By Yoshio Kuramoto, Yusuke Kato
One-dimensional quantum structures convey attention-grabbing homes past the scope of the mean-field approximation. notwithstanding, the complex arithmetic concerned is a excessive barrier to non-specialists. Written for graduate scholars and researchers new to the sector, this ebook is a self-contained account of the way to derive the unique quasi-particle photo from the precise answer of versions with inverse-square interparticle interactions. The ebook offers readers with an intuitive figuring out of tangible dynamical homes by way of unique quasi-particles that are neither bosons nor fermions. robust strategies, comparable to the Yangian symmetry within the Sutherland version and its lattice types, are defined. A self-contained account of non-symmetric and symmetric Jack polynomials is usually given. Derivations of dynamics are made more uncomplicated, and are extra concise than within the unique papers, so readers can study the physics of one-dimensional quantum platforms in the course of the least difficult version.
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Additional info for Dynamics of One-Dimensional Quantum Systems
101) which represents a left-moving quasi-particle. Quasi-particles obey the generalized statistics with the exclusion parameter λ. This can be understood by considering an (N + 2)-particle state with rapidities κ ˜1 > κ ˜2 > κ ˜ 1,0 + λ, κ ˜i = κ ˜ i−2,0 , i ∈ [3, N + 2]. 102) In this state, the rapidities κ ˜ 1 and κ ˜ 2 of quasi-particle states are subject to the exclusion rule κ ˜1 ≥ κ ˜ 2 + λ. 103) Let us now consider the one-particle addition spectrum of bosons for odd ˜ i for (N + 1)λ. Under the periodic boundary condition with N odd, κ particle states is a half odd integer but κ ˜ i,0 for the N -particle ground state is an integer.
1). 13) which shows that the Jastrow wave function is the eigenfunction with the eigenenergy E0,N . 24 Single-component Sutherland model Let us remark on the statistics. When we take an antisymmetric wave function Ψ(x1 , . . , xi , . . , xj , . . , xN ) = −Ψ(x1 , . . , xj , . . , xi , . . 3), we also obtain the eigenfunctions of the Jastrow form. 15) i Generally, all the eigenfunctions of H can be written in the form B ΦB κ = φκ + aµ φB µ. 41) µ(<κ) The eigenenergy E[κ] of ΦB κ is given by the diagonal element of H in the basis φB as κ N E[κ] = (κi − κj ) κ2i + λ i=1 1≤i
Generally, all the eigenfunctions of H can be written in the form B ΦB κ = φκ + aµ φB µ. 41) µ(<κ) The eigenenergy E[κ] of ΦB κ is given by the diagonal element of H in the basis φB as κ N E[κ] = (κi − κj ) κ2i + λ i=1 1≤i