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arXiv:cond-mat/9810308

Computing Number Fluctuations

Girish S. Setlur, Yia‐Chung Chang

Published Oct 23, 1998• 0 citations
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Abstract

Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by $ N(k,k^{'}) $ is quite an interesting object. It satisfies various elegant sum rules and is also quite singular in some respects. All these properties are brought out and a formal connection is found between this object and the momentum distribution of the interacting and non-interacting many-fermion systems. This exercise is quite general in that it does not refer to any particular hamiltonian. It is also quite useful since in an earlier preprint(cond-mat/9810043) we showed how to compute the spectral function and single-particle lifetime of homogeneous Fermi systems where the only undetermined quantity was this function $ N(k,k^{'}) $.

DOI
10.48550/arxiv.cond-mat/9810308

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Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by N(k,k′)N(k,k^{'})N(k,k′) is quite an interesting object. It satisfies various elegant sum rules and is also quite singular in some respects. All these properties are brought out and a formal connection is found between this object and the momentum distribution of the interacting and non-interacting many-fermion systems. This exercise is quite general in that it does not refer to any particular hamiltonian. It is also quite useful since in an earlier preprint(cond-mat/9810043) we showed how to compute the spectral function and single-particle lifetime of homogeneous Fermi systems where the only undetermined quantity was this function N(k,k′)N(k,k^{'})N(k,k′).
In this article, we combine the ideas introduced by us earlier in various proportions to arrive at a simple and yet powerful means of studying single-particle properties of homogeneous Fermi systems in detail without making assumptions regarding the validity or otherwise of Fermi-liquid theory. Novelties include the exact forms of the momentum distribution and spectral functions in the intermediate density regime and a dielectric function that is sensitive to significant qualitative changes in single-particle properties. This time, care is taken to include the particle-hole mode by interpreting the sum over modes as a weighted sum with the dynamical structure factor appearing as the weight.
The formalism of exact 1D quantization is reviewed in detail and applied to the spectral study of three concrete Schrödinger Hamiltonians [−d2/dq2+V(q)]±[-{\rm d}^2/{\rm d} q^2 + V(q)]^\pm[−d2/dq2+V(q)]± on the half-line {q>0}\{q>0\}{q>0}, with a Dirichlet (−-−) or Neumann (+++) condition at q=0q=0q=0. Emphasis is put on the analytical investigation of the spectral determinants and spectral zeta functions with respect to singular perturbation parameters. We first discuss the homogeneous potential V(q)=qNV(q)=q^NV(q)=qN as N→+∞N \to +\inftyN→+∞ vs its (solvable) N=∞N=\inftyN=∞ limit (an infinite square well): useful distinctions are established between regular and singular behaviours of spectral quantities; various identities among the square-well spectral functions are unraveled as limits of finite-NNN properties. The second model is the quartic anharmonic oscillator: its zero-energy spectral determinants det⁡(−d2/dq2+q4+vq2)±\det( -{\rm d}^2/{\rm d} q^2 + q^4 + v q^2)^\pmdet(−d2/dq2+q4+vq2)± are explicitly analyzed in detail, revealing many special values, algebraic identities between Taylor coefficients, and functional equations of a quartic type coupled to asymptotic v→+∞v \to +\inftyv→+∞ properties of Airy type. The third study addresses the potentials V(q)=qN+vqN/2−1V(q)=q^N+v q^{N/2-1}V(q)=qN+vqN/2−1 of even degree: their zero-energy spectral determinants prove computable in closed form, and the generalized eigenvalue problems with vvv as spectral variable admit exact quantization formulae which are perfect extensions of the harmonic oscillator case (corresponding to N=2N=2N=2); these results probably reflect the presence of supersymmetric potentials in the family above.
We demonstrate how solutions to quantum few-fermion scattering problems can be the point-of-departure of a new treatment of a generalized many-body wave function. Our focus is on a particular ansatz for the ground state wave function of a superfluid Fermi gas introduced earlier (cond-mat/0506293). Our method is perturbative in the sense that the probability amplitudes for few-fermion scattering processes are treated as small quantities; it is also NONPERTURBATIVE in the sense that whenever such scattering events occur, nonperturbative quantum few-fermion scattering physics dominates. This approach can be viewed as a new diagrammatic methodology, based on a wave function as distinct from a perturbation series in the interparticle interactions. Some generic properties of the wave function are studied, and its parameters in the Bose-Einstein condensed limit are computed beyond mean-field. These results enable us to predict many observable properties of this Fermi gas with well-controlled accuracy, such as the superfluid pairing function, the four-fermion and six-fermion correlation functions, the momentum distribution, and the two-body reduced density matrix, etc.