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.
The large-distance behaviors of the random field and random anisotropy
O(N) models are studied with the functional renormalization group in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. The random anisotropy Heisenberg
(N=3) model is found to have a phase with an infinite correlation length at low temperatures and weak disorder. The correlation function of the magnetization obeys a power law $〈\mathbf{m}({\mathbf{r}}_{1})\mathbf{m}({\mathbf{r}}_{2})〉\ensuremath{\sim}|{\mathbf{r}}_{1}\ensuremath{-}{\mathbf{r}}_{2}{|}^{\ensuremath{-}0.62\ensuremath{\epsilon}}.$ The magnetic susceptibility diverges at low fields as $\ensuremath{\chi}\ensuremath{\sim}{H}^{\ensuremath{-}1+0.15\ensuremath{\epsilon}}.$ In the random field
O(N) model the correlation length is found to be finite at the arbitrarily weak disorder for any
N>3. The random field case is studied with a simple method, based on a rigorous inequality. This approach allows one to avoid the integration of the functional renormalization-group equations.
We investigate the transition from the many-body localized phase to the ergodic thermalized phase at an infinite temperature in an
XY spin chain with
L spins, which experiences power-law decaying interactions in the form of ${V}_{ij}\ensuremath{\propto}1/{\left|i\ensuremath{-}j\right|}^{\ensuremath{\alpha}}$ (
i,j=1,...,L) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to
L=18, we systematically analyze the energy gap statistics, the half-chain entanglement entropy, and the uncertainty of the entanglement entropy of the system at different interaction exponents $\ensuremath{\alpha}$. The finite-size critical scaling allows us to determine the critical disorder strength
Wc and the critical exponent $\ensuremath{\nu}$ at the many-body-localization phase transition, as a function of the interaction exponent $\ensuremath{\alpha}$ in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$. We find that both
Wc and $\ensuremath{\nu}$ diverge when $\ensuremath{\alpha}$ decreases to a critical power of ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.16\ifmmode\pm\else\textpm\fi{}0.17$, indicating the absence of many-body localization for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. Our result differs from the analytical prediction of ${\ensuremath{\alpha}}_{c}=3/2$ found in a previous work based on a scaling argument [A. L. Burin, Phys. Rev. B 92, 104428 (2015)], but is close to the recent numerical result ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ obtained from quantum dynamics simulation [A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Phys. Rev. A 99, 033610 (2019)].
We use linear spin wave theory and series expansion method to study the diluted quantum antiferromagnetic Heisenberg model at zero temperature near the p = 1 and p =
psbQ respectively, where p is the concentration of the spins or bonds, and
psbQ is the transition point from order to disorder of the quantum spin system. For the first case, our complete calculation to the first order in x
equiv 1
− p gives a dispersion relation for long-wavelength spin waves in the form
omega(vecq) =
c(x)q+igamma(x)qsptau. Our results for
c(x) agrees with previous work, where
c(x) =
c(0)lbrack1−alphax+O(xsp2)rbrack. In particular the coefficient
alpha, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit
dto2. One major new results is that
tau =
d−1, which means that d = 2 is the lower critical dimension at which infrared divergences affect the dynamical response due to vacancies. For the second case, the critical behavior of the quantum system near
psbQ is found to be different from that of the percolation and pure systems. Thus there exists a new fixed point in the phase diagram, which specifies a new universality class. The correlation length exponents
nu (which comes from the relation
xisim(psbQ−p)spnu, where
xi is the correlation length) are estimated, which are 0.65
pm 0.2 and 0.55
pm 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that
psbQ>psbc.
We study the surface scaling behavior of a semi-infinite d-dimensional O(N) spin system in the presence of a quenched random field and random anisotropy disorders. It is known that above the lower critical dimension d(LC) = 4 the infinite models undergo a paramagnetic-ferromagnetic transition for N > N(c) (N(c) = 2.835 for the random field and N(c) =9.441 for random anisotropy). For N < N(c) and d < d(LC) there exists a quasi-long-range-order phase with a zero order parameter and a power-law decay of spin correlations. Using a functional renormalization group, we derive the surface scaling laws that describe the ordinary surface transition for d > d(LC) and the long-range behavior of spin correlations near the surface in the quasi-long-range-order phase for d < d(LC). The corresponding surface exponents are calculated to one-loop order. The obtained results can be applied to the surface scaling of periodic elastic systems in disordered media, amorphous magnets, and (3)He-A in aerogel.
We investigate the zero-temperature and finite-temperature phase transitions of quantum Ising and quantum rotor models. We here assume a long-range (falling off as ${1/r}^{d+\ensuremath{\sigma}},$ where r is the distance between two spins/rotors in units of lattice spacing) ferromagnetic interaction among the spins or rotors. We find that the long-range behavior of the interaction drastically modifies the universal critical behavior of the system. The corresponding upper critical dimension and the hyperscaling relation and exponents associated with the quantum transition are modified and, as expected, they attain values of short-range system when $\ensuremath{\sigma}=2.$ The dynamical exponent varies continuously as the parameter \ensuremath{\sigma} and is unity for $\ensuremath{\sigma}=2.$ The one-dimensional long-range quantum Ising system shows a phase transition at
T=0 for all values of \ensuremath{\sigma}. The most interesting observation is that the phase diagram for $\ensuremath{\sigma}=d=1$ shows a line of Kosterlitz-Thouless transition at finite temperature even though the
T=0 transition is a simple order-disorder transition. These finite temperature transitions are studied near the phase boundary using renormalisation group equations and a region with diverging susceptibility is located. We have also studied one-dimensional quantum rotor model which exhibits a rich and interesting transition behavior depending upon the parameter \ensuremath{\sigma}. We explore the phase diagram extending the short-range quantum nonlinear \ensuremath{\sigma} model renormalisation group equations to the present case.