We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the ν=1/2 Tao–Thouless state as the unique densest ground state.
We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the ν=1/2 Tao–Thouless state as the unique densest ground state.
We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor \nu=2/3 ν = 2 / 3 . The emergent Entangled Pauli Principle (EPP), introduced in [Phys. Rev. B 98, 161118(R) (2018)] and which defines the “DNA” of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-body M M -clustering property within any given number of Landau levels, that is, wave functions with M M th-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for the M=3 M = 3 case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of [Phys. Rev. X 1, 021015 (2011)] to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of finite bond dimension that emerge via root level entanglement.
A formalism is developed for the rigorous study of solvable fractional quantum Hall parent Hamiltonians with Landau-level mixing. The idea of organization through ``generalized Pauli principles'' is expanded to allow for root-level entanglement, giving rise to ``entangled Pauli principles.'' Through the latter, aspects of the effective field theory description become ingrained in exact microscopic solutions for a great wealth of phases for which no similar single Landau-level description is known. We discuss in detail braiding statistic, edge theory, and rigorous zero-mode counting for the Jain-221 state as derived from a microscopic Hamiltonian. The relevant root-level entanglement is found to feature an Affleck-Kennedy-Lieb-Tasaki (AKLT)-type matrix-product ground states structure associated with an emergent SU(2) symmetry.
Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.
Strong correlation among electrons under high magnetic field gives rise to an entirely new arena of emergent physics, namely fractional quantum Hall effect. Such systems have entirely different elementary degrees of freedom and generally, demand non-perturbative approaches to develop a better understanding. In the literature, there are several analytical methodologies and numerical toolkits available to study such a system. Clustering of zeros, parent Hamiltonian, off-diagonal order parameter, parton construction, matrix product states are to be named among a few of those popular methodologies in the existing literature. Most of these methods work well in the lowest Landau level or holomorphic wavefunction framework. It is, however, imperative to develop such methodology to study systems with Landau levels mixing to study more exotic as well as experimentally relevant states. In this work, we have developed particular methodologies, which denounce the traditional importance of the analytic properties of first quantized model wavefunction thereby extend the existing parent Hamiltonian, topological order-parameter, matrix product states descriptions to mixed Landau level systems. Such extension produces a deeper, compact and holistic understanding of universal physics of exotic phases in strongly correlated systems from the microscopic viewpoint, as well as produces interesting new results. Our second quantized/ non-analytic approach allows us to construct the ``entangled Pauli principle", a guidebook to extract universal/topological properties such as braiding statistics, fractional charge quantization, topological degeneracy of the ground states starting from a relatively simple many-body wavefunction, ``root pattern" of fractional quantum Hall state. Such an entangled Pauli principle can be derived from a microscopic parent Hamiltonian setting, thereby provide us a potential tool to probe the non-universal physics in quantum Hall fluids as well. Essentially, entangled Pauli principle is the ``DNA" of fractional quantum Hall states. Using this guiding principle, we have shown ground states with non-abelian excitations, such as Majorana fermion or Fibonacci fermion can be stabilized for two-particle interaction. Fibonacci fermion supports universal quantum gates, thereby a potential candidate for the topologically protected universal quantum computer. Entangled Pauli principle, along with a recently developed topological order parameter for composite fermions, gives rise to Parent Hamiltonian description for composite fermions as well.
Using an algebra of second-quantized operators, we develop local two-body parent Hamiltonians for all unprojected Jain states at filling factor n/(2np+1), with integer n and (half-)integer p. We rigorously establish that these states are uniquely stabilized and that zero mode counting reproduces mode counting in the associated edge conformal field theory. We further establish the organizing "entangled Pauli principle" behind the resulting zero mode paradigm and unveil an emergent SU(n) symmetry characteristic of the fixed point physics of the Jain quantum Hall fluid.
Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire Zk Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.