On anomalous Lieb–Robinson bounds for the Fibonacci XY chain

Abstract

Abstract. We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in |x|−v|t| is replaced by exponential decay in |x|−v|t| α with 0 < α < 1. In fact, we can characterize the values of α for which such a bound holds as those exceeding α + u , the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of [14], we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding to the XY chain via the Jordan-Wigner transformation; in our case the one-body Hamiltonian with Fibonacci potential. We can bound its dynamics by adapting techniques developed in [8, 9, 2, 4] to our purposes. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport.Along the way, we prove a new result about the one-body Fibonacci Hamiltonian: the upper transport exponent agrees with the time-averaged upper transport exponent, see Corollary 2.9. We also explain why our method does not extend to yield anomalous Lieb-Robinson bounds of power-law type for the random dimer model.

On Polynomial Lieb–Robinson Bounds for the XY Chain in a Decaying Random Field

“…We have the identity (6) a * j a j = c * j c j . In terms of the fermion operators, the Hamiltonian reads, (7) H XY n = 2C * H n C − n j=1Ṽ j where C := (c 1 , ..., c n ) T and V j := λ j 1/2 V j . The n × n matrix H n is given by…”

“…Taking adjoints, the same is also true for c * k . The bound (10) follows directly from [8,Eq. (8)] by applying (9).…”

“…Lemma 3.1. Let H n be the operator given in (8). Then there exist constants C, κ > 0 such that for all n ∈ N and all 1 ≤ j ≤ k ≤ n, we have…”

“…where > 0 is arbitary. Now, we bound the one-body propagation in terms of the many-body propagation using [8,Lm. 4.1].…”

“…For the PLR(a, b) bounds defined by (1) and (2), we only consider observables A supported at site 1. If A is supported at a site j > 1, the decaying factor is not replaced by the distance of the supports |j − k| (as would be the case in a direct polynomial generalization of the LR bound, compare [7,8]), but instead by min{j, k}/ max{j, k}. The precise statement is in Theorem 3.2.…”