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arXiv:cond-mat/9306003 · Journal of Physics A Mathematical and General

Integrable versus non-integrable spin chain impurity models

Erik Schwartz Sørensen, S Eggert, Ian K. Affleck

Published Dec 7, 1993• 64 citations
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Abstract

Recent renormalization group studies of impurities in spin-1/2 chains appear to be inconsistent with Bethe ansatz results for a special integrable model. We study this system in more detail around the integrable point in parameter space and argue that this integrable impurity model corresponds to a non-generic multi-critical point. Using previous results on impurities in half-integer spin chains, a consistent renormalization group flow and phase diagram is proposed.

DOI
10.1088/0305-4470/26/23/023

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Recent renormalization group studies of impurities in spin-1/2 chains appear to be inconsistent with Bethe ansatz results for a special integrable model. We study this system in more detail around the integrable point in parameter space and argue that this integrable impurity model corresponds to a non-generic multi-critical point. Using previous results on impurities in half-integer spin chains, a consistent renormalization group flow and phase diagram is proposed.
We consider a magnetic S = 1/2 impurity in the antiferromagnetic spin chain as a function of two coupling parameters: the symmetric coupling of the impurity to two sites in the chain J1 and the coupling between the two sites J2. By using field theory arguments and numerical calculations we can identify all possible fixed points and classify the renormalization flow between them, which leads to a nontrivial phase diagram. Depending on the detailed choice of the two (frustrating) coupling strengths, the stable phases correspond either to a decoupled spin with Curie law behavior or to a non-Fermi-liquid fixed point with a logarithmically diverging impurity susceptibility as in the two-channel Kondo effect. Our results resolve a controversy about the renormalization flow.
The authors consider an integrable SU(2)-invariant model consisting of the Heisenberg chain of arbitrary spin S (Takhtajan-Babujian model) interacting with an impurity of spin S'. The impurity is assumed to be located on the mth link of the chain and interacts only with both neighbouring sites. The starting point is a set of commuting transfer matrices, whose local weights satisfy triangular Yang-Baxter relations. The diagonalization of the transfer matrices leads to the Bethe ansatz equations for the model. The thermodynamics of the system is studied. Three situations have to be distinguished: (i) If S'=S the impurity just corresponds to one more site in the chain. (ii) If S'>S the impurity spin is only partially compensated at T=0, leaving an effective spin of (S'-S). (iii) S'
The spectral properties and phase diagram of the exact integrable spin one quantum chain introduced by Alcaraz and Bariev are presented. The model has a U(1) symmetry and its integrability is associated to an unknown R-matrix whose dependence on the spectral parameters is not of difference form. The associated Bethe ansatz equations, that fix the eigenspectra, are distinct from those associated to other known integrable spin models. The model has a free parameter tpt_ptp​. We show that at the special point tp=1t_p=1tp​=1 the model gets an extra U(1) symmetry and reduces to the deformed SU(3) Perk-Schultz model at a special value of its anisotropy q=exp⁡(i2π/3)q=\exp(i2π/3)q=exp(i2π/3) andin the presence of an external magnetic field. Our analysis is done either by solving the associated Bethe-ansatz equations or by direct diagonalization of thequantum Hamiltonian for small lattice sizes. The phase diagram is calculated by exploring the consequences of conformal invariance on the finite-size corrections of the Hamiltonian eigenspectrum. The model exhibits a critical phase ruled by c=1c=1c=1 conformal field theory separated from a massive phase by first-order phase transitions.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
Abstract We argue that the Hamiltonians for A 2 n ( 2 ) open quantum spin chains corresponding to two choices of integrable boundary conditions have the symmetries U q ( B n ) and U q ( C n ) , respectively. We find a formula for the Dynkin labels of the Bethe states (which determine the degeneracies of the corresponding eigenvalues) in terms of the numbers of Bethe roots of each type. With the help of this formula, we verify numerically (for a generic value of the anisotropy parameter) that the degeneracies and multiplicities of the spectra implied by the quantum group symmetries are completely described by the Bethe ansatz.