Quantum Many-body Model Wave Functions Based On Conformal Field Theories

Constructing model wave functions is an important aspect in the study of strongly correlated quantum many-body systems.In this thesis,our main focus is on the confor-mal field theory（CFT）approach to the construction of model wave functions,including the development of the method itself and its application to a number of specific sys-tems.These systems can be divided into two main categories:systems with non-trivial chiral topological order in two-dimensional space and quantum impurity systems in one-dimensional space.For the former,in order to obtain complete information of the topological order from the model wave functions,we focus on systems defined on the torus.As an interesting example,we construct a set of three model wave functions for an S=1 lattice spin system on the torus.These wave functions are interpreted as the topo-logically degenerate ground states of a non-abelian chiral spin liquid on the torus,they take a resonating valence bond form and are proved to be equivalent to certain chiral correlators in the SO（3）₁Wess-Zumino-Witten（WZW）CFT.By analytically calculat-ing the modular matrices of these wave functions,we identify their topological order and show that they are lattice analogues of the bosonic Moore-Read fractional quantum Hall state.In the subspace of topologically degenerate ground states on the torus,there exists a basis known as the anyon eigenbasis,which is of special importance.As the name implies,the elements in this basis are in one-to-one correspondence with the types of anyons.In order to construct the anyon eigenbasis,we propose a new approach based on chiral CFTs.In this approach,we calculate multi-point correlators of chiral con-formal fields restricted to each individual topological sector by utilising the operator formalism.By applying this approach,we construct model wave functions for the topo-logically degenerate ground states of SO（n）₁and SU（n）₁chiral spin liquids on the torus.Their topological orders are again characterised by the analytically calculated modular matrices,where the series of SO（n）₁wave functions gives an elegant realisation of the topological orders in“Kitaev’s sixteenfold way”on the torus.Quantum impurity problems constitute another important class of strongly corre-lated quantum many-body problems.We propose a series of exactly solvable models that describe a S=1/2 Kondo impurity embedded in a Luttinger liquid.These mod-els can be defined in the continuum space as well as on the lattice.Our model wave functions take the form of Jastrow products and can be expressed as chiral correlators in certain CFTs.For different choices of the Jastrow powers,these wave functions can describe either fermionic or bosonic itinerant particles.In the continuum space,these model wave functions are exact ground states of a two-component Calogero-Sutherland model with open boundary coupled to the Kondo impurity;on the lattice,their parent hamiltonians are given by those of a long-range interacting quantum impurity model of t-J type.