The formalism of bilocal field theory was applied to the calculation of the ground-state energy and the effective mass of the large polaron. This calculation illustrates the bilocal field method using a relatively realistic model of a physical system. In bilocal theory the fermion field is dependent on two spatial points (nonrelativistically). Fermi particles can then be described in terms of a center of mass and internal wave function, and field operators can be expanded in functions related to the exact one-fermion Green's function, thereby eliminating self-energy terms from the S matrix and related operators. Similarities and differences between calculated large-polaron properties using the bilocal treatment and the customary local treatment are discussed. In particular, it is found that, within the approximations made, the bilocal and local methods give the same leading term in powers of the coupling constant for the ground-state energy and effective mass in the strong-coupling limit. For weak coupling, the bilocal energy is larger (more positive) and the effective mass smaller than in the local case due to an additional kinetic energy of internal motion.
The formalism of bilocal field theory was applied to the calculation of the ground-state energy and the effective mass of the large polaron. This calculation illustrates the bilocal field method using a relatively realistic model of a physical system. In bilocal theory the fermion field is dependent on two spatial points (nonrelativistically). Fermi particles can then be described in terms of a center of mass and internal wave function, and field operators can be expanded in functions related to the exact one-fermion Green's function, thereby eliminating self-energy terms from the S matrix and related operators. Similarities and differences between calculated large-polaron properties using the bilocal treatment and the customary local treatment are discussed. In particular, it is found that, within the approximations made, the bilocal and local methods give the same leading term in powers of the coupling constant for the ground-state energy and effective mass in the strong-coupling limit. For weak coupling, the bilocal energy is larger (more positive) and the effective mass smaller than in the local case due to an additional kinetic energy of internal motion.
This thesis is concerned with the quantum mechanical system of a single particle interacting with an ideal gas of identical fermions by point interaction. In the physics literature this system is often referred to as Fermi polaron. We investigate the two-dimensional Fermi polaron. Unlike the one-dimensional case, point interactions in two or three dimensions cannot be implemented as perturbation of the quadratic form of the Laplacian. Either they are obtained as self-adjoint extensions of the Laplacian restricted to functions that vanish when the coordinates of two particles coincide, or they are constructed by a suitable limiting process. Choosing the second approach, a many-body operator with two-particle point interaction has firstly been rigorously defined by Dell'Antonio, Figari and Teta. We consider the Fermi polaron confined to a box with periodic boundary conditions and we identify a broad class of regularization schemes that approximate the Hamiltonian of the Fermi polaron as limit operator in the strong resolvent sense. The Hamiltonian is not given by a closed form, which could be conveniently used in standard variational principles. We establish a novel variational principle that characterizes all bound states, i.e. all energy eigenstates below the bottom of the spectrum of the kinetic energy. This variational principle turns out to be very useful for the following purposes. The ground state of the Fermi polaron is expected to be well approximated by the polaron and the molecule ansatz in the regime of weak and strong coupling between the impurity and the Fermi gas, respectively. In the physics literature, these two classes of trial states are used for variational computations with the (ultraviolet) regularized Hamiltonian. Although the implicit expressions for the minimal energy of both classes allow for the removal of the ultraviolet cutoff, it remains unclear whether the results are upper bounds to the ground state energy of the Fermi polaron. We show that the minimization of energy over polaron and molecule trial states can be reformulated in a natural way with the help of our variational principle. By doing so, the classes of trial states simplify considerably, and since there is no reference to regularized quantities, we can prove that the expressions for the polaron and the molecule energy in the physics literature are indeed upper bounds to the ground state energy of the Fermi polaron. As a further application of the variational principle, we prove analytically that to first order in a particle-hole expansion the molecule ansatz yields a better approximation to the ground state energy than the polaron ansatz if the coupling between the impurity and the Fermi gas is strong enough. So far, this had only been done numerically. The concluding chapter is devoted to the derivation of a lower bound to the ground state energy of the Fermi polaron in two-dimensional space. We show that the ground state energy can be bounded from below by a quantity that does not depend on the number of fermions in the Fermi gas. This result is correct under the assumption that the ratio of the mass of the impurity and the mass of a fermion exceeds 1.225. We also present a method which might yield a similar result for lower mass ratios. This method gives an estimate for the quadratic form of the regularized Hamiltonian in position space representation. In this connection, we present an inequality that bounds a singular potential of a two-dimensional Fermi gas depending only on the minimal distance between two fermions by the kinetic energy of the Fermi gas uniformly in the number of fermions. This inequality also applies to a potential with singularity 1/r^2, for which the Hardy inequality does not hold in two dimensions. Therefore, the full antisymmetry of the wave function has to be taken into account.
We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.
It is well known that two qualitatively different pictures arise in the description of the motion of an electron in an ionic crystal. In the first case, when the electron interacts weakly with the ionic lattice of the crystal, its motion is the same as the motion of a free band electron whose energy is shifted downward with respect to the bottom of the conduction band, and the effective mass is replaced by the renormalized mass (weak-coupling polaron). In the other limiting case, when the electron interacts strongly with the ionic crystal, an entire series of different self-consistent states of the electron and lattice, each of which has its own effective mass and radius (strong-coupling polaron), arises. Although the motion of an electron in an ionic crystal is described in both limiting cases with the help of the Pekar-Frohlich Hamiltonian, each case requires its own special methods of analysis. Thus, for example, in the strong-coupling theory a special form of adiabatic perturbation theory has been developed in which the trans-lational degeneracy is removed before the perturbation expansion is made. It has been possible to relate both descriptions within the framework of Feynman's varia-tional approach, in which a smooth upper bound was obtained for the energy of the ground state of the polaron for all values of the electron-phonon coupling constant a (see Fig. 1). We call attention in this connection to Ref. 6, where the thermodynamic properties are calculated with the help of a special method for performing the calculations and for finding averages of T-products. The approach developed there is more convenient for performing specific calculations than is the use of path integrals. It seems to us that it has definite advantages in the construction of general proofs concerning the choice of the approximating Hamiltonians (this is illustrated in Ref. 6 for the example of the linear Bogolyubov model), in the investigation of the characteristics of the behavior of a polaron in external fields, etc.
We have developed a new equation of motion Green's function theory for the large polaron, that is completely nonperturbative in origin, and therefore gives a unified treatment of the polaron ground state energy for all interaction strengths. The theory also incorporates translational invariance rigorously, a priori. We find the following new results. (a) Previous theories that are valid for all coupling strengths are based on a nonperturbative summation of infinite subsets of entire diagrams (equations of motion), rather than on recoil parts of all diagrams. We can thus recover lowest order results of Feynman for example, via a translationally noninvariant form of the Fock approximation, which resembles that reported in a previous work. (b) The total ground state energy we obtain in the strong coupling limit is E0 = −0.426α2 to lowest order in the coupling constant α, a result substantially below that of previous calculations, E0 = −0.106α2 (using gaussian trial functions in both cases). Our result is also below the lower bound of Lieb and Yamazaki, although we show in a following paper that our result is an upper bound. (c) The origin of the difference of our results compared to that of other strong coupling theories is several fold, and will be presented in detail in subsequent papers. However, the essence of the difference is that the phonons in the field, via their interaction with the electron, assume the role of independent degrees of freedom, in the limit of an infinite number of phonons. We then have a field theoretic problem with an infinite number of degrees of freedom which can exhibit a totally different physical behavior compared to one with a finite number of degrees of freedom. This aspect of the problem has not been given explicit recognition in all other theories, including that of Lieb and Yamazaki.
An approximate calculation is made of the polaron self-energy and effective mass as defined by Frohlich.The approach taken is a variational one based on an ex tended version of Hohler's ansatz in which the trial state function employed is an exact eigenfunction of the total wave vector of the polaron.The effective mass obtained simulates the Feynmann-Schultz result while the corresponding polaron self-energy is fairly accurate but inferior to the result of Feynman and Schultz.CONTENTS Chapters « -« T -| > £ ( l ^' (1'2) n in which n is the optical refractive index, k is the static dielectric constant, e is the electronic charge and 2 t\& is Plank's constant.The limit S->°° is to be taken with lim I = -~ S-*oo v 8tt d 3v .(1,3) A natural unit system is employed in which 2^=o)=2m=l.The first term in the Hamiltonian is the quantum mechanical operator corresponding to the kinetic energy of the electron.The second term represents the energy of the polarization field resulting from the long wave length longitudinal optical crystal modes in the absence of interaction with the electron.The last term represents the energy of interaction between the electron and the phonons.The name "polaron" has been given to the entity consisting of the Bloch conduction electron together with its accompanying nonradiative polarization field.Accordingly Frohlich defines the polaron self-energy, EQ (a), and the (ground-state) polaron effective mass, y(a), by the expansion EQ (a,ic) = Ec (a) + k 2/y(a) + 0(k4 ), (1,4)
A refined approach where the wave function is gradually improved is developed in the one-dimensional (1D) large-Frohlich-polaron problem. The results for the ground-state energy, the effective mass, and the average number of virtual phonons are obtained for a wide range of the coupling constant It, and are in agreement with those of the Feynman path-integral formalism. In addition, in the weak-coupling limit, the expansions of these observables in powers of the coupling constant are exactly calculated up to the alpha 3 -term. They are E=- alpha -0.06066 alpha 2 -0.00844 alpha 3 , m*=1+0.5 alpha +0.19194 alpha 2 +0.06912 alpha 3 , and N=0.5 alpha +0.12132 alpha 2 +0.029540 alpha 3 , which agree with the known results of the fourth-order perturbation theory.