For pt.II see ibid., vol.11, p.375 (1978). Exact renormalisation group equations (RGE) are obtained for the d-dimensional n-vector and spin-S Heisenberg versions of Dyson's hierarchical model. In the quantum mechanical case the fixed point and neighbourhood properties of the RGE are shown to be independent of S indicating universality of critical behaviour in spin magnitude. Expansions for critical exponents, analogous to the usual epsilon =4-d expansions, are obtained and the RGE solved numerically to obtain relevant scaling indices. The n to infinity or spherical limit is also discussed and 1/n corrections to critical exponents are calculated to first order in 1/n.
The goal of this thesis is to provide a practical method to calculate, in scalar field theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric and broken phases to perform high precision calculations of the Green's functions at zero momentum. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting β reach a critical value βc. We show that the round-off errors on the magnetic susceptibility grow like (βc − β) −1 near criticality. We show that the systematic errors (finite truncation and volume) can be controlled with an exponential precision and reduced to a level lower than numerical errors. We probe the numerical errors made in Renormalization Group (RG) calculations by varying slightly the resealing factor of the fields and resealing back in order to get the same (if there were no round-off errors) magnetic susceptibility. We find a significant departure from Gaussian behavior. In addition, the average value differs from the exact answer by a quantity which is of the same order as the standard deviation. We calculate the critical exponents γ and the subleading exponent Δ of Dyson's model by direct fits of the zero-momentum two-point function calculated with an Ising and a Landau-Ginzburg measure and by linearization about the non-trivial fixed point for O(N) model up to N = 5. We extend the method to the low temperature phase and show that accurate values of the connected one-, two- and three- point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We show that the leading exponents obey the scaling and hyperscaling relations. We discuss the concept of universality in terms of critical exponents, correction amplitudes and the universal coupling constant for different Landau-Ginzburg measures. We compare our accurate numerical results with those obtained from perturbative methods.
The random hierarchical model is defined by introducing random couplings between spin blocks in Dyson's original hierarchical model for continuous spins. The spin-distribution functions are now site dependent and functions of random parameters that we take to be a quadratic and quartic potential,
rl(m) and
Ul(m), respectively, for the mth spin at the
l level of the hierarchy. Renormalization-group recursion relations for the cumulant averages are derived in an expansion in ${\ensuremath{\epsilon}}^{\ensuremath{'}}=(2\ensuremath{\sigma}\ensuremath{-}1)\mathrm{ln}(2)$, where $\ensuremath{\sigma}$ is a parameter of the model and a measure of the interaction range. We find that only
rl=〈rl(m)〉, ${\ensuremath{\Delta}}_{l}=〈{[r_{l}^{}{}_{}{}^{(m)}\ensuremath{-}{r}_{l}]}^{2}〉$, and
Ul=〈Ul(m)〉 are relevant physical quantities within the ${\ensuremath{\epsilon}}^{\ensuremath{'}}$ expansion. By solving the equations for
Ul and ${\ensuremath{\Delta}}_{l}$ to $O({\ensuremath{\epsilon}}^{\ensuremath{'}})$ we obtain in general four fixed points of the recursion relations. For $\ensuremath{\sigma}>\frac{1}{2}$, there is no stable fixed point in the physical region $\ensuremath{\Delta}>~0$,
U> 0; thus the behavior of the system is still open to study. In the particular case of uniform block coupling but random spin distributions, we find a fixed point with ${U}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$, ${\ensuremath{\Delta}}^{*}\ensuremath{\approx}O({|{\ensuremath{\epsilon}}^{\ensuremath{'}}|}^{\frac{1}{2}})$ for ${\ensuremath{\epsilon}}^{\ensuremath{'}}<0$, but it is unstable.
The critical properties of a random Ising model with long-range isotropic interactions decaying as R -(d+ sigma ) are analysed by using renormalisation group methods in an expansion in epsilon '=2 sigma -d, where d is dimensionality. For epsilon '>O the critical behaviour is described by a stable fixed point O( epsilon 1 2/') in the physical region of parameter space. The crossover to short-range behaviour is analysed and takes place when sigma =2+ epsilon /106, epsilon =4-d>O. There is then a small region O< sigma -2<O( epsilon ) where the system is still dominated by long-range behaviour. The results for d=1 are compared with those obtained previously for the random hierarchical model and it is found that the two models do not show the same critical behaviour in the epsilon ' expansion.
We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h . The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h . In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L − z with a finite dynamical exponent z . The RG rules are applied numerically to chains containing L = 2 12 = 4096 spins in order to measure these critical exponents for various values of σ in the region .
Abstract We study the ground state entanglement entropy of the quantum Dyson hierarchical spin chain in which the interaction decays algebraically with the distance as . We exploit the real-space renormalisation group solution which gives the ground-state wave function in the form of a tree tensor network and provides a manageable recursive expression for the reduced density matrix of the renormalised ground state. Surprisingly, we find that at criticality the entanglement entropy obeys an area law, as opposite to the logarithmic scaling of short-range critical systems and of other non-hierarchical long-range models. We provide also some analytical results in the limit of large and small that are tested against the numerical solution of the recursive equations.