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The Journal of Chemical Physics

Many-particle quantum dynamics: An exact algorithm for correlated motion on lattices

Qing Zhang, K. Birgitta Whaley

Published Apr 1, 1992• 2 citations
Publisher

Abstract

The time dependent many-body dynamics of strongly interacting particles on a two-dimensional lattice are studied with the recently developed checkerboard propagator. Application is made to the two-dimensional infinite U Hubbard model at high concentrations, for both fermion and boson systems. The concentration, initial state, and boundary condition dependence of vacancy correlation functions are analyzed, and related to the local densities of states. Quasidissipative behavior of the correlation functions is seen beyond a short initial time scale during which single-particlelike motion is seen.

DOI
10.1063/1.462717

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The time dependent many-body dynamics of strongly interacting particles on a two-dimensional lattice are studied with the recently developed checkerboard propagator. Application is made to the two-dimensional infinite U Hubbard model at high concentrations, for both fermion and boson systems. The concentration, initial state, and boundary condition dependence of vacancy correlation functions are analyzed, and related to the local densities of states. Quasidissipative behavior of the correlation functions is seen beyond a short initial time scale during which single-particlelike motion is seen.
Correlations in many-body quantum states lead to interesting phenomena while at the same time being enormously difficult to describe theoretically. Recent experiments conducted with ultracold atoms in optical lattices allow to explore quantum many-body correlations in a controlled and tunable way. The underlying theoretical model is the single-band Hubbard model. In this thesis we investigate the one-dimensional Hubbard model by applying the newly developed time-dependent two-particle reduced density matrix method. The method is based upon the assumption that three-particle correlations are negligible. We assess this approximation for equilibrium and non-equilibrium states. We show that the method is well suited for weakly interacting and weakly correlated systems.
A technique to perform real-time quantum propagation of many-particle systems on finite lattices is presented and applied to the motion of a single hole in the Heisenberg and Ising magnets (t-J and t-Jz{\mathit{J}}_{\mathit{z}}Jz​ models) for finite two-dimensional lattices of sizes 3\ifmmode\times\else\texttimes\fi{}3 and 4\ifmmode\times\else\texttimes\fi{}4. Use of Trotter's formula together with a checkerboard decomposition allows a direct real-time calculation of the many-fermion dynamics to be made for asymptotically long times. Starting from localized initial states, the hole-density distribution in the single-hole system is evaluated as a function of time. We find statistical behavior already appears in the 4\ifmmode\times\else\texttimes\fi{}4 lattice due to the large Hilbert space of the fermion background (\ensuremath{\approxeq}105{10}^{5}105 states). For J\ensuremath{\ll}t, the hole appears to move almost incoherently in both models, with small quantum fluctuations. For larger values of the spin coupling J, the t-Jz{\mathit{J}}_{\mathit{z}}Jz​ model shows a strong tendency to localize, while incorporation of the Heisenberg spin-flip term removes this feature and introduces an increasing component of low-frequency coherent motion for larger J.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
Exact solutions of many-body problems hold a very special and significant place in condensed matter physics, helping us to understand the behaviors of interacting particles. In this thesis, we focus our attention on study the exact solutions of two interesting many-body problems in stronglycorrelated \n \nsystems. One is to calculate the exact correlation functions of \n \n1D chains of spinless fermions and hardcore bosons. The other is to calculate the single-particle dispersion relation of Hubbard model of spinless fermions on 2D square lattices.
Analytical results on the correlation functions of strongly correlated many-body systems are rare in the literature and their importance cannot be overstated. We present determinant representations for the space-, time-, and temperature-dependent correlation functions of the strongly interacting one-dimensional Hubbard model in the presence of an external trapping potential. These representations are exact and valid in both equilibrium and nonequilibrium scenarios like the ones initiated by a sudden change of the confinement potential. In addition, they can be implemented numerically very easily, significantly outperforming other numerical approaches. As applications of our results we investigate the single-particle spectral functions of systems with harmonic trapping and show that dynamical quasicondensation occurs for both fermionic and bosonic spin-1/21/21/2 systems released from a Mott insulator state.