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arXiv:2212.04352 · Physical review. B./Physical review. B

Correlated quantum dynamics of graphene clusters

François Rousse, Olle Eriksson, Magnus Ögren

Published Apr 20, 2023• 2 citations
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Abstract

Phase-space representations are a family of methods for dynamics of both bosonic and fermionic systems, that work by mapping the system's density matrix to a quasiprobability density and the Liouville-von Neumann equation of the Hamiltonian to a corresponding density differential equation for the probability. We investigate here the accuracy and the computational efficiency of one approximate phase-space representation, called the fermionic truncated Wigner approximation (fTWA), applied to the Fermi-Hubbard model. On a many-body 2D system, with hopping strength and Coulomb $U$ tuned to represent the electronic structure of graphene, the method is found to be able to capture the time evolution of first-order (site occupation) and second-order (correlation functions) moments significantly better than the mean-field, Hartree-Fock method. The fTWA was also compared to results from the exact diagonalization method for smaller systems, and in general the agreement was found to be good. The fully parallel computational requirement of fTWA scales in the same order as the Hartree-Fock method, and the largest system considered here contained 198 lattice sites.

DOI
10.1103/physrevb.107.134306

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Phase-space representations are a family of methods for dynamics of both bosonic and fermionic systems, that work by mapping the system's density matrix to a quasiprobability density and the Liouville-von Neumann equation of the Hamiltonian to a corresponding density differential equation for the probability. We investigate here the accuracy and the computational efficiency of one approximate phase-space representation, called the fermionic truncated Wigner approximation (fTWA), applied to the Fermi-Hubbard model. On a many-body 2D system, with hopping strength and Coulomb UUU tuned to represent the electronic structure of graphene, the method is found to be able to capture the time evolution of first-order (site occupation) and second-order (correlation functions) moments significantly better than the mean-field, Hartree-Fock method. The fTWA was also compared to results from the exact diagonalization method for smaller systems, and in general the agreement was found to be good. The fully parallel computational requirement of fTWA scales in the same order as the Hartree-Fock method, and the largest system considered here contained 198 lattice sites.
We introduce the parafermionic truncated Wigner approximation (pppTWA), a semiclassical phase-space framework for simulating the nonequilibrium dynamics of lattice systems with fractional exchange statistics. The method extends truncated Wigner approaches developed for bosonic and fermionic systems to Zn\mathbb{Z}_nZn​ Fock parafermions by expressing the Hamiltonian in terms of local Hubbard operators that form a closed Lie algebra. This representation leads to a Lie--Poisson phase-space formulation in which quantum dynamics is approximated by stochastic sampling of initial conditions followed by deterministic semiclassical evolution. We benchmark the approach in several settings, including single-site clock dynamics, the fully connected Zn\mathbb{Z}_nZn​ clock model, long-range Z3\mathbb{Z}_3Z3​ clock chains, and disordered Z3\mathbb{Z}_3Z3​ Fock parafermion chains. The method reproduces key features of the exact dynamics, including excitation spreading, disorder-induced suppression of transport, and the emergence of long-time imbalance plateaus. Our results demonstrate that pppTWA provides a practical tool for exploring the dynamics of parafermionic systems in regimes where exact numerical methods are limited by Hilbert-space growth.
The quench dynamics of the Hubbard model in tilted and harmonic potentials is discussed within the semiclassical picture. Applying the fermionic truncated Wigner approximation (fTWA), the dynamics of imbalances for charge and spin degrees of freedom is analyzed and its time evolution is compared with the exact simulations in a one-dimensional lattice. Quench from charge or spin density waves is considered. We show that introduction of harmonic and spin-dependent linear potentials sufficiently validates fTWA for longer times. Such an improvement of fTWA is also obtained for the higher-order correlations in terms of quantum Fisher information for charge and spin channels. This allows us to discuss the dynamics of larger system sizes and connect our discussion to the recently introduced Stark many-body localization. In particular, we focus on a finite two-dimensional system and show that at intermediate linear potential strength, the addition of a harmonic potential and spin dependence of the tilt results in subdiffusive dynamics, similar to that of disordered systems. Moreover, for specific values of harmonic potential, we observed phase separation of ergodic and nonergodic regions in real space. The latter fact is especially important for ultracold-atom experiments in which harmonic confinement can be easily imposed, causing a significant change in relaxation times for different lattice locations.
The dynamics of disordered two-dimensional systems is much less understood than the dynamics of disordered chains, mainly due to the lack of appropriate numerical methods. We demonstrate that a single-trajectory version of the fermionic truncated Wigner approximation (fTWA) gives unexpectedly accurate results for the dynamics of one-dimensional (1D) systems with moderate or strong disorder. Additionally, the computational complexity of calculations carried out within this approximation is small enough to enable studies of two-dimensional (2D) systems larger than standard fTWA. Using this method, we analyze the dynamics of interacting spinless fermions propagating on disordered 1D and 2D lattices. We find for both spatial dimensions that the imbalance exhibits a universal dependence on the rescaled time $t/{\ensuremath{\xi}}_{W}$, where in two dimensions the timescale ${\ensuremath{\xi}}_{W}$ follows a stretched-exponential dependence on disorder strength.
We review the truncated Wigner approximation (TWA) for spins as a computationally inexpensive numerical approximation method to describe interacting and / or dissipative many-body spin systems. Using the Wigner-Moyal mapping from Hilbert space to a suitable phase space, the many-body density matrix is represented by a c-number distribution, the Wigner function. The gauge freedom in continuous phase space can be exploited to find positive Wigner functions for a large class of spin states, including entangled ones. Employing different sets of correspondence rules, we derive equations of motion for the Wigner function, which, applying controlled approximations, can be mapped to stochastic differential equations. This allows a computationally inexpensive simulation of expectation values. Using a phase-space analog of the quantum regression theorem also multi-time correlations and spectra can be obtained. To illustrate the potential of the method, we benchmark the TWA for spins with some exactly solvable problems of interacting, dissipative spin systems, and then discuss its application to collective processes, such as the superradiant emission of light. Extending the TWA to imaginary time furthermore provides a tool to approximately calculate thermal and ground states of spin Hamiltonians. Finally, we show that the TWA stochastic equations can equivalently be derived within a path-integral approach, provided that the operator products in the dissipator are rigorously mapped onto the curved phase space.
A Gaussian operator representation for the many-body density matrix of fermionic systems, developed by Corney and Drummond [Phys. Rev. Lett. 93, 260401 (2004)], is used to derive approximate decoupling schemes for their dynamics. In this approach the reduced single electron density-matrix elements serve as stochastic variables which satisfy an exact Fokker-Planck equation. The number of variables scales as $\ensuremath{\sim}{N}^{2}$ rather than $\ensuremath{\sim}\text{exp}(N)$ with the basis set size, and the time-dependent Hartree-Fock approximation (TDHF) is recovered in the ``classical'' limit. An approximate closed set of equations of motion for the one- and two-particle reduced density matrices provides a direct generalization of the TDHF.
We develop and utilize the SU(3) truncated Wigner approximation (TWA) in order to analyze far-from-equilibrium quantum dynamics of strongly interacting Bose gases in an optical lattice. Specifically, we explicitly represent the corresponding Bose-Hubbard model at an arbitrary filling factor with restricted local Hilbert spaces in terms of SU(3) matrices. Moreover, we introduce a discrete Wigner sampling technique for the SU(3) TWA and examine its performance as well as that of the SU(3) TWA with the Gaussian approximation for the continuous Wigner function. We directly compare outputs of these two approaches with exact computations regarding dynamics of the Bose-Hubbard model at unit filling with a small size and that of a fully connected spin-1 model with a large size. We show that both approaches can quantitatively capture quantum dynamics on a timescale of $\ensuremath{\hbar}/(Jz)$, where JJJ and zzz denote the hopping energy and the coordination number. We apply the two kinds of SU(3) TWA to dynamical spreading of a two-point correlation function of the Bose-Hubbard model on a square lattice with a large system size, which has been measured in recent experiments. Noticeable deviations between the theories and experiments indicate that proper inclusion of effects of the spatial inhomogeneity, which is not straightforward in our formulation of the SU(3) TWA, may be necessary.