We study excited states and finite temperature properties of chains of randomly placed spins with long range interactions. To this end we extend the recently introduced strong disorder renormalization group method in real space, well suited to study bond disordered antiferromagnetic power law coupled quantum spin chains. For bond disordered, short range coupled spin chains, defined by keeping only interactions between adjacent spins, we show that the distribution of the absolute value of the couplings is the infinite randomness fixed point distribution also for the excited states. Moreover, we derive the distribution function of the sign of the couplings and find that the number of negative couplings increases with temperature T . For power law long range interaction with power exponent α we derive and solve the Master equation for the distribution function of couplings for α > 2 . The coupling amplitude is found to have the strong disorder distribution with finite width 2 α , with small corrections. We derive resulting finite temperature properties of both short and power law long ranged spin systems for α > 2 . The magnetic susceptibility is dominated by paramagnetic S = 1 / 2 -spin, resulting in a Curie law susceptibility. The distribution function of pair lengths, and the spin correlation function are found be the same as in the ground state. We find that the entanglement entropy increases logarithmically with the partition length n and has the same functional form as for the ground state of critical quantum spin chains with temperature dependent effective central charge c ¯ ( T ) , which converges at high temperature to 1 2 ln 2 , half the value it has in the ground state. We confirm that the entanglement entropy grows with time t after a quantum quench logarithmically as S ( t ) ∼ ln ( t ) / ( 2 α ) .
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power
α. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for
α>αc that the entanglement entropy grows logarithmically with time and becomes smaller with larger
α as
S(t)=Spln(t)/(2α). Here,
Sp=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large
α>αc≈1.8. For
α<αc, we find that the entanglement entropy grows as a power-law with time,
S(t)∼tγ(α) with
0<γ(α)<1 a decaying function of the interaction exponent
α.
We introduce and implement a reformulation of the strong-disorder renormalization group method in real space, well suited to study bond-disordered antiferromagnetic power-law coupled quantum spin chains. We apply it to a chain of randomly placed spins coupled by power-law long-range interaction with power $\ensuremath{\alpha}$. First, keeping only interactions between adjacent spins, we derive the Master equation and confirm that it is solved by the infinite randomness fixed-point distribution. Then, we derive the master equation for power-law long-range interaction between all spins for any anisotropy $\ensuremath{\gamma}$ ranging from the XX limit, $\ensuremath{\gamma}=0$, to the isotropic Heisenberg limit, $\ensuremath{\gamma}=1,$ which corresponds to a tight-binding chain of disordered long-range interacting fermions with long-range hopping. We thereby show that the distribution function of couplings
J smaller than renormalization scale $\mathrm{\ensuremath{\Omega}}$ flows to the strong-disorder fixed-point distribution of finite width $2\ensuremath{\alpha}.$ We find only small corrections to that distribution, which depend on power exponent $\ensuremath{\alpha}$ and coupling anisotropy $\ensuremath{\gamma}.$ As a consequence, the low-temperature magnetic susceptibility diverges with an anomalous power law. The distribution of singlet lengths
l is found to decay as ${l}^{\ensuremath{-}2}$. The entanglement entropy of a subsystem of length
n increases in the ground state logarithmically for all $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$. After a global quantum quench, the entanglement entropy increases with time logarithmically as $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.