1. Introduction 2. The Pauli Equation and its Symmetries {2.1} Gauge-Invariant Form of the Pauli Equation {2.2} Aharonov-Bohm Effect {2.3} Aharonov-Casher Effect 3. Gauge Invariance in Non-Relativistic Quantum Many-Particle Systems {3.1} Differential Geometry of the Background {3.2} Systems of Spinning Particles Coupled to External Electromagnetic and Geometric Fields {3.3} Moving Coordinates and Quantum-Mechanical Larmor Theorem 4. Some Key Effects Related to the $U(1) \times SU(2)$ Gauge Invariance of Non-Relativistic Quantum Mechanics {4.1} ``Tidal'' Aharonov-Bohm and ``Geometric'' Aharonov-Casher Effects {4.2} Flux Quantization {4.3} Barnett and Einstein-deHaas Effects {4.4} Meissner-Ochsenfeld Effect and London Theory of Superconductivity {4.5} Quantum Hall Effect 5. Scaling Limit of the Effective Action of Fermi Systems, and Classification of States of Non-Relativistic Matter 6. Scaling Limit of the Effective Action of a Two-Dimensional, Incompressible Quantum Fluid {6.1} Scaling Limit of the Effective Action {6.2} Linear Response Theory and Current Sum Rules {6.3} Quasi-Particle Excitations and a Spin-Singlet Electron Pairing Mechanism 7. Anomaly Cancellation and Algebras of Chiral Edge Currents in Two- Dimensional, Incompressible Quantum Fluids {7.1} Integer Quantum Hall Effect and Edge Currents {7.2} Edge Excitations in Spin-Polarized Quantum Hall Fluids 8. Classification of Incompressible Quantum Hall Fluids {8.1} QH Fluids and QH Lattices: Basic Concepts {8.2} A Dictionary Between the Physics of QH Fluids and the Mathematics of QH Lattices {8.3} Basic Invariants of Chiral QH Lattices (CQHLs) and their Physical Interpretations {8.4} General Theorems and Classification Results for CQHLs {8.5} Maximally Symmetric CQHLs {8.6} Summary and Physical Implications of the Classification Results1. Introduction 2. The Pauli Equation and its Symmetries {2.1} Gauge-Invariant Form of the Pauli Equation {2.2} Aharonov-Bohm Effect {2.3} Aharonov-Casher Effect 3. Gauge Invariance in Non-Relativistic Quantum Many-Particle Systems {3.1} Differential Geometry of the Background {3.2} Systems of Spinning Particles Coupled to External Electromagnetic and Geometric Fields {3.3} Moving Coordinates and Quantum-Mechanical Larmor Theorem 4. Some Key Effects Related to the $U(1) \times SU(2)$ Gauge Invariance of Non-Relativistic Quantum Mechanics {4.1} ``Tidal'' Aharonov-Bohm and ``Geometric'' Aharonov-Casher Effects {4.2} Flux Quantization {4.3} Barnett and Einstein-deHaas Effects {4.4} Meissner-Ochsenfeld Effect and London Theory of Superconductivity {4.5} Quantum Hall Effect 5. Scaling Limit of the Effective Action of Fermi Systems, and Classification of States of Non-Relativistic Matter 6. Scaling Limit of the Effective Action of a Two-Dimensional, Incompressible Quantum Fluid {6.1} Scaling Limit of the Effective Action {6.2} Linear Response Theory and Current Sum Rules {6.3} Quasi-Particle Excitations and a Spin-Singlet Electron Pairing Mechanism 7. Anomaly Cancellation and Algebras of Chiral Edge Currents in Two- Dimensional, Incompressible Quantum Fluids {7.1} Integer Quantum Hall Effect and Edge Currents {7.2} Edge Excitations in Spin-Polarized Quantum Hall Fluids 8. Classification of Incompressible Quantum Hall Fluids {8.1} QH Fluids and QH Lattices: Basic Concepts {8.2} A Dictionary Between the Physics of QH Fluids and the Mathematics of QH Lattices {8.3} Basic Invariants of Chiral QH Lattices (CQHLs) and their Physical Interpretations {8.4} General Theorems and Classification Results for CQHLs {8.5} Maximally Symmetric CQHLs {8.6} Summary and Physical Implications of the Classification Results

arXiv:cond-mat/9508062Jul 31, 199516 citations

Finite matrix model of quantum hall fluids on S2

Yi-Xin Chen, M. D. Gould, Yao-Zhong Zhang

Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.Based on Haldane's spherical geometrical formalism of the two-dimensional quantum Hall fluids, the relation between the noncommutative geometry of S2 and the two-dimensional quantum Hall fluids is exhibited. A finite matrix model on the two-sphere is explicitly constucted as an effective description of the fractional quantum Hall fluids of finite extent, and the complete sets of physical quantum states of this matrix model are determined. We also describe how the low-lying excitations in the model are constructed in terms of the quasi-particle and quasi-hole excitations. It is shown that there exists a Haldane hierarchical structure in the two-dimensional quantum Hall fluid states of the matrix model. These hierarchical fluid states are generated by the parent fluid state by condensing the quasi-particle and quasi-hole excitations level by level.

Bulletin of the Australian Mathematical SocietyAug 1, 20070 citations

Decoherence demystified : The hydrodynamic viewpoint

Kyungsun Na, Róbert E. Wyatt

The hydrodynamic formulation of quantum mechanics is used to elucidate the mechanism for decoherence, the suppression of interference effects in a system evolving from an initial coherent superposition. Analysis of time-dependent trajectory ensembles, flux maps, and elements of the stress tensor for two composite systems, in one of which the system is uncoupled to the environment, leads to the decoherence mechanism. For the uncoupled case, the quantum force acting on the fluid elements directs flux toward an attractor where the interference feature arises. For the coupled case, the classical force acting on each fluid element counters the quantum force and leads to gradual separation of the components of the initial superposition. Concomitantly, fluid stress is relieved when flux vectors diverge from a repellor in the mid-region between the separating wavepackets, thus suppressing formation of the interference feature.The hydrodynamic formulation of quantum mechanics is used to elucidate the mechanism for decoherence, the suppression of interference effects in a system evolving from an initial coherent superposition. Analysis of time-dependent trajectory ensembles, flux maps, and elements of the stress tensor for two composite systems, in one of which the system is uncoupled to the environment, leads to the decoherence mechanism. For the uncoupled case, the quantum force acting on the fluid elements directs flux toward an attractor where the interference feature arises. For the coupled case, the classical force acting on each fluid element counters the quantum force and leads to gradual separation of the components of the initial superposition. Concomitantly, fluid stress is relieved when flux vectors diverge from a repellor in the mid-region between the separating wavepackets, thus suppressing formation of the interference feature.

arXiv:quant-ph/0201108Jan 24, 20022 citations

Trion Hall effect in electron-hole double layers

Raghav Chaturvedi, Phuong X. Nguyen, Patrick Knüppel +4 more

The realization of Coulomb coupled electron-hole double layers in 2D semiconductor heterostructures has enabled the thermodynamic and transport studies of equilibrium exciton fluids without a magnetic field. By doping the exciton fluid with additional electrons/holes, an equilibrium fluid of trions - three particle bound states of electrons and holes - further emerge, providing the platform to explore new transport phenomena associated with such composite particles. Here we report the observation of a Hall effect for trions in MoSe2/WSe2 heterobilayers, which support Coulomb-coupled electron and hole fluids with tunable densities. The Hall effect arises from a Lorentz force on trions under a perpendicular magnetic field. It is manifested in both Hall drag measurements and standard Hall effect measurements on just one of the semiconductor layers. For negatively charged trions, an electron Hall effect is observed even in a hole doped WSe2 monolayer due to the presence of trion drags. The trion Hall effect also disappears when the trions are ionized at elevated temperatures and/or high trion densities. Our work opens the door for realizing quantum oscillations and the quantum Hall effect for trions.The realization of Coulomb coupled electron-hole double layers in 2D semiconductor heterostructures has enabled the thermodynamic and transport studies of equilibrium exciton fluids without a magnetic field. By doping the exciton fluid with additional electrons/holes, an equilibrium fluid of trions - three particle bound states of electrons and holes - further emerge, providing the platform to explore new transport phenomena associated with such composite particles. Here we report the observation of a Hall effect for trions in MoSe2/WSe2 heterobilayers, which support Coulomb-coupled electron and hole fluids with tunable densities. The Hall effect arises from a Lorentz force on trions under a perpendicular magnetic field. It is manifested in both Hall drag measurements and standard Hall effect measurements on just one of the semiconductor layers. For negatively charged trions, an electron Hall effect is observed even in a hole doped WSe2 monolayer due to the presence of trion drags. The trion Hall effect also disappears when the trions are ionized at elevated temperatures and/or high trion densities. Our work opens the door for realizing quantum oscillations and the quantum Hall effect for trions.

cond-mat.mes-hallCross-listedarXiv:2606.18647Jun 18, 2026

U(1) \times SU(2)

Gauge Invariance of Non-Relativistic Quantum Mechanics {4.1} ``Tidal'' Aharonov-Bohm and ``Geometric'' Aharonov-Casher Effects {4.2} Flux Quantization {4.3} Barnett and Einstein-deHaas Effects {4.4} Meissner-Ochsenfeld Effect and London Theory of Superconductivity {4.5} Quantum Hall Effect 5. Scaling Limit of the Effective Action of Fermi Systems, and Classification of States of Non-Relativistic Matter 6. Scaling Limit of the Effective Action of a Two-Dimensional, Incompressible Quantum Fluid {6.1} Scaling Limit of the Effective Action {6.2} Linear Response Theory and Current Sum Rules {6.3} Quasi-Particle Excitations and a Spin-Singlet Electron Pairing Mechanism 7. Anomaly Cancellation and Algebras of Chiral Edge Currents in Two- Dimensional, Incompressible Quantum Fluids {7.1} Integer Quantum Hall Effect and Edge Currents {7.2} Edge Excitations in Spin-Polarized Quantum Hall Fluids 8. Classification of Incompressible Quantum Hall Fluids {8.1} QH Fluids and QH Lattices: Basic Concepts {8.2} A Dictionary Between the Physics of QH Fluids and the Mathematics of QH Lattices {8.3} Basic Invariants of Chiral QH Lattices (CQHLs) and their Physical Interpretations {8.4} General Theorems and Classification Results for CQHLs {8.5} Maximally Symmetric CQHLs {8.6} Summary and Physical Implications of the Classification Results