Luca Salasnich
We study two anyons with Coulomb interaction in a uniform magnetic field $B$. By using the torus quantization we obtain the modified Landau and Zeeman formulas for the two anyons. Then we derive a simple algebraic equation for the full spectral problem up to the second order in $B$.
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One of the interesting fundamental phenomenon which was observed in the last decades is the discovery of anyons, relativistic spinning particles in $2+1$ dimensions. In contrast to three-dimensional space, indistinguishable quantum particles in two-dimensional space can, in general, have anomalous statistics [1-4]. These quasiparticles carry not only a charge $q$ also the magnetic flux $Φ_0$.
The monopole systems with hidden symmetry of the two-dimensional Coulomb problem are considered. One of them, the "charge-charged magnetic vortex" with a half-spin, is constructed by reducing the quantum circular oscillator with respect to the action of the parity operator. The other two systems are constructed by reduction from the two-dimensional complex space. The first system is a particle on the sphere in the presence of the exterior constant magnetic field (generated by Dirac's monopole located in its center). This system is dual to the massless (3+1)-dimensional particle with fixed ener…
Using finite size scaling arguments, the critical properties of a chain of interacting anyons can be extracted from the low energy spectrum of a finite system. In Phys. Rev. Lett. 98, 160409 (2007), Feiguin et al. showed that an antiferromagnetic (AFM) chain of Fibonacci anyons on a torus is in the same universality class as the tricritical Ising model, and that criticality is protected by a topological symmetry. In the present paper we first review the graphical formalism for the study of anyons on the disc and demonstrate how this formalism may be consistently extended to the study of system…
In the tensor network representation, a deformed $Z_{2}$ topological ground state wave function is proposed and its norm can be exactly mapped to the two-dimensional solvable Ashkin-Teller (AT) model. Then the topological (toric code) phase with anyonic excitations corresponds to the partial order phase of the AT model, and possible topological phase transitions are precisely determined. With the electric-magnetic self-duality, a novel gapless Coulomb state with quasi-long-range order is obtained via a quantum Kosterlitz-Thouless phase transition. The corresponding ground state is a condensate…
SU($N$) Toric Code and Nonabelian Anyons
本研究构建了二维环面上的SU($N$) Toric Code模型，推导其$N^2$个拓扑基态并以自旋网络态相干叠加显式构造，同时构造激发准粒子态，揭示非阿贝尔任意子的编织统计特性由Wigner旋转矩阵编码。
Braiding Majoranas in a linear quantum dot-superconductor array: Mitigating the errors from Coulomb repulsion and residual tunneling
该研究提出在量子点-超导体阵列中通过优化辅助量子点控制来缓解库仑排斥和残余单电子隧道对Majorana编织误差的影响，并预测实验特征。
The Aharonov-Bohm-Coulomb potentials in two dimensions may describe the interaction between two particles carrying electric charge and magnetic flux, say, Chern--Simons solitons, or so called anyons. The scattering problem for such two-body systems is extended to the relativistic case, and the scattering amplitude is obtained as a partial wave series. The electric charge and magnetic flux is ($-q$, $-ϕ/Z$) for one particle and ($Zq$, $ϕ$) for the other. When $(Zq^2/\hbar c)^2\ll 1$, and $qϕ/2π\hbar c$ takes on integer or half integer values, the partial wave series is summed up approximately t…
We discuss the problem of anyonic statistics in one and two spatial dimensions from the point of view of statistical physics. In particular we want to understand how the choice of the Bornvon Karman or the twisted periodic boundary conditions necessary in a Monte Carlo simulation to mimic the thermodynamic limit of the many body system influences the statistical nature of the particles. The particles can either be just bosons, when the configuration space is simply connected as for example for particles on a line. They can be bosons and fermions, when the configuration space is doubly connecte…
We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking gene…
Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac's Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied. In the second part of the paper we have considered the quantum theory of a charge moving in a torus kno…
Topological BF Theory construction of twisted dihedral quantum double phases from spontaneous symmetry breaking
本文提出基于连续O(2)规范场的非阿贝尔二面体量子双相构造方法，通过拓扑BF理论与任意子的对应关系，结合微观模型与重整化群分析，展示了从二面体相到U(1)库仑或手性相的直接过渡。
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