Snapshot Renormalization Group Characterizes Quantum Matter Via Self-Similar States and Scale Invariance

The study of complex quantum systems receives a significant boost from a new approach to analysing experimental data, as Laurin Brunner, Tobias Wiener, and Tiago Mendes-Santos, from the University of Augsburg and Pasqal respectively, alongside colleagues, present a novel method for characterising quantum matter. This research introduces an exact renormalization group transformation, termed SnapshotRG, which directly analyses individual configurations captured in recent simulator experiments, offering unprecedented access to the behaviour of these complex systems. The team demonstrates that datasets generated from these snapshots exhibit self-similarity during continuous phase transitions, explaining the scale-freeness observed in emerging wavefunction networks, and extending the concept of scale invariance beyond traditional measurements to encompass the full statistical structure of quantum states. This versatile tool, readily implemented with data from various numerical and simulation platforms, promises to advance the characterisation of phase transitions and critical phenomena in a wide range of materials.

Wave Function Networks Analyse Quantum Criticality

Scientists are developing innovative methods to understand complex quantum systems using data from individual quantum configurations, known as snapshots. This research focuses on wave function networks, constructed from these states, to explore how systems change under renormalization group (RG) transformations, a process that simplifies systems by removing details at short distances. The team investigates the transverse-field Ising model as a model system to reveal information about quantum phase transitions and critical behaviour. Neural quantum states approximate the ground state of a quantum system using a neural network, and wave function networks are graphs built from a series of these states.

Nodes represent individual snapshots of the quantum system, connected if similar, and the network’s structure provides insights into the system’s behaviour. The renormalization group systematically simplifies a system, focusing on long-range behaviour, and this research applies this process to the network of quantum states. The method involves simulating the quantum system using a neural network, generating snapshots, and constructing a wave function network. By applying renormalization group transformations, the system is simplified, and the resulting network structure is analysed, focusing on the degree distribution to understand how the network changes. This analysis aims to reveal how network topology reflects the quantum phase transition and critical behaviour.

SnapshotRG Renormalization of Quantum Configurations

Scientists have developed SnapshotRG, a new method for analysing complex quantum systems using data from snapshots of individual quantum configurations. This innovative approach applies an exact renormalization group (RG) transformation directly to existing datasets from simulations or experiments, bypassing computational challenges. By iteratively eliminating half of the lattice sites, SnapshotRG reduces complexity and determines the renormalized Hamiltonian without approximations. The core of the method involves masking degrees of freedom in each snapshot, representing sites being integrated out, allowing the RG transformation to be executed directly on existing datasets.

This enables researchers to study the impact of real-space RG transformations on the probability distribution from which snapshots are sampled, offering a powerful tool for analysing quantum phases and critical phenomena. SnapshotRG is not limited to real space, demonstrating equivalence to an RG operating within the dataspace of the snapshots themselves. Experiments utilize a two-dimensional simple cubic lattice as a model system, generating snapshots for SnapshotRG analysis. Each RG step involves masking alternate lattice sites, rescaling the system, and requiring coordinate rotation. The team demonstrates that the degree distribution of wave function networks, constructed from the snapshot data, remains invariant under the RG transformation at continuous phase transitions, linking microscopic data to macroscopic behaviour. SnapshotRG offers a universally applicable tool for analysing snapshot datasets.

Dataspace Scale Invariance via Measurement Snapshots

Recent research introduces the Snapshot Renormalization Group (SnapshotRG), a framework for analysing many-body quantum systems through their measurement snapshots. This data-driven approach directly analyses configurations captured in experiments or simulations, providing an exact real-space decimation procedure without requiring reformulation of the Hamiltonian. SnapshotRG reveals scale invariance not only in traditional observables like correlation functions, but extends this property to the entire dataspace through the analysis of wavefunction networks (WFNs). Researchers demonstrated SnapshotRG’s capabilities by analysing classical Ising models in two and three dimensions, as well as the 2D transverse-field Ising model.

Analysis of the degree distributions of the resulting networks revealed power-law scaling, enabling precise determination of the scaling exponent γ across different system dimensionalities and transition types. Results show a compelling connection between the network exponent γ and the critical exponent η for thermal phase transitions in both two and three dimensions, where γ ≈ 1 − η. The quantum phase transition in the 2D transverse-field Ising model deviates from this classical scaling behaviour, potentially due to finite-size effects. The team measured the power-law exponent γ to be approximately 1 − η for the thermal phase transitions, demonstrating a clear relationship between network properties and the underlying critical phenomena. This work delivers a powerful, ready-to-use tool for analysing existing and future quantum simulator data, opening new avenues for data-driven discovery in many-body physics.

SnapshotRG Reveals Self-Similarity at Phase Transitions

This work introduces the Snapshot Renormalization Group (SnapshotRG), a framework for analysing many-body quantum systems directly from measurement data. SnapshotRG operates by applying an exact decimation procedure to datasets of individual system configurations, captured as snapshots from simulations or experiments. This method functions in real space and can also be translated to an abstract dataspace of measurement configurations, offering a versatile approach to characterizing complex systems. Results demonstrate that these snapshot datasets exhibit self-similarity at continuous phase transitions, explaining the recently observed scale-freeness of wavefunction networks.

The team found evidence suggesting a relationship between the power-law exponent governing the degree distributions of wavefunction networks and the critical exponent characterizing the underlying phase transition, for the thermal phase transitions studied. However, this relationship appears to break down in quantum systems, potentially due to limitations in accessible system sizes during simulations. SnapshotRG provides a powerful, readily implementable tool for analysing data from quantum simulators and theoretical simulations, opening new avenues for data-driven discovery in many-body physics. Future research will explore how this framework can be applied to other methods for quantifying snapshot datasets, such as determining intrinsic dimension or Kolmogorov complexity. This work represents a significant step towards a more data-driven approach to understanding complex quantum systems, offering new insights into their behaviour and properties.