By Yoshio Kuramoto, Yusuke Kato

One-dimensional quantum platforms express interesting houses past the scope of the mean-field approximation. although, the advanced arithmetic concerned is a excessive barrier to non-specialists. Written for graduate scholars and researchers new to the sphere, this ebook is a self-contained account of ways to derive the unique quasi-particle photograph from the precise resolution of versions with inverse-square interparticle interactions. The booklet presents readers with an intuitive figuring out of tangible dynamical homes when it comes to unique quasi-particles that are neither bosons nor fermions. strong ideas, reminiscent of the Yangian symmetry within the Sutherland version and its lattice models, are defined. A self-contained account of non-symmetric and symmetric Jack polynomials can also be given. Derivations of dynamics are made more uncomplicated, and are extra concise than within the unique papers, so readers can examine the physics of one-dimensional quantum platforms throughout the easiest version.

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**Extra info for Dynamics of One-Dimensional Quantum Systems**

**Example text**

First we take {κi }N i as an index of one-particle states. We divide the set of one-particle states into many subsets which consist of a macroscopic number of κi (see Fig. 10). We denote the subsets by α and a representative value of one-particle state in α by κα . 127) is satisﬁed. For a given eigenstate speciﬁed by {ν(κ)}∞ κ=∞ , we introduce the average of the distribution function ν¯α = Nα , Gα Nα = ν(κ). 4 Thermodynamics Gα 49 Gα κ κα κα Fig. 10. Division of a set of one-particle states into many subsets α, α , .

132), the thermodynamic potential and other thermodynamic quantities are obtained. 135). Namely, ˜(k) includes interaction eﬀects, and is analogous to the dressed energy or renormalized energy in the thermodynamic Bethe ansatz theory [118]. 66), as shown in the next subsection. 66) by 2π/L and setting pi = 2π˜ κi /L and ki = 2πκi /L, we obtain πλ pi = ki + sgn(pi − pj ). 136) L j(=i) Henceforth, we regard sgn(0) as 0. 138) which deﬁnes a continuous function p(k). 4 Thermodynamics 51 where we have used the property 2πdN/L = ρ(p)dp = ν(k)dk.

When particles are fermions, an exact solution of the Green function is available only for an odd integer λ. 3 Quasi-holes We calculate the spectrum of one-particle removal states from the N -particle ground state. The particle number N is taken to be odd so that the ground state is non-degenerate under the periodic boundary condition. We focus on the model for bosons with even integer λ, or fermions with odd λ for simplicity. When particles are bosons and λ is even, both κi and κ ˜ i are integers in (N − 1)-particle states.