Periodic mean-field solutions and the spectra of discrete bosonic fields: Trace formula for Bose-Hubbard models

Engl, Thomas; Urbina, Juan Diego; Richter, Klaus

doi:10.1103/physreve.92.062907

Cited by 32 publications

(49 citation statements)

References 28 publications

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“…Nevertheless, a rigorous proof of the quantum chaos conjecture has so far only been possible for a much more abstract class of single-particle systems, specifically for mixing quantum graphs [15,16]. These semiclassical periodic-orbit approaches have a natural generalisation to a quantum many-body problem for bosons when the number of quanta per mode is large [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

2018

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A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges. For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc. R. Soc. London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys. Rev. Lett. 93, 014103 (2004)]. In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide a clear theoretical explanation for these observations. We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field. In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.

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Section: Introductionmentioning

confidence: 99%

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

2018

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“…The situation is even less clear for non-integrable many-body systems with simple, say clean and local, interactions, where evidence of RMT spectral correlations is abundant [17][18][19][20] but theoretical explanations are scarce. While for many-body systems of bosons with a large number of quanta per mode, or other models with small effective Planck's constant, a semiclassical reasoning may still be used [21][22][23][24], the intuition is completely lost and no methods have been known when it comes to fermionic or spin-1/2 systems. Very recently, a few steps of progress have been made.…”

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confidence: 99%

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos

Bertini¹,

Kos²,

Prosen³

2018

Phys. Rev. Lett.

292

307

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The most general and versatile defining feature of quantum chaotic systems is that they possess an energy spectrum with correlations universally described by random matrix theory (RMT). This feature can be exhibited by systems with a well defined classical limit as well as by systems with no classical correspondence, such as locally interacting spins or fermions. Despite great phenomenological success, a general mechanism explaining the emergence of RMT without reference to semiclassical concepts is still missing. Here we provide the example of a quantum many-body system with no semiclassical limit (no large parameter) where the emergence of RMT spectral correlations is proven exactly. Specifically, we consider a periodically driven Ising model and write the Fourier transform of spectral density's two-point function, the spectral form factor, in terms of a partition function of a two-dimensional classical Ising model featuring a space-time duality. We show that the self-dual cases provide a minimal model of many-body quantum chaos, where the spectral form factor is demonstrated to match RMT for all values of the integer time variable t in the thermodynamic limit. In particular, we rigorously prove RMT form factor for odd t, while we formulate a precise conjecture for even t. The results imply ergodicity for any finite amount of disorder in the longitudinal field, rigorously excluding the possibility of many-body localization. Our method provides a novel route for obtaining exact nonperturbative results in non-integrable systems.arXiv:1805.00931v3 [nlin.CD] 9 Jan 2019

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“…(2). However, as originally developed for SP [26][27][28][29][30][31][32][33] and recently extended to MB systems [34][35][36][37][38][39], there exist semiclassical techniques that adequately describe post-Ehrenfest quantum phenomena. By extending these approaches to MB commutator norms, here we develop a unifying semiclassical theory for OTOCs which bridges classical mean-field and quantum MB concepts arXiv:1805.06377v2 [cond-mat.stat-mech] 24 Sep 2018 for bosonic large-N systems.…”

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confidence: 99%

Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators

2018

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Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τE in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N -particle systems. We first show how the growth of OTOCs up to τE = (1/λ) log N is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τE, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit → 0 for fixed N , including quantum-chaotic single-and few-particle systems.

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Periodic mean-field solutions and the spectra of discrete bosonic fields: Trace formula for Bose-Hubbard models

Cited by 32 publications

References 28 publications

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos

Many-Body Quantum Interference and the Saturation of Out-of-Time-Order Correlators

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