D-dimensional Hypercubic Lattice Research Articles

Application of the linked cluster expansion to the n-component φ 4 theory

The linked cluster (high-temperature) expansion is worked out through 14th order for the O( n) symmetric φ 4 theory on a d-dimensional simple hypercubic lattice. The quantities expanded are the renormalized mass m R, the wave-function renormalization constant Z R and the renormalized coupling g R at zero momentum. The results obtained apply for all n and bare couplings, and for d = 2,3,4. Because a huge number of irreducible graphs (about 100 000) contribute to the order considered, a fully computerized approach was necessary, which we describe here in some detail. The series will be applied elsewhere to solve the 4-component model on a four-dimensional lattice.

Termite and ant diffusions in the d-dimensional lattice trapping model

The coherent medium approximation of Odagaki and Lax (1981) is generalised to the trapping model. The frequency-dependent diffusion constant in the d-dimensional hypercubic lattice is studied when the jump rate obeys a bimodal distribution. The coherent medium approximation gives the correct static diffusion constant. The imaginary part of the AC part of the diffusion constant vanishes linearly in frequency omega when d>2, as omega ln omega when d=2 and as omega d/2 when d 4, as omega 2 ln omega when d=4 and as omega d/2 when d<4. The termite limit is studied by taking the limit that one of the two jump rates (probability p) becomes infinite. In the termite diffusion the static diffusion constant is critical at p=1 and the critical exponents are the same as those for the termite diffusion in the hopping model. The ant (or ant lion) limit is defined by the limit of one jump rate being zero. The imaginary and real parts of the AC diffusion constant vanish linearly and quadratically in frequency, respectively. The critical exponent of the leading real part of the diffusion constant is one less than the corresponding exponent for the ant diffusion in the hopping model below the percolation threshold, while the leading imaginary part has the same critical exponent as those in the hopping model.

Residual entropy and validity of the third law of thermodynamics in discrete spin systems.

We study the validity of the third law of thermodynamics and the occurrence of a nonzero residual entropy in discrete spin systems. For a general classical spin system on a d-dimensional hypercubic lattice with isotropic, translationally invariant, nearest-neighbor interactions, we establish the following. (i) The necessary and sufficient condition for the third law to hold. (ii) A lower bound on the residual entropy when the third law is not valid. It is also established that the residual entropy is nonzero for all d, if it is nonzero in any dimension d&gt;1. .AE

The breakdown of dynamic scaling of the bond-diluted Ising model at the percolation threshold for hypercubic lattices

The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p.

Resistance fluctuations in randomly diluted networks

The 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 ${p}_{c}$. 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 ${M}_{q}$(x,x') of R(x,x'). When averaged over configurations of clusters, M\ifmmode\bar\else\textasciimacron\fi{}(x,x')\ensuremath{\sim}\ensuremath{\Vert}x-x'${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{\psi}}}$${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}^{(\mathrm{q})}$. We construct low-concentration series for the generalized resistive susceptibility, ${\ensuremath{\chi}}^{(q)}$, associated with M${\ifmmode\bar\else\textasciimacron\fi{}}_{q}$, from which the critical exponents \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q) are obtained. We prove that \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q) is a convex monotonically decreasing function of q, which has the special values \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(0)=${D}_{B}$, \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(1)=\ensuremath{\zeta}${\ifmmode \tilde{}\else \~{}\fi{}}_{R}$, and \ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\infty})=1/\ensuremath{\nu}. ($D sub B--- is the fractal dimension of the backbone, \ensuremath{\zeta}${\ifmmode \tilde{}\else \~{}\fi{}}_{R}$ is the usual scaling exponent for the average resistance, and \ensuremath{\nu} is the correlation-length exponent.) Using the convexity property and the accepted values of these three exponents, we construct two approximant functions for \ensuremath{\psi}(q)=\ensuremath{\psi}\ifmmode \tilde{}\else \~{}\fi{}(q)\ensuremath{\nu}, both of which agree with the series results for all q&gt;1 and with existing numerical simulations. These approximants enabled us to obtain explicit approximate forms for the multifractal functions \ensuremath{\alpha}(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 ${\ensuremath{\chi}}^{(q)}$ diverges at a lower threshold ${p}^{\mathrm{*}}$(q), which vanishes as q\ensuremath{\rightarrow}-\ensuremath{\infty}.

Series analysis of randomly diluted nonlinear resistor networks.

The behavior of a randomly diluted network of nonlinear resistors, for each of which the voltage-current relationship is \ensuremath{\Vert}V\ensuremath{\Vert}=r\ensuremath{\Vert}I${\ensuremath{\Vert}}^{\ensuremath{\alpha}}$, 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 \ensuremath{\Vert}p-${p}_{c}$${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\zeta}(\ensuremath{\alpha})}$. The exponent \ensuremath{\zeta}(\ensuremath{\alpha}) was evaluated for 0&lt;\ensuremath{\alpha}&lt;\ensuremath{\infty} and d=2, 3, 4, 5, and 6, found to decrease monotonically from the exponent describing the minimal length, at \ensuremath{\alpha}=0, via that of the linear resistance, at \ensuremath{\alpha}=1, to the exponent characterizing the singly connected bonds, \ensuremath{\xi}(\ensuremath{\infty})\ensuremath{\equiv}1. Our results agree with known results for \ensuremath{\alpha}=0 and \ensuremath{\alpha}=1, also with recent results for general \ensuremath{\alpha} at d=6-\ensuremath{\epsilon} dimensions. The second moment 〈${R}^{2}$〉 was found to diverge as 〈R${〉}^{2}$ (for all \ensuremath{\alpha} and d), indicating a scaling form for the probability distribution of R.

Migdal-Kadanoff renormalization group for theO(n) model

We perform a Migdal-Kadanoff renormalization group calculation an O(n) symmetric model on a d-dimensional hypercubic lattice, d=2, 3. We find that in two dimensions the critical fixed point disappears as n=n /sub KT/ approx. = 1.96, which is in good agreement with the exact value n /sub KT/ =2. In three dimensions the fixed point persists much longer, albeit not all the way up to infinity. Surface critical phenomena in a semiinfinite O(n) model are also considered.

Ising-like quality of the Potts model

A projection operator which maps the q-state Potts model onto an Ising model is investigated. The case of an approximate mapping between two-spin nearest-neighbour clusters on a d-dimensional hypercubic lattice is considered. In the ferromagnetic case, first-order transitions occur for q>qc(d)=1+exp(2KIc(d))>or=2, where KIc(d) is the critical Ising coupling. Among the results are qc(d) to exp(2/(d-1)) as d to 1+, qc(2)=3.41 and qc(3)=2.56. The antiferromagnetic Potts model has no long range order for q>2.

Finite-size effects in the spherical model of ferromagnetism: Antiperiodic boundary conditions.

Explicit expressions are derived for the free energy, the specific heat, and the magnetic susceptibility of a spherical model of spins on a d-dimensional hypercubical lattice, of size ${L}_{1}$\ifmmode\times\else\texttimes\fi{}${L}_{2}$\ifmmode\times\else\texttimes\fi{}...\ifmmode\times\else\texttimes\fi{}${L}_{d}$, under antiperiodic boundary conditions. The relevant scaling functions that govern the critical behavior of the system are obtained and, with the use of the asymptotic properties of these functions, various predictions of the Privman-Fisher hypothesis [Phys. Rev. B 30, 322 (1984)] on the hyperuniversality of finite systems are verified. Approach towards standard critical behavior, both for T${T}_{c}$(\ensuremath{\infty}) and Tg${T}_{c}$(\ensuremath{\infty}), is examined. In the former case, the approach generally takes place through a power law; only in some special situations does one obtain an exponential law instead. In the latter case, the approach is generally exponential, except for the susceptibility of the system which (somewhat surprisingly) displays a finite-size effect dominantly determined by the surface-to-volume ratio of the lattice.

The vibrational spectrum of Eden clusters in many dimensions

The configuration-averaged vibrational spectrum of Eden clusters with N sites on a d-dimensional hypercubical lattice is determined exactly in the limit d>>log N>>1. It is shown that the spectral dimension of these clusters is 2.

Two-step renormalisation group approach for randomly diluted Ising models

A two-step renormalisation group approach-a Migdal-Kadanoff (or decimation) transformation (MK) followed by a mean-field renormalisation group (MFRG)-has been applied to random bond Ising models on d-dimensional hypercubic lattices. Calculations of the critical couplings and percolation concentrations for two and three dimensions show much improvement, even for small size clusters, compared with the results of separate MK or MFRG calculations. Critical exponents nu T and nu p are also improved in some cases. Estimates of critical curves in the concentration-temperature plane are also obtained for the two- and three-dimensional models.

Absence of self-diffusion in directed percolation lattices

Self-diffusion in directed percolation lattices is investigated theoretically. It becomes evident that perfect trapping of a particle by dead ends causes an exponential decay of a probability distribution function and vanishment of a self-diffusion coefficient except at p =1, where p is a percolation probability. The trapping rate in d-dimensional hypercubic lattices is calculated exactly and that on the backbone of percolation clusters in a square lattice is obtained on the basis of a real-space renormalization group method.

Self-avoiding walks in wedges

The authors consider the number of self-avoiding walks confined to a subset Zd(f) of the d-dimensional hypercubic lattice Zd, such that the coordinates (x1,x2, . . .,xd) of each vertex in the walk satisfy x1>or=0 and 0<or=xk<or=fk(x1) for k=2,3, . . .,d. They show that if fk(x) to infinity as x to infinity , the connective constant of walks in Zd(f) is identical to the convective constant of walks in Zd. They also explore conditions on fk which lead to a smaller connective constant for walks in Zd(f) and, in particular, consider walks between two parallel (d-1)-dimensional hyperplanes. Finally they contrast some of these results with recent work by Grimmett on percolation on subsets of the square lattice.

High-temperature expansion for a diluted spin-glass model

The high-temperature Edwards-Anderson susceptibility series is studied for a diluted Ising spin-glass model. The model has +J, -J and 0 nearest-neighbour bonds, with probabilities c/2, c/2, 1-c respectively, on a d-dimensional hypercubic lattice. For d>or approximately=5 there is a spin-glass phase for 1>or=c>or=cp(d) where cp(d) is the bond percolation concentration. For 2 cmax(d)>c>cp(d), but probably not at c=1.

Lattice statistics of branched polymers with specified topologies

The authors study the numbers of lattice animals with specified topologies. They prove that the growth constant for animals with c cycles and nk vertices of degree k (k=3, . . ., 2d), weakly embeddable in a d-dimensional hypercubic lattice, equal to mu , the growth constant for self-avoiding walks. For some particular topologies they derive upper and lower bounds for the corresponding critical exponents and estimate the values of these exponents by deriving and analysing new exact enumeration data. They conjecture that their previous result for trees with b branches (that the exponent is gamma +b-1, where gamma is the self-avoiding walk exponent) is also valid in the more general case in which the cyclomatic index (c) is non-zero; i.e. for b>or=1, the exponent does not depend on c. They show that their results are consistent with renormalisation group arguments that the universality class of branched polymers is independent of cycle and (non-zero) branching fugacities.

On the percolation threshold for a d-dimensional simple hypercubic lattice

The authors use 1/ sigma -expressions ( sigma =2d-1) to investigate systematically a number of conjectures and approximations for the bond and site percolation thresholds of a d-dimensional simple hypercubic lattice. The technique provides convincing evidence that none of the conjectures are correct and gives a quantitative understanding of the quality of the approximations.

Lattice trees with specified topologies

The authors study the total numbers of lattice trees with specified topologies. For strongly embeddable (or site) clusters with ni branching points of degree i, they show how to prove rigorously that the growth constants exist and are all equal to the neighbour-avoiding walk limit nu . They derive some exact upper bounds for the critical exponents associated with the 'star', 'comb' and 'brush' topologies. Exact enumeration data are derived and analysed for both weak and strong embeddings of some stars, combs and brushes on the square, triangular, simple cubic and d=4 simple hypercubic lattices. Using the exact enumeration data for the general d-dimensional simple hypercubic lattice and the exact results for the interior of a Bethe lattice, they derive expansions for the growth constants in inverse powers of the dimensionality. These results are consistent with the growth constants being equal to the appropriate walk limits ( mu or nu ).

Numerical study of self-avoiding loops on d-dimensional hypercubic lattices

The loop gas in d=2, 3, 4 and 5 dimensions and with multiplicities m=0, 1, 1.5, 2, 3 and 4 is investigated by the Monte Carlo method. The critical temperatures and approximate values for the critical exponent nu corresponding to second-order phase transitions are obtained for d=2 and 3.

Lattice trees with a restricted number of branch points

The authors investigate the asymptotic behaviour of the number of trees having n vertices, weakly embeddable in the d-dimensional hypercubic lattice, with restrictions on the number (n+) of vertices with degree greater than two. They show rigorously that if n+=o(n/log n) the growth constant is equal to the corresponding quantity for self-avoiding walks. For n+ increasing linearly with n they show that the growth constant still exists and present arguments indicating that it is strictly greater than that for self-avoiding walks.

Mean-field renormalisation group transformations for geometric phase transitions

It is shown that the mean-field renormalisation group technique of Indekeu et al. (1982) may be applied directly to the self-avoiding walk (SAW) and percolation problems (without reference to the n to 0 limit of interacting spin systems). Numerical results for the SAW and bond and site percolation (both directed and undirected) problems on the square lattice, obtained by using a variety of small cells, are reported. An isotropic transformation for the percolation problems on the d-dimensional hypercubic lattice is also discussed.