Disordered Systems
Overview
Disordered quantum systems test QAtlas's ED and entanglement infrastructure in a regime where exact analytical solutions are unavailable for individual disorder realisations. The physics is governed by disorder-averaged quantities and universal properties at infinite-randomness fixed points (IRFPs).
Two disordered models are currently implemented:
- Random-bond Heisenberg chain –- verifies ground-state properties (singlet, positive entanglement) for random couplings.
- Random transverse-field Ising model –- probes the Fisher IRFP and its entanglement signatures.
Random-Bond Heisenberg Chain
Setup
The Hamiltonian is
\[H = \sum_{i=1}^{N-1} J_i\, \mathbf{S}_i \cdot \mathbf{S}_{i+1}\]
where the couplings $J_i > 0$ are drawn independently from a random distribution (e.g., uniform on $[0, 1]$ or log-uniform). OBC is used.
Verified Properties
| Property | Expected | Physical reason |
|---|---|---|
| Ground state is a singlet ($S_{\mathrm{tot}} = 0$) | Yes, for any $\{J_i > 0\}$ | Marshall sign rule: antiferromagnetic ground state is always a singlet |
| Entanglement entropy $S(l) > 0$ for $1 \leq l \leq N-1$ | Yes | Ground state is entangled across any bipartition |
| $S(l)$ varies with disorder realisation | Yes | Non-universal, sample-dependent |
These properties hold for every disorder realisation, not just on average. They serve as sanity checks that the ED machinery (Hamiltonian construction, diagonalization, entropy computation) works correctly for non-translationally-invariant systems.
Random Transverse-Field Ising Model
Setup
The Hamiltonian is
\[H = -\sum_{i=1}^{N-1} J_i\, \sigma_i^z \sigma_{i+1}^z - \sum_{i=1}^{N} h_i\, \sigma_i^x\]
where both the couplings $J_i$ and the transverse fields $h_i$ are drawn from random distributions.
Fisher Infinite-Randomness Fixed Point (IRFP)
The quantum phase transition between the ferromagnetic ($J \gg h$) and paramagnetic ($h \gg J$) phases is controlled by the Fisher IRFP when the distributions of $\ln J$ and $\ln h$ have the same mean:
\[[\ln J] = [\ln h]\]
where $[\cdot]$ denotes the disorder average. At the IRFP, the system is described by the strong-disorder renormalization group (SDRG, or Ma-Dasgupta-Hu-Fisher procedure): the strongest local coupling is decimated at each step, generating an effective random-singlet ground state.
Properties at the IRFP
| Property | Behaviour | Reference |
|---|---|---|
| Typical correlation length | $\ln \xi \sim \lvert\delta\rvert^{-1}$ (activated scaling, not power-law) | Fisher (1995) |
| Average entanglement entropy | $\overline{S(l)} = \frac{c_{\mathrm{eff}}}{3}\ln l + \text{const}$ with $c_{\mathrm{eff}} = \frac{\ln 2}{2} \cdot \frac{1}{1} \approx 0.347\ldots$ | Refael-Moore (2004) |
| Sample-to-sample fluctuations | $\mathrm{Var}[S(l)]$ does not vanish as $N \to \infty$ | Characteristic of infinite-randomness |
| Ground-state entanglement | $S(l) > 0$ for critical samples | Random-singlet structure |
What QAtlas Currently Tests
For small system sizes ($N \leq 14$), QAtlas verifies:
Critical entanglement is positive: at the IRFP tuning $[\ln J] = [\ln h]$, the ground-state entanglement entropy is non-zero for generic bipartitions, confirming the random-singlet structure.
Sample-to-sample fluctuations: repeating the calculation for multiple disorder realisations produces a spread in $S(l)$, consistent with the infinite-randomness nature of the fixed point.
Gapped phases: deep in the paramagnetic phase ($h_i \gg J_i$), $S(l) \to 0$ (product state), confirming the area law.
Future: $c_{\mathrm{eff}}$ Extraction
Extracting the effective central charge $c_{\mathrm{eff}} = \ln 2 / 2 \approx 0.347$ requires disorder-averaging $S(l)$ over many samples at the IRFP and fitting the average to $\overline{S} \propto \ln l$. This requires larger system sizes and more samples than currently implemented. It is planned as a future verification target.
QAtlas Test
# test/verification/test_disordered_heisenberg.jl
# Random-bond Heisenberg: singlet ground state
E, psi = eigen(H_random_heisenberg)
@test is_singlet(psi[:, 1])
@test all(S_l .> 0)
# Random TFIM at IRFP: positive entanglement
E, psi = eigen(H_random_tfim_irfp)
@test S_half > 0 # entanglement at midpointReferences
- 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) –- SDRG and Fisher IRFP.
- S. K. Ma, C. Dasgupta, C.-K. Hu, "Random antiferromagnetic chain", Phys. Rev. Lett. 43, 1434 (1979) –- original SDRG procedure.
- G. Refael, J. E. Moore, "Entanglement entropy of random quantum critical points in one dimension", Phys. Rev. Lett. 93, 260602 (2004) –- $c_{\mathrm{eff}} = (\ln 2)/2$ at the random-singlet fixed point.
- N. Laflorencie, "Scaling of entanglement entropy in the random singlet phase", Phys. Rev. B 72, 140408(R) (2005).
Connections
- Entanglement entropy: entanglement verification for the clean (non-disordered) case.
- TFIM: the clean model is described in TFIM; disorder adds random $J_i, h_i$.
- Heisenberg: the clean chain is described in Heisenberg; disorder adds random $J_i$.
- Calabrese-Cardy: the clean formula does not apply at the IRFP; the effective central charge $c_{\mathrm{eff}}$ replaces $c$.