Rig __ . ~ gi ELSEVIER 6 October 1997 Physics Letters A 234 (1997) 345-3.X) zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF PHYSICS LETTERS A On the spectrum of the non-Hermitian phase-difference model Nikolai M. Bogoliubov ‘, Tarek Nassar ’ zyxwvutsrqponmlkjihgfedcbaZYXW Hcl~~~ki I~stifute of Physics, ~niLiersi~ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED of ~ef~~nk~, P.O. Box 9. ~I~-#~~i4 H~l~~inki~ ~-f~lffff~f Received 6 March 1997; accepted for publication 1 July 1997 Communicated by A.P. Fordy zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPO Abstract A modified version of the phase-difference model is introduced and diagonalized by means of the algebraic Bethr ansatz. The spectrum is determined for both small and large values of the coupling constant, and the low-lying excitations are shown to exhibit a conformal profile. Applications of the model to quantum optics and growth problems are briefly discussed. 0 1997 Published by Elsevier Science B.V. PAC‘S: 05.45: 42.50 Ke,wonfs: Algebraic Bethe ansatz; Phase operators zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 1. The modified phase-difference model In the present work we introduce and solve for its eigenstates and eigenvalues the model defined by the Hamiltonian on a chain (1) *I= 1 Here c$: and 4, are the so-called exponential phase oprrutors intensively studied in quantum optics and ’ On leave of absence from St. Petersburg Department of the Steklov Mathematical Institute, Fontanka 27, St. Petersburg I9 IO I I. Russian Federation. E-mail: bogoliub@pdmi.ras.ru. ’ Laboratoire de Physique Mathtmatique, Universite de Mont- pellier If, CNRS, URA 768, 34095 Mont~llier Cedex 05, France. E-mail: n~~ssa~lpm.uni~-montp2.fr. solid state physics [l-3]. They can be defined by the following commutation relations [#“? 4:] = 7rs n flm P,* &?A= -~“~,,~ [K. &,I = eJ,,~ (2) where n;, is the vacuum projector rr, = ( IO>,{0 I ,,I and N, is the number of particles operator on the nth lattice site. The constant g is the interaction constant and the parameter y specifies the boundary condi- tions (pa+, -+ ~~41, 7rM+, + 7T,. The operators $, and +f, can be expressed in terms of the Fock number states 1 n),,, as rb,= E ln>,(n+1l,~, ?I=0 The Hamiltonian (1) is the sum of the kinetic term with equal amplitudes, describing the nearest-neigh- bor hopping from left to right. The site term is 0375.9601/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved P/I SO375~9601(97)00561-6