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arXiv:0903.3740 · Physical Review B

Entanglement entropy with localized and extended interface defects

Ferenc Iglói, Zsolt Szatmári, Yu‐Cheng Lin

Published Jul 6, 2009• 75 citations
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Abstract

The quantum Ising chain of length, $L$, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time $t>0$. For the localized defect the increase in the entropy with $\text{log}\text{ }L$ or with $\text{log}\text{ }t$ involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.

DOI
10.1103/physrevb.80.024405

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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 We study the half-chain entanglement entropy in the ground state of the spin-1/2 XX chain across an extended random defect, where the strength of disorder decays with the distance from the interface algebraically as Δ l ∼ l − κ . In the whole regime κ ⩾ 0 , the average entanglement entropy is found to increase logarithmically with the system size L as S L ≃ c e f f ( κ ) 6 ln ⁡ L + c o n s t , where the effective central charge c e f f ( κ ) depends on κ . In the regime κ < 1 / 2 , where the extended defect is a relevant perturbation, the strong-disorder renormalization group method gives c e f f ( κ ) = ( 1 − 2 κ ) ln ⁡ 2 , while, in the regime κ ⩾ 1 / 2 , where the extended defect is irrelevant in the bulk, numerical results indicate a non-zero effective central charge, which increases with κ . The variation of c e f f ( κ ) is thus found to be non-monotonic and discontinuous at κ
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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.