Harris, A. Brooks
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Publication Critical Behavior of Random Resistor Networks Near the Percolation Threshold(1978-07-01) Fisch, Ronald; Harris, A. BrooksWe use low-density series expansions to calculate critical exponents for the behavior of random resistor networks near the percolation threshold as a function of the spatial dimension d. By using scaling relations, we obtain values of the conductivity exponent μ. For d=2 we find μ=1.43±0.02, and for d=3, μ=1.95±0.03, in excellent agreement with the experimental result of Abeles et al. Our results for high dimensionality agree well with the results of ε-expansion calculations.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 Charge and Spin Ordering in the Mixed-Valence Compound LuFe2O4(2010-04-15) Harris, A. Brooks; Yildirim, TanerLandau theory and symmetry considerations lead us to propose an explanation for several seemingly paradoxical behaviors of charge ordering (CO) and spin ordering (SO) in the mixed valence compound LuFe2O4. Both SO and CO are highly frustrated. We analyze a lattice gas model of CO within mean-field theory and determine the magnitude of several of the phenomenological interactions. We show that the assumption of a continuous phase transitions at which CO or SO develops implies that both CO and SO are incommensurate. To explain how ferroelectric fluctuations in the charge-disordered phase can be consistent with an antiferroelectric-ordered phase, we invoke an electron-phonon interaction in which a low-energy (20 meV) zone-center transverse phonon plays a key role. The energies of all the zone center phonons are calculated from first principles. We give a Landau analysis which explains SO and we discuss a model of interactions which stabilizes the SO state, if it is assumed commensurate. However, we suggest a high-resolution experimental determination to see whether this phase is really commensurate, as believed up to now. The applicability of representation analysis is discussed. A tentative explanation for the sensitivity of the CO state to an applied magnetic field in field-cooled experiments is given.Publication Nuclear Magnetic Resonance and Relaxation in Solid Hydrogen(1966-05-09) Harris, A. Brooks; Hunt, Earle[No abstract included]Publication Locking of Commensurate Phases in the Planar Model in an External Magnetic Field(1991-08-01) Harris, A. Brooks; Rastelli, Enrico; Tassi, ArmandoCommensurate configuration locking is known in models like the anisotropic next-nearest-neighbor Ising model and the Frenkel-Kontorova model. We find an analogous scenario in the planar model with competing interactions when an external magnetic field is applied in the plane in which the spins lie. This model falls in the same symmetry class of the Heisenberg model with planar anisotropy. We performed a low-field, low-temperature expansion for the free energy of the model and we find phase locking energy for states with wave vectors of the form G/p where p is an integer and G is a reciprocal-lattice vector. The helix characterized by p=3 is peculiar because the commensuration energy vanishes at zero temperature. The helix corresponding to p=4 is not stable against the switching of a magnetic field that forces the spins into an up-up-down-down configuration analogous to the spin-flop phase of an antiferromagnet. For a generic commensurate value of p>4, we expect locking both at zero and finite temperature as we have verified for p=5 and 6. The consequences of our results are examined for the 3N model (a tetragonal spin lattice with in-plane competitive interactions up to third-nearest neighbors).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 High-Temperature Series for Random-Anisotropy Magnets(1990-06-01) Fisch, Ronald; Harris, A. BrooksHigh-temperature series expansions for thermodynamic functions of random-anisotropy-axis models in the limit of infinite anisotropy are presented, for several choices of the number of spin components, m. In three spatial dimensions there is a divergence of the magnetic susceptibility χM for m=2. We find Tc/J=1.78±0.01 on the simple cubic lattice, and on the face-centered cubic lattice, we find Tc/J=4.29±0.01. There is no divergence of χM at finite temperature for m≥3 on either lattice. We also give results for simple hypercubic lattices.Publication Central-Force Models Which Exhibit a Splay-Rigid Phase(1989-10-01) Wang, Jian; Harris, A. BrooksTwo models, one random the other periodic, are described which exhibit splay rigidity but are not rigid with respect to compression. The random model is based on a periodic lattice of rhombuses whose sides consist of central-force springs, which is perturbed in the following way: rhombuses can have diagonal central force struts with probability y or they can have one of the horizontal springs removed with probability x. For x,y≪1 we are led to consider a long-ranged anisotropic percolation process which is solved exactly on a Cayley tree. We show that for y/x near 2 the compressional rigidity of this system is zero but the Frank elastic constant, K, describing splay rigidity is nonzero. This is the first example of a percolation model for which this phenomenon, suggested earlier, is conclusively established. For y/x≳2 √2 the system has nonzero bulk and shear moduli. We also study the excitation spectrum for a periodic model which possesses only splay rigidity and obtain a libron dispersion relation ω=cSq, where q is the wave vector and cS∼(K/ρ)1/2, where ρ is the mass density. These results are generalized to obtain a scaling form for cS and the density of states of the random model which is valid when the correlation length for compressional rigidity becomes large.Publication Frustration and Quantum Fluctuations in Heisenberg fcc Antiferromagnets(1998-08-01) Yildirim, Taner; Harris, A. Brooks; Shender, Eugene FWe consider the quantum Heisenberg antiferromagnet on a face-centered-cubic lattice in which J, the second-neighbor (intrasublattice) exchange constant, dominates J′, the first-neighbor (intersublattice) exchange constant. It is shown that the continuous degeneracy of the classical ground state with four decoupled (in a mean-field sense) simple cubic antiferromagnetic sublattices is removed so that at second order in J′/J the spins are collinear. Here we study the degeneracy between the two inequivalent collinear structures by analyzing the contribution to the spin-wave zero-point energy which is of the form Heff/J=C0+C4σ1σ2σ3σ4(J′/J)4+O(J′/J)5, where σi specifies the phase of the ith collinear sublattice, C0 depends on J′/J but not on the σ’s, and C4 is a positive constant. Thus the ground state is one in which the product of the σ’s is −1. This state, known as the second kind of type A, is stable in the range |J′|<2|J| for large S. Using interacting spin-wave theory, it is shown that the main effect of the zero-point fluctuations is at small wave vector and can be well modeled by an effective biquadratic interaction of the form ΔEQeff=−1/2Q∑i,j[S(i)⋅S(j)]2/S3. This interaction opens a spin gap by causing the extra classical zero-energy modes to have a nonzero energy of order J′√S. We also study the dependence of the zero-point spin reduction on J′/J and the sublattice magnetization on temperature. The resulting experimental consequences are discussed.Publication High-Temperature Expansion for the Orientational Specific Heat of Solid H2 and D2(1970-03-01) Berlinsky, A. John; Harris, A. BrooksTerms up to order (Γ/kBT)5 in the high-temperature expansion of the orientational specific heat of ortho-para alloys of solid H2 or D2 are evaluated. Good agreement is obtained between theory and experiment using a Padé approximant and effective values of the quadrupolar coupling constant, Γeff/Γ0=0.83 for D2 and Γeff/Γ0=0.80 for H2, where Γ0 is the value for a rigid lattice. These values agree with other determinations of Γeff, whereas the T−2 approximation for the specific heat yields anomalously small values of Γeff.