Nonequilibrium cluster perturbation theory

The cluster perturbation theory (CPT) is one of the simplest but systematic quantum cluster approaches to lattice models of strongly correlated electrons with local interactions. By treating the interactions, in addition to the intercluster potential, as a perturbation, it is shown that the CPT can be reformulated as an all-order resummation of diagrams within standard weak-coupling perturbation theory where vertex corrections are neglected. This reformulation is shown to allow for a straightforward generalization of the CPT to the general nonequilibrium case using contour-ordered Green’s functions. By solving the resulting generalized CPT equation on the discretized Keldysh-Matsubara time contour, the transient dynamics of an essentially arbitrary initial pure or mixed state can be traced. In this way, the time-dependent expectation values of one-particle observables can be obtained within an approximation that neglects spatial correlations beyond the extension of the reference cluster. The necessary computational effort is very moderate. A detailed discussion and simple test calculations are presented to demonstrate the strengths and the shortcomings of the proposed approach. The nonequilibrium CPT is systematic and is controlled in principle by the inverse cluster size. It interpolates between the noninteracting and the atomic or decoupled-cluster limit which are recovered exactly and is found to predict the correct dynamics at very short times in a general nontrivial case. The effects of initial-state correlations on the subsequent dynamics and the necessity to extend the Keldysh contour by the imaginary Matsubara branch are analyzed carefully and demonstrated numerically. It is further shown that the approach can describe the dissipation of spin and charge to an uncorrelated bath with an essentially arbitrary number of degrees of freedom.

Figure 1

Three-branch contour $\gamma$ in the complex time plane. The upper and the lower real branches define the Keldysh contour; the imaginary branch is called the Matsubara branch.

Figure 2

Partitioning of a $D=2$ square lattice into clusters with $L_c=4$ sites each. $T_{I,\mathit{ij}}$ denotes the intracluster hopping between sites $i$ and $j$ within the same cluster $I$. $V_{\mathit{IJ},\mathit{ij}}$ is the intercluster hopping between sites $i\in I$ and $j\in J$.

Figure 3

Resummation of diagrams generated by electron-electron scattering $U$ and by scattering at the intercluster potential $V$. (a) Exact procedure: Renormalization of the free ($U=V=0$) propagator ${\mathit{G}}_0^{\prime }$ by potential scattering followed by renormalization of the $U=0$ propagator ${\mathit{G}}_0$ due to electron scattering [Eqs. (26) and (27)]. (b) CPT: Renormalization of the free ($U=V=0$) propagator ${\mathit{G}}_0^{\prime }$ by electron scattering followed by renormalization of the $V=0$ propagator ${\mathit{G}}^{\prime }$ due to potential scattering [Eqs. (29) and (30)]. (c) Self-energy diagram, second order in $U$, second order in $V$, which is not taken into account within CPT.

Figure 4

Time dependence of the site occupations in a four-site Hubbard chain for $U=0$. Results of the nonequilibrium cluster-perturbation theory (lines) are compared with the exact result (points) for different time discretizations $\Delta t$ on the Keldysh contour. The initial state is displayed schematically. At $t=0$ four electrons occupy the sites 1 and 2 while sites 3 and 4 are empty. The CPT treats the nearest-neighbor hopping between sites 2 and 3 perturbatively to all orders.

Figure 5

Time dependence of the spin-$\uparrow$ site occupations in a four-site Hubbard chain for $U=0$ as obtained by nonequilibrium cluster-perturbation theory starting from a reference system with decoupled two-site clusters and treating the nearest-neighbor hopping between sites 2 and 3 perturbatively to all orders. The initial state is the ground state of the $U=0$ chain at half filling with a local magnetic field of strength $-h$ at site 1 and $+h$ at site 2 with $h=0.2$. CPT results for large $\beta =10$ and for $\beta =0$ (i.e., neglecting the effects of the Matsubara branch). Exact results (points) are shown for comparison.

Figure 6

Time evolution as obtained from nonequilibrium CPT (red), exact-time evolution (blue), and time evolution of the reference system (green) in the strong-coupling regime ($U=8$) of a half-filled four-site Hubbard chain with a Néel initial state. Reference system: Decoupled two-site clusters. Intercluster potential: Nearest-neighbor hopping connecting sites 2 and 3.

Figure 7

Pictorial representation of the final-state Hamiltonian $H$ governing the time evolution and the Hamiltonian $B$ generating the initial state as its ground state. The system size is $i_0=L_c$. The bath is a semi-infinite linear chain starting at $i_{0,\mathrm{bath}}=L_c+1$. The coupling between system and bath $V$ is treated within CPT ($V=T$). Blue circles represent sites with $U>0$ while open red circles stand for sites with $U=0$. Initially ($t=0$) there is a ground state of a an isolated two-site Hubbard cluster perturbed by a staggered field $h$ coupling to spin or charge degrees of freedom. The field is suddenly switched off, and the coupling of the two-site cluster to the rest of the system is switched on. Due to particle-hole symmetry, the entire system (“system” plus bath) is and stays at half filling.

Figure 8

Time evolution of the average occupation $n_{\uparrow }=\langle n_{i\uparrow }\rangle =1-\langle n_{i\downarrow }\rangle$ at site $i=1$ (upper) and site $i=L_c$ (lower panel) of the system displayed in Fig. 7 after a spin excitation using a staggered field $h_{\mathrm{spin}}=0.1$ in the initial-state Hamiltonian [see Eq. (60)]. Insets: Comparison of the CPT result for $L_c=8$ with the corresponding result obtained from a calculation for an isolated system without bath. The CPT calculations have been performed for different system size $L_c$ as indicated. Further parameters: $U=2$, half filling, $T=V=1$.

Figure 9

Time evolution of the average occupation $n=\langle n_{i\uparrow }\rangle =\langle n_{i\downarrow }\rangle$ at site $i=1$ of the system displayed in Fig. 7 after a charge excitation with $h_{\mathrm{charge}}=0.1$ [see Eq. (60)]. System size $L_c$ as indicated, $U=2$, half filling, $T=V=1$.