A new method to obtain the knowledge for the ground state energy as well as one for the elementary excitation energy in an interacting many particle system is considered from the intuitive approach in the double time Green function theory.Using this method of approach the statistical diagonalization which may include properly the effects of temperature and other external fields can be performed when the Green functions are solved by an appropriate approximation, and the physical meaning of the results is considered.The application to the Heisenberg antiferromagnet is shown to lead the temperature and external magnetic field dependent results in the approximation of Tyablikov decoupling.A new method to obtain the knowledge for the ground state energy as well as one for the elementary excitation energy in an interacting many particle system is considered from the intuitive approach in the double time Green function theory.Using this method of approach the statistical diagonalization which may include properly the effects of temperature and other external fields can be performed when the Green functions are solved by an appropriate approximation, and the physical meaning of the results is considered.The application to the Heisenberg antiferromagnet is shown to lead the temperature and external magnetic field dependent results in the approximation of Tyablikov decoupling.

Proceedings of the Fujihara Memorial Faculty of Engineering Keio UniversityJan 1, 19700 citations

ON THE THEORY OF THE ENERGY SPECTRUM OF SYSTEMS OF INTERACTING PARTICLES

V.M. Yeleonskii, P. S. Zyryanov

Jan 1, 19580 citations

SPECTRA OF SYSTEMS OF INTERACTING PARTICLES

Yu. L. Klimontovich, Viktor P. Silin

Zhur. Eksptl'. i Teoret. Fiz.Aug 1, 19522 citations

Definition and rigorous derivation of effective models in many body quantum mechanics

Thomas Gamet

The object of this thesis is to study the definition and the mathematically rigorous derivation of effective models in many body quantum mechanics. In this field, it is often impossible to compte exactly physically relevant quantities - such as the ground state energy of a system - because of the large number of degrees of freedom. Hence, introducing effective models (depending only on the local density of particles for instance) allows us to compute these quantities. Then, it remains to check that the effective model is indeed an approximation of the exact model in a macroscopic limit. Besides, in some models, some physical quantities (energy, scattering amplitude...), when computed naively, appear to be infinite. One has then to renormalize the theory in order to make these divergences (that are not observed experimentally) disappear. After putting the results of this thesis into context in the introduction, we will focus on the renormalization of a bosonic quadratic Hamiltonian on the Fock space in Chapter 2. This (formal) Hamiltonian is the sum of the second quantization of a one-particle operator and the square of a field operator, and it exhibits an ultraviolet divergence (i.e. coming from local singularities). More accurately, we will try to define a self-adjoint operator, and to show that it is the limit (in the resolvent sense) of the Hamiltonian with ultraviolet cutoff when the cutoff tends to infinity. To do so, we will first diagonalize the Hamiltonian in the regular case, using Bogoliubov transformations. Then, we will prove that the diagonalized Hamiltonian (up to a renormalization of the energy and the charge) still defines a self-adjoint, and we will show the convergence of the Hamiltonian with cut-off. In a second time, we will study the semi-classical limit of a Fermi gas, in two different scalings. In Chapter 3, we will focus on the dilute limite (i.e. with an interaction range very small compared to the typical distance between particles) of an inhomogeneous (trapped with a potential V) Fermi gas, with repulsive interactions and in three dimensions. Due to the Pauli exclusion principle, two fermions of same spin can only rarely be close enough to interact. Thus, we will consider a Fermi gas with two spin components. We will then show that the ground state energy of such a system can be approximated by a Thomas-Fermi energy (in which interactions play the role of a perturbation of the interactionless energy) in the large number of particles limit. Eventually, in Chapter 4, we will study the mean-field limit (with frequent but weak interactions) of an inhomogeneous Fermi gas, with attractive interactions and in one or two dimensions this time. Once more, we will show the convergence of the ground state energy toward a Thomas-Fermi energy. Moreover, we will also prove the convergence of the ground states. More accurately, we will show that the Husimi functions of these ground states converge to a convex combination of minimizers of the Vlasov energy.The object of this thesis is to study the definition and the mathematically rigorous derivation of effective models in many body quantum mechanics. In this field, it is often impossible to compte exactly physically relevant quantities - such as the ground state energy of a system - because of the large number of degrees of freedom. Hence, introducing effective models (depending only on the local density of particles for instance) allows us to compute these quantities. Then, it remains to check that the effective model is indeed an approximation of the exact model in a macroscopic limit. Besides, in some models, some physical quantities (energy, scattering amplitude...), when computed naively, appear to be infinite. One has then to renormalize the theory in order to make these divergences (that are not observed experimentally) disappear. After putting the results of this thesis into context in the introduction, we will focus on the renormalization of a bosonic quadratic Hamiltonian on the Fock space in Chapter 2. This (formal) Hamiltonian is the sum of the second quantization of a one-particle operator and the square of a field operator, and it exhibits an ultraviolet divergence (i.e. coming from local singularities). More accurately, we will try to define a self-adjoint operator, and to show that it is the limit (in the resolvent sense) of the Hamiltonian with ultraviolet cutoff when the cutoff tends to infinity. To do so, we will first diagonalize the Hamiltonian in the regular case, using Bogoliubov transformations. Then, we will prove that the diagonalized Hamiltonian (up to a renormalization of the energy and the charge) still defines a self-adjoint, and we will show the convergence of the Hamiltonian with cut-off. In a second time, we will study the semi-classical limit of a Fermi gas, in two different scalings. In Chapter 3, we will focus on the dilute limite (i.e. with an interaction range very small compared to the typical distance between particles) of an inhomogeneous (trapped with a potential V) Fermi gas, with repulsive interactions and in three dimensions. Due to the Pauli exclusion principle, two fermions of same spin can only rarely be close enough to interact. Thus, we will consider a Fermi gas with two spin components. We will then show that the ground state energy of such a system can be approximated by a Thomas-Fermi energy (in which interactions play the role of a perturbation of the interactionless energy) in the large number of particles limit. Eventually, in Chapter 4, we will study the mean-field limit (with frequent but weak interactions) of an inhomogeneous Fermi gas, with attractive interactions and in one or two dimensions this time. Once more, we will show the convergence of the ground state energy toward a Thomas-Fermi energy. Moreover, we will also prove the convergence of the ground states. More accurately, we will show that the Husimi functions of these ground states converge to a convex combination of minimizers of the Vlasov energy.

Jun 24, 20260 citations

Discrete spectrum of the energy operator of an n-particle quantum system

S. A. Bugal'ter, G. M. Zhislin

SPhDJun 1, 19780 citations

Statistical Mechanics of Coupled Particles in the Schrödinger Representation

Robert H. Kraichnan

A perturbation treatment is presented for the operator $u(t)$ giving the evolution of the state vector of a system of $N$ particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part ${H}_{0}$ and off-diagonal part ${H}_{1}$ in the representation with $N$ -particle plane-wave states as basis. An effective pair-interaction operator ${\overline{H}}_{1}$ is defined which includes "vertex" corrections of all orders, and the off-diagonal part of $u$ is expressed exactly by an expansion which involves only ${\overline{H}}_{1}$ and the exact diagonal part of $u$ . A closed set of equations determining ${\overline{H}}_{1}$ and the diagonal part of $u$ are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of ${H}_{0}$ and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is $\mathrm{Tr}[u(\ensuremath{-}\frac{i}{\mathrm{kT}})]$. The theory simplifies greatly for very large $N$ .A perturbation treatment is presented for the operator $u(t)$ giving the evolution of the state vector of a system of $N$ particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part ${H}_{0}$ and off-diagonal part ${H}_{1}$ in the representation with $N$-particle plane-wave states as basis. An effective pair-interaction operator ${\overline{H}}_{1}$ is defined which includes "vertex" corrections of all orders, and the off-diagonal part of $u$ is expressed exactly by an expansion which involves only ${\overline{H}}_{1}$ and the exact diagonal part of $u$. A closed set of equations determining ${\overline{H}}_{1}$ and the diagonal part of $u$ are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of ${H}_{0}$ and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is $\mathrm{Tr}[u(\ensuremath{-}\frac{i}{\mathrm{kT}})]$. The theory simplifies greatly for very large $N$.

Physical ReviewNov 15, 19583 citations

A perturbation treatment is presented for the operator

u(t)

giving the evolution of the state vector of a system of

N

particles coupled by 2-body forces. The total Hamiltonian is divided into a diagonal part

{H}_{0}

and off-diagonal part

{H}_{1}

in the representation with

N

-particle plane-wave states as basis. An effective pair-interaction operator

{\overline{H}}_{1}

is defined which includes "vertex" corrections of all orders, and the off-diagonal part of

u

is expressed exactly by an expansion which involves only

{\overline{H}}_{1}

and the exact diagonal part of

u

. A closed set of equations determining

{\overline{H}}_{1}

and the diagonal part of

u

are obtained by retaining only the leading terms in the expansion. These equations include all the corrections included in the Brueckner approximation and, in addition, they contain 3-body effects, iterated to all orders, which are omitted in that approximation. No one-to-one correspondence between the eigenstates of

{H}_{0}

and those of the total Hamiltonian is appealed to, and the ground state plays no special role. The connection with statistical mechanics follows the fact that the partition function is $\mathrm{Tr}[u(\ensuremath{-}\frac{i}{\mathrm{kT}})]$. The theory simplifies greatly for very large

N