Ising SDRG Universality Class

Overview

The IsingSDRG universality class describes the critical behavior of the one-dimensional random transverse-field Ising model (RTFIM) at its infinite-randomness fixed point (IRFP). Rather than being characterized by a standard conformal field theory (CFT), the scaling behavior is governed by the Strong-Disorder Renormalization Group (SDRG) (or Ma-Dasgupta-Hu-Fisher procedure), where the ground state is an ensemble of random singlets (or random spin-pairs).

Symmetry: $\mathbb{Z}_2$ (spin-flip symmetry).

Models in this class: 1D random transverse-field Ising chain at critical tuning $[\ln J] = [\ln h]$.

Entanglement Entropy: The entanglement entropy scales logarithmically, but the prefactor is determined by the Refael-Moore effective central charge $c_{\text{eff}} = \ln(2)/2 \approx 0.34657$ rather than the clean CFT value $c = 1/2$.

Properties at the Infinite-Randomness Fixed Point (IRFP)

At the critical point of the random chain, the properties differ drastically from the clean Ising CFT:

Activated Dynamic Scaling: The characteristic energy scale $\Omega$ scales with length scale $L$ as $\ln \Omega \sim L^\psi$, with the critical exponent $\psi = 1/2$.
Effective Central Charge: The disorder-averaged entanglement entropy of a block of size $\ell$ scales as: $\overline{S(\ell)} = \frac{c_{\text{eff}}}{3} \ln \ell + \text{const}$ with $c_{\text{eff}} = \frac{\ln 2}{2} \approx 0.34657359$.
Correlation Lengths (two of them): Near criticality ($\delta \propto [\ln J] - [\ln h]$) the average correlation length is $\xi \sim \lvert\delta\rvert^{-\nu}$ with $\nu = 2$. The typical one is a different, smaller power, $\xi_{\text{typ}} \sim \xi^{1-\psi} \sim \lvert\delta\rvert^{-\nu_{\text{typ}}}$ with $\nu_{\text{typ}} = \nu(1-\psi) = 1$ — the correlation function is not self-averaging, so one length does not describe it.
Fluctuations: The entanglement entropy has non-vanishing sample-to-sample variance even in the thermodynamic limit.

QAtlas API

In QAtlas, the effective central charge of this universality class can be queried at the universality level:

using QAtlas

# Query Refael-Moore effective central charge
c_eff = QAtlas.fetch(Universality(:IsingSDRG), CentralCharge(); d=2)
# => 0.34657359027997264 (log(2.0)/2.0)

What this class deliberately does not answer

Every Calabrese–Cardy closed form — the finite-size chord, both Casimir quantities (ConformalCasimirEnergy, CasimirEnergyCorrection), Cardy's density of states, the thermal $\sinh$ form, the quench light-cone — raises an ErrorException for IsingSDRG, on every route:

QAtlas.fetch(Universality(:IsingSDRG), VonNeumannEntropy(), PBC(); ℓ=4.0, L=8.0)
# ERROR: ... the Calabrese-Cardy closed forms are consequences of conformal
#        invariance, and this universality class is not declared to be a 1+1D CFT ...

This is a refusal, not a gap in coverage: $c_{\text{eff}}$ is a logarithmic coefficient, which is strictly weaker than the conformal invariance those forms need, and the IRFP scales in an activated way ($\psi = 1/2$ above). Passing c explicitly does not route around it — what is refused is the formula, not its coefficient.

The $c_{\text{eff}}/3$ above is the two-cut coefficient ($\text{ncuts}\cdot c_{\text{eff}}/6$), i.e. a block with both edges in the bulk; one cut takes half.

Exponent table

The exact exponents, and the one quantity this class refuses, generated from the source:

AbstractQAtlas.fetch — Method

fetch(::Universality{:IsingSDRG}, ::SpatialDimension, ::Infinite) -> Int

1. The random transverse-field Ising CHAIN, so one spatial direction.

Not the d = 2 this same class takes for CriticalExponents: that one selects the 2D classical image's exponent table. Both are right and they are different numbers, which is why they now have different names.

source

AbstractQAtlas.fetch — Method

fetch(::Universality{:IsingSDRG}, ::ActivatedExponent) -> Rational{Int}

ψ = 1/2, from the random walk the strong-disorder RG maps the chain onto. The exponent of ln t_r ∼ ξ^ψ, which is what this fixed point has instead of a dynamical exponent — see the DynamicalExponent method below.

Reference: [2] Eq. (4.13), §4.1.3, which reaches it from the critical gap ε(δ=0, L) ∼ exp(−const·L^{1/2}), Eq. (4.12).

source

AbstractQAtlas.fetch — Method

fetch(::Universality{:IsingSDRG}, ::CriticalExponents; d::Int=2) -> NamedTuple

Exact exponents of the infinite-randomness fixed point of the 1D random transverse-field Ising chain.

field	value		stated at
`β`	`(3−√5)/2`	bulk magnetisation	Table 1
`x_m`	`(3−√5)/4`	its scaling dimension, `β/ν`	Table 1
`β_s`	`1`	surface magnetisation	Eq. (4.8)
`x_m_s`	`1/2`	its scaling dimension	Eq. (4.7)
`ν`	`2`	average correlation length, `ξ ∼	δ
`ν_typ`	`1`	typical correlation length — a different, smaller exponent	Eq. (4.10)
`ψ`	`1/2`	activated dynamic scaling, `ln t_r ∼ ξ^ψ`	Eq. (4.13)
`φ`	`(1+√5)/2`	cluster moment, `μ ∼	ln Ω

Everything but β and x_m is also stated in the running text, at the equation given; those two come from Table 1 alone.

Rational where the value is rational; β, x_m and φ are irrational.

α, γ, δ and η are not here. Table 1 does not carry them, and the usual scaling relations cannot supply them: those assume power-law dynamic scaling, which is exactly what this fixed point does not have.

Two different `d`

d = 2 is QAtlas's Euclidean convention (2D classical ≙ 1+1D quantum), matching fetch(Universality(:IsingSDRG), CentralCharge(); d=2). The d in AbstractQAtlas's infinite-randomness relations (ActivatedMomentGrowth, GriffithsSusceptibility, GriffithsSpecificHeat) is the spatial dimension of the chain, d = 1. Passing 2 there gives a wrong answer that does not look wrong.

source

AbstractQAtlas.fetch — Method

fetch(::Universality{:IsingSDRG}, ::DynamicalExponent)

Always throws. No finite z exists at an infinite-randomness fixed point: the gap closes as ln(1/Δ) ∼ ξ^ψ, so −d(ln Δ)/d(ln ξ) grows without bound rather than settling on a value.

This is a refusal, not a gap in the table. Returning some large z would be wrong in a way that reads as a measurement.

A finite z does exist off criticality: [2] Eq. (4.15), §4.1.3 fixes it as the positive root of [(J/h)^{1/z}]_av = 1, and Eq. (9.5), §9.1.2 has it diverging as |δ|^{−νψ} on approach — which is why nothing finite survives here. That root needs the disorder distributions, which the TFIM model does not carry, so it is not fetchable from QAtlas today.

source

References

D. S. Fisher, "Random transverse field Ising spin chains", Phys. Rev. Lett. 69, 534 (1992); "Critical behavior of random transverse-field Ising spin chains", Phys. Rev. B 51, 6411 (1995) — original SDRG solution and scaling theory.
G. Refael, J. E. Moore, "Entanglement entropy of random quantum critical points in one dimension", Phys. Rev. Lett. 93, 260602 (2004) — derivation of the effective central charge $c_{\text{eff}} = (\ln 2)/2$ at the random-singlet fixed point.
F. Iglói, C. Monthus, "Strong disorder RG approach of random systems", Phys. Rep. 412, 277 (2005), doi:10.1016/j.physrep.2005.02.006 — Table 1 (§4.1.2) collects the exponent set returned by CriticalExponents; §4.4.2 the Griffiths-phase singularities, §9.1.2 the typical-correlation and Griffiths-exponent relations for general $d$. Equation numbers cited in the source are those of the arXiv version, cond-mat/0502448.

Connections

Models: TFIM (with random couplings).
Clean counterpart: Ising — clean critical point maps to the $c=1/2$ Ising CFT.
Verification: Disordered Systems — tests verifying the random-singlet structures.