Abstract
We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class. While much insight has been gained in homogeneous systems by making use of conformal field theory techniques, our focus is on systems for which the Luttinger parameter K depends on position, and conformal invariance is broken. An important point of our analysis is that contributions stemming from the UV cutoff have to be treated very carefully, since they now depend on position. We show that such terms can be removed either by considering regularized entropies specifically designed to do so, or by tabulating numerically the cutoff, and reconstructing its contribution to the entropy through the local density approximation. We check our method numerically in the spin-1/2 XXZ spin chain in a spatially varying magnetic field, and find excellent agreement.
Highlights
Many quantum critical systems in one-dimension (1D), including various quantum gases or spin chains, belong to the universality class of Luttinger liquids [1,2,3,4,5,6,7]
We develop a general framework to compute the scaling of entanglement entropy in inhomogeneous one-dimensional quantum systems belonging to the Luttinger liquid universality class
They are ubiquitous in the 1d quantum realm because U(1) symmetry itself is ubiquitous—it is associated to particle number conservation, often naturally present in spin chains or quantum gases, and because of the limited amount of relevant perturbations—often forbidden by symmetry—that could drive the system away from the renormalization group massless fixed point
Summary
Many quantum critical systems in one-dimension (1D), including various quantum gases or spin chains, belong to the universality class of Luttinger liquids [1,2,3,4,5,6,7]. From an effective field theory (FT) perspective, Luttinger liquids are the one-dimensional (1D) systems whose low-energy description is provided by a massless free boson FT in 1 + 1D, or in other words a U(1) conformal FT [8]. They are ubiquitous in the 1d quantum realm because U(1) symmetry itself is ubiquitous—it is associated to particle number conservation, often naturally present in spin chains or quantum gases—, and because of the limited amount of relevant perturbations—often forbidden by symmetry—that could drive the system away from the renormalization group massless fixed point. Recall that the Rényi entanglement entropy of a pure state |ψ , which will always be the ground state in this paper, is defined by
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