Critical Renormalisation of Disorder Strength

• Critical Renormalisation of Disorder Strength is a framework that uses strong disorder renormalization group methods to show how quenched randomness and infinite disorder fixed points control universal scaling in quantum systems.
• The SDRG methodology iteratively decimates the strongest bonds and fields, yielding precise scaling of observables such as magnetization, correlation functions, and energy gaps.
• This framework demonstrates universal behavior across various disorder types, with implications for quantum phase transitions, Griffiths phases, and dynamics in diverse lattice architectures.

Critical Renormalisation of Disorder Strength is a framework within statistical mechanics and quantum many-body theory that addresses how quenched randomness, when subject to renormalization group (RG) transformations, fundamentally alters universal critical properties of physical systems. This concept is particularly central to disordered quantum magnets, such as the random transverse-field Ising (RTFI) model, where the disorder strength not only flows to non-trivial fixed points but also dictates correlation scaling, universality classes, and dynamical responses in both quantum and stochastic systems with discrete symmetry.

1. Infinite Disorder Fixed Points in Quantum Systems

Critical renormalization of disorder strength arises most prominently in systems governed by Infinite Disorder Fixed Points (IDFPs). In the RTFI model, the strong disorder renormalization group (SDRG) flow produces distributions of couplings and transverse fields whose variance diverges without bound; this is the infinite disorder scenario. Here, quantum fluctuations are suppressed by disorder fluctuations, rendering perturbative SDRG steps asymptotically exact. At the IDFP, rare region effects dominate all aspects of criticality, resulting in scaling forms and critical exponents that are universal and independent of microscopic disorder details.

Table: Key Critical Exponents at the 2D RTFI IDFP

Exponent	Symbol	Value (2D RTFI)	Scaling Relation
Correlation length	$\nu$	$1.24(2)$	$\xi \sim |\delta|^{-\nu}$, $\delta = \theta - \theta_c$
Magnetization	$x$	$0.982(15)$	$m \sim L^{-x}$; $\tilde{G}(r) \sim r^{-2x} e^{-r/\xi}$
Tunneling/Energy	$\psi$	$0.48(2)$	$\ln\left(\Omega_0/\Omega\right) \sim L^{\psi}$

As a consequence, physical observables such as magnetization, correlations, and energy gap statistics obey scaling forms strictly determined by the IDFP (1005.4740). This extends beyond one dimension: efficient SDRG implementations have shown that such infinite disorder scaling controls random quantum phase transitions on 2D and 3D lattices, as well as on infinite-dimensional Erdős–Rényi graphs (1010.2344).

2. SDRG Methods and Algorithmic Innovations

The implementation of critical renormalization depends on the iterative SDRG scheme, where at each step, the strongest local term (either bond $J_{ij}$ or field $h_i$) is decimated:

• Bond Decimation: Merging spins into effective clusters; update effective fields as $\tilde{h} = h_i h_j / J_{ij}$
• Field Decimation: Eliminate site; generate new bonds between remaining neighbors as $\tilde{J}_{jk} = J_{ji} J_{ik}/h_i$

Efficient algorithms exploit “local maxima” selection trees and filtering procedures to minimize irrelevant renormalizations near criticality, yielding $\mathcal{O}(L^2 \ln L)$ complexity for large 2D samples (1005.4740). In higher-$d$, further algorithmic advances based on selection theorems allow scaling to up to $10^6$ sites (1010.2344).

SDRG methods are robust to major procedural changes. For example, even when the decimation order is fixed a priori (as in Boundary SDRG), all universal features and exponents of the critical point are preserved (Monthus et al., 2012). This underscores that the flow of disorder, not the details of the RG trajectory, sets the critical physics.

3. Scaling, Universality, and Disorder Types

A core outcome is the demonstration of universality across disorder types. RTFI models with “box-$h$” (uniformly distributed) and “fixed-$h$” (deterministic) disorder converge to the same IDFP, sharing exponents $\nu$, $x$, and $\psi$, with their finite-size scaling functions also collapsing (1005.4740, 1010.2344). The implications are:

• The critical behavior is universal for all sufficiently disordered initial conditions (microscopics are irrelevant).
• The shift and width of pseudo-critical point distributions are governed by a single exponent $\nu$.
• Infinite disorder scaling persists even for coordination-randomized infinite-dimensional graphs and complex topologies.

A plausible implication is that universal infinite disorder scaling likely controls a wide class of discrete order parameter systems subject to quenched randomness, including stochastic models and classical analogs with equivalent RG structures.

4. Griffiths Phases and Dynamical Scaling

Disorder strength renormalization exerts comprehensive control over dynamical and spatial correlations both at and near criticality:

• Magnetization clusters: Effective cluster mass $\bar{\mu} \sim L^{d_f}$, $m \sim L^{-x}$
• Dynamical scaling: For the energy gap $\epsilon_L$, at criticality

$$\ln \left(\frac{\Omega_0}{\Omega}\right) \sim L^\psi \, , \quad \tilde{\gamma} = (\gamma_L-\gamma_0)L^{-\psi}$$

• Griffiths Phases:
  • Disordered phase ($\delta>0$): $\epsilon_L \sim L^{-z}$, $z$ varies continuously, and energy gap distributions shift linearly with $\ln L$.
  • Ordered phase ($\delta<0$): $\ln \epsilon_L \sim -\ln^{1/d}(L)$, with gap distributions scaling as $$\tilde{\gamma} = \gamma_L - A \ln^{1/d}(L) - \gamma_0$$ (1005.4740).

These phenomena are directly linked to rare-event statistics, with rare spatial regions controlling the asymptotic scaling, reflected in broad, non-Gaussian energy and magnetization distributions. The strong-disorder RG flow ensures that the dynamical exponent and scaling forms are dictated by the anomalously slow relaxation channels engendered by extreme local disorder fluctuation.

5. Correlated Disorder, Harris Criterion, and Exceptions

While uncorrelated disorder generically destabilizes clean critical points in violation of the Harris criterion ($d\nu < 2$ implies instability), locally correlated disorder can stabilize clean fixed points even in defiance of this rule (1011.0182). For example, when disorder is tuned such that $h_i = \exp(\delta) J_i$ (i.e., no local random-mass term), clean criticality is preserved for weak disorder—there are no Griffiths phases, and entanglement entropy remains at its clean value until a new line of fixed points is entered for larger disorder. The dynamical exponent $z$ increases continuously with disorder strength above this threshold. This demonstrates that spatial correlations in disorder can critically alter the RG flow, challenging naive expectations from the Harris criterion.

6. Generalizations and Fractal/Long-range/Correlated Structures

Critical renormalization holds in lattices of varying geometry (standard, fractal, hierarchical), with the SDRG framework adaptable to fixed cell-size RG schemes (Monthus et al., 2012). Critical exponents such as the activated exponent $\psi$, correlation exponents $\nu_{\mathrm{typ}}$, $\nu_{FS}$, and scaling forms for renormalized couplings ($\ln J_L^{\mathrm{typ}} \sim -L^\psi$) are reproduced, with rare event effects setting essential singularities in gap and magnetization distributions. The qualitative behavior—such as the similarity of RG flow on the Sierpinski gasket to one-dimensional systems, or additive surface terms in hierarchical lattices with increasing connectivity—illustrates the versatility of the infinite disorder critical point concept.

In systems with long-range correlated disorder, the effective (renormalized) disorder strength $W_\mathrm{eff} = \sqrt{S(2k)} \, W$ (with $S(2k)$ the spectral weight at twice the band momentum) determines the localization length scaling and critical exponents (Petersen et al., 2012). Here, both the critical exponent $\nu$ and fractal dimension $D$ of wavefunctions become correlation-dependent, in line with the extended Harris criterion for correlated disorder.

7. Implications and Outlook

The analysis of critical renormalization of disorder strength in quantum and stochastic models—exemplified by the 2D RTFI model—indicates that the universality class of the disordered critical point, its exponents, and scaling forms are governed by the RG flow of the entire disorder distribution, not by microscopic disorder details. SDRG-based methods have established that the infinite disorder scenario controls the phase transition in a wide parameter range, with scaling dictated by the broadening of disorder distributions rather than quantum or thermal fluctuations.

Practically, these findings enable precise extraction of exponents in large-scale simulations (1005.4740, 1010.2344), inform experimental searches for quantum Griffiths phases, and provide a theoretical framework for interpreting glassy and activated dynamics in a variety of disordered materials. The stability of IDFPs under RG flow, robustness to algorithmic variations, and universality across disorder types confirm that the critical renormalization of disorder strength is a central organizing principle in the theory of disordered quantum and stochastic systems.