Harris, A. Brooks
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Publication Resistance Fluctuations in Randomly Diluted Networks(1987-03-01) Blumenfeld, Raphael; Meir, Yigal; Aharony, Amnon; Harris, A. BrooksThe resistance R(x,x’) between two connected terminals in a randomly diluted resistor network is studied on a d-dimensional hypercubic lattice at the percolation threshold pc. When each individual resistor has a small random component of resistance, R(x,x’) becomes a random variable with an associated probability distribution, which contains information on the distribution of currents in the individual resistors. The noise measured between the terminals may be characterized by the cumulants Mq(x,x’) of R(x,x’). When averaged over configurations of clusters, M¯q(x,x’)~‖x-x’‖ψ̃(q). We construct low-concentration series for the generalized resistive susceptibility, χ(q), associated with M¯q, from which the critical exponents ψ̃(q) are obtained. We prove that ψ̃(q) is a convex monotonically decreasing function of q, which has the special values ψ̃(0)=DB, ψ̃(1)=ζ̃R, and ψ̃(∞)=1/ν. (DB is the fractal dimension of the backbone, ζ̃R is the usual scaling exponent for the average resistance, and ν is the correlation-length exponent.) Using the convexity property and the accepted values of these three exponents, we construct two approximant functions for ψ(q)=ψ̃(q)ν, both of which agree with the series results for all q>1 and with existing numerical simulations. These approximants enabled us to obtain explicit approximate forms for the multifractal functions α(q) and f(q) which, for a given q, characterize the scaling with size of the dominant value of the current and the number of bonds having this current. This scaling description fails for sufficiently large negative q, when the dominant (small) current decreases exponentially with size. In this case χ(q) diverges at a lower threshold p*(q), which vanishes as q→-∞.Publication Series Analysis of Randomly Diluted Nonlinear Resistor Networks(1986-09-01) Meir, Yigal; Blumenfeld, Raphael; Aharony, Amnon; Harris, A. BrooksThe behavior of a randomly diluted network of nonlinear resistors, for each of which the voltage-current relationship is |V|=r|I|α, is studied with use of series expansions in the concentration p of conducting bonds on d-dimensional hypercubic lattices. The average nonlinear resistance 〈R〉 between pairs of sites separated by the percolation correlation length, scales as |p-pc|−ζ(α). The exponent ζ(α) was evaluated for 0<α<∞ and d=2, 3, 4, 5, and 6, found to decrease monotonically from the exponent describing the minimal length, at α=0, via that of the linear resistance, at α=1, to the exponent characterizing the singly connected bonds, ξ(∞)≡1. Our results agree with known results for α=0 and α=1, also with recent results for general α at d=6-ε dimensions. The second moment 〈R2〉 was found to diverge as 〈R⟩2 (for all α and d), indicating a scaling form for the probability distribution of R.Publication Magnetization Measurements of Antiferromagnetic Domains in Sr2Cu3O4Cl2(2001-03-15) Parks, Beth; Kastner, Marc A; Kim, Youngjune; Harris, A. Brooks; Chou, Fangcheng; Entin-Wohlman, Ora; Aharony, AmnonThe Cu3O4 layer in Sr2Cu3O4Cl2 is a variant of the square CuO2 lattice of the high-temperature superconductors, in which the center of every second plaquette contains an extra Cu2+ ion. Whereas the ordering of the spins in the ground-state and the spin-wave excitations of this frustrated spin system are both well understood, we find peculiar behavior resulting from antiferromagnetic domain walls. Pseudodipolar coupling between the two sets of Cu2+ ions results in a ferromagnetic moment, the direction of which reflects the direction of the antiferromagnetic staggered moment, allowing us to probe the antiferromagnetic domain structure. After an excursion to the high fields (>1 T), as the field is lowered, we observe the growth of domains with ferromagnetic moment perpendicular to the field. This gives rise to a finite domain wall susceptibility at small fields, which diverges near 100 K, indicating a phase transition. We also find that the shape of the sample influences the domain-wall behavior.Publication Distribution of the Logarithms of Currents in Percolating Resistor Networks. I. Theory(1993-03-01) Aharony, Amnon; Blumenfeld, Raphael; Harris, A. BrooksThe distribution of currents, ib, in the bonds b of a randomly diluted resistor network at the percolation threshold is investigated through a study of the moments of the distribution P^(i2) and the moments of the distribution P(y), where y=-lnib2. For q>qc the qth moment of P^(i2), Mq (i.e., the average of i2q), scales as a power law of the system size L, with a multifractal (noise) exponent ψ̃(q)-ψ̃(0). Numerical data indicate that qc is negative, but becomes small for large L. Assuming that all derivatives ψ̃(q) exist at q=0+, we show that for positive integer k the kth moment, μk, of P(y) is given by μk=(α0 lnL)k{1+[kC1+1/2k(k-1)D1] (lnL)−1+O[(lnL)−2]}, where α0 and D1 (but not C1) are universal constants obtained from ψ̃(q). A second independent argument, requiring an assumed analyticity property of the asymptotic multifractal function, f(α), leads to the same equation for all k. This latter argument allows us to include finite-size corrections to f(α), which are of order (lnL)−1. These corrections must be taken into account in interpreting numerical studies of P(y). We note that data for P(-lni2) seem to show power-law behavior as a function of i2 for small i. Values of the exponents are directly related to the values of qc, and the numerical data in two dimensions indicate it to be small (but probably nonzero). We suggest, in view of the nature of the finite-size corrections in the expression for μk, that the asymptotic regime may not have been reached in the numerical work. For d=6 we find that Mq(L)~(lnL)θ(q), where θ(q)→∞ for q→qc=-1/2.Publication Ordering Due to Quantum Fluctuations in Sr2Cu3O4Cl2(1999-07-26) Kim, Youngjune; Aharony, Amnon; Birgeneau, Robert J; Chou, Fangcheng; Entin-Wohlman, Ora; Erwin, Ross W; Greven, Martin; Harris, A. Brooks; Kastner, Marc A; Korenblit, I. Ya; Lee, Youngsu; Shirane, GenSr2Cu3O4Cl2 has CuI and CuII subsystems, forming interpenetrating S=1/2 square lattice Heisenberg antiferromagnets. The classical ground state is degenerate, due to frustration of the intersubsystem interactions. Magnetic neutron scattering experiments show that quantum fluctuations cause a two dimensional Ising ordering of the CuII's, lifting the degeneracy, and a dramatic increase of the CuI out-of-plane spin-wave gap, unique for order out of disorder. The spin-wave energies are quantitatively predicted by calculations which include quantum fluctuations.Publication Spin Structures of Tetragonal Lamellar Copper Oxides(1994-06-06) Yildirim, Taner; Harris, A. Brooks; Entin-Wohlman, Ora; Aharony, AmnonThe spin Hamiltonian of tetragonal lamellar antiferromagnets is shown to contain several novel anisotropies. Symmetry allows bond-dependent anisotropic exchange interactions, which lead to (a) interplane mean-field coupling and (b) an in-plane anisotropy which vanishes classically but arises from quantum zero point energy (QZPE). A similar QZPE involving the interplane isotropic interaction prefers collinear spins. Adding also diploar anisotropy, the competition between all these effects explains for the first time the spin structures of many cuprates.Publication Magnetic Structure of the Jahn-Teller System LaTiO3(2005-04-19) Schmitz, Robert; Entin-Wohlman, Ora; Aharony, Amnon; Harris, A. Brooks; Müller-Hartmann, ErwinWe investigate the effect of the experimentally observed Jahn-Teller distortion of the oxygen octahedra in LaTiO3 on the magnetic exchange. We present a localized model for the effective hopping between nearest-neighbor Ti ions and the intrasite Coulomb interactions, based on a nondegenerate orbital ground state due to the static crystal field. The latter corresponds to an orbital order which has recently been confirmed experimentally. Using perturbation theory we calculate, in addition to the Heisenberg coupling, antisymmetric (Dzyaloshinskii-Moriya) and symmetric anisotropy terms of the superexchange spin Hamiltonian, which are caused by the spin-orbit interaction. Employing this spin Hamiltonian, we deduce that at low temperatures the spins have predominantly a G-type antiferromagnetic ordering along the crystallographic a axis, accompanied by a weak ferromagnetic moment along the c axis and by a weak A-type antiferromagnetic moment along the b axis. The first two components are found to be in good agreement with experiment.Publication Localization Length Exponent in Quantum Percolation(1995-03-13) Chang, Iksoo; Lev, Zvi; Harris, A. Brooks; Adler, Joan; Aharony, AmnonConnecting perfect one-dimensional leads to sites i and j on the quantum percolation (QP) model, we calculate the transmission coefficient Tij(E) at an energy E near the band center and the averages of ΣijTij, Σijr2ijTij, and Σijr4ijTij to tenth order in the concentration p. In three dimensions, all three series diverge at pq=0.36+0.01−0.02, with exponents γ=0.82+0.10−0.15, γ+2ν, and γ+4ν. We find ν=0.38±0.07, differing from “usual” Anderson localization and violating the bound ν≥2/d of Chayes et al. [Phys. Rev. Lett. 57, 2999 (1986)]. Thus, QP belongs to a new universality class.Publication Anisotropic Spin Hamiltonians Due to Spin-Orbit and Coulomb Exchange Interactions(1995-10-01) Yildirim, Taner; Harris, A. Brooks; Aharony, Amnon; Entin-Wohlman, OraHere we correct, extend, and clarify results concerning the spin Hamiltonian ℋS used to describe the ground manifold of Hubbard models for magnetic insulators in the presence of spin-orbit interactions. Most of our explicit results are for a tetragonal lattice as applied to some of the copper oxide lamellar systems and are obtained within the approximation that ℋS consists of a sum of nearest-neighbor bond Hamiltonians. We consider both a "generic" model in which hopping takes place from one copper ion to another and a "real" model in which holes can hop from a copper ion to an intervening oxygen 2p band. Both models include orbitally dependent direct and exchange Coulomb interactions involving two orbitals. Our analytic results have been confirmed by numerical diagonalizations for two holes occupying any of the 3d states and, if applicable, the oxygen 2p states. An extension of the perturbative scheme used by Moriya is used to obtain analytic results for ℋS up to order t2 (t is the matrix of hopping coefficients) for arbitrary crystal symmetry for both the "generic" and "real" models. With only direct orbitally independent Coulomb interactions, our results reduce to Moriya’s apart from some minor modifications. For the tetragonal case, we show to all orders in t and λ, the spin-orbit coupling constant, that ℋS is isotropic in the absence of Coulomb exchange terms and assuming only nearest-neighbor hopping. In the presence of Coulomb exchange, scaled by K, the anisotropy in ℋS is biaxial and is shown to be of order Kt2λ2. Even when K=0, for systems of sufficiently low symmetry, the anisotropy in ℋS is proportional to t6λ2 when the direct on-site Coulomb interaction U is independent of the orbitals involved and of order t2λ2 otherwise. These latter results apply to the orthorhombic phase of La2CuO4.Publication Yildirim et al. Reply(1995-04-03) Yildirim, Taner; Harris, A. Brooks; Entin-Wohlman, Ora; Aharony, AmnonA Reply to the Comment by S. Skanthakumar, J. W. Lynn, and I. W. Sumarlin, Phys. Rev. Lett. 74, 2842 (1995).