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The Randomly Diluted Quantum Heisenberg Antiferromagnet at Zero Temperature

Chuncheng Wan

Published Jan 1, 1991• 0 citations
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Abstract

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 = $p\\sb{Q}$ respectively, where p is the concentration of the spins or bonds, and $p\\sb{Q}$ 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 $\\equiv$ 1 $-$ p gives a dispersion relation for long-wavelength spin waves in the form $\\omega(\\vec q)$ = $c(x)q + i\\gamma(x)q\\sp\\tau$. Our results for $c(x)$ agrees with previous work, where $c(x)$ = $c(0)\\lbrack 1 - \\alpha x + O(x\\sp2)\\rbrack$. In particular the coefficient $\\alpha$, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit $d\\to 2$. One major new results is that $\\tau$ = $d - 1$, 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 $p\\sb{Q}$ 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 $\\nu$ (which comes from the relation $\\xi\\sim(p\\sb{Q} - p)\\sp\\nu$, where $\\xi$ is the correlation length) are estimated, which are 0.65 $\\pm$ 0.2 and 0.55 $\\pm$ 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that $p\\sb{Q} > p\\sb{c}$.

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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.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
Generalizing the recent work of Collins, the wave-vector-dependent susceptibility and the static correlation function for a randomly diluted Heisenberg paramagnet on a Bravais lattice with quenched-site, or nearest-neighbor exchange-bond, dilution is studied by high-temperature series-expansion techniques. The first five coefficients for the spin correlation function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ and susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ are calculated for arbitrary $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, the sign of the exchange and the spin magnitude and the magnetic site ccc and bond ppp concentrations. Numerical results for these functions are presented for various spins and for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ along the symmetry directions (1,1,1) and (1,0,0). Because of the generality of the present series, their length is rather limited, and as such they are not suitable for a careful analysis of the critical region. Nonetheless, to get a qualitative feel for the accuracy of the present results near the transition region, we have used Pad\'e approximants for determining the correlation length $\ensuremath{\xi}(T)$ as a function of the reduced temperature $\ensuremath{\epsilon}=\frac{[T\ensuremath{-}{T}_{C(N)}(\mathrm{random})]}{{T}_{C(N)}(\mathrm{random})}$. In this manner, we estimate that the numerical accuracy of our results should be reasonable as long as the system is moderately above the transition temperature, e.g., $\ensuremath{\epsilon}\ensuremath{\gtrsim}0.5$.
The three-dimensional quenched random bond diluted $({J}_{1}\ensuremath{-}{J}_{2})$ quantum Heisenberg antiferromagnet is studied on a simple-cubic lattice. Using extensive stochastic series expansion quantum Monte Carlo simulations, we perform very long runs for an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ lattice up to L=48L=48L=48. By employing a standard finite-size scaling method, the numerical values of the N\'eel temperature are determined with high precision as a function of the coupling ratio r=J2/J1r={J}_{2}/{J}_{1}r=J2​/J1​. Based on the estimated critical exponents, we find that the critical behavior of the considered model belongs to the pure classical three-dimensional O(3)O(3)O(3) Heisenberg universality class.
The dynamic critical behavior of one-dimensional random spin systems in a field at zero temperature is investigated by a generalized transfer-matrix scaling technique. Three spin models are considered: Heisenberg in a longitudinal field, XY in a transverse field, and Ising in a transverse field, with either spin-glass or random-field disorder. Near the transition, i.e., for small values of the reduced field \ensuremath{\Delta}=(H-Hc{H}_{c}Hc​), where Hc{H}_{c}Hc​ is the critical field, one finds that the dispersion relation for the low-frequency \ensuremath{\omega}, small-wave-vector k spin-wave excitations takes the scaling form \ensuremath{\omega}=kz{k}^{z}kzf(\ensuremath{\Delta}/kcphi{k}^{\mathrm{cphi}}kcphi), and exact results are given for the dynamic exponent z and crossover exponent cphi. This form contains a crossover of the frequency \ensuremath{\omega} between two different asymptotic behaviors: kz{k}^{z}kz for \ensuremath{\Delta}\ensuremath{\ll}kcphi{k}^{\mathrm{cphi}}kcphi and ${h}^{\ensuremath{\delta}}$ for \ensuremath{\Delta}\ensuremath{\gg}kcphi{k}^{\mathrm{cphi}}kcphi, where the field exponent is \ensuremath{\delta}=z/cphi. In the case of the Heisenberg systems the spin-glass disorder gives rise to the nontrivial dynamic exponent z=(3/2, whereas in the random-field case it is the exponent associated to the field that becomes nontrivial, taking the value (4/3. For the transverse XY systems the random-field disorder implies nontrivial values for both the dynamic and field exponents, (3/2 and (3/4, respectively, whereas the spin-glass disorder does not affect the dynamics of the system, which behaves like a pure transverse XY ferromagnet. Finally, for the transverse Ising systems neither the random field nor the spin-glass disorder affects the dynamics, which is the same as for a pure transverse Ising ferromagnet.
The quantum renormalization-group technique recently developed by Chakravarty and the present author is used to study weak random couplings in quantum-mechanical and classical nearest-neighbor Heisenberg ferromagnets in D=2+\ensuremath{\epsilon} dimensions and at low temperatures T. Momentum-shell recursion relations are obtained by considering the renormalization-group transformations of the cumulants of the probability distribution of the random exchange couplings. From a one-loop calculation it is found that, if the second cumulants ${\ensuremath{\sigma}}^{2}$ of the random couplings are smaller than a critical value ${\ensuremath{\sigma}}_{c}^{2}$, the only modification due to randomness in the low-temperature behavior of the order-parameter correlation length \ensuremath{\xi} in two dimensions is a renormalization of the spin stiffness ${\ensuremath{\rho}}_{s}$ according to ${\ensuremath{\rho}}_{s}$(X)=(1-X${)}^{\ensuremath{\varphi}}$${\ensuremath{\rho}}_{s}$(0), where X=${\ensuremath{\sigma}}^{2}$/${\ensuremath{\sigma}}_{c}^{2}$ and \ensuremath{\varphi}=(1/3 within the one-loop approximation. At zero temperature \ensuremath{\xi} diverges for with ${\ensuremath{\nu}}_{x}$=1/D. Precisely at X=1 the critical behavior of the system is characterized by new exponents; the dynamical critical exponent z is given by z=2+\ensuremath{\varphi}/${\ensuremath{\nu}}_{x}$. For D(1-\ensuremath{\varphi})>2 a new phase appears in the T-X phase diagram, and it is argued that this phase is characterized by spin-glass order.