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...
