Properties of minimum spanning trees and fractional quantum Hall states Thomas Sundal Jackson 2010 This dissertation consists of work done on two disjoint problems. In the first two chapters I discuss fractal properties of average-case solutions to the random minimal spanning tree (MST) problem: given a graph with costs on the edges, the MST is the spanning tree minimizing the sum of the total cost of the chosen edges. In the random version the costs are quenched random variables. I solve the random MST problem on the Bethe lattice with appropriate boundary conditions and use the results to infer fractal dimensions in the mean-field approximation. I find that connected components of the MST in a window have dimension D = 6, which establishes the upper critical dimension dc = 6. This contradicts a value dc = 8 proposed previously in the literature; I correct the argument that led to this value. I then develop an exact low-density expansion for the random MST on a finite graph and use it to develop an expansion for the MST on critical percolation clusters. I prove this perturbation expansion is renormalizable around dc = 6. Using a renormalization-group approach, I calculate the fractal dimension Dp of paths on the latter MST to first order in e = 6− d for d ≤ 6, with the result Dp ∼ 2− e/7. In the final chapter, I investigate the correspondence between wavefunctions in the fractional quantum Hall effect obtained as blocks of a conformal field theory (CFT) versus those defined as zero-energy eigenstates of projection Hamiltonians, specifically one which forbids three particles to come together in one of two linearly-independent states of relative angular momentum six and all states of lesser relative angular momentum. I construct zero-energy states from amplitudes of superconformal currents using a result due to Simon. The counting of edge excitations of these states agrees with the character formula for the superconformal Kac vacuum module at generic central charge c, which implies this Hamiltonian is gapless for all c. I attempt to obtain a rational theory by projecting out additional states, focusing on the M(3, 8) and tricritical Ising CFTs. Properties of minimum spanning trees and fractional quantum Hall states A DISSERTATION PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF YALE UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY THOMAS SUNDAL JACKSON DISSERTATION DIRECTOR: NICHOLAS READ
Properties of minimum spanning trees and fractional quantum Hall states Thomas Sundal Jackson 2010 This dissertation consists of work done on two disjoint problems. In the first two chapters I discuss fractal properties of average-case solutions to the random minimal spanning tree (MST) problem: given a graph with costs on the edges, the MST is the spanning tree minimizing the sum of the total cost of the chosen edges. In the random version the costs are quenched random variables. I solve the random MST problem on the Bethe lattice with appropriate boundary conditions and use the results to infer fractal dimensions in the mean-field approximation. I find that connected components of the MST in a window have dimension D = 6, which establishes the upper critical dimension dc = 6. This contradicts a value dc = 8 proposed previously in the literature; I correct the argument that led to this value. I then develop an exact low-density expansion for the random MST on a finite graph and use it to develop an expansion for the MST on critical percolation clusters. I prove this perturbation expansion is renormalizable around dc = 6. Using a renormalization-group approach, I calculate the fractal dimension Dp of paths on the latter MST to first order in e = 6− d for d ≤ 6, with the result Dp ∼ 2− e/7. In the final chapter, I investigate the correspondence between wavefunctions in the fractional quantum Hall effect obtained as blocks of a conformal field theory (CFT) versus those defined as zero-energy eigenstates of projection Hamiltonians, specifically one which forbids three particles to come together in one of two linearly-independent states of relative angular momentum six and all states of lesser relative angular momentum. I construct zero-energy states from amplitudes of superconformal currents using a result due to Simon. The counting of edge excitations of these states agrees with the character formula for the superconformal Kac vacuum module at generic central charge c, which implies this Hamiltonian is gapless for all c. I attempt to obtain a rational theory by projecting out additional states, focusing on the M(3, 8) and tricritical Ising CFTs. Properties of minimum spanning trees and fractional quantum Hall states A DISSERTATION PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF YALE UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY BY THOMAS SUNDAL JACKSON DISSERTATION DIRECTOR: NICHOLAS READ
We consider spaces of trial wave functions for ground states and edge excitations in the fractional quantum Hall effect that can be obtained in various ways. In one way, functions are obtained by analyzing the entanglement of the ground-state wave function, partitioned into two parts. In another, functions are defined by the way in which they vanish as several coordinates approach the same value, or by a projection-operator Hamiltonian that enforces those conditions. In a third way, functions are given by conformal blocks from a conformal field theory (CFT). These different spaces of functions are closely related. The use of CFT methods permits an algebraic formulation to be given for all of them. In some cases, we can prove that there is a ground state, a Hamiltonian, and a CFT such that, for any number of particles, all of these spaces are the same. For such cases, this resolves several questions and conjectures: it gives a finite-size bulk-edge correspondence, and we can use the analysis of functions to construct a projection-operator Hamiltonian that produces those functions as zero-energy states. For a model related to the N=1 superconformal algebra, the corresponding Hamiltonian imposes vanishing properties involving only three particles; for this we determine all the wave functions explicitly. We do the same for a sequence of models involving the M(3,p) Virasoro minimal models that has been considered previously, using results from the literature. We exhibit the Hamiltonians for the first few cases of these. The techniques we introduce can be applied in numerous examples other than those considered here.
Finding the minimum spanning tree (MST) of a graph is an important task in computer vision, as it enables a sparse and low-cost representation of connectivity among elements (such as superpixels, points, or regions), which is useful for tasks such as segmentation, reconstruction, and clustering. In this work, we propose and evaluate a fully quantum pipeline for computing MSTs using the FALQON algorithm, a feedback-based quantum optimization method that does not require classical optimizers. We construct a Hamiltonian formulation whose ground-state energy encodes the MST of a graph and compare different FALQON strategies: (i) time rescaling (TR-FALQON) and (ii) multi-driver configurations. To avoid domain-specific biases, we adopt graphs with random weights and show that the FALQON variants exhibit significant differences in ground-state fidelity. We discuss the relevance of this approach for computer vision problems that naturally yield graph representations, and experimental results on synthetic instances together with a small demonstrative study on image segmentation illustrate both the potential and the current limitations of the method. Our numerical simulations on randomly weighted graphs show that standard one drive FALQON, although it reduces the expected energy, fails to concentrate amplitude in the MST solution. The multi drive variant succeeds in redistributing probability mass toward the ground state so that the MST appears among the most probable outcomes, and TR FALQON applied over multi drive produces the best results with faster convergence, lower final energy, and the highest solution state probability or fidelity in our tested instances. These improvements were observed on small synthetic graphs, underscoring both the promise of multi drive controls with temporal rescaling and the need for further scaling and hardware validation.
The Hilbert spaces of the edge excitations of several ``paired'' fractional quantum Hall states, namely, the Pfaffian, Haldane-Rezayi, and 331 states, are constructed and the states at each angular momentum level are enumerated. The method is based on finding all the zero-energy states for those Hamiltonians for which each of these known ground states is the exact, unique, zero-energy eigenstate of lowest angular momentum in the disk geometry. For each state, we find that, in addition to the usual bosonic charge-fluctuation excitations, there are fermionic edge excitations. The wave functions for each case have a similar form, related to Slater determinants, and the edge states satisfy a ``projection rule,'' that the parity of the number of fermions added to the edge equals the parity of the charge added. The edge states can be built out of quantum fields that describe the fermions, in addition to the usual scalar bosons (or Luttinger liquids) that describe the charge fluctuations. The fermionic fields in the Pfaffian and 331 cases are a noninteracting Majorana (i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state, the field is an anticommuting scalar. For this system, we exhibit a chiral Lagrangian that has manifest SU(2) symmetry, but breaks Lorentz invariance, because of the breakdown of the spin-statistics connection implied by the scalar nature of the field and the positive-definite norm on the Hilbert space. Finally, we consider systems on a cylinder, where the fluid has two edges, and construct the sectors of zero-energy states, discuss the projection rules for combining states at the two edges, and calculate the partition function for each edge excitation system at finite temperature in the thermodynamic limit. The corresponding theory for the hierarchy and its generalizations is also given. It is pointed out that the conformal field theories for the edge states are examples of orbifold constructions. Two appendixes contain technical details. \textcopyright{} 1996 The American Physical Society.
In the theory of the fractional quantum Hall effect, much attention is paid to the correspondence between fractional quantum Hall wave functions and conformal blocks in certain rational conformal field theories (CFT). This correspondence is powerful, enabling the calculation of the fractional statistics of a state that would be difficult if not impossible to calculate directly from the wave functions. But it is, in general, conjectural, remaining without microscopic justification in many cases of interest, and involves heavy mathematical machinery. We detail an alternative method to calculate Abelian and non-Abelian fractional statistics, the coherent state method. The method relies on assumptions which are independent of those underlying the CFT correspondence, so it serves as an independent check of results for the statistics where they exist, and an alternative source of results when a CFT is not known. We show how the coherent state method can be used to derive the statistics of several increasingly complicated trial wave functions: ν=1/2 Laughlin, Moore-Read, and k=3 Read-Rezayi. We discuss implications of our method for a possible notion of braiding statistics of the Gaffnian state, a "nonunitary" quantum Hall state for which the ramifications of the CFT correspondence are less well understood. We go on to derive formulas for the counting of zero modes of all these states on the torus.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.
The fractional quantum Hall (FQH) effect plays a prominent role in the study of topological phases of matter and of strongly correlated electron systems in general. FQH systems have been demonstrated to show many interesting novel properties such as fractional charges, and are believed to harbor even more intriguing phenomena such as fractional statistics. However, there remain many interesting questions to be addressed in this regime. The work reported in this thesis aims to push the envelope of our understanding of the low-energy properties of FQH states using microscopic principles. In the first part of the thesis, we present a systematic perturbative approach to study excitations in the thin cylinder/torus limit of the quantum Hall states. The approach is applied to the Haldane-Rezayi and Gaffnian quantum Hall states, which are both expected to have gapless excitations in the usual two-dimensional thermodynamic limit. For the Haldane-Rezayi state, we confirm that gapless excitations are present also in the ``one-dimensional'' thermodynamic limit of an infinite thin cylinder, in agreement with earlier considerations based on the wave functions alone. In contrast, we identify the lowest excitations of the Gaffnian state in the thin cylinder limit, and conclude that they are gapped, using a combination of perturbative and numerical means. We discuss possible scenarios for the cross-over between the two-dimensional and the one-dimensional thermodynamic limit in this case. In the second part of the thesis, we study the low energy spectral properties of positive center-of-mass conserving two-body Hamiltonians as they arise in models of FQH states. Starting from the observation that positive many-body Hamiltonians must have ground state energies that increase monotonously in particle number, we explore what general additional constraints can be obtained for two-body interactions with "center-of-mass conservation" symmetry, both in the presence and absence of particle-hole symmetry. We find general bounds that constrain the evolution of the ground state energy with particle number, and in particular constrain the chemical potential at T=0. Special attention is given to Hamiltonians with zero modes, in which case similar bounds on the first excited state are also obtained, using a duality property. In this case, in particular an upper bound on the charge gap is also obtained. We further comment on center-of-mass and relative-decomposition in disk geometry within the framework of second quantization.