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arXiv:0810.3788 · Journal of Statistical Mechanics Theory and Experiment

Entanglement in spin chains with gradients

Viktor Eisler, Ferenc Iglói, Ingo Peschel

Published Feb 3, 2009• 72 citations
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Abstract

We study solvable spin chains where either fields or couplings vary linearly in space and create a sandwich-like structure of the ground state. We find that the entropy of entanglement between two halves of a chain varies logarithmically with the interface width. After quenching to a homogeneous critical system, the entropy grows logarithmically in time in the XX model, but quadratically in the transverse Ising chain. We explain this behaviour and indicate generalizations to other power laws.

DOI
10.1088/1742-5468/2009/02/p02011

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We study solvable spin chains where either fields or couplings vary linearly in space and create a sandwich-like structure of the ground state. We find that the entropy of entanglement between two halves of a chain varies logarithmically with the interface width. After quenching to a homogeneous critical system, the entropy grows logarithmically in time in the XX model, but quadratically in the transverse Ising chain. We explain this behaviour and indicate generalizations to other power laws.
The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean and a random part, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion S∼ln⁡ln⁡(L){\cal S}\sim \ln\ln(L)S∼lnln(L), where LLL is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance lll from the contact point as ∼l−κ\sim l^{-\kappa}∼l−κ. For κ<1/2\kappa<1/2κ<1/2, the entropy scales as S(κ)≃(1−2κ)S(κ=0){\cal S}(\kappa) \simeq (1-2\kappa){\cal S}(\kappa=0)S(κ)≃(1−2κ)S(κ=0), while for κ≥1/2\kappa \ge 1/2κ≥1/2, when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.
Abstract The entanglement entropy in clean, as well as in random quantum spin chains has a logarithmic size-dependence at the critical point. Here, we study the entanglement of composite systems that consist of a clean subsystem and a random subsystem, both being critical. In the composite, antiferromagnetic XX-chain with a sharp interface, the entropy is found to grow in a double-logarithmic fashion S ∼ ln ln ⁡ ( L ) , where L is the length of the chain. We have also considered an extended defect at the interface, where the disorder penetrates into the homogeneous region in such a way that the strength of disorder decays with the distance l from the contact point as ∼ l − κ . For κ < 1 / 2 , the entropy scales as S ( κ ) ≃ ln ⁡ 2 ( 1 − 2 κ ) 6 ln ⁡ L , while for κ ⩾ 1 / 2 , when the extended interface defect is an irrelevant perturbation, we recover the double-logarithmic scaling. These results are explained through strong-disorder RG arguments.
By means of free fermionic techniques we study the time evolution of the entanglement entropy S(t)\mathcal{S}(t)S(t) of a block of spins in the random transverse-field Ising chain after a sudden change of the parameters of the Hamiltonian. We consider global quenches, when the parameters are modified uniformly in space, as well as local quenches, when two disconnected blocks are suddenly joined together. For a noncritical final state, the dynamical entanglement entropy is found to approach a finite limiting value for both types of quenches. If the quench is performed to the critical state, the entropy grows for an infinite block as $\mathcal{S}(t)\ensuremath{\sim}\mathrm{ln}\mathrm{ln}t$. This type of ultraslow increase is explained through the strong disorder renormalization group method.
We study zero-temperature XX chains and transverse Ising chains and join an initially separate finite piece on one or on both sides to an infinite remainder. In both critical and non-critical systems we find a typical increase of the entanglement entropy after the quench, followed by a slow decay towards the value of the homogeneous chain. In the critical case, the predictions of conformal field theory are verified for the first phase of the evolution, while at late times a step structure can be observed.
Abstract. Entanglement between different regions in momentum space is studied for ground states of some spin-chain Hamiltonians: the XY model, the Ising model in a transverse field (ITF) and the XXZ models. In the XY and ITF cases, entanglement only takes place between states with opposite momenta. Thus, an anisotropy in the interaction induces entanglement in the momentum pairs. In the ITF case, the ferromagnetic phase is characterized by a total entropy between left- and right-moving modes which is independent of the external field. This result characterizes the Ising phase transition in momentum space. In the critical XXZ case, we provide evidence that the maximal entropy between energy modes around the Fermi point grows logarithmically with the system size, with a prefactor which depends on the compactification radius. The slow growth of the entanglement in Fourier space with the system size lies at the origin of the success of the renormalization techniques in momentum space.
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c eff , defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.