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arXiv:1906.03503 · Physical review. B./Physical review. B

Logarithmic entanglement growth in two-dimensional disordered fermionic systems

Yang Zhao, Jesko Sirker

Published Jul 22, 2019• 7 citations
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Abstract

We investigate the growth of the entanglement entropy ${S}_{\mathrm{ent}}$ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case, we show that an intermediate weak localization regime exists in which ${S}_{\mathrm{ent}}(t)$ grows logarithmically in time $t$ before Anderson localization sets in. For the case of binary bond disorder near the percolation transition, we find additive logarithmic corrections to area and volume laws as well as a scaling at long times, which is consistent with an infinite randomness fixed point.

DOI
10.1103/physrevb.100.014203

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We investigate the growth of the entanglement entropy Sent{S}_{\mathrm{ent}}Sent​ following global quenches in two-dimensional free fermion models with potential and bond disorder. For the potential disorder case, we show that an intermediate weak localization regime exists in which Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) grows logarithmically in time ttt before Anderson localization sets in. For the case of binary bond disorder near the percolation transition, we find additive logarithmic corrections to area and volume laws as well as a scaling at long times, which is consistent with an infinite randomness fixed point.
We numerically investigate the growth of the entanglement entropy Sent(t){S}_{\mathrm{ent}}(t)Sent​(t) in time ttt, after a global quench from a product state, in quantum chains with various kinds of disorder. The main focus is, in particular, on fermionic chains with bond disorder. In the noninteracting case at criticality we numerically test recent predictions by the real-space renormalization group for the entanglement growth in time, the maximal entanglement as a function of block size, and the decay of a density-wave order parameter. We show that multiprecision calculations are required to reach the scaling regime and perform such calculations for specific cases. For interacting models with binary bond disorder we present data based on infinite-size density matrix renormalization group calculations and exact diagonalizations. We obtain numerical evidence of a many-body localized phase in bond-disordered systems where ${S}_{\mathrm{ent}}(t)\ensuremath{\sim}lnt$ seems to hold. Our results for bond disorder are contrasted with the well-studied case of potential disorder.
Random fractal lattices provide a geometrically disordered setting in which quantum correlations can be shaped by noninteger dimensionality rather than onsite randomness. We investigate the entanglement properties of noninteracting fermions on random fractal lattices generated by a stochastic growth algorithm. By varying the growth parameter and adding missing links with probability ppp, we tune the Hausdorff and spectral dimensions while keeping the system free of onsite disorder. For ground states at different fillings, we compute the bipartite entanglement entropy of subregions defined by graph distance and analyze its scaling with subsystem size. Over a broad parameter range, we find robust power-law behavior governed primarily by the Hausdorff dimension, consistent with a generalized area law and without the logarithmic enhancement familiar from Euclidean free fermions. We also study entanglement growth following a global quench from an uncorrelated checkerboard state and uncover an asymptotic scaling collapse in which the subsystem-size dependence is governed by the Hausdorff dimension, while the temporal evolution is governed by the spectral dimension. The resulting dynamics are logarithmically slow over an extended intermediate-time window. These results show that geometric randomness alone can generate both nontrivial ground-state entanglement structure and slow quantum-information spreading in free-fermion systems.
We study theoretically and numerically the entanglement entropy of the ddd-dimensional free fermions whose one-body Hamiltonian is the Anderson model. Using the basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩$ of the ddd dimension cube $\mathrm{\ensuremath{\Lambda}}$ of side length lll admits the area law scaling $⟨{S}_{\mathrm{\ensuremath{\Lambda}}}⟩\ensuremath{\sim}{l}^{(d\ensuremath{-}1)},l\ensuremath{\gg}1$, even in the gapless case, thereby manifesting the area law in the mean for our model. For d=1d=1d=1 and $l\ensuremath{\gg}1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not self-averaging, i.e., has nonvanishing random fluctuations even if $l\ensuremath{\gg}1$.
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 ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=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≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.
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Free fermion models with an operator entanglement growing as O(t)O(\sqrt{t})O(t​)
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