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Journal of the Physical Society of Japan

Anyons in Commensurate Flux Phase

Yasumasa Hasegawa, Osamu Narikiyo, Kazuhiro Kuboki, Hidetoshi Fukuyama

Published Mar 15, 1990• 32 citations
Publisher

Abstract

Anyons are constructed in the commensurate flux phases in the t - J model using slave boson mean-field theory. Fluctuations of the local constraint and the mean-field order parameters, which couple to both fermions and slave bosons, are treated as the gauge field. The coupling to the fermion creates the Chern-Simons term for the gauge field, which changes the statistics of bosons into semions.

DOI
10.1143/jpsj.59.822

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The extended t - J model has been studied in the slave boson scheme. It is found that there exists some range of parameters where the chiral spin state is stabilized within a mean field approximation. The Chern-Simons Lagrangian for the gauge field, which represents the fluctuation of the phases of the order parameters and the local constraint, is obtained by integrating out spin fermion fields as in the undoped case [Wen, Wilczek and Zee: Phys. Rev. B 39 (1989) 11413]. The coefficient of this term turns out to depend on doping. Due to the coupling to this Chern-Simons gauge field, hole bosons are transmuted into anyons.

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The flux phase solutions in the t-J model, the gauge theories with Chern-Simons term, promising new exciting physics, are so far obtained only within slave-boson (or fermion) approaches, where local gauge invariance is not taken into account exactly. We inspect the possibility to do this, mainly, in the example of the t-J model. The analysis of different parametrizations in the functional representation shows that (i) calculations which take into account constraints for the number of on-site-available states are describing quasiparticles in terms of the wrong local statistics, and contain gauge non-invariant objects; (ii) application of the projection technique in the slave-boson(fermion) representation reproduces the correct statistics, and is exactly equivalent to the conventional diagram technique for Hubbard operators. In the language of slave bosons this means that the Fourier transformation leading to Bloch states should be made only after accounting constraints in the measure of functional integral.

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We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:

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The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter $θ$. $θ= 0, π$ correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, $θ$ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase $θ$ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the $ν= 1/3$ fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.

arXiv:2210.02530•Oct 5, 2022•5 citations
Anyons are constructed in the commensurate flux phases in the t - J model using slave boson mean-field theory. Fluctuations of the local constraint and the mean-field order parameters, which couple to both fermions and slave bosons, are treated as the gauge field. The coupling to the fermion creates the Chern-Simons term for the gauge field, which changes the statistics of bosons into semions.
The extended t - J model has been studied in the slave boson scheme. It is found that there exists some range of parameters where the chiral spin state is stabilized within a mean field approximation. The Chern-Simons Lagrangian for the gauge field, which represents the fluctuation of the phases of the order parameters and the local constraint, is obtained by integrating out spin fermion fields as in the undoped case [Wen, Wilczek and Zee: Phys. Rev. B 39 (1989) 11413]. The coefficient of this term turns out to depend on doping. Due to the coupling to this Chern-Simons gauge field, hole bosons are transmuted into anyons.
The extended t - J model, which has both transfer integral and exchange interaction between next-nearest neighbors besides those between nearest neighbors, has been investigated by the mean field approximation in the slave-boson scheme. Especially, the staggered flux state and the chiral spin state have been investigated. For the latter state, anyons are derived by use of the result of the Chern-Simons term for the doped case. It is shown that the prefactor of the Chern-Simons term and thus the fractional statistics depend on the degree of doping.
We extend Abelian and non-Abelian bosonization formulas found in a previous paper, in combination with coherent-state methods, to present a systematic derivation of the slave-fermion, slave-boson, and semion representations of the two-dimensional t-J model in path-integral form. The slave-fermion and slave-boson representations are shown to arise from two different gauge fixing constraints introduced in the path-integral representation of the bosonized t-J model. For each representation we discuss the approximations leading to a mean-field theory. In the mean-field theory based on the semion representation, the holons are shown to be ``Dirac semions.''
The flux phase solutions in the t-J model, the gauge theories with Chern-Simons term, promising new exciting physics, are so far obtained only within slave-boson (or fermion) approaches, where local gauge invariance is not taken into account exactly. We inspect the possibility to do this, mainly, in the example of the t-J model. The analysis of different parametrizations in the functional representation shows that (i) calculations which take into account constraints for the number of on-site-available states are describing quasiparticles in terms of the wrong local statistics, and contain gauge non-invariant objects; (ii) application of the projection technique in the slave-boson(fermion) representation reproduces the correct statistics, and is exactly equivalent to the conventional diagram technique for Hubbard operators. In the language of slave bosons this means that the Fourier transformation leading to Bloch states should be made only after accounting constraints in the measure of functional integral.
We summarize a study of an Abelian gauge theory in 2+1 dimensions, the gauge field being coupled to nonrelativistic Fermions. The Action for the gauge field is a combination of the Maxwell term and a Chern-Simons (CS) term. We study the limit of vanishing Fermions ’ charge, keeping fixed the gauge field mass induced by the CS term. By considering a closed surface, in particular a torus to keep translational invariance, we show that the Fermions do not decouple completely from the gauge field, and in fact they behave as Anyons treated in the-translationally invariant formulation of the- Mean Field approximation. We describe the exact solution of this limiting case. This lecture is based on ref.[1] and on previous work done on the problem of a translational invariant formulation of the Anyon mean field theory and its properties with respect to superfluidity and possibly superconductivity (for a general review of the previous work see ref.[2]). Actually, to discuss those issues in a proper way it is necessary to introduce a suitable mathematical formalism:
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θθθ. θ=0,πθ= 0, πθ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θθθ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θθθ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν=1/3ν= 1/3ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.