In describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations has been developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator
G, defined in the presence of a nonlocal external scalar field
U. Approximations for
G(U) are obtained by replacing the
G2(U) which appears in these equations by various functionals of
G(U). If the approximation for
G2(U) satisfies certain simple symmetry conditions, then the
G(U) thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L$, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal
f-sum rule.Several examples of conserving approximations are described. The Hartree approximation,
G2(U)=G(U)G(U), generates the random-phase approximation for
L. The Hartree-Fock approximation for
G(U) leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for
G(U) is obtained by expanding the self-energy to first order in the many-particle scattering matrix
T(U). This
T is obtained by summing ladder diagrams in which the sides of the ladder are composed of
G(U)′s. The resulting
L equation, which involves coefficients proportional to
∣T∣2, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential,
Vs(U). This potential is obtained by summing bubbles composed of two
G(U)′s. The resulting
L equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to
∣Vs∣2.
This paper investigates the criteria for maintenance of the macroscopic conservation laws of number, momentum, and energy by approximate two-particle correlation functions in many-body systems. The methods of generating such approximations are the same as in a previous paper. However, the derivations of the conservation laws given here clarify both why the approximation method works and the connection between the macroscopic conservation laws and those at the vertices.Conserving nonequilibrium approximations are based on self-consistent approximations to the one-particle Green's function. The same condition that ensures that the nonequilibrium theory be conserving also ensures that the equilibrium approximation has the following properties. The several common methods for determining the partition function from the one-particle Green's function all lead to the same result. When applied to a zero-temperature normal fermion system, the approximation procedure maintains the Hugenholtz-Van Hove theorem. Consequently, the self-consistent version of Brueckner's nuclear matter theory obeys this theorem.
Many useful properties of dilute Bose gases at ultra-low temperature are predicted precisely by the product-state ansatz, in which all particles are in the same quantum state. As particle-particle correlations become important, however, the product ansatz begins to fail. We consider a new set of states, which constitute a natural generalization of the product ansatz; the particle-correlated state of
N=l×n identical particles is derived by symmetrizing the
n-fold product of an
l-particle quantum state. The particle-correlated states can be simulated efficiently for large
N, because their parameter spaces, which depend on
l, do not grow with
n. Here we pay special attention to the pure-state case for
l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions for bosons, which we call pair-correlated states (PCS), were introduced by Leggett [Rev. Mod. Phys.
73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves the reduced (marginal) density matrices (RDMs)---these are the correlation functions---associated with PCS in time
O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large
N limit. To test the efficacy of our theory, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-
N approximate form. The relative errors of the ground state energy for both cases are within
10−5 for
N=1000 particles over the entire parameter region from a single condensate to a Mott insulator. Moreover, we present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.
A procedure is proposed to obtain the exact self-energy in the many-body problem. This algorithm is based on the formal analysis to reach the exact theory by repeated applications of an operator F to an arbitrarily chosen input self-energy represented as a functional of the dressed Green's function. The operator F is so defined that the microscopic conservation law for particle number is satisfied. The rigorous self-energy is obtained by the solution of an eigenfunction of F. Particular attention is paid to the relation between the present procedure and the Baym-Kadanoff framework of conserving approximations. By simplifying the procedure in F with use of the generalized Ward identity, we suggest a practical method to implement this algorithm rather easily in actual systems. In order to suggest future directions to improve on this practical method, the recently developed mean-field theory for the Hubbard model in the limit of high spatial dimensions is also discussed in the context of our theory.
The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.
Many useful properties of dilute Bose gases at ultralow temperature are predicted precisely by the (mean-field) product-state Ansatz, in which all particles are in the same quantum state. Yet, in situations where particle-particle correlations become important, the product Ansatz fails. To include correlations nonperturbatively, we consider a new set of states: the particle-correlated state of $N=l\ifmmode\times\else\texttimes\fi{}n$ bosons is derived by symmetrizing the
n-fold product of an
l-particle quantum state. Quantum correlations of the
l-particle state ``spread out'' to any subset of the
N bosons by symmetrization. The particle-correlated states can be simulated efficiently for large
N, because their parameter spaces, which depend on
l, do not grow with
n. Here we formulate and develop in great detail the pure-state case for
l=2, where the many-body state is constructed from a two-particle pure state. These paired wave functions, which we call pair-correlated states (PCS), were introduced by A. J. Leggett [Rev. Mod. Phys. 73, 307 (2001)] as a particle-number-conserving version of the Bogoliubov approximation. We present an iterative algorithm that solves for the reduced (marginal) density matrices (RDMs), i.e., the correlation functions, associated with PCS in time
O(N). The RDMs can also be derived from the normalization factor of PCS, which is derived analytically in the large-
N limit. To test the efficacy of PCS, we analyze the ground state of the two-site Bose-Hubbard model by minimizing the energy of the PCS state, both in its exact form and in its large-
N approximate form, and comparing with the exact ground state. For
N=1000, the relative errors of the ground-state energy for both cases are within ${10}^{\ensuremath{-}5}$ over the entire parameter region from a single condensate to a Mott insulator. We present numerical results that suggest that PCS might be useful for describing the dynamics in the strongly interacting regime.