Inverse participation ratios in the XX spin chain

Tsukerman, Emmanuel

doi:10.1103/physrevb.95.115121

Cited by 9 publications

(5 citation statements)

References 10 publications

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“…The maxima of random waves and chaotic eigenstates have also been considered [15,40]. Eigenstate statistics have often been characterized through the inverse participation ratio, extensively over several decades for single-particle systems [17,[41][42][43][44] and more recently also for many-body systems [23,34,36,[45][46][47][48][49][50]. Generalizing the inverse participation ratio, eigenstate statistics has also been studied through the Shannon and Rényi entropies [45,[51][52][53][54][55][56][57].…”

Section: Introductionmentioning

confidence: 99%

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

2019

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Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems -the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction -the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic", in the sense that their eigenfunction statistics deviate from random matrix theory. arXiv:1905.03099v2 [cond-mat.stat-mech]

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Section: Introductionmentioning

confidence: 99%

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

2019

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“…More importantly, the Schmidt number as a metric of entanglement allowed us to characterize the winding number as topological invariant in the SSH model. We found a strong relation between the Schmidt number and the inverse participation ratio that quantifies the degree of delocalization of the wave function on the system [46] because in the topological phase transitions both quantities are maximal. This represents a new paradigm towards the understanding the behavior of these topological materials properties and also opens the possibility to explore the hybrid-nature entangled states studied, as well as their potential application in quantum information processing.…”

Section: Discussionmentioning

confidence: 87%

“…A measure of localization of the wave function is defined by the inverse participation ratio (IPR) [46]. It is a simple way to quantify how many states a particle is distributed over when there is some intrinsic uncertainly about where is the particle.…”

Section: Inverse Participation Ratiomentioning

confidence: 99%

Geometrical phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings interaction

Navarro-Labastida¹,

Domínguez-Serna²,

Rojas³

2021

Preprint

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We study the hybrid atoms-cell site entanglement in one-dimensional Su-Schrieffer-Heeger (SSH) topological insulator with first, second neighbor-hopping interaction and intra-cell modulation, in space representation of finite chains. We determine the geometric phases by the Resta electric polarization and the entanglement in the atomic basis by the Schmidt number as metric of the degree of non-classicity, the maximally entangled quantum states are characterized by the inverse participation ratio IPR. A relation between entanglement and the topological phase transitions (TPT) are found, since the Schmidt number presented critical points of maximally entangled (ME) states just in the singularities of the change of the winding number for the nearest states to the zero mode energy and shifted for higher energy state.We present the general conditions that has to be satisfy to produce a ME hybrid Bell states. We obtain that the states with second neighbors, with two phases, shows a higher degree of entanglement than the first order hopping. We obtain parameter conditions with broad regions for non null topological phases that can have maximum entanglement. We find a robust relation between entanglement and the inverse participation ratio (IPR) for the the eigenstates with ME which are associated to maximum delocalization in both models. The amplitude distribution for each atom are spatially localized in cell site showing a bunching in cell sets. The intra-cell modulation provides a better connection between entanglement and electric polarization with a periodical driven period. The study of topological properties through entanglement is an complementary tool for determining macroscopic properties and analyzing more complex systems with many-body particles or higher dimensions.

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“…4(b), the rightmost case with g = 1 is topological equality of that in Fig. 4(a), where the two pairs of corner states have relatively large inverse participation ratio (IPR = i | ψ|i | 4 for a normalized state |ψ with i | ψ|i | 2 = 1) [84][85][86][87][88] because of their comparatively high degree of localization and the acuteangle corner state is more localized than the obtuse-angle one. Furthermore, the spectrum evolves with a decreasing g. The edge gap formed between the (green-blue) edge states moves downwards as a whole, in which course, the (green) obtuse-angle corner states remain in the middle of the gap, indicating the stability of the obtuse-angle corner state in two senses: (i) its energy position relative to the edge gap, and (ii) its degree of localization; both of which are effectively unaffected.…”

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confidence: 86%

Topological Corner States in Graphene by Bulk and Edge Engineering

Zeng¹,

Chen²,

Ren³

et al. 2022

Preprint

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Two-dimensional higher-order topology is usually studied in (nearly) particle-hole symmetric models, so that an edge gap can be opened within the bulk one. But more often deviates the edge anticrossing even into the bulk, where corner states are difficult to pinpoint. We address this problem in a graphene-based Z2 topological insulator with spin-orbit coupling and in-plane magnetization both originating from substrates through a Slater-Koster multi-orbital model. The gapless helical edge modes cross inside the bulk, where is also located the magnetization-induced edge gap. After demonstrating its second-order nontriviality in bulk topology by a series of evidence, we show that a difference in bulk-edge onsite energy can adiabatically tune the position of the crossing/anticrossing of the edge modes to be inside the bulk gap. This can help unambiguously identify two pairs of topological corner states with nonvanishing energy degeneracy for a rhombic flake. We further find that the obtuse-angle pair is more stable than the acute-angle one. These results not only suggest an accessible way to "find" topological corner states, but also provide a higher-order topological version of "bulk-boundary correspondence".

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Inverse participation ratios in the XX spin chain

Cited by 9 publications

References 10 publications

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

Geometrical phases and entanglement in real space for 1D SSH topological insulator: effects of first and second neighbor-hoppings interaction

Topological Corner States in Graphene by Bulk and Edge Engineering

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