Universal corrections to the entanglement entropy in gapped quantum spin chains: A numerical study

Abstract

We carry out a numerical study of the bi-partite entanglement entropy in the gapped regime of two paradigmatic quantum spin chain models: the Ising chain in an external magnetic field and the anti-ferromagnetic XXZ model. The universal scaling limit of these models is described by the massive Ising field theory and the SU (2)-Thirring (sine-Gordon) model, respectively. We may therefore exploit quantum field theoretical results to predict the behaviour of the entropy. We numerically confirm that, in the scaling limit, corrections to the saturation of the entropy at large region size are proportional to K0(2mr) where m is a mass scale (the inverse correlation length) and r the length of the region under consideration. The proportionality constant is simply related to the number of particle types in the universal spectrum. This was originally predicted in [1,2] for two-dimensional quantum field theories. Away from the universal region our numerics suggest an entropic behaviour following quite closely the quantum field theory prediction, except for extra dependencies on the correlation length. Introduction and discussion. Entanglement is a fundamental property of the state of a quantum system. In the context of quantum computation, it is a crucial resource [3]. Conceptually, it characterizes the structure of quantum fluctuations in a more universal way than other widely studied objects such as correlation functions. Developing theoretical measures of entanglement is therefore important to further understand the structure of quantum states, a problem of particular interest and difficulty for quantum many-body systems. A popular measure of entanglement is the bi-partite entanglement entropy. It measures the entanglement between two complementary sets of observables in a quantum system [4]. Interestingly, the entanglement entropy exhibits universal behaviour near quantum critical points: it has features which do not depend on the details of the model but rather on its universality class. In the last decade, this property has made the study of the entanglement entropy a very active field of research.

Boundary fidelity and entanglement in the symmetry protected topological phase of the SSH model

“…This type of scaling behaviour has recently been confirmed numerically for certain spin models [34].…”

“…Apart from a recent numerical study of spin models [34] this is the second independent numerical confirmation of this formula for a different class of gapped models. Again, SPT and trivial phase show exactly the same properties.…”

Symmetry resolved entanglement in integrable field theories via form factor bootstrap

“…(ϑ, n, α) = p ± D (ϑ, n, ϑ)P (ϑ, n, 0) satisfies the parity and cyclic property Eqs. ( 62) and ( 63) and the equivalent of (49), i.e., (64). Furthermore, p ± D (ϑ, n, α) does not introduce any additional poles on the physical sheet.…”

U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap

“…Although all matrix elements are in principle computable, the multi-point correlation functions at large distances are generically dominated by the first few sets of form factors. This property has been exploited in integrable QFTs (IQFTs) to obtain predictions for the entanglement entropy in many different physical circumstances [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75].…”