Long-range hopping in the confined electronic states of atomically constructed Cs chains on InSb(110)
Niek. M. M. Aarts, Anna M. H. Krieg, Emil Sierda, Anna Reinhold, Danis Badrtinov, Yann in ’t Veld, Brian Kiraly, Elze J. Knol, Malte Rösner, Daniel Wegner, Alexander Ako Khajetoorians
Solid-state quantum simulation is a powerful approach in which the material properties of a tunable platform are used to emulate user-defined model Hamiltonians. One of the biggest challenges in materials-based quantum simulation is understanding how to exactly map material-specific changes onto model parameters in a given Hamiltonian. Here, we study the electronic structure resulting from tailored quasi-1D atomic Cs chains constructed atom by atom, using scanning tunneling microscopy and spectroscopy with the aim to uncover how Cs adatoms on InSb can be used for quantum simulation. We study the dependence of the Cs-induced confined electronic states as a function of various parameters of the chain structure. We find that the lowest-level states can be well described by a 1D confinement potential, determined by the chain length, and that the critical parameter determining the depth of the attractive potential is the Cs density. We relate the observations to a lattice model Hamiltonian with exponential long-range hopping, and compare the calculated local density of states with the experimental observations, which reproduce the lowest energy states well. We find deviations from the 1D-confinement potential description for the higher-energy states, which we detail and speculate on its origin. These studies are critical for developing this material platform toward quantum simulation of electronic structure.
Solid-state quantum simulation is a powerful approach in which the material properties of a tunable platform are used to emulate user-defined model Hamiltonians. One of the biggest challenges in materials-based quantum simulation is understanding how to exactly map material-specific changes onto model parameters in a given Hamiltonian. Here, we study the electronic structure resulting from tailored quasi-1D atomic Cs chains constructed atom by atom, using scanning tunneling microscopy and spectroscopy with the aim to uncover how Cs adatoms on InSb can be used for quantum simulation. We study the dependence of the Cs-induced confined electronic states as a function of various parameters of the chain structure. We find that the lowest-level states can be well described by a 1D confinement potential, determined by the chain length, and that the critical parameter determining the depth of the attractive potential is the Cs density. We relate the observations to a lattice model Hamiltonian with exponential long-range hopping, and compare the calculated local density of states with the experimental observations, which reproduce the lowest energy states well. We find deviations from the 1D-confinement potential description for the higher-energy states, which we detail and speculate on its origin. These studies are critical for developing this material platform toward quantum simulation of electronic structure.
A long-term goal of quantum simulation is to faithfully reproduce the low-energy Hamiltonians of correlated quantum materials, establishing a direct connection between solid-state experiment, electronic-structure theory, and programmable many-body systems. Here, we introduce an integrated quantum-simulation architecture based on fermionic alkaline-earth atoms that enables programmable control over the key ingredients of multi-orbital material Hamiltonians. Long-lived clock states encode synthetic orbitals, their differential polarizabilities generate orbital-selective tunneling, and contact interactions provide multi-orbital Hubbard and Hund couplings. Spatially structured coherent driving on the ultra-narrow clock transition further realizes symmetry-selective inter-orbital hybridization, while the clock detuning independently controls orbital energy offsets. In this way, the spatial symmetries of solid-state orbitals can be encoded directly into the effective Hamiltonian, even though the atoms themselves remain in conventional s-wave Wannier orbitals of the lowest optical-lattice bands. We demonstrate the resulting integrated control using the four-band low-energy Hamiltonian of the bilayer nickelate La3Ni2O7 as a demanding benchmark. Self-consistent calculations show that the microscopic parameters obtained from first-principles modeling can be reproduced within a single experimentally accessible lattice configuration and independently programmed over substantial parameter ranges. Our results establish a general route for translating material-derived multi-orbital Hamiltonians into programmable ultracold-atom quantum simulators.
Feynman path-integral quantum Monte Carlo (QMC) simulations and an analytic many-body approach are used to study the ground state properties of one-dimensional (1D) chains in the theoretical framework of model Hamiltonians of the Hubbard type. The QMC algorithm is employed to derive position-space quantities, while band structure properties are evaluated by combining QMC data with expressions derived in momentum (k) space. Bridging link between both representations is the quasi-chemical approximation (QCA). Electronic charge fluctuations <(Δn 2 i )> and the fluctuations of the magnetic local moments <(Δs 2 i )> are studied as a function of the on-site density <n i > and correlation strength, which is given by the ratio between two-electron interaction and kinetic hopping. Caused by the non-analytic behaviour of the chemical potential μ = ∂E/∂<n i > (with E denoting the electronic energy), strict 1D systems with an on-site density <n i > of 1·0 do not exhibit the properties of a conductor for any non-zero interaction beyond the mean-field approximation. The QMC simulations lead to straightforward access to the probabilities P i (n) of finding n = 0, 1, 2 electrons at the ith lattice site. The P i (n) elements allow to calculate the enhancement factors on the electron spin susceptibility χ, effective electronic mass m* and Knight shift κ. m* is enhanced by a bandwidth renormalization factor D -1 0, κ by an element ηK mapping the additional localization of the correlated electrons in the presence of an external magnetic field B and χ by the product D -1 0 ηK. Available experimental data are discussed in the light of the present theoretical findings.
Tuning spin-orbit coupling is essential in controlling both spin and charge in confined semiconductor nanostructures, yet it is rarely a truly controllable parameter. Here, we show control over the spin-orbit Hamiltonian in quantum dots and the resulting quantum states by tailoring the confinement potential with atomic-scale precision. Using scanning tunnelling microscopy and spectroscopy, we pattern individual Cs ions into designer quantum dot structures on the surface of indium antimonide, in which electrons from a two-dimensional electron gas are confined with chosen in-plane electric-field gradients. We then quantify the atomic level structure, both spatially resolving the orbital character of the electronic states and their magnetic-field evolution. We demonstrate that the level structure, including the induced zero-field splitting, can be tailored by the designed geometry of the local electric fields. These effects can be described using a Hamiltonian that allows consistent treatment of the confinement-induced spin-orbit coupling beyond the conventional Bychkov-Rashba description. This Hamiltonian is derived from a multiband k.p model and takes the energy dependence of the relevant physical parameters into account. Such precise control of spin-orbit coupling in semiconductor quantum dots is relevant to quantum and spintronic technologies.
Strongly interacting electron systems can provide insight into quantum many-body phenomena, such as Mott insulating behavior and spin liquidity, facilitating semiconductor optimization. The Fermi-Hubbard model is the prototypical model used to study such systems. Recent research, however, has shown that the extended Fermi-Hubbard model, which accounts for long-range interactions, is more accurate, especially for systems far from half-filling. In this study, we use the extended Fermi-Hubbard model to mathematically analyze charge transport through a lattice of quantum dots. One-dimensional chains with spinless electrons and source-drain bias are observed, focusing on the transition between the ground state and the first excited state. Level repulsion decreases the expected energy levels of anticrossings as the hopping onto the chain tends to the hopping within the chain. The distribution of charge density along the chain is characterized in terms of the hopping, nuclear, and Coulomb parameters and novel plasmonic behavior is analyzed. Minor perturbations in electron transport are identified, corresponding to the one-dimensional nature of the observed systems. This research will lead to a better understanding of electron behavior in silicon-doped semiconductors, like the formation of correlation-induced band gaps, and open the door to using the extended Fermi-Hubbard model as a more accurate alternative to study quantum many-body systems.
We study properties of ultracold quantum gases trapped in 1D and under the influence of an underlying optical lattice. We analyze both fermionic and bosonic systems following different numerically exact approaches, like quantum Monte Carlo for soft-core bosons and two-species fermions, and a recently developed exact approach for HCB. The exposition is organized as follows. In Chap. 2, we study single species degenerated fermions on 1D lattices. We show that a splitting of the system takes place in a region of the spectrum with eigenstates that have a non-vanishing weight only in a fraction of the system. We also show that if on top of the lattice an alternating potential is introduced, doubling the original periodicity, an additional slicing' of the system can be achieved. The width and number of such regions can be controlled in a given energy range by the amplitude of the new modulation. By filling these systems the ground state may exhibit insulating regions due to the presence of local gaps, which can be observed in the local density of states. As a complement to these 1D results we present in Appendix A results obtained in 2D lattices. In Chap. 3 we review in some detail the two quantum Monte Carlo (QMC) techniques that are relevant to our study of confined soft-core bosons and two-species fermions on optical lattices, the worldline QMC and the zero temperature projector QMC, respectively. The basis of the classical Monte Carlo method are also presented, included a proof of the Central Limit theorem (Appendix B) that establishes its foundations. Properties of soft-core bosons confined on optical lattices are presented in Chap. 4. QMC simulations show that due to the presence of the slowly varying confining potential the concept of commensurability, typical for periodic systems, loses its meaning. Mott domains appears for a continuous range of fillings and always coexist with superfluid phases, as it was mentioned before. The latter feature is reflected by the behavior of the global compressibility, which never vanishes. A local compressibility, proportional to the variance of the local density, is defined in order to characterize the local Mott-insulating phases. Finally, a phase diagram for trapped systems is also presented. In Chap. 5 we study the ground state properties of two-components fermions confined on optical lattices. Like for the bosonic case, Mott domains also appear for a continuous range of fillings and always coexist with compressible phases. We define a local order parameter (that we denominate local compressibility), which characterizes the Mott insulating phases in an unambiguous way. By means of this local compressibility, we study in detail the interphase between the metallic and insulating region finding that critical behavior sets in revealing a new critical exponent. Furthermore, the behavior of the local compressibility and the variance of the density are found to be universal in this case, independently of the confining potential and the strength of the interaction, as usual in critical phenomena. We present in Chap. 6 a recently developed exact numerical approach that allows to study ground state properties of HCB confined on 1-D lattices. This exact treatment is applied to study the off-diagonal behavior of the one-particle density matrix (OPDM) and related quantities in the equilibrium case. We find that the OPDM decays as a power-law x^-1/2 for large-x, irrespective of the confining potential chosen. The power-law above is shown to determine the scaling of the occupation of the lowest natural orbital in the thermodynamic limit. This scaling and its finite size corrections are also studied for arbitrary powers of the confining potential. In addition, we find a power-law decay of the NO occupations at low densities, and show that its exponent is also universal. The low density limit in the lattice, equivalent to continuous systems, is also analyzed in detail. The approach above is generalized in Chap. 7 to study the non-equilibrium dynamics of HCB in 1-D configurations with an underlying lattice. The presence of the lattice enables the creation of pure Fock states of HCB (a HCB per lattice site) where there is no coherence in the system. We show that quasi-long range correlations develop in the equal-time-one-particle density matrix when such states are allowed to evolve freely, and that they lead to the formation of quasi-condensates of HCB at finite momentum. In addition,we obtain an universal power-law describing the population of the quasi-condensate as a function of time. Finally, we discuss how such systems can be used to create atom lasers with a wave-length that can be controlled through the lattice parameter.
Physics at the level of an atom is dominated by laws of quantum mechanics. Often, this is entangled with a high complexity in behavior of the systems at that length scale. Unravelling the properties of a material at the atomic level is, therefore, a challenging task that easily supersedes current computational capabilities. A route to circumvent this problem is found in physical realization of simpler quantum systems that are representative of the complex quantum systems one is interested in. These simpler physical systems, unlike their more complex counterparts, can actually be measured and information about the complex system, otherwise inaccessible, gained. This thesis describes experimental work focusing mainly on the property of magnetism in spin chains. To mimic these complex systems, we employ a scanning tunneling microscope (STM) to build atomic chains on solid state surfaces and probe their magnetic properties. The intrinsic strength of STM in building and testing structures with single atom precision makes STM a great candidate for simulation of complex quantum systems. In addition to STM having a role of a quantum simulator, I present work supporting STM as a control device determining the very existence of the magnetic excitations of the atom it measures. Finally, I present experimental findings that suggest we are able to probe the magnetic excitations of the atom with subatomic resolution. In summary, this thesis work presents STM as a powerful probing and control tool for studies on quantum magnetism at the level of a single atom.