A classical spherical random-field Hamiltonian with long-range (power-law) interactions is investigated by means of the replica theory. Both ferromagnetic and anti-ferromagnetic interactions are considered. The use of continuous variables instead of Ising variables in the spherical version of the model allows one to calculate the free energy exactly. The existence of an equilibrium phase transition is investigated based on the replica-symmetric solution. The results are applied to the Coulomb-glass model of interacting localized electrons in a disordered solid. This model is shown not to have an equilibrium phase transition for spatial dimensions D<or=4. For D>4 the model has a phase transition to an ordered phase; however, it does not have a phase transition to a 'glassy' phase.
In the present paper we analyze the critical properties of a quantum spherical spin-glass model with short-range random interactions. Since the model allows for rigorous detailed calculations, we can show how the effective partition function calculated with the help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0;\ensuremath{\omega};\ensuremath{-}\ensuremath{\omega})$ and a strictly short-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k}{\ensuremath{\omega}}_{1}{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The mean-field part
WMF coincides with previous results. The short-range part
WSR describes a phase transition in a
Q3-field theory, where the fluctuating fields depend on a space variable $\stackrel{P\vec}{r}$ and times ${\ensuremath{\tau}}_{1}$ and ${\ensuremath{\tau}}_{2}$. This we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the critical dimensionality as
dc=5 at very low temperatures due to the dimensionality shift
Dc=dc+1=6. We then perform an ${ϵ}^{\ensuremath{'}}$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We study the one-dimensional Dyson hierarchical model in the presence of a random field. This is a long range model where the interaction scales with the distance in a power-law-like form, $J(r)\ensuremath{\sim}{r}^{\ensuremath{-}\ensuremath{\rho}}$, and we can explore mean-field and non-mean-field behavior by changing $\ensuremath{\rho}$. We analyze the model at
T=0 and we numerically compute the non-mean-field critical exponents for Gaussian disorder. We also compute an analytic expression for the critical exponent $\ensuremath{\delta}$, and give an interesting relation between the critical exponents of the disordered model and the ones of the pure model, which seems to break down in the non-mean-field region. We finally compare our results for the critical exponents with the expected ones in
D-dimensional short range models and with the ones of the straightforward one-dimensional long range model.
We analyze the critical properties of a quantum spherical spin-glass model with random inverse power-law interactions ${r}^{\ensuremath{-}(d+\ensuremath{\sigma})}$. It was shown in a previous publication that the effective partition function calculated with help of the replica method for the spin-glass fluctuating fields ${Q}_{\ensuremath{\alpha}\ensuremath{\gamma}}(\stackrel{P\vec}{k},{\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2})$ separates into a mean-field contribution for the ${Q}_{\ensuremath{\alpha}\ensuremath{\alpha}}(0,\ensuremath{\omega},\ensuremath{-}\ensuremath{\omega})$ and a strictly long-range partition function for the fields ${Q}_{\ensuremath{\alpha}\ensuremath{\ne}\ensuremath{\gamma}}(\stackrel{P\vec}{k},{\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{2})$. Here $\ensuremath{\alpha},\ensuremath{\gamma}=1,\dots{},n$ are replica indices. The long-range part
WLR describes a phase transition in a
Q3-field theory that we analyze using the renormalization group with dimensional regularization and minimal subtraction of dimensional poles. By generalizing standard field theory methods to our particular situation, we can identify the upper critical dimensionality as ${d}_{c}=\frac{5\ensuremath{\sigma}}{2}$ at very low temperatures due to the dimensionality shift ${D}_{c}={d}_{c}+\frac{\ensuremath{\sigma}}{2}=3\ensuremath{\sigma}$. We then perform an ${ϵ}^{\ensuremath{'}}={d}_{c}\ensuremath{-}d$ expansion to order one loop to calculate the critical exponents by solving the renormalization-group equations.
We present a numerical study of an Ising spin glass with hierarchical interactions--the hierarchical Edwards-Anderson model with an external magnetic field (HEA). We study the model with Monte Carlo (MC) simulations in the mean-field (MF) and non-mean-field (NMF) regions corresponding to d ≥ 4 and d < 4 for the d-dimensional ferromagnetic Ising model respectively. We compare the MC results with those of a renormalization-group (RG) study where the critical fixed point is treated as a perturbation of the MF one, along the same lines as in the -expansion for the Ising model. The MC and the RG method agree in the MF region, predicting the existence of a transition and compatible values of the critical exponents. Conversely, the two approaches markedly disagree in the NMF case, where the MC data indicates a transition, while the RG analysis predicts that no perturbative critical fixed point exists. Also, the MC estimate of the critical exponent ν in the NMF region is about twice as large as its classical value, even if the analog of the system dimension is within only ~2% from its upper-critical-dimension value. Taken together, these results indicate that the transition in the NMF region is governed by strong non-perturbative effects.
We investigate the balanced
M=4,
p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range
p-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for
σ=0,0.25, and 0.55. For these values of the long-range exponent
σ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the
λ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of
λ≡ω2/ω1<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of
λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent
σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of
λ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature
TK in three dimensions might be zero and the complete absence of phase transitions in structural glasses.