Entanglement Entropy in a Boundary Impurity Model

Abstract

Boundary impurities are known to dramatically alter certain bulk properties of 1 + 1 dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling/bosonization scheme. For a Luttinger liquid of length 2L and UV cut off ǫ, the boundary impurity correction (δSimp) to the bulk logarithmic entanglement entropy (Sent ∝ ln L/ǫ) scales as δSimp ∼ yr ln L/ǫ, where yr is the renormalized backscattering coupling constant. In this way, bulk entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case (g < 1), δSimp diverges (negatively) suggesting that the bulk entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow. PACS numbers:Quantum field theories describe coupled quantum oscillators and, therefore, even the ground state of noninteracting fields may exhibit quantum correlations over a long range. Consider the ground state of the free electromagnetic field. An electric field amplitude E at a point x is quantum correlated with an amplitude E ′ at a point x ′ in that there are contributions to the ground state wavefunction containing such products as . . . |E x |E ′ x ′ . . .. Since the ground state contains superpositions of such states, it is said to be entangled-even though it is a free field. If, in a d-dimensional spacetime, a spatial box of volume L d−1 is formed, it follows that the degrees of freedom which reside exclusively in the box will appear to be in a mixed state. The degree of mixing may be characterized by the von Neumann entropy, S = −trρ ln ρ, where the reduced density matrix ρ has been formed by tracing over the degrees of freedom exterior to the volume L d−1 .Geometric or entanglement entropy formed in this fashion was introduced in the context of black hole quantum mechanics and Hawking-Bekenstein entropy [1], where it was found that entanglement entropy is not an extensive quantity but, rather, scales as the area of the bounding surface, S ∝ L d−2 [2]. This highly suggestive result is believed to bear some relation to holographic principle proposals [3]. In condensed matter physics, the role of entanglement entropy in understanding quantum critical phenomena [4] and quantum phase transitions [5] has recently been emphasized. 1 + 1 dimensional conformal field theories-which describe critical spin chains, Luttinger liquids and other massless theories-have pointlike bounding surfaces; remarkably, the entanglement entropy was shown to depend universally upon the central charge of the theory and to diverge logarithmically with the length of the subsystem [6,7,8,9]. Specifically, the entropy is given by S = c 3 ln L/ǫ where c is the central charge. Studies of entanglement entropy have, so far, been restricted to homogeneous models. [For a recent exception, see [14].] However, consider the artificial introduction of an inhomogeneity at one...

Critical entanglement ofXXZHeisenberg chains with defects

“…One then wonders how the entanglement behaves in such a case and how the logarithmic law (1) is affected. This question was first raised by Levine [4] who used bosonization and found results to lowest order in the impurity strength which are consistent with the general picture. The simpler case of a free-fermion hopping model, corresponding to the XX spin chain, was treated numerically in Ref.…”

“…An analogous formula with c/6 replaced by c/3 holds for a segment of length L in an infinite chain. The logarithmic divergence is a particular signature of the criticality and can be related to analogous universal contributions to the free energy of critical two-dimensional systems with conical shape [2,3,4]. It has been verified numerically for a number of quantum chains [5,6] and derived analytically for free fermions hopping on a chain [7].…”

Fano resonances and entanglement entropy

“…An example is given by critical lattice models where single defective links separate the two subsystems. This can lead to a modified prefactor for the logarithm varying continuously with the defect strength in models with free fermions [6][7][8][9] , while for interacting electrons the defect either renormalizes to a cut or to the homogeneous value in the L → ∞ scaling limit 10,11 . In the more general context of a conformal interface separating two CFTs the effective central charge has been calculated recently and was shown to depend on a single parameter 12 .…”

Entanglement and boundary critical phenomena

“…Recently much work has been done to understand entanglement in quantum many-body systems. In particular, the behavior of various entanglement measures at or near a quantum phase transition [1] has received a lot of attention [2,3,4,5,6,7,8,9]. These entanglement measures include the von Neumann entropy and the single-copy entanglement, among others.…”

“…For a system in a pure state |ψ (e.g. the ground state) that is partitioned into two subsystems A and B, the von Neumann entropy is S 1 ≡ −Tr A ρ A log 2 ρ A where ρ A = Tr B |ψ ψ| is the reduced density matrix for A, and the single-copy entanglement is S ∞ ≡ − log 2 λ 1 , where λ 1 is the largest eigenvalue of ρ A .Studies of the von Neumann entropy for quantum spin chains [3,4,5,6,7,8] have revealed that its dependence on the size ℓ of the block A is very different for noncritical and critical systems. For the former, the von Neumann entropy increases logarithmically with ℓ until it saturates when ℓ becomes of order the correlation length ξ, while for the latter (having ξ = ∞) it diverges logarithmically with ℓ.…”