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Autonomous phase discovery

Shiyu Zhou, Yuxuan Zhang, Sebastian Wetzel, Roger Melko, Xiu-Zhe Luo

arXiv:2609.33802v1quant-phcond-mat.str-elcs.LG

TL;DR

Quantum phase discovery lacks a universal way to explore Hamiltonian spaces without phase-specific prior knowledge. This work uses differentiable solvers and unsupervised learning to steer an adaptive search, and reports that trajectory-based exploration can find small phase sectors missed by same-budget random sampling.

  • Problem

    No single formalism provides a universal phase diagnostic without phase-specific inputs, while autonomous discovery without prior knowledge or extensive parameter sweeps remains unestablished.

    Source: §II · p. 3

  • Method

    A differentiable solver and autoencoder map ground states into a latent space whose differentiable distance guides a feedback-loop search through Hamiltonian parameters.

    Source: §I · p. 2

  • Results

    The system detects distinct phases through latent clusters and gradient divergence, and its trajectory search can identify small phase sectors missed by same-budget random sampling.

    Source: A. Protocol · p. 7, §V · p. 9

  • Takeaways & Limitations

    The system provides a route to exploring Hamiltonian space without phase labels, order parameters, or predetermined parameter sweeps.

    Source: §V · p. 9

    The comparison is limited to the simple harmonic families considered: generic coefficients reach substantially more sectors than symmetric closed-form families.

    Source: §3 · p. 20

Abstract

from arXiv · show

Understanding quantum phases of matter has long relied on physicists' intuition and mathematical tools such as symmetry and topology. Remarkably successful as these approaches have been, they provide no universal way to explore a Hamiltonian space whose organizing principle is not known in advance. In this work, we introduce a fully autonomous system combining differentiable programming and unsupervised learning for quantum phase discovery. The search evaluates ground-state data along an adaptive trajectory rather than on a predetermined parameter grid. We demonstrate the system with three different solvers and benchmark it against random sampling at equal ground-state-evaluation budgets. On a generalized cluster chain hosting up to $200$ distinct phases, the search finds up to $25$ more phases at the same budget, and matches random sampling given thirty times its budget. On a $50$-parameter Chern insulator, it reaches sectors not obtained by the simple harmonic constructions considered here, in a family whose inverse problem remains open, while recovering all sectors found by sampling. Our results establish autonomous, gradient-driven exploration of Hamiltonian space as a practical route to discovering quantum phases without phase labels or a prescribed target phase.

I. INTRODUCTION

Existing phase-discovery approaches rely on phase-specific knowledge or predetermined parameter sweeps, leaving no universal recipe for exploring unknown phases. This work combines differentiable solvers with unsupervised learning in a feedback loop that guides exploration through Hamiltonian space without phase labels or a prescribed sweep.

I. INTRODUCTION

  • No universal diagnostic explores Hamiltonian space without phase-specific inputs, and a fixed-resolution parameter grid scales exponentially with parameter count.Symmetry, topology, and entanglement tools are tied to particular classes of phases and require prior physical intuition.Source: §I · p. 1
  • Supervised methods require known phase labels, while unsupervised methods remove labels but still rely on predetermined parameter sweeps.The autonomous framework instead starts from data sampled locally around an arbitrary point and evaluates along its discovery trajectory.Source: Figure 1 · p. 1, §I · p. 1
  • A diverging gradient signals a phase-boundary gap closing and triggers a boosted-rate jump, while a vanishing gradient triggers autoencoder retraining on newly explored data.The loss is minimized by gradient descent on Hamiltonian parameters.Source: §I · p. 2
  • The proposed system combines differentiable programming and unsupervised phase detection to guide a closed-loop search without phase labels or predetermined parameter sweeps.Gradients pass through the solver to the Hamiltonian parameters, while latent-space distances guide the search toward states distinct from training data.Source: §I · p. 2
  • The paper proceeds from system design to tests on three interacting models and equal-evaluation-budget comparisons with random sampling on a cluster chain and a 50-parameter Chern insulator.The three models include a transverse-field XXZ chain with five phases, including a gapless Luttinger liquid.Source: §I · p. 2

II. THE AUTONOMOUS SYSTEM

The autonomous system combines a differentiable Hamiltonian solver, an unsupervised autoencoder, and a latent-space pathfinder to search for phases without a universal phase-specific diagnostic. Gradients guide parameter updates, while boundary crossings and stalled searches trigger distinct responses.

II. THE AUTONOMOUS SYSTEM

  • The system integrates a differentiable solver, an unsupervised autoencoder phase detector, and a pathfinder that directs searches toward unknown phases.The solver’s gradients propagate through the final loss back to the Hamiltonian parameters.Source: §II · p. 3
  • The study fixes the interaction terms and leaves searches that vary interaction types, locations, or spin numbers to future work.The paper notes that phases could in principle depend on those additional choices.Source: §II · p. 3
  • The pathfinder maximizes latent-space distance from the training-cluster centroid, backpropagating through the encoder and differentiable solver to update Hamiltonian parameters.The loss is scaled by the square root of the latent dimension.Source: §II · pp. 3–4
  • Training data are sampled near a starting point within a radius chosen to keep them inside one phase, and their latent representations define the cluster centroid.The autoencoder can use full wavefunctions from ED or ITE, or ground-state observable expectations from DMRG.Source: §II · pp. 3–4
  • A diverging gradient signals an approaching gap-closing boundary; the search reverts to its last finite-gradient step and kicks forward to cross it.The gap-closing signature is sharpest for diagonalization-based solvers such as ED and DMRG.Source: §II · p. 4
  • When gradients vanish over a sustained window, the system treats the current phase as explored, adds nearby ground-state data, and retrains the autoencoder.The updated centroid lets the pathfinder resume from the stalled point.Source: §II · p. 4

III. DEMONSTRATIONS ACROSS MODELS AND SOLVERS

The system is demonstrated on three models using exact diagonalization, imaginary time evolution, and density matrix renormalization group. It traverses all phases in each model, including five phases in the transverse-field XXZ model.

III. DEMONSTRATIONS ACROSS MODELS AND SOLVERS

  • In XXZ, the DMRG trajectory visits five distinct phase clusters, from z-directional ferromagnetic order through paramagnetic, y-directional Néel, and Luttinger-liquid phases to z-directional Néel order.The trajectory is shown in Hamiltonian parameter space and projected onto the first two autoencoder latent-space principal components.Source: Figure 3 · p. 5
  • The system traverses all phases in three demonstrations: two phases each in TFIM and Z2 gauge theory, and five in transverse-field XXZ.The demonstrations use ED, ITE, and DMRG, respectively.Source: §III · p. 5

A. TFIM

In the TFIM demonstration, trajectories starting in either phase cross the boundary and move toward the opposite phase. The system also shows this behavior across multiple starting points.

A. TFIM

  • All three TFIM runs cross the phase boundary and find the opposite phase, including trajectories starting in both the symmetry-breaking and paramagnetic phases.The two runs starting in the symmetry-breaking phase move deep into the paramagnetic phase.Source: A. TFIM · p. 5
  • For the trajectory starting at g0 = −0.3, the system drives parameters from the symmetry-breaking cluster into the paramagnetic regime.The projected trajectory uses the first two latent-space principal components.Source: A. TFIM · pp. 5–6

B. Z2 lattice gauge model

In the Z2 lattice gauge theory, all three searches cross between deconfined and confined phases, with two starting points reaching opposite field-sign regions of the confined phase. The latent-space clusters connect continuously at the transition.

B. Z2 lattice gauge model

  • All three Z2 searches cross the phase boundary; the two starts in the deconfined phase reach confined regions with opposite transverse-field signs.The two regions are related by a unitary transformation that maps λ to −λ.Source: B. Z2 lattice gauge model · p. 6
  • The first two latent-space principal components capture 22.6% of the reference variance, and the phase clusters connect continuously at the boundary.The passage contrasts this continuous connection with the discrete jump observed for the TFIM and relates it to the continuous confinement-deconfinement transition.Source: B. Z2 lattice gauge model · p. 6

C. XXZ model

The transverse-field XXZ chain tests phase discovery in a two-dimensional parameter space with five phases. The search uses autoencoder bootstrapping to choose new directions and identifies all five phases, including the gapless Luttinger liquid.

C. XXZ model

  • In the two-dimensional (∆, h) space, bootstrapping retrains the autoencoder on visited and newly found phases so the search can pursue another unfamiliar direction.The model’s updated latent representation records visited phases, and the pathfinder loss gradient points toward regions unfamiliar to that representation.Source: C. XXZ model · p. 6
  • The demonstration uses DMRG and local expectation values ⟨σz_i⟩ as ground-state inputs.These inputs contain local information and distinguish most of the XXZ phases in this demonstration.Source: C. XXZ model · p. 6
  • The Luttinger liquid lies along the h = 0 line, so the search registers it through a gradient-divergence event rather than settling inside a finite-area region.The closing gap between the ground and first excited states causes the loss gradient to diverge and triggers a “kick.”Source: C. XXZ model · p. 7
  • The search finds all five phases of the transverse-field XXZ model, including the gapless Luttinger liquid.The trajectory proceeds from the z-directional ferromagnetic phase through the paramagnetic phase, then reaches the y- and z-directional Néel phases.Source: C. XXZ model · pp. 6–7

IV. PHASE DISCOVERY AT SCALE

The paper scales autonomous phase discovery to free-fermionic models with many parameters and phases. Against random sampling at equal evaluation budgets, its coverage advantage grows with coupling dimension in the cluster benchmark and with query budget in the Chern benchmark.

IV. PHASE DISCOVERY AT SCALE

  • The system is tested on a generalized cluster-state chain with up to 200 phases and a 50-parameter Chern insulator with an unknown number of accessible phases.The benchmarks compare phase coverage against random sampling.Source: §IV · p. 7
  • The autonomous system outperforms random sampling, with its coverage advantage increasing with coupling dimension in the cluster benchmark and query budget in the Chern benchmark.Both comparisons use phase coverage as the metric.Source: §IV · p. 7
  • The benchmarks extend earlier demonstrations in parameter spaces of at most two dimensions and with at most five phases.The scaled tests use free-fermionic models with substantially more parameters and phases.Source: §IV · p. 7

A. Protocol

The protocol compares autonomous search with random sampling using the same number of ground-state evaluations. It measures how many distinct phases each method covers, while noting that trajectory-based exploration can reach small sectors that random sampling may miss.

A. Protocol

  • Each query is one ground-state evaluation, and random sampling draws parameter vectors from a rotation-invariant Gaussian measure.Phase coverage is evaluated using winding number or Chern number, respectively.Source: A. Protocol · p. 7
  • A phase with probability mass v under the sampling distribution is missed by N independent samples with probability (1 − v)^N.Random sampling becomes unlikely to resolve sectors with v ≪ 1/N.Source: A. Protocol · p. 7
  • Trajectory-based search can enter a small phase sector by crossing from a neighboring region, and the benchmarks identify sectors missed by equal-budget random sampling.The sector’s probability mass under the sampling distribution need not limit whether the trajectory reaches it.Source: A. Protocol · p. 7
  • Random sampling alone does not identify the phases represented by its samples; the comparison measures coverage without adding unsupervised post-processing.Winding and Chern numbers are used only to evaluate the number of distinct phases covered.Source: A. Protocol · p. 7

B. Cluster chain with up to 200 phases

The cluster-chain benchmark systematically increases both parameter dimension and phase count, then compares autonomous search with random sampling at equal evaluation budgets. The search covers more phases as dimension grows and can reach sectors absent from extensive random sampling.

B. Cluster chain with up to 200 phases

  • The benchmark varies d from 20 to 200, with d setting both the number of couplings and the number of winding-number phases.The closed chain uses length L = 500; its phases are characterized by winding numbers ω = 0, …, d − 1.Source: B. Cluster chain with up to 200 phases · pp. 7–8
  • At equal budgets, autonomous search covers 5–25 more phases than random sampling for d ≥ 40, and 104 evaluations already exceed random sampling’s coverage at 3 × 105.At d = 20, both methods find all available phases; the advantage persists across budgets from 104 to 3 × 105 evaluations.Source: Figure 4 · p. 8, B. Cluster chain with up to 200 phases · p. 8
  • For the closed-chain benchmark, winding number is evaluated after the search and is not provided to the autonomous system.The ground state is specified by Majorana correlations, which form the encoder input.Source: B. Cluster chain with up to 200 phases · p. 8
  • At d = 140, four autonomous-search seeds at a budget of 105 contain 21 winding sectors absent from four million random samples.The search can reach these small sectors by traversing neighboring parameter-space regions rather than relying on direct sampling.Source: B. Cluster chain with up to 200 phases · p. 8

C. A 50-parameter Chern insulator

The generalized QWZ model provides a 50-parameter setting with unknown accessible Chern sectors and no general recipe for constructing a point with a prescribed Chern number. At equal evaluation budgets, autonomous search reaches more sectors than random sampling, with its advantage concentrated in rare sectors.

C. A 50-parameter Chern insulator

  • The generalized QWZ family has 50 parameters and permits a broad range of Chern sectors, but their accessible set is not enumerated and no general construction targets a prescribed Chern number.Evaluating the Chern number at a parameter point is straightforward; the inverse construction remains open.Source: C. A 50-parameter Chern insulator · p. 9
  • At 3 × 10^5 evaluations, autonomous search finds a median 207 distinct Chern sectors versus 139 for random sampling; across five runs it reaches 283 versus 160 at equal total budgets.The five-run totals use 1.5 × 10^6 evaluations per method, and every sector found by random sampling is also found by autonomous search.Source: C. A 50-parameter Chern insulator · p. 9
  • The autoencoder receives normalized Bloch vectors sampled on a 32 × 32 momentum grid, while Chern numbers are withheld from it and computed separately on a finer 768 × 768 grid for coverage evaluation.Source: C. A 50-parameter Chern insulator · p. 9
  • With an additional 10^6 samples, random sampling reaches 168 sectors, compared with 283 for autonomous search.Source: C. A 50-parameter Chern insulator · p. 9
  • Both methods saturate the four probability-mass decades resolved by the 10^6-sample reference, while autonomous search gains mainly in the two rarest bins, below probability mass 10^-5.The comparison is resolved by each Chern sector’s probability mass under the Gaussian sampling measure and averages five seeds.Source: Figure 5 · p. 9, C. A 50-parameter Chern insulator · p. 9

V. DISCUSSION & OUTLOOK

The discussion frames phase discovery as gradient-guided exploration of Hamiltonian space without phase labels, order parameters, or a predetermined sweep. It highlights distinct signals for different phase types while noting limits from connectivity, evaluation cost, and solver scale.

V. DISCUSSION & OUTLOOK

  • The system casts discovery as gradient descent through a differentiable solver in an autoencoder latent space, without phase labels, order parameters, or a predetermined parameter sweep.Source: §V · p. 9
  • Symmetry-broken phases cluster in latent space, whereas the zero-measure gapless Luttinger liquid is detected through gradient divergence rather than a finite-area phase region.Source: §V · p. 9
  • A trajectory can enter rare sectors from neighboring regions, but its walk is restricted to phases connected to its starting phase; combining random starting points with outward walks remains untested.Source: §V · p. 10
  • The search spends on the order of a thousand ground-state evaluations per added sector, and the authors identify visited-point repulsion and retraining as targets for improvement.They suggest broader random-pass training, richer phase representations, and label-aware repulsion where labels are available.Source: §V · p. 10
  • Differentiable neural-network solvers are proposed as a way to relax the cost and size constraints of exact diagonalization, density-matrix renormalization group, and imaginary-time evolution.Source: §V · p. 10

Appendix A: Principal Component Analysis of the Latent Space

The appendix describes how latent-space trajectories are projected for visualization and specifies shared discovery-loop settings alongside selected per-run exceptions.

Appendix A: Principal Component Analysis of the Latent Space

  • Because latent spaces have dimension much greater than two, trajectories are visualized by projection onto the two dominant principal components of a reference set.The centered reference representations are decomposed by singular value decomposition; the resulting axes are fixed by that reference set, and trajectories are projected onto them.Source: Appendix A · p. 13
  • The reported retained embedding information is the sum of the first two explained variance ratios, with reference sets drawn from each experiment’s final model training data.Source: Appendix A · p. 13

Appendix B: Hyperparameters

  • Runs share a fully connected ReLU autoencoder and a discovery loop in which divergent gradients trigger a boosted-rate kick and stalled searches trigger retraining on prior data plus a fresh local circle.Wavefunction inputs use infidelity loss and XXZ observable vectors use mean squared error; run-specific settings include a history-aware XXZ centroid and no jump in the long-horizon benchmark.Source: Appendix B · p. 13

Appendix C: Benchmark configuration

This appendix documents benchmark labeling and configuration, including the generalized cluster chain’s ground-state representation and phase labels, and examines the Chern-number inverse problem using signed counts and harmonic constructions. It notes that simple harmonic families reach many Chern sectors but do not provide a general inverse construction, while the full set of realizable sectors remains unknown.

Appendix C: Benchmark configuration

  • Successive-halving selection used 27 → 9 → 3 → 1 trials, and held-out-seed re-evaluation reduced one finalist from +9 in-bracket to −1.All Section IV numbers use seed families not used in selection; at H = 16, two of the 50 Chern couplings are invisible to the 32 × 32 input grid.Source: Appendix C · p. 14
  • At 768^2, lattice Chern labels agree with the reference on 98.2% of points, with the same agreement rate for search visits and baseline Gaussian draws.The residual label error is therefore symmetric between the two methods.Source: Appendix C · p. 14

Appendix D: Majorana ground-state correlations and phase characterization

  • Winding numbers provide exact cluster-chain phase labels only for evaluating coverage, while the autoencoder receives the Majorana correlation vector g rather than a phase label.The g representation retains correlations over all separations, including cluster-string correlations, without selecting a local order parameter.Source: Appendix D · p. 14, §2 · p. 16, §4 · p. 17

3. Ground-state correlations from the polar factor

  • For a nonsingular quadratic benchmark Hamiltonian, ground-state correlations are the orthogonal polar factor of C: singular values are flattened while their eigenvectors are retained.At a gap closing, a zero-mode occupation must be specified separately because the ground state is not uniquely selected by this expression.Source: §3 · p. 16

5. Why g completely specifies the Gaussian ground state

  • The real vector g completely determines the Gaussian ground state within this translation-invariant, bipartite Majorana benchmark, but this reduction does not represent a generic interacting chain.The representation depends on quadratic Gaussian structure, translation invariance, and bipartite Majorana coupling.Source: §5 · p. 18, §6 · p. 18

1. Reduction to a signed lattice count

  • The Chern number is a signed checkerboard sum over zeros of dx and dy where dz > 0, and the degree is bounded by |C| ≤ 512, without implying every value is realizable.For H = 16, the bound allows 1025 integer values, from −512 to 512.Source: §1 · p. 19

2. Sign patterns at fixed harmonics

  • At H = 16, fixed-harmonic sign patterns have at most approximately 2 × 10^38 realizable cells; sampling 18 dz parameters on one fixed grid found 110 Chern numbers spanning [−69, 58].Not every assignment of signs on the zero grid is realizable, though membership can be tested by linear feasibility.Source: §2 · p. 19

3. Simple harmonic constructions and their limitations

  • Unit-coefficient ansätze with one, two, and three active harmonics realize 33, 99, and 174 distinct Chern numbers, respectively, far below the 1025 values allowed by the bound.Adding a fourth harmonic yields little further increase; harmonic combinations do not follow a simple additive rule.Source: §1 · p. 19, §3 · p. 20
  • At equal evaluation counts, generic coefficients increase observed Chern values from 88 spanning [−176, 160] to 123 spanning [−256, 256] with two harmonics, and from 159 spanning [−159, 153] to 195 with three.With equal coefficients, symmetry makes the d_x and d_y root sets coincide and restricts checkerboard sign patterns.Source: §3 · p. 20
  • Generic coefficients still require a search, and the resulting Chern number must be evaluated afterward; no general construction maps a target Chern number to a coupling vector.The benchmark’s simple constructions are reference points, not the baseline for the scaling comparison, which compares autonomous search with random sampling at matched query budgets.Source: §3 · p. 20, §4 · p. 20

4. Relation to the sectors found by autonomous search

  • Pooling five autonomous-search runs yields 283 distinct Chern sectors, 151 shared with the one-, two-, and three-harmonic constructions and 132 not reached by them.The latter sectors have explicit realizations from the autonomous search.Source: §4 · p. 20
  • The autonomous search reaches |C| = 192, within the range accessible to simple constructions, while locating a broader collection of sectors across that range.It supplies explicit coupling vectors for many Chern values not produced by the simple harmonic ansätze.Source: §4 · p. 20

5. Complexity and open questions

  • The complete set of realizable Chern sectors and a general map from target Chern number to parameter vector remain unknown, motivating the generalized QWZ model as an open-ended discovery benchmark.The forward map is efficiently evaluable, but checkerboard-grid geometry and mass-function sign patterns cannot be chosen independently.Source: §5 · p. 21

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