The investigation of the dynamics of quantum many-body systems is a concerted effort involving computational studies of mathematical models and experimental studies of material samples. Some commonalities of the two tracks of investigation are discussed in the context of the quantum spin dynamics of low-dimensional magnetic systems, in particular spin chains. The study of quantum fluctuations in such systems at equilibrium amounts to exploring the spectrum of collective excitations and the rate at which they are excited from the ground state by dynamical variables of interest. The exact results obtained via Bethe ansatz or algebraic analysis (quantum groups) for a select class of completely integrable models can be used as benchmarks for numerical studies of nonintegrable models, for which computational access to the spectrum of collective excitations is limited.
The investigation of the dynamics of quantum many-body systems is a concerted effort involving computational studies of mathematical models and experimental studies of material samples. Some commonalities of the two tracks of investigation are discussed in the context of the quantum spin dynamics of low-dimensional magnetic systems, in particular spin chains. The study of quantum fluctuations in such systems at equilibrium amounts to exploring the spectrum of collective excitations and the rate at which they are excited from the ground state by dynamical variables of interest. The exact results obtained via Bethe ansatz or algebraic analysis (quantum groups) for a select class of completely integrable models can be used as benchmarks for numerical studies of nonintegrable models, for which computational access to the spectrum of collective excitations is limited.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In these lectures, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exists which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In this Lecture Note, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.
Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one-dimensional integrable models are often well understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one-dimensional strongly correlated model, i.e., the antiferromagnetic spin-21 XXZ chain with the Ising anisotropy, via the form-factor formulas. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multiparticle excitations based on the commensurate N\'eel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multiparticle antibound state excitations can be generalized to nonintegrable spin systems.
In the present thesis we perform a step towards understanding the effec-
tiveness of Hamiltonian perturbation theory in quantum dynamics, which is made possible by the Hamiltonian framing of quantum mechanics. To such a purpose, we focus our attention on quantum spin chains, namely on the isotropic Heisenberg ferromagnet, which is well known to be equivalent, via the Jordan-Wigner transformation, to a system of interacting fermions, and is also known to be exactly integrable through the Bethe ansatz. The interaction term of the Hamiltonian, of order four in fermionic creation and annihilation operators, is treated as a perturbation in the regime of small
excitations: small number of reversed spins or, equivalently, small number of
fermions per site. The calculation of the first order normal form Hamiltonian
amounts to erasing all the nonresonant interaction terms, and allows to draw
interesting conclusions on the dynamics of the system, e.g. to determine ap-
proximately conserved quantities. Moreover, we get a good correspondence
between our perturbative energy spectrum and the exact one computed via
the Bethe ansatz, the approximation being of course the more accurate the
smaller is the number of fermions per site. The present method, here tested
in an exactly solvable case, can be applied to any nonintegrable system of
weakly interacting fermions (and/or bosons).
In this thesis, we investigated the evolution of low-dimensional quantum many-body systems through the lenses of their dynamics. The main tools we used are unitary dynamics and kinetic theory. The interest in this class of systems derives from the fact that they are dominated by the effects of interactions between fundamental constituents. In one dimension, the situation is even more dramatic: even the smallest interaction leads to a description that cannot be cast into a small perturbation of the initial model. Even though, a particular set of systems still allows for exact solutions; hence, they are deemed integrable and they allow for an infinite number of conserved currents. A simple way, that can also be performed experimentally, to drive many-body systems outside of the equilibrium is called quantum quench. It consists in switching instantaneously the parameters of the theory after the system is at the equilibrium and measuring an observable after the driving has occurred. Our contributions to the field amount to providing a full characterization of the evolution of the observables of an integrable system after a global quantum quench. We were also able to extend these results, valid in principle for relativistic theories, to the Lieb-Liniger model, prominent for the description of experimental setups, by the means of non-relativistic limit. For two-dimensional models, interactions are not changing the system from its foundations, although there are unexpected effects. A paradigmatic example is graphene: in a two-dimensional system where carriers are charged and massless, namely their dispersion relation does not have any gap and particles and holes can be freely excited from the thermal state, it is predicted that they are interacting via Coulomb force hundreds of times more than normal electrons in void. Hence, the system behaves in a completely different way from electrons in quantum electrodynamics: for instance, there are additional modes which are called plasmons that are also massless but neutral. They represent collective excitations of the model, as oscillations in the local density of charge. This system is not integrable but still strongly interacting; solving it as we do for one-dimensional systems is not possible. The role of conservation laws is a common theme of this thesis. In the example above exposed, total charge, momentum and energy are conserved in the pristine model, but in presence of impurities and defects they dissipate. The evolution of such quantities can though be derived exactly by the means of hydrodynamics. We were able to derive the equations that rule their evolution in time for any electronic system with two bands touching at the Dirac point. They are generalizations of the celebrated Navier-Stokes equations, and they differ from them only outside the compressible regime. We are able to show that the transport properties in the hydrodynamic regime strongly differ from the usual ballistic picture popularized by Drude and his model; this is captured for instance by violations of the Wiedemann-Franz ratio. Moreover, we prove that the role of interaction is prominent in the viscosity of these fluids.
The investigation of many-body quantum systems stands as a cornerstone in contemporary physics research. These systems exhibit a richness of phenomena, characterized by the emergence of collective behavior, which leads to the formation of quasi-particle excitations with occasionally unconventional statistics, alongside the occurrence of quantum phase transitions. Their significance extends further as they serve as models for understanding the behavior of real quantum materials, underlining the necessity for their comprehension.In the realm of theoretical physics, exact solutions are often elusive for many-body systems and are limited to a few select cases or toy models. However, real materials are often more complicated with additional dominant interactions beyond solvable limits. Consequently, the study and understanding of generic Hamiltonians necessitate resorting to computational simulations. Utilizing computational resources becomes indispensable, albeit challenging, as the Hilbert space expands exponentially with the number of spins, demanding substantial memory and computational bandwidth.However, recent advancements in quantum computing present a promising avenue for tackling these challenges. Quantum computers offer the potential to simulate many-body quantum systems with minimal overhead, overcoming the limitations of classical computational techniques. Although current quantum computing platforms operate within the confines of the NISQ era -noisy intermediate-scale quantum- they hold immense promise for the exact modelling of many-body systems.In this doctoral research, our objective is to investigate the predictive capabilities of quantum computers through simulating and benchmarking quantum many-body systems relevant to magnetic materials. We showcase three different projects on studying frustrated magnetism: 1) simulating non-equilibrium dynamics of frustrated Ising models on an analog quantum annealer, 2) studying frustrated Rydberg atom quench dynamics on a neutral atom platform, and 3) simulating the Kitaev honeycomb model and probing its topological order on a digital quantum computer. By leveraging the potential of quantum computing in this domain, this research aims to advance both theoretical understanding and practical applications in quantum physics.