Physics – Condensed Matter – Disordered Systems and Neural Networks
Scientific paper
2011-10-14
2012 J. Phys. A: Math. Theor. 45, 095002
Physics
Condensed Matter
Disordered Systems and Neural Networks
22 pages, v2=final version
Scientific paper
10.1088/1751-8113/45/9/095002
For the quantum Ising model with ferromagnetic random couplings $J_{i,j}>0$ and random transverse fields $h_i>0$ at zero temperature in finite dimensions $d>1$, we consider the lowest-order contributions in perturbation theory in $(J_{i,j}/h_i)$ to obtain some information on the statistics of various observables in the disordered phase. We find that the two-point correlation scales as : $\ln C(r) \sim - \frac{r}{\xi_{typ}} +r^{\omega} u$, where $\xi_{typ} $ is the typical correlation length, $u$ is a random variable, and $\omega$ coincides with the droplet exponent $\omega_{DP}(D=d-1)$ of the Directed Polymer with $D=(d-1)$ transverse directions. Our main conclusions are (i) whenever $\omega>0$, the quantum model is governed by an Infinite-Disorder fixed point : there are two distinct correlation length exponents related by $\nu_{typ}=(1-\omega)\nu_{av}$ ; the distribution of the local susceptibility $\chi_{loc}$ presents the power-law tail $P(\chi_{loc}) \sim 1/\chi_{loc}^{1+\mu}$ where $\mu$ vanishes as $\xi_{av}^{-\omega} $, so that the averaged local susceptibility diverges in a finite neighborhood $0<\mu<1$ before criticality (Griffiths phase) ; the dynamical exponent $z$ diverges near criticality as $z=d/\mu \sim \xi_{av}^{\omega}$ (ii) in dimensions $d \leq 3$, any infinitesimal disorder flows towards this Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=2)=1/3$ and $\omega(d=3) \sim 0.24$) (iii) in finite dimensions $d > 3$, a finite disorder strength is necessary to flow towards the Infinite-Disorder fixed point with $\omega(d)>0$ (for instance $\omega(d=4) \simeq 0.19$), whereas a Finite-Disorder fixed point remains possible for a small enough disorder strength. For the Cayley tree of effective dimension $d=\infty$ where $\omega=0$, we discuss the similarities and differences with the case of finite dimensions.
Garel Thomas
Monthus Cecile
No associations
LandOfFree
Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Random Transverse Field Ising Model in dimension $d>1$ : scaling analysis in the disordered phase from the Directed Polymer model will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-522273