Spectrum and Transition Rates of the XX Chain Analyzed Via Bethe Ansatz

Spectrum and transition rates of the XX chain analyzed via Bethe ansatz Daniel Biegel,1 Michael Karbach,1, 2 Gerhard Müller,2 and Klaus Wiele3 1 Bergische Universität Wuppertal, Fachbereich Naturwissenschaften, Physik, D-42097 Wuppertal, Germany 2 Department of Physics, University of Rhode Island, Kingston RI 02881-0817, USA 3 Bergische Universität Wuppertal, Fachbereich Physik, D-42097 Wuppertal, Germany (Dated: October 5, 2018) 1 As part of a study that investigates the dynamics of the s = 2 XXZ model in the planar regime |∆| < 1, we discuss the singular nature of the Bethe ansatz equationsforthecase∆ = 0(XX model). We identify the general structure of the Bethe ansatz solutions for the entire XX spectrum, which include states with real and complex magnon momenta. We discuss the relation between the spinon or magnon quasiparticles (Bethe ansatz) and the lattice fermions (Jordan-Wigner representation). We present determinantal expressions for transition rates of spin fluctuation operators between Bethe wave functions and reduce them to product expressions. We apply the new formulas to two-spinon transition rates for chains with up to N = 4096 sites. I. INTRODUCTION relationship between the magnon, spinon, and lattice fermion quasiparticles is important for the interpretation The key to a meaningful interpretation of experimen- of the excitation spectrum via dynamical probes as re- tal or computational data for the low-temperature dy- alized experimentally or computationally. Furthermore, namics of quantum many-body systems is a thorough recent advances in calculating transition rates via Bethe 6,7,8,9,10 understanding of the nature of the physical vacuum and ansatz offer opportunities to extend the list of the dynamically relevant collective excitations including exact results for dynamical properties of the XX chain. their quasiparticle composition. In completely integrable In Sec. II we analyze the singularities of the XXZ Bethe ansatz equations for ∆ 0. In Sec. III we discuss systems, the quasiparticle configurations that produce → particular collective excitations can be investigated close the mapping between the spinon and fermion composi- up. The identity of the former is preserved by conser- tions of the XX spectrum. The two-spinon spectrum of vation laws notwithstanding their mutual interaction.1,2 the planar XXZ model including the XX limit is dis- Integrable Hamiltonians that depend on continuous pa- cussed in Sec. IV. In Sec. V transition rate formulas for rameters make it possible to observe how the physical the XXZ model are introduced and further processed for vacuum transforms gradually and, occasionally, changes the XX limit. The two-spinon part of the dynamic spin abruptly across a quantum phase transition. Along the structure factor S−+(q,ω) is discussed in Sec. VI. way, the configurations of quasiparticles in the collective excitations are subject to change as well. Recently, we investigated the metamorphosis of the II. BETHE ANSATZ EQUATIONS physical vacuum and the dynamically relevant quasipar- ∗ 1 ticles (magnons, spinons, ψ, ψ ) of the 1D s = 2 Heisen- We consider Hamiltonian (1.1) for even N and periodic berg antiferromagnet (XXX model) by varying the (in- boundary conditions over the range 0 ∆ 1 of the ≤ ≤ tegrability preserving) external magnetic field.3,4 Intro- anisotropy parameter. In the invariant subspace with z- ducing instead a uniaxial exchange anisotropy also pre- component of the total spin Sz = N/2 r, all eigenstates T − serves integrability. The parameter ∆, which controls the are represented by r interacting magnons. The magnon 1 5 anisotropy in the 1D s = 2 XXZ model, momenta ki are conserved in the scattering processes and are determined by the Bethe ansatz equations:11 N . y N x x y z z r H = SnSn+1 + SnSn+1 + ∆SnSn+1 ∆ , (1.1) i(ki+kj ) iki − 4 iNki 1+ e 2∆e n=1 e = − , i =1, . , r. X −1+ ei(ki+kj ) 2∆eikj 6 affects the physical vacuum and the quasiparticle config- Yj=i − urations differently. In both situations, the Bethe ansatz (2.1) arXiv:cond-mat/0309247v1 [cond-mat.str-el] 10 Sep 2003 is an ideal framework for studying quasiparticles, their The energy and wave number of any such r-magnon state transformations and their interactions. depend on the magnon momenta alone: The main focus in this paper is on a technical point of r r considerable importance in the study of the XXZ model, E = (cos k ∆) , k = k . (2.2) i − i namely the identification of the general structure of the i=1 i=1 Bethe ansatz solutions of all eigenstates for the case ∆ = X X 0 (XX model). This will facilitate tracking all XXZ At first glance it looks as if Eqs. (2.1) are drastically Bethe ansatz solutions across the planar regime, ∆ < 1. simplified in the limit ∆ 0 with all magnon momenta At ∆ = 0 all states can be characterized| as| non- restricted to solutions of→eiNki = ( 1)r−1. However, a − interacting composites of fermions. Understanding the closer look reveals that magnon pairs with momenta ki + 2 kj π for ∆ 0 are a common occurrence. This equations at ∆ = 0 afford a universal treatment of the opens→ the door to→ nontrivial real and complex solutions singularities for arbitrary N. of Eqs. (2.1). The singular behavior is related to level In the absence of any critical pair of rapidities, degeneracies. Such level crossings, which also occur at Eqs. (2.6) decouple and yield the solutions other values of ∆, have been traced back to a realization −1 of the sl2 loopalgebra symmetry at ∆ = (q+q )/2 = 1 πIi 2π 2N 12,13,14 6 ± y = tan k = π I (2.9) with q = 1 for N 2. i N ⇔ i − N i In the following, we≥ use the anisotropy parameter . for i = 1,...,r, which are all real. The energy of any γ =arccos∆ (0 γ π/2) (2.3) such regular state is ≤ ≤ 15 and transform the magnon momenta into the rapidities r y−1 y r E = i − i = cos k . (2.10) − −1 i i=1 yi + yi i=1 . γ ki X X y = tan cot , i =1, . , r. (2.4) i 2 2 B. Real critical pairs The Bethe ansatz equations (2.1) thus become N r Now let us assume that among the set of rapidities c2yi + i c1(yi yj ) + i(1 yiyj ) = − − (2.5) y1,...,yr, there is one critical pair, c2yi i c1(yi yj ) i(1 yiyj ) − j=6 i − − − Y 0 0 . yj yj∗ =1 kj + kj∗ = π (mod 2π), (2.11) for i =1, . , r, with c1 = cot γ and c2 = cot(γ/2). Tak- ⇔ ing the logarithm yields and that it is a real pair. Substituting the ansatz r 0 yi yj y = y + c δ , i =1, . , r, (2.12) Nφ(c y )=2πI + φ c − i =1, . , r, i i 1 i 2 i i 1 1 y y 6 i j Xj=i − into Eqs. (2.6) and taking the limit c 0 then yields the (2.6) 1 . non-critical and critical rapidities from→ successive orders with φ(x) = 2 arctan(x). Expressions (2.2) for energy in a c -expansion. The non-critical rapidities y , i = j, j∗ and wave number now read 1 i 6 are given by Eqs. (2.9) as in regular solutions. The c1- r −1 expansion of Eqs. (2.6) leads to the following equation 2 yi yi 0 0 E = −1 −1− , (2.7) determining the critical rapidities yj ,yj∗: − (c2y ) + c2y c2 + c2 i=1 i i X r − σj 2π − 2 σj yj k = πr Ii. (2.8) φ(y )= φ . (2.13) − N j i=1 N " 1 ξj # ! X − The trigonometric Bethe ansatz equations (2.6) have the ± 1 0 0 Here we have introduced y = (y y ∗ ) and we use advantage that each solution is characterized by a set j 2 j ± j of (integer or half-integer) Bethe quantum numbers Ii. r + . 2 yj sin ki These discriminating markers are used to count and clas- ξ = − , (2.14) j + −1 sify the solutions and are essential for numerical algo- N ∗ (yj ) sin ki i=6 j,j − rithms designed to find solutions. X y+ 2 j . Ij −Ij∗ sgn = (Ij + Ij∗ ), σj = ( 1) . (2.15) ξj ! −N − A. XX-limit The non-critical ki are from (2.9). Criticality implies − Here we describe the general structure of the solutions (y+)2 (y )2 = 1. Note that the contributions of the j − j of the Bethe ansatz equations (2.6) in the limit ∆ 0, critical rapidities to the energy (2.10) cancel out. Equa- → implying c1 0 and c2 1. There exist regular so- tion (2.15) implies that the Bethe quantum numbers of lutions and singular→ solutions.→ The latter are character- the critical pair must satisfy ′ ized by the occurrence of pairs of rapidities yi,yi with ′ the property yiyi = 1. For any such critical pair, the Ij + Ij∗ = N/2. (2.16) argument of φ on the right-hand side of Eqs. (2.6) is in- | | determinate and must, therefore, be treated as a limit The generalization of this solution to the case of s real process. We shall see that some limiting singular solu- critical pairs, tions are real while others are complex. An important 0 0 fact is that the simplified structure of the Bethe ansatz y y ∗ =1, l =1, . s, (2.17) jl jl 3 with Bethe quantum numbers satisfying I +I ∗ = N/2 D. Generic case jl jl is straightforward. The non-critical rapidities| are| as in Eq. (2.9) and the critical pairs are determined from Suppose we have t critical pairs of complex conjugate solutions, y = u + iv = y∗,...,y = u + iv = y∗ , − σjl 1 1 1 2 2t−1 t t 2t ∗ 2 σjl y − jl and s pairs of critical real solutions, l = jl, jl 2t+ φ(yjl )= φ , l =1,...,s (2.18) J { }⊂{ N "1 ξjl # ! 1,...,r , l =1,...,s.

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