We study the details of the U(3)--O(4) quantum phase transition in the U(4) vibron model. Both asymptotic analysis in the classical limit and rigorous calculations for finite boson number systems indicate that a second-order phase transition is still there even for the systems with boson number
N ranging from tens to hundreds. Two kinds of effective order parameters, including
E1 transition ratios $B(E1:{2}_{1}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$ and $B(E1:{0}_{2}\ensuremath{\rightarrow}{1}_{1})/B(E1:{1}_{1}\ensuremath{\rightarrow}{0}_{1})$, and the energy ratios
E21/E02 and
E31/E02 are proposed to identify the second-order phase transition in experiments. We also found that the critical point of phase transition can be approximately described by the E(3) symmetry, which persists even for moderate
N 10 protected by the scaling behaviors of quantities at the critical point. In addition, a possible empirical example exhibiting roughly the E(3) symmetry is discussed.
We develop an analytical theory for generic disorder-driven quantum phase transitions. We apply this formalism to the superconductor-insulator transition and we briefly discuss the applications to the order-disorder transition in quantum magnets. The effective spin-
21 models for these transitions are solved in the cavity approximation which becomes exact on a Bethe lattice with large branching number
K⪢1 and weak dimensionless coupling
g⪡1. The characteristic feature of the low-temperature phase is a large self-formed inhomogeneity of the order-parameter distribution near the critical point $K\ensuremath{\ge}{K}_{c}(g)$, where the critical temperature
Tc of the ordering transition vanishes. We find that the local probability distribution
P(B) of the order parameter
B has a long power-law tail in the region where
B is much larger than its typical value
B0. Near the quantum-critical point, at $K\ensuremath{\rightarrow}{K}_{c}(g)$, the typical value of the order parameter vanishes exponentially, ${B}_{0}\ensuremath{\propto}{e}^{\ensuremath{-}C/[K\ensuremath{-}{K}_{c}(g)]}$ while the spatial scale
Ninh of the order parameter inhomogeneities diverges as ${[K\ensuremath{-}{K}_{c}(g)]}^{\ensuremath{-}2}$. In the disordered regime, realized at
K<Kc(g) we find actually two distinct phases characterized by different behavior of relaxation rates. The first phase exists in an intermediate range of ${K}^{\ensuremath{\ast}}(g)<K<{K}_{c}(g)$. It has two regimes of energies: at low excitation energies, $\ensuremath{\omega}<{\ensuremath{\omega}}_{d}(K,g)$, the many-body spectrum of the model is discrete, with zero-level widths, while at $\ensuremath{\omega}>{\ensuremath{\omega}}_{d}$ the level acquire a nonzero width which is self-generated by the many-body interactions. In this phase the spin model provides by itself an intrinsic thermal bath. Another phase is obtained at smaller $K<{K}^{\ensuremath{\ast}}(g)$, where all the eigenstates are discrete, corresponding to full many-body localization. These results provide an explanation for the activated behavior of the resistivity in amorphous materials on the insulating side near the superconductor-insulator transition and a semiquantitative description of the scanning tunneling data on its superconductive side.
After a quench of transverse field, the asymptotic long-time state of the Ising model displays a transition from a ferromagnetic phase to a paramagnetic phase as the post-quench field strength increases, which is revealed by the vanishing of the order parameter defined as the averaged magnetization over time. We estimate the critical behavior of the magnetization at this nonequilibrium phase transition by using the mean-field approximation. In the vicinity of the critical field, the magnetization vanishes as the inverse of a logarithmic function, which is significantly distinguished from the critical behavior of order parameter at the corresponding equilibrium phase transition, i.e. a power-law function.