The density matrix renormalization group (DMRG) has been extended to study quantum phase transitions on random graphs of fixed connectivity. As a relevant example, we have analysed the random Ising model in a transverse field. If the couplings are random, the number of retained states remains reasonably low even for large sizes. The resulting quantum spin-glass transition has been traced down for a few disorder realizations, through the careful measurement of the entanglement entropy and some selected observables: spatial correlations, energy gap and spin-glass susceptibility, among others.
We study the anisotropic Heisenberg spin-glass model in a three-dimensional hierarchical lattice (designed to approximate the cubic lattice), within a real-space renormalization-group approach. Two different initial probability distributions for the exchange interaction (
Jij), Gaussian and uniform, are used, with zero mean and width
Jˉ. The
(kT/Jˉ)×Δ0 phase diagram is obtained, where
T is the temperature,
Δ0 is the first moment of the probability distribution for the uniaxial anisotropy, and
k is the Boltzmann constant. For the Ising model (
Δ0=1), there is a spin-glass phase at low temperatures (high
Jˉ) and a paramagnetic phase at high temperatures (low
Jˉ). For the isotropic Heisenberg model (
Δ0=0), our results indicate no spin-glass phase at finite temperatures. The transition temperature between the spin-glass and paramagnetic phase decreases with
Δ0, as expected, but goes to zero at a finite value of the anisotropy parameter, namely
Δ0=Δc∼0.59. Our results indicate that the whole transition line, between the paramagnetic and the spin-glass phases, for
Δc≤Δ0<1, belongs to the same universality class as the transition for the Ising spin glass.
We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards–Anderson model to discuss the existence of the spin glass (SG) phase. We derive the RG equations under the replica symmetric (RS) ansatz and the resulting flow diagram indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that this failure at the MCP is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.