Time-dependent properties of one-dimensional spin-systems : a DMRG-study
In this thesis we have applied the DMRG method to calculate dynamical and time-dependent properties of one-dimensional spin systems. Two different kinds of DMRG algorithms have been applied, and improved or generalized. In the correction vector DMRG used to calculate dynamical structure factors, the most time-consuming step is to solve a system of complex linear equations A|x>=|b> with A and |b> given where the matrix A is non-hermitian. In this work we have used the so-called "GMRES"-algorithm so solve for real and imaginary part of |x> simultaneously. Since this avoids squaring the matrix it leads to a speed-up of the algorithm. We have applied this improved correction vector DMRG to a spin-$1$ Heisenberg chain in a strong external magnetic field. The results of this study are summarized below. The second kind of DMRG algorithm that has been used in this thesis is the recently developed adaptive time-dependent DMRG (adaptive t-DMRG). It allows us to calculate the time-evolution of a given quantum many-body state under the action of a certain time-independent system hamiltonian. In its original formulation the adaptive t-DMRG is restricted to hamiltonians containg only short range interactions because the time-evolution operator has to be split up into local operators via a Suzuki-Trotter decomposition. In this work we have implemented a generalization of this algorithm to hamiltonians containing arbitrary interactions basd on ideas by Steven R. White. Several different time-evolution algorithms and sets of target states have been used and compared. The optimal choice among the algorithms tested is a Krylov subspace algorithm allowing for extremely long time steps and hence leading to short calculation times. In this form the adaptive t-DMRG shows at least comparable performance on systems where also Suzukmi-Trotter algorithms could be used. We have used the adaptive t-DMRG to calculate the time-evolution of a single electronic spin in a semiconductor quantum dot due to the hyperfine interaction with the nuclear spins of the underlying lattice. The findings are summarized below. The results of the applications of these different DMRG algorithms presented in this work can be summarized as follows: A Tomonaga-Luttinger liquid (TLL) is one of the paradigmatic concepts in solid state physics. Although it is believed to exist in several different systems its observation is difficult and most of the experimental evidence has been obtained indirectly. One reason for this is that the excitations forming a TLL are gapless and thus their contributions to dynamical structure factors are at extremely low energies. Higher energy features might be easier to measure. A spin-1 Heisenberg chain is a well-known example of a gapped antiferromagnet that, when driven by a magnetic field, undergoes a quantum phase transition to a TLL. In this regime the spectral function, apart from the TLL-continuum, exhibits a second band at higher energies. It is formed by excitations that act as mobile impurities and interact strongly with the excitations of the underlying Tomonaga-Luttinger liquid. Due to this strong interaction their contribution to dynamical structure factors are believed to bear TLL-properties like a power-law decay and a non-universal edge-exponent. In this work we have shown that the high-energy continuum emerging in the longitudinal dynamical structure factor contains an edge singularity similar to the one existing in the low-energy response. We have demonstrated that this high energy part of the spectrum for energies of the order of the Haldane gap Delta is dominated by this power law edge singularity. We have studied the behavior of the corresponding exponent alpha' and its edge frequency omega as functions of the external magnetic field. The behavior of the exponent alpha' as a function of the magnetic field is very different from the one of the TLL edge exponent alpha of the low-energy continuum. While alpha grows with increasing field alpha' decreases. We have also been able to calculate the effective interaction strength between the Sz=0 magnons forming the high-energy continuum and the Sz=1 magnons forming the underlying TLL. The field of quantum information processing and quantum computation has been very active in recent years. Many proposals of solid state realizations of qubits have been made one of the most promising of which is to use the spin of a single electron confined in a semiconductor quantum dot as natural qubit. To actually perform quantum computing several pre-requisites have to be fulfilled. One of the most important is a long decohenerce time of the qubit states such that as many gate operations as possible can be performed. Thus a deeper understanding if the processes leading to decoherence is indispensable. We have investigated the time-evolution and the decoherence behavior of a single electronic spin in a semiconductor quantum dot due to the hyperfine interaction with the nuclear spins of the underlying lattice. In general the time-evolution strongly depends on the initial state chosen. Several such initial states have been studied. The quantity we have analyzed is the long-time limit of the electron spin as a function of an external magnetic field. We have found that it runs through a minimum as a function of the polarization of the nuclear spin system when the magnetic field is kept constant due to energy conservation . The same is true when the magnetic field B is changed at fixed nuclear polarization. In this case the value of B where the minimum is reached strongly depends on the polarization of the nuclear spin bath. One way to extend the decoherence time for electronic states in a quantum dot is to store these states in the nuclear spin bath, which has a much larger intrinsic decoherence time. The coupling between the nuclear spins and the electron induces oscillations in S^z(t). If all coupling constants between the electronic and the nuclear spins were equal and the nuclear spin bath was fully polarized the electronic state would be completely encoded in the nuclear spins after half a period. At that point the electron is ejected from the dot. After some time a new electron, which is in down state too, is injected into the dot. After another half oscillation the electronic state is fully recovered. Real baths are not fully polarized and the coupling constants are not equal. This leads to a loss of fidelity F. We have investigated the fidelity of this protocol as a function of the polarization of the nuclear spins. This has been done for two different distributions of coupling constants. Both sets of coupling constants qualitatively show the expected behavior. The corresponding fidelities decrease with decreasing nuclear polarization. Nevertheless strong differences can be realized. They refer to the slope of the mentioned decrease and the maximum reachable fidelity at full nuclear polarization. For one set of couplings we have investigated the dependence of the fidelity on the number of nuclear spins $L$ for fixed polarization of the nuclear spin bath. Contrary to what is generally assumed we find a rather strong L-dependence. The fidelity of the protocol decreases roughly linearly with increasing number of nuclear spins. For real quantum dots with approximately 100000 nuclei this is a serious problem. For an experimental realization a very large nuclear spin polarization is needed which is not reachable with today's technology.
info:eu-repo/classification/ddc/530, Physik
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