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Quantum Many-Body Scars as Temporal Phase Closure of the Existence Equation

Jae-Ahn Shin

Published Mar 30, 2026• 6 citations
Publisher

Abstract

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)

DOI
10.5281/zenodo.19327776

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Helical Rydberg Scars as Mixed-Dimensional Phase Closure of the Existence Equation

Jae-Ahn Shin

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)

•Mar 30, 2026•6 citations

Physics and mathematics of quantum many-body scars and beyond

Keita Omiya

This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.

•Jan 1, 2026•0 citations

Engineering Quantum Many-Body Scars through Lattice Geometry

Erick Parra Verde, Kevin P. Mours, Johannes Zeiher +2 more

Quantum many-body scars enable persistent non-ergodic dynamics in otherwise thermalizing systems, yet their stabilization typically relies on fine-tuned initial states or engineered Hamiltonian perturbations. Here we show that lattice geometry alone can serve as a powerful and experimentally accessible control knob for inducing and enhancing scarring. By transforming a one-dimensional chain into a quasi-one-dimensional triangle-decorated lattice, we find that the fully polarized state -- normally thermalizing in the PXP model -- exhibits pronounced fidelity revivals, slow entanglement growth, and strong overlap with a tower of weakly entangled eigenstates. We trace this behavior to a geometry-induced restructuring of the constrained Hilbert space, whereby the adjacency graph decomposes into hypercube subgraphs that enforce coherent population transfer and stabilize an emergent approximate $\mathrm{su}(2)$ algebra. We propose a direct implementation in programmable arrays of tweezer-trapped Rydberg atoms, where the triangle-decorated geometry can be realized using spatial light modulators and the resulting scarring dynamics probed via time-resolved measurements of excitation density. Our results establish lattice connectivity as a design principle for engineering non-ergodic dynamics in constrained quantum systems.

cond-mat.quant-gasCross-listedarXiv:2605.05296•May 8, 2026•0 citations

Quantum Many-Body Scars beyond the PXP Model in Rydberg Simulators

Aron Kerschbaumer, Marko Ljubotina, Maksym Serbyn +1 more

Persistent revivals recently observed in Rydberg atom simulators have challenged our understanding of thermalization and attracted much interest to the concept of quantum many-body scars (QMBSs). QMBSs are non-thermal highly excited eigenstates that coexist with typical eigenstates in the spectrum of many-body Hamiltonians, and have since been reported in multiple theoretical models, including the so-called PXP model, approximately realized by Rydberg simulators. At the same time, questions of how common QMBSs are and in what models they are physically realized remain open. In this Letter, we demonstrate that QMBSs exist in a broader family of models that includes and generalizes PXP to longer-range constraints and states with different periodicity. We show that in each model, multiple QMBS families can be found. Each of them relies on a different approximate su(2) algebra, leading to oscillatory dynamics in all cases. However, in contrast to the PXP model, their observation requires launching dynamics from weakly entangled initial states rather than from a product state. QMBSs reported here may be experimentally probed using Rydberg atom simulator in the regime of longer-range Rydberg blockades.

arXiv:2410.18913•Physical Review Letters•Apr 22, 2025•17 citations

Asymmetric decay of quantum many-body scars in XYZ quantum spin chains

Dhiman Bhowmick, Vir B. Bulchandani, Wen Wei Ho

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 $S \to \infty$, 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.

arXiv:2505.05435•May 8, 2025•0 citations

Locality forces equal energy spacing of quantum many-body scar towers

Nicholas O'Dea, Lei Gioia, Sanjay Moudgalya +1 more

Quantum many-body scars are non-thermal eigenstates embedded in the spectra of otherwise non-integrable Hamiltonians. Paradigmatic examples often appear as quasiparticle towers of states, such as the maximally ferromagnetic spin-1/2 states, also known as Dicke states. A distinguishing feature of quantum many-body scars is that they admit multiple local "parent" Hamiltonians for which they are exact eigenstates. In this work, we show that the locality of such parent Hamiltonians strongly constrains the relative placement of these states within the energy spectrum. In particular, we prove that if the full set of Dicke states are exact eigenstates of an extensive local Hamiltonian, then their energies must necessarily be equally spaced. Our proof builds on recent results concerning parent Hamiltonians of the $W$ state, together with general algebraic structures underlying such quasiparticle towers. We further demonstrate that this equal spacing property extends to local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and expander graphs. Hamiltonians with $k$-local interactions and a bounded number of interaction terms per site are also encompassed by our proof. On the same classes of graphs, we additionally establish equal spacing for towers constructed from multi-site quasiparticles on top of product states. For the towers considered here, an immediate corollary of the equal spacing property is that any state initialized entirely within the quantum many-body scar manifold exhibits completely frozen entanglement dynamics under any local Hamiltonian for which those scars are exact eigenstates. Overall, our results reveal a stringent interplay between locality and the structure of quantum many-body scars.

quant-pharXiv:2601.14206•Jan 21, 2026•0 citations
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)
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)
This thesis investigates the anomalous dynamics observed in a Rydberg atom experiment in the broadest sense: a strongly interacting spin chain, without integrability or disorder, exhibits coherent many-body revivals ensuing only from specific initial states. This non-thermal behavior is attributed to the presence of quantum many-body scars (QMBS), a novel class of ergodicity breaking phenomena. Unlike other non-thermalizing systems such as integrable models and many-body localization, the mechanism of ergodicity breaking in QMBS is quite subtle, relying on a (vanishingly) small subset of anomalous eigenstates, known as scar states: one could observe ergodicity breaking only if initial states have appreciable overlap with these scar states. In other words, systems hosting QMBS are indistinguishable from typical chaotic, thermalizing systems, except for scar states. While the theoretical understanding of QMBS has advanced through the investigation of various toy quantum models, ranging from the Affleck-Kennedy-Lieb-Tasaki (AKLT) model to the Hubbard model, these insights do not fully resolve the mysterious oscillations in the Rydberg experiment: crucially the wavefunction corresponding to the experiment, and its relation to many other toy models in the literature remain open questions. Furthermore, beyond mathematical curiocity, the observability of non-thermal dynamics suggests that QMBS may yield distinct physical effects. the nature of which remains to be determined. This dissertation addresses these issues by solving two concrete problems: (i) the mathematical characterization of scar states in the PXP model, the effective model for the experiment, and (ii) the identification of physical consequences of QMBS in both short- and long-time dynamics. In the first part, we clarify the algebraic structure underlying the scar states of the PXP model and derive analytic ansatz for these wavefunctions. The proposed forms make explicit the hidden algebraic structure of the PXP scars and demonstrate how this structure is shared with many other QMBS Hamiltonians, including the AKLT model. The second part investigates dynamical signatures and robustness. Given the fragility of many-body eigenstates to perturbations, identifying which scar characteristics survive realistic perturbations is crucial. In closed systems, we show that out-of-time-ordered correlators (OTOCs) and the associated Lyapunov exponent can distinguish QMBS from generic chaotic states, at least within a semiclassical regime. Specifically, the OTOC growth associated with scar states is parametrically slower and remains slow under small ``scar-breaking'' perturbations. For open systems governed by non-Hermitian dynamics, we introduce a novel non-equilibrium phase, the QMBS phase, where the long-time wavefunction is localized in a small corner of the Hilbert space. We further identify a first-order transition between the QMBS phase and a thermal phase, where the wavefunction becomes delocalized.
Quantum many-body scars enable persistent non-ergodic dynamics in otherwise thermalizing systems, yet their stabilization typically relies on fine-tuned initial states or engineered Hamiltonian perturbations. Here we show that lattice geometry alone can serve as a powerful and experimentally accessible control knob for inducing and enhancing scarring. By transforming a one-dimensional chain into a quasi-one-dimensional triangle-decorated lattice, we find that the fully polarized state -- normally thermalizing in the PXP model -- exhibits pronounced fidelity revivals, slow entanglement growth, and strong overlap with a tower of weakly entangled eigenstates. We trace this behavior to a geometry-induced restructuring of the constrained Hilbert space, whereby the adjacency graph decomposes into hypercube subgraphs that enforce coherent population transfer and stabilize an emergent approximate su(2)\mathrm{su}(2)su(2) algebra. We propose a direct implementation in programmable arrays of tweezer-trapped Rydberg atoms, where the triangle-decorated geometry can be realized using spatial light modulators and the resulting scarring dynamics probed via time-resolved measurements of excitation density. Our results establish lattice connectivity as a design principle for engineering non-ergodic dynamics in constrained quantum systems.
Persistent revivals recently observed in Rydberg atom simulators have challenged our understanding of thermalization and attracted much interest to the concept of quantum many-body scars (QMBSs). QMBSs are non-thermal highly excited eigenstates that coexist with typical eigenstates in the spectrum of many-body Hamiltonians, and have since been reported in multiple theoretical models, including the so-called PXP model, approximately realized by Rydberg simulators. At the same time, questions of how common QMBSs are and in what models they are physically realized remain open. In this Letter, we demonstrate that QMBSs exist in a broader family of models that includes and generalizes PXP to longer-range constraints and states with different periodicity. We show that in each model, multiple QMBS families can be found. Each of them relies on a different approximate su(2) algebra, leading to oscillatory dynamics in all cases. However, in contrast to the PXP model, their observation requires launching dynamics from weakly entangled initial states rather than from a product state. QMBSs reported here may be experimentally probed using Rydberg atom simulator in the regime of longer-range Rydberg blockades.
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-SSS 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→∞S \to \inftyS→∞, 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.
Quantum many-body scars are non-thermal eigenstates embedded in the spectra of otherwise non-integrable Hamiltonians. Paradigmatic examples often appear as quasiparticle towers of states, such as the maximally ferromagnetic spin-1/2 states, also known as Dicke states. A distinguishing feature of quantum many-body scars is that they admit multiple local "parent" Hamiltonians for which they are exact eigenstates. In this work, we show that the locality of such parent Hamiltonians strongly constrains the relative placement of these states within the energy spectrum. In particular, we prove that if the full set of Dicke states are exact eigenstates of an extensive local Hamiltonian, then their energies must necessarily be equally spaced. Our proof builds on recent results concerning parent Hamiltonians of the WWW state, together with general algebraic structures underlying such quasiparticle towers. We further demonstrate that this equal spacing property extends to local Hamiltonians defined on arbitrary bounded-degree graphs, including regular lattices in any spatial dimension and expander graphs. Hamiltonians with kkk-local interactions and a bounded number of interaction terms per site are also encompassed by our proof. On the same classes of graphs, we additionally establish equal spacing for towers constructed from multi-site quasiparticles on top of product states. For the towers considered here, an immediate corollary of the equal spacing property is that any state initialized entirely within the quantum many-body scar manifold exhibits completely frozen entanglement dynamics under any local Hamiltonian for which those scars are exact eigenstates. Overall, our results reveal a stringent interplay between locality and the structure of quantum many-body scars.