Fermionic Coherent State Representation Research Articles

Master equation approach to transient quantum transport in nanostructures

In this review article, we present a non-equilibrium quantum transport theory for transient electron dynamics in nanodevices based on exact master equation derived with the path integral method in the fermion coherent-state representation. Applying the exact master equation to nanodevices, we also establish the connection of the reduced density matrix and the transient quantum transport current with the Keldysh nonequilibrium Green functions. The theory enables us to study transient quantum transport in nanostructures with back-reaction effects from the contacts, with non-Markovian dissipation and decoherence being fully taken into account. In applications, we utilize the theory to specific quantum transport systems, a variety of quantum decoherence and quantum transport phenomena involving the non-Markovian memory effect are investigated in both transient and stationary scenarios at arbitrary initial temperatures of the contacts.

Fermionic-mode entanglement in non-Markovian environment

We evaluate the non-Markovian effects on the entanglement dynamics of a fermionic system interacting with two dissipative vacuum reservoirs. The exact solution of density matrix is derived by utilizing the Feynman–Vernon influence functional theory in the fermionic coherent state representation and the Grassmann calculus, which are valid for both the fermionic and bosonic baths, and their difference lies in the dependence of the parity of the initial states. The fermionic entanglement dynamics is presented by adding an additional restriction to the density matrix known as the superselection rules. Our analysis shows that the usual decoherence suppression schemes implemented in qubits systems can also be achieved for systems of identical fermions, and the initial state proves its importance in the evolution of fermionic entanglement. Our results provide a potential way to decoherence controlling of identical fermions.

Efficient simulation of Grassmann tensor product states

Recently, the Grassmann-tensor-entanglement renormalization group(GTERG) approach was proposed as a generic variational approach to study strongly correlated boson/fermion systems. However, the weakness of such a simple variational approach is that generic Grassmann tensor product states(GTPS) with large inner dimension $D$ will contain a large number of variational parameters and be hard to be determined through usual minimization procedures. In this paper, we first introduce a standard form of GTPS which significantly simplifies the representations. Then we describe a simple imaginary-time-evolution algorithm to efficiently update the GTPS based on the fermion coherent state representation and show all the algorithm developed for usual tensor product states(TPS) can be implemented for GTPS in a similar way. Finally, we study the environment effect for the GTERG approach and propose a simple method to further improve its accuracy. We demonstrate our algorithms by studying a simple 2D free fermion system on honeycomb lattice, including both off-critical and critical cases.

Fermionic stochastic Schrödinger equation and master equation: An open-system model

This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath, to the models in which the system operators commute with the fermion bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schr\"{o}dinger equation (SSE), called fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced density operator for the open system can be recovered from the average of the solutions to the QSD equation over the Grassmann-type noise. By employing the exact fermionic QSD equation, we can derive the corresponding exact master equation. The power of our approach is illustrated by the applications of our stochastic approach to several models of interest including the one-qubit dissipative model, the coupled two-qubit dissipative model, the quantum Brownian motion model and the N-fermion model coupled to a fermionic bath. Different effects caused by the fermionic and bosonic baths on the dynamics of open systems are also discussed.

Newton–Leibniz integration for ket–bra operators (III)—Application in fermionic quantum statistics

The Newton–Leibniz integration over Dirac’s ket–bra operators introduced in Ref. [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480–494] is generalized to Newton–Leibniz–Berezin integration over fermionic ket–bra projection operators, the corresponding technique of integration within an ordered product (IWOP) of Fermi operators is proposed which is then used to develop fermionic quantum statistics. The generalized partition function formula of multi-mode quadratic interacting fermion is derived via the fermionic coherent state representation and the IWOP technique. The two-mode fermionic squeezing operators and their group property studied by their fermionic coherent state representation as well as fermionic permutation operator are also deduced in this way. Thus Dirac’s symbolic method for Fermi system can also be developed, which exhibits Bose-Fermi supersymmetry in this aspect.

A group property for the coherent state representation of fermionic squeezing operators

For the two-mode fermionic squeezing operators we find that their coherent state projection operator representation make up a loyal representation, which is homomorphic to an SO(4) group, though the fermionic coherent states are not mutual orthogonal. Thus the result of successively operating with many fermionic squeezing operators on a state can be equivalent to a single operation. The fermionic squeezing operators are mappings of orthogonal transformations in Grassmann number pseudo-classical space in the fermionic coherent state representation.

GENERALIZED FERMI PARTITION FUNCTION OBTAINED VIA COHERENT STATE REPRESENTATION OF THE DENSITY OPERATOR

We generalize the well-known partition function of Fermi harmonic oscillator (H =ω(f+f + ½))[Formula: see text] to [Formula: see text] for Fermi system described by the n-mode fermionic Hamiltonian [Formula: see text]. This result is derived via constructing the fermionic coherent state representation of the density operator [Formula: see text] in which [Formula: see text] is identified as exp {-β Γ Π}.

Fermionic Coherent State Representation for Generalized Fermionic Bogolyubov Transformation

We show that the generalized fermionic Bogolyubov transformation generator [Formula: see text], where Λ is an even n×n matrix, has the manifest fermionic coherent state representation [Formula: see text] The normally ordered form of V(Λ) can be obtained through the technique of integration within an ordered product of fermionic operators. This work is a supersymmetric partner of Ref. 1.

Normally ordered unitary operator for multimode squeezed fermion states

We derive normally ordered unitary operators for generalizing multimode squeezed fermion states. The operators are considered as images corresponding to a pseudo-classical Grassmann numbers' SO(2 n) orthogonal transformation in the n-mode fermionic coherent state representation. The derivation can be facilitated by the newly developed technique of integration within an ordered product of operators for the fermionic system.

Similarity transformation for Fermi operators

The normally ordered Hilbert space image of the complex linear similarity transformation in Grassmann number phase space is derived in fermionic coherent state representation. Though keeping the anticommutator of Fermi operators invariant, the quantum mechanical images of classical transformations are generally not unitary. Remarkably, although kets and bras produced by the non-unitary similarity transformations are not Hermitian conjugates, they still form a complete basis. The derivation of the normally ordered operator which engenders the similarity transformation is facilitated by virtue of the technique of integration within order products.

Spin Depolarization of a Quantum Particle; Fermionic Path Integral Approach

The spin depolarization of a quantum particle in random magnetic field is treated in terms of a path integral in the spin states as well as the translational motion. The path integral of the spin operator, represented in terms of Fermion operators, is performed in the Fermion coherent state representation. An expression for the spin depolarization is obtained for an arbitrary magnetic field distribution. A particular case, in which the magnetic field is oriented in the z -direction and the magnitude is Gaussianly distributed, is considered and the spin decays exponentially in time.

Normally ordered Fermi operator realization of the SUn group

The mapping of the SUn group transformation in Grassmann number space into unitary normally ordered and antinormally ordered Fermi operator realizations in Hilbert space are investigated. The unitary Fermi operators are evaluated in fermion coherent state representation using the ‘‘integration within ordered product’’ technique for a fermionic system. Some new Fermi operator identities are thus obtained.

A new two-mode fermion coherent state representation

A technique of integration within the ordered product for a fermion system is introduced. In terms of the technique, a new two-mode fermion coherent state representation is found. Some properties and applications of the new representation are derived.

Mapping of a classical transformation in Grassmann number space to a Fermi unitary operator.

The formalism of mapping a classical canonical transformation to quantum unitary operators for a boson system is generalized to fermion systems. In terms of the ``integration within an ordered product'' technique for fermion systems, the transition of a classical transformation in Grassmann number space to a quantum image is manifestly evaluated in the fermion coherent-state representation. The normally ordered unitary operators that generate the Bogolyubov-Valatin transformation and the SU(2) rotation are derived in this way.

Microscopic theory of large-amplitude collective motions based on theSO(2N+1) lie algebra of the fermion operators

A microscopic theory of large-amplitude collective motions is proposed on the basis of theSO(2N+1) Lie algebra of the fermion operators. Following the idea of Kuriyama-Yamamura, a canonical condition for collective variables is imposed under the adoption of anSO(2N+1) wave function. We thus obtain constraints governing the collective and «independent-particle» degrees of freedom. Furthermore, we get equations to determine a collective path on theSO(2N) group manifold by requiring the condition that the “angerous» term should vanish. An approximate solution of our equations is shown to be the well-known RPA one in the small-amplitude limit. Approximate forms of theSO(2N+1) energy and density matrices are also given. Our theory is in striking contrast with the Kuriyama-Yamamura theory using the fermion coherent-state representation. The latter forces us to introduce inevitably anticommuting Grassmann numbers.

A Microscopic Theory of Collective and Independent-Particle Motions

A microscopic theory of collective and independent·particle motion in many· fermion system is developed in the framework of the classical theory. The basic idea is an extention of the conventional time-dependent Hartree-Fock method with the use of fermion coherent state representation. Equation to determine collective path and constraints to govern the collective and the independent-particle variables are given. Hamiltonian and other physical quantities given with the original fermion variables are rewritten to the forms expressed in terms of the collective and the independent-particle variables.