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Journal of the Physical Society of Japan

Off-Resonance Effect on Electron Tunneling through a Hubbard Gap

Ryuichi Ugajin

Published Feb 15, 1997• 7 citations
Publisher

Abstract

Electronic states in a Hubbard chain under an overall confining potential, where electron tunneling through a Hubbard gap is seen, are investigated by the Lanczos method. The numerical results suggest that off-resonance between the ends of the chain reduces tunneling amplitude through a Hubbard gap in the center of the chain, when the confining potential is asymmetric. This asymmetry can induce electron tunneling through a Hubbard gap even in a chain with an odd number of electrons.

DOI
10.1143/jpsj.66.425

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Electronic states in a Hubbard chain under an overall confining potential, where electron tunneling through a Hubbard gap is seen, are investigated by the Lanczos method. The numerical results suggest that off-resonance between the ends of the chain reduces tunneling amplitude through a Hubbard gap in the center of the chain, when the confining potential is asymmetric. This asymmetry can induce electron tunneling through a Hubbard gap even in a chain with an odd number of electrons.
The canonical BCS wave function is tested for the attractive Hubbard model. Results are presented for one dimension, and are compared with the exact solutions by the Bethe ansatz and the results from the conventional grand canonical BCS approximation, for various chain lengths, electron densities, and coupling strengths. While the exact ground state energies are reproduced very well both by the canonical and grand canonical BCS approximations, the canonical method significantly improves the energy gaps for small systems and weak coupling. The ``parity'' effect emerges naturally in our canonical results, as a positive gap for an even number of electrons and a negative ``gap'' for an odd number of electrons. Furthermore, we find a ``super-even'' effect: the energy gap oscillates as a function of even electron number, depending on whether the number of electrons is 4m4m4m or 4m+2(m4m+2 (m4m+2(m integer). Such oscillations as a function of electron number should be observable with tunneling measurements in ultrasmall metallic grains.
We study the effects of electron correlation on transport through a small interacting system connected to reservoirs using an effective Hamiltonian which describes the free quasiparticles of a Fermi liquid. The effective Hamiltonian is defined microscopically with the value of the self-energy at $\ensuremath{\omega}=0.$ Specifically, we apply the method to a Hubbard chain of finite size N (=1,2,3,…),(=1,2,3,\dots{}),(=1,2,3,…), and calculate the self-energy within the second order in U in the electron-hole-symmetric case. When couplings between the chain and the reservoirs on the left and right are small, the conductance for even N decreases with increasing N, showing a tendency toward a Mott-Hubbard insulator. This is caused by the off-diagonal element of the self-energy, and this behavior is qualitatively different from that in the special case examined in previous work. We also study the effects of the asymmetry in the two couplings. While a perfect transmission due to the Kondo resonance occurs for any odd N in the symmetric coupling, the conductance for odd N decreases with increasing N in the case of asymmetric coupling.
Quantum tunneling is at the heart of many low-temperature phenomena. In strongly correlated lattice systems, tunneling is responsible for inducing effective interactions, and long-range tunneling substantially alters many-body properties in and out of equilibrium. We observe resonantly enhanced long-range quantum tunneling in one-dimensional Mott-insulating Hubbard chains that are suddenly quenched into a tilted configuration. Higher-order tunneling processes over up to five lattice sites are observed as resonances in the number of doubly occupied sites when the tilt per site is tuned to integer fractions of the Mott gap. This forms a basis for a controlled study of many-body dynamics driven by higher-order tunneling and demonstrates that when some degrees of freedom are frozen out, phenomena that are driven by small-amplitude tunneling terms can still be observed.
We study two (very) weakly coupled Hubbard chains in the half-filled case, and especially the situation where the intrachain Mott scale m is much larger than the (bare) single-electron interchain hopping ${t}_{\ensuremath{\perp}}.$ First, we find that the divergence of the intrachain umklapp channel at the Mott transition results in the complete vanishing of the single-electron interchain hopping: this is significant of a strong confinement of coherence along the chains. Excitations are usual charge fermionic solitons and spinon-(anti)spinon pairs of the Heisenberg chain. Then, we show rigorously how the tunneling of spinon-(anti)spinon pairs produces an antiferromagnetic interchain exchange of the order of ${J}_{\ensuremath{\perp}}{=t}_{\ensuremath{\perp}}^{2}/m.$ In the ``confined'' phase and in the far infrared, the system behaves as a pure spin ladder. The final result is an insulating ground state with spin-gapped excitations exactly as in the opposite ``delocalized'' limit (i.e., for rather large interchain hoppings) where the two-leg ladder is in the well-known insulating $D\ensuremath{-}\mathrm{Mott}$ phase. Unlike materials with an infinite number of coupled chains (Bechgaard salts), the confinement/deconfinement transition at absolute zero is here a simple crossover: no metallic phase is found in undoped two-leg ladders. This statement might be generalized for N-leg ladders with N=3,4,…N=3,4,\dots{}N=3,4,… (but not too large).