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arXiv:1309.7470 · Physical Review B

Critical exponents of the random field hierarchical model

Giorgio Parisi A, Jacopo Rocchi

Published Jul 17, 2014• 7 citations
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Abstract

We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, $J(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\rho}}$, and we can explore mean-field and non-mean-field behavior by changing $\ensuremath{\rho}$. We analyze the model at $T=0$ and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent $\ensuremath{\delta}$, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in $D$-dimensional short range models and with the ones of the straightforward one-dimensional long range model.

DOI
10.1103/physrevb.90.024203

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We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, J(r)\ensuremath∼r\ensuremath−\ensuremathρJ(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\rho}}J(r)\ensuremath∼r\ensuremath−\ensuremathρ, and we can explore mean-field and non-mean-field behavior by changing \ensuremathρ\ensuremath{\rho}\ensuremathρ. We analyze the model at T=0T=0T=0 and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent \ensuremathδ\ensuremath{\delta}\ensuremathδ, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in DDD-dimensional short range models and with the ones of the straightforward one-dimensional long range model.
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.
We study analytically the equilibrium properties of the spherical hierarchical model in the presence of random fields. The expression for the critical line separating a paramagnetic from a ferromagnetic phase is derived. The critical exponents characterising this phase transition are computed analytically and compared with those of the corresponding D -dimensional short-range model, leading to conclude that the usual mapping between one dimensional long-range models and D -dimensional short-range models holds exactly for this system, in contrast to models with Ising spins. Moreover, the critical exponents of the pure model and those of the random field model satisfy a relationship that mimics the dimensional reduction rule. The absence of a spin-glass phase is strongly supported by the local stability analysis of the replica symmetric saddle-point as well as by an independent computation of the free-energy using a renormalization-like approach. This latter result enlarges the class of random field models for which the spin-glass phase has been recently ruled out.
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 ννν between the two methods across the entire range of power-law decays 1/2<σ<11/2 < σ< 11/2<σ<1. We further benchmark our results against Monte Carlo simulations and analytical expansions near the upper boundary of the nontrivial regime, σ≲1σ\lesssim 1σ≲1.
We study the long-distance behavior of the O(N)O(N)O(N) model in the presence of random fields and random anisotropies correlated as $\ensuremath{\sim}1∕{x}^{d\ensuremath{-}\ensuremath{\sigma}}$ for large separation xxx using the functional renormalization group. We compute the fixed points and analyze their regions of stability within a double $\ensuremath{\epsilon}=d\ensuremath{-}4$ and $\ensuremath{\sigma}$ expansion. We find that the long-range disorder correlator remains analytic but generates short-range disorder whose correlator develops the usual cusp. This allows us to obtain the phase diagrams in $(d,\ensuremath{\sigma},N)$ parameter space and compute the critical exponents to first order in $\ensuremath{\epsilon}$ and $\ensuremath{\sigma}$. We show that the standard renormalization group methods with a finite number of couplings used in previous studies of systems with long-range correlated random fields fail to capture all critical properties. We argue that our results may be relevant to the behavior of $^{3}\mathrm{He}\text{\ensuremath{-}}\mathrm{A}$ in aerogel.
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 .
Random-field models are considered with long-range exchange, J(r) approximately 1/r d+ sigma , and/or long-range correlated random fields, (h(r)h(r')) approximately 1/ mod r-r' mod d- rho . A general scaling analysis is presented, in which the existence of three independent critical exponents is emphasized. Four different universality classes are identified, according to whether the fixed point controlling the critical behaviour is of short-range or long-range character for the exchange and/or the random-field correlations. The lines in the rho - sigma plane at which the long-range couplings first become relevant are expressed in terms of the exponents eta and eta of the corresponding short-range problem: long-range exchange is relevant for sigma <2- eta , long-range field correlations for rho >2 eta - eta . The scaling results are confirmed and amplified by explicit calculations of three independent exponents of the n-component spin model, correct to order 1/n. The results reveal errors in published expansions around the upper critical dimension.