Abstract Quantum dots placed along a vibrating nanotube provide a quantum simulation platform that can directly address the electron-phonon interaction. This offers promising prospects for the search of new quantum materials and the study of strong correlation effects. As this platform is naturally operated by coupling the dots to an electronic reservoir, state preparation is straightforwardly achieved by driving into the steady state. Here we show that for intermediate electron-phonon coupling strength, the system with spin-polarized quantum dots undergoes a Peierls transition into an insulating regime which exhibits charge-density wave order in the steady state as a consequence of the competition between electronic Coulomb repulsive interactions and phonon-induced attractive interactions. The transport phenomena can be directly observed as fingerprints of electronic correlations. We also present powerful methods to numerically capture the physics of such an open electron-phonon system at large numbers of phonons. Our work paves the way to study and detect correlated electron-phonon physics in the nanotube quantum simulator with current experimentally accessible techniques.
Abstract Quantum dots placed along a vibrating nanotube provide a quantum simulation platform that can directly address the electron-phonon interaction. This offers promising prospects for the search of new quantum materials and the study of strong correlation effects. As this platform is naturally operated by coupling the dots to an electronic reservoir, state preparation is straightforwardly achieved by driving into the steady state. Here we show that for intermediate electron-phonon coupling strength, the system with spin-polarized quantum dots undergoes a Peierls transition into an insulating regime which exhibits charge-density wave order in the steady state as a consequence of the competition between electronic Coulomb repulsive interactions and phonon-induced attractive interactions. The transport phenomena can be directly observed as fingerprints of electronic correlations. We also present powerful methods to numerically capture the physics of such an open electron-phonon system at large numbers of phonons. Our work paves the way to study and detect correlated electron-phonon physics in the nanotube quantum simulator with current experimentally accessible techniques.
Quantum simulations can provide new insights into the physics of strongly correlated electronic systems. A well-studied system, but still open in many regards, is the Hubbard-Holstein Hamiltonian, where electronic repulsion is in competition with attraction generated by the electron-phonon coupling. In this context, we study the behavior of four quantum dots in a suspended carbon nanotube and coupled to its flexural degrees of freedom. The system is described by a Hamiltonian of the Hubbard-Holstein class, where electrons on different sites interact with the same phonon. We find that the system presents a transition from the Mott insulating state to a polaronic state, with the appearance of pairing correlations and the breaking of the translational symmetry. These findings will motivate further theoretical and experimental efforts to employ nanoelectromechanical systems to simulate strongly correlated systems with electron-phonon interactions.
We develope a method of mapping quantum nonequilibrium steady-state to an effective equilibrium system and present an algorithm to calculate electron-transport using an equilibrium technique. A systematic implementation of boundary conditions in steady-state nonequilibrium is made in the statistical operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ constructed from scattering state operators. We explicitly demonstrate the equivalence of this method to nonequilibrium Green function techniques for a noninteracting quantum dot model. In electron-phonon coupled quantum dot systems, we formulate an algorithm to construct the statistical bias operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Y}$ and perform a full many-body calculation with the quantum Monte Carlo technique. The results coherently demonstrate various transport behaviors such as phonon dephasing, $I\ensuremath{-}V$ staircase, and phonon-assisted tunneling phenomena. This formulation makes the existing computational quantum many-body techniques applicable to quantum steady-state nonequilibrium problems, which will complement the theories based on the diagrammatic approach.
Suspended carbon nanotubes hosting electrostatically defined quantum dots allow for exceptionally strong and tunable electromechanical coupling as well as mechanical modes that can reach the quantum ground state of motion simply by cryogenic cooling. This makes them a unique platform for quantum simulation of electron-phonon coupling. Here, we propose an experimentally realisable setup with two such carbon nanotubes in parallel, each hosting four quantum dots. Our system not only exhibits phonon-mediated electron-electron attraction, but also supports a robust, maximally entangled Bell phase at mesoscopic scales shared across the subsystems. These features highlight its potential as a simulator of strongly correlated quantum systems.
The Hubbard-Holstein model is one of the simplest to incorporate both electron-electron and electron-phonon interactions. In one dimension at half filling, the Holstein electron-phonon coupling promotes on-site pairs of electrons and a Peierls charge-density wave, while the Hubbard on-site Coulomb repulsion U promotes antiferromagnetic correlations and a Mott insulating state. Recent numerical studies have found a possible third intermediate phase between Peierls and Mott states. From direct calculations of charge and spin susceptibilities, we show that (i) as the electron-phonon coupling is increased, first a spin gap opens, followed by the Peierls transition. Between these two transitions, the metallic intermediate phase has a spin gap, no charge gap, and properties similar to the negative-U Hubbard model. (ii) The transitions between Mott/intermediate and intermediate/Peierls states are of the Kosterlitz-Thouless form. (iii) For larger U, the two transitions merge at a tricritical point into a single first-order Mott/Peierls transition. In addition, we show that an intermediate phase also occurs in the quarter-filled model.
The quantum phase transition between a repulsive Luttinger liquid and an insulating Peierls state is studied in the framework of the one-dimensional spinless Holstein model. We focus on the adiabatic regime but include the full quantum dynamics of the phonons. Using continuous-time quantum Monte Carlo simulations, we track, in particular, the dynamic charge structure factor and the single-particle spectrum across the transition. With increasing electron-phonon coupling, the dynamic charge structure factor reveals the emergence of a charge gap and a clear signature of phonon softening at the zone boundary. The single-particle spectral function evolves continuously across the transition. Hybridization of the charge and phonon modes of the Luttinger liquid description leads to two modes, one of which corresponds to the coherent polaron band. This band acquires a gap upon entering the Peierls phase, whereas the other mode constitutes the incoherent, high-energy spectrum with backfolded shadow bands. Coherent polaronic motion is a direct consequence of quantum lattice fluctuations. In the strong-coupling regime, the spectrum is described by the static, mean-field limit. Importantly, whereas finite electron density in general leads to screening of polaron effects, the latter reappear at half-filling due to charge ordering and lattice dimerization.
Abstract In pursuit of a practical quantum advantage 1 , analogue quantum systems provide an invaluable way to simulate the physics of quantum materials 2–4 , quantum systems out of equilibrium 5,6 or interaction-induced localization 7 . Notable recent progress to realize such systems has been achieved in ultracold atoms 8–12 , superconducting circuits 13–15 and twisted van der Waals materials 16–19 . However, so far, these platforms have struggled to simulate large-scale strongly interacting fermionic systems at low temperatures, at which electronic correlations dominate materials properties and numerical simulations remain restricted in accuracy and scope 20,21 . Here we demonstrate the realization of a new platform consisting of large-scale 2D arrays of sub-nanometre precision-engineered atom-based quantum dots (15,000 sites) to simulate strongly interacting, low-temperature physics. By observing a metal–insulator (MI) transition on a 2D square lattice of atom-based quantum dots, we demonstrate independent and precise control of the on-site interaction U and tunnelling t . Magneto-transport measurements further indicate the formation of an insulating state driven by Mott–Hubbard/Anderson physics and promising signatures of correlated electron physics. These precision-engineered analogue quantum simulators provide a unique platform to simulate quantum materials on arbitrary 2D lattices and to explore many unanswered questions in the formation of quantum magnetism, interacting topological quantum matter and unconventional superconductivity.