System Of Noninteracting Fermions Research Articles

Density and physical current density functional theory

AbstractFor a system of N electrons in an external scalar potential v(r) and external vector potential A(r), we prove that the wave function ψ is a functional of the gauge invariant ground state density ρ(r) and ground state physical current density j(r), and a gauge function α(R) (with R = r1,…,rN):ψ=ψ [ρ,j,α]. It is the presence of the gauge function α(R) that ensures the wave function functional is gauge variant. We prove this via a unitary transformation and by a proof of the bijectivity between the potentials {v(r),A(r)} and the ground state properties {ρ(r),j(r)}. Thus, the natural basic variables for the system are the gauge invariant ρ(r) and j(r). Because each choice of gauge function corresponds to the same physical system, the choice of α(R) = 0 is equally valid. As such, we construct a {ρ(r),j(r)} functional theory with the corresponding Euler equations for the density ρ(r) and physical current density j(r) , together with the constraints of charge conservation and continuity of the current. With the assumption of existence of a system of noninteracting fermions with the same ρ(r) and j(r) as that of the electrons, we provide the equations describing this model system, the definitions being within the framework of Kohn–Sham theory in terms of energy functionals of {ρ(r),j(r)} and their functional derivatives. A special case of the {ρ(r),j(r)} functional theory is the magnetic‐field density‐functional theory of Grayce and Harris. We discuss and contrast our work with the paramagnetic current‐ and density‐functional theory of Vignale and Rasolt in which the variables are the gauge invariant ground state density ρ(r), and vorticity ν(r) = ∇× (jp(r)/ρ(r)) , where jp(r) is the paramagnetic current density. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010

On the local virial theorems for linear and isotropic harmonic oscillator potentials in d dimensions

For the system of noninteracting fermions in a one-body potential , the local virial theorems (LVT) are relations, at a given point in space, between this potential, kinetic energy and particle densities. It was recently shown (Brack et al 2010 J. Phys. A: Math. Theor. 43 255204) that for d-dimensional linear and also for isotropic harmonic oscillator potentials these LVTs are exactly satisfied. We present alternative and simple proofs of these theorems, by consideration of the canonical or Bloch density matrix and its relation to the kinetic energy density. The explicit analytical forms of the Bloch density matrix are used for the above-mentioned potentials to achieve the proofs. For the case of linear potential, we obtain a more general result for the so-called semilocal virial theorem, and for the harmonic oscillator potential case we derive a new relationship between the diagonal part of the canonical bloch density and the kinetic energy density.

Entropic bond-descriptors of molecular information systems in local resolution

The communication theory of the chemical bond is extended to the local description of electron distributions in molecules. The key concepts of the molecular information channel in local resolution as well as its average-“noise” (conditional-entropy) and information-flow (mutual-information) descriptors are introduced. For a given electron density the information propagation in molecular communication system is compared for the Hartree system of non-interacting spin-less particles, the Kohn–Sham system of non-interacting fermions, and the real molecular system of interacting electrons, in order to separate the effects due to the exchange and Coulomb correlation. The stockholder partition of the molecular electron density into pieces attributed to atoms-in-molecules (AIM) is used to explore the effects due to the inter- and intra-atomic scattering of the electron probability in the local description. In this atomic resolution of a diatomic molecule several illustrative molecular information systems are investigated, which differ in the admissible level of the information scattering between infinitesimal local volume elements. First, the parallel arrangement of the AIM sub-channels, which allows only for the intra-atomic non-local probability scattering, is examined and the relevant grouping-rules are established for combining the atomic entropy/information data into the bond indices of the molecule as a whole. Next, the vertical Hirshfeld channel, admitting only the inter-atomic, local scattering of the electron probability, is used to probe the entropy non-additivity in both the molecular and promolecular systems. Finally, the truly non-local channel of independent atoms in the Hartree limit is discussed, to extract differences in the entropy/information bond descriptors due to the inter-atomic probability scattering.

Local effective potential theory: Nonuniqueness of potential and wave function

In local effective potential energy theories such as the Hohenberg-Kohn-Sham density functional theory (HKS-DFT) and quantal density functional theory (Q-DFT), electronic systems in their ground or excited states are mapped to model systems of noninteracting fermions with equivalent density. From these models, the equivalent total energy and ionization potential are also obtained. This paper concerns (i) the nonuniqueness of the local effective potential energy function of the model system in the mapping from a nondegenerate ground state, (ii) the nonuniqueness of the local effective potential energy function in the mapping from a nondegenerate excited state, and (iii) in the mapping to a model system in an excited state, the nonuniqueness of the model system wave function. According to nondegenerate ground state HKS-DFT, there exists only one local effective potential energy function, obtained as the functional derivative of the unique ground state energy functional, that can generate the ground state density. Since the theorems of ground state HKS-DFT cannot be generalized to nondegenerate excited states, there could exist different local potential energy functions that generate the excited state density. The constrained-search version of HKS-DFT selects one of these functions as the functional derivative of a bidensity energy functional. In this paper, the authors show via Q-DFT that there exist an infinite number of local potential energy functions that can generate both the nondegenerate ground and excited state densities of an interacting system. This is accomplished by constructing model systems in configurations different from those of the interacting system. Further, they prove that the difference between the various potential energy functions lies solely in their correlation-kinetic contributions. The component of these functions due to the Pauli exclusion principle and Coulomb repulsion remains the same. The existence of the different potential energy functions as viewed from the perspective of Q-DFT reaffirms that there can be no equivalent to the ground state HKS-DFT theorems for excited states. Additionally, the lack of such theorems for excited states is attributable to correlation-kinetic effects. Finally, they show that in the mapping to a model system in an excited state, there is a nonuniqueness of the model system wave function. Different wave functions lead to the same density, each thereby satisfying the sole requirement of reproducing the interacting system density. Examples of the nonuniqueness of the potential energy functions for the mapping from both ground and excited states and the nonuniqueness of the wave function are provided for the exactly solvable Hooke's atom. The work of others is also discussed.

Class of exactly soluble models of one-dimensional spinless fermions and its application to the Tomonaga-Luttinger Hamiltonian with nonlinear dispersion

It is shown that for some special values of Hamiltonian parameters the Tomonaga-Luttinger model with nonlinear dispersion is unitary equivalent to the system of noninteracting fermions. For such parameter values the density-density propagator of the Tomonaga-Luttinger Hamiltonian with nonliner dispersion can be found exactly. In a generic situation the exact solution can be used as a reference point around which a perturbative expansion in orders of certain irrelevant operators may be constructed.

Critical properties of the XXZ model with long-range interactions on the double chain

The XXZ model ( s = 1 2 ) in a transverse field on a double chain with a uniform long-range interaction among the z components of the spins is considered. The nearest-neighbour interactions are restricted to the components in the xy plane and to the spins within the same chain leg, such that the Hamiltonian is given by H = - ∑ m = 1 2 J m ∑ j = 1 N ( S m , j x S m , j + 1 x + S m , j y S m , j + 1 y ) - ( I / N ) ∑ m , n = 1 2 ∑ j , k = 1 N S m , j z S n , k z - h ∑ m = 1 2 ∑ j = 1 N S m , j z , where N is the number of sites of the lattice and m , n ( m , n = 1 , 2 ) label the chain legs. The model is solved exactly by introducing the Jordan–Wigner and integral Gaussian transformations, which map the Hamiltonian in a non-interacting fermion system and corresponds to an extension of the model recently studied by the authors for a single chain. The equation of state is obtained in closed form, and the critical classical (at T > 0 ) and quantum (at T = 0 ) behaviours can be determined exactly. The quantum critical surface is determined in the space generated by the transverse field and interaction parameters, and the crossover lines separating the different critical regimes are also obtained. It is also shown that, differently from the results obtained for the single chain, the system can present multiple quantum transitions.

Ground-state properties of a one-dimensional system of dipoles

A one-dimensional (1D) Bose system with dipole-dipole repulsion is studied at zero temperature by means of a quantum Monte Carlo method. It is shown that, in the limit of small linear density, the bosonic system of dipole moments acquires many properties of a system of noninteracting fermions. At larger linear densities, a variational Monte Carlo calculation suggests a crossover from a liquidlike to a solidlike state. The system is superfluid on the liquidlike side of the crossover and is normal deep on the solidlike side. Energy and structural functions are presented for a wide range of densities. Possible realizations of the model are 1D Bose atomic systems, with permanent dipoles or dipoles induced by static field or resonance radiation; or indirect excitons in coupled quantum wires; etc. We propose parameters of a possible experiment and discuss manifestations of the zero-temperature quantum crossover.

Efficient generation of graph states for quantum computation

We present an entanglement generation scheme which allows arbitrary graph states to be efficiently created in a linear quantum register via an auxiliary entangling bus (EB). The dynamical evolution of the EB is described by an effective non-interacting fermionic system undergoing mirror-inversion in which qubits, encoded as local fermionic modes, become entangled purely by Fermi statistics. We discuss a possible implementation using two species of neutral atoms stored in an optical lattice and find that the scheme is realistic in its requirements even in the presence of noise.

The extended Thomas–Fermi kinetic energy density functional with position-dependent effective mass in one dimension

The point canonical transformations map the Schrodinger equation with constant mass to a wave equation with a position-dependent effective mass. Using such a technique we derive, for a one-dimensional inhomogeneous system of noninteracting fermions with density ρ(x) and spatially dependent effective mass distribution m(x), the semiclassical kinetic energy density functional τ(ρ) in the so-called extended Thomas–Fermi model up to order 2. For a given position-dependent mass, we compare numerically the total semiclassical kinetic energy with its exact quantum mechanical counterpart. The qualitative agreement is excellent.

State arbitrariness of the noninteracting fermion model in quantal density functional theory

AbstractIn quantal density functional theory (Q‐DFT), any nondegenerate or degenerate ground or excited state of a system of electrons may be mapped to one of noninteracting fermions such that the equivalent density, energy, and ionization potential are obtained. As these model fermions are noninteracting, their effective potential energy, a local (multiplicative) operator, is the same for a particular state of the model system. The state of the noninteracting fermions, however, is arbitrary in that they could be in a ground or excited state. Hence, within Q‐DFT, there exist, in principle, an infinite number of local effective potential energy functions that can generate the same density as that of the interacting electrons. In this article, we demonstrate this state arbitrariness by mapping a (nondegenerate) ground state of the Hookean atom to two different noninteracting fermion systems: one in a singlet ground state (1s2) and the other in a singlet first excited state (1s2s). In each case, the density and energy determined are equivalent to that of the Hookean atom in the ground state, with the highest occupied eigenvalues of the model system being the negative of the ionization potential. The contrast with the mapping within traditional Kohn–Sham and generalized adiabatic connection Kohn–Sham density functional theories is also made. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Energy absorption in time-dependent unitary random matrix ensembles: Dynamic versus anderson localization

We consider energy absorption in an externally driven complex system of noninteracting fermions with the chaotic underlying dynamics described by the unitary random matrices. In the absence of quantum interference, the energy absorption rate W(t) can be calculated with the help of the linear-response Kubo formula. We calculate the leading two-loop interference correction to the semiclassical absorption rate for an arbitrary time dependence of the external perturbation. Based on the results for periodic perturbations, we make a conjecture that the dynamics of the periodically driven random matrices can be mapped onto the one-dimensional Anderson model. We predict that, in the regime of strong dynamic localization, W(t)∝-ln(t)/t 2 rather than decaying exponentially.

Specific heat anomalies of non-interacting fermions with multifractal energy spectra

In this work, we compute the specific heat of a system of non-interacting fermions whose energy spectrum presents a self-similar character. The critical attractor of the logistic map is used to generate a multifractal energy spectrum. We study the temperature dependence of the specific heat for distinct band fillings. At low temperatures the specific heat presents anomalies which evolve from log-periodic oscillations at low band fillings to more complex patterns which strongly depends on the actual position of the Fermi energy. We relate the character of these anomalies with some scaling exponents which characterize the discrete scale invariance of the energy spectrum.

Operator-based truncation scheme based on the many-body fermion density matrix

In [S. A. Cheong and C. L. Henley, cond-mat/0206196 (2002)], we found that the many-particle eigenvalues and eigenstates of the many-body density matrix $\rho_B$ of a block of $B$ sites cut out from an infinite chain of noninteracting spinless fermions can all be constructed out of the one-particle eigenvalues and one-particle eigenstates respectively. In this paper we developed a statistical-mechanical analogy between the density matrix eigenstates and the many-body states of a system of noninteracting fermions. Each density matrix eigenstate corresponds to a particular set of occupation of single-particle pseudo-energy levels, and the density matrix eigenstate with the largest weight, having the structure of a Fermi sea ground state, unambiguously defines a pseudo-Fermi level. We then outlined the main ideas behind an operator-based truncation of the density matrix eigenstates, where single-particle pseudo-energy levels far away from the pseudo-Fermi level are removed as degrees of freedom. We report numerical evidence for scaling behaviours in the single-particle pseudo-energy spectrum for different block sizes $B$ and different filling fractions $\nbar$. With the aid of these scaling relations, which tells us that the block size $B$ plays the role of an inverse temperature in the statistical-mechanical description of the density matrix eigenstates and eigenvalues, we looked into the performance of our operator-based truncation scheme in minimizing the discarded density matrix weight and the error in calculating the dispersion relation for elementary excitations. This performance was compared against that of the traditional density matrix-based truncation scheme, as well as against a operator-based plane wave truncation scheme, and found to be very satisfactory.

On Ehrenfest's theorem

AbstractEhrenfest's theorem for a system of electrons in a time‐dependent external force is the quantal analogue to Newton's second law of motion for interacting classical particles. The theorem is in terms of the averaged force and position of the electrons. Here we draw the quantal analogue to the equation of motion for the individual classical particle and thereby describe the internal forces experienced by each electron and the evolution of these forces with time. This explanation is based on the description of Schrödinger theory in terms of fields and their quantal sources. The quantum mechanical average of the internal fields and their averaged torques, taken over all the electrons, vanish as they must. However, unlike classical physics, the vanishing of the averaged internal fields cannot be attributed solely to Newton's third law but is additionally a consequence of quantum mechanics. The field perspective thus leads to a new derivation of Ehrenfest's theorem. Furthermore, the theorem can be expressed in terms of the response of the electrons to the external field as described by a field representative of the electronic current density. In addition, the vanishing of the averaged torque of the internal fields leads to a quantal torque–angular momentum equation analogous to that of classical physics. The structure of the internal fields is demonstrated for both a ground and an excited state of the analytically solvable Hooke's atom. The concept of the internal field is extended via quantal density functional theory to the S system of noninteracting Fermions with density equivalent to that of the Schrödinger theory of electrons. By invoking Ehrenfest's theorem and the quantal torque–angular momentum relationship, sum rules for the S system are derived. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

Bosonic excitations in random media

We consider classical normal modes and non-interacting bosonic excitations in disordered systems. We emphasise generic aspects of such problems and parallels with disordered, non-interacting systems of fermions, and discuss in particular the relevance for bosonic excitations of symmetry classes known in the fermionic context. We also stress important differences between bosonic and fermionic problems. One of these follows from the fact that ground state stability of a system requires all bosonic excitation energy levels to be positive, while stability in systems of non-interacting fermions is ensured by the exclusion principle, whatever the single-particle energies. As a consequence, simple models of uncorrelated disorder are less useful for bosonic systems than for fermionic ones, and it is generally important to study the excitation spectrum in conjunction with the problem of constructing a disorder-dependent ground state: we show how a mapping to an operator with chiral symmetry provides a useful tool for doing this. A second difference involves the distinction for bosonic systems between excitations which are Goldstone modes and those which are not. In the case of Goldstone modes we review established results illustrating the fact that disorder decouples from excitations in the low frequency limit, above a critical dimension $d_c$, which in different circumstances takes the values $d_c=2$ and $d_c=0$. For bosonic excitations which are not Goldstone modes, we argue that an excitation density varying with frequency as $\rho(\omega) \propto \omega^4$ is a universal feature in systems with ground states that depend on the disorder realisation. We illustrate our conclusions with extensive analytical and some numerical calculations for a variety of models in one dimension.

Semiclassical results in the linear response theory

We consider a quantum system of non-interacting fermions at temperature T, in the framework of linear response theory. We show that semiclassical theory is an appropriate framework to describe some of their thermodynamic properties, in particular through asymptotic expansions in ℏ (Planck constant) of the dynamical susceptibilities. We show how the closed orbits of the classical motion in phase space manifest themselves in these expansions, in the regime where T is of the order of ℏ.

Some exact results for a trapped quantum gas at finite temperature

We present closed analytical expressions for the particle and kinetic energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For d=2 and 3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the |l|-degeneracy in a harmonic oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.

Relevance of inter-composite-fermion interaction to the edge Tomonaga-Luttinger liquid.

It is shown that Wen's effective theory correctly describes the Tomonaga-Luttinger liquid at the edge of a system of noninteracting composite fermions. However, the weak residual interaction between composite fermions is found to be a relevant perturbation. The filling factor dependence of the Tomonaga-Luttinger parameter is estimated for interacting composite fermions in a microscopic approach and satisfactory agreement with experiment is achieved. It is suggested that the electron field operator may not have a simple representation in the effective one-dimensional theory.

Two-dimensional random-bond Ising model, free fermions, and the network model

We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width $M$ is proportional to $M^{3}$, and not exponential in $M$ as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour $\pm J$ RBIM, in which bonds are independently antiferromagnetic with probability $p$, and ferromagnetic with probability $1-p$. Studying temperatures $T\geq 0.4J$, we obtain precise coordinates in the $p-T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small $p$, and determine critical exponents at the multicritical point.

Quantal density functional theory of excited states.

We explain by quantal density functional theory the physics of mapping from any bound nondegenerate excited state of Schrödinger theory to an S system of noninteracting fermions with equivalent density and energy. The S system may be in a ground or excited state. In either case, the highest occupied eigenvalue is the negative of the ionization potential. We demonstrate this physics with examples. The theory further provides a new framework for calculations of atomic excited states including multiplet structure.