An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Luck, J. M.

doi:10.1088/1751-8113/49/11/115303

Cited by 15 publications

(46 citation statements)

References 47 publications

(74 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1), is significantly smaller than ln (2). A very similar phenomenon has already been observed in the much simpler situation of a tight-binding particle in a random potential with binary disorder [7], where the hybridization of degenerate molecular states has been shown to result in a nontrivial asymptotic return probability Q ≈ 0.373 in the limit of an infinitely strong disorder.…”

Section: Gapped Phasesupporting

confidence: 67%

“…It can range from T min = 1 (all the eigenstates spread maximally over the whole basis) to T max = D (each eigenstate matches exactly a single basis state). This quantity was originally introduced [7] from a dynamical point of view, as the trace of the matrix Q whose entries Q ab are the timeaverage probabilities to go from state |a at the initial time to state |b at time t. If the system is prepared in state |a at t = 0, the probability to observe it in state |b at time t is P ab (t) = | b|ψ(t) | 2 , i.e., P ab (t) = m,n e i(En−Em)t b|m m|a a|n n|b .…”

Section: Generalitiesmentioning

confidence: 99%

“…In the case of a degenerate multiplet of eigenstates |n 1 , · · · , |n d , the corresponding contribution to T (as derived from the dynamical point of view [7]) should . For open boundary conditions and a generic value of ∆, it turns out that the only degeneracies in the spectrum of Eq.…”

Section: B Remarks On Degeneraciesmentioning

confidence: 99%

See 2 more Smart Citations

Inverse participation ratios in the XXZ spin chain

2016

View full text Add to dashboard Cite

We investigate numerically the inverse participation ratios in a spin-1/2 XXZ chain, computed in the "Ising" basis (i.e., eigenstates of $\sigma^z_i$). We consider in particular a quantity $T$, defined by summing the inverse participation ratios of all the eigenstates in the zero magnetization sector of a finite chain of length $N$, with open boundary conditions. From a dynamical point of view, $T$ is proportional to the stationary return probability to an initial basis state, averaged over all the basis states (initial conditions). We find that $T$ exhibits an exponential growth, $T\sim\exp(aN)$, in the gapped phase of the model and a linear scaling, $T\sim N$, in the gapless phase. These two different behaviors are analyzed in terms of the distribution of the participation ratios of individual eigenstates. We also investigate the effect of next-nearest-neighbor interactions, which break the integrability of the model. Although the massive phase of the non-integrable model also has $T\sim\exp(aN)$, in the gapless phase $T$ appears to saturate to a constant value.Comment: 8 pages, 7 figures. v2: published version (one figure and 3 references added, several minor changes

show abstract

Section: Gapped Phasesupporting

confidence: 67%

Section: Generalitiesmentioning

confidence: 99%

See 1 more Smart Citation

Inverse participation ratios in the XXZ spin chain

2016

View full text Add to dashboard Cite

show abstract

“…Finally, the case of a tight-binding particle on a finite electrified chain is considered in section 5. This situation is somehow a deterministic analogue of the case considered in [7], where the ladder of Wannier-Stark states replaces Anderson localized states. It is however richer, as it features two successive scaling regimes in the small-field region.…”

Section: Introductionmentioning

confidence: 97%

“…This quantity has been put forward as a measure of the degree of equilibration of the full system if launched from a typical basis state. This line of thought has been illustrated by means of a detailed study of a single tight-binding particle on a finite segment of N sites in a random potential [7]. The main findings concern the regime of a weak disorder, where T exhibits a finite value T = T (0) ≈ 3/2 in the ballistic regime, testifying good equilibration, a linear growth with the sample size (T ≈ N Q) in the localized regime, testifying poor equilibration, and a universal scaling law of the form…”

Section: Introductionmentioning

confidence: 99%

Equilibration properties of small quantum systems: further examples

Luck¹

2017

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

It has been proposed to investigate the equilibration properties of a small isolated quantum system by means of the matrix of asymptotic transition probabilities in some preferential basis. The trace T of this matrix measures the degree of equilibration of the system prepared in a typical state of the preferential basis. This quantity may vary between unity (ideal equilibration) and the dimension N of the Hilbert space (no equilibration at all). Here we analyze several examples of simple systems where the behavior of T can be investigated by analytical means. We first study the statistics of T when the Hamiltonian governing the dynamics is random and drawn from a distribution invariant under the group U(N ) or O(N ). We then investigate a quantum spin S in a tilted magnetic field making an arbitrary angle with the preferred quantization axis, as well as a tight-binding particle on a finite electrified chain. The last two cases provide examples of the interesting situation where varying a system parameter -such as the tilt angle or the electric field -through some scaling regime induces a continuous transition from good to bad equilibration properties.

show abstract

Krylov localization and suppression of complexity

Rabinovici

Sánchez-Garrido

Shir

et al. 2022

J. High Energ. Phys.

View full text Add to dashboard Cite

Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

show abstract

An investigation of equilibration in small quantum systems: the example of a particle in a 1D random potential

Cited by 15 publications

References 47 publications

Inverse participation ratios in the XXZ spin chain

Inverse participation ratios in the XXZ spin chain

Equilibration properties of small quantum systems: further examples

Krylov localization and suppression of complexity

Contact Info

Product

Resources

About