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Physical review. B, Condensed matter

Thermodynamics of dilute XY quantum chains

Michael F. Thorpe, Sasuke Miyazima

Published Dec 1, 1981• 13 citations
Publisher

Abstract

It is shown that the specific heat and susceptibility (${\ensuremath{\chi}}^{\mathrm{zz}}$) of the dilute $\mathrm{XY}$ $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}$ chain can be calculated by summing over the finite segments. At low temperatures the specific heat is determined by segments with an even number of spins, whereas the susceptibility is determined by segments with an odd number of spins. At low temperatures the specific heat behaves as ${(\frac{{T}_{0}}{T})}^{\frac{5}{4}}\mathrm{exp}[\ensuremath{-}2{(\frac{{T}_{0}}{T})}^{\frac{1}{2}}]$, where ${T}_{0}$ is a characteristic temperature in the dilute spin system.

DOI
10.1103/physrevb.24.6686

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Physical review. B, Condensed matter•Apr 1, 1986•5 citations
It is shown that the specific heat and susceptibility (\ensuremathχzz{\ensuremath{\chi}}^{\mathrm{zz}}\ensuremathχzz) of the dilute XY\mathrm{XY}XY spin\ensuremath−12\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}\frac{1}{2}spin\ensuremath−21​ chain can be calculated by summing over the finite segments. At low temperatures the specific heat is determined by segments with an even number of spins, whereas the susceptibility is determined by segments with an odd number of spins. At low temperatures the specific heat behaves as (T0T)54exp[\ensuremath−2(T0T)12]{(\frac{{T}_{0}}{T})}^{\frac{5}{4}}\mathrm{exp}[\ensuremath{-}2{(\frac{{T}_{0}}{T})}^{\frac{1}{2}}](TT0​​)45​exp[\ensuremath−2(TT0​​)21​], where T0{T}_{0}T0​ is a characteristic temperature in the dilute spin system.
A full description of the previously proposed method of calculating low temperature specific heat and susceptibility of quantum spin chains are given. By the use of internal energy as the expansion variable the domain of convergence of the high temperature series is remarkably extended. The low temperature properties are obtained by continuing it in a natural way to polynomials in internal energy, whose form is suggested by the spin wave theory. The method is checked in the S =1/2 XY model and is then applied to the Heisenberg model of S =1/2, 1, 3/2 and 2. The results are satisfactory in ferromagnets and reasonable in the S =1/2 antiferromagnet. In antiferromagnets of S ≧1 the results suggest inadequacy of our polynomial, which is, presumably, due to the existence of an energy gap.
We consider the quantum XY spin chain with quasi-periodic two-valued exchange couplings and a uniform transverse field. The equivalence of the system with a free fermionic model permits a detailed analysis of its thermodynamics. The zero-temperature magnetization is a Cantor function of the applied field. The zero-field specific heat and susceptibility have a power-law behaviour at low temperature, modulated by periodic amplitudes. The critical exponents interpolate continuously between those of the periodic and of the random model.
The energy, the specific heat, and the magnetic susceptibility of the bond-diluted S=(1/2) Heisenberg ferromagnet on a two-dimensional square lattice are studied using a Monte Carlo sampling of the terms contributing to the partition function. The full lattice is studied in detail for sizes up to 20\ifmmode\times\else\texttimes\fi{}20, and evidence is presented to suggest that the low-temperature correlation length behaves as \ensuremath{\xi}\ensuremath{\sim}${e}^{\ensuremath{\nu}\ensuremath{\beta}J}$ with \ensuremath{\nu}\ensuremath{\sim}1.91\ifmmode\pm\else\textpm\fi{}0.06. The temperature dependence of the susceptibility, however, is not quite a simple exponential in \ensuremath{\beta}J. Knowledge of the correlation length and finite-size scaling allows the determination of the prefactor. Our conclusion is that the low-temperature susceptibility behaves as ${T}^{\mathrm{\ensuremath{-}}1}$${e}^{2\ensuremath{\nu}\ensuremath{\beta}J}$. Square lattices of size ranging from 6\ifmmode\times\else\texttimes\fi{}6 to 12\ifmmode\times\else\texttimes\fi{}12 are studied for kB{k}_{B}kB​T/J ranging from 0.1 to 3 and for several values of the bond probability p. Not surprisingly, the most drastic thermodynamic behavior near the critical bond-percolation probability is the sharp increase in the low-temperature susceptibility for p>pc{p}_{c}pc​. This corresponds to much larger correlation effects in the percolating lattice than below the percolation threshold. We also compare the thermodynamics at p=pc{p}_{c}pc​ to the scaling theories of Lubensky and Stanley et al. which suggest that the correlations at the percolation threshold are predominantly one dimensional.