ed matter. To study QPs and QPTs, systems should include rich quantum phase diagram. In this sense,
corresponding quantum spin models should have strong quantum fluctuation, strong geometric frustration, complicated spin-spin
exchange or orbital degrees of freedom, which induces a variety of spontaneous symmetry breaking (SSB) or hidden spontaneous
symmetry breaking. QPs induced by SSB can be characterized by local order parameters, a concept from Landau-Ginzburg-Wilson
paradigm (LGW). However, there is also a novel class of topological QPs beyond LGW which has aroused one’s great interest since
the finding of Haldane phase. Such QPs could only be characterized by topological long-range nonlocal string correlation order
parameters instead of local order parameters. In this paper, we will investigate spin-1/2 quantum compass chain model (QCC) with
orbital degrees of freedom in x, y and z components whose prototype is quantum compass model including novel topological QPs
beyond LGW. Consequently, one can also anticipate the existence of novel topological QPs in QCC. However, very few attentions have
been paid to the QPs and QPTs for QCC, which deserves a further investigation. By using infinite time evolving block decimation in
the presentation of matrix product states, we study the QPs and QPTs of QCC. To characterize QPs and QPTs of QCC, ground state
energy, local order parameter, topological long-range nonlocal string correlation order parameters, critical exponent, correlation length
and central charge are calculated. The results show the phase diagram of QCC including local antiferromagnetic phase, local stripe
antiferromagnetic phase, oscillatory odd Haldane phase and monotonic odd Haldane phase. QPTs from oscillatory odd Haldane phase
to local stripe antiferromagnetic phase and from local antiferromagnetic phase to monotonic odd Haldane phase are continuous; On the
contrary, QPTs from local stripe antiferromagnetic phase to local antiferromagnetic phase and from oscillatory odd Haldane phase to
monotonic odd Haldane phase are discontinuous. The crossing point where line of continuous QPTs meets with line of discontinuous
QPTs is multiple critical point. The critical exponents of local antiferromagnetic order parameter, local stripe antiferromagnetic order
parameter, topological long-range nonlocal oscillatory odd string correlation order parameter and topological long-range nonlocal
monotonic odd string correlation order parameter β = 1=8 and the central charges c = 1=2 at the critical points show that the QPTs
from local phases to nonlocal phases belong to the Ising-type universality class.
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