We consider the stochastic dynamics of the pure and random ferromagnetic Ising model on the hierarchical diamond lattice of branching ratio
K with fractal dimension
df=(ln(2K))/ln2. We adapt the Real Space Renormalization procedure introduced in our previous work [C. Monthus and T. Garel, J. Stat. Mech. P02037 (2013)] to study the equilibrium time
teq(L) as a function of the system size
L near zero-temperature. For the pure Ising model, we obtain the behavior $t_{eq}(L) \sim L^α e^{β2J L^{d_s}} $ where
ds=df−1 is the interface dimension, and we compute the prefactor exponent
α. For the random ferromagnetic Ising model, we derive the renormalization rules for dynamical barriers
Beq(L)≡(lnteq/β) near zero temperature. For the fractal dimension
df=2, we obtain that the dynamical barrier scales as $ B_{eq}(L)= c L+L^{1/2} u$ where
u is a Gaussian random variable of non-zero-mean. While the non-random term scaling as
L corresponds to the energy-cost of the creation of a system-size domain-wall, the fluctuation part scaling as
L1/2 characterizes the barriers for the motion of the system-size domain-wall after its creation. This scaling corresponds to the dynamical exponent
ψ=1/2, in agreement with the conjecture
ψ=ds/2 proposed in [C. Monthus and T. Garel, J. Phys. A 41, 115002 (2008)]. In particular, it is clearly different from the droplet exponent
θ≃0.299 involved in the statics of the random ferromagnet on the same lattice.
We consider the stochastic dynamics of Ising ferromagnets (either pure or random) near zero temperature. The master equation satisfying detailed balance can be mapped onto a quantum Hamiltonian which has an exact zero-energy ground state representing the thermal equilibrium. The largest relaxation time
teq governing the convergence towards this Boltzmann equilibrium in finite-size systems is determined by the lowest non-vanishing eigenvalue
E1=1/teq of the quantum Hamiltonian
H. We introduce and study a real-space renormalization procedure for the quantum Hamiltonian associated to the single-spin-flip dynamics of Ising ferromagnets near zero temperature. We solve explicitly the renormalization flow for two cases. (i) For the one-dimensional random ferromagnetic chain with free boundary conditions, the largest relaxation time
teq can be expressed in terms of the set of random couplings for various choices of the dynamical transition rates. The validity of these RG results in
d=1 is checked by comparison with another approach. (ii) For the pure Ising model on a Cayley tree of branching ratio
K, we compute the exponential growth of
teq(N) with the number
N of generations.
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 .
The Dyson hierarchical version of the quantum Ising chain with Long-Ranged power-law ferromagnetic couplings J ( r ) ∝ r −1− σ and pure or random transverse fields is studied via real-space renormalization. For the pure case, the critical exponents are explicitly obtained as a function of the parameter σ , and are compared with previous results of other approaches. For the random case, the RG rules are numerically applied and the critical behaviors are compared with previous strong disorder renormalization results.
For the one-dimensional long-ranged Ising spin-glass with random couplings decaying with the distance r as J ( r ) ∼ r − σ and distributed with the Lévy symmetric stable distribution of index 1 < μ ≤ 2 (including the usual Gaussian case μ = 2), we consider the region σ > 1/ μ where the energy is extensive. We study two real space renormalization procedures at zero temperature, namely a simple box decimation that leads to explicit calculations, and a strong disorder decimation that can be studied numerically on large sizes. The droplet exponent governing the scaling of the renormalized couplings J L ∝ L θ μ ( σ ) is found to be whenever the long-ranged couplings are relevant θ μ ( σ ) ≥ −1. For the statistics of the ground state energy over disordered samples, we obtain that the droplet exponent θ μ ( σ ) governs the leading correction to extensivity of the averaged value . The characteristic scale of the fluctuations around this average is of order , and the rescaled variable is Gaussian distributed for μ = 2, or displays the negative power-law tail in 1/(− u ) 1+ μ for u → − ∞ in the Lévy case 1 < μ < 2. Finally we apply the zero-temperature renormalization procedure to the related Dyson hierarchical spin-glass model where the same droplet exponent appears.
We investigate the critical behavior of the one-dimensional Ising model with long-range interactions using the functional renormalization group in the local potential approximation (LPA), and compare our findings with Dyson's hierarchical model (DHM). While the DHM lacks translational invariance, it admits a field-theoretical description closely resembling the LPA, up to minor but nontrivial differences. After reviewing the real-space renormalization group approach to the DHM, we demonstrate a remarkable agreement in the critical exponent
ν between the two methods across the entire range of power-law decays
1/2<σ<1. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime,
σ≲1.