We discuss the quantum mechanics of particles of arbitrary statistics on an infinite cylinder with and without a magnetic field perpendicular to the surface. In the presence of a magnetic field, the translational symmetry along the cylinder is broken down to a discrete one by the Aharonov-Bohm effect. For interacting fermions in a strong field we get an effective one-dimensional lattice model that in a limit can be mapped on an Ising chain. We also show that a system of anyons on a cylinder are, in a certain limit, closely related to the one-dimensional (integrable) Sutherland model. By order of magnitude estimates we demonstrate that none of these effects are likely to be experimentally observed with present techniques.
We discuss the quantum mechanics of particles of arbitrary statistics on an infinite cylinder with and without a magnetic field perpendicular to the surface. In the presence of a magnetic field, the translational symmetry along the cylinder is broken down to a discrete one by the Aharonov-Bohm effect. For interacting fermions in a strong field we get an effective one-dimensional lattice model that in a limit can be mapped on an Ising chain. We also show that a system of anyons on a cylinder are, in a certain limit, closely related to the one-dimensional (integrable) Sutherland model. By order of magnitude estimates we demonstrate that none of these effects are likely to be experimentally observed with present techniques.
To study the manifestation of the Aharonov-Bohm effect in many-body systems we consider the statistical mechanics of the Gross-Neveu model on a ring (1+1 dimensions) and on a cylinder (2+1 dimensions) with a thin solenoid coinciding with the axis. For such systems with a non-trivial magnetic flux (θ) many thermodynamical observables, such as the order parameter, induced current and virial coefficients, display periodic but non-analytic dependence on θ. In the 2+1 dimensional case we further find that there is an interval of θ∈(1/3,2/3) (modulo integers) where parity is always spontaneously broken, independent of the circumference of the cylinder. We show that the mean-field character of the phase transitions is preserved to the leading order in 1/N, by verifying the θ-independence of all the critical exponents. The precise nature of the quasi-particle, locally fermion-like and globally anyon-like, is illuminated through the calculation of the equal-time commutator and the decomposition of the propagator into a sum over paths classified by winding numbers.
In two-dimensional space, the topological coupling arising between charged particles with attached magnetic fluxes, via the Aharonov–Bohm effect, causes distinguishable particles to effectively acquire quantum statistics: a nontrivial quantum phase is generated when a particle of one species encircles one of a different species. We discuss a number of exact and numerical results concerning distinguishable particles with fractional statistics. Just like for anyons, the spectrum of such particles confined to the lowest Landau level of a strong magnetic field can be found exactly. Then a system of particles in the presence of static magnetic fluxes (equivalent to particles with infinite mass) is considered, and the low-lying states of one particle in the presence of two fluxes and two particles in the presence of one flux are analyzed.
We study the motion of a quantum charged particle, constrained on the surface of a cylinder, in the presence of a radial magnetic field. When the spin of the particle is neglected, the system essentially reduces to an infinite family of simple harmonic oscillators, equally spaced along the axis of the cylinder. Interestingly enough, it can be used as a quantum Fourier transformer, with convenient visual output. When the spin-1/2 of the particle is taken into account, a non-conventional perturbative analysis results in a recursive closed form for the corrections to the energy and the wavefunction, for all eigenstates, to all orders in the magnetic moment of the particle. A simple two-state system is also presented, the time evolution of which involves an approximate precession of the spin perpendicularly to the magnetic field. A number of plots highlight the findings while several three-dimensional animations have been made available on the web.
To gain a better understanding of the interplay between frustrated long-range interactions and zero-temperature quantum fluctuations, we investigate the ground-state phase diagram of the transverse-field Ising model with algebraically decaying long-range Ising interactions on quasi-one-dimensional infinite-cylinder triangular lattices. Technically, we apply various approaches including low- and high-field series expansions. For the classical long-range Ising model, we investigate cylinders with an arbitrary even circumference. We show the occurrence of gapped stripe-ordered phases emerging out of the infinitely degenerate nearest-neighbor Ising ground-state space on the two-dimensional triangular lattice. Further, while cylinders with circumferences 6, 10, 14, etc., are always in the same stripe phase for any decay exponent of the long-range Ising interaction, the family of cylinders with circumferences 4, 8, 12, etc., displays a phase transition between two different types of stripe structures. For the full long-range transverse-field Ising model, we concentrate on cylinders with circumference four and six. The ground-state phase diagram consists of several quantum phases in both cases including an x-polarized phase, stripe-ordered phases, and clock-ordered phases which emerge from an order-by-disorder scenario already present in the nearest-neighbor model. In addition, the generic presence of a potential intermediate gapless phase with algebraic correlations and associated Kosterlitz-Thouless transitions is discussed for both cylinders.
This thesis addresses the dimensional reduction of two-dimensional (2D) Abelian anyons under strong anisotropic confinement, studied both at the microscopic many-body level and at the mean-field/almost-bosonic level.Anyons are quasi-particles with fractional statistics that interpolate between bosons and fermions, characterized by an exchange phase e^{i·π·α} for α in (0, 1). The theoretical concept of anyons dates back to the work of Leinaas and Myrheim in 1977, while convincing experimental evidence for their existence emerged only in 2020, through their realization in fractional quantum Hall systems. Investigating their behavior under confinement not only clarifies the connections between one- and two-dimensional anyons, but also offers new insights into the mathematical structure of anyonic systems.At the microscopic level, we consider N anyons in the magnetic gauge picture with a harmonic trapping potential that freezes the motion in the longitudinal (y) direction. We rigorously prove that, after subtracting the frozen longitudinal contribution, the associated 2D energy Γ-converges to that of the one-dimensional (1D) Tonks-Girardeau (TG) gas. This establishes a precise mathematical connection between 2D anyons and 1D impenetrable bosons, and, via the Bose-Fermi mapping, to a free Fermi gas. Importantly, this 1D limit model is independent of the exchange parameter α at leading order, indicating that the effect of the magnetic gauge field is gauged away up to o(1) corrections. Equivalently, the long-range Aharonov-Bohm interactions collapse into a hard-core exclusion constraint in this dimensional reduction limit.At the mean-field level, in the average-field regime (N → ∞ with α proportional to N^{-1}), the 2D anyon system is described by the Chern-Simons-Schrödinger (CSS) functional, a nonlinear magnetic Schrödinger model with a self-consistent gauge field. Performing the same anisotropic dimensional reduction limit, the resulting 1D effective model is described by a quintic nonlinear Schrödinger (NLS) equation, whose quintic nonlinearity stems from the magnetic self-consistent gauge in the CSS model. This dimensional reduction is established both at the level of ground-state energies and, under suitable well-posedness assumptions on the CSS dynamics, at the level of time evolution. Unlike the TG gas, the effective coupling constant g in the quintic NLS retains a dependence on the exchange parameter α through g = π·β² for β proportional to α·N.Taken together, these results yield two distinct effective 1D descriptions arising from different orders of limits. Crucially, the mean-field/almost-bosonic limit and the dimensional reduction (strong confinement) limit do not commute: the TG gas limit is α-independent, whereas the quintic NLS limit keeps an α-dependence through β proportional to α·N. This non-commutativity highlights the subtle interplay between scaling limits and confinement in anyonic systems. Future work might investigate whether a critical regime separates these two situations.
Under second-order degenerate perturbation theory, we show that the physics of N particles with arbitrary spin confined in a one-dimensional trap in the strongly interacting regime can be described by superexchange interaction. An effective spin-chain Hamiltonian (non-translationally-invariant Sutherland model) can be constructed from this procedure. For spin-1/2 particles, this model reduces to the non-translationally-invariant Heisenberg model, where a transition between Heisenberg antiferromagnetic (AFM) and ferromagnetic (FM) states is expected to occur when the interaction strength is tuned from the strongly repulsive to the strongly attractive limit. We show that the FM and the AFM states can be distinguished by two different methods: the first is based on their distinct responses to a spin-dependent magnetic gradient, and the second is based on their distinct momentum distributions. We confirm the validity of the spin-chain model by comparison with results obtained from several unbiased techniques.