Polarizability and dynamic structure factor of the one-dimensional Bose gas near the Tonks-Girardeau limit at finite temperatures

Alexander Yu. Cherny and Joachim Brand

Phys. Rev. A 73, 023612 – Published 17 February 2006

Abstract

Correlation functions related to the dynamic density response of the one-dimensional Bose gas in the model of Lieb and Liniger are calculated. An exact Bose-Fermi mapping is used to work in a fermionic representation with a pseudopotential Hamiltonian. The Hartree-Fock and generalized random phase approximations are derived and the dynamic polarizability is calculated. The results are valid to first order in $1 ∕ γ$ , where $γ$ is Lieb-Liniger coupling parameter. Approximations for the dynamic and static structure factor at finite temperature are presented. The results preclude superfluidity at any finite temperature in the large- $γ$ regime due to the Landau criterion. Due to the exact Bose-Fermi duality, the results apply for spinless fermions with weak $p$ -wave interactions as well as for strongly interacting bosons.

Received 22 July 2005

DOI:https://doi.org/10.1103/PhysRevA.73.023612

Authors & Affiliations

Alexander Yu. Cherny^1,2 and Joachim Brand¹

¹Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
²Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia

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Vol. 73, Iss. 2 — February 2006

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Images

Figure 1
(Color online) (a) The isothermal speed of sound $v_{T}$ of Eq. (21) versus the inverse coupling constant $γ^{- 1}$ in the HF approximation for different temperatures. The quantity $v_{0} \equiv ℏ π n ∕ m$ is the speed of sound of the TG gas at zero temperature. The speed of sound becomes zero at some critical value of $γ$ , below which the HF solution becomes unstable (see discussion in Sec. 3.) (b) The solid (black) line shows the speed of sound in the HF approximation at $T = 0$ given by Eq. (22), and the dashed (violet) line shows the exact speed of sound in the Lieb-Liniger model [2] for comparison.Reuse & Permissions
Figure 2
(Color online) The excitation spectrum and the DSF at $γ = 13$ for various temperatures. The upper and lower thin (blue) lines show the dispersions $ω_{+} (q)$ and $ω_{-} (q)$ of Eq. (35), respectively, limiting the elementary excitations of the Lieb-Liniger model at $T = 0$ . The dimensionless value of the rescaled DSF $S (q, ω) q ε_{F} ∕ (k_{F} N)$ from Eqs. (40, 43) is shown in shades of gray between zero (white) and 1.0 (black). Here $k_{F} = π n$ and $ε_{F} = ℏ^{2} k_{F}^{2} ∕ (2 m)$ . The dotted (red) line indicates a $δ$ -function contribution at $ω_{0} (q)$ . At nonzero temperatures, the $δ$ -function contribution washes out and becomes a part of the continuum.Reuse & Permissions
Figure 3
(Color online) The DSF $S (q, ω) ε_{F} ∕ N$ at interaction strength $γ = 13.33$ as a function of $ω$ at $q = 0.4 k_{F}$ and $q = 2 k_{F}$ for various temperatures. The solid (black) line shows the RPA result (40) or (43), the dashed (red) line shows the first order expansion in $γ^{- 1}$ (41) or (44), and the dotted (blue) line shows the DSF of the TG limit $(γ = \infty)$ for comparison. The circular (violet) dots on the $x$ axis denote $ω_{\pm}$ . The first order expansion always has a divergence to $\pm \infty$ near $ω_{\pm}$ at $T = 0$ , respectively, and the maximum is shifted to $ω_{+}$ at finite $T$ . The RPA result, however, shows an enhancement of low-energy excitation near $ω_{-}$ at small $q$ . Note the unphysical negative values of the DSF in the first order expansion near $ω_{-}$ , in particular for the umklapp excitations at $q = 2 k_{F}$ and $ω$ close to zero.Reuse & Permissions
Figure 4
(Color online) Contour plot of $A (q) 2 m ω_{0} (q) ∕ (ℏ N q^{2})$ , which shows the importance of the $δ$ -function contribution. Values are given in 10 contours between 0 (white) and 1 (black). Between the thick (red) lines there is no discrete contribution because $χ$ given by Eq. (28) has no poles. Left of this region, there is a discrete part with $ω_{0} < ω_{-}$ and right of it there is one with $ω_{0} > ω_{+}$ . Above the dashed (green) line the RPA breaks down due to an instability of the HF ground state as discussed in Sec. 3.Reuse & Permissions
Figure 5
(Color online) The static structure factor $S (q)$ as a function of momentum for $γ = 13.3$ and various temperatures. The solid (black) line shows the RPA result obtained by numerical integration from Eqs. (40) or (43). The dashed (red) line shows the first order expansion (51) in $γ^{- 1}$ with Eqs. (52, 53) at zero temperature or Eqs. (57, 58) at nonzero temperatures. The dotted (blue) line shows the static structure factor in the TG limit $(γ = \infty)$ (52) or (57) for comparison. Note the unphysical cusp at $q = 2 k_{F}$ in the first order expansion of the static structure factor at zero temperature and some traces of it at small temperature.Reuse & Permissions

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