  In this paper, we study global quantum discord (GQD) in infinite-size spin
chains. For this purpose, in the framework of matrix product states (MPSs), we
propose an effective procedure to calculate GQD (denoted as Gn) for consecutive
n-site subchains in infinite chains. For a spin-1/2 three-body interaction
model, whose ground state can be exactly expressed as MPSs, We use the
procedure to study Gn with n up to $24$. Then for a spin-1/2 XXZ chain, we
firstly use infinite time-evolving block decimation (iTEBD) algorithm to obtain
the approximate wavefunction in the from of MPSs, and then figure out Gn with n
up to $18$. In both models, Gn shows an interesting linear growth as the
increase of n, that is, Gn = k*n+b. Moreover, in non-critical regions the slope
$k$ of Gn converges very fast, while in critical regions it converges
relatively slow, and the behaviors are explained in a clear physical picture
with the short-range and long-range correlations. Based on these results, we
propose to use Gn/n to describe the global correlations in infinite chains.
Gn/n has twofold physical meanings. Firstly, it can be regarded as "global
discord per site", very similar to "energy per site" or "magnetization per
site" in quantum magnetic systems. Secondly, Gn/n (when n is large enough)
describes the quantum correlation between a single site and an (n-1)-site
block. Then we successfully apply our theory to an exactly soluble
infinite-size spin XY chain which is beyond the matrix product formula, and the
Hamiltonian can reduce to the transverse-field Ising model and the XX model.
The relation between GQD and quantum phase transitions in these models is
discussed.
