Stringent numerical test of the Poisson distribution for finite quantum integrable Hamiltonians

Abstract

Using a class of exactly solvable models based on the pairing interaction, we show that it is possible to construct integrable Hamiltonians with a Wigner distribution of nearest-neighbor level spacings. However, these Hamiltonians involve many-body interactions and the addition of a small integrable perturbation very quickly leads the system to a Poisson distribution. Besides this exceptional case, we show that the accumulated distribution of an ensemble of random integrable two-body pairing Hamiltonians is in perfect agreement with the Poisson limit. These numerical results for quantum integrable Hamiltonians provide a further empirical confirmation of the work of Berry and Tabor in the semiclassical limit. The concept of quantum chaos still lacks a clear definition. The main ideas in this field have been obtained using the semiclassical approximation for quantum systems having a classical analog. In a seminal paper, Bohigas et al. [1] conjectured that the fluctuation properties of generic quantum systems, which in the classical limit are fully chaotic, coincide with those of random matrix theory (RMT). This conjecture is strongly supported by experimental data, numerical calculations, and analytical work based on semiclassical arguments. For a generic quantum integrable system, Berry and Tabor [2] showed that under very general conditions the spectral fluctuations in the semiclassical limit are well described by the Poisson statistics; i.e., the successive energy levels are uncorrelated.The analysis of spectral fluctuations provides an essential tool in the study of quantum chaos. Moreover, as RMT was introduced to explain the fluctuation properties of manybody systems like the atomic nucleus, it is usually considered that the level statistics establishes a link between manybody systems and the semiclassical picture. Actually, if the semiclassical limit is not valid, comparison of the system spectral fluctuations with those predicted by RMT is the main criterion to decide whether the system dynamic is regular or chaotic. When spectral fluctuations fall very near the RMT predictions the quantum system is considered fully chaotic. On the contrary, if they follow closely enough the Poisson statistic, the system is considered regular. Therefore, the concepts of chaotic and regular quantum systems are not well established since they rely on results that have only been checked in the semiclassical limit.The concept of integrability in classical mechanics was well defined after the work of Liouville in the 19th century. A classical Hamiltonian system is integrable if it has a complete set of independent integrals of motion commuting with respect to the Poisson brackets. The total number of integrals of motion should be half of the dimension of the phase space to assure completeness. In quantum mechanics, the concept of integrability is usually derived from an extension of the Liouville definition. A quantum system is said to be integrable if it is possible to define a complete set of Hermitian operators, the integral...

Классическая И Квантовая Интегрируемость Гамильтонианов Без Состояний Рассеяния

“…Поскольку эти гамильтонианы являются интегри-руемыми, мы получаем дополнительное подтверждение гипотезы Берри-Табора без использования квазиклассического предела. Этот подход следует подходу Реланьо с соавторами [9], однако проверка, которую мы выполняем, не является численной. Нами получено также подтверждение того, что найденные этими авторами исклю-чения из гипотезы Берри-Табора не являются общими случаями.…”

“…Например, Реланьо с соавторами [9] для того, чтобы провести численную про-верку гипотезы Берри-Табора, использовали точно решаемые модели Ричардсона-Годена, квантовая интегрируемость которых установлена без обращения к квази-классическому пределу. Было найдено несколько исключений из гипотезы Берри-Табора, однако с помощью численных методов эти авторы показали, что небольшие интегрируемые возмущения восстанавливают спектр Пуассона.…”

Integrable matrix theory: Level statistics

“…In contrast, chaotic systems exhibit Wigner-Dyson statistics, with level repulsion P (s) ∝ s 2 or s at small s. Moreover, level statistics are often used as a litmus test for quantum integrability even though there are integrable models that fail this test, e.g. the reduced BCS model [5] (which is a particular linear combination of commuting Gaudin Hamiltonians). In the present work, we quantify when and why Poisson statistics occur in quantum integrable models, while also characterizing exceptional (non-Poisson) behavior.…”

“…It is generally believed that the energy levels of integrable systems [1] follow a Poisson distribution [2][3][4][5][6][7][8]. For example, the probability that a normalized spacing between adjacent levels lies between s and s + ds is expected to be P (s)ds = e −s ds.…”

Rotationally invariant ensembles of integrable matrices

“…RMT does not, however, capture the typical behavior observed in exactly solvable many-body models, such as e.g. Poisson level statistics [7][8][9][10][11][12][13]. Though there exist matrix ensembles (e.g.…”