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Physical review. B, Condensed matter

Exact time-dependent propagation of vacancy motion in thet-Jlimit of the two-dimensional Hubbard Hamiltonian

Qing Zhang, K. Birgitta Whaley

Published May 1, 1991• 13 citations
Publisher

Abstract

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-${\mathit{J}}_{\mathit{z}}$ 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}${10}^{5}$ 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-${\mathit{J}}_{\mathit{z}}$ 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.

DOI
10.1103/physrevb.43.11062

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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.
The effective Hamiltonian, obtained from the Hubbard model in the strong-coupling limit, is diagonalized exactly for a periodic two-dimensional square lattice of ten sites. We obtain the ground state of the system for any filling of the lattice and the KKK-space excitation spectrum for a single hole. Within our finite-size system, it is found that for very small coupling ratio tU\frac{t}{U}Ut​, each hole creates a local ferromagnetic environment at least of the size of our system, whereas for increasing tU\frac{t}{U}Ut​, the holes tend to form clusters, induced by antiferromagnetic spin correlations. A range of tU\frac{t}{U}Ut​ exists for which pairing of the holes may be possible.
Abstract Tremendous recent progress in the quantum simulation of the Hubbard model paves the way to controllably study doped antiferromagnetic Mott insulators. Motivated by these experimental advancements, we numerically study the real-time dynamics of a single hole created in an antiferromagnet on a square lattice, as described by the t–J model. Initially, the hole spreads ballistically with a velocity proportional to the hopping matrix element. At intermediate to long times, the dimensionality as well as the spin background determine the hole dynamics. A hole created in the ground state of a two dimensional (2D) quantum antiferromagnet propagates again ballistically at long times but with a velocity proportional to the spin exchange coupling, showing the formation of a magnetic polaron. We provide an intuitive explanation of this dynamics in terms of a parton construction, which leads to a good quantitative agreement with the numerical tensor network state simulations. In the limit of infinite temperature and no spin exchange couplings, the dynamics can be approximated by a quantum random walk on a Bethe lattice with coordination number z ̃ = 4 . Adding Ising interactions corresponds to an effective disordered potential, which can dramatically slow down the hole propagation, consistent with subdiffusive dynamics. The study of the hole dynamics paves the way for understanding the microscopic constituents of this strongly correlated quantum state.
Numerical calculations on two-dimensional spin-1/2 XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schr\"odinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving \ensuremath{\approxeq}107{10}^{7}107 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
Numerical calculations on two-dimensional spin-½ XXZ Heisenberg models are presented. The computational technique entails solving the time-dependent Schrödinger equation. The time propagation of wave functions is performed employing Trotter-Suzuki product formulas. The algorithms are numerically stable and are used to study models involving ≈10^7 states. With this approach both the thermodynamic properties as well as the many-particle dynamics can be obtained. Densities of states and zero-temperature dynamic magnetic structure factors are presented. For the isotropic Heisenberg antiferromagnet thermodynamic properties are computed for lattices up to 26 sites.
The ground state of the two-dimensional Hubbard model on a square lattice is studied in the large-U limit in a half-filled band with a dynamical hole. For a 4\ifmmode\times\else\texttimes\fi{}4 lattice with hopping t=1 and coupling JJc{J}_{c}Jc​\ensuremath{\approxeq}0.075, the ground state has zero momentum (k) and spin 15/2 (ferromagnetic background). For J\ensuremath{\ge}Jc{J}_{c}Jc​ the ground state is degenerate with nonzero k and spin 1/2, in agreement with recent variational calculations. The nonzero k of the ground state comes from a nontrivial phase in the overlap of the spin states before and after a hole move. For small lattices we show that the effective Hamiltonian of the hole is that of a particle moving in a magnetic field. We also find that many of the features of the weak-coupling region carry over continuously to the strong-coupling region.