Continuum limit Tonks-Girardeau matrix elements. Part I. The ground state and the uniform density state

An Addendum to this article is available

Abstract

The Tonks-Girardeau model is a quantum mechanical model of N impenetrable bosons in 1+1 dimensions. A Vandermonde determinant provides the exact N -particle wave function of the ground state, or equivalently the matrix elements with respect to position eigenstates. We consider the large N limit of these matrix elements. We present a binning prescription which calculates the leading terms of the matrix elements in a time which is independent of N, and so is suitable for this limit. In this sense, it allows one to solve for the ground state of a strongly coupled continuum quantum field theory in the field eigenstate basis. As examples, we calculate the matrix elements with respect to states with uniform density and also states consisting of two regions with distinct densities.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].

  2. [2]

    L. Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres, Phys. Rev. 50 (1936) 955.

    ADS  Article  Google Scholar 

  3. [3]

    M. Girardeau, Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension, J. Math. Phys. 1 (1960) 516.

    ADS  MathSciNet  Article  Google Scholar 

  4. [4]

    E.H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. 1. The General solution and the ground state, Phys. Rev. 130 (1963) 1605 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    D.J. Gross, Meron Configurations in the Two-Dimensional O(3) σ-model, Nucl. Phys. B 132 (1978) 439 [INSPIRE].

  6. [6]

    V.A. Fateev, I.V. Frolov and A.S. Shvarts, Quantum Fluctuations of Instantons in the Nonlinear σ-model, Nucl. Phys. B 154 (1979) 1 [INSPIRE].

  7. [7]

    I. Affleck, Mass Generation by Merons in Quantum Spin Chains and the O(3) σ Model, Phys. Rev. Lett. 56 (1986) 408 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. [8]

    M. Brockmann and J.-M. Stéphan, Universal terms in the overlap of the ground state of the spin-1/2 XXZ chain with the Néel state, J. Phys. A 50 (2017) 354001 [arXiv:1705.08505] [INSPIRE].

  9. [9]

    M. Cohen, The Energy Spectrum of Excitations in Liquid Helium, Ph.D. Thesis, Caltech (1956) [https://resolver.caltech.edu/CaltechETD:etd-03192004-153651].

  10. [10]

    L.D. Faddeev and L.A. Takhtajan, Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, J. Sov. Math. 24 (1984) 241 [INSPIRE].

  11. [11]

    L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, in Relativistic gravitation and gravitational radiation. Proceedings, School of Physics, Les Houches, France, 26 September–6 October 1995, pp. 149–219 (1996) [hep-th/9605187] [INSPIRE].

  12. [12]

    J. De Nardis, B. Wouters, M. Brockmann and J.-S. Caux, Solution for an interaction quench in the Lieb-Liniger Bose gas, Phys. Rev. A 89 (2014) 033601 [arXiv:1308.4310].

  13. [13]

    M. Brockmann, J. De Nardis, B. Wouters and J.-S. Caux, A Gaudin-like determinant for overlaps of Néel and XXZ Bethe states, J. Phys. A 47 (2014) 145003 [arXiv:1401.2877].

    ADS  MathSciNet  Article  Google Scholar 

Download references

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jarah Evslin.

Additional information

ArXiv ePrint: 1906.00683

Rights and permissions

This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Evslin, J., Mohammed, H., Liu, H. et al. Continuum limit Tonks-Girardeau matrix elements. Part I. The ground state and the uniform density state. J. High Energ. Phys. 2019, 248 (2019). https://doi.org/10.1007/JHEP10(2019)248

Download citation

Keywords

  • Bethe Ansatz
  • Integrable Field Theories
  • Field Theories in Lower Dimensions
  • Lattice Integrable Models