Nonstandard Hubbard model and electron pairing

M. Zendra, F. Borgonovi, G. L. Celardo, and S. Gurvitz

Phys. Rev. B 109, 195137 – Published 10 May 2024

Abstract

We present a nonstandard Hubbard model applicable to arbitrary single-particle potential profiles and interparticle interactions. Our approach involves a treatment of Wannier functions, free from the ambiguities of conventional methods and applicable to finite systems without periodicity constraints. To ensure the consistent evaluation of Wannier functions, we develop a perturbative approach, utilizing the barrier penetration coefficient as a perturbation parameter. With the defined Wannier functions as a basis, we derive the Hubbard Hamiltonian, revealing the emergence of density-induced and pair tunneling terms alongside standard contributions. Our investigation demonstrates that long-range interparticle interactions can induce a mechanism for repulsive particle pairing. This mechanism relies on the effective suppression of single-particle tunneling due to density-induced tunneling. Contrary to expectations based on the standard Hubbard model, an increase in interparticle interaction does not lead to an insulating state. Instead, our proposed mechanism implies the coherent motion of correlated electron pairs, similar to bound states within a multiwell system, resistant to decay from single-electron tunneling transitions. These findings carry significant implications for various phenomena, including the formation of flat bands, the emergence of superconductivity in twisted bilayer graphene, and the possibility of a metal-insulator transition.

Received 21 September 2023
Revised 22 April 2024
Accepted 23 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.195137

Physics Subject Headings (PhySH)

Flat bands Metal-insulator transition

Extended Hubbard model Hubbard model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

M. Zendra ^1,2,3,*, F. Borgonovi ^1,3, G. L. Celardo ^4,5, and S. Gurvitz ^6,†

¹Dipartimento di Matematica e Fisica and Interdisciplinary Laboratories for Advanced Materials Physics, Università Cattolica del Sacro Cuore, via della Garzetta 48, 25133 Brescia, Italy
²Institute for Theoretical Physics, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium
³Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy
⁴Dipartimento di Fisica e Astronomia, Università di Firenze, via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy
⁵Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via Bruno Rossi 1, 50019 Sesto Fiorentino, Firenze, Italy
⁶Department of Particle Physics and Astrophysics, Weizmann Institute of Science, 76100 Rehovot, Israel

^*Corresponding author: matteo.zendra@unicatt.it
^†Corresponding author: shmuel.gurvitz@weizmann.ac.il

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Issue

Vol. 109, Iss. 19 — 15 May 2024

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Images

Figure 1
(a) Symmetric double-well potential $V (x) = V_{1} (x) + V_{2} (x)$ , with lattice depth $V_{0}$ . Dashed lines represent the first two lowest-band energy levels $E_{1, 2}$ . (b),(c) Single-well potentials $V_{1, 2} (x)$ , with classical turning points $\mp \bar{x}$ , so that $V_{1, 2} (\mp \bar{x}) = {\bar{E}}_{0}$ , the single-site ground state energy. The separation point $x_{0}$ is defined so that $V_{1} (x_{0}) = V_{2} (x_{0}) = 0$ .
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Figure 2
(a) First three exact eigenfunctions $ψ_{1} (x)$ (blue curve), $ψ_{2} (x)$ (red curve), and $ψ_{3} (x)$ (green curve) of a square triple-well potential. (b) Symmetric square triple-well potential. The three wells have width $L$ and depth $V_{0}$ and are separated by barriers of width $b$ , where $x_{1, 2}$ are the separation points. Dashed colored lines represent the first three exact energy levels of the system $E_{1, 2, 3}$ , corresponding to the eigenfunctions shown in panel (a), which read $E_{1} = - 4.171$ , $E_{2} = - 3.897$ , and $E_{3} = - 3.545$ . Parameters: $L = 2$ , $b = 0.5$ , and $V_{0} = 5$ , in arbitrary units.
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Figure 3
Left-, middle-, and right-well WFs for the square triple-well potential shown with dashed gray lines. Red solid curves correspond to exact calculations in Eq. (9), blue dashed curves show our analytical results obtained with the TPA in Eqs. (24), and black dashed curves show the orbital functions $Φ_{0}^{(1, 2, 3)} (x)$ in Eqs. (22). Parameters: $L = 2$ , $b = 0.5$ , and $V_{0} = 5$ , in arbitrary units.
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Figure 4
DT amplitude $Ω_{1}$ (blue curve) and PT amplitude $Ω_{2}$ (red curve) as a function of the interaction range $\bar{d}$ for a symmetric square double-well potential. Parameters: $L = 2$ , $b = 0.5$ , $V_{0} = 5$ , and $Ω_{0} = - 0.22$ . Interaction strength $V_{δ} = 1$ , in arbitrary units.
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Figure 5
(a) Coherent motion of two interacting electrons with parallel spins in a symmetric square triple-well potential, corresponding to the PT process. Black dashed lines correspond to the single-site ground state energy $E_{0}$ , while $\bar{d}$ is the interaction range. (b) Occupancy probabilities $P_{L M} (t)$ (red curve) and $P_{L R} (t)$ (blue curve) for $L = 2$ , $b = 0.5$ , $V_{0} = 5$ , and $Ω_{0} ≃ - 0.22$ . (c) Occupancy probabilities $P_{L M} (t)$ (red curve) and $P_{L R} (t)$ (blue curve) for $L = 4$ , $b = 1$ , $V_{0} = 5$ , and $Ω_{0} ≃ - 0.0167$ . Interaction strength $V_{δ} = 3$ and interaction range $\bar{d} / L = 2$ , so that $\bar{U} ≃ 0.35$ , $Ω_{1} ≃ 0.0125$ , and $Ω_{2} ≃ 0.0012$ in (b) and $\bar{U} ≃ 0.19$ , $Ω_{1} ≃ 0.0027$ , and $Ω_{2} ≃ - 6.4 \times 10^{- 6}$ in (c), in arbitrary units.
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Figure 6
(a) Occupancy probabilities $P_{L M} (t)$ and (b) occupancy probabilities $P_{L R} (t)$ obtained with the nonstandard Hubbard model (red curves) and the extended Hubbard model (green curves), for the case of complete single-particle tunneling suppression. Parameters: $L = 2$ , $b = 0.1$ , $V_{0} = 1.1$ , and $Ω_{0} ≃ - 0.32$ . Interaction strength $V_{δ} = 22$ and interaction range $\bar{d} / L = 2$ , so that $\bar{U} ≃ 1.54$ , $Ω_{1} ≃ - Ω_{0}$ , and $Ω_{2} ≃ 0.22$ , in arbitrary units.
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