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Physical Review Letters

Exact Solution of a Model of Localization

A. Brooks Harris

Published Jul 26, 1982• 33 citations
Publisher

Abstract

The exact solution is presented for the "susceptibility," $\ensuremath{\chi}$ (the number of sites covered by the maximally extended eigenfunction), for the zero-energy solutions of a hopping model on a randomly dilute Cayley tree. If $p$ is the concentration, then $\ensuremath{\chi}\ensuremath{\sim}{({p}^{*}\ensuremath{-}p)}^{\ensuremath{-}1}$ with ${p}^{*}\ensuremath{\sim}{p}_{c}{e}^{\frac{1}{{\ensuremath{\xi}}_{1}}}$, where ${p}_{c}$ is the critical percolation concentration and ${\ensuremath{\xi}}_{1}$ the one-dimensional localization length. This result is argued to hold for the dilute quantum Heisenberg antiferromagnet at zero temperature.

DOI
10.1103/physrevlett.49.296

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The exact solution is presented for the "susceptibility," \ensuremathχ\ensuremath{\chi}\ensuremathχ (the number of sites covered by the maximally extended eigenfunction), for the zero-energy solutions of a hopping model on a randomly dilute Cayley tree. If ppp is the concentration, then \ensuremathχ\ensuremath∼(p∗\ensuremath−p)\ensuremath−1\ensuremath{\chi}\ensuremath{\sim}{({p}^{*}\ensuremath{-}p)}^{\ensuremath{-}1}\ensuremathχ\ensuremath∼(p∗\ensuremath−p)\ensuremath−1 with p∗\ensuremath∼pce1\ensuremathξ1{p}^{*}\ensuremath{\sim}{p}_{c}{e}^{\frac{1}{{\ensuremath{\xi}}_{1}}}p∗\ensuremath∼pc​e\ensuremathξ1​1​, where pc{p}_{c}pc​ is the critical percolation concentration and \ensuremathξ1{\ensuremath{\xi}}_{1}\ensuremathξ1​ the one-dimensional localization length. This result is argued to hold for the dilute quantum Heisenberg antiferromagnet at zero temperature.
We use linear spin wave theory and series expansion method to study the diluted quantum antiferromagnetic Heisenberg model at zero temperature near the p = 1 and p = psbQp\\sb{Q}psbQ respectively, where p is the concentration of the spins or bonds, and psbQp\\sb{Q}psbQ is the transition point from order to disorder of the quantum spin system. For the first case, our complete calculation to the first order in x equiv\\equivequiv 1 −-− p gives a dispersion relation for long-wavelength spin waves in the form omega(vecq)\\omega(\\vec q)omega(vecq) = c(x)q+igamma(x)qsptauc(x)q + i\\gamma(x)q\\sp\\tauc(x)q+igamma(x)qsptau. Our results for c(x)c(x)c(x) agrees with previous work, where c(x)c(x)c(x) = c(0)lbrack1−alphax+O(xsp2)rbrackc(0)\\lbrack 1 - \\alpha x + O(x\\sp2)\\rbrackc(0)lbrack1−alphax+O(xsp2)rbrack. In particular the coefficient alpha\\alphaalpha, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit dto2d\\to 2dto2. One major new results is that tau\\tautau = d−1d - 1d−1, which means that d = 2 is the lower critical dimension at which infrared divergences affect the dynamical response due to vacancies. For the second case, the critical behavior of the quantum system near psbQp\\sb{Q}psbQ is found to be different from that of the percolation and pure systems. Thus there exists a new fixed point in the phase diagram, which specifies a new universality class. The correlation length exponents nu\\nunu (which comes from the relation xisim(psbQ−p)spnu\\xi\\sim(p\\sb{Q} - p)\\sp\\nuxisim(psbQ−p)spnu, where xi\\xixi is the correlation length) are estimated, which are 0.65 pm\\pmpm 0.2 and 0.55 pm\\pmpm 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that psbQ>psbcp\\sb{Q} > p\\sb{c}psbQ>psbc.
Series expansions are used to treat the randomly diluted spin-1/2 Heisenberg antiferromagnet at zero temperature. A series is obtained at zero temperature in powers of the concentration for many correlation functions and for the correlation length from which static, dynamic, and crossover exponents are estimated. The correlation length exponent is found to be 0.77±0.10 in two dimensions. The critical concentration for the appearance of long-range order is indistinguishable from the percolation threshold.
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The method of high-temperature series expansions is used to study the dependence of T c on the concentration p of interactions in a randomly diluted S= 1 / 2 Heisenberg model. It is found that, for all three cubic lattices, the interval of p over which the Pade approximants to the series converge extends further towards the critical probability for percolation than for the corresponding site diluted model. In this interval the critical curve is linear and extrapolates to intersect the T=0 boundary at a value of p marginally above the critical probability for bond percolation. The anomalous dependence of the susceptibility exponent gamma on p which has recently been reported for the site diluted model is not observed for the model studied here.