On stability of gapped states in quantum spin systems
Stabiliteit van toestanden met gap in kwantum spinsystemen
Schütz, Marius; R0462874
Quantum spin systems are simple models for quantum many body physics involving an unbounded number of intricately interacting degrees of freedom. Their complexity gives rise to interesting phenomena, offers many challenging problems in mathematical physics, and also presents a resource of entanglement for prospective quantum computation. Ground states are important for understanding systems at low temperatures, but of course they usually cannot be obtained exactly in such models. Restricting to ground states that are gapped (uniformly in the volume), as we do in this thesis, allows to elaborate on the general structure of these states. Ground states along uniformly gapped paths of Hamiltonians share similar properties on large length scales. This stability property constitutes the basis for the common definition of gapped ground state phases (two ground states are in the same phase if such a connecting gapped path of Hamiltonians exists between them) and can be described transparently by Hastings’ spectral flow. It allows to link different states from the same phase through quasi-local transformations, and one central motivation for the work in this thesis is to improve and extend the spectral flow technique and the stability principle behind it. For impurity models we show that, after activation of a localized and hence bounded perturbation which preserves a gapped sector in the spectrum, the gapped eigenstates can be related to the unperturbed ones through exponentially local maps. In contrast, the locality in the spectral flow’s standard construction is only sub-exponential. For weakly coupled spin systems, we construct an exponentially quasi-local dressing transformation which is applicable for general weak local perturbation acting throughout the entire volume. Our construction, which is based on a renormalization procedure that progressively removes frustration in the Hamiltonian, moreover works also for perturbations which are not self-adjoint. This extension is relevant rather for studying stationary states of quantum or classical Markov dynamics originating from local gapped stochastic generators (better known as Lindbladians in the quantum case) in place of ground states from gapped Hamiltonians. Weakly coupled quantum Markov dynamics are examined in detail and we establish a uniqueness regime where the thermodynamic limit of stationary states is unique and independent of boundary conditions. At last, we analyze classical restrictions of quantum equilibrium and ground states to Abelian sub-algebras generated by single-site observables. For general high temperature systems and at low and zero temperature for a class of weakly coupled spin systems with unique gapped ground state we show that these induced classical spin states are Gibbs measures. These findings imply quantum large deviation results which are novel in the case of low temperature and ground states. The occurrence of non-Gibbsian classical restrictions can be viewed as sign for non-local dependencies in the underlying quantum state, and we give an example for that in the ground state of the transverse field Ising model.