Probing quantum scars and weak ergodicity breaking through quantum complexity

Bhattacharjee, Budhaditya; Sur, Samudra; Nandy, Pratik

doi:10.1103/physrevb.106.205150

Cited by 48 publications

(15 citation statements)

References 105 publications

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“…We only discuss operator complexity in this paper. For state complexity and its applications, the reader is directed to [20,45,46,50].…”

Section: Lanczos Algorithm and Krylov Complexitymentioning

confidence: 99%

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A Lanczos approach to the Adiabatic Gauge Potential

Bhattacharjee¹

2023

Preprint

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The Adiabatic Gauge Potential (AGP) is the generator of adiabatic deformations between quantum eigenstates. There are many ways to construct the AGP operator and evaluate the AGP norm. Recently, it was proposed that a Gram-Schmidt-type algorithm can be used to explicitly evaluate the expression of the AGP [1]. We employ a version of this approach by using the Lanczos algorithm to evaluate the AGP operator in terms of Krylov vectors and the AGP norm in terms of the Lanczos coefficients. It has the advantage of minimizing redundancies in evaluating the nested commutators in the analytic expression for the AGP operator. The algorithm is used to explicitly construct the AGP operator for some simple systems. We derive an integral transform relation between the AGP norm and the autocorrelation function of the deformation operator. We present a modification of the Variational approach to derive the regulated AGP norm. Using this, we approximate the AGP to varying degrees of success. Finally, we compare and contrast the quantum-chaos-probing capacities of the AGP and K-complexity in view of the Operator Growth Hypothesis.

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“…We only discuss operator complexity in this paper. For state complexity and its applications, the reader is directed to [20,45,46,50].…”

Section: Lanczos Algorithm and Krylov Complexitymentioning

confidence: 99%

“…been found in real-life examples while studying the state complexity of many-body scars [46]. The Hamiltonian of such a system is given by…”

Section: Chaotic: Bn ∼ Nmentioning

confidence: 99%

A Lanczos approach to the Adiabatic Gauge Potential

Bhattacharjee¹

2023

Preprint

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“…Starting from by some initial state ⟨w 0 and u 0 ⟩, one follows the following recursive algorithm [58]: The algorithm recasts the Hamiltonian into the form (C.1) (and thereby (2.12)). Therefore, one can choose an initial state (in a lattice system, this can be thought of as a ground state of some arbitrary Hamiltonian, see [59] for an example) and evolve this with the non-Hermitian Hamiltonian. In the case of a Hermitian Hamiltonian, the tridiagonalized matrix form of the Hamiltonian is usually known to have real, but nonzero, diagonal coefficients [57,60].…”

Section: Appendix: a Generalized Version Of Spread Complexitymentioning

confidence: 99%

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Bhattacharya¹,

Nandy²,

Nath³

et al. 2023

Preprint

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Continuing the previous initiatives [1,2], we pursue the exploration of operator growth and Krylov complexity in dissipative open quantum systems. In this paper, we resort to the bi-Lanczos algorithm generating two bi-orthogonal Krylov spaces, which individually generate non-orthogonal subspaces. Unlike the previously studied Arnoldi iteration, this algorithm renders the Lindbladian into a purely tridiagonal form, thus opening up a possibility to study a wide class of dissipative integrable and chaotic systems by computing Krylov complexity at late times. Our study relies on two specific systems, the dissipative transverse-field Ising model (TFIM) and the dissipative interacting XXZ chain. We find that, for the weak coupling, initial Lanczos coefficients can efficiently distinguish integrable and chaotic evolution before the dissipative effect sets in, which results in more fluctuations in higher Lanczos coefficients. This results in the equal saturation of late-time complexity for both integrable and chaotic cases, making the notion of late-time chaos dubious.

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“…The minimisation of the spread complexity in the Krylov basis requires extracting the Lanczos coefficients associated to the Hamiltonian of the model 8 . A previous discussion in the literature [117] attempted to approximate the broken su (2) symmetry within the scar subspace of the parent Hamiltonian using a quantum-deformed su (2) q symmetry and associated Lanczos coefficients. However, as we explicitly demonstrate in this work one needs to be cautious both in the terms of capturing the symmetry breaking via su (2) q and interpreting its physical implications.…”

Section: Introductionmentioning

confidence: 99%

Quantum state complexity meets many-body scars

Nandy,

Mukherjee,

Bhattacharyya

et al. 2024

J. Phys.: Condens. Matter

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Scar eigenstates in a many-body system refers to a small subset of non-thermal ﬁnite energy density eigenstates embedded into an otherwise thermal spectrum. This novel non-thermal behaviour has been seen in recent experiments simulating a one-dimensional PXP model with a kinetically-constrained local Hilbert space realized by a chain of Rydberg atoms. We probe these small sets of special eigenstates starting from particular initial states by computing the spread complexity associated to time evolution of the PXP hamiltonian. Since the scar subspace in this model is embedded only loosely, the scar states form a weakly broken representation of the Lie Algebra. We demonstrate why a careful usage of the Forward Scattering Approximation (FSA), instead of any other method, is required to extract the most appropriate set of Lanczos coefﬁcients in this case as the consequence of this approximate symmetry. Only such a method leads to a well deﬁned notion of a closed Krylov subspace and consequently, that of spread complexity. We show this using three separate initial states, namely |ℤ2〉,|ℤ3〉 and the vacuum state, due to the disparate classes of scar states hosted by these sectors. We also discuss systematic methods of remedying the imperfections in the FSA setup stemming from these approximate symmetries.

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Probing quantum scars and weak ergodicity breaking through quantum complexity

Cited by 48 publications

References 105 publications

A Lanczos approach to the Adiabatic Gauge Potential

A Lanczos approach to the Adiabatic Gauge Potential

On Krylov complexity in open systems: an approach via bi-Lanczos algorithm

Quantum state complexity meets many-body scars

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