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Quantifying and improving the accuracy of the Matrix Product Operator Renormalization Group of random spin chains

Christophe Chatelain

Published Jul 19, 2018• 1 citation
OpenAlex

Abstract

A relatively simple real-space renormalization procedure, due to Ma and Dasgupta and known as Strong-Disorder Renormalization Group (SDRG), provides in principle exact results on the critical properties at the infinite-randomness fixed point of random spin chains. Numerically, SDRG can be efficiently implemented as a renormalization of Matrix Product Operators (MPO-RG). By considering larger blocks than SDRG, MPO-RG was recently used to compute non-critical quantities that are inaccessible to SDRG. In this work, we study the accuracy of the approach and propose two simple and fast improvements. We observe that the accuracy on the ground state energy is improved by a factor at least equal to 4.

Related papers

Improved matrix product operator renormalization group: application to the N -color random Ashkin–Teller chain

Christophe Chatelain

Abstract Strong-disorder renormalization group (SDRG), despite being a relatively simple real-space renormalization procedure, provides in principle exact results on the critical properties at the infinite-randomness fixed point of random quantum spin chains. Numerically, SDRG can be efficiently implemented as a renormalization of matrix product operators (MPO-RG). By considering larger blocks than SDRG, MPO-RG was recently used to compute non-critical quantities of finite chains that are inaccessible to SDRG. In this work, the accuracy of this approach is studied and two simple and fast improvements are proposed. The accuracy on the ground state energy is improved by a factor at least equal to four for the random Ising chain in a transverse field. Finally, the proposed algorithms are shown to yield Binder cumulants of the three-color random Ashkin–Teller chain that are compatible with a second-order phase transition while a first-order one is predicted by the original MPO-RG algorithm.

arXiv:1807.08984•Journal of Statistical Mechanics Theory and Experiment•Sep 1, 2019•2 citations

Griffiths-McCoy Singularities in Random Quantum Spin Chains: Exact Results through Renormalization

Ferenc Iglói, Róbert Juhász, Péter Lajkó

The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.

arXiv:cond-mat/0005375•Physical Review Letters•Feb 12, 2001•49 citations

Machine Learning the Strong Disorder Renormalization Group Method for Disordered Quantum Spin Chains

A. Ustyuzhanin, J. Vahedi, S. Kettemann

We train machine learning algorithms to infer the entanglement structure of disordered long-range interacting quantum spin chains by learning from the strong disorder renormalisation group (SDRG) method. The system consists of $S=1/2$-quantum spins coupled by antiferromagnetic power-law interactions with decay exponent $α$ at random positions on a one-dimensional chain. Using SDRG as a physics-informed teacher, we compare a Random Forest classifier as a classical baseline with a graph neural network (GNN) that operates directly on the interaction graph and learns a bond-ranking rule mirroring the SDRG decimation policy. The GNN achieves a disorder-averaged pairing accuracy close to one and reproduces the entanglement entropy $S(\ell)$ in excellent quantitative agreement with SDRG across all subsystem sizes and interaction exponents. RG flow heat maps confirm that the GNN learns the sequential decimation hierarchy rather than merely fitting final-state observables. Finite-temperature entanglement properties are incorporated via the SDRGX framework through a two-stage strategy, using the zero-temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.

cond-mat.dis-nnCross-listedarXiv:2603.05164•Machine Learning Science and Technology•Mar 6, 2026•0 citations

RENORMALIZATION-GROUP INVESTIGATION OF THE S = 1 RANDOM ANTIFERROMAGNETIC HEISENBERG CHAIN

Péter Lajkó

We introduce variants of the Ma-Dasgupta renormalization-group (RG) approach for random quantum spin chains, in which the energy-scale is reduced by decimation built on either perturbative or non-perturbative principles. In one non-perturbative version of the method, we require the exact invariance of the lowest gaps, while in a second class of perturbative Ma-Dasgupta techniques, different decimation rules are utilized. For the S = 1 random antiferromagnetic Heisenberg chain, both type of methods provide the same type of disorder dependent phase diagram, which is in agreement with density-matrix renormalization-group calculations and previous studies.

arXiv:0903.0462•International Journal of Modern Physics C•Dec 1, 2006•1 citation

Real-Space Renormalization Group Study of Effects of Anisotropy on S=1 Random Antiferromagnetic Chain

Shigeru Koikegami, Synge Todo, Hajime Takayama

We investigate S=1 antiferromagnetic quantum spin chain, whose exchange couplings are strongly disordered. By the real-space renomalization group method, introduced by Ma, Dasgupta, and Hu, the renormalization group flows are analyzed numerically in a plain of the anisotoropy of the exchange coupling vs. the staggered magnetic field. As the result, the Heisenberg point, which has a zero average of the exchange coupling anisotropy, is specified as the unstable fixed point against the anisotropy.

arXiv:cond-mat/0106289•Jun 15, 2001•0 citations

Adaptive density matrix renormalization group for disordered systems

J. C. Xavier, José A. Hoyos, E. Miranda

We propose a simple modification of the density matrix renormalization-group (DMRG) method in order to tackle strongly disordered quantum spin chains. Our proposal, akin to the idea of the adaptive time-dependent DMRG, enables us to reach larger system sizes in the strong disorder limit by avoiding most of the metastable configurations, which hinder the performance of the standard DMRG method. We benchmark our adaptive method by revisiting the random antiferromagnetic XXZ spin-1/2 chain for which we compute the random-singlet ground-state average spin-spin correlation functions and von Neumann entanglement entropy. We then apply our method to the bilinear-biquadratic random antiferromagnetic spin-1 chain tuned to the antiferromagnet and gapless highly symmetric SU(3) point. We find the new result that the mean correlation function decays algebraically with the same universal exponent $\ensuremath{\phi}=2$ as the spin-1/2 chain. We then perform numerical and analytical strong-disorder renormalization-group calculations, which confirm this finding and generalize it for any highly symmetric $\mathrm{SU}(N$) random-singlet state.

arXiv:1809.04029•Physical review. B./Physical review. B•Nov 13, 2018•9 citations
A relatively simple real-space renormalization procedure, due to Ma and Dasgupta and known as Strong-Disorder Renormalization Group (SDRG), provides in principle exact results on the critical properties at the infinite-randomness fixed point of random spin chains. Numerically, SDRG can be efficiently implemented as a renormalization of Matrix Product Operators (MPO-RG). By considering larger blocks than SDRG, MPO-RG was recently used to compute non-critical quantities that are inaccessible to SDRG. In this work, we study the accuracy of the approach and propose two simple and fast improvements. We observe that the accuracy on the ground state energy is improved by a factor at least equal to 4.
Abstract Strong-disorder renormalization group (SDRG), despite being a relatively simple real-space renormalization procedure, provides in principle exact results on the critical properties at the infinite-randomness fixed point of random quantum spin chains. Numerically, SDRG can be efficiently implemented as a renormalization of matrix product operators (MPO-RG). By considering larger blocks than SDRG, MPO-RG was recently used to compute non-critical quantities of finite chains that are inaccessible to SDRG. In this work, the accuracy of this approach is studied and two simple and fast improvements are proposed. The accuracy on the ground state energy is improved by a factor at least equal to four for the random Ising chain in a transverse field. Finally, the proposed algorithms are shown to yield Binder cumulants of the three-color random Ashkin–Teller chain that are compatible with a second-order phase transition while a first-order one is predicted by the original MPO-RG algorithm.
The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.
We train machine learning algorithms to infer the entanglement structure of disordered long-range interacting quantum spin chains by learning from the strong disorder renormalisation group (SDRG) method. The system consists of S=1/2S=1/2S=1/2-quantum spins coupled by antiferromagnetic power-law interactions with decay exponent ααα at random positions on a one-dimensional chain. Using SDRG as a physics-informed teacher, we compare a Random Forest classifier as a classical baseline with a graph neural network (GNN) that operates directly on the interaction graph and learns a bond-ranking rule mirroring the SDRG decimation policy. The GNN achieves a disorder-averaged pairing accuracy close to one and reproduces the entanglement entropy S(ℓ)S(\ell)S(ℓ) in excellent quantitative agreement with SDRG across all subsystem sizes and interaction exponents. RG flow heat maps confirm that the GNN learns the sequential decimation hierarchy rather than merely fitting final-state observables. Finite-temperature entanglement properties are incorporated via the SDRGX framework through a two-stage strategy, using the zero-temperature GNN to generate the RG flow and sampling thermal occupations from the canonical ensemble, yielding results in agreement with both numerical SDRGX and analytical predictions without retraining.
We introduce variants of the Ma-Dasgupta renormalization-group (RG) approach for random quantum spin chains, in which the energy-scale is reduced by decimation built on either perturbative or non-perturbative principles. In one non-perturbative version of the method, we require the exact invariance of the lowest gaps, while in a second class of perturbative Ma-Dasgupta techniques, different decimation rules are utilized. For the S = 1 random antiferromagnetic Heisenberg chain, both type of methods provide the same type of disorder dependent phase diagram, which is in agreement with density-matrix renormalization-group calculations and previous studies.
We propose a simple modification of the density matrix renormalization-group (DMRG) method in order to tackle strongly disordered quantum spin chains. Our proposal, akin to the idea of the adaptive time-dependent DMRG, enables us to reach larger system sizes in the strong disorder limit by avoiding most of the metastable configurations, which hinder the performance of the standard DMRG method. We benchmark our adaptive method by revisiting the random antiferromagnetic XXZ spin-1/2 chain for which we compute the random-singlet ground-state average spin-spin correlation functions and von Neumann entanglement entropy. We then apply our method to the bilinear-biquadratic random antiferromagnetic spin-1 chain tuned to the antiferromagnet and gapless highly symmetric SU(3) point. We find the new result that the mean correlation function decays algebraically with the same universal exponent $\ensuremath{\phi}=2$ as the spin-1/2 chain. We then perform numerical and analytical strong-disorder renormalization-group calculations, which confirm this finding and generalize it for any highly symmetric SU(N\mathrm{SU}(NSU(N) random-singlet state.