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Tieftemperatureigenschaften einer dotierten Spin S=1 Kette : integrables Modell

Constantin Sobiella

Published Jan 1, 2002• 0 citations
Publisher

Abstract

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.

DOI
10.15488/6106

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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.
The low-temperature magnetization process of antiferromagnetic spin- SSS chains doped with mobile spin- $(S\ensuremath{-}1/2)$ carriers is studied in an exactly solvable model. For sufficiently high magnetic fields the system is in a metallic phase with a finite gap for magnetic excitations. In this phase, which exists for a large range of carrier concentrations xxx, the zero-temperature magnetization is determined by xxx alone. This leads to plateaus in the magnetization curve at a tunable fraction of the saturation magnetization. The critical behavior at the edges of these plateaus is studied in detail.
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.
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.
We consider the one-dimensional t-J model, which consists of spin-1/2 electrons on a lattice with nearest neighbor hopping t constrained by the excluded multiple occupancy of the lattice sites and spin-exchange J between neighboring sites. The model is integrable at the supersymmetric point, J=2t. We extend the model by introducing an impurity of arbitrary spin S that interacts with the electrons on the neighboring sites without destroying the integrability. The lattice model is defined by the scattering matrices via the quantum inverse scattering method. The interaction Hamiltonian between the impurity and the itinerant electrons is only explicitly constructed in the continuum limit. The discrete Bethe ansatz equations diagonalizing the model are derived and the solutions are classified according to the string hypothesis. The thermodynamic Bethe ansatz equations are derived and the impurity free energy is obtained for arbitrary bandfilling as a function of temperature and external magnetic field. The properties of the impurity depend on one coupling parameter. The impurity can localize up to one itinerant electron and has in general mixed valent properties. The integer valent low T small H fixed point of the impurity corresponds to an asymptotically free spin S, while if either T or H (or both) become large the impurity behaves like an asymptotically free spin (S+).
In this work we consider several exactly solvable models of magnetic impurities in critical quantum antiferromagnetic spin chains and multichannel Kondo impurities. Their ground-state properties are studied, and the finite set of nonlinear integral equations which exactly describe the thermodynamics of the models is constructed. We obtain several analytical low-energy expressions for the temperature, magnetic field, and frequency dependences of important characteristics of exactly solvable disordered quantum spin models and disordered multichannel Kondo impurities with essential many-body interactions. We show that the only low-energy parameter that gets renormalized is the velocity of the low-lying excitations (or the effective crossover scale connected with each impurity); the others appear to be universal. In our study several kinds of strong disorder important for experiments were used. Some of them produce low divergences in certain characteristics of our strongly disordered critical systems (compared with finite values for the homogeneous case or a single impurity). For weak disorder, or for narrow distributions of the local Kondo temperatures, our exact results reveal the presence of Kondo screening of disordered ensembles of magnetic impurities by low-lying excitations of the host. We point out that our results qualitatively coincide with the data of experiments on real disordered quasi-one-dimensional antiferromagnetic systems and with the similar behavior of some heavy metallic alloys.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.