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arXiv:2104.11925 · Physical Review A

Interacting-fermion dynamics in Majorana phase space

Ria Rushin Joseph, P. D. Drummond, L. Rosales-Zárate

Published Dec 22, 2021• 2 citations
arXivHTMLPDF

Abstract

The problem of fermion dynamics is studied using the $Q$ function for fermions. This is a probabilistic phase-space representation, which we express using Majorana operators and a phase-space variable of real antisymmetric matrices. We consider a general Hamiltonian with quadratic and quartic Majorana interactions and arbitrary couplings. Our model includes the Majorana-Hubbard and Fermi-Hubbard Hamiltonians, as well as general quantum field theories of interacting fermions. The relevant coefficients are calculated for a number of both local and nonlocal Hamiltonian models. We also give a detailed example of a four-Majorana interaction. Using the Majorana $Q$ function we derive a generalized Fokker-Planck equation, with results for the drift and diffusion terms. The diffusion term is shown to be traceless, which has a dynamical interpretation as a forward-backward stochastic process. We prove that an initial pure state has trajectories that remain on a surface of pure states. This approach leads to a model of quantum measurement in terms of an ontology with real vacuum fluctuations.

DOI
10.1103/physreva.104.062208

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The problem of fermion dynamics is studied using the QQQ function for fermions. This is a probabilistic phase-space representation, which we express using Majorana operators and a phase-space variable of real antisymmetric matrices. We consider a general Hamiltonian with quadratic and quartic Majorana interactions and arbitrary couplings. Our model includes the Majorana-Hubbard and Fermi-Hubbard Hamiltonians, as well as general quantum field theories of interacting fermions. The relevant coefficients are calculated for a number of both local and nonlocal Hamiltonian models. We also give a detailed example of a four-Majorana interaction. Using the Majorana QQQ function we derive a generalized Fokker-Planck equation, with results for the drift and diffusion terms. The diffusion term is shown to be traceless, which has a dynamical interpretation as a forward-backward stochastic process. We prove that an initial pure state has trajectories that remain on a surface of pure states. This approach leads to a model of quantum measurement in terms of an ontology with real vacuum fluctuations.
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.
We obtain a positive probability distribution or Q -function for an arbitrary fermionic many-body system. This is different to previous Q -function proposals, which were either restricted to a subspace of the overall Hilbert space, or used Grassmann methods that do not give probabilities. The fermionic Q -function obtained here is constructed using normally ordered Gaussian operators, which include both non-interacting thermal density matrices and BCS states. We prove that the Q -function exists for any density matrix, is real and positive, and has moments that correspond to Fermi operator moments. It is defined on a finite symmetric phase-space equivalent to the space of real, antisymmetric matrices. This has the natural SO (2 M ) symmetry expected for Majorana fermion operators. We show that there is a physical interpretation of the Q -function: it is the relative probability for observing a given Gaussian density matrix. The distribution has a uniform probability across the space at infinite temperature, while for pure states it has a maximum value on the phase-space boundary. The advantage of probabilistic representations is that they can be used for computational sampling without a sign problem.
We introduce a positive phase-space representation for fermions, using the most general possible multimode Gaussian operator basis. The representation generalizes previous bosonic quantum phase-space methods to Fermi systems. We derive equivalences between quantum and stochastic moments, as well as operator correspondences that map quantum operator evolution onto stochastic processes in phase space. The representation thus enables first-principles quantum dynamical or equilibrium calculations in many-body Fermi systems. Potential applications are to strongly interacting and correlated Fermi gases, including coherent behavior in open systems and nanostructures described by master equations. Examples of an ideal gas and the Hubbard model are given, as well as a generic open system, in order to illustrate these ideas.
We use the Gaussian Phase-Space Representation to solve the real-time dynamic of interacting fermions in 1D, 2D, and 3D systems. The method is exact up to a spiking point, which represents a limit on the practical simulation time. The spiking can be delayed, and the practical simulation time extended, by adjusting the gauges of the representation, resulting in different equivalent stochastic differential equations. Here, we work on the so-called diffusion gauge and propose an algorithm to find efficiently new implementations of the noise terms. Compared with our initial results [F. Rousse \textit{et al.} 2024, J. Phys. A: Math. Theor. \textbf{57}, 015303], the new method achieves a significantly longer practical simulation time and can be applied to significantly larger systems.
Majorana phase-space representations for fermions map fermionic many-body physics into a distribution over one of the Cartan symmetric spaces of Lie group theory. The representation is in terms of $2M\ifmmode\times\else\texttimes\fi{}2M$ complex antisymmetric matrices, which generate the Gaussian Majorana operators. Here we show how this expansion can be utilized to calculate quantities arising in quantum thermodynamics and quantum information. Purity and the linear entropy are calculated, as well as the quantum fidelity between two general fermionic states, with numerical examples for pure states. We describe the geometrical properties of the phase space, and show that the overlap between two Gaussian Majorana states depends on the product of their antisymmetric matrices. Fermionic phase space is divided up into two orthogonal subspaces of different number parity, whose matrix representations differ by an orthogonal reflection in the phase space.
Abstract We formulate and discuss the explicit computation of dynamic correlation functions in open quadratic fermionic systems which are driven and dissipated by Lindblad jump processes that are linear in canonical fermionic operators. Dynamic correlators are interpreted in terms of a local quantum quench where the pre-quench state is a non-equilibrium steady state, i.e. a fixed point of the Liouvillian. As an example, we study the XY spin 1/2 chain and quadratic Majorana chains with boundary Lindblad driving, whose dynamics exhibit asymmetric (skewed) light cone behaviour. We also numerically treat the two-dimensional XY model and the XY spin chain with additional Dzyaloshinskii–Moriya interactions. The latter exhibits a new non-thermal phase transition which can be understood in terms of bifurcations of the quasi-particle dispersion relation. Finally, considering in some detail the periodic quadratic fermionic ring with dissipation at a single (arbitrary) site, we present analytical expressions for first order corrections (in the strength of dissipation) to the spectrum and non-equilibrium steady state (NESS) correlation functions.