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 ) .
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 $S(t)\ensuremath{\sim}ln(t)/(2\ensuremath{\alpha})$.
Entanglement measures are useful tools in characterizing otherwise unknown quantum phases and indicating transitions between them. Here we examine the concurrence and entanglement entropy in quantum spin chains with random long-range couplings, spatially decaying with a power-law exponent $\ensuremath{\alpha}$. Using the strong disorder renormalization group (SDRG) technique, we find by analytical solution of the master equation a strong disorder fixed point, characterized by a fixed point distribution of the couplings with a finite dynamical exponent, which describes the system consistently in the regime $\ensuremath{\alpha}>\frac{1}{2}$. A numerical implementation of the SDRG method yields a power-law spatial decay of the average concurrence, which is also confirmed by exact numerical diagonalization. However, we find that the lowest-order SDRG approach is not sufficient to obtain the typical value of the concurrence. We therefore implement a correction scheme which allows us to obtain the leading-order corrections to the random singlet state. This approach yields a power-law spatial decay of the typical value of the concurrence, which we derive both by a numerical implementation of the corrections and by analytics. Next, using numerical SDRG, the entanglement entropy (EE) is found to be logarithmically enhanced for all $\ensuremath{\alpha}$, corresponding to a critical behavior with an effective central charge
c=ln(2), independent of $\ensuremath{\alpha}$. This is confirmed by an analytical derivation. Using numerical exact diagonalization (ED), we confirm the logarithmic enhancement of the EE and a weak dependence on $\ensuremath{\alpha}$. For a wide range of partition size
l, the EE fits a critical behavior with a central charge close to
c=1, which is the same as for the clean Haldane-Shastry model with a power-law-decaying interaction with $\ensuremath{\alpha}=2$. Only for small $l\ensuremath{\ll}L$, in a range which increases with the number of spins
N, we find deviations which are rather consistent with the strong disorder fixed point central charge
c=ln(2). Furthermore, we find using ED that the concurrence shows power-law decay, albeit with smaller power exponents than obtained by SDRG. We also present results obtained with DMRG and find agreement with ED for sufficiently small $\ensuremath{\alpha}<2$, whereas for larger $\ensuremath{\alpha}$ DMRG tends to underestimate the entanglement entropy and finds a faster decaying concurrence.
We investigate the magnetic susceptibility $\ensuremath{\chi}(T)$ of quantum spin chains of
N=1280 spins with power-law long-range antiferromagnetic couplings as a function of their spatial decay exponent $\ensuremath{\alpha}$ and cutoff length $\ensuremath{\xi}$. The calculations are based on the strong disorder renormalization method which is used to obtain the temperature dependence of $\ensuremath{\chi}(T)$ and distribution functions of couplings at each renormalization step. For the case with only algebraic decay $(\ensuremath{\xi}=\ensuremath{\infty})$ we find a crossover at ${\ensuremath{\alpha}}^{*}=1.066$ between a phase with a divergent low-temperature susceptibility $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ to a phase with a vanishing $\ensuremath{\chi}(T\ensuremath{\rightarrow}0)$ for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$. For finite cutoff lengths $\ensuremath{\xi}$, this crossover occurs at a smaller ${\ensuremath{\alpha}}^{*}(\ensuremath{\xi})$. Additionally, we study the localization of spin excitations for $\ensuremath{\xi}=\ensuremath{\infty}$ by evaluating the distribution function of excitation energies, and we find a delocalization transition that coincides with the opening of the pseudogap at ${\ensuremath{\alpha}}_{c}={\ensuremath{\alpha}}^{*}$.
Quantum information theoretical measures are useful tools for characterizing quantum dynamical phases. However, employing them to study excited states of random spin systems is a challenging problem. Here, we report results for the entanglement entropy (EE) scaling of excited eigenstates of random XX antiferromagnetic spin chains with long-range (LR) interactions decaying as a power law with distance with exponent $\ensuremath{\alpha}$. To this end, we extend the real-space renormalization group technique for excited states (RSRG-X) to solve this problem with LR interactions. For comparison, we perform numerical exact diagonalization (ED) calculations. From the distribution of energy-level spacings, as obtained by ED for up to $N\ensuremath{\sim}18$ spins, we find indications of a delocalization transition at ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ in the middle of the energy spectrum. With RSRG-X and ED, we show that for $\ensuremath{\alpha}>{\ensuremath{\alpha}}^{*}$ the EE of excited eigenstates retains a logarithmic divergence similar to the one observed for the ground state of the same model, while for $\ensuremath{\alpha}<{\ensuremath{\alpha}}^{*}$ EE displays an algebraic growth with the subsystem size $l, {S}_{l}\ensuremath{\sim}{l}^{\ensuremath{\beta}}$, with $0<\ensuremath{\beta}<1$. We find that ${\ensuremath{\alpha}}^{*}\ensuremath{\approx}1$ coincides with the delocalization transition ${\ensuremath{\alpha}}_{c}$ in the middle of the many-body spectrum. An interpretation of these results based on the structure of the RG rules is proposed, which is due to rainbow proliferation for very long-range interactions $\ensuremath{\alpha}\ensuremath{\ll}1$. We also investigate the effective temperature dependence of the EE, allowing us to study the half-chain EE of eigenstates at different energy densities, where we find that the crossover in EE occurs at ${\ensuremath{\alpha}}^{*}<1$.
The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties 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 $\ensuremath{\alpha}$. Using a dynamical version of the strong disorder renormalization group we find for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\ensuremath{\alpha}$ as $S(t)={S}_{p}ln(t)/(2\ensuremath{\alpha})$. Here, ${S}_{p}=2ln2\ensuremath{-}1$. We present results of numerical exact diagonalization calculations for system sizes up to $N\ensuremath{\sim}16$ spins, in good agreement with the analytical results for sufficiently large $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.8$. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the entanglement entropy grows as a power law with time, $S(t)\ensuremath{\sim}{t}^{\ensuremath{\gamma}(\ensuremath{\alpha})}$ with $0<\ensuremath{\gamma}(\ensuremath{\alpha})<1$ a decaying function of the interaction exponent $\ensuremath{\alpha}$.
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.