Critical entanglement ofXXZHeisenberg chains with defects

Abstract

We study the entanglement properties of anisotropic open spin one-half Heisenberg chains with a modified central bond. The entanglement entropy between the two half-chains is calculated with the density-matrix renormalization method (DMRG). We find a logarithmic behaviour with an effective central charge c ′ varying with the length of the system. It flows to one in the ferromagnetic region and to zero in the antiferromagnetic region of the model. In the XX case it has a non-universal limit and we recover previous results.The consideration of entanglement properties has brought a new element into the study of manyparticle quantum states. Entanglement is related to a division of the system into two parts and can be quantified via the reduced density matrix ρ and the entropy S = −tr(ρ ln ρ) connected with it. This quantity is particularly interesting for the ground state of critical one-dimensional systems. For example, if one cuts an open chain into two halves of length L, conformal invariance predicts the universal formwhere c is the central charge in the conformal classification. 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]. For this system it can also be proven by putting proper bounds on S [8, 9]. Since the entanglement entropy measures the mutual coupling of the two parts of a system in the wave function, a defect at their boundary should have a strong influence. In one dimension, defects show particularly interesting features in Luttinger liquids, i.e. in strongly correlated fermionic systems. Their effective strength then depends on the sign of the interaction [10,11] and goes to zero for attraction while it diverges for repulsion, as the system size increases. This has been checked in various further studies, see e.g. [12,13,14,15,16,17]. 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. [18]. In this case, the logarithmic behaviour was found to persist, but with a prefactor c ef f which depends continuously on the defect strength.In the present paper we present a numerical study of this problem for the case of a planar XXZ spin chain, which has c = 1 and is the lattice version of a Luttinger model. We treat open chains with one modified bond in the middle and use density-matrix renormalization (DMRG) [19,20]. This method is ideally suited for such a study since the calc...

Fano resonances and entanglement entropy

“…The coefficient of the entropy therefore renormalizes to the value 1/6 corresponding to a semiinfinite chain with an open boundary. Note, that a similar renormalization behavior was found for interacting electrons in the Luttinger liquid regime bisected by a hopping defect 10,11 .…”

“…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 in many-body systems

“…They find a logarithmic behavior with an effective central charge varying with the length of the system. The numerical simulations of (Zhao et al, 2006) show that by going from the antiferromagnetic to the ferromagnetic case the effective central charge grows from zero to one in agreement with (Levine, 2004). The combined presence of interaction between the excitation and a local impurity modifies in an important way the properties of a onedimensional system.…”

“…The entanglement entropy of one-dimensional systems is affected by the presence of impurities in the bulk (Levine, 2004;Zhao et al, 2006) or aperiodic couplings (Igloi et al, 2007). In these cases the entanglement entropy has the same form as in Eq.…”

The partition function zeroes of quantum critical points

“…Firstly, in the vicinity of the two critical points at φ = 0 and φ = 0.9 the logarithm of the zeros density follows the second order scaling fit of (28). We tabulate our fitting results to (28) with recent DMRG and exact diagonalization studies [45][47] [49]. Furthermore, the system size at which our analysis appears to be come unreliable directly corresponds to the range of volumes that can be treated in these recent studies, which indicates a potential common dominant source of systematic error in the numerical roundoff errors from diagonalization.…”