Tunneling Hamiltonian Framework

• The tunneling Hamiltonian framework is a method that divides a many-body Hamiltonian into separate subsystems linked by a tunneling barrier.
• It captures both standard (Bardeen-type) tunneling and additional correlated inelastic processes arising from electron–electron interactions.
• This approach enables precise modeling of electronic transport in nanoscale conductors by integrating single-particle and two-particle contributions.

The tunneling Hamiltonian framework provides a foundational method for describing quantum tunneling in many-body systems, with special emphasis on electronic transport, correlation effects, and inelastic processes in nanoscale and mesoscopic conductors. It formalizes the separation of a full interacting Hamiltonian into subsystems—typically labeled as "left" and "right" leads or regions—connected via a tunneling barrier. A complete analysis requires proper treatment of electron-electron interactions and their projection onto subsystem degrees of freedom, ultimately yielding both single-particle and correlated-tunneling operators. The framework not only recovers the standard (Bardeen-type) tunneling Hamiltonian in the weak coupling limit, but also naturally generates additional correlated tunneling terms that control experimentally observed inelastic and many-body effects.

1. Hamiltonian Projection Methodology

The starting point is the exact many-body Hamiltonian in second quantization, incorporating both kinetic and electron–electron interaction terms. The system is partitioned into "left" (L) and "right" (R) subsystems via projection operators $P_L$ and $P_R$, which extract the Hilbert subspaces corresponding to eigenstates localized in each spatial region. The field operator is decomposed as

$$\Psi_\sigma(x) \simeq \Psi_{L,\sigma}(x) + \Psi_{R,\sigma}(x) + O(T),$$

where the projected operators are defined as $$\Psi_{L,\sigma}(x) = P_L^\dagger \Psi_\sigma(x) P_L$$ and analogously for $R$. The full Hamiltonian is thereby expressed as

$H = H_L + H_R + H_T + O(T^2),$

where $H_L$ and $H_R$ act in the left and right subspaces, and $H_T$ is the tunneling Hamiltonian connecting the two. In the weak tunneling ($T_{kk'}$ small) limit, corrections $O(T^2)$ are negligible, yielding the canonical tunneling Hamiltonian form: $$H_T = \sum_{k, k', \sigma} \left[ T_{kk'} a^\dagger_{L, k, \sigma} a_{R, k', \sigma} + \text{H.c.} \right].$$ Overlap corrections due to nonorthogonality of basis states between the regions are absorbed as a renormalization of $T_{kk'}$.

2. Interaction Projection and Correlated Tunneling

When projecting the full electron–electron interaction onto the L/R subspaces, new cross terms emerge beyond the standard tunneling picture. The projected two-body interaction contains, beyond intra-subsystem interactions, inter-subsystem contributions that describe genuinely correlated (two-particle) tunneling processes: $$\sum_\sigma \int \! dx \, dx' \, U(x, x') \left\{ \Psi^\dagger_{L,\sigma}(x) \Psi_{R,\sigma}(x)[n_L(x') + n_R(x')] + \text{H.c.} \right\}.$$ Such terms physically represent interaction-induced processes where electron tunneling is accompanied by a density (or spin) fluctuation in either subsystem. These correlated (or inelastic) tunneling processes are crucial for capturing the rich structure seen in experimental transport measurements. Theoretically, these terms directly arise from the structure of the interaction Hamiltonian under projection, and would be absent if subsystems were treated completely independently.

3. Determination of Tunneling Current

Within the linear response regime, the total tunneling current contains additive contributions:

• Single-particle tunneling:

$$I_1 = 2\pi e |T|^2 \sum_\sigma \int \! d\omega \, \rho_{L,\sigma}(x, \omega + eV)\, \rho_{R,\sigma}(x, \omega)\, [n_F(\omega) - n_F(\omega + eV)],$$

where $\rho_{L/R,\sigma}$ are local densities of states.

• Correlated (two-particle) tunneling:

$$I_2 = 2e U^2\, \rho_L(\varepsilon_F)\, \rho_R(\varepsilon_F) \int \! d\omega\, d\omega'\, S_R(x, \omega - \omega') \{\ldots\} + \text{density–density term},$$

where $S_R$ is the spin–spin spectral function and the curly braces include combinations of Fermi functions relating occupation and excitation energies. The second term stems directly from the correlated-tunneling operators present in the projected Hamiltonian and involves two-particle susceptibilities (spin-spin, density-density).

4. Experimental Manifestations and Correlation Signatures

Several experimental systems reveal clear signatures of correlated tunneling:

• Inelastic Tunneling Spectroscopy (IETS/STM-IETS): Spectral features in $I$–$V$ characteristics of single magnetic atoms and finite spin chains are observed that are inconsistent with single-particle LDOS; they match spin excitation energies (e.g., singlet–triplet transitions) and are unambiguously traced to the correlated-tunneling term.
• Anomalous conductance features: Anomalies such as the "0.7 anomaly" in quantum point contacts are hypothesized to arise from correlated tunneling effects induced by interactions.
• Density and spin susceptibility coupling: The dependence of correlated tunneling on local susceptibilities provides a direct, theory–experiment link between inelastic conductance features and the underlying many-body excitation spectrum.

The table below summarizes current determinants and their observables:

Term	Physical Quantity	Experimental Signature
$I_1$	Single-particle LDOS	Baseline conductance; no structure at inelastic gaps
$I_2$	Two-particle susceptibilities (spin/density)	Steps/peaks at excitation energies; inelastic STM

5. Implications for Theoretical and Experimental Transport

The tunneling Hamiltonian framework, with full inclusion of correlated tunneling, provides:

• A rigorous microscopic justification for empirical extensions of the tunneling current to account for inelastic processes.
• A natural integration of many-body correlation functions (e.g., spin susceptibilities) into transport observables, linking electronic structure theory with mesoscopic transport.
• A method to bridge strong electron correlations with experimental tunneling spectroscopy, enabling extraction of excitation spectra and correlation energy scales.

The projection methodology and its associated operator structure are broadly applicable, including in the analysis of quantum dots, nanowires, and correlated heterostructures. Subsequent research has extended these ideas to systems with superconducting correlations and hybrid quantum devices.

6. Conceptual and Methodological Advances

The formalism developed demonstrates:

• That the standard (Bardeen-type) tunneling Hamiltonian is an approximation valid in the weak-coupling limit, with higher-order corrections arising systematically from the many-body projection.
• That tunneling currents, especially in correlated and low-dimensional systems, cannot generally be described by LDOS alone; explicit two-particle terms are required.
• The critical importance of correctly projecting electron–electron interactions to account for physically observable inelastic tunneling phenomena, with implications for both fundamental studies and advanced device design.

In conclusion, the tunneling Hamiltonian framework described in (1007.1238) provides both the mathematical structure and physical insight necessary to understand and predict electron transport in correlated non-superconducting systems, especially where inelastic and many-body effects are non-negligible.