Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0711.0823 · Physical Review A

Density correlations in ultracold Fermi systems within the exact Richardson solution

Simon Staudenmayer, Wolfgang Belzig, Christoph Bruder

Published Jan 17, 2008• 7 citations
arXivHTMLPDF

Abstract

We discuss the occupation number correlations in an ultracold system of interacting fermionic atoms. For a system with a special energy-level distribution, viz. two multiply degenerate levels, explicit expressions for the correlation functions are derived in a canonical approach using the exact ground state wave function of the reduced BCS Hamiltonian. We evaluate the correlators numerically for different interaction strengths and find analytical expressions in some limiting cases. Due to the underlying fermionic nature of the pairs the occupations are predominantly anticorrelated and their statistics is a multinomial distribution.

DOI
10.1103/physreva.77.013612

Related papers

Density correlations in ultracold atomic Fermi gases

Wolfgang Belzig, Christian Schroll, Christoph Bruder

We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.

arXiv:cond-mat/0412269•Physical Review A•Jun 12, 2007•22 citations

Analytic calculation of ground-state properties of correlated fermions with the Gutzwiller wave function

Walter Metzner, D. Vollhardt

A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of ${n}_{\mathrm{k}}$ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.

Physical review. B, Condensed matter•May 1, 1988•178 citations

Ground-state correlations of one-dimensional quantum many-body systems

Bill Sutherland

We study the correlations in a family of ground-state wave functions for one-dimensional quantum many-body systems. For a particular ordering of the M particles, these wave functions are given as the absolute value of the free-fermion ground-state wave function, raised to an arbitrary power \ensuremath{\lambda}. For other permutations of the particles, we use the particle statistics---boson or fermion---to construct the wave function. This prescription works equally well on the continuum line of length L or on the discrete lattice of size N. The theory of random matrices singles out the ``special'' values of \ensuremath{\lambda}=1/2, 1, and 2, and provides techniques for calculating all correlations. Results obtained in this work include analytic expressions for the lattice momentum distributions, accurate numerical results for all momentum distributions and their critical exponents, and explicit results for the lattice pair-correlation functions. We emphasize that this is only for the stated special values of \ensuremath{\lambda}. Finally, these wave functions are evaluated as candidates for variational wave functions.

Physical review. B, Condensed matter•Jan 1, 1992•22 citations

Correlations in a confined gas of harmonically interacting spin-polarized fermions

F. Brosens, J. T. Devreese, L. F. Lemmens

For a fermion gas with equally spaced energy levels, the density and the pair correlation function are obtained. The derivation is based on the path integral approach for identical particles and the inversion of the generating functions for both static responses. The density and the pair correlation function are evaluated explicitly in the ground state of a confined fermion system with a number of particles ranging from 1 to 220, and filling the Fermi level completely.

arXiv:cond-mat/9805379•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Aug 1, 1998•28 citations

New superfluid states - a first glance

George Kastrinakis

We introduce a new fermionic variational wavefunction, generalizing the Bardeen-Cooper-Schrieffer (BCS) wavefunction, which is suitable for interacting multi-species spinful systems and sustaining superfluidity. Applications range from quark matter to the high temperature superconductors. A wide class of Hamiltonians, comprising interactions and hybridization of arbitrary momentum dependence between different fermion species, can be treated in a comprehensive manner. This is the case, as both the intra-species and the inter-species interactions are treated on equally rigorous footing, which is accomplished via the introduction of a new quantum index attached to the fermions. The index is consistent with known fermionic physics, and allows for heretofore unaccounted for fermion-fermion correlations. We have derived the finite temperature version of the theory, thus obtaining the renormalized quasiparticle dispersion relations, and we discuss the appearance of charge and spin density wave order. We present numerical solutions for two electron species in 2 dimensions. Based on these solutions, we show that, for equivalent spin up and down fermions, the Fermi occupation factor (per spin) equals 1/2 deep in the Fermi sea. This constitutes a unique experimental prediction of the theory, both for the normal and superfluid states. Interestingly, this result, obtained in the thermodynamic limit, is consistent with Fermi occupation factor (in-)equalities for finite systems of electrons, derived (in a different context) by Borland and Dennis, J. Phys. B {\bf 5}, 7 (1972) and by Altunbulak and Klyachko, Commun. Math. Phys. {\bf 282}, 287 (2008).

•Jan 16, 2009•0 citations

Correlation functions for interacting fermions in the Gutzwiller ansatz

Florian Gebhard, D. Vollhardt

We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.

Physical review. B, Condensed matter•Oct 1, 1988•77 citations
We discuss the occupation number correlations in an ultracold system of interacting fermionic atoms. For a system with a special energy-level distribution, viz. two multiply degenerate levels, explicit expressions for the correlation functions are derived in a canonical approach using the exact ground state wave function of the reduced BCS Hamiltonian. We evaluate the correlators numerically for different interaction strengths and find analytical expressions in some limiting cases. Due to the underlying fermionic nature of the pairs the occupations are predominantly anticorrelated and their statistics is a multinomial distribution.
We investigate density fluctuations in a coherent ensemble of interacting fermionic atoms. Adapting the concept of full counting statistics, well known from quantum optics and mesoscopic electron transport, we study second-order as well as higher-order correlators of density fluctuations. Using the mean-field BCS state to describe the whole interval between the BCS limit and the Bose-Einstein condensate (BEC) limit, we obtain an exact expression for the cumulant-generating function of the density fluctuations of an atomic cloud. In the two-dimensional case, we obtain a closed analytical expression. Poissonian fluctuations of a molecular condensate on the BEC side are strongly suppressed on the BCS side. The size of the fluctuations in the BCS limit is a direct measure of the pairing potential. We also discuss the BEC-BCS crossover of the third cumulant and the temperature dependence of the second cumulant.
A new analytic approach to the evaluation of ground-state properties of Hubbard-type models in terms of the Gutzwiller variational wave function is presented. It is based on the observation that expectation values in terms of this wave function may be expressed by sums over different lattice sites. This makes the application of Wick's theorem and the resulting contractions extremely simple, since the latter involve only anticommuting numbers as in a Grassmann algebra. Expressions for the momentum distribution ${n}_{\mathrm{k}\ensuremath{\sigma}}$ and the Hubbard interaction in terms of a power series in a particular correlation parameter are derived which are valid for all dimensions. An explicit diagrammatic evaluation of the coefficients is described. In one dimension these coefficients may be determined to all orders which yields an approximation-free calculation of nk{n}_{\mathrm{k}}nk​ and ground-state energy E for arbitrary density n and interaction strength U. In the case of a half-filled band and large U, hitherto unexpected nonanalyticities are found. The results allow for the first approximation-free assessment of the properties of the Gutzwiller wave function. It is shown that the well-known Gutzwiller approximation may be derived diagrammatically, too. The approximation is seen to yield the exact results for expectation values in terms of this wave function in the limit of infinite dimensionality.
We study the correlations in a family of ground-state wave functions for one-dimensional quantum many-body systems. For a particular ordering of the M particles, these wave functions are given as the absolute value of the free-fermion ground-state wave function, raised to an arbitrary power \ensuremath{\lambda}. For other permutations of the particles, we use the particle statistics---boson or fermion---to construct the wave function. This prescription works equally well on the continuum line of length L or on the discrete lattice of size N. The theory of random matrices singles out the ``special'' values of \ensuremath{\lambda}=1/2, 1, and 2, and provides techniques for calculating all correlations. Results obtained in this work include analytic expressions for the lattice momentum distributions, accurate numerical results for all momentum distributions and their critical exponents, and explicit results for the lattice pair-correlation functions. We emphasize that this is only for the stated special values of \ensuremath{\lambda}. Finally, these wave functions are evaluated as candidates for variational wave functions.
We introduce a new fermionic variational wavefunction, generalizing the Bardeen-Cooper-Schrieffer (BCS) wavefunction, which is suitable for interacting multi-species spinful systems and sustaining superfluidity. Applications range from quark matter to the high temperature superconductors. A wide class of Hamiltonians, comprising interactions and hybridization of arbitrary momentum dependence between different fermion species, can be treated in a comprehensive manner. This is the case, as both the intra-species and the inter-species interactions are treated on equally rigorous footing, which is accomplished via the introduction of a new quantum index attached to the fermions. The index is consistent with known fermionic physics, and allows for heretofore unaccounted for fermion-fermion correlations. We have derived the finite temperature version of the theory, thus obtaining the renormalized quasiparticle dispersion relations, and we discuss the appearance of charge and spin density wave order. We present numerical solutions for two electron species in 2 dimensions. Based on these solutions, we show that, for equivalent spin up and down fermions, the Fermi occupation factor (per spin) equals 1/2 deep in the Fermi sea. This constitutes a unique experimental prediction of the theory, both for the normal and superfluid states. Interestingly, this result, obtained in the thermodynamic limit, is consistent with Fermi occupation factor (in-)equalities for finite systems of electrons, derived (in a different context) by Borland and Dennis, J. Phys. B {\bf 5}, 7 (1972) and by Altunbulak and Klyachko, Commun. Math. Phys. {\bf 282}, 287 (2008).
We make use of a recently developed diagrammatic theory to calculate correlation functions for interacting fermions of Hubbard-type models in terms of the Gutzwiller wave function. Of the eleven nontrivial correlation functions involving the spin, density, empty and doubly occupied sites, and local Cooper pairs, four are shown to be independent. They are expressed as a power series in a suitably chosen correlation parameter, whose terms are represented diagrammatically. In one dimension these terms may be evaluated to arbitrary order by employing symmetry relations. This allows for an analytic, approximation-free calculation of the correlation functions for arbitrary momentum, particle density, and interaction strength. In the atomic limit the momentum-dependent spin-correlation function shows an antiferromagnetic divergence at half filling in all dimensions. In one dimension the behavior is in very good agreement with all exact analytic and numerical results for the antiferromagnetic Heisenberg chain. Hole-hole correlations also compare very well with exact results. However, correlations between holes and doubly occupied sites appear insufficient. Superconducting correlations involving on-site, singlet Cooper pairs are suppressed. The results allow for an analytic evaluation of the ground-state energy of a large class of extended Hubbard models in terms of the Gutzwiller wave function. Thus they provide exact upper bounds for their ground-state energies.