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Entanglement in spin chains and lattices with long-range interactions

Wolfgang Dür, Ludwig Hartmann, M. Hein, H. J. Briegel

Published Jul 9, 2004• 1 citation
OpenAlex

Abstract

We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.

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We investigate entanglement properties of N initially disentangled spins, embedded in a ring or d-dimensional lattice. The spins interact via some long-range Ising-type interaction. A description in terms of generalized valence bond solids allows us, for arbitrarily large N, to efficiently calculate reduced density operators of up to ten particles. From these density operators we can compute many quantities, for example the bipartite entanglement S_L between a block of L neighboring spins and the remaining system, lower and upper bounds on the localizable entanglement, and higher order correlation functions. We vary the distance dependence of the interaction and investigate how this variation affects the scaling of the entropy S_L as L grows. We provide a sufficient condition when S_L saturates. In other cases, we find that the entanglement length diverges. For special configurations, we determine S_L analytically for all L in the limit N to infinity.
We consider N initially disentangled spins, embedded in a ring or d-dimensional lattice of arbitrary geometry, which interact via some long-range Ising-type interaction. We investigate relations between entanglement properties of the resulting states and the distance dependence of the interaction in the limit N-->infinity. We provide a sufficient condition when bipartite entanglement between blocks of L neighboring spins and the remaining system saturates and determine S(L) analytically for special configurations. We find an unbounded increase of S(L) as well as diverging correlation and entanglement length under certain circumstances. For arbitrarily large N, we can efficiently calculate all quantities associated with reduced density operators of up to ten particles.
We consider systems of interacting spins and study the entanglement that can be localized, on average, between two separated spins by performing local measurements on the remaining spins. This concept of localizable entanglement (LE) leads naturally to notions like entanglement length and entanglement fluctuations. For both spin-1∕21∕21∕2 and spin-1 systems, we prove that the LE of a pure quantum state can be lower bounded by connected correlation functions. We further propose a scheme, based on matrix-product states and the Monte Carlo method, to efficiently calculate the LE for quantum states of a large number of spins. The virtues of LE are illustrated for various spin models. In particular, characteristic features of a quantum phase transition such as a diverging entanglement length can be observed. We also give examples for pure quantum states exhibiting a diverging entanglement length but finite correlation length. We have numerical evidence that the ground state of the antiferromagnetic spin-1 Heisenberg chain can serve as a perfect quantum channel. Furthermore, we apply the numerical method to mixed states and study the entanglement as a function of temperature.
We calculate the reduced density matrix of a block of integer spin- S 's in a q -deformed valence-bond-solid (VBS) state. This matrix is diagonalized exactly for an infinitely long block in an infinitely long chain. We construct an effective Hamiltonian with the same spectrum as the logarithm of the density matrix. We also derive analytic expressions for the von Neumann and Rényi entanglement entropies. For blocks of finite length, we calculate the eigenvalues of the reduced density matrix by perturbation theory and numerical diagonalization. These results enable us to describe the effects of finite-size corrections on the entanglement spectrum and entropy in this generalized VBS model.
Weighted graph states naturally arise when spin systems interact via an Ising-type interaction. First, we abstractly define the class of weighted graph states and demonstrate its computational accessibility. We show how reduced density matrices of a small number of spins (≈10) can be computed from arbitrarily large systems using weighted graph techniques and projected entangled pair techniques, and we discuss various entanglement measures accessible from these reduced density matrices. Second, we apply these findings to spin chains and lattices with long-range interactions and analytically derive area laws for the scaling of block-wise entanglement. Then, we turn to disordered spin systems, spin gases, which are connected to random weighted graph states and which share their entanglement properties. Finally, we use a spin gas as a bath that introduces decoherence in single as well as multipartite spin systems. The microscopic, exact decoherence model we obtain can operate in different regimes and exhibit non-Markovian features as well as spatially correlated noise effects.
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 $\ensuremath{\alpha}$.
Here we evaluate the many-body entanglement properties of a generalized SU(nnn) valence-bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for an elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and R\'enyi entropies of a block, localizable entanglement, and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU(222) case.