Strong Disorder Renormalization Group

• SDRG is a real-space method that decimates dominant couplings to capture low-energy physics in strongly disordered quantum and classical systems.
• It employs iterative perturbative renormalization to derive universal scaling laws for observables like susceptibility and entanglement entropy in both short- and long-range models.
• SDRG extends to finite temperatures and excited states, offering insights into quantum criticality and the effects of extreme randomness in spin chains.

The strong disorder renormalization group (SDRG) is a real-space renormalization technique that asymptotically captures the low-energy, large-scale physics of quantum and classical disordered systems in regimes of extreme randomness. It achieves this by recursively identifying and perturbatively "decimating" the highest local energy scales (such as couplings or fields) and generating effective interactions among the surviving degrees of freedom. SDRG is asymptotically exact at infinite-randomness fixed points in one dimension and remains remarkably accurate in a broad class of higher-dimensional and long-range-interacting systems. In particular, recent work extends SDRG to bond-disordered antiferromagnetic quantum spin chains with power-law long-range interactions, allowing the analysis of excited states, finite-temperature properties, and universal observables such as susceptibility and entanglement (Kettemann, 22 Sep 2025).

1. Model Classes and Hamiltonians

SDRG has been systematically applied to disordered quantum spin chains with both short-range and long-range couplings. The canonical short-range model is the random-bond XX chain: $$H = \sum_i J_{i,i+1} \left( S^x_i S^x_{i+1} + S^y_i S^y_{i+1} \right),$$ where each coupling $J_{i,i+1}>0$ is drawn from a broad probability distribution $P(J)$. In the long-range case, all pairs can interact with power-law decaying couplings: $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$ with $r_i$ denoting site positions, possibly a random configuration to induce bond disorder (Kettemann, 22 Sep 2025).

Additional extensions include models with arbitrary anisotropy (XXZ, Heisenberg), arbitrary spin $S$, and higher-dimensional analogs, as well as generalizations to classical systems (e.g., contact processes, random elastic networks) (Iglói et al., 2018, Monthus et al., 2010, Juhász, 2023).

2. SDRG Decimation Rules and Algorithm

SDRG operates iteratively at each step using the following key protocol:

1. Selection of the maximal scale: At energy scale $\Omega$, find the largest surviving coupling,

$\Omega = \max_{i<j} |J_{ij}|.$

1. Decimation: The pair $(i,j)$ corresponding to $\Omega$ forms a singlet (in the $J_{i,i+1}>0$0 XX model), and is removed from the active set.
2. Renormalization of the remaining couplings: New effective couplings are generated via second-order perturbation theory,

$J_{i,i+1}>0$1

which, in the nearest-neighbor case, reduces to $J_{i,i+1}>0$2 for adjacent $J_{i,i+1}>0$3.

1. Sign rules: In the ground-state SDRG, all couplings retain positive sign for the XX chain. In the excited-state or finite-temperature variants ("SDRG-X"), negative signs may arise, determined by thermal or state occupations,

$J_{i,i+1}>0$4

The occupation probabilities $J_{i,i+1}>0$5 follow from the Boltzmann weights over the four two-site eigenstates (Kettemann, 22 Sep 2025).

2. Iteration: The process repeats, with renormalized couplings, until all degrees of freedom are paired.

This procedure underlies both zero-temperature ground-state analysis and excited-state/finite-temperature extensions (SDRG-X) (Kettemann, 22 Sep 2025). For long-range models, all pairwise couplings are updated at each step, not just near-neighbors. The master equation for the joint distribution of coupling magnitudes and signs encodes the SDRG flow.

3. Universal Fixed Points and Master Equations

SDRG flows the system to universal fixed point distributions:

• Short-range XX chain (infinite randomness fixed point, IRFP):

$J_{i,i+1}>0$6

In terms of logarithmic variables $J_{i,i+1}>0$7, this reads $J_{i,i+1}>0$8. As $J_{i,i+1}>0$9, $P(J)$0, resulting in extremely broad distributions and the random-singlet ground state. All remaining energy scales become widely separated (Kettemann, 22 Sep 2025).

• Long-range chain ($P(J)$1): The fixed point is of finite width,

$P(J)$2

Here, the width saturates at $P(J)$3, with small corrections for $P(J)$4 (Kettemann, 22 Sep 2025).

The exact master equation for the (sign-labeled) joint distribution $P(J)$5 at finite temperature is of the form: $P(J)$6 with creation terms reflecting all possible pairings and thermal occupations, and where sign randomness is introduced at $P(J)$7 (Kettemann, 22 Sep 2025).

4. Finite Temperature and Excited-State SDRG

The SDRG-X extends the renormalization procedure to finite energy-density states and finite temperature ensembles. At each decimation, the occupation of the four two-spin eigenstates is sampled according to their Boltzmann weight: $P(J)$8 For $P(J)$9, the population of unentangled triplet states grows, increasing the number of negative couplings under RG. The amplitude distribution, however, remains at the fixed-point form of the corresponding SDRG (IRFP or finite width), while the sign distribution evolves, interpolating from fully positive at low $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$0 to equally weighted at high $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$1 (Kettemann, 22 Sep 2025).

The master equation for sign-resolved distributions involves convolutions over all possible parent coupling signs and decimation outcomes, allowing for quantitative calculation of thermodynamic observables at any temperature.

5. Physical Observables: Susceptibility, Entanglement, Concurrence

SDRG provides analytic predictions for universal thermodynamic and quantum information observables:

Magnetic susceptibility $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$2:

• Short-range:

$$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$3

• Long-range ($$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$4):

$$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$5

Entanglement entropy $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$6:

In the random-singlet phase, singlets crossing a partition of length $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$7 each contribute $$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$8, yielding

$$H = \sum_{i<j} J_{ij} ( S^x_i S^x_j + S^y_i S^y_j ),\qquad J_{ij} = J_0\,|r_i - r_j|^{-\alpha},\quad \alpha > 0,$$9

reminiscent of a logarithmic scaling with effective central charge $r_i$0.

Concurrence $r_i$1:

Average two-site entanglement at separation $r_i$2 is given by the singlet probability,

$r_i$3

At finite $r_i$4, only thermal occupations of singlet and entangled triplet bonds contribute to distributed entanglement (Kettemann, 22 Sep 2025).

6. Regimes of Validity and Limitations

The SDRG method and the above universality results are robust for short-range chains and for long-range chains with $r_i$5. For $r_i$6, the simple two-spin decimation scheme breaks down: higher-order clusters and nontrivial multi-spin correlations proliferate, and explicit treatment of generated pseudo-spin couplings is necessary. Furthermore, SDRG as formulated addresses the XX or bond-disordered models; the inclusion of transverse fields, non-bipartite disorder, or more general Hamiltonians can require additional extensions beyond the canonical protocol (Kettemann, 22 Sep 2025).

Finite temperature and finite-energy SDRG extend the applicability to excited states, but the amplitude laws remain essentially unchanged, while sign randomness and correlation functions acquire nontrivial corrections.

7. Physical Insights and Broader Context

SDRG for strongly disordered spin chains yields the following physical pictures:

• Short-range random XX/XXX chains: Flow to the infinite-randomness fixed point, random-singlet ground state, universal low-temperature thermodynamics, and ultra-broad coupling distributions.
• Long-range random chains ($r_i$7): Exhibit a finite-width random-singlet phase with universal but $r_i$8-dependent thermodynamic scaling, notably new power laws in $r_i$9.
• Finite-$S$0 and excited-state SDRG: Thermally generated negative bonds and broad sign distributions, yet essentially unchanged amplitude distributions.

The SDRG framework unifies the understanding of strong randomness phases and transitions in both quantum and classical disordered systems, and forms the basis for analytic and numerical studies of random spin chains, quantum criticality, Griffiths phases, entanglement scaling, and quantum information propagation (Kettemann, 22 Sep 2025, Iglói et al., 2018).

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For further technical derivations and extensions, see the foundational works of Fisher (Phys. Rev. B 50, 3799), Pekker et al. (Phys. Rev. X 4, 011052), and recent developments in long-range interacting chain analysis (Kettemann, 22 Sep 2025, Kettemann, 13 Jan 2025, Ustyuzhanin et al., 5 Mar 2026).