Inverse participation ratios in the XXZ spin chain

Misguich, Grégoire; Pasquier, Vincent; Luck, J. M.

doi:10.1103/physrevb.94.155110

Cited by 22 publications

(16 citation statements)

References 22 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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“…This result is in stark contrast to the participation ratios summing over all the eigenstates in Ref. [53]. The WYSI can be used as an efficient measure to quantify quantum coherence (QC), but it is rarely adopted to detect QPTs.…”

Section: Numerical Resultsmentioning

confidence: 87%

Quantum phase transitions in spin-1 XXZ chains with rhombic single-ion anisotropy

2018

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We explore numerically the inverse participation ratios in the ground-state of one-dimensional spin-1 XXZ chains with the rhombic single-ion anisotropy. By employing the techniques of density matrix renormalization group, effects of the rhombic single-ion anisotropy on various information theoretical measures are investigated, such as the fidelity susceptibility, the quantum coherence and the entanglement entropy. Their relations with the quantum phase transitions are also analyzed. The phase transitions from the Y-Néel phase to the large-Ex or the Haldane phase can be well characterized by the fidelity susceptibility. The second-order derivative of the ground-state energy indicates all the transitions are of second order. We also find that the quantum coherence, the entanglement entropy, the Schmidt gap and the inverse participation ratios can be used to detect the critical points of quantum phase transitions. Results drawn from these quantum information observables agree well with each other. Finally we provide a ground-state phase diagram as functions of the exchange anisotropy ∆ and the rhombic single-ion anisotropy E.

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“…Recently in [MPL16], Misguich, Pasquier and Luck began a numerical study of the so-called Inverse Participation Ratios (IPRs) in the spin-1/2 XXZ chains. IPRs give a measure of localization of the system with respect to a preferential basis.…”

Section: Introductionmentioning

confidence: 99%

“…

We continue the study of the Inverse Participation Ratios (IPRs) of the XXZ Heisenberg spin chain initiated by Misguich, Pasquier and Luck (2016) by focusing on the case of the XX Heisenberg Spin Chain. For the ground state, Misguich et al note that calculating the IPR is equivalent to Dyson's constant term ex-conjecture.

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confidence: 99%

Inverse participation ratios in the XX spin chain

Tsukerman

2017

Phys. Rev. B

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Abstract. We continue the study of the Inverse Participation Ratios (IPRs) of the XXZ Heisenberg spin chain initiated by Misguich, Pasquier and Luck (2016) by focusing on the case of the XX Heisenberg Spin Chain. For the ground state, Misguich et al. note that calculating the IPR is equivalent to Dyson's constant term ex-conjecture. We express the IPRs of excited states as an apparently new "discrete" Hall inner product. We analyze this inner product using the theory of symmetric functions (Jack polynomials, Schur polynomials, the standard Hall inner product and ωq,t) to determine some exact expressions and asymptotics for IPRs. We show that IPRs can be indexed by partitions, and asymptotically the IPR of a partition is equal to that of the conjugate partition. We relate the IPRs to two other models from physics, namely, the circular symplectic ensemble of Dyson and the DysonGaudin two-dimensional Coulomb lattice gas. Finally, we provide a description of the IPRs in terms of a signed count of diagonals of permutohedra.

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Inverse participation ratios in the XXZ spin chain

Cited by 22 publications

References 22 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

Quantum phase transitions in spin-1 XXZ chains with rhombic single-ion anisotropy

Inverse participation ratios in the XX spin chain

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