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arXiv:1901.06971 · Physical review. B./Physical review. B

Many-body delocalization dynamics in long Aubry-André quasiperiodic chains

Elmer V. H. Doggen, A. D. Mirlin

Published Sep 13, 2019• 80 citations
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Abstract

We study quench dynamics in an interacting spin chain with a quasiperiodic on-site field, known as the interacting Aubry-Andr\'e model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to $L=50$. We find that the choice of periodicity $\mathrm{\ensuremath{\Phi}}$ of the quasiperiodic field influences the dynamics. For $\mathrm{\ensuremath{\Phi}}=(\sqrt{5}\ensuremath{-}1)/2$ (the inverse golden ratio) and interaction $\mathrm{\ensuremath{\Delta}}=1$, the model most frequently considered in the literature, we obtain the critical disorder ${W}_{c}=4.8\ifmmode\pm\else\textpm\fi{}0.5$ in units where the noninteracting transition is at $W=2$. At the same time, for periodicity $\mathrm{\ensuremath{\Phi}}=\sqrt{2}/2$ we obtain a considerably higher critical value, ${W}_{c}=7.8\ifmmode\pm\else\textpm\fi{}0.5$. Finite-size effects on the critical disorder ${W}_{c}$ are much weaker than in the purely random case. This supports the enhancement of ${W}_{c}$ in the case of a purely random potential by rare ``ergodic spots,'' which do not occur in the quasiperiodic case. Further, the data suggest that the decay of the antiferromagnetic order in the delocalized phase is faster than a power law.

DOI
10.1103/physrevb.100.104203

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We study quench dynamics in an interacting spin chain with a quasiperiodic on-site field, known as the interacting Aubry-Andr'e model of many-body localization. Using the time-dependent variational principle, we assess the late-time behavior for chains up to L=50L=50L=50. We find that the choice of periodicity \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ of the quasiperiodic field influences the dynamics. For \ensuremathΦ=(5\ensuremath−1)/2\mathrm{\ensuremath{\Phi}}=(\sqrt{5}\ensuremath{-}1)/2\ensuremathΦ=(5​\ensuremath−1)/2 (the inverse golden ratio) and interaction \ensuremathΔ=1\mathrm{\ensuremath{\Delta}}=1\ensuremathΔ=1, the model most frequently considered in the literature, we obtain the critical disorder Wc=4.8\ifmmode±\else\textpm\fi0.5{W}_{c}=4.8\ifmmode\pm\else\textpm\fi{}0.5Wc​=4.8\ifmmode±\else\textpm\fi0.5 in units where the noninteracting transition is at W=2W=2W=2. At the same time, for periodicity \ensuremathΦ=2/2\mathrm{\ensuremath{\Phi}}=\sqrt{2}/2\ensuremathΦ=2​/2 we obtain a considerably higher critical value, Wc=7.8\ifmmode±\else\textpm\fi0.5{W}_{c}=7.8\ifmmode\pm\else\textpm\fi{}0.5Wc​=7.8\ifmmode±\else\textpm\fi0.5. Finite-size effects on the critical disorder Wc{W}_{c}Wc​ are much weaker than in the purely random case. This supports the enhancement of Wc{W}_{c}Wc​ in the case of a purely random potential by rare ``ergodic spots,'' which do not occur in the quasiperiodic case. Further, the data suggest that the decay of the antiferromagnetic order in the delocalized phase is faster than a power law.
We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-Andr\'e and the Fibonacci models. We find that stronger many-body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model's localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potentials.
We theoretically study the quench dynamics for an isolated Heisenberg spin chain with a random on-site magnetic field, which is one of the paradigmatic models of a many-body localization transition. We use the time-dependent variational principle as applied to matrix product states, which allows us to controllably study chains of a length up to L=100L=100L=100 spins, i.e., much larger than $L\ensuremath{\simeq}20$ that can be treated via exact diagonalization. For the analysis of the data, three complementary approaches are used: (i) determination of the exponent $\ensuremath{\beta}$ which characterizes the power-law decay of the antiferromagnetic imbalance with time; (ii) similar determination of the exponent ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ which characterizes the decay of a Schmidt gap in the entanglement spectrum; and (iii) machine learning with the use, as an input, of the time dependence of the spin densities in the whole chain. We find that the consideration of the larger system sizes substantially increases the estimate for the critical disorder Wc{W}_{c}Wc​ that separates the ergodic and many-body localized regimes, compared to the values of Wc{W}_{c}Wc​ in the literature. On the ergodic side of the transition, there is a broad interval of the strength of disorder with slow subdiffusive transport. In this regime, the exponents $\ensuremath{\beta}$ and ${\ensuremath{\beta}}_{\mathrm{\ensuremath{\Lambda}}}$ increase, with increasing LLL, for relatively small LLL but saturate for $L\ensuremath{\simeq}50$, indicating that these slow power laws survive in the thermodynamic limit. From a technical perspective, we develop an adaptation of the ``learning by confusion'' machine-learning approach that can determine Wc{W}_{c}Wc​.
We investigate the localization properties of a spin chain with an antiferromagnetic nearest-neighbor coupling, subject to an external quasiperiodic on-site magnetic field. The quasiperiodic modulation interpolates between two paradigmatic models, namely the Aubry-André and the Fibonacci models. We find that stronger many- body interactions extend the ergodic phase in the former, whereas they shrink it in the latter. Furthermore, the many-body localization transition points at the two limits of the interpolation appear to be continuously connected along the deformation of the quasiperiodic modulation. As a result, the position of the many-body localization transition depends on the interaction strength for an intermediate degree of deformation. Moreover, in the region of parameter space where the single-particle spectrum contains both localized and extended states, many-body interactions induce an anomalous effect: weak interactions localize the system, whereas stronger interactions enhance ergodicity. We map the model’s localization phase diagram using the decay of the quenched spin imbalance in relatively long chains. This is accomplished employing a time-dependent variational approach applied to a matrix product state decomposition of the many-body state. Our model serves as a rich playground for testing many-body localization under tunable potential
Anderson localization, where quenched disorder leads to the localization of single-particle wave functions, has been studied extensively. However, the effect of interaction and non-random potentials on many-body localization (MBL) has gained attention recently. We investigate the dynamics of the interacting Aubry-André model to assess the stability of the many-body localized (MBL) phase. Our motivation to consider the Aubry–André potential as the disorder potential is to eliminate the avalanche mechanism, which often destabilizes MBL in systems with quenched disorder. We analyze the time evolution and the system-size scaling across a range of disorder strengths for the entanglement entropy, number entropy, and the Hartley number entropy. In the non-interacting case, our results confirm an Anderson localization transition at a critical disorder strength Dc = 2, which is consistent with earlier studies. However, when interactions are introduced, the number entropy appears to grow unboundedly as SN ~ ln ln t in the thermodynamic limit, even for large disorder. Meanwhile, the entanglement entropy grows logarithmically, S ~ ln t, and both entropies saturate in a correlated manner at finite times td ~ e^(L·D). These findings suggest that using a quasiperiodic potential to remove the avalanche mechanism does not stabilize true MBL behavior. Instead, the long-time dynamics are qualitatively similar to those found in systems with random disorder. In this thesis, we first introduce the fundamental concepts of thermalization, quantum quench, and symmetry-resolved entropies, then provide a study of localization, specifically in the Aubry-André model. Finally, we place our numerical results in the broader context of the current debate on the stability of many-body localized phases.
Slow delocalization in strongly disordered many-body quantum chains can originate from two distinct mechanisms: rare resonances and avalanches, rooted in the energy statistics, or an energy-independent combinatorial path-entropy obstruction in operator space. Quasiperiodic potentials eliminate rare regions with similar potentials that are thought to be responsible for avalanche instabilities and modify the statistics of resonances, thus allowing us to delineate the two mechanisms. To do so, we numerically study the entanglement entropy S(t)S(t)S(t), number entropy SN(t)S_N(t)SN​(t), and Hartley number entropy SH(t)S_H(t)SH​(t) following a quantum quench in the interacting Aubry-André chain with disorder strength DDD. We find S(t)∼μln⁡tS(t)\simμ\ln tS(t)∼μlnt, SN(t)∼νln⁡ln⁡tS_N(t)\simν\ln\ln tSN​(t)∼νlnlnt with growth rates μ∼ν∼D−γμ\simν\sim D^{-γ}μ∼ν∼D−γ and γ≈3γ\approx 3γ≈3, the same behavior that was previously found in the case of quenched disorder. Furthermore, the saturation values SH(t→∞)S_H(t\to\infty)SH​(t→∞) continue to grow with system size LLL as ln⁡L\ln LlnL, a growth that persists for D≳8D\gtrsim 8D≳8 where MBL was argued to be stable. The similarity of the quasi-periodic and quenched disorder cases points to a common origin of the slow particle transport that does not rely on rare-region avalanches and persists despite strongly modified resonance statistics.
While global quantum quench has been extensively used in the literature to understand the localization-delocalization transition for the one-dimensional quantum spin chain, the effect of geometric quench (which corresponds to a sudden change of the geometry of the chain) in the context of such transitions is yet to be well understood. In this work, we investigate the effect of geometric quench in the Aubry-Andr\'e model, which supports localization-delocalization transition even in one dimension. We study the spreading of the entanglement and the site occupation with time and find many interesting features that can be used to characterize localization-delocalization transition. We observe that geometric quench causes a power-law type growth of the entanglement entropy in the delocalized phase in contrast to the linear growth which is found in the global quench studies. We also find that the saturation values in the many-body localized (MBL) phase obey area law in contrast to the usual volume law, which is a signature feature of the MBL phase in the context of global quench. This area law can also be understood from the long time site-occupation profile in the MBL phase.