We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyons. Long-range coupling interaction is a fundamental character of this knot lattice model. The short-range coupling models, suc...We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyons. Long-range coupling interaction is a fundamental character of this knot lattice model. The short-range coupling models, such as the Ising model,Hamiltonian model of quantum Hall effect, fermion pairing model, Kitaev honeycomb lattice model, and so on, are the short-range coupling cases of this knot lattice model. The long-range coupling knot lattice model bears Abelian and nonAbelian anyons, and shows integral and fractional filling states like the quantum Hall system. The fusion rules of anyons are explicitly demonstrated by braiding crossing states. The eigenstates of quantum models can be represented by a multilayer link lattice pattern whose topology is characterized by the linking number. This topological linking number offers a new quantity to explain and predict physical phenomena in conventional quantum models. For example, a convection flow loop is introduced into the well-known Bardeen–Cooper–Schrieffer fermion pairing model to form a vortex dimer state that offers an explanation of the pseudogap state of unconventional superconductors, and predicts a fractionally filled vortex dimer state. The integrally and fractionally quantized Hall conductance in the conventional quantum Hall system has an exact correspondence with the linking number in this multi-knot lattice model. The real-space knot pattern in the topological insulator model has an equivalent correspondence with the Lissajous knot in momentum space. The quantum phase transition between different quantum states of the quantum spin model is also directly quantified by the change of topological linking number, which revealed the topological character of phase transition. Circularized photons in an optical fiber network are a promising physical implementation of this multi-knot lattice, and provide a different path to topological quantum computation.展开更多
Gentile statistics describes fractional statistical systems in the occupation number representation.Anyon statistics researches those systems in the winding number representation.Both of them are intermediate statisti...Gentile statistics describes fractional statistical systems in the occupation number representation.Anyon statistics researches those systems in the winding number representation.Both of them are intermediate statistics between Bose–Einstein and Fermi–Dirac statistics.The second quantization of Gentile statistics shows a lot of advantages.According to the symmetry requirement of the wave function and the property of braiding,we give the general construction of transformation between anyon and Gentile statistics.In other words,we introduce the second quantization form of anyons in an easier way.This construction is a correspondence between two fractional statistics and gives a new description of anyon.Basic relations of second quantization operators,the coherent state and Berry phase are also discussed.展开更多
Non-Abelian anyons can emerge as fractionalized excitations in two-dimensional systems with topological order. One important example is the Moore–Read fractional quantum Hall state. Its quasihole states are zero-ener...Non-Abelian anyons can emerge as fractionalized excitations in two-dimensional systems with topological order. One important example is the Moore–Read fractional quantum Hall state. Its quasihole states are zero-energy eigenstates of a parent Hamiltonian, but its quasiparticle states are not. Both of them can be modeled on an equal footing using the bipartite composite fermion method. We study the entanglement spectrum of the cases with two or four non-Abelian anyons. The counting of levels in the entanglement spectrum can be understood using the edge theory of the Moore–Read state, which reflects the topological order of the system. It is shown that the fusion results of two non-Abelian anyons is determined by their distributions in the bipartite construction.展开更多
Anyons can be used to realize quantum computation, because they are two-level systems in two dimensions. In this paper, we propose a scheme to simulate single-qubit gates and CNOT gate using Abelian anyons in the Kita...Anyons can be used to realize quantum computation, because they are two-level systems in two dimensions. In this paper, we propose a scheme to simulate single-qubit gates and CNOT gate using Abelian anyons in the Kitaev model. Two pairs of anyons （six spins） are used to realize single-qubit gates, while ten spins are needed for the CNOT gate. Based on these quantum gates, we show how to realize the Grover algorithm in a two-qubit system.展开更多
We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electric and magnetic fields in a specific configuration.The electric and magne...We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electric and magnetic fields in a specific configuration.The electric and magnetic fields we applied contain a magnetic and two electric fields.We find that when the atom is cooled down to the limit of the negligibly small kinetic energy,the atom behaves like an anyon because its angular momentum takes fractional values.The fractional part of the angular momentum is determined by both the magnetic and one of the electric fields.Roles electric and magnetic fields played are analyzed.展开更多
In order to analyze a limiting case of the 1D delta anyon model,the coupling strength of the δ interactionc is modified to become a function of the anyonic parameter κ.A pedagogic derivation of the solution for this...In order to analyze a limiting case of the 1D delta anyon model,the coupling strength of the δ interactionc is modified to become a function of the anyonic parameter κ.A pedagogic derivation of the solution for this modifiedmodel using the method of anyon-boson mapping plus Bethe ansatz is presented.The limiting case as κ→π andsimultaneously c→0,which was previously neglected,is analyzed.Some unexpected properties of this limiting case arediscovered.The BAEs are compared with previous results.展开更多
We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model...We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model is helpful to understand the toric code of anyons in a more symmetric way. On the other hand, there is a direct relation between this toric honeycomb model and a boundary coupled Ising chain array in a square lattice via Jordan-Wigner transformation. We discuss the equivalence between these two models in the low-lying sector and realize these anyon excitations in a conventional fermion system. The analysis for the ground state degeneracy in the last section can also be thought of as a complementarity of our previous work [Phys. A： Math. Theor. 43 （2010） 105306].展开更多
In this paper,starting with the well known U(1)Chern-Simons Lagrangian and the covariant derivative ofa complex scalar matter field,we give a detailed discussion of some topological properties of anyons.We show that t...In this paper,starting with the well known U(1)Chern-Simons Lagrangian and the covariant derivative ofa complex scalar matter field,we give a detailed discussion of some topological properties of anyons.We show that the'basic'charge carried by anyons has an inner structure and can be decomposed in terms of the Chern-Simons couplingand the gauge coupling constants of the theory.Also some incorrect results obtained in the literature are revised.展开更多
The note gives a watertight confirmation of the E-infinity Cantorian theory results for ordinary and dark cosmic energy density of the universe and respectively. The computation is fundamentally based on a golden mean...The note gives a watertight confirmation of the E-infinity Cantorian theory results for ordinary and dark cosmic energy density of the universe and respectively. The computation is fundamentally based on a golden mean fusion function that goes back to the highly original anyon proposal of F. Wilczek.展开更多
Jean Anyon(1941-2013)is a critical pedagogical researcher and social activist in the United States.All her life,she devoted herself to taking education as a breakthrough,exposing the ugliness of capitalist countries a...Jean Anyon(1941-2013)is a critical pedagogical researcher and social activist in the United States.All her life,she devoted herself to taking education as a breakthrough,exposing the ugliness of capitalist countries and promoting the equality of society and education.She uses Marxist theory to analyze the relationship between social class and school knowledge,hidden curriculum;education and mainstream ideology;and conflicts and resistances in education,emphasizing the theoretical and practical significance of Marxism in today’s education.In order to find the deepest problem of education,Jean Anyon demonstrates the relationship between school education and macroeconomic policy from the perspective of political economics;the dialectical relationship between education reform and economic reform;and explains the reasons for education failure by quoting David Harvey’s concept of“deprivation accumulation”.Facing the cruel situation,Jean Anyon did not lose confidence,but found the“radical possibility”and put forward the concept of social change centered on education reform.展开更多
Topological flat bands have attracted significant interest across various branches of physics,where synthetic gauge fields are typically considered an essential prerequisite.Numerous mechanisms have been proposed for ...Topological flat bands have attracted significant interest across various branches of physics,where synthetic gauge fields are typically considered an essential prerequisite.Numerous mechanisms have been proposed for implementing these fields,including magnetic fields on electrons,differential optical paths for photons,and strain-induced effective magnetic fields,among others.In this work,we introduce a novel approach to generating synthetic gauge fields through quantum statistics and demonstrate their effectiveness in realizing anyonic topological flat bands.Notably,we discover that a pair of strongly interacting anyons can induce square-root topological flat bands within a lattice model that remains dispersive and topologically trivial for a single particle.To validate our theoretical predictions,we experimentally simulate the quantum statistics-induced topological flat bands and square-root topological boundary states by mapping the eigenstates of two anyons onto modes in electric circuits.Our findings not only open a new pathway for creating topological flat bands but also deepen our understanding of anyonic physics and the underlying principles of flat-band topology.展开更多
Majorana zero modes(MZMs)have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC).In addit...Majorana zero modes(MZMs)have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC).In addition to the Majorana scheme,some non-Majorana quasiparticles obeying non-Abelian statistics,including topological Dirac fermionic modes,have also been proposed and therefore become new candidates for TQC.In this review,we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes,emphasizing the recent progress on topological Dirac fermionic modes.A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry,which can be realized in a number of platforms,including the one-dimensional topological insulator,higher-order topological insulator,and spin superconductor.This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin,such as superconductivity-free and larger gaps.Therefore,it provides a new avenue for investigating non-Abelian physics and possible TQC.展开更多
<正> By using the gauge-invariance,a Klein-Gordon’s type equation for anyons in external fields is constructed.The equation can be solved exactly in a constant electric and magnetic field,as well as in a quanti...<正> By using the gauge-invariance,a Klein-Gordon’s type equation for anyons in external fields is constructed.The equation can be solved exactly in a constant electric and magnetic field,as well as in a quantized electromagnetic field.The analytic forms of solutions are also given in detail.展开更多
We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the ...We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than h is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 27r-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant No.11304062)
文摘We proposed an entangled multi-knot lattice model to explore the exotic statistics of anyons. Long-range coupling interaction is a fundamental character of this knot lattice model. The short-range coupling models, such as the Ising model,Hamiltonian model of quantum Hall effect, fermion pairing model, Kitaev honeycomb lattice model, and so on, are the short-range coupling cases of this knot lattice model. The long-range coupling knot lattice model bears Abelian and nonAbelian anyons, and shows integral and fractional filling states like the quantum Hall system. The fusion rules of anyons are explicitly demonstrated by braiding crossing states. The eigenstates of quantum models can be represented by a multilayer link lattice pattern whose topology is characterized by the linking number. This topological linking number offers a new quantity to explain and predict physical phenomena in conventional quantum models. For example, a convection flow loop is introduced into the well-known Bardeen–Cooper–Schrieffer fermion pairing model to form a vortex dimer state that offers an explanation of the pseudogap state of unconventional superconductors, and predicts a fractionally filled vortex dimer state. The integrally and fractionally quantized Hall conductance in the conventional quantum Hall system has an exact correspondence with the linking number in this multi-knot lattice model. The real-space knot pattern in the topological insulator model has an equivalent correspondence with the Lissajous knot in momentum space. The quantum phase transition between different quantum states of the quantum spin model is also directly quantified by the change of topological linking number, which revealed the topological character of phase transition. Circularized photons in an optical fiber network are a promising physical implementation of this multi-knot lattice, and provide a different path to topological quantum computation.
基金supported by the Fundamental Research Funds for the Central Universities Grant No.2020JKF306 and NSFC Grant No.11675119。
文摘Gentile statistics describes fractional statistical systems in the occupation number representation.Anyon statistics researches those systems in the winding number representation.Both of them are intermediate statistics between Bose–Einstein and Fermi–Dirac statistics.The second quantization of Gentile statistics shows a lot of advantages.According to the symmetry requirement of the wave function and the property of braiding,we give the general construction of transformation between anyon and Gentile statistics.In other words,we introduce the second quantization form of anyons in an easier way.This construction is a correspondence between two fractional statistics and gives a new description of anyon.Basic relations of second quantization operators,the coherent state and Berry phase are also discussed.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11804107)。
文摘Non-Abelian anyons can emerge as fractionalized excitations in two-dimensional systems with topological order. One important example is the Moore–Read fractional quantum Hall state. Its quasihole states are zero-energy eigenstates of a parent Hamiltonian, but its quasiparticle states are not. Both of them can be modeled on an equal footing using the bipartite composite fermion method. We study the entanglement spectrum of the cases with two or four non-Abelian anyons. The counting of levels in the entanglement spectrum can be understood using the edge theory of the Moore–Read state, which reflects the topological order of the system. It is shown that the fusion results of two non-Abelian anyons is determined by their distributions in the bipartite construction.
基金Supported by the National Natural Science Foundation of China under Grant No. 10874098the National Basic Research Program of China under Grant Nos. 2009CB929402, 2011CB9216002the Specialized Research Fund for the Doctoral Program of Education Ministry of China under Grant No. 20060003048
文摘Anyons can be used to realize quantum computation, because they are two-level systems in two dimensions. In this paper, we propose a scheme to simulate single-qubit gates and CNOT gate using Abelian anyons in the Kitaev model. Two pairs of anyons （six spins） are used to realize single-qubit gates, while ten spins are needed for the CNOT gate. Based on these quantum gates, we show how to realize the Grover algorithm in a two-qubit system.
基金the National Natural Science Foundation of China(Grant No.11465006),20200981-SIP-IPN,and the CONACyT(Grant No.288856-CB-2016).
文摘We show that it is possible to simulate an anyon by a trapped atom which possesses an induced electric dipole moment in the background of electric and magnetic fields in a specific configuration.The electric and magnetic fields we applied contain a magnetic and two electric fields.We find that when the atom is cooled down to the limit of the negligibly small kinetic energy,the atom behaves like an anyon because its angular momentum takes fractional values.The fractional part of the angular momentum is determined by both the magnetic and one of the electric fields.Roles electric and magnetic fields played are analyzed.
基金National Fundamental Research Program of China under Grant No.2001CB309310National Natural Science Foundation of China under Grant No.60573008
文摘In order to analyze a limiting case of the 1D delta anyon model,the coupling strength of the δ interactionc is modified to become a function of the anyonic parameter κ.A pedagogic derivation of the solution for this modifiedmodel using the method of anyon-boson mapping plus Bethe ansatz is presented.The limiting case as κ→π andsimultaneously c→0,which was previously neglected,is analyzed.Some unexpected properties of this limiting case arediscovered.The BAEs are compared with previous results.
基金Supported by National Natural Science Foundation of Chinathe National Program for Basic Research of MOST of Chinathe Key Lab of Frontiers in Theoretical Physics of CAS and a Fund From CAS
文摘We study an anyon model in a toric honeycomb lattice. The ground states and the low-lying excitations coincide with those of Kitaev toric code model and then the excitations obey mutual semionic statistics. This model is helpful to understand the toric code of anyons in a more symmetric way. On the other hand, there is a direct relation between this toric honeycomb model and a boundary coupled Ising chain array in a square lattice via Jordan-Wigner transformation. We discuss the equivalence between these two models in the low-lying sector and realize these anyon excitations in a conventional fermion system. The analysis for the ground state degeneracy in the last section can also be thought of as a complementarity of our previous work [Phys. A： Math. Theor. 43 （2010） 105306].
基金supported by National Natural Science Foundation of China and the Doctoral Education Fund of the Ministry of Education of China
文摘In this paper,starting with the well known U(1)Chern-Simons Lagrangian and the covariant derivative ofa complex scalar matter field,we give a detailed discussion of some topological properties of anyons.We show that the'basic'charge carried by anyons has an inner structure and can be decomposed in terms of the Chern-Simons couplingand the gauge coupling constants of the theory.Also some incorrect results obtained in the literature are revised.
文摘The note gives a watertight confirmation of the E-infinity Cantorian theory results for ordinary and dark cosmic energy density of the universe and respectively. The computation is fundamentally based on a golden mean fusion function that goes back to the highly original anyon proposal of F. Wilczek.
文摘Jean Anyon(1941-2013)is a critical pedagogical researcher and social activist in the United States.All her life,she devoted herself to taking education as a breakthrough,exposing the ugliness of capitalist countries and promoting the equality of society and education.She uses Marxist theory to analyze the relationship between social class and school knowledge,hidden curriculum;education and mainstream ideology;and conflicts and resistances in education,emphasizing the theoretical and practical significance of Marxism in today’s education.In order to find the deepest problem of education,Jean Anyon demonstrates the relationship between school education and macroeconomic policy from the perspective of political economics;the dialectical relationship between education reform and economic reform;and explains the reasons for education failure by quoting David Harvey’s concept of“deprivation accumulation”.Facing the cruel situation,Jean Anyon did not lose confidence,but found the“radical possibility”and put forward the concept of social change centered on education reform.
基金supported by the National Key R&D Program of China under Grant No.2022YFA1404900the National Natural Science Foundation of China under Grant No.12422411.
文摘Topological flat bands have attracted significant interest across various branches of physics,where synthetic gauge fields are typically considered an essential prerequisite.Numerous mechanisms have been proposed for implementing these fields,including magnetic fields on electrons,differential optical paths for photons,and strain-induced effective magnetic fields,among others.In this work,we introduce a novel approach to generating synthetic gauge fields through quantum statistics and demonstrate their effectiveness in realizing anyonic topological flat bands.Notably,we discover that a pair of strongly interacting anyons can induce square-root topological flat bands within a lattice model that remains dispersive and topologically trivial for a single particle.To validate our theoretical predictions,we experimentally simulate the quantum statistics-induced topological flat bands and square-root topological boundary states by mapping the eigenstates of two anyons onto modes in electric circuits.Our findings not only open a new pathway for creating topological flat bands but also deepen our understanding of anyonic physics and the underlying principles of flat-band topology.
基金financially supported by the Innovation Program for Quantum Science and Technology(Grant No.2021ZD0302400)the National Natural Science Foundation of China(Grant No.11974271)+2 种基金the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDB28000000)the National Basic Research Program of China(Grant No.2015CB921102)the China Postdoctoral Science Foundation(Grant No.2021M690233)。
文摘Majorana zero modes(MZMs)have been intensively studied in recent decades theoretically and experimentally as the most promising candidate for non-Abelian anyons supporting topological quantum computation(TQC).In addition to the Majorana scheme,some non-Majorana quasiparticles obeying non-Abelian statistics,including topological Dirac fermionic modes,have also been proposed and therefore become new candidates for TQC.In this review,we overview the non-Abelian braiding properties as well as the corresponding braiding schemes for both the MZMs and the topological Dirac fermionic modes,emphasizing the recent progress on topological Dirac fermionic modes.A topological Dirac fermionic mode can be regarded as a pair of MZMs related by unitary symmetry,which can be realized in a number of platforms,including the one-dimensional topological insulator,higher-order topological insulator,and spin superconductor.This topological Dirac fermionic mode possesses several advantages compared with its Majorana cousin,such as superconductivity-free and larger gaps.Therefore,it provides a new avenue for investigating non-Abelian physics and possible TQC.
基金Project supported in part by the National Natural Science Foundation of China.
文摘<正> By using the gauge-invariance,a Klein-Gordon’s type equation for anyons in external fields is constructed.The equation can be solved exactly in a constant electric and magnetic field,as well as in a quantized electromagnetic field.The analytic forms of solutions are also given in detail.
基金supported by the National Natural Science Foundation of China (Grant No. 11075099)
文摘We study quantum classical correspondence in terms of the coherent wave functions of a charged particle in two-dimensional central-scalar potentials as well as the gauge field of a magnetic flux in the sense that the probability clouds of wave functions are well localized on classical orbits. For both closed and open classical orbits, the non-integer angular-momentum quantization with the level space of angular momentum being greater or less than h is determined uniquely by the same rotational symmetry of classical orbits and probability clouds of coherent wave functions, which is not necessarily 27r-periodic. The gauge potential of a magnetic flux impenetrable to the particle cannot change the quantization rule but is able to shift the spectrum of canonical angular momentum by a flux-dependent value, which results in a common topological phase for all wave functions in the given model. The well-known quantum mechanical anyon model becomes a special case of the arbitrary quantization, where the classical orbits are 2π-periodic.
