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Physical review. B, Condensed matter

Improved upper bounds on the critical temperature of the 1n2 ising spin chain

J. O. Vigfusson

Published Sep 1, 1986• 9 citations
Publisher

Abstract

Recent results on the upper-bound property of the magnetization in certain approximations to ferromagnetic spin systems are here used to derive a rigorous upper bound for the critical temperature of the one-dimensional ferromagnetic $\frac{1}{{n}^{2}}$ Ising spin system. We find that ${\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})$. The best previous bound was 0.360. A further improvement by the same technique is possible.

DOI
10.1103/physrevb.34.3466

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Recent results on the upper-bound property of the magnetization in certain approximations to ferromagnetic spin systems are here used to derive a rigorous upper bound for the critical temperature of the one-dimensional ferromagnetic 1n2\frac{1}{{n}^{2}}n21​ Ising spin system. We find that \ensuremathβcJ\ensuremath≥0.441(\ensuremathβc=1kTc){\ensuremath{\beta}}_{c}J\ensuremath{\ge}0.441({\ensuremath{\beta}}_{c}=\frac{1}{k{T}_{c}})\ensuremathβc​J\ensuremath≥0.441(\ensuremathβc​=kTc​1​). The best previous bound was 0.360. A further improvement by the same technique is possible.
We prove rigorous lower and upper bounds for the mass gap of the ferromagnetic spin 1/2 XXZ chain. The two bounds coincide asymptotically in the Ising limit Δ→∞Δ\to\inftyΔ→∞. Near the isotropic point, Δ=1Δ=1Δ=1, the estimates are good enough to determine the critical behaviour of the mass gap unambiguously. The derivation does not rely on exact solutions.
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Magnon bound states emerging in one-dimensional (1D) spin chains still lack a rigorous understanding. In this Rapid Communication we show that the length-1 spin strings significantly dominate the critical properties of spinons, magnons, and free fermions in the 1D antiferromagnetic spin-1/2 chain. Using the Bethe ansatz solution, we analytically calculate the scaling functions of the thermal and magnetic properties of the model, providing a rigorous understanding of the quantum criticality of spinons. It turns out that the double maxima in specific heat elegantly mark two crossover temperatures fanning out from the critical point, indicating three quantum phases: the Tomonaga-Luttinger liquid (TLL), the quantum critical, and fully polarized ferromagnetic phases. For the TLL phase, the Wilson ratio RW=4Ks{R}_{W}=4{K}_{s}RW​=4Ks​ remains almost temperature independent, where Ks{K}_{s}Ks​ is the Luttinger parameter. Furthermore, by applying our results, we precisely determine the quantum scalings and critical exponents of all magnetic properties in the ideal 1D spin-1/2 antiferromagnet Cu(C4H4N2)(NO3)2\mathrm{Cu}(\mathrm{C}{}_{4}\mathrm{H}{}_{4}\mathrm{N}{}_{2})(\mathrm{NO}{}_{3}){}_{2}Cu(C4​H4​N2​)(NO3​)2​, recently studied by Kono et al. [Phys. Rev. Lett. 114, 037202 (2015)]. We further find that the magnetization maximum used in experiments is not a good quantity to map out the finite-temperature TLL phase boundary.
A long-range Ising chain, which has ferromagnetic 1/r 3 type interactions, has been studied by exact numerical calculation of thermodynamic properties of the sequence of systems of N spins (N<or=30). By expressing the sequence in terms of a series and using extrapolation methods such as the Pade approximant, the authors have found a consistent way to obtain critical temperatures which is accurate to 0.4% for the very long-range equivalent neighbour model. The authors have estimated the high-temperature susceptibility and the low- temperature long-range correlations of the infinite chain and thence have made estimates for the critical exponents gamma and beta . Although the nature of the phase transition appears to be classical for 1<s<or approximately=1.3, it is certainly not classical for 1.6<s<2; in particular, as s approaches 2 from below, gamma increases regularly to about 2 or greater and beta decreases regularly to near 0.