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Register a new userFrom chaos to many-body localization: some introductory notes
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Added by
Levan Chotorlishvili L
Publication date
2019
fields
Physics
and research's language is
English
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Staring from the kicked rotator as a paradigm for a system exhibiting classical chaos, we discuss the role of quantum coherence resulting in dynamical localization in the kicked quantum rotator. In this context, the disorder-induced Anderson localization is also discussed. Localization in interacting, quantum many-body systems (many-body localization) may also occur in the absence of disorder, and a practical way to identify its occurrence is demonstrated for an interacting spin chain.
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Characterizing states of matter through the lens of their ergodic properties is a fascinating new direction of research. In the quantum realm, the many-body localization (MBL) was proposed to be the paradigmatic ergodicity breaking phenomenon, which extends the concept of Anderson localization to interacting systems. At the same time, random matrix theory has established a powerful framework for characterizing the onset of quantum chaos and ergodicity (or the absence thereof) in quantum many-body systems. Here we numerically study the spectral statistics of disordered interacting spin chains, which represent prototype models expected to exhibit MBL. We study the ergodicity indicator $g=log_{10}(t_{rm H}/t_{rm Th})$, which is defined through the ratio of two characteristic many-body time scales, the Thouless time $t_{rm Th}$ and the Heisenberg time $t_{rm H}$, and hence resembles the logarithm of the dimensionless conductance introduced in the context of Anderson localization. We argue that the ergodicity breaking transition in interacting spin chains occurs when both time scales are of the same order, $t_{rm Th} approx t_{rm H}$, and $g$ becomes a system-size independent constant. Hence, the ergodicity breaking transition in many-body systems carries certain analogies with the Anderson localization transition. Intriguingly, using a Berezinskii-Kosterlitz-Thouless correlation length we observe a scaling solution of $g$ across the transition, which allows for detection of the crossing point in finite systems. We discuss the observation that scaled results in finite systems by increasing the system size exhibit a flow towards the quantum chaotic regime.
Linking thermodynamic variables like temperature $T$ and the measure of chaos, the Lyapunov exponents $lambda$, is a question of fundamental importance in many-body systems. By using nonlinear fluid equations in one and three dimensions, we prove that in thermalised flows $lambda propto sqrt{T}$, in agreement with results from frustrated spin systems. This reveals an underlying universality and provides evidence for recent conjectures on the thermal scaling of $lambda$. We also reconcile seemingly disparate effects -- equilibration on one hand and pushing systems out-of-equilibrium on the other -- of many-body chaos by relating $lambda$ to $T$ through the dynamical structures of the flow.
We study many-body chaos in a (2+1)D relativistic scalar field theory at high temperatures in the classical statistical approximation, which captures the quantum critical regime and the thermal phase transition from an ordered to a disordered phase. We evaluate out-of-time ordered correlation functions (OTOCs) and find that the associated Lyapunov exponent increases linearly with temperature in the quantum critical regime, and approaches the non-interacting limit algebraically in terms of a fluctuation parameter. OTOCs spread ballistically in all regimes, also at the thermal phase transition, where the butterfly velocity is maximal. Our work contributes to the understanding of the relation between quantum and classical many-body chaos and our method can be applied to other field theories dominated by classical modes at long wavelengths.
We examine the many-body localization (MBL) phase transition in one-dimensional quantum systems with quenched randomness and short-range interactions. Following recent works, we use a strong-randomness renormalization group (RG) approach where the phase transition is due to the so-called avalanche instability of the MBL phase. We show that the critical behavior can be determined analytically within this RG. On a rough $textit{qualitative}$ level the RG flow near the critical fixed point is similar to the Kosterlitz-Thouless (KT) flow as previously shown, but there are important differences in the critical behavior. Thus we show that this MBL transition is in a new universality class that is different from KT. The divergence of the correlation length corresponds to critical exponent $ u rightarrow infty$, but the divergence is weaker than for the KT transition.
Recent years have seen an increasing interest in quantum chaos and related aspects of spatially extended systems, such as spin chains. However, the results are strongly system dependent, generic approaches suggest the presence of many-body localization while analytical calculations for certain system classes, here referred to as the ``self-dual case, prove adherence to universal (chaotic) spectral behavior. We address these issues studying the level statistics in the vicinity of the latter case, thereby revealing transitions to many-body localization as well as the appearance of several non-standard random-matrix universality classes.
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