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arXiv:2304.08034

Grassmann phase space dynamics of strongly-correlated fermion

Hassan Al-Hamzawi, Alessandro Principi, Leone Di Mauro Villari

Published Apr 17, 2023• 0 citations
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Abstract

We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Annals of Physics 370, 12 (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker-Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.

DOI
10.48550/arxiv.2304.08034

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We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Annals of Physics 370, 12 (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker-Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.
We discuss the numerical implementation of two related representations of fermionic density matrices which have been introduced in Dalton et al. (2016). In both of them, the density matrix is expanded in a basis of Bargmann coherent states with weights given by the two phase space distributions. We derive the equations of motion for the distributions when imaginary time evolution is generated by the Hubbard Hamiltonian. One of them is a Grassmann Fokker–Planck equation that can be re-cast into a remarkably simple Itô form involving solely complex variables. In spite of this simple form, we demonstrate that complications arise in numerically computing the expectation value of any observable. These are due to exponential growth in the matrix elements of the stochastic propagator, delicate numerical sensitivity in performing primitive linear algebra operations, and the re-appearance of a sign problem.
Abstract This paper concerns the application of Grassmann phase space theory (GSPT) to treat the dynamical evolution of systems of identical fermions, such as ultracold gases of fermionic atoms. Phase space theory (which originated from quantum optics) is increasing in importance since it overcomes certain issues associated with other theoretical methods, such as Greens functions, variational methods, quantum-Monte-Carlo equations, etc. In phase-space theory quantum states are represented by quasi-probability distribution functions of phase space variables associated with canonical system operators—such as annihilation, creation operators. Evolution is described via a Fokker-Planck equation for the distribution function, which is equivalent to Ito stochastic equations for (time dependent) stochastic phase space variables. Quantum correlation functions given as averages of products of phase space variables over the quasi-probability distributions then become stochastic averages of products of stochastic phase space variables. In GSPT, the phase space variables are Grassmann numbers, but as computer representation of g-numbers is difficult, Grassmann phase space methods were regarded as being computationally inaccessible. However, previous work using the un-normalised B distribution shows that computer representation of Grassmann variables is unnecessary. Stochastic averages of products for quantum correlation functions at later times are related linearly to stochastic averages at earlier times via stochastic matrices only involving c-numbers. Thus, GSPT calculations of quantum correlation functions now only involve c-number computations. This paper presents the first correct numerical calculation of a quantum correlation function for a fermionic system using stochastic methods based on Grassmann phase space theory, namely the time dependence of the coherence between two Cooper pair states in a four-mode fermion system, where the short and finite time solutions can be compared to known exact results. Good agreement between the stochastic and exact results is found, showing that GPST is a valid approach for treating fermionic systems. The treatment of time evolution involves a novel use of the eigenvalues and biorthogonal column eigenvectors of a stochastically determined c-number matrix M and its transpose. Other topics of interest in ultra-cold fermi gases for which the GSPT could be applied are highlighted, such as the strong interaction regime for the BEC/BCS crossover achieved using magnetically tuned Feshbach resonance techniques.
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.
Abstract Fermionic phase space representations are a promising method for studying correlated fermion systems. The fermionic Q-function and P-function have been defined using Gaussian operators of fermion annihilation and creation operators. The resulting phase-space of covariance matrices belongs to the symmetry class D, one of the non-standard symmetry classes. This was originally proposed to study mesoscopic normal-metal-superconducting hybrid structures, which is the type of structure that has led to recent experimental observations of Majorana fermions. Under a unitary transformation, it is possible to express these Gaussian operators using real anti-symmetric matrices and Majorana operators, which are much simpler mathematical objects. We derive differential identities involving Majorana fermion operators and an antisymmetric matrix which are relevant to the derivation of the corresponding Fokker–Planck equations on symmetric space. These enable stochastic simulations either in real or imaginary time. This formalism has direct relevance to the study of fermionic systems in which there are Majorana type excitations, and is an alternative to using expansions involving conventional Fermi operators. The approach is illustrated by showing how a linear coupled Hamiltonian as used to study topological excitations can be transformed to Fokker–Planck and stochastic equation form, including dissipation through particle losses.
Abstract The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker–Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems.
Phase space distribution functional methods utilising c-number and Grassmann fields for bosons and fermions respectively are being developed for theoretical treatments of degenerate quantum gases with large fermionic and bosonic atom numbers. The Jaynes-Cummings model where a single mode bosonic system and a two mode fermionic system are coupled provides a simple testbed for phase space methods involving Grassmann variables. The Fokker-Planck equation for the distribution function is shown to be equivalent to six coupled equations for c-number bosonic functions. Numerical solutions enable well-known effects such as collapses/revivals of the atomic population difference to be demonstrated.