Universality in the onset of quantum chaos in many-body systems

Letter

Universality in the onset of quantum chaos in many-body systems

Tyler LeBlond, Dries Sels, Anatoli Polkovnikov, and Marcos Rigol

Phys. Rev. B 104, L201117 – Published 30 November 2021

Abstract

We show that the onset of quantum chaos at infinite temperature in two many-body one-dimensional lattice models, the perturbed spin-1/2 XXZ and Anderson models, is characterized by universal behavior. Specifically, we show that the onset of quantum chaos is marked by maxima of the typical fidelity susceptibilities that scale with the square of the inverse average level spacing, saturating their upper bound, and that the strength of the integrability- or localization-breaking perturbation at these maxima decreases with increasing system size. We also show that the spectral function below the “Thouless” energy (in the quantum-chaotic regime) diverges when approaching those maxima. Our results suggest that, in the thermodynamic limit, arbitrarily small integrability- or localization-breaking perturbations result in quantum chaos in the many-body quantum systems studied here.

Received 30 December 2020
Accepted 16 November 2021

DOI:https://doi.org/10.1103/PhysRevB.104.L201117

Physics Subject Headings (PhySH)

Anderson localization Eigenstate thermalization Quantum statistical mechanics

Integrable systems

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Tyler LeBlond¹, Dries Sels^2,3, Anatoli Polkovnikov ⁴, and Marcos Rigol ¹

¹Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
²Center for Computational Quantum Physics, Flatiron Institute, New York, New York 10010, USA
³Department of Physics, New York University, New York, New York 10003, USA
⁴Department of Physics, Boston University, Boston, Massachusetts 02215, USA

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Vol. 104, Iss. 20 — 15 November 2021

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Images

Figure 1
Typical fidelity susceptibility $χ_{typ}^{}$ (scaled to exhibit collapse in the quantum-chaotic regime) vs the integrability-breaking parameter $Δ^{'}$ for observables ${\hat{U}}_{n}$ (a) and ${\hat{K}}_{nn}$ (b) in clean periodic chains. To calculate $χ_{typ}^{}$ and $ω_{H}^{}$ , we average over the central 50% of the eigenstates in the even- $Z_{2}$ sector in each total quasimomentum sector considered. For $L < 24$ , we report the weighted average over all $k \neq (0, π)$ sectors, while for $L = 24$ we report results for the $k = π / 2$ sector. Circles on the $y$ axis show $χ_{typ}^{}$ at the integrable point ( $Δ^{'} = 0$ ), and diamonds show the maximal $χ_{typ}^{*}$ (at $Δ^{' *} = - b / 2 a$ ) obtained from polynomial fits $a x^{2} + b x + c$ (black solid lines about the maxima). The dotted lines on the right of the first peaks are a guide for the eye and depict $Δ^{' - 2.55}$ behavior. Inset in (a): $χ_{typ}^{*}$ vs $ω_{H}^{}$ for both observables, along with the results of power-law fits. Inset in (b): $Δ^{' *}$ vs $ω_{H}$ for both observables (the values of $Δ^{' *}$ overlap). The dotted line depicts $ω_{H}^{0.39}$ behavior.
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Figure 2
Spectral functions in clean periodic chains with $L = 24$ for observables ${\hat{U}}_{n}$ [(a) and (b)] and ${\hat{K}}_{nn}$ [(c) and (d)] over 2 decades of the integrability-breaking parameter $Δ^{'}$ [see labels at the top and legends in (b) and (d)]. In (a) and (c), the top insets show $F_{O} = {(ω / Δ^{'})}^{2} {| f_{O} (ω) |}^{2}$ vs $ω / Δ^{'}$ at $Δ^{'} = 1.58 \times 10^{- 2}$ , while the bottom insets show $| f_{O} {(ω) |}^{2}$ vs $ω / Δ^{'}$ at $Δ^{'} = 1.58 \times 10^{- 1}$ , for the three largest chains studied. The insets in (b) and (d) show $| f_{O}^{p} {(ω) |}^{2}$ vs $Δ^{'}$ , where $| f_{O}^{p} {(ω) |}^{2}$ is the value of $| f_{O} {(ω) |}^{2}$ at the plateaus in the main panels (and for other values of $Δ^{'}$ for which $| f_{O} {(ω) |}^{2}$ is not shown). The dotted lines are a guide for the eye and depict $Δ^{' - 2.55}$ behavior. All computations were done as for Fig. 1.
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Figure 3
Spectral functions in disordered periodic chains with $L = 18$ for observables ${\hat{K}}_{n}$ [(a) and (b)] and ${\hat{U}}_{n}$ [(c) and (d)] over 2 decades of the interaction strength $Δ$ [see labels at the top, and legends in (b) and (d)]. In (a) and (c), the top insets show $F_{O} = {(ω / Δ^{'})}^{2} {| f_{O} (ω) |}^{2}$ vs $ω / Δ^{'}$ at $Δ^{'} = 1.58 \times 10^{- 2}$ , while the bottom insets show $| f_{O} {(ω) |}^{2}$ vs $ω / Δ^{'}$ at $Δ^{'} = 1.58 \times 10^{- 1}$ , for the three largest chains studied. The insets in (b) and (d) show $| f_{O}^{p} {(ω) |}^{2}$ vs $Δ$ , where $| f_{O}^{p} {(ω) |}^{2}$ is the value of $| f_{O} {(ω) |}^{2}$ at the plateaus in the main panels (and for other values of $Δ$ for which $| f_{O} {(ω) |}^{2}$ is not shown). The dotted lines are a guide for the eye and depict $Δ^{' - 2.4}$ behavior. To calculate $| f_{O} {(ω) |}^{2}$ , we average over the central 50% of the eigenstates in each chain and then over disorder realizations (200 for $L \leq 16$ , 100 for $L = 17$ , and 50 for $L = 18$ ).
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Figure 4
Typical fidelity susceptibility $χ_{typ}^{}$ (scaled to exhibit collapse in the quantum-chaotic regime) vs the interaction strength $Δ$ for observables ${\hat{K}}_{n}$ (a) and ${\hat{U}}_{n}$ (b) in disordered periodic chains. Circles on the $y$ axis show $χ_{typ}^{}$ at the Anderson-localized point ( $Δ = 0$ ), and diamonds show the maximal $χ_{typ}^{*}$ (at $Δ^{*} = - b / 2 a$ ) obtained from polynomial fits $a x^{2} + b x + c$ (black solid lines about the maxima). The inset in (a) shows $χ_{typ}^{*}$ vs $ω_{H}^{}$ for both observables, along with the results of power-law fits. The error bars are the (propagated) standard deviation of the average over disorder realizations (see Ref. [71] for details) at the value of $Δ$ (for which we carried out a calculation) that is closest to $Δ^{*}$ . The inset in (b) shows $Δ^{*}$ vs $ω_{H}$ for both observables. The dotted line depicts $ω_{H}^{0.28}$ behavior. All computations were done as for Fig. 3.
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