Fermionic Ising glasses with BCS pairing interaction. Tricritical behaviour

We have examined the role of the BCS pairing mechanism in the formation of the magnetic moment and henceforth a spin glass (SG) phase by studying a fermionic Sherrington-Kirkpatrick model with a local BCS coupling between the fermions. This model is obtained by using perturbation theory to trace out the conduction electrons degrees of freedom in conventional superconducting alloys. The model is formulated in the path integral formalism where the spin operators are represented by bilinear combinations of Grassmann fields and it reduces to a single site problem that can be solved within the static approximation with a replica symmetric ansatz. We argue that this is a valid procedure for values of temperature above the de Almeida-Thouless instability line. The phase diagram in the T-g plane, where g is the strength of the pairing interaction, for fixed variance J ² /N of the random couplings J_ij, exhibits three regions: a normal paramagnetic (NP) phase, a spin glass (SG) phase and a pairing (PAIR) phase where there is formation of local pairs.The NP and PAIR phases are separated by a second order transition line g=g _c (T) that ends at a tricritical point T ₃ =0.9807J, g ₃ =5,8843J, from where it becomes a first order transition line that meets the line of second order transitions at T _c =0.9570J that separates the NP and the SG phases. For T<T _c the SG phase is separated from the PAIR phase by a line of first order transitions. These results agree qualitatively with experimental data in . Received 14 May 1998     

Mixed spin transverse Ising model with longitudinal random crystal field interactions

The effect of a longitudinal random crystal field interaction on the phase diagrams of the mixed spin transverse Ising model consisting of spin-1/2 and spin-1 is investigated within the finite cluster approximation based on a single-site cluster theory. In order to expand a cluster identity of spin-1, we transform the spin-1 to spin-1/2 representation containing Pauli operators. We derive the state equations applicable to structures with arbitrary coordination number N. The phase diagrams obtained in the case of a honeycomb lattice (N=3) and a simple-cubic lattice (N=6), are qualitatively different and examined in detail. We find that both systems exhibit a variety of interesting features resulting from the fluctuation of the crystal field interactions. Received: 13 February 1998 / Accepted: 17 March 1998     

Stability conditions for fermionic Ising spin-glass models in the presence of a transverse field

The stability of a spin-glass (SG) phase is analyzed in detail for a fermionic Ising SG (FISG) model in the presence of a magnetic transverse field Γ. The fermionic path integral formalism, replica method and static approach have been used to obtain the thermodynamic potential within one step replica symmetry breaking ansatz. The replica symmetry (RS) results show that the SG phase is always unstable against the replicon. Moreover, the two other eigenvalues λ_± of the Hessian matrix (related to the diagonal elements of the replica matrix) can indicate an additional instability to the SG phase, which enhances when Γ is increased. Therefore, this result suggests that the study of the replicon cannot be enough to guarantee the RS stability in the present quantum FISG model, especially near the quantum critical point. In particular, the FISG model allows changing the occupation number of sites, so one can get a first order transition when the chemical potential exceeds a certain value. In this region, the replicon and the λ_± indicate instability problems for the SG solution close to all ranges of a first order boundary.     

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits us to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix or operator representation. In particular, special dualities such as exact dimensional reduction, emergent and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the ?₂ Higgs model is dual to the extended toric code model in any number of dimensions. Non-local transformations such as dual variables and Jordan–Wigner dictionaries are algorithmically derived from the local mappings of bond algebras. This permits us to establish a precise connection between quantum dual and classical disorder variables. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long-standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.     

Twist free energy in a spin glass

The field theory of a short range spin glass with Gaussian random interactions, is considered near the upper critical dimension six. In the glassy phase, replica symmetry breaking is accompanied with massless Goldstone modes, generated by the breaking of reparametrization invariance of a Parisi type solution. Twisted boundary conditions are thus imposed at two opposite ends of the system in order to study the size dependence of the twist free energy. A loop-expansion is performed to first order around a twisted background. It is found, as expected but it is non trivial, that the theory does renormalize around such backgrounds, as well as for the bulk. However two main differences appear, in comparison with simple ferromagnetic transitions: (i) the loop expansion yields a (negative) anomaly in the size dependence of the free energy, thereby lifting the lower critical dimension to a value greater than two (ii) the free energy is lowered by twisting the boundary conditions. This situation is common in spin glasses, reflecting the non-positivity of mode multiplicity in replica symmetry breaking, but its physical meaning is still unclear. Received 12 April 2002 / Received in final form 30 July 2002 Published online 19 November 2002     

Change of the scaling behavior of the end-to-end square distance in a two-dimensional polydisperse system

The simulation of a two-dimensional, broadly polydisperse, living polymers system at high concentration reveals an unusual conformational behaviour for the longer chains. Unlike in three dimensions, the longer chains are not swollen but are squeezed by the smaller chains. This observation is discussed in terms of a two dimensional solvent- polymer mixture whose solvent particules are larger than the polymer monomers. Received: 13 December 1996 / Revised: 16 March 1998 / Accepted: 27 March 1998     

The effect of stiffness in wormlike micelles

The effect of stiffness in a 2D living polymer system is investigated by Monte-Carlo simulation in a canonical ensemble. As the flexibility decreases, the mean chain contour length decreases and goes through a local maximum. The mean end to end square distance shows a non-monotonic behaviour due to the coil-to-rod transition and the decrease in chain contour length. Near the maximum of chain ordering in the bulk, the chain length distribution adapts itself to increase the configurational entropy. With the parameters used in this simulation, it seems that the effect of the stiffness for high stiffness is to decrease as in the isotropic case, since the ordering decreases again. Received: 16 September 1997 / Revised: 27 June 1998 / Accepted: 29 June 1998     

Phase diagram of the two-dimensional Ising model with random competing interactions

An Ising model with ferromagnetic nearest-neighbor interactions J₁ (J₁>0) and random next-nearest-neighbor interactions [+J₂ with probability p and −J₂ with probability (1−p); J₂>0] is studied within the framework of an effective-field theory based on the differential-operator technique. The order parameters are calculated, considering finite clusters with n=1,2, and 4 spins, using the standard approximation of neglecting correlations. A phase diagram is obtained in the plane temperature versus p, for the particular case J₁=J₂, showing both superantiferromagnetic (low p) and ferromagnetic (higher values of p) orderings at low temperatures.     

Antiferromagnetic Ising model in scale-free networks

The antiferromagnetic Ising model in uncorrelated scale-free networks has been studied by means of Monte Carlo simulations. These networks are characterized by a connectivity (or degree) distribution P(k) ∼k^-γ. The disorder present in these complex networks frustrates the antiferromagnetic spin ordering, giving rise to a spin-glass (SG) phase at low temperature. The paramagnetic-SG transition temperature T_c has been studied as a function of the parameter γ and the minimum degree present in the networks. T_c is found to increase when the exponent γ is reduced, in line with a larger frustration caused by the presence of nodes with higher degree.     

On the properties of small-world network models

We study the small-world networks recently introduced by Watts and Strogatz [Nature 393, 440 (1998)], using analytical as well as numerical tools. We characterize the geometrical properties resulting from the coexistence of a local structure and random long-range connections, and we examine their evolution with size and disorder strength. We show that any finite value of the disorder is able to trigger a “small-world” behaviour as soon as the initial lattice is big enough, and study the crossover between a regular lattice and a “small-world” one. These results are corroborated by the investigation of an Ising model defined on the network, showing for every finite disorder fraction a crossover from a high-temperature region dominated by the underlying one-dimensional structure to a mean-field like low-temperature region. In particular there exists a finite-temperature ferromagnetic phase transition as soon as the disorder strength is finite. [0.5cm] Received 29 March 1999 and Received in final form 21 May 1999     

Scaling and infrared divergences in the replica field theory of the Ising spin glass

Replica field theory for the Ising spin glass in zero magnetic field is studied around the upper critical dimension d=6. A scaling theory of the spin glass phase, based on Parisi's ultrametrically organised order parameter, is proposed. We argue that this infinite step replica symmetry broken (RSB) phase is nonperturbative in the sense that amplitudes of scaling forms cannot be expanded in term of the coupling constant w². Infrared divergent integrals inevitably appear when we try to compute amplitudes perturbatively, nevertheless the -expansion of critical exponents seems to be well-behaved. The origin of these problems can be traced back to the unusual behaviour of the free propagator having two mass scales, the smaller one being proportional to the perturbation parameter w² and providing a natural infrared cutoff. Keeping the free propagator unexpanded makes it possible to avoid producing infrared divergent integrals. The role of Ward-identities and the problem of the lower critical dimension are also discussed. Received 23 December 1998 and Received in final form 23 March 1999     

The effect of competition between the frequency of the field and the frequency of the spin flipping on the kinetics of Ising metamagnet

We consider the effect of a variable representing the competition between the frequency of the field and the frequency of the spin flipping (Ω) on the dynamics of the metamagnetic Ising model in a cubic lattice under the presence of a time varying (oscillating) external magnetic field. The system is modelled with a formalism of master equation at a mean-field level. The time averaged staggered magnetization (Ms) acts as the order parameter and divides temperature field plane into three regions: anti-ferromagnetic, paramagnetic and coexistence of anti-ferromagnetic and paramagnetic phases. It is observed that the topology of the dynamical phase diagram depends strongly on Ω as well as the ratio between interlayer and intralayer couplings.     

Exact results of a dimerized S=1 Ising chain with single-ion anisotropy

The dimerized spin-1 Ising chain with both longitude and transverse single-ion anisotropies D_z and D_x is solved exactly by means of a mapping to the spin- Ising chain with the alternating transverse fields and the Jordan-Wigner transformation. The analytical expressions of the quasi-particles’ spectra Λ_k, the minimal energy gap Δ₀ for exciting a fermion quasi-particle, the minimal energy gap Δ_h for exciting a hole, and the ground-state energy E_g are obtained. The phase diagram of the ground state is also given. The results show that the system exhibits a series of quantum phase transitions depending on the dimerization strength of the crystal fields, while the quantum critical points are determined exactly.     

Frustrated Blume-Emery-Griffiths model

A generalised integer S Ising spin glass model is analysed using the replica formalism. The bilinear couplings are assumed to have a Gaussian distribution with ferromagnetic mean . Incorporation of a quadrupolar interaction term and a chemical potential leads to a richer phase diagram with transitions of first and second order. The first order transition may be interpreted as a phase separation, and contrary to what has been argued previously, it persists in the presence of disorder. Finally, the stability of the replica symmetric solution with respect to fluctuations in replica space is analysed, and the transition lines are obtained both analytically and numerically. Received 13 January 1997     

Analytical solution of 1D Ising-like systems modified by weak long range interaction

It is well-known that 1D systems with only nearest neighbour interaction exhibit no phase transition. It is shown that the presence of a small long range interaction treated by the mean field approximation in addition to strong nearest neighbour interaction gives rise to hysteresis curves of large width. This situation is believed to exist in spin crossover systems where by the deformation of the spin changing molecules, an elastic coupling leads to a long range interaction, and strong bonding between the molecules in a chain compound leads to large values for nearest neighbour interaction constants. For this interaction scheme an analytical solution has been derived and the interplay between these two types of interaction is discussed on the basis of experimental data of the chain compound which exhibits a very large hysteresis of 50 K above RT at 370 K. The width and shape of the hysteresis loop depend on the balance between long and short range interaction. For short range interaction energies much larger than the transition temperature the hysteresis width is determined by the long range interaction alone. Received 26 November 1998     

Quantum fluctuations in a scale-free network-connected Ising system

We study the effect of quantum fluctuations in an Ising spin system on a scale-free network of degree exponent γ>5 using a quantum Monte Carlo simulation technique. In our model, one can adjust the magnitude of the magnetic field perpendicular to the Ising spin direction and can therefore control the strength of quantum fluctuations for each spin. Our numerical analysis shows that quantum fluctuations reduce the transition temperature T_c of the ferromagnetic-paramagnetic phase transition. However, the phase transition belongs to the same mean-field type universality class both with and without the quantum fluctuations. We also study the role of hubs by turning on the quantum fluctuations exclusively at the nodes with the most links. When only a small number of hub spins fluctuate quantum mechanically, T_c decreases with increasing magnetic field until it saturates at high fields. This effect becomes stronger as the number of hub spins increases. In contrast, quantum fluctuations at the same number of “non-hub” spins do not affect T_c. This implies that the hubs play an important role in maintaining order in the whole network.     

Critical behavior of sound attenuation in a metamagnetic Ising system

In this study, the sound attenuation coefficient of a spin- metamagnetic Ising system is calculated by the method of thermodynamics of irreversible processes. The behavior of sound attenuation near the phase transition temperatures is analyzed according to various values of phenomenological rate coefficients (γij). For all γm and γs values it is found that sound attenuation peaks occur below TN(H) and depend on frequency ω and the value of the off-diagonal rate coefficient γ. On the other hand, the critical behavior of the sound attenuation in the hydrodynamic regime is obtained analytically via the critical exponents. Moreover, the behavior of the sound attenuation as a function of frequency is also investigated and ω² dependence is observed for the attenuation coefficient. These results are in a good agreement with ultrasonic investigations of magnetic systems.     

The spin-2 antiferromagnet on the Bethe lattice

The complete phase diagrams of the antiferromagnetic spin-2 Blume-Capel Ising system is studied on the Bethe lattice by the use of exact recursion relations. In order to specify the states of the system, i.e. the different spin configurations, the ground state phase diagram is obtained on the (H/|J|, D/|J|) plane corresponding to the reduced external magnetic and crystal fields, respectively. As a result, the thermal change of the order-parameters, the magnetisations belonging to the two sublattice system, was investigated to obtain the full phase diagrams of the system on the (H/|J|, kT/|J|) planes. The behavior of the order-parameters with respect to the external magnetic field was also studied for the given values of D/|J|. Besides the interesting thermal and external magnetic field change of the sublattice magnetisations, the system also exhibits interesting critical behaviors including first- and second-order phase transitions, therefore, triciritical points and the reentrant behavior. The calculations are carried out for the coordination number q=4, corresponding to the square lattice, only.     

Nucleation and hysteresis in Ising model: classical theory versus computer simulation

We have studied the nucleation in the nearest neighbour ferromagnetic Ising model, in different (d) dimensions, by extensive Monte-Carlo simulation using the heat-bath dynamics. The nucleation time () has been studied as a function of the magnetic field (h) for various system sizes in different dimensions (d=2,3,4). The logarithm of the nucleation time is found to be proportional to the power (-(d-1)) of the magnetic field (h) in d dimensions. The size dependent crossover from coalescence to nucleation regime is observed in all dimensions. The distribution of metastable lifetimes are studied in both regions. The numerical results are compared and found to be consistent with the classical theoretical predictions. In two dimensions, we have also studied the dynamical response to a sinusoidally oscillating magnetic field. The reversal time is studied as a function of the inverse of the coercive field. The applicability of the classical nucleation theory to study the hysteresis and coercivity has been discussed. Received: 21 January 1998 / Accepted: 17 March 1998