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Journal of Physics Condensed Matter

The Heisenberg chain with impurities: crossover from Fermi-liquid to non-Fermi-liquid behaviour

P. Schlottmann

Published Jan 1, 1999• 1 citation
Publisher

Abstract

We consider an integrable SU(2)-invariant model consisting of a Heisenberg chain of spins S (the Takhtajan-Babujian model) interacting with a finite concentration c of impurity spins S ´. The thermodynamic Bethe- ansatz equations are stated for this model. The ground-state equations are analysed as a function of c , the magnetic field and the coupling parameter (impurity rapidity p 0 ) of the impurities to the lattice. In zero field the ground state is a singlet for finite c , but becomes non-Fermi-liquid-like as c 0 for S ´< S . Two rapidity bands play a role at T = 0 corresponding to strings of length 2 S and 2 S ´, respectively. The van Hove singularities of the empty bands define two critical fields, H c ( c , p 0 ) and H s ( c , p 0 ), at which the susceptibility diverges. H c tends to zero as c 0 giving rise to a crossover from non-Fermi-liquid behaviour for H > H c to Fermi-liquid-like behaviour for H < H c . The spectrum of elementary excitations and the finite-size corrections to the ground-state energy are calculated, and used to discuss the asymptotic behaviour of spin-spin correlation functions for long times and large distances.

DOI
10.1088/0953-8984/11/24/304

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We consider an integrable SU(2)-invariant model consisting of a Heisenberg chain of spins S (the Takhtajan-Babujian model) interacting with a finite concentration c of impurity spins S ´. The thermodynamic Bethe- ansatz equations are stated for this model. The ground-state equations are analysed as a function of c , the magnetic field and the coupling parameter (impurity rapidity p 0 ) of the impurities to the lattice. In zero field the ground state is a singlet for finite c , but becomes non-Fermi-liquid-like as c 0 for S ´< S . Two rapidity bands play a role at T = 0 corresponding to strings of length 2 S and 2 S ´, respectively. The van Hove singularities of the empty bands define two critical fields, H c ( c , p 0 ) and H s ( c , p 0 ), at which the susceptibility diverges. H c tends to zero as c 0 giving rise to a crossover from non-Fermi-liquid behaviour for H > H c to Fermi-liquid-like behaviour for H < H c . The spectrum of elementary excitations and the finite-size corrections to the ground-state energy are calculated, and used to discuss the asymptotic behaviour of spin-spin correlation functions for long times and large distances.
We consider the Heisenberg chain of spin 1/2 with a concentration x of impurity spins S. The thermodynamic Bethe ansatz equations are stated for this model. The ground-state equations are analyzed as a function of x and magnetic field. In zero field the ground state is a singlet for finite x, but non-Fermi-liquid-like for x\ensuremath{\rightarrow}0. Two energy bands play a role at T=0. The Van Hove singularities of the empty bands define two critical fields, Hc1{\mathit{H}}_{\mathit{c}1}Hc1​(x) and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​(x), at which the susceptibility diverges. The non-Fermi-liquid behavior as x\ensuremath{\rightarrow}0 arises from nonanalyticities as Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ vanishes.
An impurity of spin S′=1/2 is introduced in the Babujian-Takhtajian Heisenberg model of spin S in one dimension. The impurity is located on the mth link of the chain and interacts with both neighboring sites. The interaction between impurity and Heisenberg chain is of a special type so that the integrability is preserved. The diagonalization of the transfer matrices leads to the Bethe ansatz equations of the model. The thermodynamics of the system is studied. For ferromagnetic coupling in the Heisenberg chain, the impurity is locked into the critical behavior of the lattice, i.e., at low temperatures the specific heat is proportional to T1/2 (ferromagnetic magnons) and the susceptibility diverges as T−2 (with logarithmic corrections). For antiferromagnetic coupling and S =1/2 the impurity is just one more site in the chain. If S ≳1/2 the collective properties of impurity and lattice give rise to critical behavior in complete analogy to the overcompensated n-channel Kondo problem. The zero-temperature entropy is finite in zero-field, but zero if the field is nonzero, giving rise to an essential singularity at T=H=0. As a consequence a two-peak structure arises in the specific heat in a small but finite field.
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 numerical solution of the Bethe ansatz equations of an integrable SU(2)-invariant generalization of the spin-S antiferromagnetic Heisenberg chain (Takhtajian-Babujian model) with a spin-S' impurity in zero magnetic field is presented. The entropy, specific heat and susceptibility of an impurity of spin S' are obtained numerically as a function of the spin S of the chain. 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 (undercompensated) at T=0, leaving an effective spin of (S'-S); (iii) if S'
The thermodynamics of finite open antiferromagnetic XXZ chains is studied using field theory, Bethe ansatz and quantum Monte Carlo methods. For the susceptibility a parameter-free result as a function of the number of sites L and temperature T beyond the scaling limit is derived. The limiting cases ( J being the exchange constant), where the boundary correction shows a logarithmically suppressed Curie behaviour, and , where a crossover to the ground state behaviour takes place, are discussed in detail. On the basis of this analysis we present a simple formula for the averaged susceptibility of a spin-1/2 chain doped with non-magnetic impurities. We show that the effective Curie constant has a highly non-trivial temperature dependence and shows scaling in the low temperature limit. Finally, corrections due to intrachain and interchain couplings and implications for experiments on Sr 2 Cu 1− x Pd x O 3+δ and related compounds are discussed.
In this thesis the low temperature properties of an exactly solvable anti-ferromagnetic spin S = 1 chain doped with mobile spin S = 1 =2 impurities will be considered.The doped spin S = 1 chain is a generalization of the super-symmetric t-J model and interpolates between the isotropic spin S = 1 =2 HEISENBERG model and the spin S = 1 TAKHTAJAN-BABUJIAN chain for the completely doped case and vanishing impurity concentration x h = 0 respectively.Within the framework of the Quantum Inverse Scattering Method the doped spin S = 1 chain can be built from a gl(2j1)-invariant solution of the graded YANG-BAXTER equation corresponding to the atypical [1] + -representation of the super-algebra gl(2j1).The associated transfer matrix is diagonalized by means of a nested algebraic BETHE ansatz.Using a generalized fusion procedure the Hamiltonian and it's eigenvalues are constructed.By applying TAKAHASHI's string hypothesis we deduce in the third part the thermodynamic BETHE ansatz equations.Performing the low temperature limit we can determine ground state properties and the phase diagram depending on an external magnetic field H and the impurity concentration x h .The system shows spin-charge separation.We also examine the magnetic properties of the model.For sufficiently high magnetic fields and finite impurity concentration we find a new metallic phase with a finite gap for the magnetic excitations.In this phase the magnetization curve shows a plateau that is only a fraction of the saturation magnetization and can be tuned by the impurity concentration x h .The critical behavior near the edges of the plateau is also analyzed.To study the dynamical properties e.g. the low lying magnetic excitations we calculate the Finite Size Correction of the model.Using predictions of Conformal Field Theory we can identify the effective low temperature theories as well as the scaling dimensions which determine the asymptotic behavior of the correlation functions at criticality.In the plateau phase no magnetic excitations are found to exist.