Quantum-liquid regimes for spin chains coupled to phonons: Phonon density wave versus magnetic order
We consider a chain of localized spins, coupled to phonons. Recently this problem has been solved exactly for a ``basic model,'' a family of spin-phonon Hamiltonians HBM characterized by one parameter (coupling constant K), and a zero-T first-order phase transition from the magnetic (ferro or antiferro) state at low couplings to the nonmagnetic state with a phonon density wave at high couplings was found. Here we probe the general case, constructing an effective Hamiltonian H for low-energy degrees of freedom by means of regular expansion in deviations δH=H-HBM of the general Hamiltonian H from that of the basic model. In linear approximation in δH the problem appears to be exactly solvable as well, due to an infinite number of conservation laws. If K is far enough from the critical value Kc, then the character of the basic model solution is not altered. In the vicinity of Kc the magnetic state is dramatically reconstructed: Here the ground state is a gapless magnetic quantum liquid, consisting of mobile singlet spin-phonon complexes and unbound spins. The fraction of singlets increases gradually upon approaching Kc, and the magnetic order parameter gradually vanishes. Thus we have here a partial screening of spins by phonons without formation of a phonon density wave. The latter appears only at K=Kc in the first-order phase transition. Corrections, quadratic in δH, destroy the integrability of the system, but outside a narrow critical region around Kc they only lead to an opening of a small gap in the spectrum of the quantum liquid. The behavior of the system within the critical region is an open question. Most likely the continuous magnetic phase transition at K=Kc becomes a first-order one, but close to second order. The relevance of our results for three-dimensional systems and possible applications to compounds with anomalously weak magnetism are briefly discussed.