Quantum Quenches that Resemble Operator Growth

The gist: Growth quenches are local quenches that may gradually destabilize a false vacuum in certain kinetic constrained quantum lattice models, and this study points out a formal analogy with the dynamics of a local operator in the Heisenberg picture.

How it works

The paper establishes an analogy between growth quenches and operator growth dynamics by mapping the time evolution of local operators to growth quenches in the doubled Hilbert space of operators, which is known as vectorization or Choi-Jamio lkowski map The identity operator plays the role of the false vacuum, and a local operator is defined as a growth quench initial state that differs from the identity/vacuum only at a few sites This mapping relates physically distinct quantities such as return amplitude with Green function, expectation value with outof-time order correlation functions, conserved quantities with Fock space cage states, and so on

The recursion method is applied to growth quenches by mapping any quantum dynamics to the quantum mechanics of a 1D semiinfinite “Krylov chain”, defined by the Lanczos coefficients Key results in this context generalize directly to growth quenches with little change in the proof The hypothesis conjectures the linear growth of Lanczos coefficients in generic quenches, stating that diagonal coefficients grow as am ∼ νm (diagonal), and off-diagonal ones as bm ∼ αm (off-diagonal)

Krylov Localization and Fock Localization

The paper investigates localization in Krylov space, finding that when ν > 2α, the dynamics in the Krylov space is localized This Krylov space localization implies a localization in the Fock space because the quench dynamics fails to spread therein and is confined by quantum coherent effect When ν < 2α, the spread complexity grows exponentially, K(t) ∼ e t√4α 2−ν squared This exponential growth rate is bounded by a growth quench “Lyapunov exponent” λL ≤ √4α squared − ν squared when ν < 2α

In large N solvable models inspired by Sachdev-Ye-Kitaev (SYK) models, the bound on the Lyapunov exponent is saturated in large-q SYK grow quench The Krylov/Fock localization here can be understood as a confinement by a linear potential (in the Krylov space) For the all-to-all mean field model, Fock localization is the generic behavior

Conserved Charges and Transport

In 1D growth quenches in the East-West model, Fock space cage states are interpreted as conserved charges These conserved quantities have ballistic transport in the pure East-West model due to current conservation When adding hopping with a fine-tuned amplitude g = 1/√2, the quench dynamics appears to be partially localized due to an almost flat band This partial localization is due to a flat band of Fock space bound states

Numerical and Solvable Models

The paper presents results from solvable large N models where the critical points coincide and the bound on Lyapunov exponent is saturated in (and only in) the large-q limit Numerical studies of finite q, large N SYK growth quench show a Krylov localization critical point at (U/J)c ≈ 1.125, or U/J ≈ 1.378, at which point 2α = ν In the deformed East-West model at g = 1/√2, the conserved quantity G˜(x, t) stays away from 0 and essentially maintains its phase as t increases, which is strong evidence for partial localization

Conclusion

The results show that the growth quench dynamics are localized in general and slowly and intermediately delocalized at a special point in deformed East-West models The existence of Fock localization due to linear onsite potential in the Krylov space is a novelty here The paper suggests that growth quenches are a promising approach to realize non-equilibrium coherent phenomena in many-body systems

How it works

The paper studies growth quenches, which are local quantum quenches in kinetic constrained models that have the global effect of destabilizing a false vacuum It imports the recursion method from operator growth to growth quenches and generalizes key results: general hypothesis on linear growth of Lanczos coefficients, and bound on Lyapunov exponent from spread complexity The novelty here is the possibility of Krylov and Fock space localization due to linear onsite potential (in the Krylov space), which we then realize in mean field models

Conserved Charges and Transport

The paper points out that in 1D growth quenches in (deformed) East-West models, Fock cage states are interpreted as conserved charges These conserved quantities have ballistic transport in the pure East-West model due to current conservation

Deformed East West: Partial Localization

The paper identifies a “magic point” g = 1/√2 where the growth quench dynamics is partially localized This partial localization is due to a flat band of Fock space bound states

Conclusion

The paper shows a simple solvable growth quench where Fock space localization leads to a ballistic transport of conserved charge Such a mechanism is also at play in the deformed East-West model at hopping amplitude g = 1/√2 (Sec. VII D) The Fock localization is only partial, and moreover, a flat band gives rise to partial physical localization

Acknowledgments

I thank Maurizio Fagotti, Ewan McCulloch, Maksym Serbyn, Thomas Scaffidi, and Lenart Zadnik for helpful discussions, and valuable feedback on a preliminary version of the manuscript The pre-print “Nonthermal dynamics protected by destructive interference in chaotic spin chains”, by Aron Kerschbaumer, Jean-Yves Desaules, and Maksym Serbyn, is appearing on the same arXiv posting [122]

Appendix A: Proof of bounds

In analogy with operators, we define the support of a state to be supp(Ψ⟩) = U This notion of state support reduces to the usual notion of operator support under the correspondence The moment bound can be improved for the reason that we briefly recall [48, 87, 88]

Appendix B: Moments from Lanczos

The basic point is that when m is large, and if am, bm depend smoothly on m, as in (33), the sum over Motzkin paths of (B2) can be evaluated by a saddle point method The minimization of the action can be in principle carried out using the Euler-Langrange equation

Appendix C: A solvable 1D growth quench

The Hamiltonian is H2 = Xj [Xj (nj+1 − nj−1) + Unj] This tower of conserved quantities is reminiscent of the infinite sequence of conserved charges in non-interacting and integrable systems The Krylov space will have q-depend energies, so we have bands with non-trivial dispersion relation, and thus ballistic transport of the conserved quantity (with m = 1)

Appendix D: Numerical study

The numerical study shows that the data is obtained by a time-evolved bond decimation (TEBD) simulation<ref:2605.

1. Precise prediction of localization transitions in quantum systems: The model can predict a localization transition at the threshold (2α/ν)c = ±1 in Krylov space and Fock space, allowing AI to identify parameter regimes where dynamics switch from delocalized to localized behavior based on calculated Lyapunov exponents.

2. Characterization of transport properties in kinetic systems: The system can predict ballistic transport for the pure East-West model (g = 0) and partial localization with a fine-tuned amplitude g = 1/√2, enabling AI to distinguish between current conservation effects and hopping-induced localization.

3. Modeling of non-equilibrium coherent phenomena: The AI system can simulate the growth quench dynamics in all-to-all mean field models, which is shown to generate multi-particle states with volume-law entanglement, allowing it to study and predict non-equilibrium coherent phenomena.

4. Computationally efficient simulation of many-body dynamics: By using the recursion method adapted from operator growth, the system can compute Lanczos coefficients efficiently and estimate Lyapunov exponents, which are related to spread complexity as a bound on scrambling.

5. Identification of conserved quantities as physical observables: The AI can identify Fock space cage states in 1D models as analogues to conserved charges, allowing it to predict their transport behavior, such as the ballistic transport of the conserved quantity in the pure East-West model.