2016
DOI: 10.1088/1751-8113/49/37/375101
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Particle-time duality in the kicked Ising spin chain

Abstract: Previously, we demonstrated that the dynamics of kicked spin chains possess a remarkable duality property. The trace of the unitary evolution operator for N spins at time T is related to one of a non-unitary evolution operator for T spins at time N . Using this duality relation we obtain the oscillating part of the density of states for a large number of spins. Furthermore, the duality relation explains the anomalous short-time behavior of the spectral form factor previously observed in the literature.

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Cited by 104 publications
(112 citation statements)
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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%
“…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%
“…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%
“…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%