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Physical review. B, Condensed matter

Curie and non-Curie behavior of impurity spins in quantum antiferromagnets

David G. Clarke, Thierry Giamarchi, Boris I. Shraiman

Published Sep 1, 1993• 65 citations
Publisher

Abstract

We investigate a number of cases of an impurity spin S coupled to a D-dimensional antiferromagnetic (AFM) background. The qualitative behavior of the impurity susceptibility depends upon the sublattice symmetry of the coupling, whether S is integer or half-integer, and the spectral density of the low-lying AFM excitations. When the AFM background is ordered we show that the appropriate model to consider is a non-Abelian spin-boson model. Within this model, we argue that the symmetrically coupled half-integer spin impurity retains the Curie divergence of its susceptibility for D\ensuremath{\ge}2. In the case of a D=1 spin-1/2 AFM chain the sublattice symmetric impurity behaves in the same way as the Kondo spin in the two-channel Kondo model and thus has a non-Curie ln(${\mathit{T}}_{\mathrm{*}}$/T) divergent susceptibility. This latter case is investigated via bosonization at the Emery-Kivelson strong-coupling line.

DOI
10.1103/physrevb.48.7070

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We investigate a number of cases of an impurity spin S coupled to a D-dimensional antiferromagnetic (AFM) background. The qualitative behavior of the impurity susceptibility depends upon the sublattice symmetry of the coupling, whether S is integer or half-integer, and the spectral density of the low-lying AFM excitations. When the AFM background is ordered we show that the appropriate model to consider is a non-Abelian spin-boson model. Within this model, we argue that the symmetrically coupled half-integer spin impurity retains the Curie divergence of its susceptibility for D\ensuremath{\ge}2. In the case of a D=1 spin-1/2 AFM chain the sublattice symmetric impurity behaves in the same way as the Kondo spin in the two-channel Kondo model and thus has a non-Curie ln(T∗{\mathit{T}}_{\mathrm{*}}T∗​/T) divergent susceptibility. This latter case is investigated via bosonization at the Emery-Kivelson strong-coupling line.
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.
By making use of the Lanczos and the density-matrix renormalization-group methods, we study the zero-temperature properties of a spin-1/2 antiferromagnetic Heisenberg chain system with a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins. A comparison is made with the results for a magnetic impurity coupled only to the nearest-neighbor spin in the chain. The binding energy and the spin correlations are calculated. The matrix elements of the impurity spin operators are also calculated between several low-energy states. By carefully analyzing finite-size effects, we show that the dynamical structure factor of the impurity contains a \ensuremath{\delta}-function contribution at zero frequency which does not vanish in the thermodynamic limit for the case of symmetrical coupling. This indicates a partial decoupling of the impurity spin from the spins in the chain, and therefore the impurity susceptibility is predicted to show a Curie behavior. This contradicts the two-channel Kondo behavior suggested from analytic considerations.
We calculate the impurity susceptibility for a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins in an antiferromagnetic Heisenberg spin-1/2 chain. This is done by extending White's density-matrix renormalization group (DMRG) technique to finite temperatures. It is found that the impurity susceptibility satisfies Curie law at low temperatures, and that the Curie constant decreases rapidly with increasing coupling constant between the impurity spin and the spins in the chain. For strong couplings, a logarithmic dependence of the susceptibility on temperature emerges. The Curie law behavior agrees with the DMRG analysis at zero temperature but deviates from the prediction of conformal field theory.
We study the effect of an isolated impurity on the low-energy properties of a half-odd-integer-spin Heisenberg antiferromagnetic chain using both numerical and conformal field theory techniques. The impurity corresponds to the substitution of a magnetic ion by a different ion with the same or different spin, or else to the coupling of a magnetic impurity to a single spin on the chain. Depending on which kind of impurity is present, the low-energy behavior corresponds either to a ``healing'' of the defect or else to a severing of the chain at the impurity location. In both cases there is a coupling-dependent length scale over which magnetic screening takes place. Analogies with the Kondo effect are elucidated.
Using a one-dimensional spin-orbital model as a typical example of quantum spin systems with richer symmetries, we study the effect of an isolated impurity on its low-energy dynamics in the gapless phase through bosonization and renormalization-group methods. In the case of internal impurities, depending on the symmetry, the boundary fixed points can be either an open chain with a residual spin or (and) orbital triplet left behind, or a periodic chain. However, these two fixed points are indistinguishable in the sense that in both cases, the lead-correction-to-scaling boundary operators (LCBO) only show Fermi-liquid-like corrections to thermodynamical quantities (except the possible Curie-like contributions from the residual moments in the former cases). In the case of external (Kondo) impurities, the boundary fixed points, depending on the sign of orbital couplings, can be either an open chain with an isolated orbital doublet due to Kondo screening or an intermediate fixed point with the same LCBO as that of the two-channel Kondo problem. Comparison with the Kondo effect in one-dimensional Heisenberg spin chain and multiband Hubbard models is also made.
Dynamical properties of the impurity spin-12\frac{1}{2}21​ in two-dimensional (2D) and quasi-2D Heisenberg antiferromagnets (AFs) at $T\ensuremath{\geqslant}0$ are discussed. The specific case of an impurity coupled symmetrically to two neighboring host spins is considered. The specific feature of this problem is that the defect is degenerate (frustrated) being located in zero molecular field. It is shown that this problem can be described by the spin-boson model without a tunneling term and with a more complex interaction. We demonstrate that the effect of the host system on the defect is completely described by the spectral function. It is found within the spin-wave approximation that for not too small $\ensuremath{\omega}$ the spectral function is proportional to ${\ensuremath{\omega}}^{2}∕{J}^{3}$, where JJJ is the exchange constant between the host spins. The defect dynamical susceptibility is derived using Abrikosov's pseudofermion technique and diagrammatic expansion. The calculations are performed within the fourth order of the dimensionless coupling parameter fff. It is found that transverse impurity susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(\ensuremath{\omega})$ has a Lorenz peak, with the width proportional to f4J(T∕J)3{f}^{4}J{(T∕J)}^{3}f4J(T∕J)3 which disappears at T=0T=0T=0, and a nonresonant term. The latter term diverges logarithmically as $\ensuremath{\omega},\phantom{\rule{0.2em}{0ex}}T\ensuremath{\rightarrow}0$. The static susceptibility $\ensuremath{\chi}(0)$ has the free-spin-like contribution 1∕(4T)1∕(4T)1∕(4T), and a logarithmic correction proportional to f2ln(J∕T){f}^{2}\mathrm{ln}(J∕T)f2ln(J∕T). The influence of finite concentration of the defects nnn on the low-temperature properties of AFs is also investigated. A logarithmic correction to spin-wave velocity of the form $n{f}^{4}\mathrm{ln}\ensuremath{\mid}J∕\ensuremath{\omega}\ensuremath{\mid}$ and an anomalous damping of spin waves proportional to $n{f}^{4}\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}$ are obtained. The results of the present paper can be applied to other systems with a frustrated impurity in which the spectral function is proportional to ${\ensuremath{\omega}}^{2}$.