Kavli Affiliate: Joel E. Moore
| First 5 Authors: Yi J. Zhao, Samuel J. Garratt, Joel E. Moore, ,
| Summary:
We develop an excited-state real-space renormalization group (RSRG-X)
formalism to describe the dynamics of conserved densities in randomly
interacting spin-$frac{1}{2}$ systems. Our formalism is suitable for systems
with $textrm{U}(1)$ and $mathbb{Z}_2$ symmetries, and we apply it to chains
of randomly positioned spins with dipolar $XX+YY$ interactions, as arise in
Rydberg quantum simulators and other platforms. The formalism generates a
sequence of effective Hamiltonians which provide approximate descriptions for
dynamics on successively smaller energy scales. These effective Hamiltonians
involve “superspins”: two-level collective degrees of freedom constructed
from (anti)aligned microscopic spins. Conserved densities can then be
understood as relaxing via coherent collective spin flips. For the well-studied
simpler case of randomly interacting nearest-neighbor $XX+YY$ chains, the
superspins reduce to single spins. Our formalism also leads to a numerical
method capable of simulating the dynamics up to an otherwise inaccessible
combination of large system size and late time. Focusing on disorder-averaged
infinite-temperature autocorrelation functions, in particular the local spin
survival probability $overline{S_p}(t)$, we demonstrate quantitative agreement
in results between our algorithm and exact diagonalization (ED) at low but
nonzero frequencies. Such agreement holds for chains with nearest-neighbor,
next-nearest-neighbor, and long-range dipolar interactions. Our results
indicate decay of $overline{S_p}(t)$ slower than any power law and feature no
significant deviation from the $sim 1/ log^2(t)$ asymptote expected from the
infinite-randomness fixed-point of the nearest-neighbor model. We also apply
the RSRG-X formalism to two-dimensional long-range systems of moderate size and
find slow late-time decay of $overline{S_p}(t)$.
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