Thermodynamics and Dynamics of Disordered Quantum Spin Systems with Long Range Interactions

本项目的目的是系统地研究具有局域量子自由度的无序系统的热力学和动力学性质，例如通过长程相互作用耦合的自旋和隧道系统。这种系统在真实材料中随处可见，比如含磁性杂质的金属、掺杂半导体和玻璃系统。施主自旋作为量子计算机的量子比特被广泛研究。然而，它们的相干控制和读出将需要对这些系统的热力学和动力学性质有详细的了解。无序和长程相互作用的结合使这成为理论物理中一个具有挑战性的开放问题。当调节系统参数，如自旋密度、长程相互作用的幂定律和局域场强时，人们可以预期从局域量子激发到离域全局激发的转变。我们打算使用和发展的方法是对实空间重整化群(RSRG)方法的修改，结合对激发能分布、关联函数和纠缠熵的标度分析。利用数值有限尺寸标度分析，我们将确定量子相变的临界参数。我们将研究对RSRG的修正，这将使我们能够量化RSRG方法的精度，并计算典型的关联函数，如对任意距离r的共生性。我们将把这些结果与密度矩阵重整化群方法的张量网络扩展得到的计算进行比较。然后，我们计划将分析扩展到混合铁磁和反铁磁耦合的模型。将研究扩展到更高的维度，将使我们能够对真实系统的性质进行建模，并分析现有和未来的实验结果。接下来，我们研究了量子猝灭后的动力学。首先，我们确定量子保真度，即微扰开启前后自旋系统基态之间的标量积，作为自旋数目N的函数。然后，我们发展了响应理论和动态SDRG方法的组合，其中RG规则被修改以考虑三重态耦合到其他自旋的事实。由此，我们得到了长程耦合无序AFM自旋系统的自旋分量的暂态动力学，并可以研究量子猝灭后扰动的驰豫动力学和传播。

The purpose of this project is a systematic study of the thermodynamic and dynamic properties of disordered systems with local quantum degrees of freedom, such as spins and tunneling systems which are coupled by long range interactions. Such systems are ubiquitous in real materials, like metals with magnetic impurities, doped semiconductors and glassy systems. Donor spins are studied intensively as Qubits for quantum computers. However, their coherent control and readout will require a detailed understanding of the thermodynamic and dynamic properties of such systems. The combination of disorder and long range interactions makes this a challenging open problem of theoretical physics. One can expect a transition from localized quantum excitations to delocalized global excitations when tuning the system parameters such as the spin density, the power law of the long-range interactions and the local field strength. The methods we intend to use and develop to this end are modifications of the real space renormalisation group (RSRG) method in combination with a scaling analysis of the distribution of excitation energies, correlation functions and entanglement entropy. Employing numerical finite size scaling analysis, we will determine the critical parameters of the quantum phase transitions. We will study corrections to the RSRG, which will allow us to quantify the accuracy of the RSRG method and to calculate typical correlation functions such as the concurrence, for any distance r. We will compare these results with calculations obtained by a tensor network extension of the density matrix renormalization group method. We then plan to extend the analysis to models with mixed ferro- and antiferro-coupling. Extending the study to higher dimensions will enable us to model the properties of real systems and to analyze existing and future experimental results. Next, we study the dynamics after quantum quenches. First, we determine the quantum fidelity, the scalar product between the ground state of a spin system before and after a perturbation has been turned on, as function of the number of spins N. Then, we develop a combination of response theory and a dynamic variant of the SDRG method, where the RG rules are modified to account for the fact that triplet states couple to other spins. Thereby, we obtain the transient dynamics of the spin components for long range coupled disordered AFM spin systems and can study the relaxation dynamics and propagation of disturbances after quantum quenches.