted curve is the analytical fit using $n_0-n_1{J}_1\left(2Bt\right)/\left(Bt\right)$ with $n_0=1.013, n_1=0.354$, and $B=1.825$. (Right) The histograms of the energy-pair difference $D\left(\omega \right)\sim {\sum }_{n,m;n\ne m}\delta (\omega -(E_m-E_n))$, with the many-body eigenenergy $E_i$ computed in the CAS(4,8) truncated space. The normalization is chosen to satisfy $\int d\omega D\left(\omega \right)=1$.

Figure 7

(a) Time dependence of impurity occupation for $N_o=N_e=20, U=0$, and $\mu =\pm 2,\pm 1,\pm 0.5$. Initial states are chosen as the ground state of different impurity potential $\mu =-1$ and $U=2$. $n_{\mathrm{imp}}$ approaches to the expectation value (blue dashed lines) of the noninteracting Hamiltonian. (b) Time dependence of impurity occupation for $N_o=N_e=20, U=0$, and $\mu =2$. Three initial states are chosen as the many-body ground states of $U=2,10,20$ and $\mu =-U/2$.

Figure 8

(a) Evolution of impurity occupation for the Hamiltonian of $N_o=N_e=8, U=20$, and $\mu =-10$. The initial state is the ground state of $N_o=N_e=8, U=0$, and $\mu =-10$. The exact (black), CAS(4,8) (red), and RAS(4,$-1$;4,1) (blue) results are shown. For the long-time limit, RAS(4,$-1$;4,1) behaves better than CAS(4,8), although for $t<10$ the latter appears to be closer to the exact result. (b) Time dependence of impurity occupation for $N_o=N_e=20, (U,\mu )=(2,-1)$, and $(U,\mu )=(20,-10)$. The initial states are the noninteracting ground state of $\mu =2$ [for $(U,\mu )=(2,-1)$] and $\mu =-1$ [for $(U,\mu )=(20,-10)$], respectively. Results from CAS(4,8) and CAS(6,12) are shown.