Symmetry Breaking in Matrix-Product States

We consider matrix-product states for the transverse-field Ising chain of finite and infinite size $N$ and small matrix sizes $D=2-8$. The matrices are variationally optimized using several methods. For finite $N$, below the critical field, there are energy minimums for symmetric as well as symmetry-broken states. The energies cross at a field strength $h_c(N,D)$; thus the transition is first-order in this approximation. However, for $N \to \infty$ the transition is continuous for any $D$. We find that the asymptotic critical behavior is then always mean-field like (the magnetization exponent $\beta=1/2$), but a window of the exactly known power-law scaling ($\beta=1/8$) emerges as $D$ increases. We point out that even if the energy is optimized to the level of double precision ($\approx 10^{-12}$ relative error) there is significant finite-size smoothing of the magnetization curve. Higher precision is required to access the asymptotic critical behavior.