Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived.They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N -particle system and the corresponding Bogoliubov-Duhamel inner product.The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to N for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian.The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter.
Infinite sets of inequalities which generalize all the known inequalities that can be used in the majorization step of the Approximating Hamiltonian method are derived.They provide upper bounds on the difference between the quadratic fluctuations of intensive observables of a N -particle system and the corresponding Bogoliubov-Duhamel inner product.The novel feature is that, under sufficiently mild conditions, the upper bounds have the same form and order of magnitude with respect to N for all the quantities derived by a finite number of commutations of an original intensive observable with the Hamiltonian.The results are illustrated on two types of exactly solvable model systems: one with bounded separable attraction and the other containing interaction of a boson field with matter.
Recently, new thermodynamic inequalities have been obtained, which set bounds on the quadratic fluctuations of intensive observables of statistical mechanical systems in terms of the Bogoliubov - Duhamel inner product and some thermal average values. It was shown that several well-known inequalities in equilibrium statistical mechanics emerge as special cases of these results. On the basis of the spectral representation, lower and upper bounds on the one-sided fidelity susceptibility were derived in analogous terms. Here, these results are reviewed and presented in a unified manner. In addition, the spectral representation of the symmetric two-sided fidelity susceptibility is derived, and it is shown to coincide with the one-sided case. Therefore, both definitions imply the same lower and upper bounds on the fidelity susceptibility.
Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N d d -dimensional bosons for large N. The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form (N−1)−1Ndβv(Nβ⋅) (N-1)-1Ndβv(Nβ·) for β∈[0,4d1) β∈[0,14d) . We derive a sequence of N-body functions which approximate the true many-body dynamics in L2(RdN) L2(RdN) -norm to arbitrary precision in powers of N−1 N-1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
We investigate families of generalized mean-field theories that can be formulated using the Peierls-Bogoliubov inequality. For test Hamiltonians describing mutually non-interacting subsystems of increasing size, the thermodynamics of these mean-field-type systems approaches that of the infinite, fully interacting system except in the immediate vicinity of their respective mean-field critical points. Finite-size scaling analysis of this mean-field critical behaviour allows us to extract the critical exponents of the fully interacting system. It turns out that this procedure amounts to the coherent anomaly method (CAM) proposed by Suzuki (1987), which is thus given a clear interpretation in terms of conventional renormalization group ideas. Moreover, given the geometry of approximating systems, we can identify the family of approximants which is optimal in the sense of the Peierls-Bogoliubov inequality. In the case of the 2D Ising model it turns out that, surprisingly, this optimal family gives rise to a spurious singularity in the thermodynamic functions.
A simple model of a boson system is considered for which the exact Schr\"odinger equation reduces to a difference equation which can be solved numerically. Application of the Bogoliubov method and comparison of its results with the numerical results then yield some insight into the validity of the Bogoliubov approximations. The simple model considered has, in the absence of interactions, only a zero energy state and two states of unit energy available to each boson. Initially there is assumed to be no interaction between bosons in the degenerate excited states and all existing interactions are assumed to be repulsive. The ground-state energy is calculated in the Bogoliubov approximation, with and without the depletion effect considered, and in another numerical approximation. These results and the results of the numerical solution of the exact Schr\"odinger equation converge to the Bogoliubov result with depletion ignored when the number of particles in the system approaches infinity. It is surprising to note that, for the interaction strengths considered, the Bogoliubov result is within 3% of the numerical result for as few as 32 particles in the system, and within 10% for only 4 particles. A modification of the above system is considered in which there exists an additional two-body interaction between particles in the degenerate excited states which may be attractive or repulsive. It is shown that the ground-state energy, with this additional interaction present, deviates from the Bogoliubov value linearly with the strength of the added interaction, i.e., $W={W}_{\mathrm{Bog}}+\ensuremath{\alpha}F$ (with F the strength parameter of the interaction). The factor $\ensuremath{\alpha}$ is found to be small and constant over a wide range of F and in this range the deviation of W from the Bogoliubov value is small. There is, however, a transition point FT, beyond which $\ensuremath{\alpha}$ is constant and large and the usual Bogoliubov approximation is invalid for interactions more attractive than that characterized by FT. A Bogoliubov-like approximation is shown to be quite accurate in this region. It is then shown that $F_{T}^{\ensuremath{-}1}$ is a linear function of the number of particles in the system.
We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity.Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian.We prove an energy expansion, including the two-mode Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), and a variance bound for the number of particles falling inside each potential well.The latter is a signature of a correlated ground state in that it violates the central limit theorem.1. Introduction 1885 2. Main statements 1889 3. Mapping to the space of excitations 1897 4. Bounds on the two-mode Hamiltonian 1903 5. Derivation of the Bogoliubov Hamiltonian and reduction to right and left modes 1910 6.A priori estimates on the ground state of H N 1926 7. Shifted Hamiltonians and lower bound 1932 8. Proof of the main results 1939 Appendix A. The one-body Hartree problem 1949 Appendix B. Estimates and identities in the two-mode space 1951