The scaling law of the arrival time of spin systems that present pretty good transmission

TL;DR

This paper investigates how the time for pretty good transmission in spin chains scales with system parameters, revealing it depends on the number of irrational eigenvalues and can vary with coupling configurations.

Contribution

It demonstrates that the scaling exponent is a power law of the number of irrational eigenvalues, not just chain length, and provides explicit examples showing how couplings affect this exponent.

Findings

The exponent scales as a power law of irrational eigenvalues.

Couplings influence the transmission time scaling exponent.

For centrosymmetric chains, the exponent is at most half the chain length.

Abstract

The pretty good transmission scenario implies that the probability of sending one excitation from one extreme of a spin chain to the other can reach values arbitrarily close to the unity just by waiting a time long enough. The conditions that ensure the appearance of this scenario are known for chains with different interactions and lengths. Sufficient conditions for the presence of pretty good transmission depend on the spectrum of the Hamiltonian of the spin chain. Some works suggest that the time $t_{\varepsilon}$ at which the pretty good transmission takes place scales as $1/(|\varepsilon|)^{f(N)}$, where $\varepsilon$ is the difference between the probability that a single excitation propagates from one extreme of the chain to the other and the unity, while $f(N)$ is an unknown function of the chain length. In this paper, we show that the exponent is not a simple function of the chain length but a power law of the number of linearly independent irrational eigenvalues of the one-excitation block of the Hamiltonian that enter into the expression of the probability of transmission of one excitation. We explicitly provide examples of a chain showing that the exponent changes when the couplings between the spins change while the length remains fixed. For centrosymmetric spin chains the exponent is at most $N/2$.