Lower Critical Dimension Research Articles

ISING SPIN GLASS: RECENT PROGRESS IN THE FIELD THEORY APPROACH

A loop expansion around Parisi's replica symmetry breaking mean field theory is constructed, in zero field. We obtain the equation of state (and associated Parisi's solution) below the upper critical dimension du=6, and, in particular, explicit corrections in εlnt and (εlnt)2 with t=(Tc−T)/Tc and ε=6−d. This allows us to verify that standard scaling is satisfied with β=1+ε/2+0(ε2). We have also investigated the transverse (replicon) correlation function, particularly at zero overlap. Near du, G00 (p) ~ t/P4 (for p~0) is the most obvious obstacle to a meaningful theory in d=3. If we make the twofold assumption that (i) scaling applies, and (ii) the replicon propagator is dominated by the spectrum of the (small) transverse masses, we obtain the softer behavior G00(p) ~ (1/p2–η). (t/p1/v)β and a prediction for a soft replicon spectrum in tγk2γ/β γ/β instead of tk2 at du. We have checked tγ to one loop and work is in progress to check k2γ/β to the same order. Taking the above divergence in p−(d+2−η)/2 as the leading divergence defines the lower critical dimension d ℓ by d ℓ=2−η (d ℓ). Known values of η (at d=6, from the ε expansion at Tc, or from numerical work at d=4, 3) are compatible with d ℓ ~ 2.5±.3.

New critical-point exponent and new scaling laws for short-range Ising spin glasses.

We investigate the chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass. Some of our analytic and numerical results, obtained by the Migdal-Kadanoff renormalization scheme but argued to have general validity, differ from those of the scaling and droplet theory. First, chaos is present also at and above the critical temperature. Second, between ${\mathit{d}}_{\mathit{l}}$=2.46 (the lower critical dimension) and ${\mathit{d}}_{+}$=3.4, the chaos in the critical region is characterized by a new critical exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathit{c}}$ that has hitherto escaped attention.

Lower critical dimensions for superconducting long-range order in type-II superconductors.

It is shown that phase fluctuations destroy off-diagonal long-range order (phase coherence) in the Meissner phase below two dimensions, below four dimensions in the conventional Abrikosov flux-lattice phase, and below three dimensions in the limit of infinite Ginzburg ratio \ensuremath{\kappa}. The destruction of long-range order in the flux lattice is due to phase changes induced by shear motions of the flux lines. The phase coherence decays over a long length scale (of order millimeters) in three-dimensional systems.

Monte Carlo study of the bimodal three-state Potts glass.

Employing Monte Carlo simulations, we compute the spin-glass susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{SG}}$(T) of the three-state Potts glass model on a simple-cubic lattice for various temperatures and lattice sizes ranging from L=4 to 10. We use the discrete \ifmmode\pm\else\textpm\fi{}J distribution for the bonds. Comparing our results with a recent high-temperature series expansion, we find a systematic deviation at lower temperatures, which cannot be explained by finite-size effects in our data. The low-temperature behavior of ${\mathrm{\ensuremath{\chi}}}_{\mathrm{SG}}$(T) is compatible with d = 3 being the lower critical dimension of this model.

Vortex-glass transition in three dimensions.

We investigate the possibility of a vortex-glass transition in a disordered type-II superconductor in a magnetic field in three dimensions by numerical studies of a simplified model. Monte Carlo simulations at finite temperature and domain-wall renormalization-group calculations at T=0 indicate that d=3 is just above the lower critical dimension ${\mathit{d}}_{\mathit{l}}$, though the possibility that ${\mathit{d}}_{\mathit{l}}$=3 cannot be definitely ruled out. A comparison is made with XY and Ising spin glasses. The (effective) correlation-length exponent \ensuremath{\nu} and dynamical exponent z are in fairly good agreement with experiment.

Computer simulation of models for orientational glasses

Monte Carlo studies of two- and three-dimensional lattice models where quadrupoles interact with a nearest-neighbor Gaussian coupling are reviewed. None of these models has a thermodynamic glass phase transition at non-zero temperature like the Ising spin glass: rather, phase transitions at zero temperature occur that exhibit a dynamical freeze-in spread out over a wide temperature range and are characterized by a strongly non-exponential relaxation. The time-dependent glass order parameter, q(t), decays with time, t, compatible with a stretched exponential decay q( t) ∼ exp [− ( t/ τ) y ] with a strongly temperature-dependent exponent. While the static glass ‘susceptibility’ for isotropic orientational glasses diverges as x G ∼ T − γ 0 as T → 0, for the three-dimensional three-state Potts glass an exponential divergence is found, 1n x G ∼ T −2, implying that the system is at its lower critical dimension.

General validity of Jastrow-Laughlin wave functions

We construct a class of interacting-boson Hamiltonians whose exact ground-state wave functions are of Jastrow form. These Hamiltonians generally have both two- and three-body interactions; however, we show that the three-body interaction does not affect the long-wavelength physics. This enables us to deduce that (a) for Coulomb interacting bosons at T=0 the lower critical dimension is ${\mathit{d}}_{\mathit{c}}$=2; (b) in two dimensions, the ground-state wave function has the form of the modulus of the Laughlin wave function and exhibits algebraic long-range order; and (c) for short-range repulsions, we obtain a simple expression for the sound velocity. We also show that the Laughlin wave function is the generic ground-state wave function for fermions in a magnetic field corresponding to a filling factor of \ensuremath{\nu}=1/q.

Finite-size-scaling study of the simple cubic three-state Potts glass: Possible lower critical dimension d=3.

For small lattices with linear dimension L ranging from L=3 to L=8 we obtain the distribution function P(q) of the overlap q between two real replicas of the three-state Potts-glass model with symmetric nearest-neighbor interaction with a Gaussian distribution. A finite-size-scaling analysis suggests a zero-temperature transition to occur with an exponentially diverging correlation length ${\ensuremath{\xi}}_{\mathrm{SG}}$\ensuremath{\sim}exp(C/${\mathit{T}}^{\mathrm{\ensuremath{\sigma}}}$). This implies that d=3 is the lower critical dimension.

Critical exponents of the vortex glass to order epsilon 3.

The \ensuremath{\epsilon} expansion of the critical exponents of the vortex glass has been obtained to O(${\mathrm{\ensuremath{\epsilon}}}^{3}$) where \ensuremath{\epsilon}=6-d. A Pad\'e resummation of the series provides tentative evidence that the lower critical dimension might be greater than three.

Theory of self-avoiding walks on percolation fractals

A phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self-avoiding walks (SAW) is used to derive a new expression for the Flory exponent describing the average radius of gyration of SAWs on fractals. We focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of SAWs. The mean-field version of our new formula corresponds to the Aharony and Harris expression, who used the standard Flory approach for its derivation.

Random exchange and random field ising systems: Static and dynamic critical behavior

Theoretical and experimental studies of random magnets have generated considerable interest and controversy during the last decade. This critical review attempts to clarify experimental aspects of the REIM and, particularly, the RFIM problems in dilute Ising antiferromagnets. It has been shown that REIM to RFIM crossover scaling governs both static and dynamic critical behavior in all d = 3 systems (Fe x Zn 1− x F 2, Mn x Zn 1− x F 2 and Fe x Mg 1− x Cl 2), as well as nonequilibrium behavior, with a universal exponent φ = 1.42 ± 0.03 which satisfies the Aharony inequality φ#62; γ, the REIm susceptibility exponent. Extreme critical slowing down is observed in the d = 3 RFIM case. The behavior of RFIM systems at the lower critical dimension d ℓ = 2 differs in almost every respect from that d = 3 #62; d ℓ, contrary to what was claimed in the earliest neutron experiments and is still popularly believed.

Lower critical dimensions of stochastic nonequilibrium ferromagnetic Ising systems.

The existence of ordered phases in stochastic Ising dynamical systems with short-range ``ferromagnetic'' interactions is discussed with the emphasis on open questions. Various examples of models are presented; some of these do and some others do not undergo positive temperature (i.e., nonzero noise) phase transitions in two and three dimensions. The examples suggest that such equilibrium notions as lower critical dimension may not be uniquely definable for these nonequilibrium systems. The existence of temporally periodic states is also briefly discussed.

Renormalization-group analysis of the crumpling transition in large d.

The crumpling transition of D-dimensional elastic membranes is studied for large dimensions of embedding space d. The magnetic and thermal critical exponents ${y}_{m}$ and ${y}_{t}$ for isotropic membranes are obtained in a 1/d expansion. Both exponents vanish simultaneously at a lower critical dimension ${D}_{\mathrm{LC}=2\mathrm{\ensuremath{-}}2/\mathrm{d}+\mathrm{O}(1/{d}^{2}}$). We also consider the role of self-avoidance and of elastic anisotropies.

Zero-temperature scaling for Potts spin glasses.

The zero-temperature scaling approach is applied to the three-state Potts spin glass. Our results suggest that the \ifmmode\pm\else\textpm\fi{}J model has a lower critical dimensionality (LCD) greater than 3, whereas the Gaussian model has a LCD slightly less than 3. The T=0 scaling exponents are estimated in D=2 and D=3.

Hartree version of the Lifshitz NAC point

In the frame of a simple Hartree theory it is shown that “unusual” phenomena observed in the vicinity of the multicritical Lifshitz NAC point (nematic-smectic A-smectic C) are a reflection of the crossover between fluctuation regimes with different lower critical dimensionalities.

Phase diagrams for random-bond D-vector models

The mean field renormalization group is applied to study critical temperatures of the quenched (F, AF and SG phases) and annealed (F and AF phases) random bond D -vector model with short range interactions. The critical temperature against concentration phase diagrams are analytically calculated for general spin dimensionality and coordination number. The Ising ( D = 1), Heisenberg ( D = 3) and D → ∞ cases are analyzed for several values of the competition parameter. Reentrancies and limiting slope of the ferro- and antiferromagnetic boundaries are studied in detail. There is a critical value of the competition parameter for the existence of reentrancies in these lines. A mapping is applied to study the phase diagram of the antiferromagnetic Ising model with random decorating competing D -vector bond spins. It is found that for a fixed competing parameter there is a lower critical dimensionality D c for the stability of the ferromagnetic and spin glass phases. In all diagrams reentrancies are more prominent for higher dimensionalities. The present results are equivalent to the Bethe-Peierls approximation.

Ising Model in Correlated Random Fields on Kagomé Lattice

An Ising model on a Kagome lattice is solved exactly under correlated random fields which take values ± h . There exists a critical field h c =4 J ( J >0 is the exchange integral) and a phase transition for the system with h < h c , but there is no phase transition for the system with h ≥ h c . This result is consistent with the lower critical dimensionality obtained by the domain energy argument.

Absence of Spin Glass Ordering in Some Random Spin Systems

We discuss the improvement of the imperfect argument of Schuster for the absence of spin glass ordering at and below four dimensions in short-range continuous spin systems, and propose a condition for his method to be justified. Assuming satisfaction of the above condition, we apply the method to long-range interactions to reproduce the phase diagram drawn by a renormalization group approach. In particular the three-dimensional RKKY system is found to be at the lower critical dimension. Effects of exchange and uniaxial anisotropies are discussed. The latter anisotropy is shown not to induce transverse spin glass ordering if the system is not ordered in the , x y -plane by the exchange interaction.

Phase transition in the 3d random field Ising model

We show that the three-dimensional Ising model coupled to a small random magnetic field is ordered at low temperatures. This means that the lower critical dimension,dl for the theory isdl≦2, settling a long controversy on the subject. Our proof is based on an exact Renormalization Group (RG) analysis of the system. This analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of Imry and Ma. The RG acts in the space of Ising models and in the space of random field distributions, driving the former to zero temperature and the latter to zero variance.

Monte Carlo study of the Potts glass with nearest-neighbour random Gaussian interaction

The three-state Potts model with nearest-neighbour random Gaussian interaction on the simple cubic lattice is investigated by Monte Carlo simulation. Both static quantities (e.g. glass ordering susceptibility and the correlation function) and dynamic quantities (the analogue of the time-dependent Edwards-Anderson order parameter q(t) for Potts spins) are obtained. As for related models for orientational glasses, it is found that q(t) is consistent with the Kohlrausch law q(t) varies as exp(-(t/ tau )y) for a wide range of temperatures, with a strongly temperature-dependent exponent y, with y becoming very small as the temperature T to 0. The relaxation time tau increases dramatically as T is lowered; thus the system could only be equilibrated for temperatures where the correlation length is rather small. Since the critical region has not been reached, it cannot be distinguished whether the critical temperature Tc is non-zero or at T=0. If Tc=0, the divergences of tau and the ordering susceptibility are probably exponential, i.e. the system is then at its lower critical dimensionality.