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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205 [INSPIRE].
L. Tonks, The Complete Equation of State of One, Two and Three-Dimensional Gases of Hard Elastic Spheres, Phys. Rev. 50 (1936) 955.
M. Girardeau, Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension, J. Math. Phys. 1 (1960) 516.
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].
D.J. Gross, Meron Configurations in the Two-Dimensional O(3) σ-model, Nucl. Phys. B 132 (1978) 439 [INSPIRE].
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].
I. Affleck, Mass Generation by Merons in Quantum Spin Chains and the O(3) σ Model, Phys. Rev. Lett. 56 (1986) 408 [INSPIRE].
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].
M. Cohen, The Energy Spectrum of Excitations in Liquid Helium, Ph.D. Thesis, Caltech (1956) [https://resolver.caltech.edu/CaltechETD:etd-03192004-153651].
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].
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].
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].
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].
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
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1906.00683
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
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
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)248