Particle-time duality in the kicked Ising spin chain

Akila, Maram; Waltner, Daniel; Gutkin, Boris; Guhr, Thomas

doi:10.1088/1751-8113/49/37/375101

Cited by 119 publications

(126 citation statements)

References 31 publications

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“…Observing that (8) couples only "spins" on neighbouring sites in both t and L directions, the partition function (7) can be written both as the trace of a transfer matrix propagating in the time direction and as the trace of a transfer matrix propagating in the space direction. This reveals the known duality transformation of the kicked Ising model [32,33]. The transfer matrix in the time direction is clearly given by U KI [h], while the transfer matrix "in space",Ũ KI [h j ], is given by the same algebraic form (2,3) exchanging J and J but acting on a spin chain of t sites.…”

mentioning

confidence: 85%

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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mentioning

confidence: 85%

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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“…But as its matrix dimension (2j + 1) N × (2j + 1) N grows exponentially with N , a direct calculation of the spectrum ofÛ is impossible, e.g., even the propagatorÛ T for N = 19 spins at j = 1 has a matrix dimension of 10 9 × 10 9 . Luckily, recently developed duality relations [28,45] provide the solution and make possible, for the first time, a semiclassical analysis of genuine many-body orbits. The crucial ingredient is the exact identity…”

Section: Explosion Of Dimension and Duality Relationmentioning

confidence: 99%

“…[40,46] we generalize this duality approach, developed for j = 1/2 in Ref. [45], to j 1. The dual "particle-number-evolution" operator is a product as well,Ŵ =Ŵ IŴK .…”

Section: Explosion Of Dimension and Duality Relationmentioning

confidence: 99%

Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins

et al. 2017

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The field of quantum chaos originated in the study of spectral statistics for interacting many-body systems, but this heritage was almost forgotten when single-particle systems moved into the focus. In recent years new interest emerged in many-body aspects of quantum chaos. We study a chain of interacting, kicked spins and carry out a semiclassical analysis that is capable of identifying all kinds of genuine many-body periodic orbits. We show that the collective many-body periodic orbits can fully dominate the spectra in certain cases.

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“…Since we are interested in the semiclassical limit of the Hamiltonian, a different idea is needed to reduce the complexity of the Hilbert space. To this end we employ a recently developed approach, see [30,31,20], to evaluate traces of the time evolution operator for kicked chain-like systems with local interactions. This method, for discrete maps and integer times T , is based on the exact duality relation…”

Section: Introductionmentioning

confidence: 99%

“…On a more fundamental level, by virtue of its low dimension the operator W is much better suited for studies of large scale many-body spectral fluctuations in comparison to the original time evolutionÛ . Originally, the duality approach has been developed for the Kicked Ising Chain model with a fixed spin quantum number of j = 1/2 [31]. In the present paper we focus on a natural extension of this setting to an arbitrary j.…”

Section: Introductionmentioning

confidence: 99%

Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains

et al. 2018

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While plenty of results have been obtained for single-particle quantum systems with chaotic dynamics through a semiclassical theory, much less is known about quantum chaos in the many-body setting. We contribute to recent efforts to make a semiclassical analysis of many-body systems feasible. This is nontrivial due to both the enormous density of states and the exponential proliferation of periodic orbits with the number of particles. As a model system we study kicked interacting spin chains employing semiclassical methods supplemented by a newly developed duality approach. We show that for this model the line between integrability and chaos becomes blurred. Due to the interaction structure the system features (non-isolated) manifolds of periodic orbits possessing highly correlated, collective dynamics. As with the invariant tori in integrable systems, their presence lead to significantly enhanced spectral fluctuations, which by order of magnitude lie in-between integrable and chaotic cases.

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Particle-time duality in the kicked Ising spin chain

Cited by 119 publications

References 31 publications

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

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

Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins

Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains

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