Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain

Marlon Brenes, John Goold, and Marcos Rigol

Phys. Rev. B 102, 075127 – Published 19 August 2020

Abstract

We study the matrix elements of local operators in the eigenstates of the integrable XXZ chain and of the quantum-chaotic model obtained by locally perturbing the XXZ chain with a magnetic impurity. We show that, at frequencies that are polynomially small in the system size, the behavior of the variances of the off-diagonal matrix elements can be starkly different depending on the operator. In the integrable model we find that, as the frequency $ω \to 0$ , the variances are either nonvanishing (generic behavior) or vanishing (for a special class of operators). In the quantum-chaotic model, on the other hand, we find the variances to be nonvanishing as $ω \to 0$ and to indicate diffusive dynamics. We highlight which properties of the matrix elements of local operators are different between the integrable and quantum-chaotic models independently of the specific operator selected.

Received 25 May 2020
Revised 16 July 2020
Accepted 30 July 2020

DOI:https://doi.org/10.1103/PhysRevB.102.075127

Physics Subject Headings (PhySH)

Eigenstate thermalization Nonequilibrium statistical mechanics Quantum chaos

1-dimensional spin chains

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Marlon Brenes ¹, John Goold ¹, and Marcos Rigol ²

¹School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland
²Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA

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Vol. 102, Iss. 7 — 15 August 2020

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Images

Figure 1
Diagonal matrix elements of (a) and (b) ${\hat{σ}}_{N / 2}^{z}$ and (c) and (d) ${\hat{T}}^{NN}$ in the eigenstates (a) and (c) of the (integrable) XXZ and (b) and (d) of the (nonintegrable) single-impurity models for $Δ = 0.55$ (main panels) and $Δ = 1.1$ (insets) for different chain sizes $N$ . The black lines correspond to the microcanonical averages (within windows with $δ ε_{n} = 0.008$ ) for the largest chain ( $N = 20$ ). We plot the matrix elements vs the energy density $ε_{n}$ , defined as $ε_{n} : = E_{n} - E_{min} / E_{max} - E_{min}$ , where $E_{n}$ is the $n th$ energy eigenvalue and $E_{min}$ ( $E_{max}$ ) is the ground-state (highest) energy eigenvalue.
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Figure 2
Average (a) and (b) $\bar{| {[{\hat{σ}}_{N / 2}^{z}]}_{n m} |^{2}}$ and (c) and (d) $\bar{| {[{\hat{T}}^{NN}]}_{n m} |^{2}}$ vs $ω$ in the (a) and (c) XXZ and (b) and (d) single-impurity models for $Δ = 0.55$ (main panels) and $Δ = 1.1$ (insets) for different chain sizes $N$ . The matrix elements were computed within a small window of energy around $\bar{E} \approx 0$ of width $0.05 ε$ , where $ε : = E_{max} - E_{min}$ denotes the bandwidth. The averages in $ω$ were calculated in windows with $δ ω = 0.1$ .
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Figure 3
Probability distribution of the off-diagonal matrix elements of (a) and (b) ${\hat{σ}}_{N / 2}^{z}$ and (c) and (d) ${\hat{T}}^{NN}$ in the eigenstates of (a) and (c) ${\hat{H}}_{XXZ}$ and (b) and (d) ${\hat{H}}_{SI}$ . The matrix elements used to compute the distributions were selected within a small window of energy around $\bar{E} \approx 0$ of width $0.05 ε$ , where $ε : = E_{max} - E_{min}$ denotes the bandwidth, and $ω \leq 0.05$ .
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Figure 4
(a) and (b) $Γ_{{\hat{σ}}_{N / 2}^{z}}$ and (c) and (d) $Γ_{{\hat{T}}^{NN}}$ vs $ω$ [see Eq. (6)] in the (a) and (c) XXZ and (b) and (d) single-impurity models for $Δ = 0.55$ (main panels) and $Δ = 1.1$ (insets) for different chain sizes $N$ . The horizontal line in (b) and (d) marks $π / 2$ . The matrix elements were computed using the same energy window as in Fig. 2, while the coarse-graining parameter was chosen to be $δ ω = 0.05$ .
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Figure 5
Scaled variances of the off-diagonal matrix elements of (a) and (b) ${\hat{σ}}_{N / 2}^{z}$ , (c) and (d) ${\hat{T}}^{NN}$ , and (e) and (f) $\hat{T}$ in the (a), (c), and (e) XXZ and (b), (d), and (f) single-impurity models for different chain sizes $N$ . The main panels show results for $Δ = 0.55$ , while the insets show results for $Δ = 1.1$ . The matrix elements were computed within a small window of energy around $\bar{E} \approx 0$ of width $0.075 ɛ$ . For the binned averages, we used $δ ω = 0.06$ for the integrable (left) and $δ ω = 5 \times 10^{- 4}$ for the quantum-chaotic (right) models, such that smooth curves are obtained that are robust against changes in $δ ω$ .
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