Snapshot renormalization group for quantum matter

TL;DR

This paper introduces SnapshotRG, an exact real-space RG that operates directly on measurement snapshots and can also be viewed as an RG in snapshot configuration space. It bypasses Hamiltonian reformulation by mapping decimation to the marginalization of the quantum state, allowing RG analysis to be performed directly on data from experiments or neural quantum state simulations. The authors show that at continuous phase transitions the resulting wave function networks exhibit scale-free degree distributions that remain invariant under SnapshotRG, linking RG flow to universal critical behavior in both classical and quantum settings. They find that classical exponents align with known relations suggesting gamma-like behavior, while quantum cases show deviations, highlighting open questions and the utility of snapshot-driven RG for data-driven discovery.

Abstract

Recent advances in quantum simulator experiments enable unprecedented access to quantum many-body states through snapshot measurements of individual many-body configurations. Here, we introduce an exact renormalization group (RG) transformation that can be directly applied to any such snapshot dataset. Our SnapshotRG operates in real space, but can also be directly translated to an RG in the abstract dataspace of measurement configurations, providing a framework for the characterization of quantum many-body systems on a more general level. We demonstrate that snapshot datasets in dataspace exhibit self-similarity at continuous phase transitions, providing an explanation for the recently observed scale-freeness of so-called wavefunction networks. As a consequence, scale invariance extends beyond traditional low-order correlation functions to encompass the full statistical structure of quantum states as contained in their snapshot datasets. Our SnapshotRG can be readily implemented with snapshot data generated by numerical method such as neural quantum states or any quantum simulation platform, offering a versatile tool for characterizing quantum phase transitions and critical phenomena in quantum matter.

Figures (4)

• Figure 1: Workflow of the SnapshotRG method on a 2D simple cubic lattice, illustrated for two consecutive RG steps. Snapshots are obtained from either simulations or experiments and processed through iterative SnapshotRG steps. Each SnapshotRG step acts as a mask on individual snapshots, effectively removing every second site. After the first RG transformation, the system becomes a simple cubic lattice with rescaled lattice constant rotated by 45°, requiring every second RG step to be performed in a rotated coordinate system. The degree distribution of the WFN, calculated at each SnapshotRG step independelty, remains invariant at phase transitions but changes away from critical points.
• Figure 2: Effect of SnapshotRG transformation on rescaled spin-spin correlation $C(d) = \frac{1}{L^2}\sum_i\langle s_i s_{i+d}\rangle$ of the classical 2D Ising model. $\lambda=\sqrt{2}$ is the rescaling factor and $\eta=\frac{1}{4}$ is the critical exponent that describes the power-law decay of the correlation function at the critical point. At the critical temperature $T_c$ the SnapshotRG leaves the correlation invariant while for higher temperatures there is a RG flow to infinite temperature.
• Figure 3: SnapshotRG transformation on the degree distribution $P_k$ for the 2D Ising model in the presence of perturbations. Initial system size before the SnapshotRG is $L_x\times L_y = 256\times 256$. Number of samples for all curves is $N_r = 5\cdot10^5$. a) Symmetry-preserving perturbation. The degree distribution remains invariant under the SnapshotRG. For the perturbed system the temperature is chosen such that the system is located at the phase transition point. b) Symmetry-breaking perturbation. The degree distribution of the perturbed system exhibits a different shape without clear power-law and ceises to be an invariant under the SnapshotRG. The perturbed system is chosen at the critical temperature of the unperturbed system.
• Figure 4: The power-law of the degree distribution at critical points for a) 2D classical Ising model (square lattice with $L = 256$), b) 3D classical Ising model (cubic lattice with $L= 40$), and c) 2D quantum transverse field Ising model (square lattice with $L = 16$). For all lines we took $N_r=10^6$ snapshots. The extracted exponents are a)$\gamma = -0.79 \pm 0.02$, b)$\gamma = -0.90 \pm 0.01$, and c)$\gamma = -1.32 \pm 0.01$. Reference power-law curves with exponent $\gamma = 1 - \eta$ are included for comparison, where $\eta$ represents the corresponding critical exponent of each system. For panel c), additional reference curves using the $\eta$ values from both two- and three-dimensional classical Ising models are displayed, demonstrating that neither provides agreement with the quantum system behavior.