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arXiv:cond-mat/0310517 · Physical Review B

Boundary contributions to specific heat and susceptibility in the spin- 12 XXZ chain

Akira Furusaki, Takashi Hikihara

Published Mar 24, 2004• 29 citations
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Abstract

Exact low-temperature asymptotic behavior of boundary contribution to specific heat and susceptibility in the one-dimensional spin-$\frac{1}{2}$ $\mathrm{XXZ}$ model with exchange anisotropy $1/2<\ensuremath{\Delta}<~1$ is analytically obtained using the Abelian bosonization method. The boundary spin susceptibility is divergent in the low-temperature limit. This singular behavior is caused by the first-order contribution of a bulk leading irrelevant operator to boundary free energy. The result is confirmed by numerical simulations of finite-size systems. The anomalous boundary contributions in the spin isotropic case are universal.

DOI
10.1103/physrevb.69.094429

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Exact low-temperature asymptotic behavior of boundary contribution to specific heat and susceptibility in the one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model with exchange anisotropy 1/2<\ensuremathΔ< 11/2<\ensuremath{\Delta}<~11/2<\ensuremathΔ< 1 is analytically obtained using the Abelian bosonization method. The boundary spin susceptibility is divergent in the low-temperature limit. This singular behavior is caused by the first-order contribution of a bulk leading irrelevant operator to boundary free energy. The result is confirmed by numerical simulations of finite-size systems. The anomalous boundary contributions in the spin isotropic case are universal.
We investigate the low-temperature thermodynamics of the spin-1/2 Heisenberg chain with open ends. On the basis of boundary conformal field theory arguments and numerical density matrix renormalization group calculations, it is established that in the isotropic case the impurity susceptibility exhibits a Curie-like divergent behavior as the temperature decreases, even in the absence of magnetic impurities. A similar singular temperature dependence is also found in the boundary contributions of the specific heat coefficient. In the anisotropic case, for 1/2<Delta<1, these boundary quantities still show a singular temperature dependence obeying a power law with an anomalous dimension. Experimental consequences will be discussed.
We calculate the bulk and boundary parts of the free energy for an open spin-1/2 XXZ -chain in the critical regime by bosonization. We identify the cut-off independent contributions and determine their amplitudes by comparing with Bethe ansatz calculations at zero temperature T . For the bulk part of the free energy we find agreement with Lukyanov's result (1998 Nucl. Phys. B 522 533). In the boundary part we obtain a cut-off independent term which is linear in T and determines the temperature dependence of the boundary susceptibility in the attractive regime for . We further show that at particular anisotropies where contributions from irrelevant operators with different scaling dimensions cross, logarithmic corrections appear. We give explicit formulae for these terms at those anisotropies where they are most important. We verify our results by comparing with extensive numerical calculations based on a numerical solution of the T = 0 Bethe ansatz equations, the finite temperature Bethe ansatz equations in the quantum transfer matrix formalism, and the density matrix renormalization group applied to transfer matrices.
The magnetic susceptibility of the one-dimensional Hubbard model with open boundary conditions at arbitrary filling is obtained from field theory at low temperatures and small magnetic fields, including leading and next-leading orders. Logarithmic contributions to the bulk part are identified as well as algebraic-logarithmic divergences in the boundary contribution. As a manifestation of spin-charge separation, the result for the boundary part at low energies turns out to be independent of filling and interaction strength and identical to the result for the Heisenberg model. For the bulk part at zero temperature, the scale in the logarithms is determined exactly from the Bethe ansatz. At finite temperature, the susceptibility profile as well as the Friedel oscillations in the magnetisation are obtained numerically from the density-matrix renormalisation group applied to transfer matrices. Agreement is found with an exact asymptotic expansion of the relevant correlation function.
We present a brief review on the recent theoretical developement about boundary effects in one-dimensional quantum spin systems focusing on the application of boundary conformal field theory. We discuss about low-temperature behaviors of the boundary contributions of the spin susceptibility and the specific heat coefficient. The anomalous behaviors at boundaries are deeply related with finite-temperature corrections of the boundary entropy or the ground state degeneracy, which yields a highly sensitive response of spin excitations in the vicinity of open boundaries.
We study the zero- and the finite-temperature behavior of the integrable spin-1/2 XXZ periodic chain with an impurity by the algebraic and thermal Bethe ansatz methods. We evaluate the local magnetization on the impurity site at zero temperature analytically and derive the impurity susceptibility exactly from it. In the graphs of the impurity specific heat versus temperature, we show how the impurity spin becomes more liberated from the bulk many-body effect as the exchange coupling between the impurity spin and other spins decreases and that at low temperature it couples strongly to them such as in the Kondo effect. Thus, we observe not only the crossover behavior from the high- to the low-temperature regime, but another from the N -site chain to the ( N − 1)-site chain with a free impurity spin. We also show that the estimate of the Wilson ratio at a given low temperature is independent of the impurity parameter if its absolute value is small enough with respect to the temperature and the universality class is described by the XXZ anisotropy in terms of the dressed charge.
The Kondo effect in a one-dimensional spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ model in the gapless XY\mathrm{XY}XY regime $(\ensuremath{-}1<\ensuremath{\Delta}<~1)$ is studied both analytically and numerically. In our model an impurity spin (S=1/2)(S=1/2)(S=1/2) is coupled to a single spin in the XXZ\mathrm{XXZ}XXZ spin chain. Perturbative renormalization-group (RG) analysis is performed for various limiting cases to deduce low-energy fixed points. It is shown that in the ground state the impurity spin is screened by forming a singlet with a spin in the host XXZ\mathrm{XXZ}XXZ chain. In the antiferromagnetic side $(0<\ensuremath{\Delta}<~1)$ the host chain is cut into two semi-infinite chains by the singlet. In the ferromagnetic side $(\ensuremath{-}1<\ensuremath{\Delta}<0),$ on the other hand, the host XXZ\mathrm{XXZ}XXZ chain remains as a single chain through ``healing'' of a weakened bond in the low-energy (long-distance) limit. The density-matrix renormalization-group method is used to study the size scaling of finite-size energy gaps and the power-law decay of correlation functions in the ground state. The numerical results are in good agreement with the predictions of the RG analysis. Low-temperature behaviors of specific heat and susceptibility are also discussed.