Browsing by Author "Barthel, Thomas"

Quantum entanglement lies at the heart of modern physics and pervades various research fields. In the field of quantum many-body physics, celebrated area and log-area laws have been established for the entanglement entropy of ground states. However, there exists a long-standing question regarding the transition of eigenstate entanglement entropy from the ground state to the highly excited states. Our study fills this gap and elucidates a crossover behavior with universal scaling properties. In the study of quantum matters, knowledge about the entanglement structure can be used to guide the design of tensor network state simulations. For example, we may iteratively eliminate short-range entanglement in a so-called entanglement renormalization scheme so that the entangled ground state is mapped to a product state and then resolved exactly. This idea can be adapted to hybrid quantum-classical algorithms and speed up the simulation of strongly correlated quantum many-body systems.

In the first part of this dissertation, we investigate eigenstate entanglement scaling in quantum many-body systems and characterize the crossover from the ground-state entanglement regime at low energies and small subsystem sizes to extensive volume laws at high energies or large subsystem sizes. We first establish a weak eigenstate thermalization hypothesis (ETH) for translation-invariant systems, argue that the entanglement entropies of (almost) all energy eigenstates are described by a single crossover function whenever the (weak) ETH applies, and point out the universal scaling properties in the quantum critical regime. We then comprehensively confirm these scaling properties by analyzing large classes of quantum many-body systems. Particularly, we give the eigenstate entanglement scaling functions in analytical form for critical one-dimensional systems based on conformal field theory and for $d$-dimensional fermionic systems with Fermi surfaces. For $d=1,2,3$ non-interacting fermions, the scaling functions are numerically verified, and for $d=1,2,3$ harmonic lattice models (free scalar field theory), they are numerically determined. ETH is confirmed with Monte Carlo methods by sampling energy eigenstates or squeezed states for fermions or bosons with $d=1,2$. We also probe and confirm the described scaling properties and the applicability of the ETH in integrable and non-integrable interacting spin-1/2 chains by using exact diagonalization. All the evidence appearing here strongly suggests the existence of crossover functions. Their transition from ground-state scaling to extensive scaling, as well as the universal scaling properties in quantum-critical regimes, are generic.

In the second part of this dissertation, we present a quantum-classical tensor network state algorithm for condensed matter systems. First, we describe this algorithm, which is based on the multi-scale entanglement renormalization ansatz (MERA) and gradient-based optimizations. Due to its narrow causal cone, the algorithm can be implemented on noisy intermediate-scale quantum (NISQ) devices and still describe large systems. We show that the number of required qubits is independent of the system size, increasing only to logarithmic scaling when using quantum amplitude estimation to speed up gradient evaluations. Translational invariance can be used to make the computational cost square-logarithmic with respect to the system size and to describe the thermodynamic limit. The method is particularly attractive for ion-trap devices with ion shuttling capabilities. We then demonstrate it numerically for MERA with Trotterized disentanglers and isometries and find that the computational cost of such MERA quantum eigensolvers is substantially lower than that of the corresponding classical algorithms. In particular, numerical results in various strongly-correlated quantum magnet models show that it has a polynomial quantum advantage over the classical approach. In the experimental implementation, small angles in the employed two-qubit quantum gates are advantageous. We find that, by adding an angle penalty term to the energy functional, the average absolute values of the angles can be moderately reduced without significantly affecting the energy accuracies. Finally, we propose that the Trotter-type circuit in each tensor can be replaced by a parallel random circuit. However, this replacement does not seem to result in further gains as long as the tensor-network bond dimensions are small.

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.