Novel Tensor Network Methods for Interacting Quantum Matter and Its Dynamical Response
The simulation of interacting quantum matter remains challenging. The Hilbert space dimension required to describe the physics grows exponentially with the system size, yet many interesting collective phenomena emerge only for large enough systems. Tensor network methods allow for the simulation of quantum many-body systems by reducing the effective number of degrees of freedom in controlled approximations. For one-dimensional lattice models, algorithms employing matrix product states (MPS) are currently regarded as the most powerful numerical techniques. For example, density matrix renormalization group (DMRG) algorithms can be used to efficiently compute precise approximations for ground states of local Hamiltonians. The computation of finite-temperature properties and dynamical response functions is more challenging, yet crucial for a complete understanding of the physics and for comparisons with experiments.
In the first part of this dissertation, we introduce and demonstrate novel matrix product state techniques. First, we present an improved version of the minimally entangled typical thermal states (METTS) algorithm, a sampling approach for the simulation of thermal equilibrium. Our modification allows the use of symmetries in the MPS operations, which renders the algorithm significantly more efficient and makes the finite-temperature simulation of previously inaccessible models possible. Then, we introduce a new technique utilizing infinite matrix product states (iMPS) with infinite boundary conditions for the computation of response functions. These quantities are of great importance as their Fourier transform yields spectral functions or dynamic structure factors, which give detailed insights into the low-lying excitations of a model and can be directly compared to experimental data. Our improved algorithm allows to significantly reduce the number of required time-evolution runs in the simulations.
In the second part of this dissertation, we study the physics of the bilinear-biquadratic spin-1 chain in detail. Our new scheme for the simulation of response functions enables us to compute high-resolution dynamic structure factors for the model, which we use as a starting point to explore the low-lying excitations in all quantum phases of the rich phase diagram. Comparing our numerical data to exact results and field-theory approximations, we gain insights into the nature of the relevant excitations. In the Haldane phase, the model can be mapped to a continuum field theory, the non-linear sigma model (NLSM). We find that the NLSM does not capture the influence of the biquadratic term correctly and gives only unsatisfactory predictions for the relevant physical quantities. However, several features in the Haldane phase can be explained by a non-interacting approximation for two- and three-magnon states. Moving into the extended critical phase, we explain the observed contraction of the multi-soliton continua from the Uimin-Lai-Sutherland point by comparison with a field-theory description. In addition, we discover new excitations at higher energies and find that their dispersions are described by simple cosine-functions in the purely biquadratic limit. We characterize them as elementary one-particle excitations and relate them to the integrable Temperley-Lieb chain. The Temperley-Lieb chain can also be used to describe the physics at the opposite biquadratic point, which places the model in the gapped dimerized phase. Here, the excitation spectrum is related to that of an anisotropic spin-1/2 chain. In the ferromagnetic phase, the two-magnon excitations can be computed exactly and contain bound and resonant states in addition to two-particle continua.
Finally, we address the extraction of spectral functions or dynamic structure factors from real-time response functions computed with MPS techniques. In the time evolution, the computation costs grow with time, hence the response function can only be evaluated up to some maximum time. As the spectral functions are obtained by Fourier transform from the time to the frequency domain, this limits the frequency resolution of the result. Here, we introduce and discuss new approaches for the extraction of dynamic structure factors from the limited response-function data.
matrix product states
strongly correlated systems
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