A method for calculating the spin of the ground quantum state of nonrelativistic electrons and distance between energy levels of quantum states differing in the spin magnitude from first principles is proposed. The approach developed is free from the one-electron approximation and applicable in multielectron systems with allowance for all spatial correlations. The possibilities of the method are demonstrated by the example of calculating the energy gap between spin states in model ellipsoidal quantum dots with a harmonic confining field. The results of computations by the Monte Carlo method point to high sensitivity of the energy gap to the break of spherical symmetry of the quantum dot. For three electrons, the phenomenon of inversion has been revealed for levels corresponding to high and low values of the spin. The calculations demonstrate the practical possibility to obtain spin states with arbitrarily close energies by varying the shape of the quantum dot, which is a key condition for development prospects in technologies of storage systems based on spin qubits.
A method for calculating the spin of the ground quantum state of nonrelativistic electrons and distance between energy levels of quantum states differing in the spin magnitude from first principles is proposed. The approach developed is free from the one-electron approximation and applicable in multielectron systems with allowance for all spatial correlations. The possibilities of the method are demonstrated by the example of calculating the energy gap between spin states in model ellipsoidal quantum dots with a harmonic confining field. The results of computations by the Monte Carlo method point to high sensitivity of the energy gap to the break of spherical symmetry of the quantum dot. For three electrons, the phenomenon of inversion has been revealed for levels corresponding to high and low values of the spin. The calculations demonstrate the practical possibility to obtain spin states with arbitrarily close energies by varying the shape of the quantum dot, which is a key condition for development prospects in technologies of storage systems based on spin qubits.
It is well known that the use of the primitive second-order propagator in path-integral Monte Carlo calculations of many-fermion systems leads to the sign problem. This work will show that by using the similarity-transformed Fokker-Planck propagator, it is possible to solve for the ground state of a large quantum dot, with up to 100 polarized electrons, without solving the sign problem. These similarity-transformed propagators naturally produce rotational symmetry-breaking ground-state wave functions previously used in the study of quantum dots and quantum Hall effects. However, instead of localizing the electrons at positions that minimize the potential energy, this derivation shows that they should be located at positions that maximize the bosonic ground-state wave function. Further improvements in the energy can be obtained by using these as initial wave functions in a ground-state path-integral Monte Carlo calculation with second- and fourth-order propagators.
The time-dependent quantum Monte Carlo method for fermions is introduced and applied in the calculation of the entanglement of electrons in one-dimensional quantum dots with several spin-polarized and spin-compensated electron configurations. The rich statistics of wave functions provided by this method allow one to build reduced density matrices for each electron, and to quantify the spatial entanglement using measures such as quantum entropy by treating the electrons as identical or distinguishable particles. Our results indicate that the spatial entanglement in parallel-spin configurations is rather small, and is determined mostly by the spatial quantum nonlocality introduced by the ground state. By contrast, in the spin-compensated case, the outermost opposite-spin electrons interact like bosons, which prevails their entanglement, while the inner-shell electrons remain largely at their Hartree-Fock geometry. Our findings are in close correspondence with the numerically exact results, wherever such comparison is possible.
On the basis of the path integral representation of the partition function of a quantum spin system, which is derived by Lieb with use of the Bloch coherent state representation, we propose a new Monte Carlo method for quantum spin systems. In this method, the magnitude S of the spin can be varied freely, because S appears in the path integral only as a parameter. The approximation used in this method can be systematically improved by the 1/n-expansion, where n is the additional dimension in the path integral representation. The formulation and some properties of the method are discussed. Simple examples of the application of the method are also presented.
In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter, which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. 7, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and nonextensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.
In this thesis I present a theory of a macroscopic singlet-triplet qubit in quantum dots embedded in nanowires, each containing 4 electrons and together simulating an artficial Haldane gap material. A Haldane gap material exhibits a 4-fold degenerate ground state separated by an energy gap from excitations. The ground state is equivalent to a degenerate spin-singlet and -triplet state. The 4 degenerate states exhibit the characteristics of spins-1/2 localized on either end of the chain. These states may be used as a coded qubit for quantum information processing. Using the effective mass approximation, I calculate single-particle energy levels of one and two quantum dots in a quantum wire. Using these energy levels I compute the Coulomb matrix elements of the interacting Hamiltonian. Using configuration interaction I demonstrate that the ground state of a quantum dot with 4 electrons is a spin-1 state. I then show that the two dot system behaves approximately like two spin-1 objects interacting via an antiferromagnetic Heisenberg Hamiltonian. While the Heisenberg model is approximate, the two dots have a spin-0 ground-state, indicating antiferromagnetic coupling. I then present a simpler spin model to illustrate the physical parameters which control this interaction. Finally, I present a brief solution to the Heisenberg Hamiltonian for finite spin-chains, and show how one can manipulate the singlet-triplet combined ground state of the spin-chain via localized magnetic field, realizing a singlet-triplet qubit in a macroscopic semiconductor device.
In the last years, the electron spin in quantum dots has become a most promising candidate for the realization of qubits, the basic element for quantum computation. In quantum dot spin qubits the exchange interaction implements a fundamental two qubit gate, which originates in the Coulomb interaction and the Pauli principle. Compared to single qubit gates, the exchange based gates are much faster and easier to control locally, motivating the solely exchange energy based quantum computation. The control is based on the exponential sensitivity of the exchange energy on the interparticle distance. Manipulation then can proceed, for example, shifting the single particle states electrically or compressing them magnetically. The practical manipulation schemes require quantitative knowledge of the exchange energy. The spin-orbit interaction plays an important role in this nano-system. Indeed the usual exchange interaction changes its magnitude, while a new term, called anisotropic exchange, appears. In this thesis we propose an accurate effective spin Hamiltonian for modeling spin-orbit-induced spin dynamics in the presence of the magnetic field. We have also performed numerically exact calculations of the isotropic and anisotropic exchange coupling in realistic two-electron GaAs coupled quantum dots in the presence of both Dresselhaus and Bychkov-Rashba spin-orbit interactions. The numerics allow us to establish the goodness of our model and to see the limits of previous statements. Furthermore we find that in zero magnetic field the second-order spin-orbit effects are absent at all interdot couplings.