Effects of finite-range interactions on the one-electron spectral properties of one-dimensional metals: Application to Bi/InSb(001)

José M. P. Carmelo, Tilen Čadež, Yoshiyuki Ohtsubo, Shin-ichi Kimura, and David K. Campbell

Phys. Rev. B 100, 035105 – Published 8 July 2019

Abstract

We study the one-electron spectral properties of one-dimensional interacting electron systems in which the interactions have finite range. We employ a mobile quantum impurity scheme that describes the interactions of the fractionalized excitations at energies above the standard Tomonga-Luttinger liquid limit and show that the phase shifts induced by the impurity describe universal properties of the one-particle spectral function. We find the explicit forms in terms of these phase shifts for the momentum dependent exponents that control the behavior of the spectral function near and at the $(k, ω)$ -plane singularities where most of the spectral weight is located. The universality arises because the line shape near the singularities is independent of the short-distance part of the interaction potentials. For the class of potentials considered here, the charge fractionalized particles have screened Coulomb interactions that decay with a power-law exponent $l > 5$ . We apply the theory to the angle-resolved photo-electron spectroscopy (ARPES) in the highly one-dimensional bismuth-induced anisotropic structure on indium antimonide Bi/InSb(001). Our theoretical predictions agree quantitatively with both (i) the experimental value found in Bi/InSb(001) for the exponent $α$ that controls the suppression of the density of states at very small excitation energy $ω$ and (ii) the location in the $(k, ω)$ plane of the experimentally observed high-energy peaks in the ARPES momentum and energy distributions. We conclude with a discussion of experimental properties beyond the range of our present theoretical framework and further open questions regarding the one-electron spectral properties of Bi/InSb(001).

Received 18 December 2018
Revised 1 April 2019

DOI:https://doi.org/10.1103/PhysRevB.100.035105

Physics Subject Headings (PhySH)

Quantum correlations, foundations & formalism

Strongly correlated systems

Angle-resolved photoemission spectroscopy Electron-correlation calculations Luttinger liquid model Non-Fermi-liquid theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

José M. P. Carmelo^1,2,3,4, Tilen Čadež^5,3, Yoshiyuki Ohtsubo^6,7, Shin-ichi Kimura^6,7, and David K. Campbell ¹

¹Boston University, Department of Physics, 590 Commonwealth Ave, Boston, Massachusetts 02215, USA
²Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139, USA
³Center of Physics of University of Minho and University of Porto, P-4169-007 Oporto, Portugal
⁴Department of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, Portugal
⁵Beijing Computational Science Research Center, Beijing 100193, China
⁶Graduate School of Frontier Biosciences, Osaka University, Suita 565-0871, Japan
⁷Department of Physics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan

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Vol. 100, Iss. 3 — 15 July 2019

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Images

Figure 1
Sketch of the $s$ (spin) and $c$ and $c^{'}$ (charge) branch lines in the one-electron removal spectral function of the lattice electronic correlated models discussed in this paper. The soft grey region refers to the small spectral-weight distribution continuum whereas the darker grey regions below the branch lines typically display more weight. In the actual spectral function, Eq. (2), this applies to $k$ subdomains for which the exponents that control the line shape near those lines are negative. The lack of spectral weight in some of the figure $(k, ω)$ -plane regions is imposed by kinematical constraints.
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Figure 2
$tan (Φ) = - (Δ a / \tilde{a}) cot (π / [l - 2])$ , Eqs. (6) and (11), as a function of the renormalized charge parameter ${\tilde{ξ}}_{c}$ for the electronic density $n_{e} = 0.176$ , interaction $u = U / 4 t = 0.30$ , and integer quantum numbers $l = 6 - 12$ used in Sec. 5 for Bi/InSb(001). For ${\tilde{ξ}}_{c} \to 1 / 2$ and at ${\tilde{ξ}}_{c} = {\tilde{ξ}}_{c}^{⊘} = 1 / ξ_{c} = 0.805, tan (Φ)$ reads $cot (π / (l - 2))$ and 0, respectively, and at both ${\tilde{ξ}}_{c} = {\tilde{ξ}}_{c}^{⊖ -} = 0.857$ and ${\tilde{ξ}}_{c} = {\tilde{ξ}}_{c}^{⊖ +} = 1.166$ it is given by $- cot (π / (l - 2))$ . The MQIM-HO is not valid near ${\tilde{ξ}}_{c} = 1$ at which $Δ a / \tilde{a} = \infty$ and the corresponding scattering problem does not refer to the unitary limit. The finite-range effects are more pronounced for ${\tilde{ξ}}_{c} \in] 1 / 2, {\tilde{ξ}}_{c}^{⊘} [$ when $Δ a / \tilde{a} < 0$ and $tan (Φ) > 0$ .
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Figure 3
(a) Raw Bi/InSb(001) ARPES data for $ℏ ω = 15$ eV. (b) An ARPES EDC at $k = 0 Å^{- 1}$ (c) ARPES MDC at $ω = - 0.05$ eV. (d) ARPES EDC from $k = - 0.16$ (bottom) to $+ 0.16$ (top) $Å^{- 1}$ . The thick line is the normal-emission spectrum ( $k = 0 Å^{- 1}$ ). (e) and (f) Second-derivative ARPES images. Derivation was made along momentum in (e) and energy in (f). Circles and error bars in (e) indicate the MDC peak positions. Solid and dashed lines overlaid in (e) are the theoretical $s$ (red), $c$ (light blue), and $c$ ' (black) branch lines for $u = U / 4 t = 0.30, t = 1.22$ eV, and electronic density $n_{e} = 0.176$ . Only for the solid-line $k$ ranges in (e) for which the exponents are negative in Fig. 4 and Figs. 5 and 6 of Appendix pp1 can they be seen in the ARPES image. (ARPES from the same experimental data as in Refs. [17, 18]).
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Figure 4
The exponents, Eq. (A3) of Appendix pp1, in the spectral function, Eq. (2), that control the line shape near the theoretical $c, c^{'}$ , and $s$ branch lines in Fig. 3, respectively, associated with the experimentally observed high-energy Bi/InSb(001) ARPES MDC and EDC peaks [17, 18]. They are here plotted as a function of the momentum $k$ within the MQIM-HO for (a) $l = 6$ and (b) $l = 12$ at $u = 0.30, n_{e} = 0.176$ , and several ${\tilde{ξ}}_{c}$ and $α = {(2 - {\tilde{ξ}}_{c}^{2})}^{2} / (8 {\tilde{ξ}}_{c}^{2})$ values. The black solid lines refer to the bare limit, ${\tilde{ξ}}_{c} = ξ_{c}$ ( $ξ_{c} = 1.242$ and $α_{0} = 0.017$ ). The black dashed and the dashed-dotted lines correspond to $α < 1 / 8$ and $α > 1 / 8$ values, respectively. Moreover, ${\tilde{ξ}}_{c} = 0.805$ , 0.857, and 1.166 refer to ${\tilde{ξ}}_{c}^{⊘} = 1 / ξ_{c}, {\tilde{ξ}}_{c}^{⊖ -}$ , and ${\tilde{ξ}}_{c}^{⊖ +}$ , respectively. The ${\tilde{ξ}}_{c}$ values of the lines whose negative exponents ranges agree with the experimentally observed high-energy ARPES $(k, ω)$ -plane MDC and EDC peaks in Fig. 3 are those for which the $c^{'}$ branch-line exponent crosses zero between $k / π = 0$ and $k / π \approx 0.07$ . For $l = 6$ and 12, this refers to the small ${\tilde{ξ}}_{c}$ subintervals $α = 0.610 - 0.633$ and $α = 0.674 - 0.700$ , respectively. (Such limiting values are given in Tables 4 and 5 for all $l = 6 - 12$ integers).
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Figure 5
The same exponents as in Fig. 4 for (a) $l = 8$ and (b) 10. The choice of the ${\tilde{ξ}}_{c}$ intervals corresponding to the lines whose negative exponents ranges agree with the experimentally observed high-energy ARPES $(k, ω)$ -plane MDC and EDC peaks in Fig. 3 obeys the same criterion as in that figure. For $l = 8$ and 10, such intervals whose limiting values are given in Tables 4 and 5 are $α = 0.660 - 0.680$ and $0.670 - 0.695$ , respectively.
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Figure 6
The same exponents as in Figs. 4 and 5 for (a) $l = 7$ , (b) 9, and (c) 11. The choice of the ${\tilde{ξ}}_{c}$ intervals corresponding to the lines whose negative exponents ranges agree with the experimentally observed high-energy ARPES $(k, ω)$ -plane MDC and EDC peaks in Fig. 3 obeys the same criterion as in Fig. 4. For $l = 7$ , 9, and 11 such intervals whose limiting values are given in Tables 4 and 5 are $α = 0.640 - 0.662, 0.665 - 0.691$ , and $0.672 - 0.699$ , respectively.
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