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We develop a method to efficiently calculate trial wave functions for quantum Hall systems which involve projection onto the lowest Landau level. The method essentially replaces lowest Landau level projection by projection onto the $M$ lowest eigenstates of a suitably chosen hamiltonian acting within the lowest Landau level. The resulting energy projection is a controlled approximation to the exact lowest Landau level projection which improves with increasing $M$. It allows us to study projected trial wave functions for system sizes close to the maximal sizes that can be reached by exact diagonalization and can be straightforwardly applied in any geometry. As a first application and test case, we study a class of trial wave functions first proposed by Girvin and Jach, which are modifications of the Laughlin states involving a single real parameter. While these modified Laughlin states probably represent the same universality class exemplified by the Laughlin wave functions, we show by extensive numerical work for systems on the sphere and torus that they provide a significant improvement of the variational energy, overlap with the exact wave function and properties of the entanglement spectrum.
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We study lattice wave functions obtained from the SU(2)$_1$ Wess-Zumino-Witten conformal field theory. Following Moore and Reads construction, the Kalmeyer-Laughlin fractional quantum Hall state is defined as a correlation function of primary fields. By an additional insertion of Kac-Moody currents, we associate a wave function to each state of the conformal field theory. These wave functions span the complete Hilbert space of the lattice system. On the cylinder, we study global properties of the lattice states analytically and correlation functions numerically using a Metropolis Monte Carlo method. By comparing short-range bulk correlations, numerical evidence is provided that the states with one current operator represent edge states in the thermodynamic limit. We show that the edge states with one Kac-Moody current of lowest order have a good overlap with low-energy excited states of a local Hamiltonian, for which the Kalmeyer-Laughlin state approximates the ground state. For some states, exact parent Hamiltonians are derived on the cylinder. These Hamiltonians are SU(2) invariant and nonlocal with up to four-body interactions.
We use conformal field theory to construct model wavefunctions for a gapless interface between latti
We investigate the nature of the plasma analogy for the Laughlin wave function on a torus describing the quantum Hall plateau at $ u=frac{1}{q}$. We first establish, as expected, that the plasma is screening if there are no short nontrivial paths around the torus. We also find that when one of the handles has a short circumference -- i.e. the thin-torus limit -- the plasma no longer screens. To quantify this we compute the normalization of the Laughlin state, both numerically and analytically. For the numerical calculation we expand the Laughlin state in a Fock basis of slater-determinants of single particle orbitals, and determine the Fock coefficients of the expansion as a function of torus geometry. In the thin torus limit only a few Fock configurations have non-zero coefficients, and their analytical forms simplify greatly. Using this simple limit, we can reconstruct the normalization and analytically extend it back into the 2D regime. We find that there are geometry dependent corrections to the normalization, and this in turn implies that the plasma in the plasma analogy is not screening when in the thin torus limit. Further we obtain an approximate normalization factor that gives a good description of the normalization for all tori, by extrapolating the thin torus normalization to the thick torus limit.
We conjecture that the counting of the levels in the orbital entanglement spectra (OES) of finite-sized Laughlin Fractional Quantum Hall (FQH) droplets at filling $ u=1/m$ is described by the Haldane statistics of particles in a box of finite size. This principle explains the observed deviations of the OES counting from the edge-mode conformal field theory counting and directly provides us with a topological number of the FQH states inaccessible in the thermodynamic limit- the boson compactification radius. It also suggests that the entanglement gap in the Coulomb spectrum in the conformal limit protects a universal quantity- the statistics of the state. We support our conjecture with ample numerical checks.
Interfaces between topologically distinct phases of matter reveal a remarkably rich phenomenology. We study the experimentally relevant interface between a Laughlin phase at filling factor $ u=1/3$ and a Halperin 332 phase at filling factor $ u=2/5$. Based on our recent construction of chiral topological interfaces in [Nat. Commun. 10, 1860 (2019)], we study a family of model wavefunctions that captures both the bulk and interface properties. These model wavefunctions are built within the matrix product state framework. The validity of our approach is substantiated through extensive comparisons with exact diagonalization studies. We probe previously unreachable features of the low energy physics of the transition. We provide, amongst other things, the characterization of the interface gapless mode and the identification of the spin and charge excitations in the many-body spectrum. The methods and tools presented are applicable to a broad range of topological interfaces.
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