Evidence Paper III of the Existence Equation series. Quantum many-body scars on Rydberg ladders have been studied with Néel and vacuum initial states. This paper identifies a previously unstudied phenomenon: the selective dominance of the helical initial configuration — excitation alternating between legs — over all other product-state initializations, and an accompanying emergent chiral degeneracy. Input: 3 items. Nothing else. #InputDescription 1Two-leg ladderL rungs (2L sites), periodic boundary conditions 2Longitudinal blockadeNo two adjacent sites on same leg simultaneously excited 3Transverse exclusion + flipNo two sites on same rung simultaneously excited; flip if and only if both constraints satisfied No Hamiltonian written. No coupling constants. No symmetry imposed. The two constraints decide. Output: The constraint selects the helix. LD‖T−H‖F₁ (helix)F₁ (Néel)Selectivity|ΔF| (chirality)T_rev 61990.000.9110.3152.9×10⁻¹⁵3.22 81,1550.000.8580.2223.9×10⁻¹⁵3.99 106,7270.000.8260.1515.5×10⁻¹⁶4.98 The constraint-built transition matrix and the mathematical ladder Hamiltonian are the same object — to machine precision, at every system size tested. Why this matters: The selectivity ratio increases with system size (2.9× → 3.9× → 5.5×), meaning the constraint's preference for the helix strengthens in larger systems. The helical and anti-helical states (+q and −q winding) produce identical dynamics to 10⁻¹⁵ precision — an emergent chiral degeneracy that was not imposed by any symmetry. Within the Existence Equation framework, when longitudinal blockade prevents spatial spreading along each leg and transverse exclusion prevents independent dynamics across legs, the deviation field has no option but to spiral between both dimensions. A helical quantum scar is the trace left by deviation that cannot spread along either available axis and must spiral between them. The structural trilogy: EP IEP IIEP III ClosureTemporalSpatialHelical ResultScarsVorticesSpiral EmergentPeriodicityTopologyChirality Chirality is not broken. It is born. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
Evidence Paper III of the Existence Equation series. Quantum many-body scars on Rydberg ladders have been studied with Néel and vacuum initial states. This paper identifies a previously unstudied phenomenon: the selective dominance of the helical initial configuration — excitation alternating between legs — over all other product-state initializations, and an accompanying emergent chiral degeneracy. Input: 3 items. Nothing else. #InputDescription 1Two-leg ladderL rungs (2L sites), periodic boundary conditions 2Longitudinal blockadeNo two adjacent sites on same leg simultaneously excited 3Transverse exclusion + flipNo two sites on same rung simultaneously excited; flip if and only if both constraints satisfied No Hamiltonian written. No coupling constants. No symmetry imposed. The two constraints decide. Output: The constraint selects the helix. LD‖T−H‖F₁ (helix)F₁ (Néel)Selectivity|ΔF| (chirality)T_rev 61990.000.9110.3152.9×10⁻¹⁵3.22 81,1550.000.8580.2223.9×10⁻¹⁵3.99 106,7270.000.8260.1515.5×10⁻¹⁶4.98 The constraint-built transition matrix and the mathematical ladder Hamiltonian are the same object — to machine precision, at every system size tested. Why this matters: The selectivity ratio increases with system size (2.9× → 3.9× → 5.5×), meaning the constraint's preference for the helix strengthens in larger systems. The helical and anti-helical states (+q and −q winding) produce identical dynamics to 10⁻¹⁵ precision — an emergent chiral degeneracy that was not imposed by any symmetry. Within the Existence Equation framework, when longitudinal blockade prevents spatial spreading along each leg and transverse exclusion prevents independent dynamics across legs, the deviation field has no option but to spiral between both dimensions. A helical quantum scar is the trace left by deviation that cannot spread along either available axis and must spiral between them. The structural trilogy: EP IEP IIEP III ClosureTemporalSpatialHelical ResultScarsVorticesSpiral EmergentPeriodicityTopologyChirality Chirality is not broken. It is born. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
Evidence Paper I of the Existence Equation series. Quantum many-body scars — persistent, non-thermalizing revivals observed in constrained quantum systems — are conventionally explained by analyzing the spectral properties of the PXP Hamiltonian. This paper demonstrates that the PXP Hamiltonian is not a modeling choice but a geometric consequence of a single constraint: nearest-neighbor exclusion on a one-dimensional chain. Input: 3 items. Nothing else. #InputDescription 11D chainL sites, each with two states (|0⟩ and |1⟩) 2ConstraintNo two adjacent sites simultaneously in |1⟩ 3Flip ruleAt each site, flip if and only if constraint is satisfied No Hamiltonian written. No projection operators. No Pauli matrices. No quantum-mechanical formula invoked. The constraint decides. Output: The PXP Hamiltonian — derived, not assumed. LHilbert dim D‖T_constraint − H_PXP‖Revival T/T_∞First peak F₁Thermal 1/D 8550.000.780.7961.8 × 10⁻² 101440.000.860.7686.9 × 10⁻³ 123770.000.980.7432.7 × 10⁻³ 149870.000.980.7191.0 × 10⁻³ 162,5840.000.980.6953.9 × 10⁻⁴ 186,7650.000.980.6731.5 × 10⁻⁴ 2017,7110.000.980.6545.6 × 10⁻⁵ 2246,3680.000.980.6362.2 × 10⁻⁵ 24121,3930.000.990.6188.2 × 10⁻⁶ The constraint-built transition matrix and the textbook PXP Hamiltonian are the same object — to machine precision, at every system size tested, up to D = 121,393. Why this matters: The conventional direction of inference is: Hamiltonian → spectrum → structure. This paper reverses it: constraint → structure → dynamics → symmetry. The constraint is the cause; the Hamiltonian is the consequence. Within the Existence Equation framework, scars have a structural explanation. The nonlinear term α|Ψ|²Ψ enforces hard-core exclusion in the strong-coupling limit, condensing the field into a binary manifold with nearest-neighbor exclusion. The smoothing operator ∇²Ψ cannot act — spatial spreading is blocked. The deviation field has only one remaining option: it must reorganize temporally, circulating along closed trajectories in phase space. A quantum many-body scar is the temporal trace of deviation forced to close on itself. In the standard view, scars are anomalous because they violate thermalization. In the constraint view, scars are expected because spatial smoothing is blocked and temporal reorganization is the only remaining option. The absence of a scar — thermalization despite complete spatial obstruction — would be the anomaly requiring explanation. Spectral evidence: At L = 14 (D = 987), only 22 of 987 eigenstates carry significant Néel overlap. These scar states form a tower distributed symmetrically across the spectrum. Their participation entropy lies systematically below the thermal bulk (Page value), confirming that they are localized near the closed temporal trajectory — not spread across the full Hilbert space. Revival convergence: The revival period converges toward T_∞ ≈ 4.72/Ω (the thermodynamic limit from Turner et al., 2018), reaching 98.6% at L = 24. First revival peaks exceed the thermal expectation 1/D by three to five orders of magnitude. The system does not thermalize. It returns. The structural principle: Constraint → structure → symmetry, never the reverse. This principle extends across the full Evidence Program: the same mechanism that produces scars here (temporal phase closure) produces FQHE topological degeneracy (EP II), helical chirality (EP III), non-commutative algebra (EP IV), dark matter (EP V), the Tsirelson bound (EP VI), and Josephson physics (EP VII) — each as a structural consequence of the same equation. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
We consider the spectrum of a U(1) quantum link model where gauge fields are realized as S=1/2 spins and demonstrate a new mechanism for generating quantum many-body scars (high-energy eigenstates that violate the eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body dynamics with local constraints has attracted much attention due to the recent discovery of nonergodic behavior in quantum simulators based on Rydberg atoms. Lattice gauge theories provide natural examples of constrained systems since physical states must be gauge invariant. In our case, the Hamiltonian H=O_{kin}+λO_{pot}, where O_{pot} (O_{kin}) is diagonal (off-diagonal) in the electric flux basis, contains exact midspectrum zero modes at λ=0 whose number grows exponentially with system size. This massive degeneracy is lifted at any nonzero λ but some special linear combinations that simultaneously diagonalize O_{kin} and O_{pot} survive as quantum many-body scars, suggesting an "order-by-disorder" mechanism in the Hilbert space. We give evidence for such scars and show their dynamical consequences on two-leg ladders with up to 56 spins, which may be tested using available proposals of quantum simulators. Results on wider ladders point towards their presence in two dimensions as well.
Quantum many-body scars are atypical energy eigenstates of chaotic quantum many-body systems that prevent certain special non-equilibrium initial conditions from thermalizing. We point out that quantum many-body scars exist for any nearest-neighbor spin-S XYZ quantum spin chain, and arise in the form of an infinite family of highly excited yet nonentangled product-state eigenstates, which define periodic textures in spin space. This set of scars, discovered originally by Granovskii and Zhedanov in 1985, encompasses both the experimentally relevant 'spin helices' for XXZ chains and more complicated helix-like states constructed from Jacobi elliptic functions for generic XYZ chains. An appealing feature of Granovskii-Zhedanov scars is that they are well-defined in the semiclassical limit S→∞, which allows for a systematic and analytical treatment of their dynamical instability to perturbations of the Hamiltonian. Using time-dependent spin-wave theory, we predict that upon perturbing along certain directions in Hamiltonian space, Granovskii-Zhedanov scars exhibit a dramatic asymmetry in their decay: depending on the sign of the perturbation, the decrease of their contrast is either slow and linear, or fast and exponential in time. This asymmetry can be traced to the absence (presence) of imaginarity in the spectrum of the Bogoliubov Hamiltonian governing quantum fluctuations about the scar, which corresponds to the absence (presence) of a non-zero Lyapunov exponent for the limiting classical trajectory. Numerical simulations using matrix product states (MPS) and infinite time-evolving block decimation (iTEBD) confirm that our prediction remains valid even far from the semiclassical limit. Our findings challenge existing theories of how quantum-many body scars relax.
Hilbert space fragmentation (HSF) is a mechanism for generating quantum many-body scar (QMBS), which provides a route to weakly break ergodicity. The zero-energy QMBSs widely exist across various systems due to the intertwining of chiral symmetry and spatial inversion symmetry. In this work, we study the phenomenology of the zero-energy QMBS under the interplay between the chiral symmetry and pseudo HSF, where the Hilbert space is approximately fragmented into different blocks. We consider a model of tilted chain of interacting spinless fermions with periodically varying tunneling strength. At small tunneling strength and under resonance condition, the system is described by an effective model with chiral symmetry and pseudo HSF. We find that the interplay between the two gives rise to a highly localized zero-energy QMBS when the particle number is even. We identify a simple product state to signalize the zero-energy QMBS, which gives rise to unusual scarred dynamics. The fidelity oscillates around a fixed value without decaying instead of showing the usual collapse and revival in common scarred systems. We show that the signature of the zero-energy QMBS can also be captured by the original Hamiltonian. Our results uncover a new scar phenomenon and provide an example that does not need the intertwining of chiral and spatial symmetries to support zero-energy QMBS.
Quantum many-body scars in Rydberg atom arrays have thus far only been observed on bipartite lattices, leaving open the question of whether and how they survive frustration, and what the appropriate initial states are that lead to nonthermal dynamics. We introduce a graph-theoretic framework to find suitable candidates for scarring on arbitrary lattices. Our framework predicts two distinct mechanisms: type-I scars generalize the bipartite case by using locally entangled states to overcome mild frustration, while type-II scars exploit strong frustration to pin part of the lattice, leaving the remainder to oscillate freely. We numerically demonstrate both mechanisms and uncover an exponential family of scarred trajectories on the hexagonal lattice that can encode information protected from thermalization. Our results establish scarring as a generic feature of Rydberg systems beyond one dimension and provide an experimentally accessible route to systematically probing non-thermal dynamics in quantum simulators.
The Wallstrom condition---the requirement that the phase of the wave function be single-valued up to integer multiples of 2πℏ---has been a foundational obstacle in all attempts to derive quantum mechanics from stochastic or hydrodynamic dynamics since Wallstrom's 1994 proof that the Madelung equations alone are insufficient to recover the Schrödinger equation. In this paper, we provide a complete derivation of the Wallstrom condition from the axioms of the Constraint-Dispersion Unified Framework (CDUFD). The derivation follows the CDUFD methodology of solution-space construction and natural constraint-driven reduction: we initialize the solution space at the electroweak critical point with both Ising (amplitude) and XY (phase) fluctuation modes as built-in degrees of freedom of the SO(8)-embedded order parameter; we then demonstrate that A3 critical dynamics and A4 topological quantization jointly eliminate Ising domain walls while preserving XY vortices through their critical lifetime disparity. The argument is built upon three robust pillars: (i) SO(8) group structure guarantees a non-vanishing XY coupling g=0; (ii) A4 topological quantization guarantees integer vortex winding numbers; (iii) the electroweak critical point provides a finite, large physical cutoff ξ(δ=0.0102) that prevents complete infrared suppression of the XY sector even in the dangerously irrelevant case. Together with the numerically testable premise that the effective XY coupling at this cutoff is strong enough to sustain a percolating vortex network, the surviving vortices, spanning the entire system via divergent correlation length, impose global phase quantization on all closed contours---precisely the Wallstrom condition. Combined with the previously established Madelung transformation from A2 Langevin dynamics, this completes the rigorous emergence of the full Schrödinger equation from CDUFD axioms. The sole remaining numerical premise is explicitly isolated as a verifiable open problem.